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1Modern Theories of Continuum

Models

1.1 The Physical Model

Jacopo Tomasi

1.1.1 Introduction

As the title indicates, this chapter focuses on methodological problems relating to thedescription of phenomena of chemical interest occurring in solution, using methods inwhich a part of the whole material system is described by continuum models.

The inclusion in the book of this introductory section has been motivated by theremarkable advances of continuum methods. Their extension to more complex propertiesand to more complex systems makes it necessary to have a more detailed understandingof the way in which physical concepts have to be further developed to continue thispromising line of investigation. The relatively simple procedures in use for three decadesto obtain with a limited computational effort the numerical values of some basic prop-erties, such as the solvation energy of a solute in very dilute solution, are no longersufficient.

To appreciate the basic reasons why continuous models are so versatile and promisingfor more applications, however, we have to consider again the simple systems and thesimple properties mentioned above. The best way to gain this initial appreciation is tocontrast the procedures given by discrete and continuum methods to obtain the solvationenergy in a very dilute solution.

Continuum Solvation Models in Chemical Physics: Theory and Applications Edited by B. Mennucci and R. Cammi© 2007 John Wiley & Sons, Ltd

COPYRIG

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ATERIAL

2 Continuum Solvation Models in Chemical Physics

1.1.2 Solvation Energy

The Discrete ApproachThe material model consists of a large assembly of molecules, each well characterized andinteracting according to the theory of noncovalent molecular interactions. Within thisframework, no dissociation processes, such as those inherently present in water, nor othercovalent processes are considered. This material model may be described at differentmathematical levels. We start by considering a full quantum mechanical (QM) descriptionin the Born–Oppenheimer approximation and limited to the electronic ground state. TheHamiltonian in the interaction form may be written as:

H tot�rM� rS� = HM�rM�+ HS�rS�+ HSS�rS�+ HMS�rM� rS� (1.1)

In extremely dilute solutions only a single solute molecule M is sufficient and so HM

refers to a single molecule only. The number of solvent molecules S is in principleinfinite, but the physics of the system is sufficiently well described by a finite, albeitlarge, number n of S units.

The third term of the Hamiltonian, HSS, represents the interactions between suchmolecules, and the last term, HMS the interactions between M and the n solvent molecules.The coordinates �rM� rS� apply to both electrons and nuclei. Nuclear coordinates have tobe explicitly considered, because the mobility of solvent molecules is a very importantfactor in liquid systems, and changes in their internal geometry, due to the intermolecularinteractions, may also play a role.

The formulation of the Hamiltonian given in Equation (1.1) has introduced considerablesimplifications in the formulation of the problem (the existence of specific moleculesand their persistence has been acknowledged) but the computational problem remainsformidable. Approximations are unavoidable.

The system is described as an assembly of interacting molecules whose motions aregoverned, in a semiclassical approximation, by a potential energy surface (PES) ofextremely large dimensions related to the positions of all the nuclei of the system, internalnuclear motions within single molecule being for the moment still allowed. The approachused for the characterization of small clusters, i.e. searching first for the minimum energyconformation of the PES, cannot be used here. The physics of solvation is remarkablydifferent. Solvation energy and related properties (solvent effects on the solute geometryare an example) are averaged properties and we are compelled to perform a suitableaverage upon the energies corresponding to all the accessible conformations of the wholemolecular system.

Statistical thermodynamics gives us the recipes to perform this average. The mostappropriate Gibbsian ensemble for our problem is the canonical one (namely theisochoric–isothermal ensemble N,V,T). We remark, in passing, that other ensembles suchas the grand canonical one have to be selected for other solvation problems). To deter-mine the partition function necessary to compute the thermodynamic properties of thesystem, and in particular the solvation energy of M which we are now interested in, of acomputer simulation is necessary [1].

We do not enter into the description of Monte Carlo of Molecular Dynamics methods,as these details are not important for our discussion. There are other more general aspectsof computer simulations to consider here.

Modern Theories of Continuum Models 3

(1) These averaging procedures introduce macroscopic parameters, temperature and density whichare not present in the QM formulation of the problem given by the Hamiltonian of Equa-tion (1.1). The use of macroscopic parameters is necessary for the description of molecularsystems in a condensed phase, whether one uses a discrete or continuum approach.

(2) The use of a thermodynamic description leads to a more precise definition of the energy weare seeking. The correct choice is the Helmholtz free energy A, directly defined in the (N,V,T)ensemble, which in liquids may be replaced by the Gibbs free energy G, which is formallyrelated to the isothermal–isobaric ensemble (N,P,T) corresponding more to the usual conditionsof physico-chemical measurements in solution. This remark on the thermodynamic status ofthe solvation energy is important for several reasons we shall discuss later. We anticipate oneof them, namely that the molecular properties we can put in the form of a molecular responsemust be expressed as partial derivatives of the free energy, a condition often neglected in thecalculation of properties based on discrete models.

(3) The use of thermodynamically averaged solvent distributions replaces the discrete descriptionwith a continuum distribution (expressed as a distribution function). The discrete descrip-tion of the system, introduced at the start of the procedure, is thus replaced in the finalstage by a continuous distribution of statistical nature, from which the solvation energy maybe computed. Molecular aspects of the solvation may be recovered at a further stage, espe-cially for the calculation of properties, but a new, less extensive, average should again beapplied.

The need for computer simulations introduces some constraints in the description ofsolvent–solvent interactions. A simulation performed with due care requires millions ofmoves in the Monte Carlo method or an equivalent number of time steps of elementarytrajectories in Molecular Dynamics, and each move or step requires a new calculation ofthe solvent–solvent interactions. Considerations of computer time are necessary, becausemethodological efforts on the calculation of solvation energies are motivated by the needto have reliable information on this property for a very large number of molecules ofdifferent sizes, and the application of methods cannot be limited to a few benchmarkexamples. There are essentially two different strategies.

The first strategy maintains the QM description of the solvent molecules but reducestheir number and adopts a different description for other molecules (often adoptinga continuum distribution) to take account of bulk effects in the calculation. TheseQM simulation methods, of which the first and most frequently used is the Car–Parrinello method [2], are in use since several years, and have largely passed thestage of benchmark examples. This strategy is the most satisfactory under the formalaspects we have at present, and will surely be employed more and more with increasingcomputer power, but will certainly not completely replace, in the foreseeable future, otherstrategies.

The second strategy we mention in this rapid survey replaces the QM description ofthe solvent–solvent and solute–solvent with a semiclassical description. There is a largevariety of semiclassical descriptions for the interactions involving solvent molecules,but we limit ourselves to recall the (1,6,12) site formulation, the most diffuse. Theinteraction is composed of three terms defined in the formula by the inverse power of thecorresponding interaction term (1 stays for coulombic interaction, 6 for dispersion and12 for repulsion). Interactions are allowed for sites belonging to different molecules andare all of two-body character (in other words all the three- and many-body interactionsappearing in the cluster expansion of the HSS and HMS terms of the Hamiltonian (1.1)


are neglected). The interaction energy is thus expressed as a sum of terms with thegeneral formula

AKi B

Km

rKim

where Ai is the ith site of molecule A and Bm the mth site of molecule B. The numericalvalues of the coefficients are empirically defined, with starting guesses from QM calcu-lations on the dimer and then refined with a variety of methods. This simple form of theinteraction potential is appropriate to perform the numerical simulations leading to thenumerical expression of the thermally averaged distributions.

The continuation of the strategy presents at this point a bifurcation. The solute M maybe described with a semiclassical procedure similar to that used for solvent molecules,or with a QM approach. The first method is often called classical (or semiclassical)MM description [3], the second a combined QM/MM approach [4]. The physics of thefirst method is rather elementary, but notwithstanding this it opened the doors to ourpresent understanding of the solvation of molecules. The second method is markedlymore accurate, because the QM description of the solute has the potential of taking intoaccount subtler solvent effects, such as the solvent polarization of the solute electronicpolarization and the changes in geometry within M.

A different approach to mention here because it has some similarity to QM/MM iscalled RISM–SCF [5]. It is based on a QM description of the solute, and makes use ofsome expressions of the integral equation of liquids (a physical approach that for reasonsof space we cannot present here) to obtain in a simpler way the information encoded inthe solvent distribution function used by MM and QM/MM methods. Both RISM–SCFand QM/MM use this information to define an effective Hamiltonian for the solute andboth proceed step by step in improving the description of the solute electronic distributionand solvent distribution function, which in both methods are two coupled quantities.There is in this book a contribution by Sato dedicated to RISM-SCF to which the readeris referred. Sato also includes a mention of the 3D-RISM approach [6] which introducesimportant features in the physics of the model. In fact the simulation-based methodswe have thus far mentioned use a spherically averaged radial distribution function, ��r�instead of a full position dependent function ��r� expression. For molecules of irregularshape and with groups of different polarity on the molecular periphery the examinationof the averaged ��r� may lead to erroneous conclusions which have to be corrected insome way [7]. The 3D version we have mentioned partly eliminates these artifacts.

The use of radial distribution functions is one of the costs paid by simulations methodsto the high computational cost of this approach. The ever increasing availability ofcomputer power has allowed a sizable portion of these shortcomings to be eliminated.In a few years the description of the QM part of QM/MM applications has progressedfrom a rather crude semiempirical description to ab initio levels now sufficiently accu-rate to describe with reasonable accuracy solvent effects on molecular properties andreaction mechanisms. A greater availability of computer power has also permitted theintroduction of some improvements in the formulation of the site–site potentials webriefly characterized above.


The original force field greatly reduced the number of degrees of freedom to monitorduring the simulation and the number of elements in the many-body problem introducedwith the Hamiltonian (1.1). Any improvement will inevitably increase the number ofdegrees of freedom and the number of interaction terms, rapidly leading to unmanageableexpressions. For this reason the improvement of the force field have proceeded slowly,following the increment of computer power.

In the original force field the internal geometry of the molecule was kept fixed;until now calculations with flexible potential are a rarity. For standard solvents, inwhich molecule are small and comparatively rigid, this defect is less important than theneglect or incomplete description of polarization effects, In the first QM/MM formulationspolarization sites (one for each solvent molecule) were introduced. This effect wasexpressed in terms of induced dipole moments (one per site) computed as product ofa site isotropic polarizability multiplied by an electric field vector generated by all thecharge and induced dipoles present in the system. The complexity of the calculation isthus considerably increased because these new terms are not of a two-body character asare the original (1,6,12) terms, and have to be computed iteratively. This commendableeffort continues, introducing more than a single polarization site for a molecule, but thefinal (practical) solution of the problem has not yet been reached. This formulation ofthe problem in fact neglects several aspects of the physics of the phenomenon, whichfurther analyses have shown to be important, and the error in the description of thepolarization response that this methodology gives is of the order of 10–20 %. This erroris to a large extent due to the absence of some coupling terms, but the situation ismore complex, also including parameters which change from case to case (the natureof the solvent, the presence of a net charge on the solute, the macroscopic parametersT and P, etc).

The Continuum ApproachWe report in this subsection a discussion on some aspects of continuum solvation (CS)methods which seems to us useful to examine how the physics of solvation is describedby these models. Other contributions in the book will give more details about theirmethodology, implementation and use. We consider our recent review [8] to be anappropriate text to complement what is said here.

The Hamiltonian for the basic formulation of the problem, to be compared with thatgiven in Equation (1.1), may be written in the following form:

H tot�rM� = HM�rM�+ HMS�rM� (1.2)

The solvent coordinates �rS� do not appear in Equation (1.2) and this is the basic differencebetween discrete and continuum models.

The Hamiltonian H tot�rM� is an effective Hamiltonian, written as two separate termsin Equation (1.2) to facilitate comparison with Equation (1.1) but in actual calculationsit is convenient to treat H tot as a whole, because its structure is very similar to that ofHM in vacuo, in the Hartree–Fock (HF) or Density Functional Theory (DFT) formalism.The passage at higher levels of the QM theory follows the same lines as for isolatedmolecules.


Actually HMS is a sum of different interaction operators each related to an interactionwith a different physical origin. The coupling between interactions is ensured by theiterative solution of the pseudo HF (or DFT) solution of the whole Schrödinger equation.These operators are expressed in terms of solvent response functions based on an averagedcontinuous solvent distribution. They will be symbolically indicated with the symbolsQx��r� �r ′� where �r is a position vector and x stands for one of the interactions. We shallexamine later the form of some of the operators, which actually are the kernels of integralequations.

In contrast with discrete methods, the thermal average is introduced in the continuumapproach at the beginning of the procedure. Computer information on the distributionfunctions and related properties could be used (and in some cases are actually used),but in the standard formulation the input data only include macroscopic experimentalbulk properties, supplemented by geometric molecular information. The physics of thesystem permits the use of this approximation. In fact the bulk properties of the solventare slightly perturbed by the inclusion of one solute molecule. The deviations from thebulk properties (which become more important as the mole ratio increases) are small andcan be considered at a further stage of the development of the model.

In the standard continuum solvation model, exemplified by the Polarizable ContinuumModel (PCM) we developed in Pisa [9], the solute–solvent interaction energies aredescribed by four Qx operators, each having a clearly defined physical nature. Eachterm gives a contribution to the solvation energy which has the nature of a free energy.The free energy of M in solution is thus defined as the sum of these four terms,supplemented by a fifth describing contributions due to thermal motions of the molecularframework;

G�M� = Gcav +Gel +Gdis +Grep +Gtm (1.3)

The order of contribution given in Equation (1.3) corresponds to the best order in whicha sequence of ‘charging processes’ could be performed. A ‘charging processes’ basicallyis an integration performed with respect to an appropriate parameter running from zeroto the final value which couples a given distribution with a potential function. At the endof the charging process the distribution is modified and used for the following chargingprocess. The best sequence is that in which the residual couplings are minimized. Inab initio PCM three charging processes are unified and described by the solution ofthe Schrödinger equation, thus avoiding the problem of coupling a sequence of separatecharge processes. Only the first, namely that giving the cavity formation energy, is treatedseparately. The last contribution, describing thermal motions of the solute, is composedof different terms and is treated in a different manner.

In spite of the unification of different processes in the calculations, each term will herebe separately presented and commented on.

Cavity formation energyThe first charging process is related to the formation in the pure solvent of a voidcavity having the appropriate shape and size to accommodate the solute. The electronicproperties of the solute are not used here, only the geometrical nuclear parameters


are employed to define the correct shape and size. The reversible work spent to formthe cavity is exerted against the forces giving cohesion to the liquid. Calculations areperformed at a given temperature T and a given solvent density. There are differentmethods, using different solvent parameters, to compute this contribution to the solvationenergy. We simply mention two methods: the oldest based on the surface tension of thesolute [10], and the newest based on the use of information theory methods [11], withoutgiving details, to focus our attention on the method used in PCM and in other variantsof the continuum solvation approach. This method is based on the scaled particle theory(SPT), an integral equation method which is simple and effective. The formulation givenby Pierotti [12], extended to cavities of molecular shapes according to a suggestion givenby Claverie [13], is adopted in PCM. The parameter characterizing the solvent is thesolvent equivalent radius. The expression for Gcav is analytical for a spherical cavityand semi-analytical for cavities defined in terms of atomic solute spheres. We started touse SPT derived cavity formation energies in 1981 [14], with many initial perplexitiesabout the physical correctness of the use of hard spheres also for solvents exhibitinghydrogen bonds or irregular shapes. Fortunately, the cavity formation energy is a term(the only one in the whole expression (1.3)) for which an independent validation of itsnumerical value is possible. There are at present a sufficiently large number of results,obtained with semiclassical simulations with accurate force field potentials, showing thatthe SPT approach gives good results for a large variety of solvents and cavity sizesand shapes.

The formal operator Qcav is not included in the Hamiltonian (1.2). In the BO approx-imation we are using, this term is constant as long the geometry of the molecule isunchanged. From this point of view it may be assimilated into the nuclear repulsion Vnn

of a single molecule, again in the BO approximation.The cavity formation charging process produces an important change in the solvent

distribution. After the charging the portion of space within the cavity has zero density.In the outer space the solvent density can be kept constant assuming the cavity volumeis infinitely small with respect to the bulk.

Electrostatic energyIn ab initio formulations this charging process includes the whole molecular density aswell as the electric polarization of the solvent, starting from noninteracting nuclei andelectrons that will compose the molecule. This is a variant with respect to the traditionalview of first defining with QM calculations the molecular density in vacuo, and then ofpassing to a different version of the charging process to activate mutual solute–solventpolarization effects. The QM procedure normally adopted follows the first strategy, witha single charging process; the traditional strategy which decouples the charging process isnecessary when one has to compute the solvation energy given as the difference betweenthe free energies of the molecule in solution and in vacuo. When the explicit evaluationof the solvation energy is not required, the traditional procedure may be considered tobe a waste of computer time, because two geometry optimizations are required. The twostrategies lead to the same result, and people wishing to know in advance the structureof the isolated molecule and to look at the changes in geometry and electro-distributionproduced by the solvent obviously perform two sets of calculations.


The medium response function for Gel is the polarization function �P. In the simplestformulation of PCM we are at present considering, the following formulation of thepolarization function is used:

�P = �−14�

�E (1.4)

were �E is the electric field generated directly or via the apparent charges spread onthe cavity surface, and � is the scalar permittivity, constant over the whole body of thesolvent. The basic electrostatic relation one has to satisfy is given by the Poisson equation.We shall have to reconsider the expression for �P, because most of the physics of solutionnot yet considered in this preliminary presentation is related to the appropriate definitionof �P. In all systems and for all the properties and phenomena the electrostatic componentis the most sensitive to changes in the system and to the quality of the description.The utmost care must be taken to have a reliable description of electrostatic solventeffects.

Repulsion energyThis term is physically related to the electron exchange contributions appearing wheninteractions among molecules are described at the QM level. The description of thiscontribution has been extensively examined for small discrete systems. In CS modelsthere are no discrete representations of solvent molecules, but from the wide experienceon dimers and small clusters it is possibly to justify the expression used in PCM whereit is introduced a Qrep��r� operator based on the solvent density, the number density ofelectron pairs in the solvent, the normal component at the cavity surface of the electricfield generated by the solute and an overlap function. The resulting operator is oneelectron in character and it is inserted in the Hamiltonian (1.2) under the form of adiscretized surface integral, each belonging to a specific portion (tessera) of the closedsurface [15]. The physics of this interaction has perhaps to be reconsidered to accuratelydescribe high pressure effects on solvation.

Dispersion energyThe dispersion contribution to the interaction energy in small molecular clusters has beenextensively studied in the past decades. The expression used in PCM is based on theformulation of the theory expressed in terms of dynamical polarizabilities. The Qdis��r� �r ′�operator is reworked as the sum of two operators, mono- and bielectronic, both based onthe solvent electronic charge distribution averaged over the whole body of the solvent.For the two-electron term there is the need for two properties of the solvent (its refractiveindex ns, and the first ionization potential) and for a property of the solute, the averagetransition energy �M . The two operators are inserted in the Hamiltonian (1.2) in the formof a discretized surface integral, with a finite number of elements [15].

The procedure we have outlined for these three terms of Equation (1.3) is of the abinitio type, with the form used for HF (or DFT) calculation for an isolated moleculewith the addition of a few new operators, all expressed as one-electron integrals over theexpansion basis set (also the two-body dispersion contribution is reduced to the combina-tion of two one-electron integrals). We remark that all the elements of the solute–solventinteraction, cavity formation excluded, are expressed as discretized contributions on the


cavity surface, computed at the same positions. The whole computational framework hasa compact form, without detriment of the description of the physical effects.

We resume here the nature and number of macroscopic parameters used in this versionof PCM: the temperature T , the density �S of the solvent, its permittivity (here reduced toa constant T dependent �), and its refractive index nS. Among the constitutive parametersthere is the hard sphere radius of the solvent molecule and its first ionization potential IS.

When the thermal motion contributions Gtm (on which we do not enter into details)are added we have an ‘absolute’ value of the free energy of M in the given solvent. Thereference state is given by the unperturbed solvent and the amount of noninteracting elec-trons and nuclei necessary to form M. By making the difference with the ‘absolute’ freeenergy of M in vacuo computed with the same QM procedure (the reference state is givenby the necessary amount of noninteracting electrons and nuclei) we have an estimate ofthe free energy of solvation Gsol�M�. Comparison with experimental values shows thatthe results are quite good for larges classes of systems (solutes and solvents). The limitedcases in which this agreement is only fair will be considered in the following section.

With this statement we conclude our summary of a long and complex journey alongformal considerations, models for partial contributions to the energy and developmentsof computational procedures. No experimental values or well established computationalresults are available for the separate components (apart from cavity formation energy).However, we have to consider that this empirical evidence of good values of solvationenergies for large classes of systems (solutes and solvents) is nothing more that anencouragement to proceed further in the construction of models based on well definedphysical bases. The energy is not too sensitive a property and casual compensationsamong errors of different sign could have improved the results.

The approach we have considered presents some features which recommend it forfurther extensions. Firstly, it is an ab initio method with a low computational cost. Acalculation a solution with a good basis set has a computational cost lower that doublethe analogous calculations for the isolated molecules, and the ratio of computational costsbecomes even more favourable in passing to higher levels of the QM theory.

Secondly, all the features of modern quantum chemistry can be easily implemented inthis model. For example, the standard sequence of molecular calculations often adoptedfor a better characterization of the molecule (HF, DFT, MP2, CCSD, CCSD(T)) could beadopted (see also the contribution by Cammi in this book). As shown in other chapters ofthis book, analytical expressions for the derivatives necessary for geometry optimizationsand calculations of response properties are now available; the interpretative tools in usefor characterizing electronic structures can be employed.

The last aspect we stress is the flexibility of the method. Simplified versions areabundant, and they have an important role in computational chemistry, but in this chapterwe consider extensions and refinements which introduce in the model other aspects ofthe physics of solvation.

1.1.3 The Solvent Around the Solute

Several possible refinements of the continuum model can be examined using againinfinitely dilute solutions. In the basic model we have used a uniform distribution of the


solvent, characterized by a constant value of the permittivity. Intuition suggests that localdisturbances to this description are more probable near the solute, and there are goodreasons to think that such disturbances have a measurable effect on some properties of thesolvent. We remark that the agreement with experimental solvation energy data is quitegood in general, but there are classes of systems in which a greater deviation has beenobserved. We could try to examine the extent to which this partial disagreement in thesolvation energy is due to a local disturbance of the solvent, but surely other cases of localdisturbance are not visible in the solvation energy, a property relatively insensitive tosmall changes in the interaction potential. To look at these cases, more sensitive indicatorsare needed, and they are given by other properties, mostly of spectroscopic origin. Thereis a large variety of phenomena to consider in this section, not all completely understood,related to a large variety of effects, all amenable to the physics of interacting molecularsystems, some of general occurrence, others with a character of chemical specificity. Aclear cut classification is not possible because often different effects are intermingled,and our exposition will not be systematic but limited to some aspects of greater physicalinterest. More systematic analyses will be found in other chapters of the book.

Nonlinearities in the Dielectric ResponseAmong factors of general occurrence we have omitted in the description of the basic CSmodel, some are related to refinements of the dielectric theory. The charge distribution ofalmost all solutes gives rise to strong electric fields. These fields are stronger for chargedspecies, especially those of small size such as atomic ions, but they are also present forneutral molecules exhibiting anisotropies in the charge distributions of chemical groupsnear the periphery of the molecule. The case of ions has been largely explored, but weshall also consider the case of neutral solutes.

The occurrence of strong permanent fields may disturb the linear response regime inthe dielectric response we have so far employed. The standard treatment of nonlineardielectric response is based on the expansion of the dielectric displacement function �Din powers of the electric field �E, generally interrupted at the first correction:

�D = �E+4� �P = ��+4��3�E2��E (1.5)

This expression introduces the third order susceptibility of the medium, a quantity not easyto be accurately determined for the small portions of solvent in which the nonlinearityeffect is sizeable. In addition we remark that with the favourable exception of atomic ionswhich have a spherical symmetry, the solvent layer in question has an irregular shape(not directly amenable to the molecular shape because the chemical groups responsiblefor nonlinearities are not regularly placed on the molecular surface). For this reason thewhole tensorial expression of �3� with a position dependent formulation, should be used.

The origin of the effect here represented by �3� can be derived from modelisticconsiderations. Solvent molecules are mobile entities and their contribution to the dielec-tric response is a combination of different effects: in particular the orientation of themolecule under the influence of the field, changes in its internal geometry and its vibra-tional response, and electronic polarization. With static fields of moderate intensity allthe cited effects contribute to give a linear response, summarized by the constant value� of the permittivity. This molecular description of the dielectric response of a liquid is


locally modified by a strong molecular field: firstly a saturation in the response with anonlinearity reducing the actual permittivity with respect to that obtained in the linearformulation (still valid at larger distances); secondly a displacement of the first shellmolecules toward the solute. Liquids are remarkably incompressible, and a collectivedisplacement with a local increase of density requires an appreciable amount of workagainst molecular repulsions. However, this effect is possible (measurements in solutionare generally performed at fixed pressure), and it is called electrostriction. A third effectis related to possible anisotropies in the molecular polarizability; this contribution is alsopositive.

In conclusion the contribution to the dielectric response given by the third ordersusceptibility has different sources with opposite signs. Molecular simulations on ions insolution show that both dielectric saturation and electrostriction effects are presumablypresent and that for ions with a high charge density electric saturation predominates. Thissuggestion is in agreement with the general consensus that dielectric saturation is thefirst element to consider in the description of nonlinearities.

In spite of the remarkable difficulty in defining a detailed model, the number ofcomputational codes introducing dielectric nonlinearity, especially in the form of dielec-tric saturation, is quite abundant. We quote here the main approaches; more details canbe found in the already quoted review [8].

Layered modelsThe solvent is described as a set of onion-like shells with increasing values of �, constantwithin each shell. The layers approach gained some popularity in the late 1970s, generallyapplied to semiclassical descriptions of the solute. The electrostatic part has analyticalsolutions for cavities of regular shape (spheres, ellipsoids) but its use is also possiblefor irregular cavity shapes and for QM descriptions of the solute. Applications of theapproach in this more general formulation have been formulated and used for old versionsof PCM, with appreciable results (this is an example of the flexibility of continuummodels) [16]. We remark that at each layer separation there are boundary electrostaticconditions equivalent to those present in the single cavity model. Several published papersneglect this coupling, and the error may be sizeable. A correct application leads to anincrease of the computational times, and for this reason the approach has been abandonedin PCM because there are more efficient ways to describe the saturation phenomenon.The layered model in PCM has not been abandoned, however, and it has been adopted inmore specialized approaches addressing specific phenomena, such as the nonequilibriumsolvation, electron transfer reactions, and phenomena related to the behaviour of theliquid in phase separations. A case deserving mention is that of solvation in supercriticalliquids in which the standard sequence of values of the dielectric constant in the layers,from lower to higher values, has been reversed to describe electrostriction effects [17].

Position dependent dielectric constantThis model has been, and still is, widely used especially for some specific applications.An older use is in the description of dielectric saturation effects around ions. The originis the Debye model, not completely satisfying and thus subjected over the years to manyvariants. The spherical symmetry of the problem suggests the use of a distance dependentfunction ��r�. The functions belonging to this family are often called ‘sigmoidal functions’because their spatial profile starts from a low value and increases monotonically to reach


the bulk value with a sigmoidal shape. The definition of these functions is empirical; thecontribution of computer simulations to the validation of these functions has been minimalbecause the longitudinal component of ��k� (calculations are generally performed inreciprocal space) has, at least in dipolar liquids, a nonmonotonic shape, and the portionof the function at high k values, the most important for the definition of solvent effectson the energy, is rarely computed, and the available data have a low numerical reliability.The ��r� functions are frequently employed for large molecular systems of biologicalinterest, to screen the coulombic interactions between the point charges used in thesemodels. Position dependent models are also in use for interfaces of a planar type, underthe form of ��z� functions, where z is the Cartesian coordinate perpendicular to the phaseseparation surface (see the contribution of Corni and Frediani in this book).

Electric saturation effects in the description of neutral solutes in polar media have beenstrongly advocated by Sandberg et al. [18], who worked out a complete continuum abinitio solvation code containing the ��r� feature and published results of good qualityfor a large number of solutes. Sandberg et al. remark that PCM calculations do notneed corrections for electric saturation, this being due, in their opinion, to the cavityPCM uses.

We also quote the proposal, made by Luo and Tucker [19], of a model using a dielectricfunction with dependence of the dielectric constant on the electric field acting on thegiven position, used for supercritical liquids, in which the solvent density is particularlysensitive to the local value of external electric fields. Emphasis is given in this model toelectrostriction effects.

This mention of a family of solvents with particular physical properties prompt usto remark that there are other solvents with special physical quantities requiring somemodifications in the methodological formulation of basic PCM. We cite, among others,liquid crystals in which the electric permittivity has an intrinsic tensorial character, andionic solutions. Both solvents are included in the IEF formulation of the continuummethod [20] which is the standard PCM version.

Nonlocal dielectric constantThe dielectric theory may be expressed in a nonlocal form based on the definition of thesusceptibility and permittivity in a form that makes these physical quantities the kernelof appropriate integral equations.

The formal definitions of the nonlocal operators and � can be expressed in the formof their application to a generic F�r� function:

�F��r� =∫

d3r�r� r ′�F�r ′� (1.6a)

��F��r� =∫

d3r��r� r ′�F�r ′� (1.6b)

The expression for the polarization is given by

�P�r� =∫

d3�r ′�r� r ′��E�r ′� (1.7)

which shows that the permittivity depends on the field felt at all positions of themedium.


The nonlocal dielectric theory has as a special case the standard local theory. Its fullerformulation permits the introduction in a natural way of statistical concepts, such as thecorrelation length which enters as a basic parameter in the susceptibility kernel . Forbrevity we do not cite many other features making this approach quite useful for thewhole field of material systems, not only for solutions.

What is of interest here is the description of nonlinear dielectric effects with a linearprocedure. Nonlinear dielectrics were introduced in the theory of liquids by Dogonatzeand Kornyshev in the 1970s [21]; the reformulation of the theory in more recent yearsby Basilevsky [22] permits its insertion in the whole machinery of the PCM version ofthe CS method. The reader is also referred to the contribution of Basilevsky and Chuevdedicated to non-local dielectric solvation models.

Specific Solute–Solvent InteractionsInteractions between the solute and solvent molecule are always present in solution.Their nature depends on the chemical constitution of the interacting partners, and therules of interaction are the same of those studied in simpler molecular clusters. However,there is an important difference between the same M–S interaction in the gas phaseand in solution. In the gas phase the geometry of M–S tends to correspond to the mostfavourable conformation, and to disrupt the M–S association it is necessary to expendthe energy corresponding to the stabilization energy of the dimer. In solution there iscompetition between the S molecule interacting with the solute and with other solventmolecules. These interactions may disturb the most favourable conformation of M–S and,more importantly, change the nature of the disruption of S from a dissociation to a anact of replacement. These are naïve considerations, but it is convenient to recall thembecause in our opinion they are often neglected.

An example of the application of this different nature of molecular interactions insolution concerns an aspect we have already mentioned, without comment. Among theenergy terms collected into the Gtm term there is the contribution due to the rotationof M. This contribution is certainly not equal to that of the freely rotating molecule invacuo, but it is even more erroneous to assimilate it into the contributions of a rotorimpeded by a barrier equal to that, for example, of a hydrogen bond, the existence ofwhich has been inferred from the chemical composition of the system. During the rota-tion the hydrogen bond assumed to be present at a given moment will be deformed andreplaced by other molecular interactions, quite frequently of a similar nature. A param-eterization of the rotational contribution to the free energy has to be based on otherparameters. This error has been repeated in several of the early attempts at modellingliquid systems.

Solute–solvent local interactions may play a role in several aspects of the solva-tion effects. The analysis is delicate because finer aspects of the physics of interactingmolecules have to be introduced.

Let us start with a complement to the naïve considerations exposed few lines above.An important aspect of the local interactions in condensed media subjected to thermalaveraging is their persistence. The persistence is clearly related to the strength of theinteraction, but it is also related to the collective effects of the nearby molecules. Thepersistence times span a wide range: from the short times corresponding to librations of


the molecule to very long times. We limit our considerations here to short and intermediatepersistence times, typical of neutral solutes.

When we examine the response properties of the solute, attention has to be paid tocomparing the persistence of these local interactions with the time necessary to measurethe property. Also measurement times may span a very large interval, depending to theproperty one is measuring and the technique one is using.

Let us consider again the solvation energy, which is a response property. All thestandard experimental methods to measure solvation energy require long times. Withinsuch times almost all the local interactions are mediated, losing to a great extent thespecificity exhibited for example in a Monte Carlo simulation addressing the definition ofthe minimal internal energy of the solvation cluster. Only a limited number of interactionsof particular strength remain to have an effect on the averaged solvent distribution.This is the case for hydrogen bonding and the effect on the distribution function is thereason for the often repeated remark that continuum methods are unable to describehydrogen bond effects. Actually this is not true, since for many years it has been wellestablished [23] that the energy of hydrogen bonds is well described by the combinationof the electrostatic, repulsion and dispersion terms also used in continuum solvationmethods, and this is a fortiori true for the deformed hydrogen bond description given forthe averaged solvent. The errors given by calculations that are sometimes performed tosupport this claim are, to the best of our knowledge, due to a poor implementation of thecontinuum model [24]. These hydrogen bond interactions do, however, influence otherproperties. We now examine some examples. Solvent effects are comparatively greateron the vibrational properties of the solute group involved in the hydrogen bond. Thecontinuum method gives a fairly good description of the vibrational solvent shift, but notsufficient to reach spectroscopic accuracy. The same holds for the corresponding intensity.We remark that this small error on these vibrations has no effect of the vibrationalcomponent of Gtm, because their contribution to the energy of the relevant distributionfunction is completely negligible. However, there is a small contribution to the zero pointenergy.

There are a number of other molecular properties that may be affected by these persis-tent interactions. The more studied properties so far are the electronic excitation energyof a chromophore involved in the permanent interaction, and the magnetic shieldingof atoms (notably O and N) directly involved in this interaction, but all the proper-ties exhibiting a local character (for example the nuclear quadrupole resonance) may besubject to similar persistent interactions.

Persistent interactions are not limited to hydrogen bonds. We mention for examplethose appearing in solutions of molecules with a terminal C=O or C≡N group dissolvedin liquids such as acetone or dimethylsulfoxide. These solvents prefer at short distancesan antiparallel orientation which changes at greater distances to a head-to-tail preferredorientation. The local antiparallel orientation is somewhat reinforced by the interactionwith the terminal solute group and it is detected by the PCM calculation of nuclearshielding and vibrational properties. Recent experimental correlation studies [25] haveconfirmed the orientational behaviour of these solvents found in an indirect way fromcontinuum calculations. The physical effect found in this class of solvent–solute pairsseems to be due to dispersion forces.


Calculations show that the main contribution to the solvent effect on these propertiesis described by the standard CS method, but there is often a missing part. The entity andpercentage weight of this part may change noticeably when the molecular frameworkof the solute is changed. This is an indirect hint that all the solute charge distributionis in some way involved. Calculations also show that by including in the solute a smallnumber of solvent molecules (i. e. going from M to MSn. with n = 1�2�3 according tothe case) the continuum method gives fully satisfactory results.

The study of this problem is an example of the usefulness of CS ab initio methods. Itis computationally easy to repeat calculations of wavefunction, energy and all the abovementioned properties for MSn solutes with an increasing number n of solvent moleculesand to determine at what n value the saturation for this effect is reached. Calculationson MSn systems show other interesting aspects of the problem. The n S molecules mustbe inserted in the solvent as a supermolecule. In fact MM descriptions or Hartree QMdescriptions (without exchange) have no effect on this correction. The quality of thewavefunction seems not to be important for the correction (it is important, however, forthe main calculation of the property); calculations with an ONIOM scheme [26] with thesolvent molecules kept at a low HF description gives the same accurate description asthe full high level QM calculations [24].

These empirical findings show that something is missing in the physics we are using.Analyses of the M wavefunctions seem to indicate that in the cases of a missing contri-bution to the property there is a flow of electrons from M to S. We have arrived at a pointwhich touches on some basic simplifications taken for granted in all theories regardingweak interactions between molecules. The basis for these continuum models, as well asfor the QM/MM methods, is given by the application of the perturbation theory approachto the description of noncovalent interactions. It is worth examining the evolution ofthese theories. The first steps were taken by Debye around 1920, the theory recast ina QM form in 1927, and developed and refined for some decades, until it was recog-nized in the middle of the 1970s that a discarded contribution, namely that related tothe complete antisymmetry of electrons in the interacting system, was essential. In thefollowing 30 years the perturbation theory was reworked and refined again within thismodified theoretical background. It now seems that the extension to more accurate calcu-lations of response properties leads to a critical examination of another of the basic tenetsof the standard noncovalent interaction theory, i.e. that the amount of electronic chargewithin each interaction partner has to be kept fixed in defining the interaction.

Chemists are well aware that strong molecular interactions may be accompanied by aflow of electron charge but the evidence they present has been disregarded by physicists.The latter consider this evidence not to represent legitimate noncovalent interactions, withthe additional remark that in the case of very small electron transfers the polarizationcontribution is able to describe such small effects.

The problem we have raised seems to be of methodological relevance and to requireattention. From the computational point of view the strategy of using MSn clusters wehave outlined may be accepted as a reasonable provisional compromise. We recall whatwe have already said, i.e. that the whole cluster has to be considered as a unique super-molecule, and we add that the problem of extracting from a supermolecule a true molec-ular observable is not yet fully resolved. In conclusion it may be said that for response


properties of solutes exhibiting permanent interactions, active in the measurement, gooddescriptions are possible, but with a blur in the finest details.

1.1.4 Dynamical Aspects of Solvation

We have so far considered static aspects of the solvation phenomena. This is a stronglimitation, because dynamical aspects are always present and they often play the dominantrole. Our selection of topics to consider in this section will be however severely reducedwith respect to the number of phenomena of relevance to the section’s title. The varietyis too great. A few considerations will justify our reduction.

Firstly, the time scales: phenomena in which the molecular aspect of the solute–solventinteractions is the determinant aspect (a subject central to this book) span about 15 ordersof magnitude, and such a sizeable change of time scale implies a change of methodology.Secondly, the variety of scientific fields in which the dynamical behaviour of liquids isof interest: to give an example friction in hydrodynamics and in biological systems hasto be treated in different ways.

All types of time evolution are present in dynamical solvation effects. It is difficult,and perhaps not convenient, to define a general formulation of the Hamiltonian which canbe used to treat all the possible cases. It is better to treat separately more homogeneousfamilies of phenomena. The usual classification into three main types: adiabatic, impulsiveand oscillatory, may be of some help. The time dependence of the phenomenon mayremain in the solute, and this will be the main case in our survey, but also in the solvent;in both cases the coupling will oblige us to consider the dynamic behaviour of thewhole system. We shall limit ourselves here to a selection of phenomena which will beconsidered in the following contributions for which extensions of the basic equilibriumQM approach are used, mainly phenomena related to spectroscopy. Other phenomenawill be considered in the next section.

Nonequilibrium Aspect of Spectroscopic PhenomenaIn going from static to dynamic descriptions we have to introduce an explicit depen-dence on time in the Hamiltonian. Both terms of the Hamiltonian (1.2) may exhibittime dependence. We limit our attention here to the interaction term. Formally, timedependence may be introduced by replacing the set of response operators collected intoQ�r� r ′� with Q�r� r ′� t� and maintaining the decomposition of this operator we presentedin Section 1.1.2. For simplicity we reduce Q�r� r ′� t� to the dielectric component under theform �P�r� t�. With this simplification we discard both dielectric nonlocality and nonelec-trostatic terms, which actually play a role in dynamical processes, especially dispersionand nonlocality.

The basic aspects of the theory of the behaviour of dielectrics in time dependentelectric fields have been known for a long time. We recall some elements useful for ourdiscussion.

We start with the time dependent polarization function �P�t�. This quantity may beexpressed in the form of an integral equation:

�P�t� =∫ t

−�dt′Q�t− t′��E�t′� (1.8)


where the kernel Q�t− t′� is the solvent response function and �E�t′� is the Maxwell field.In the case of an external sinusoidally varying electric field it is easy to obtain from �P�t�the frequency dependent permittivity �� which is a complex function

�� = �′��+ i�′′�� (1.9)

Both �′�� (called the frequency dependent dielectric constant) and �′′�� (called theloss factor) play a role in our applications of the theory.

In continuum methods we have to use the �� function of pure liquids. Both compo-nents of �� can be experimentally measured and can also be computed with theoreticalmethods, but it is convenient to introduce here the physical structure of the �� spec-trum. The intensity of the dielectric absorption is proportional to the imaginary part of��. The spectrum consists of separate absorption bands, with moderate overlap andseparated by regions of very low intensity (the ‘transparent’ regions). The harmonicdecomposition of the spectrum into normal modes shows the dominance of a limitednumber of classes, each having correlation ranges of approximately the same value. Asimplified model consists in using a single collective mode per class. Of course morerefined descriptions are possible, and for some phenomena they are necessary. We shallnot use these refinements, limiting ourselves to stating that models exist that try todescribe better the regions in which there is overlap between classes and models givinga description of the ‘transparent’ regions.

The microscopic origin of the collective modes has been identified since a long time.They are reported here with the corresponding typical correlation times (CT): reorienta-tion modes (this is the so-called Debye region, CT > 10−12 s), libration modes (rotationsimpeded by collisions, CT = 10−13 s), atomic motions (vibrations, CT = 10−14 s), elec-tronic motions �CT = 10−16 s�. When the frequency of the external field increases, thevarious components of the polarization we have introduced here become progressively nolonger active, because the corresponding motions of the solute lag behind the variationof the electric field.

These considerations have to be applied to phenomena in which the ‘external’ fieldhas its origin in the solute (or, better, in the response of the solute to some stimulus).The characteristics of this field (behaviour in time, shape, intensity) strongly depend onthe nature of the stimulus and on the properties of the solute. The analysis we havereported of the behaviour of the solvent under the action of a sinusoidal field can herebe applied to the Fourier development of the field under examination. It may happenthat the Fourier decomposition will reveal a range of frequencies at which experimentaldeterminations are not available: to have a detailed description of the phenomena anextension of the �� spectrum via simulations should be made. It may also happen thatthe approximation of a linear response fails; in such cases the theory has to be revisited.It is a problem similar to the one we considered in Section 1.1.2 for the description ofstatic nonlinear solvation of highly charged solutes.

Current applications have so far avoided these more detailed formulations of thedielectric relaxation, and the scheme of decomposition into collective modes is simplifiedto two terms only, which here we denote as ‘fast’ and ‘slow’

�P ≈ �Pfast + �Pslow (1.10)


This partition is known under two names, Pekar and Marcus, and it may actually beexpressed in two ways, with different couplings between the various components (seeref. [8]). The two decomposition schemes are equivalent in the linear response regime.This two-mode partition is used for a wide variety of phenomena, characterized by asudden change in the solute charge distribution (electrons as well as nuclei). We givesome examples: a sudden change of state in the solute (electronic, but also vibrational),intermolecular electron and energy transfer, and proton transfer. These examples maybe extended to other phenomena, and the examples given may also be partitioned intoseveral classes for which the physics of the problem suggests different ways of usingthe basic approach. This partition is appropriate to characterize the initial nonequilib-rium step of many phenomena, such as those occurring in the spectroscopic domain(but also at intermediate stages, such as the rapid step of proton transfer in chemicalreactions). To proceed further in the description of a phenomenon one has to replacethe two-mode description with a more appropriate model. An example will clarify thisdiscussion.

The electronic transition of a solute is a sudden phenomenon followed by other dynam-ical stages, with different exit channels. According to QM a sudden perturbation (due toa photon in this case) gives rise to nonzero amplitudes for a manifold of states. This alsohappens for molecules in solution.

The first quantity to be computed is the lowest vertical transition energy. Almostall CS methods (including PCM which probably was the first to do it at ab initioQM level) use a two-mode approximation with the slow component of the polarizationvector determined on the ground state electronic distribution ��Pslow�GS�� and the fastone using the electronic distribution of the excited state of interest ��Pfast�EX��. This fastcomponent is based only on the electronic dielectric relaxation of the solvent and has tobe determined with an iterative process which also modifies the effective Hamiltonianin use. As a consequence the two wavefunctions, � (GS) and � (EX), are computed withtwo different Hamiltonians. The same happens for the other states in the manifold createdby the sudden perturbation. The conclusion is that the amplitudes of such states mustbe described by an expression more complex than that used in the standard formulationfor molecules in vacuo. The QM description of molecules in condensed phases is rich inproblems of this type. We stress that the physical basis of the description is correct: theorigin of the differences with respect to the standard picture is due to the use of effectiveHamiltonians, a feature we cannot abandon. We briefly mention a mathematical problemrelated to the definition of determinants in CI procedures addressing the improvementof the wavefunctions (ground as well as excited states). This is a question of marginalrelevance in our rapid discussion, and the mention of the problem, for which a reasonablesolution is possible, is sufficient: more details can be found in the contribution byMennucci. Let us to continue the discussion of the fate of the electronic excitation. Weselect the channel that after the initial vertical excitation leads to a fluorescent emission.This spectroscopic signal has been widely studied because it leads to information about therelaxation of the solvent. The other modes of dielectric relaxation become progressivelyactive after the excitation and the effects are measured by the time resolved fluorescentStokes shift (TDFSS). A detailed analysis of these phenomena is given in the contributionby Ladanyi; here we shall merely make some general comments.


The sequence of the observed frequencies, resolved on the time scale, may be regroupedin a form giving a quantity S�t� which may be related to a time correlation functionCE�t� which represents the ensemble average of solvent fluctuations.

S�t� ≡ ��t�−��

��0�−��= E�t�−E��

E�0�−E��(1.11)

CE�t� ≡ < E�0� >< E�t� >

< � E�2 >

The relationship between spectroscopic and statistical functions has been exploited for avariety of phenomena related in different ways to the dynamical response of the medium.We cite as examples spectral line broadening, photon echo spectroscopy and phenomenarelated to TDFSS we are examining here. A variety of methods are used for these studiesand we add here methods based on ab initio CS. The basic model is actually the same forall the methods in use: ab initio CS has the feature, not yet implemented in other methods,of using a detailed QM description of the solute properties, allowing a description ofeffects due to specificities of the solute charge distribution.

The expression of the S�t� function contains the combination of three terms, twoof which, E�0� and E��, correspond to the differences of energy with respect tothe ground state, computed in the vertical transition approximation using respectivelythe two-mode nonequilibrium and the equilibrium formulations. The third term, E�t�,which gives the shape of the correlation function, and which is generally drawn fromexperimental measurements, may be computed in the continuum framework making use ofan auxiliary function expressed as an integral over the whole range of frequencies � wherethe integrand is a function of the imaginary part of �� [27]. We thus obtain an expressionin which the continuum method requires the knowledge of another bulk property of thesolvent, the spectrum of ��. There are experimental determinations of portions of thisspectrum for a sizeable number of solvent, and there are empirical analytical formulaewhich describe well, or passably well, the portions at low and intermediate frequencies,while for the portions at high frequency, shown from calculations to be essential for thedetermination of the fastest steps of the relaxation process, the best way to proceed isto drawn information from accurate MD simulations. We remark that the �� spectrumis to a good approximation a property of the solvent alone, and so, once determined, itmay be used for many solutes.

The formulation of the method we have sketched, thus far applied with some approx-imations, may in principle also be applied to nonpolar solvents. However, there arepractical difficulties to overcome. The mode analysis in nonpolar solvents is less devel-oped and experimental data on the dielectric spectra are scarcer. The solution of usingcomputed values of �� for the whole spectrum is expensive and computationally deli-cate. The best way is perhaps to develop for apolar solvents a variant of the reductionof Q�r� r ′� t� that we have introduced for polar solvents, which takes into account thatin nonpolar solvents the interaction is dominated by nonelectrostatic terms. The refor-mulation of the theory has not yet been attempted, at least by our group, but in recentversions of the continuum ab initio solvation methods there are the elements to developand test this new implementation.


In our discussion about the TDFSS we have not made mention of the relaxation ofthe solute after the vertical excitation. This relaxation occurs in all cases, except foratomic solutes. Relaxation times are of the same order of magnitude as those activein the first stages of the relaxation of the solvent, so the two processes are coupled.TDFSS measurements have been used mostly to study the dynamical behaviour ofliquids, and for this reason the solutes used in experiments are generally quite rigid.In nature (and in laboratories) there are many examples of relaxation phenomena inwhich the characterizing part is given by the solute geometry relaxation. We remarkthat in some cases solvent effects on the relaxation of the excited state geometry arebetter modelled, to a first approximation, in terms of the solute viscosity [28] also in thepresence of permanent dipoles. We are here touching on an aspect of great importancein the description of the dynamical evolution of molecular systems in condensed phases,that of motions in the presence of stochastic fluctuations. We shall consider this aspectin the following section, making use of the Langevin equation approach.

1.1.5 Interactions between Solutes

The whole body of chemistry is essentially based on the exploitation of interactionsbetween molecules in a liquid phase. There is an enormous wealth of empirical evidenceabout the influence of solvents on chemical reactions. Chemists actively exploit thisbody of evidence in many ways, according to different strategies based on their expe-rience and tuned to their needs. Rarely does a new synthesis start with a preliminaryaccurate theoretical study. However, there is a progressively increasing trend of usingcomputational tools even in the start-up stage of novel syntheses. Computer derivedestimates of the solvent influence on some parameters, essentially relating to chemicalequilibria and reaction rates, give hints on the definition of an opportune strategy for thesynthesis.

A good number of the computational tools of this sort rely on the use of continuumdescriptions of the solvent, and for this reason they have to be mentioned here. Forpragmatic reasons researchers tend to adopt low cost methods. Reduction of computationalcost is achieved by simplifications in the description of the physics of phenomena involvedin the reaction process. The confidence gained with more accurate studies on reactionprocesses helps in this reduction of the physics, which is accompanied by a strongparameterization to increase the reliability of the computed parameters. For the solvationenergy, to give an example, there are procedures specialized for given classes of solvents(nonpolar, polar, water), for specific classes of solutes, with different types of moleculardescriptor, starting from models with a single descriptor, such as molecular volumeor area, progressing then to more complex models combining e.g. molecular volumeand noncovalent solute–solvent interactions or volume and dipole-driven electrostaticinteractions. This variety of models, of which we have given just a few examples,found their justifications in the results obtained with the methods we have introduced inSection 1.1.1 of this contribution.

Because this contribution is dedicated to the physics of solvation and not to compu-tational issues, we do not add other comments on these methods, except to remark thata full understanding of the basic justifications of such methods is necessary to avoidmisunderstandings and erroneous conclusions in their use.


Detailed and accurate descriptions of reaction mechanisms, however, have beenperformed for several years, in some cases with the inclusion of solvent effects. Inthis section we shall briefly examine some aspects of the solvation physics related tothe chemical reaction mechanisms; a more general discussion on chemical reactions insolution is given in the contribution by Truhlar and Pliego.

We start by considering the simple extension of the basic material model considered inSection 1.1.1: an infinite isotropic solution, containing as solute just the minimal numberof molecules involved in the reaction. For simplicity we consider a bimolecular reaction,giving rise after the chemical interaction two different molecules:

A +B → C+D (1.12)

This simplification of the model eliminates some preliminary aspects of the processwhich sometimes have considerable importance, such as the processes bringing intocontact separate reactions partners. We shall return later to this point for reactions insolution but let us consider first reactions in gas phase.

Noncovalent interactions between the two separate molecules define, in the gas phaseanalogue of this reactive system, the preferential channels of approach (in the simplercases there is just one channel leading to the reaction) with shape and strength determinedonly by these interactions. As a general rule, these channels carry the reactants to astationary point on the potential energy surface called the initial reaction complex.

In solution things are more complex. The reaction partners are no longer free in theirtranslational motion as they are in the gas phase; they have to move in a condensedmedium, and their motion is governed by other physical phenomena which for economyof exposition we shall not consider in detail. It is sufficient to recall that the physicalmodels for the most important terms, Brownian motions, diffusion forces, are expressedin their basic form using a continuum description of the medium.

Both isolated partners of the reaction (1.12) are solvated, and we may consider, forsimplicity, that during an initial stage of mutual approach they both maintain theirequilibrium solvation shell, as described in Section 1.1.2. To reach the intimate contactcorresponding to the initial reaction complex defined for the in vacuo reaction, thetwo solvation shells must be distorted and strongly rearranged. In solution there are nosimple association processes, but more complex processes in which there is a replacementof molecular associations. The modelling of this process is not immediate. Solute–solvent interaction energies are often of comparable strength, the entropy contributionsare considerably greater in solution than in vacuo, and so the description cannot be limitedto the comparison of the relative strength of the bimolecular interactions involved in thischange of molecular interactions. The consequences may be remarkable. Well knownexamples are given by bimolecular association processes. These reactions, simpler tostudy than the standard reactions where bond are broken and formed, presented some‘surprises’ in the first accurate studies performed some years ago. A typical example isthat of the association of two amide molecules. In vacuo a stabilizing interaction supportedby hydrogen bonds (one or two, according to the channel and the nature of substituentgroups in the amide) leads to a remarkable stability of the dimer. In water this typeof interaction is destabilizing, and is replaced by a feeble �–� interaction leading to acompletely different dimer geometry. The reason is that the water–amide H-bond strength


is comparable with that of the amide–amide H-bond, and entropy changes strongly hinderthe formation of an H-bond association between amide molecules.

In addition, owing their chemical nature, reactive groups in reacting molecules oftenexhibit local solvent interactions stronger than other portions of the same molecule. Thisfact may shift the initial complex contact to another molecular group with a less stronglocal solvation, inducing modifications of the reaction mechanism with respect to the gasphase analogue.

The computational evidence supporting these general considerations is so far scarce,because to do it the examination of rather complex bimolecular systems is necessary,performed with care and good accuracy. The considerable computational cost suggestswaiting for more powerful computers. The problem is well known to people undertakingchemical syntheses; the search for the most appropriate solvents is to a large extentrelated to such differential interactions. Even greater is the indirect evidence comingfrom reactions occurring in living organisms; the admirable machinery of biochemicalreactions exploits the complex nature of the medium, which cannot be assimilated to bulkisotropic water, to enhance or to hinder reaction mechanisms using a variety of physicaleffects.

Let us return to the examination of reaction mechanisms. For reactions in vacuothe methodology to study the steps following the formation of the initial complex arenowadays sufficiently standardized, to a first approximation. The basic concept in useis that of the potential energy surface (PES). This is not a true physical concept, beingrelated to an approximation in the mathematical machinery of formulation of the quantummechanical problem, but the Born–Oppenheimer approximation on which the PES isbased is remarkably accurate and stable and so we may accept the PES as a physicalingredient of the theory. The definition of the family of PESs for an isolated system isunequivocal. We shall consider here cases in which the attention may be limited to asingle PES: that of the electronic ground state. The starting point for the characterizationof the mechanism is the search for the stationary point corresponding to the top of thereaction barrier (the transition state, TS). The search for this stationary point is still almostan art, but it is feasible and the validation of the result is based on precise mathematicalalgorithms. The formal definition of the reaction path (RP), a one-dimensional nonlinearcoordinate connecting the initial complex of reagents, TS and the final complex ofproducts, is standardized in a quite acceptable form. The definition leads to the definitionof the computational strategy which starts from the geometry of the TS and proceeds withperforming calculations along the two directions defined by the coordinate correspondingto the descent from the TS [29].

No additional physical concepts are necessary for this static definition of the mech-anism. The strategy is well defined and relatively simple to apply to reactions with asimple PES form, i.e. surfaces with a single TS. Actually the topological structure ofthe surface may be more complex, with several TSs defining accessory stationary points,some of which correspond to intermediates along the RP, others defining alternativeroutes.

Turning now to the mechanisms in solution, the same strategy apparently seems to beapplicable. However, there are important differences making its application more difficult.One complication is related to an approximation adopted in the gas phase model which wehave not mentioned in introducing the PES concept. The quantity to use in defining the


geometrical evolution of the system in a reaction is the free energy and not the energy. Inthe BO approximation both quantities depend parametrically on the nuclear coordinatesand can be described as a hypersurface in nuclear coordinate space R. The approximationwe have mentioned consists in neglecting entropic contributions in the definition of thegeometries corresponding to TS and RP. This is an acceptable simplification for systemsin vacuo, but it is not acceptable for systems in solution. To pass from internal energyto free energy there are no conceptual problems but major computational problems formethods based on discrete descriptions of the solvent. Umbrella sampling simulationsand constrained molecular dynamics methods, now in use, rely on the previous definitionin vacuo of a one-dimensional RP on which point by point a free energy profile iscomputed. Actually the TS in vacuo may be quite different from the TS in solution.A possible alternative to define the lowest free energy path is the use of a method inwhich appropriate collective variables are introduced [30]. This RP is then used in aset of umbrella sampling simulations. No analytical derivative methods are in use fordiscrete solvent models.

Things are much simpler in continuum methods. Continuum methods in fact directlygive free energies which can be collected in a function G�R� (which could be alsocalled the FES) continuous over the R space and computationally well defined at everypoint of this space (as it is for the PES function in vacuo) In continuum models thereare computational codes enabling the analytical calculation of derivatives (see also thecontribution by Cossi and Rega in this book) necessary for the definition of TS andRP. We shall thus limit ourselves to the examination of G�R� obtained with continuummethods.

As remarked before there are aspects of the early stages of the reaction which it is notconvenient to describe with the G�R� formalism. The approach of the two molecules Aand B entering into reaction is modulated and impeded by interactions with the solvent,which at large distances are little affected by A–B interactions. The physical keys for thisinitial stage of the reaction are given by Brownian motions and diffusion phenomena, twoimportant chapters in the physics of solution, amply studied, originally formulated withcontinuum descriptions of the solvent, and for which modern continuum methods mightgive important contributions. For economy in the discussion we shall not treat thesethemes in this contribution, limiting ourselves to the core of the reaction, the descriptionof which is based on the G�R� function.

Let us suppose we have obtained by an analysis of G�R� a description of the wholeRP in solution making use of the appropriate analytical derivatives. The examinationof evolution of the system along the RP starting from the initial complex shows aninitial region in which the main effects are to be assigned to conformational changes,accompanied by moderate electronic polarization and changes in the internal geometryof the chemical groups. The decomposition of the forces acting on the nuclei of the QMsubsystem (a mathematical procedure that may be performed with tools developed forthe semiclassical analysis [31] of isolated molecules and easily inserted into continuumsolvation codes [32]) shows that the net solvation force component for some groups of themolecule pushes the group towards the completion of the reaction, while for other groupsof the molecule a counteracting effect can occur: the local solvation forces act againstthe completion of the reaction. On the whole there is a distortion of the mechanism withrespect to that found in the absence of solvation forces.


Near the TS minor changes in the nuclear geometry are accompanied by markedchanges in the electronic distribution; new bonds are formed and others broken in thisregion. An analogous change in the relative evolution of nuclear and electronic compo-nents also happens in vacuo. The differences with respect to reaction in vacuo remain inthe solvent, which always plays a role, in some cases quite specific. The specific role ofthe solvent is evident in reactions in water in which an H atom is transferred from onegroup to another; in these cases the H transfer is mediated by a bridge of a few watermolecules, acting as a catalyst. These water molecules must be inserted in the portion ofthe system described at the QM level and thus in the definition of the free energy hyper-surface G�R� on an enlarged R space. This is just an example, the best studied example,but the active role of solvent molecules has also been found in other cases. Other solventmolecules, not only water, may play a specific role in the reaction. There is no gener-ally accepted terminology, and we use here a term we coined years ago: that of activelyassisting solvent molecules [32].

The enlarging of the R space to include actively assisting solvent molecules is a delicateproblem. The cases in which the assisting molecules may be defined in position andnumber at the level of the initial complex are rare. The empirical solution often adoptedis that of obtaining an approximate description of the TS without assisting molecules, andthen of adding here, after an accurate analysis, a single solvent molecule in a position inwhich it may exert an assisting role. This computational task is easy for simple cases, butwhen the assisting role is exerted by two or more molecules the procedure of insertionhas to be repeated on a G�R� surface becoming progressively more flat. It is worthremarking that this procedure has been initially applied to studies of reaction mechanismswith models in which the solvent was described in terms of a few discrete molecules: theaddition of the first active solvent molecule is in this case an easy task, but the additionof more active molecules is more difficult, because the added molecules prefer to interactwith other portions of the solute. This optimization artefact rarely occurs in continuumsolvation methods, because the solvation of other portions of the molecule is alreadyensured by the continuum reaction potential.

Dynamical Aspects of Chemical ReactionsIn describing the PES-based approach for molecules in the gas phase we added the remarkthat the picture of the reaction mechanism we have described was static. The same remarkalso holds for the description of reactions in solution. In neglecting dynamical aspects wehave greatly simplified the tasks of describing and interpreting the reaction mechanism,and at the same time we have lost aspects of the reaction that could be important.

Let us consider first the in vacuo cases. Dynamical aspects of the reaction in vacuomay be recovered by resorting to calculations of semiclassical trajectories. A cluster ofindependent representative points, with accurately selected classical initial conditions,are allowed to perform trajectories according to classical mechanics. The reaction path,which is a static semiclassical concept (the best path for a representative point withinfinitely slow motion), is replaced by descriptions of the density of trajectories. A widelyemployed approach to obtain dynamical information (reaction rate coefficients) is basedon modern versions of the Transition State Theory (TST) whose original formulationdates back to 1935. Much work has been done to extend and refine the original TST.


Among the numerous features added to the method we mention the concept of a dividingsurface (DS) which separates reactant and product regions in the R space. The DS hasto be determined dynamically with one of the proposed procedures. We do not givemore details of the complex set of TST procedures thus far developed, each addingnew features and new approximations. This methodological activity, pursued by severalresearchers, has been guided by the activity of Truhlar, initiated in the late 1970s andcontinuing today. We shall refer to this large body of methodological study with theacronym VTST (variational TST). We do not give here more details on VST which is aquite complex and detailed method. Additional aspects of VTST will be considered laterin the context of reactions in solution.

The dynamics of reactions in solution must include an appropriate description of thesolvent dynamics. To simplify this problem we start with some considerations supportedby intuition and by some concepts described in the preceding sections. In the initial stagesof the reaction the characteristic time is given by the nuclear motions of the solute, largeenough to allow the use of the adiabatic perturbation approximation for the description ofmotions. In practice this means that the evolution of the system in time may be describedwith a time independent formalism, with the solvent reaction potential equilibrated ateach time step for the appropriate geometry of the solute.

Near the TS things change. The rapid evolution of the light components of the system(electrons and H atoms involved in a transfer process) makes the adiabatic approximationquestionable. Also the sudden time dependent perturbation we introduced in Section 1.1.3to describe solvent effects on electronic transitions is not suitable. We are consideringhere an intermediate case for which the time dependent perturbation theory does notprovide simple formulae to support our intuitive considerations. Other descriptions haveto be defined.

An important physical feature which has to be recovered in these descriptionsis related to the influence that dynamical solute–solvent interactions have when thesolute passes from the reactant to the product region of G�R�. The solvent moleculesinvolved are subject to thermal random motions and cannot be categorized as assistingmolecules.

There are different approaches to the description of these dynamical interactions leadingto different computational strategies. We shall briefly examine the two most commonlyused approaches. A description of the evolution of the system near the TS is given bythe VTST. The most complete description of the method has been given by Truhlarand co-workers [33]; in this book there is a good synopsis by Truhlar and Pliego. Thedynamical correlation between solute and solvent molecules is described in VTST interms of trajectories which are scattered back, contributing in this way to the definitionof the dividing surface (DS). The introduction of the DS concept has an importantmethodological relevance because it changes the dimensionality of the critical quantityof the theory. In fact the TS is defined as a single point on the G�R� surface, whileDS is a surface with �3N − 1� dimensions. This fact, certainly important for reactionsin vacuo, assumes a greater importance in solutions, where the free energy landscapeat the discrete molecular level exhibits a large number of geometrical configurationsquasi-degenerate in energy, all capable of acting as a watershed between reactants andproducts (this also happens with the reduction of solvent degrees of freedom introducedby the continuum approximation; the explicit assisting solvent molecules are sufficient to


introduce a sizeable number of quasi-degenerate configurations). The concept of a singleTS point is untenable in almost all chemical reactions in solution.

The VTST briefly summarized here has been implemented in a computational codewhich contains many other features [34]. Among them we cite those related to thedescription of tunnel effects, to which much attention has been paid in the developmentof the method (to emphasize this aspect the acronym VTST/OMT has been used, whereOMT stays for optimized multidimensional tunnelling). We have not paid attention inthe preceding pages to tunnelling effects, which are of extreme importance in molecularbiology, but also present and important in many other reactions. Having a code able todescribe in an optimized way this physical feature of solutions will in the near future be anecessary requisite for the study of reactions in solution. VTST/OMT also contains manyother features. It is a complex code in which a good portion of the complexity is due tothe effort of defining suitable approximations with the scope of reducing computationalcosts without losing a clear identification of the thermodynamic characteristics of all thepartial quantities introduced. We are confident that the continued development of theprocedure will lead to codes that are simpler to use, but the final goal of having codescontaining all the features considered in VTST/OMT, and as easy to use as those nowavailable for the construction of PES in vacuo, seems to us still distant.

The other approach we are considering here is based on a description of the dynamicalinteractions occurring after the passage of the TS (or better of the DS divide) in terms ofan additional force of a frictional type related to the time correlation of a random force.This formulation was introduced by Kramers in 1940 [35], in the form of a Langevinequation. The Langevin equation, proposed in 1908 just to treat the above mentionedBrownian motions, has had a tremendous impact on the study of all phenomena in physicsexhibiting both fluctuations and irreversibility. In the study of solutions the Kramersformulation was later (1980) extended by Grote and Hynes [36] who introduced a timedependence in the friction coefficient. This was the beginning of the family of GeneralizedLangevin Equations (GLE) on which much work has been done. We remark that GLand GLE procedures are typically limited to a single coordinate, interpreted as the RPcoordinate. The extension to a few more coordinates is possible, but the development ofa computational protocol to treat with these procedures the many dimensional problemfor polyatomic molecules with many degrees of freedom is a hard task. The greatmerit of GLE studies is the insight they give on the basic nonequilibrium aspects ofsimple reactions. Another way of introducing nonequilibrium effects in the dynamicalequation is given by the addition to the reaction coordinate a solvent coordinate swhich measures deviations from the equilibrium distribution of the solvent, followingthe approach pioneered in 1956 by Marcus [37]. This coordinate describes with a singleparameter the dynamical participation of solvent molecules. The definition of the solventcoordinate s given by Zusman [38] is based on the continuum solvation model, withthe two-mode decomposition we have introduced in Equation (1.10). The dynamicalcoordinate is essentially related to Pslow.

To complete this short discussion of the dynamics of reactions we remark thatcontinuum models play an important role in the dynamical procedures. The basic under-lying static description G�R� is more easily developed, simple molecular models apart,with a continuum solvation code, and it is more easily extended to include the solventassisting molecules. Continuum models easily give the vibrations and the elements of


the Hessian matrix (second order partial derivatives with respect to nuclear coordinates)necessary for a topological characterization of the points on the hypersurface. In thedynamical part continuum models may also play a role, and some comments have beengiven in the preceding pages; here we add that the introduction of noise is possible, evenif not yet fully explored. With these remarks we do not claim that the whole compu-tational machinery can be reduced to continuum calculations. A judicious combinationof different approaches is probably the best choice. We are at present at a stage in thedevelopment of the computational models in which it is still necessary to obtain a furtherinsight on the numerical stability and computational effectiveness of the models in useto describe the various physical effects. Our ultimate goal is, in our opinion, to use thisincreased knowledge to establish methods and computational protocols that are simplerto use, at the cost of some well selected simplifications in the description of the physicalmodel.

References

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[2] R. Car and M. Parrinello, Phys. Rev. Lett., 55 (1985) 2471.[3] See for example: W. Damm, A. Frontera, J. Tirado-Rives and W. L. Jørgensen, J. Comput.

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[4] A. Warshel, M. Levitt, J. Mol. Biol. 103 (1976) 227; J. Gao, Rev. Comput. Chem., 7 (1995) 115.[5] S. Ten-no, F. Hirata and S. Kato, J. Chem. Phys., 100 (1994) 7443.[6] H. Sato, A. Kovalenko and F. Hirata, J. Chem. Phys., 112 (2000) 9463.[7] F. M. Floris, A. Tani and J, Tomasi, Chem.Phys., 169 (1993) 11.[8] J. Tomasi, B. Mennucci and R. Cammi, Chem. Rev., 105 (2005) 2999.[9] S. Miertuš, E. Scrocco and J. Tomasi, Chem. Phys., 55 (1981) 117.

[10] H. H. Ulig, J. Phys. Chem., 41 (1937) 1215.[11] G. Hummer, S. Garde, A. E. Garcia, M. E. Paulaitis and L. R. Pratt, Proc. Natl. Acad. Sci.,

USA, 93 (1996) 8951.[12] R. A. Pierotti, Chem. Rev., 76 (1976) 712.[13] F. Vigne’-Maeder and P. Claverie, J. Am. Chem. Soc., 109 (1987) 24.[14] R. Bonaccorsi, C. Ghio and J. Tomasi, The effect of the solvent on electronic transitions

and other properties of molecular solutes, in R. Carbo (ed.), Current Aspects of QuantumChemistry, Elsevier, Amsterdam, 1982, p. 407.

[15] C. Amovilli and B. Mennucci, J. Phys. Chem. B, 101 (1997) 1051.[16] J. Tomasi, G. Alagona, R. Bonaccorsi and C. Ghio, A theoretical model for solvation with

some applications to biological sistems, in Z. B. Maksic (ed.), Modelling of Structure andProperties of Molecules, Ellis-Horwood, Chichester, 1987, p. 330.

[17] C. Pomelli and J. Tomasi, J. Phys. Chem. A, 101 (1997) 3561.[18] L. Sandberg, R. Casemyr and O. Edholm, J. Phys. Chem. B, 106 (2002) 7889.[19] H. Luo and S. C. Tucker, J. Phys. Chem., 100 (1995) 11165.[20] E. Cancès, B. Mennucci and J. Tomasi, J. Chem. Phys., 107 (1997) 3032.[21] R. R. Dogonadze and A. A. Kornishev, J. Chem. Soc. Faraday Trans., 2, 70 (1974) 1121.[22] V. Basilevsky and D. F. Parsons, J. Chem. Phys., 105 (1996) 3734.


[23] G. Alagona, c. Ghio, R. Cammi and J. Tomasi, A Reappraisal of the hydrogen bonding inter-action obtained by combining energy decomposition analyses and counterpoise corrections,in J. Maruani (ed.), Moleculaes in Physics, Chemistry, Biology, Vol. II, Kluwer, Dordrecht,1988, p. 507.

[24] J. Tomasi, Theor. Chem. Acc., 112 (2004) 184.[25] S. E. McLain, A. K. Soper and A. Luzar, J. Chem. Phys., 124 (2006) 074502.[26] T. Vreven and K. Morokuma, J. Comput. Chem., 21 (2000) 1419.[27] (a) C. P. Hsu, X. Song and R. A. Marcus, J. Phys. Chem. B, 101 (1997) 2546; (b) M. Caricato,

F. Ingrosso, B. Mennucci and J. Tomasi, J. Chem. Phys., 122 (2005) 154501.[28] A. Espagne, D. H. Paik, P. Changenet-Barret, M. M. Martin and A. H. Zewail, Chem. Phys.

Chem., 7 (2006) 1717.[29] H. B. Schlegel, Reaction path following, in Encyclopedia of Computational Chemistry, Vol.

4, John Wiley & Sons, Ltd, Chichester, 1998, p. 2432.[30] B. Ensing, A. Laio, M. Parrinello and M. L. Klein, J. Phys. Chem. B, 109 (2005) 6676.[31] G. Alagona, R. Bonaccorsi, C. Ghio, R. Montagnani and J. Tomasi, Pure Appl. Chem., 60

(1988) 231.[32] E. L. Coitino, J. Tomasi and O. N. Ventura, J. Chem. Soc., Faraday Trans., 90 (1994) 1745.[33] D. G. Truhlar, J. Gao, M. Garcia-Viroca, C. Alhambra, J. Corchado, M. L. Sanchez and T.

D. Poulsen, Int J Quantum Chem., 100 (2004) 1136. and references cited therein.[34] Polyrate 9.6, http://comp.chem.umn.edu/polyrate/[35] H. A. Kramers, Physica 7 (1940) 284.[36] R. F. Grote and J. T. Hynes, J. Chem. Phys., 76 (1980) 2715.[37] R. A. Marcus, J. Chem. Phys., 24 (1956) 966.[38] I. Zusman, Chem. Phys., 49 (1980) 295.

1.2 Integral Equation Approaches for Continuum Models

Eric Cancès

1.2.1 Introduction

The integral equation approach is a general purpose numerical method for solving math-ematical problems involving linear partial differential equations with piecewise constantcoefficients. It is commonly used in various fields of science and engineering, such asacoustics, electromagnetism, solid and fluid mechanics, � � �

In the context of implicit solvent models, several numerical methods based on inte-gral equations (DPCM, COSMO, IEF, � � � ) have been proposed for calculating reactionpotentials and energies.

In Section 1.2.3 an integral representation of the reaction potential is derived, underthe assumption that the molecular charge distribution is entirely supported inside thecavity C. This representation is then used to reformulate the reaction field energy

ER��′� =

∫R3

�′��r�VR��r�d�r

as an integral on the interface � = �C:

ER��′� =

∫�� V ′

M (1.13)

where V ′M is the potential generated by the charge distribution �′ in the vacuum, i.e.

V ′M��r� =

∫R3

�′��r ′��r −�r ′� d�r ′ (1.14)

The surface charge � is a solution of an integral equation on � , that is of an equation ofthe form

∀�s ∈ ��∫�kA��s��s ′��s ′�d�s ′ = b��s� (1.15)

where kA is the Green kernel of some integral operator A and where the left-hand sideb� depends linearly on the charge distribution �. The various integral equation methodsunder examination in this chapter correspond to different choices for A and b�. Forinstance, the original version of COSMO [1] is obtained with kA��s��s′� = 1/��s−�s′� andb��s� = −f��

∫R3 ��r ′�/��s−�r ′� d�r ′, with f�� = ��−1�/��+0�5�.

DPCM and IEF are exact (and therefore equivalent) as long as the solute charge liescompletely inside the cavity, whereas COSMO is only asymptotically exact in the limitof large dielectric constants. If there is some escaped charge, i.e. if some part of thecharge distribution is supported outside the cavity, all these methods are approximations.The error generated by the fact that, in QM calculations, the electronic tail of the solutenecessarily spreads outside the cavity, is discussed in Section 1.2.4.


The usual discretization methods for integral equations (collocation vs Galerkin,boundary elements) are presented in Section 1.2.5.

Section 1.2.6 is concerned with geometry optimization, and more generally with thecalculation of observables involving derivatives with respect to the shape of the cavity(shape derivatives).

Lastly, the extensions of the standard implicit solvent model to more sophisticatedsettings (liquid crystals, ionic solvents, metallic surfaces, � � � ) are briefly dealt with insection 1.2.7.

1.2.2 Representation Formula for the Poisson Equation

All the integral equation methods discussed in this chapter are based on an integralrepresentation of the reaction potential. Let us state this point precisely.

Consider a function W � R3 −→ R satisfying⎧⎪⎨⎪⎩

−W = 0 inC

−W = 0 outside C

W −→ 0 at infinity

(1.16)

Let �n��s� be the outward pointing normal vector at �s ∈ � . We now assume that thefollowing limits exist for all �r ∈ �

Wi��r� = lim�→0+

W��r −��n��s�� W

�n

∣∣∣∣i

= lim�→0+

Wi��r�−W��r −��n��s��

(1.17)

We��r� = lim�→0+

W��r +��n��s�� W

�n

∣∣∣∣e

= lim�→0+

W��r +��n��s��−We��r��

(1.18)

Note that the existence of these limits does not imply that the function W nor its normalderivative are continuous across � . On the other hand, they ensure that the jump

�W��s� �= Wi��s�−We��s�

of W at �s ∈ � is well-defined, and that so is the jump of its normal derivative[�W

�n

]��s� �= �W

�n

∣∣∣∣i

��s�− �W

�n

∣∣∣∣e

��s�

We can now state a representation formula for W : for all �r �∈ �

W��r� =∫�

1

4��r −�s ′�[�W

�n

]��s ′�d�s ′ −

∫�

�

�n�s ′

(1

4��r −�s ′�)�W��s ′�d�s ′ (1.19)

where we have used the notation

�

�n�s ′

(1

4��r −�s ′�)

= ��s ′

(1

4��r −�s ′�)

· �n��s ′� = ��r −�s ′� · �n��s ′�4��r −�s ′�3


The integral representation (1.19) implies that it is sufficient to know the jumps of Wand �W/�n at the crossing of the interface � to know W everywhere in R

3 \� .The function W , being a priori discontinuous at the crossing of � , does not have a

well-defined value on � . On the other hand, the following representation formula holdsfor every �s ∈ � :

Wi��s�+We��s�2

=∫�

1

4��s−�s ′�[�W

�n

]��s ′�d�s ′ −

∫�

�

�n�s ′

(1

4��s−�s ′�)�W��s ′�d�s ′

(1.20)Similarly for all �s ∈ � ,

12

(�W

�n

∣∣∣∣i

+ �W

�n

∣∣∣∣e

)��s� =

∫�

�

�n�s

(1

4��s−�s ′�)[

�W

�n

]��s ′�d�s ′

−∫�

�2

�n�s�n�s ′

(1

4��s−�s ′�)�W��s ′�d�s ′

(1.21)

with

�

�n�s

(1

4��s−�s ′�)

= ��s

(1

4��s−�s ′�)

· �n��s� = − ��s−�s ′� · �n��s�4��s−�s ′�3

and

�2


(1

4��s−�s ′�)

= �n��s� · �n��s ′�4��s−�s ′�3 +3

��s ′ − s� · �n��s�� s− s ′� · �n��s ′��4��s−�s ′�3 �

The integral representation formulae (1.20) and (1.21) suggest to introduce the integraloperators S�D�D∗ and N defined for � � � → R and �s ∈ � by

�S��s� =∫�

1

��s−�s ′��s′�d�s ′ (1.22)

�D��s� =∫�

�

�n�s ′

(1

��s−�s ′�)��s ′�d�s ′ (1.23)

�D∗��s� =∫�

�

�n�s

(1

��s−�s ′�)��s ′�d�s ′ (1.24)

�N��s� =∫�

�2


(1

��s−�s ′�)��s ′�d�s ′ (1.25)

When the interface � is regular (C1 at least), the Green kernels of the operators S�Dand D∗ exhibit integrable singularities: they behave as 1/��s−�s′� when �s′ goes to �s (for��s−�s′� ·n�s� ∼ ��s−�s′� ·n�s′ � ≤ ��s−�s′�2 when �s′ is close to �s). On the other hand, the Greenkernel of the operator N is hypersingular (it behaves as 1/��s−�s′�3 when �s′ is close to �s)so that the notations (1.21) and (1.25) are only formal: the integral∫

�

�2


(1

��s−�s ′�)��s ′�d�s ′


has to be given the sense of a Cauchy principal value [2].The operators S�D and D∗ play a central role in the usual ASC methods (DPCM,

COSMO, IEF). As the operator N does not appear in these methods, we will not furtherdetail its properties. Regarding the operators S�D and D∗, they satisfy the following threeproperties:

• Property 1: on L2��, the operator S is self-adjoint, and D∗ is the adjoint of D.• Property 2: DS = SD∗.• Property 3: denoting by Hs�� the Sobolev space of index s ∈ R [3], the applications

S � Hs�� → Hs+1��

and

�−D∗ � Hs�� → Hs�� for −2� < � < +�

are bicontinuous isomorphisms for any s ∈ R. We will comment on the practical consequencesof these properties in the end of Section ID. At this point, let us only mention that the functionalspace H0�� coincide with L2��, and that for s ∈ N

∗� Hs�� is the set of functions which are inL2�� and whose surface derivatives of orders lower than or equal to s all are in L2��. BesidesHs+1�� ⊂ Hs�� for all s ∈ R, and the larger s, the more regular the functions of Hs��.

The first two properties are algebraic in nature and are used in the formal derivationof the various ASC equations. The third property is concerned with functional analysis.As it is of no use for the formal derivation of ASC methods, it is rarely reported in thechemistry literature. However, it has direct consequences on the comparative numericalperformances of the various ASC methods (see Section 1.2.5).

In the special case of a spherical cavity, the operators S�D�D∗ and N have simpleexpressions. Assume for simplicity that � is the unit sphere S2. A function u defined on� = S2 can then be expended on the spherical harmonics Ym

l �� (see e.g. [4]):

u�� =+�∑l=0

∑−l≤m≤l

uml Ym

l ��

where uml are complex numbers. Recall that the spherical harmonics form a Hilbert basis

of L2�S2� so that, in particular,

∫S2

Yml Ym′

l′ =∫ �

0

∫ 2�

0Ym

l ��Ym′l′ �� sin � d� d� = ll′ mm′

The operators S�D�D∗ and N turn out to be diagonal in this basis:

for � = S2 (the unit sphere)

⎧⎪⎨⎪⎩Su = ∑+�

l=0

∑−l≤m≤l

4�2l+1 um

l Yml

Du = D∗u = − 12Su

Nu = −4�∑+�

l=0

∑−l≤m≤l

l�l+1�2l+1 um

l Yml


Note that D =D∗ for spherical cavities only. Still in this basis, the Sobolev spaces Hs�S2�have a nice, simple, definition

Hs�S2� ={u =

+�∑l=0

∑−l≤m≤l

uml Ym

l such that �u�2Hs =

+�∑l=0

∑−l≤m≤l

�l+1�2s�uml �2 < +�

}

The properties of the operators S�D and D∗ listed above can then be easily establishedin the special case when � is the unit sphere.

For more details on the properties of the operators S�D�D∗ and N , and in particularon their relation with Calderon projectors, we refer to ref. [2].

We conclude this mathematical section with the useful definitions of single-layer anddouble-layer potentials.

A single-layer potential is a function W which can be written as

W��r� =∫�

��s ′��r −�s ′� d�s ′� ∀�r ∈ R

3 \� (1.26)

with � ∈ H−1/2��. A single layer potential fulfils Equations (1.16) and the limitsdefined by Equations (1.17) and (1.18) exist. By identification with the representationformula (1.19), one finds

�W� = 0 and[�W

�n

]= 4�� (1.27)

This implies in particular that the potential W is continuous across � (and therefore onR

3), and that Equation (1.26) also holds true for �r ∈ � . In other words, � is solution tothe integral equation

S� = W

A double-layer potential is a function W which can be written as

W�x� =∫�

�

�ny

(1

�x−y�)

p�y�dy� ∀x ∈ R3 \�

with p∈H1/2��. A single-layer potential fulfils Equations (1.16) and the limits defined byEquations (1.17) and (1.18) exist. By identification with the representation formula (1.19),one finds

�W� = 4�p and[�W

�n

]= 0

A double-layer potential is continuous on R3 \� but exhibits a discontinuity across the

interface � . The density p is a solution to the integral equation

Np = −�W

�n


1.2.3 Reaction Field Energies of Interior Charges

The reaction potential VR is defined as VR = V −VM where V is the unique solution to

−�� · ��r� ��V��r�� = 4��M��r� (1.28)

vanishing at infinity, and where

VM��r� =∫R3

�M��r ′��r −�r ′� d�r ′

denotes the potential generated by �M in the vacuum. As ��r� = 1 in C and ��r� = �outside C, and as, in this section, �M is assumed to be supported inside C, one has⎧⎪⎨⎪⎩

−V = 4��M in C

−V = 0 outside C

V −→ 0 at infinity

Likewise, the potential VM also satisfies⎧⎪⎨⎪⎩−VM = 4��M in C

−VM = 0 outside C

VM −→ 0 at infinity

Hence VR = V −VM is such that⎧⎪⎨⎪⎩−VR = 0 in C

−VR = 0 outside C

VR −→ 0 at infinity

In QM calculations, �M is the sum of the nuclear contribution (a linear combination ofpoint charges located inside C) and of a regular function (the electronic density), that, inthis section, is assumed to be supported in C. It then follows from standard functionalanalysis results [3] that for such �M , the limits VR�i � VR�e � �VR/�n�i, and �VR/�n�edefined by Equations (1.17) and (1.18) exist, and VR is continuous across � . We thusinfer from the representation formula (1.19) that

∀�r ∈ �3� VR��r� =∫�

��s ′��r −�s ′� d�s ′ (1.29)

where

� = 14�

[�VR

�n

]


The reaction potential VR is therefore a single-layer potential. In order to calculate theapparent surface charge (ASC) distribution � , one makes use on the one hand of therelations

�VR

�n

∣∣∣∣i

− �VR

�n

∣∣∣∣e

= 4��

12

[�VR

�n

∣∣∣∣i

+ �VR

�n

∣∣∣∣e

]= D∗�

and on the other hand of the jump condition (see e.g. ref. [5])

0 = �V

�n

∣∣∣∣i

− ��V

�n

∣∣∣∣e

= �VR

�n

∣∣∣∣i

− ��VR

�n

∣∣∣∣e

+ �1− ��VM

�n

This leads to the integral equation(2�

�+1�−1

−D∗)� = �VM

�n(1.30)

Equation (1.30) is nothing but the DPCM equation [6, 7]. The existence and uniquenessof the solution � of Equation (1.30) is ensured by property 3 stated in Section 1.2.2.

The reaction field energy ER��′� can then, as announced, be written as an integral

over � :

ER��′� =

∫R3

�′��r�VR��r�d�r

=∫R3

�′��r�(∫

�

��s ′��r −�s ′� d�s ′

)d�r

=∫��s ′�

(∫R3

�′��r��r −�s ′� d�r

)d�s ′

=∫��s ′�V ′

M��s ′�d�s ′ (1.31)

The various IEF equations can be derived from the DPCM Equation (1.30) as follows.Multiplying Equation (1.30) by S on the left-hand side, we get

S

(2�

�+1�−1

−D∗)� = S

�VM

�n(1.32)

Using the commutation relation SD∗ = DS, we also have(2�

�+1�−1

−D

)S� = S

�VM

�n(1.33)


Applying the representation formula (1.20) to the function W defined by W��r� = 0 if �ris in C and W��r� = VM��r� if �r is outside C, we find that for all �s ∈ � ,

12VM��s� = −

∫�

1

4��s−�s ′��VM

�n��s ′�d�s ′ +

∫�

�

�n�s ′

(1

4��s−�s ′�)VM��s ′�d�s ′ (1.34)

The above relation can be rewritten, using the integral operators S and D, as

2�VM = −S�VM

�n+DVM (1.35)

Combining Equations (1.32), (1.33) and (1.35) it is possible to construct a whole familyof ASC equations, including the original IEF equation [8–10][

�2� −D�S+ 1�S�2� +D∗�

]� = −�2� −D�VM − 1

�S�VM

�n(1.36)

and the IEFPCM [11], also called SS(V)PE [12, 13], equation

S

[2�

�+1�−1

−D∗]� = −�−1

��2� −D�VM (1.37)

Equation (1.37) was obtained independently by Mennucci et al. [11] and byChipmann [12].

Note that the integral operators involved in the IEF and IEFPCM equations are in factthe same, up to a multiplicative constant, and are symmetric:(

S

[2�

�+1�−1

−D∗])∗

=[

2��+1�−1

−D∗]∗

S∗

=[

2��+1�−1

−D

]S

= S

[2�

�+1�−1

−D∗]

On the other hand, the integral operator of the DPCM Equation (1.30) is not symmetric.Finally, the COSMO model introduced in ref. [1] can be recovered as follows. First,

the IEFPCM Equation (1.37) can be rewritten as⎧⎨⎩S� = −VM(2�

�+1�−1

−D∗)� = �−1

��2� −D∗��

(1.38)

The COSMO model is an approximation of Equations (1.38) consisting in solving exactlythe first of Equations (1.38) and in replacing the second equation by

� � �−1�+k

�


where k is an empirical parameter. In the special case when � is the unit sphere, thesecond of Equations (1.38) can be solved analytically:

�ml = ��−1�2

��+1�

⎡⎢⎢⎣ 1+ 12l+1

1+ �−1�+1

12l+1

⎤⎥⎥⎦ �ml

The optimal value for k is k= 1 for l= 0 and k= 2 for l= +�. On the other hand, numer-ical simulations on real molecular systems seem to show that, depending on the chargeand shape of the system, the optimal value for k is between k = 0 and k = 1/2. Thediscrepancy between theoretical arguments and numerical results might originate in theescaped charge problem, that is addressed in the following section.

1.2.4 The Escaped Charge Problem

As underlined above, there is no approximation in the integral representation (1.31) ofthe reaction field energy, provided (i) the charge distribution � is entirely supportedinside the cavity C and (ii) � is computed using the DPCM Equation (1.30), the IEFEquation (1.36) or the IEFPCM Equation (1.37).

If condition (i) is not satisfied, the integral equation method presented in the previoussection needs to be modified. Proceeding as above, it is easy to show that the totalelectrostatic potential V solution to Equation (1.28) can be decomposed as

V��r� = V intM ��r�+ 1

�V ext

M ��r�+∫�

�a��s ′��r −�s ′� d�s ′

where

V intM ��r� =

∫C

��r ′��r −�r ′� d�r ′� V ext

M ��r�∫R3\C

��r ′��r −�r ′� d�r ′

and where �a is an apparent surface charge that can be obtained by solving someintegral equation involving the operators S� D, and/or D∗, as well as the potentials V int

M

and V extM and/or their normal derivatives. There is therefore no theoretical obstacle in

formulating an exact integral equation method in the presence of escaped charge. Inclassical molecular dynamics, this program can be easily realized. The main practicaldifficulty arising in quantum chemistry (in particular with gaussian basis sets) is thatthere is no convenient way to compute the potentials V int

M and V extM . For this reason,

quantum chemistry calculations are usually performed using the equations derived underthe assumption that the charge distribution is entirely supported inside the cavity. Theerror due to the escaped charge is either neglected or corrected by some empirical rule. Itis important to note that, whereas the DPCM, IEF and IEFPCM are exact (and thereforeequivalent) when there is no escaped charge, they are non-equivalent approximationsin the presence of escaped charge. Theoretical arguments [12], confirmed by numericalsimulations, show that the IEFPCM method behaves very much better than the DPCMmethod in the presence of escaped charge.


The simplest method to evaluate the magnitude of the error due to the escaped chargeconsists in computing the amount of escaped charge by means of Gauss’s theorem.Denoting by Q = ∫

R3 � the total charge, the escaped charge is

Qs = Q−∫C� = Q+ 1

4�

∫CVM��r�d�r = Q+ 1

4�

∫�

�VM

�n��s�d�s

If Qs/Q exceeds a few percent, it is likely that the calculation will not be very reliable.A more elaborate procedure consists in establishing error estimates. For instance, it isproved in ref. [14] that the exact reaction field energy ER�� and the IEFPCM estimateof it, denoted by EIEFPCM

R ��, satisfy

ER�� ≤ EIEFPCMR �� (1.39)

and

ER�� ≥ EIEFPCMR ��− �−1

�

[65

(4�3

)1/3

�1/3maxQ

5/3s −�S−1V ext

M �V extM ��

](1.40)

where �max = supR3\C ��. These inequalities are optimal (they reduce to equalities) if thecharge distribution � is entirely supported in C. Inequality (1.39) means that the IEFPCMmethod provides an upper bound of the exact reaction field energy. In practice, the lowerbound (1.40) can be estimated using calculations performed on the interface � [14].

1.2.5 Discretization Methods

The usual numerical methods for solving integral equations can be classified in twogroups: the collocation methods and the Galerkin methods.

Let us detail each approach for the example of the generic integral equation

A� = g (1.41)

where the unknown � belongs to Hs��, where the right-hand side g is in Hs′��,and where the integral operator A ∈ L�Hs��Hs′�� is characterized by the Greenkernel kA��s��s′�:

�A��s� =∫�kA��s��s ′��s ′�d�s ′� ∀�s ∈ �

Let us consider a mesh �Ti�1≤i≤n on � , that, in a first step, will be considered as drawnon the curved surface � ; let us denote by �si a representative point of the element Ti

(e.g. its ‘centre’). The P0 collocation and Galerkin methods for solving Equation (1.41)provide two approximations of � in the space Vh of piecewise constant functions whoserestriction to each element Ti is constant:

• in the collocation method, �c is the element of the Vh solution to∫�kA��si��s ′��c��s ′�d�s ′ = g��si�� ∀1 ≤ i ≤ n


• while in the Galerkin method, �g is the element of Vh satisfying

∀� ∈ Vh�∫�

(∫�kA��s��s ′��g��s ′�d�s ′

)��s�d�s =

∫�g��s� ��s�d�s (1.42)

These two methods lead to the matrix equations

�A�c · ��c = �g�c and �A�g · ��g = �g�g

where

�A�cij =

∫Tj

kA��si��s ′�d�s ′� �g�ci = g��si�

�A�gij =

∫Ti

(∫Tj

kA��s��s ′�d�s ′)

d�s� �g�gi =

∫Ti

g

��ci and ��

gi denoting the values of � on Ti under the collocation and Galerkin approx-

imations, respectively. The collocation method is more natural and easier to implement(at least at first sight); for these reasons, it is often used in apparent surface charge calcu-lations; on the other hand, the Galerkin method leads to a symmetric linear system whenthe operator A is itself symmetric, which may appreciably simplify the numerical resolu-tion of the linear system [15,16]. Let us remark incidentally that in the Galerkin setting,�D∗�g

ij = �D�gji. This symmetry is broken with the collocation method: �D∗�c

ij �= �D�cji.

The approximation methods described above belong to the class of boundary elementmethods (BEMs). BEMs follow the same lines as finite element methods (FEMs). Inboth cases, the approximation space is constructed from a mesh. The terminology FEM isusually restricted to the case when the equation to be solved is set on some domain of theambient space, whereas BEM implicitly means that the equation is set on the boundary ofsome domain of the ambient space. In most applications, FEMs are used to solve partialdifferential equations involving local differential operators. On the other hand, BEMsare often used to solve integral equations involving nonlocal operators. In the context ofimplicit solvent models, two options are open: either solve the (local) partial differentialEquation (1.28), complemented with convenient boundary conditions, by FEM on a 3Dmesh, or solve one of the (nonlocal) integral equations derived in Section 1.2.3, by BEMon a 2D mesh. In the former case, the resulting linear system is very large, but sparse.In the latter case, it is of much lower size, but full. The particular instances of BEMdescribed above are the simplest ones: on each element Ti of the mesh, the functionsof the approximation space are constant. In other words, they are polynomials of order0, hence the terminology P0 BEM. It is possible to further improve the accuracy ofthe approximation, while keeping the same mesh, by refining the description of the testfunctions on each Ti. In Pk BEM, the functions of the approximation space are continuouson � and such that their restriction to each Ti is piecewise polynomial of total degreelower than or equal to k in some local map (see ref. [2] for instance).

In many applications, a polyhedral approximation � of the surface � is used; it isobtained by considering the Ti as planar tesserae (Figure 1.1).


Points on the molecularsurface Molecular surface

Gauss points on Ti

Curved trianglePlanar trianglei T

Figure 1.1 Polyhedral approximation of a molecular surface.

This approximation makes easier the computation of the coefficients of the matrices

�S�gij =

∫Ti

∫Tj

1

��s−�s ′� d�s d�s ′ and �D�gij =

∫Ti

(∫Tj

�

�n�s ′

(1

��s−�s ′�)

d�s ′)

d�s

It is indeed to be noticed that the function

fS��s� =∫T

1

��s−�s ′� d�s ′

has an analytical expression when T is a planar triangle, which allows an inexpensiveevaluation of the inner integral

∫Tj

. Similarly, the function

fD��s� =∫T

�

��s ′

(1

��s−�s ′�)

d�s ′

which corresponds to the solid angle formed by the geometric element T and the centre�s [2] also admits a simple analytical expression for �s ∈ R

3 when T is planar. Let us noticethat in this case, fD��s� = 0 for any �s ∈ T ; therefore the diagonal elements �D�c

ii and �D�gii

are all equal to zero under this geometric approximation. In the Galerkin approximation,the outer integration can be performed with an adaptive Gaussian integration method [17],the number of integration points depending on the distance and relative orientation of theelements Ti and Tj .

The error induced by the polyhedral approximation can be estimated as follows [18]:

• for the resolution of S� = g,

�� − � �P−1�H−1/2�� ≤ C h3/2 ��H2��

• for the resolution of ��+D∗�� = g� −2� < � < +�,

�� − � �P−1�L2�� ≤ C h��H1��


where � denotes the exact solution of the integral equation on the exact surface �� theexact solution of the integral equation on � , h = max diam�Ti� the characteristic sizeof the sides of the polyhedron � �P the orthogonal projection on � (which defines aone-to-one application from � to � when h is small enough), and C a constant.

Let us remark incidentally that the van der Waals, solvent-accessible and solvent-excluded molecular surfaces commonly used in apparent surface charge calculations, canbe discretized without resorting to a polyhedral approximation. Indeed, these surfacesare made of pieces of spheres and tori and it is therefore possible to mesh and computeintegrals on the molecular surfaces since analytical local maps are available [19].

As a matter of illustration, let us write in detail the numerical algorithm for computingER��

′� with the PCM model (1.30) and (1.31) and the Galerkin approximation withP0 planar boundary elements:

1. Mesh an approximation of the cavity surface � with planar triangles.2. Assemble the matrix

�A�gij =

[2�

�+1�−1

−D∗]g

ij

= 2��+1�−1

area�Ti�area�Tj�− �D�gji

= 2��+1�−1

area�Ti�area�Tj�−∫Tj

(∫Ti

�

�n�s ′

(1

��s−�s ′�)

d�s ′)

d�s

by analytical (or numerical) integration on Ti and numerical integration on Tj .3. Assemble the right-hand side

�g�gi =

∫Ti

�VM

�n

by numerical integration.4. Solve the linear system

�A�g��g = �g�g (1.43)

5. Compute ER��′� by the approximation formula

ER��′� � E

appR ��′� =

n∑i=1

��gi

∫Ti

V ′M

the integrals∫Ti

being calculated numerically.

Recall that when the charge densities � and �′ are composed of point charges, dipoles,or gaussian–polynomial functions, analytical expressions of the potential VM and thenormal derivatives �VM/�n are available.

It can be proved that this numerical method is of order 1 in h = max diam�Ti�. Asmentioned above, higher order methods can be obtained by first using curved tesseraeinstead of planar triangles and then increasing the degree of the polynomial approximationon each tessera (P1 or P2 BEM [2]).


The transposition to the above algorithm to the COSMO framework is straightforward.On the other hand, the extention to IEF-type methods require some attention. Indeed, adirect transposition of the above algorithm to the IEFPCM framework leads to the matrixelements

�A�gij = 2�

�+1�−1

�S�ij −∫Ti

d�s∫�

d�s ′∫Tj

d�s ′′ 1

��s−�s ′��

�n�s ′

(1

��s ′ −�s ′′�)

For practical calculations, the integral over � has to be discretized, which introduces anadditional numerical error. An alternative consists in applying the Galerkin approximationto system (1.38), which is equivalent to Equation (1.37). The discretized apparent surfacecharge �� is obtained by solving successively the linear systems

�S�g ��g = −�VM�g (1.44)

�B�g� ��

g = �−1�

�B�g� ��g (1.45)

with

��B�g��ij = 2�

�+1�−1

area�Ti�area�Tj�− �D�gji and �B�� = lim

�→+��B��

The computational efficiency of an integral equation method is related to the size, thestructure and the conditioning of the linear systems to be solved. Recall that there arebasically two strategies to solve an N ×N linear system of the form Ax = b. The firstoption is to store the matrix A and to invert it by a direct method, such as the LUdecomposition or the Choleski algorithm [15] (the latter algorithm being restricted to thecase when A is symmetric, positive definite). The second option is to solve the linearsystem Ax = b by an iterative method [16], such as the conjugate gradient algorithm(if A is symmetric, positive definite), or the GMRes or BiCGStab algorithms (in thegeneral case). Iterative methods only require the calculation of matrix–vector productsand scalar products. For large systems, the first option is not tractable: the memoryoccupancy scales as N 2 and the computational time as N 3. The linear systems associatedwith the COSMO, DPCM, IEF and IEFPCM methods enjoy a remarkable property thatmake iterative methods very efficient: as the corresponding matrices A originate fromintegral operators involving the Poisson kernel 1/��r� or its derivatives, it is possible tocompute matrix–vector products Ay for y ∈ R

N , without even assembling the matrix A, inN log N elementary operations, by means of Fast Multipole Methods (FMMs) [20,21].The number of conjugate gradient, GMRes or BiCGStab iterations depends on the onehand on the quality of the initial guess, and on the other hand on the conditioning of thelinear system. Recall that the conditioning parameter of an invertible matrix A for the� · �2 norm defined by �A�2= supx∈RN �Ax� / �x� (� · � denoting the euclidian normon R

N ) is the real number �2�A� =�A�2�A−1�2. If A is symmetric, definite positive,�2�A� = �N�A�/�1�A� where 0 < �1�A� ≤ · · · ≤ �N�A� are the eigenvalues of A. Thelarger �2�A�, the larger the number of iterations. If A is symmetric, it can indeed be


proved that the sequence �xk� generated by the conjugate gradient algorithm with initialguess x0 converges to the solution x to Ax = b, and that one has the error estimate

�xk −x�A ≤ 2

(√�2�A�−1√�2�A�+1

)k

�x0 −x�A

where �y�A= �Ay� y�1/2. Note that �2�A� ≥ 1 and that �2�A� = 1 if and only if A is theidentity matrix, up to a multiplicative constant. Not surprisingly, the conjugate gradientalgorithm converges in a single iteration in the latter case. For completeness, let us alsomention that the conjugate gradient converges in at most N iterations.

It follows from the above arguments that the efficiencies of the various integralequation methods under examination are directly related to the conditioning parametersof the matrices �S� and ��−D∗�. It is at that point that the functional analysis propertiesof the underlying operators S and �−D∗ come into play. Indeed, as �−D∗ is for all� > −2� an isomorphism on L2�� and as the P0 BEM test functions are in L2��, theconditioning parameter of the matrix �2��+1�/��−1�−D∗�g is bounded independentlyof N. Consequently, the number of iterations needed to solve the PCM Equation (1.30)or Equation (1.45) in the P0 BEM Galerkin approximation does not dramatically vary ifthe mesh is refined. On the other hand, while the operator S is bounded from L2�� toL2�� S−1 maps L2�� onto H−1�� and is therefore an unbounded operator on L2��.This implies that the larger eigenvalue of �S�g is bounded independly of the size of themesh, and that the smallest eigenvalue of �S�g goes to zero when the mesh is refined.Hence, the conditioning of �S�g goes to infinity when the mesh is refined. This problemis encountered with the COSMO, IEF and IEFPCM methods. In order to prevent theiterative algorithm from breaking down in the limit of large molecular systems and/orfine mesh, preconditioning techniques are needed [16].

In the special case of spherical cavities and regular meshes, analytical estimates of theconditioning parameters of �S�g and �2��+1�/��−1�−D∗�g are available: �2��S�

g� �N 1/2 and �2��2��+1�/��−1�−D∗�g� � 2�/��+1�.

1.2.6 Derivatives and Geometry Optimization

For molecular systems in the vacuum, exact analytical derivatives of the total energywith respect to the nuclear coordinates are available [22] and lead to very efficientlocal optimization methods [23]. The situation is more involved for solvated systemsmodelled within the implicit solvent framework. The total energy indeed contains reactionfield contributions of the form ER��

′�, which are not calculated analytically, but arereplaced by numerical approximations E

appR ��′�, as described in Section 1.2.5. We

assume from now on that both the interface � and the charge distributions � and �′depend on n real parameters ��1� · · · ��n�. In the geometry optimization problem, the�i are the cartesian coordinates of the nuclei. There are several nonequivalent ways toconstruct approximations of the derivatives of the reaction field energy with respect tothe parameters ��1� · · · ��n�:

1. One way consists in first calculating analytically the derivatives ��/��i�ER��′� of the exact

reaction field energy, and then approximating ��/��i�ER��′�, yielding the quantities denoted

by ��/��i�ER��′��app.


2. A second way consists in calculating the derivatives ��/��i�EappR ��′� of the approxi-

mated energy EappR ��′�. This second approach can be subdivided into three methods:

��/��i�EappR ��′� can be computed (i) by finite differences, (ii) by deriving analytically the

discrete equations used for the calculation of EappR ��′�, (iii) by automatic differentiation [24].

Although (ii) and (iii) are theoretically equivalent, they are not in practice: they correspond totwo dramatically different implementations of a single mathematical formalism.

The main practical difficulty in optimizing the geometry of solvated molecules arisesfrom the fact that Eapp

R ��′� is not, in general, a continuous function of the parameters �i.Discontinuities are indeed introduced by the mesh generator. Efficient, robust geometryoptimization procedures for solvated molecules are still to be designed.

Let us conclude this section by providing an expression of the analytical derivative

�

��i

�ER��′��

at ��1� · · · ��n� = ��∗1� · · · ��∗

n� valid in the case when � and �′ are supported inside thecavity. Let us denote by � �= ��∗

1� · · · ��∗n�, and denote for all s ∈ � by

�U i� ·n��s� = d

dtd[�s� ��∗

1� · · · ��∗i−1��

∗i + t� �∗

i+1� · · · ��∗n�]∣∣∣∣

t=0

the velocity field generated by an infinitesimal variation of ith parameter. In the previousexpression, d��s� ��·�� denotes the signed distance between �s and ��·�:

d�x��·�� ={

− inf ��y−x�� y ∈ ��·� if x ∈ R3\ ⊆ �·�

+ inf ��y−x�� y ∈ ��·� if x ∈⊆ �·�

The analytical derivative formula then reads [25]

�

��i

�ER��′�� =

∫��V ′

M

��i

+∫�� ′ �VM

��i

+∫�

�

4��Ui

� ·n� (1.46)

with

� = 16�2�

�−1� � ′ + ��−1��V��V

′�� (1.47)

��V��s� denoting the projection of the vector �V��s� on the tangent plane to � at �s. Inthe limit � = +�, one has [26]

�� = 16�2�� ′

The integral ∫�

��

4��Ui

� ·n� =∫�

4�� ′�U i� ·n�


then has a simple physical interpretation:∫�

4�� ′�U i� ·n� is the virtual power of the

electrostatic pressure p = 4�� ′ exerted on the walls of a perfect conductor [5]. Whenthe permitivity � is high (which is typically the case for water) the approximate analyticalderivative formula

�

��i

ER��′� �

∫�� ′ �VM

��i

+∫��V ′

M

��i

+ 4��

�−1

∫�� ′�U i

� ·n�

is reasonably accurate [27].

1.2.7 Beyond the Standard Dielectric Model

The range of application of the integral equation method is not limited to the standarddielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals),weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),� � � However, it is required that the electrostatic equation outside the cavity is linear,with constant coefficients. For instance, liquid crystals and weak ionic solutions can bemodelled by the electrostatic equations

−div ��LC ·�V� = 4�� (1.48)

and

−div ��PB�V�+ �PB�2V = 4�� (1.49)

respectively. In Equation (1.48), �LC is a 3×3 symmetric positive definite matrix whoseeigenvectors correspond to the principal axes of the liquid crystal. Equation (1.49) is alinearization of the nonlinear Poisson–Boltzmann equation

−div ��PB�V�+ �PB sinh��2V� = 4�� (1.50)

and is valid for weak ionic solutions, in the limit when �2V is small (� is the Debyelength of the ionic solution). It is important to note that the integral equation method isnot appropriate for strong ionic solutions, since Equation (1.50) is nonlinear.

One can associate with any linear electrostatic equation with constant coefficient,formally denoted by LeV = 4�� (Le is a differential operator with constant coefficients),a function Ge��r� called the Green kernel of the operator Le/4� and defined by

LeGe = 4� 0

where 0 is the Dirac distribution. In particular the Green kernels for Equations (1.48)and (1.49) read

Ge��r� =⎧⎨⎩�det �LC�

−1/2 ��−1LC�r� �r�−1/2 for Equation (1.48)

exp�−��r�� PB��r��−1 for Equation (1.49)


In the special cases when �LC is the identity matrix, and when �PB = 1 and � = 0, bothEquations (1.48) and (1.49) reduce to the Poisson equation −V = 4��, and Ge��r� =��r�−1 (��r�−1 is the Green function of the operator −/4�).

When the linear isotropic dielectric medium used in the standard model is replaced witha linear homogeneous medium with Green kernel Ge, and when the charge distributionis entirely supported inside the cavity, the reaction potential inside the cavity still has asimple integral representation:

∀�r ∈ C� VR��r� =∫�

��s ′��r −�s ′� d�s ′ (1.51)

The apparent surface charge � involved in the above expression satisfies the integralequation

��2� −De�S+Se�2� +D∗�� = −�2� −De�VM −Se

�VM

�n

where S and D∗ are given by Equations (1.22) and (1.24) and where Se and De

are defined by similar formulae as S and D, replacing ��s −�s′�−1 with Ge��s −�s′� and��/�n�s′��s−�s′�−1 with �� ·��s′Ge��r−�r ′�� ·n�s′ respectively. An important difference betweenthe integral representation formulae (1.29) (standard model) and Equation (1.51) is thatEquation (1.29) is valid on the whole space R

3 whereas Equation (1.51) only holds trueinside the cavity. The reaction field energy of two charge distributions � and �′ bothsupported inside the cavity can nevertheless be obtained remarking that

ER��′� =∫

IR3�′VR =

∫C�′VR

=∫C�′��r�

(∫�

��s ��r −�s � d�s

)d�r

=∫��s�

(∫C

� ′��r��r −�s� d�r

)d�s

=∫��s�

(∫IR3

� ′��r��r −�s � d�r

)d�s

=∫��s�V ′

M��s�d�s

Lastly, let us mention that the integral equation method applies mutatis mutandisto the case of multiple cavities (i.e. to the case when C has several connectedcomponents). This situation is encountered when studying chemical reactionsin solution.

References

[1] A. Klamt and G. Schüürman, COSMO: A new approach to dielectric screening in solventswith expressions for the screening energy and its gradient, J. Chem. Soc. Perkin Trans.,2 (1993) 799.


[2] W. Hackbusch, Integral Equations – Theory and Numerical Treatment, BirkhäuserVerlag, (1995).

[3] E. H Lieb and M. Loss, Analysis, 2nd edn, American Mathematical Society, New York, (2001).[4] E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational

quantum chemistry: a primer, in Ph. Ciarlet and C. Le Bris (eds), Handbook of NumericalAnalysis. Volume X: Special Volume: Computational Chemistry, Elsevier, Amsterdam, (2003),pp 3–270.

[5] L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, Butterworth-Heinemann, (1999).

[6] S. Miertuš, E. Scrocco and J. Tomasi, Electrostatic interaction of a solute with a continuum.A direct utilization of ab initio molecular potentials for the prevision of solvent effects, Chem.Phys., 55 (1981) 117.

[7] J. Tomasi and M. Persico, Molecular interactions in solution: An overview of methods basedon continuous distribution of solvent, Chem. Rev., 94 (1994) 2027.

[8] E. Cancès and B. Mennucci, New applications of integral equation methods for solvationcontinuum models: ionic solutions and liquid crystals, J. Math. Chem., 23 (1998) 309.

[9] E. Cancès, B. Mennucci and J. Tomasi, A new integral equation formalism for the polariz-able continuum model: theoretical background and applications to isotropic and anisotropicdielectrics, J. Chem. Phys., 107 (1997) 3032.

[10] B. Mennucci, E. Cancès and J. Tomasi, Evaluation of solvent effects in isotropic andanisotropic dielectrics, and in ionic solutions with a unified integral equation method: theo-retical bases, computational implementation and numerical applications, J. Phys. Chem. B,101 (1997) 10506.

[11] B. Mennucci, R. Cammi and J. Tomasi, Excited states and solvatochromic shifts within anonequilibrium solvation approach: A new formulation of the integral equation formalismmethod at the self-consistent field, configuration interaction, and multiconfiguration self-consistent field level, J. Chem. Phys., 109 (1998) 2798.

[12] D. M. Chipmann, Reaction field treatment of charge penetration, J. Chem. Phys., 112(2000) 5558.

[13] E. Cancès and B. Mennucci, Comment on: Reaction field treatment of charge penetration,J. Chem. Phys., 114 (2001) 4744.

[14] E. Cancès and B. Mennucci, The escaped charge problem in solvation continuum models,J. Chem. Phys., 115 (2001) 6130.

[15] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press,Ithaca, NY, (1996).

[16] Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn, Society for Industrial andApplied Mathematics (2003).

[17] P. J. Davis and I. Polonsky, in M. Abramowitz and I. A. Stegun (eds), Handbook of Mathe-matical Functions, Dover Publications, New York, Chapter 25, (1965) pp 875–924.

[18] J. C. Nédélec and J. Planchard, Une méthode variationelle d’éléments finis pour la résolutiond’un problème extérieur dans R

3, RAIRO 7 (1973) 105.[19] R. J. Zauhar and R. S. Morgan, Computing the electric potential of biomolecules: applications

of a new method of molecular surface triangulation, J. Comput. Chem., 11 (1990) 603.[20] L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace

equation in three dimensions, Acta Numerica 6 (1997) 229.[21] G. Scalmani, V. Barone, K. N. Kudin, C. S. Pomelli, G. E. Scuseria and M. J. Frisch,

Achieving linear-scaling computation cost for the polarizable continuum model of solvation,Theoret. Chem. Acc., 111 (2004) 90.

[22] J. A. Pople, R. Krishnan, H. B. Schlegel and J. S. Binkley, Derivative studies in Hartree–Fockand Møller–Plesset theories, Int. J. Quantum Chem., 13 (1979) 225.


[23] P. Y. Ayala and P. B. Schlegel, A combined method for determining reaction paths, minimaand transition state geometries, J. Chem. Phys., 107 (1997) 375.

[24] G. Corliss, C. Faure, A. Griewank, L. Hascoet and U. Naumann, (eds), Automatic Differen-tiation of Algorithms, from Simulation to Optimization, Springer, Heidelberg, (2001).

[25] E. Cancès and B. Mennucci, Analytical derivatives for geometry optimization in solvationcontinuum models I: Theory, J. Chem. Phys., 109 (1998) 249.

[26] E. Cancès, PhD Thesis, Ecole Nationale des Ponts et Chaussées (in French), (1998).[27] E. Cancès, B. Mennucci and J. Tomasi, Analytical derivatives for geometry optimization in

solvation continuum models II: Numerical applications, J. Chem. Phys., 109 (1998) 260.[28] S. Corni and J. Tomasi, Excitation energies of a molecule close to a metal surface, J. Chem.

Phys., 117 (2002) 7266.

1.3 Cavity Surfaces and their Discretization

Christian Silvio Pomelli

1.3.1 Introduction

In a previous contribution in this book, Cancès has presented the formal background of theintegral equation methods for continuum models and has shown how the correspondingequations can be solved using numerical methods. In this chapter the specific aspects ofthe implementation of such numerical algorithms within the framework of the PolarizableContinuum Model (PCM) [1] family of methods will be considered.

As described in the previous contributions by Cancès and by Tomasi, in such a familyof methods the solvent effects on the molecular solutes are evaluated by introducing a setof apparent charges representing the polarization of the dielectric medium. These chargesare obtained by solving integral equations defined on the domain of the boundary of thecavity which hosts the molecular solute. The solution of such equations can be dividedin two main steps.

The first step defines a molecule–solvent boundary from the molecular geometry andsome solvent-related quantities. This boundary is then discretized in a finite number ofsmall elements called tesserae. This step is independent of the molecular structure theoryin use (MM, DFT, MP2, etc.).

The second step solves the integral equations using the boundary elements previouslyintroduced. The result of this second step is the evaluation of the various contributions ofdifferent physical origin (electrostatic, repulsion, dispersion, cavitation) which determinethe solvent reaction field. This second step depends (at least for the electrostatic part) onthe level of description of the molecular structure.

The main scope of this chapter is to give some numerical and computational detailsof the machinery that is under the surface of modern continuum solvation models andespecially those belonging to the PCM family. Knowledge of the details of the boundarypartitioning into elements can help one to avoid numerical troubles especially with large(or complex) molecular systems. A smart choice of the method used to solve discretizedintegral equations can lead to valuable savings in CPU time and hard disk usage and canpermit calculations to be performed on large solvated systems with limited computationalresources.

This chapter is divided into three main parts: one presents and comments the mainaspects related to the definition of the solute cavity and the solvent–solute boundary, thesecond focuses on the numerical techniques to obtain boundary elements while the thirdpart describes the main numerical procedures to solve the integral equations.

1.3.2 The Cavity and its Surface

In continuum solvation methods the molecular cavity is the portion of space within thesurrounding medium (solvent) that is occupied by the solute molecule: the boundary ofthe molecular cavity is called molecular surface.

There are several models to define the molecular cavities and their surfaces. Histori-cally, the first models proposed were based on the simplest three-dimensional geometrical


shapes: the sphere [2] and the ellipsoid [3]. The radius of the sphere, or the ellipsoidaxes, are given as parameters and they are empirically based on the extension in space ofthe molecule. These simple models, which disregard many of the stereochemical detailsof the molecule, are still in use as they allow an analytical solution of the electrostaticequations defining the solvent reaction field.

A completely different definition is based on the isodensity surface [4], i.e. the surfaceconstituted by the set of points having a specified electronic density value (given as aparameter).

The most common way to define molecular cavities, however, is to use a set ofinterlocking spheres centred on the atoms constituting the molecular solute (Figure 1.2).Based on such a definition of the cavity, we can define different molecular surfaces:

Figure 1.2 Definitions of cavities based on interlocking spheres. In black (dashed) thespheres centred on atoms A and B, in red the SAS, in cyan the shared parts of VWS and SES.In green the concave part of SES. In blue the crevice part of VWS. In black (dotted) somepositions of tangent solvent probes (see Colour Plate section).

(i) The van der Waals surface (VWS) is defined as the surface obtained from a set of interlockingspheres, each centred on an atom or group of atoms and having as radius the correspondingvan der Waals radius. Several compilations of van der Waals radii [5, 6] are reported in theliterature. The VWS is commonly used to calculate the cavitation contribution to the solvationfree energy, namely the energy required to build a void cavity inside the medium (see also thechapter by Tomasi).

(ii) The solvent-accessible surface (SAS) [7] is defined as the surface determined by the set ofpoints described by the centre of a spherical solvent probe rolling on the VWS: the radiusof the solvent probe is related to the dimensions and the nature of the solvent. From thisdefinition it turns out that the SAS is equivalent to a VWS in which the radius of the solventprobe is added to each atomic radius. The SAS is commonly used to calculate the short-range(dispersive and repulsive) contributions to the solvation free energy.

(iii) The solvent-excluded surface (SES) [8] is defined as the surface determined by the set ofthe tangent (or contact) points described by a spherical solvent probe rolling on the VWS.This surface delimits the portion of space in which the solvent probe cannot enter withoutintersecting the VWS. The SES appears as the VWS in which the crevices correspondent tosphere–sphere intersection are smoothed; the convex part of the SES is shared with the VWSand is called the contact surface, whereas the part of the surface which is not shared withthe VWS is concave and is called the re-entrant surface. The region of the space, which isenclosed in the SES but not in VWS, is called the solvent-excluded volume.


The Solvent-excluded VolumeAs described above VWS and SAS are easily defined as sets of spheres centred on atoms.This definition, however, does not apply to SES; in this case in fact, the pair of surfacesdelimiting the boundary between the excluded volume and the solvent cannot be definedusing spheres. There are several algorithms which translate the abstract definition of theSES into a complex solid composed of simple geometrical objects from which the surfacecan be easily tessellated.

The first and most famous algorithm to calculate the SES has been proposed byConnolly [9]: in this algorithm a set of points on the surface of the solvent sphericalprobe is acquired by rolling the sphere on the VWS and it is further organized in a meshto build the tessellation. The rolling and sampling procedures has been improved overthe years so to give an optimal meshing.

The package of Connolly, named MSDOT, is widely used in molecular modeling forvisualization of molecules (especially in the field of biochemistry and molecular biology),ESP fitting, and docking but it has been rarely used in combination with continuumsolvation methods [10]. In its modern formulation, the Connolly surface presents a fullanalytical tessellation [11] but the reliability of it and of its differentiability has neverbeen tested with PCM-like calculations.

As a matter of fact, in the field of molecular modelling and molecular graphics there areseveral algorithms to calculate the molecular volume and surface and to visualize them,but the number of tesserae needed to produce a good graphical rendering is larger thanthat needed for the solution of the PCM equations and none of the rendering/modelling-oriented methods yields a differentiable tessellation.

Completely different approaches are DefPol and BLMOL. In DefPol [12] a giant poly-hedron with triangular faces, built around the whole molecule, is deformed until its verticeslie on the molecular surface. This latter is described by a shape function different fromzero only in the space inside the molecular cavity. The shape function is a combina-tion of terms related to single atomic spheres supplemented by terms related to pairs ortriples of spheres. The multiple sphere terms take account of the solvent-excluded volume.DefPol can also be used for VWS and SAS, simply by skipping the calculation of two-and three-sphere terms. The method is fast from the numerical point of view, but it isaffected by serious numerical problems in computing derivative terms and to be appliedto oblong and nonconvex molecular shapes. For these reasons, it is currently not in use.BLMOL [13] is a specialized version of a very general tessellated surfaces packagecalled BLSURF [14]. The BLMOL package partitions the SES in patches and trian-gulates each of them by using an advancing front algorithm. Each patch represents aconnected portion of the surface with homogenous curvature properties (e.g. a fragmentof an atomic sphere, a portion of torus generate by the rolling of the solvent probewhile tangent to two spheres, etc.). BLMOL requires a dimension of the single triangulartesserae very small with respect to that commonly used in this context; these characteris-tics and the fact that it is not freely available limit its use. Also, the BLMOL tessellationis in principle differentiable but its derivatives have never been implemented.

The last method which will be considered here is the GEPOL, which was first elabo-rated in Pisa by Tomasi and Pascual-Ahuir [15]. GEPOL will be presented in two steps:in this section we will treat the excluded volume filling, whereas the definition of surfaceelements will be given in the next section.


The GEPOL ApproachIn GEPOL the excluded volume is approximated by a set of supplementary (or ‘added’)spheres, which are defined through a recursive algorithm. The spheres centred on atomsconstitute the first generation of sphere. For each pair of spheres, for which rAB <RA +RB +2RS where rAB is the distance between the atoms and RA�RB�RS are the radiiof atomic and solvent probe spheres, one or more spheres are added. The centre of thenew spheres lies on the segment joining the centres of the two generating spheres and theposition and the radius of the spheres are chosen in such a way as to maximize the solvent-excluded volume filled by the new sphere. This procedure is repeated recursively withthe inclusion of the newly generated spheres in the pair-search procedure: in principlethis process should not terminate as it tries to fill a concave space with convex objects.Its termination is determined by two tests, namely:

1. If the radius of the generated sphere is less than a given threshold, such a sphere is not addedto the sphere set.

2. If the generated sphere overlaps the existing spheres too much, it is not added to the sphere set.A geometrical parameter is used to decide if this condition is verified, and several versions ofthis test have been proposed over the years.

In Figure 1.3 some examples of ‘added’ sphere patterns are illustrated. It is evidentthat the number, position and radius of these spheres change with the change of themolecular geometry. The space filling procedure has been upgraded over the years,so to efficiently handle large molecular systems, such as proteins [16], to account formolecular symmetry [17,18] and to reduce the computational complexity from quadraticto linear [19] by using lists of nearby spheres.

(a) (b) (c)

Figure 1.3 Generation of GEPOL added spheres. (a) For two close spheres a single sphereintersecting with the two parent ones is generated. (b) For farther spheres, first a spherethat does not intersect with the two parent spheres is generated, then two ‘third generation’spheres are added between the second generation sphere and each of the two first generationspheres. (c) For any pair of spheres with a large separation, small spheres very overlappedwith the primitive ones are generated. This last case occurs only with very loose thresholds forthe termination tests. In each case all the added spheres are tangential to the solvent probespheres tangential to both the atomic spheres.


The definition of excluded volume in GEPOL which is exact only if we consider aninfinite generation of supplementary spheres, replaces the complex geometrical structureof torus and curvilinear prisms used in BLSURF, Connolly and DEFPOL by simplyextending the set of atomic spheres. This aspect is very important from the computationalpoint of view, because it allows an easy development and implementation of well-definedtessellations.

1.3.3 The Surface Tessellation

In order to be suitable in the application of the boundary element method (BEM)procedures required to build the reaction field, a molecular surface must be tessellated.A tessellation is a partition of a surface in subsets named tesserae each with a surface areaa, a sampling point �s and a unit outward vector n at the sampling point. The tessellationelements �a��s� n� are the main quantities used to solve the BEM equations.

A differentiable tessellation is defined in such a way that it is possible to analyticallycalculate derivatives with respect to the molecular geometry. A tessellation is well definedwhen the tessellation related quantities and their derivatives are stable from the numericalpoint of view. The kinds of partitions of the surface area that lead to a well definedtessellation are one of the main issues of this contribution and will be discussed in thenext section.

Tessellation of SpheresThe partition of the sphere surface is a well known topic in geometry [20]. Apart from themathematical speculation, this problem is very important in modern computer graphicsfor the rendering of spherical objects. An important remark is that for the computation ofthe reaction field even at high numerical accuracy it is sufficient to partition a surface intoa number of elements noticeably smaller than that used in any modern rendering package.In particular, the various versions of GEPOL that have been released through the yearsuse geodesic partition schemes based on polyhedra inscribed into a sphere. The originalversion exploits a 60 tesserae partition scheme based on a pentakisdodecahedron for allthe spheres [16]. A flexible partition scheme has been introduced by using some basicpolyhedra, in which the original triangular faces are partitioned through an equilateraldivision procedure [21] (see Figure 1.4 for details). The equilateral division procedure

Figure 1.4 Equilateral division of a triangle. From left to right, divisions of order N = 2�3�6.Each side of the original triangle is divided in N equal parts (in the case of spherical trianglesthe sides are circumference equatorial arcs). A segment (or an arc) is traced from eachdivision point to the corresponding point on another side, so that the final result is a divisionof the original triangle in N2 triangles.


replaces each original triangle of the polyhedron with N 2 triangles, being N the orderof equilateral division, so that if M is the original number of polyhedron faces, the finalone is MN 2.

There are two ways of using this flexible partition scheme, (i) the same partition ofthe surface is used for each sphere (TsNum), or (ii) a number of tesserae proportional tothe sphere surface (TsAre) is used (see Figure 1.5).

Figure 1.5 Molecular cavity for H2CO using the TsNum = 60 option (left) and theTsAre = 0�2 option (right). Both the cavities respect the C2v symmetry of the molecule.

The TsAre option is nowadays the default option in some widely used computationalpackages. Details on the benefits of the TsAre scheme are reported in the subsectionabout GEPOL numerical stability.

Quantum mechanical computational packages use the molecular symmetry in order toreduce the computational effort. This feature can be used if the point sampling of thecavity surface respects the molecular symmetry. A way of obtaining this requirementconsists in partitioning each sphere surface by respecting the molecular symmetry [17]:this can be obtained by using basic polyhedra which subtend the same point group of themolecule, so that the resulting cavity partition is invariant under any geometrical transfor-mation that belongs to the molecular symmetry group. In this way a symmetry-reducedcavity, containing only ‘unique by symmetry’ tesserae is obtained (this procedure issimilar to the ‘petite list’ of orbitals used in symmetry-adapted ab initio calculations [22]).

Partition of Intersecting SpheresWhen two or more spheres intersect, some of their tesserae are cut to exclude the portionof their surface that lies inside the other spheres. In GEPOL, this cutting procedure testswhether a tessera intersects a sphere surface (excluding the sphere to which the tesserabelongs) and cuts the part of the tessera that lies inside it, so that for any tessera–sphereintersection a part of the tessera is cut away. If the entire tessera lies inside the sphere,it is completely removed from the tesserae list. Such a procedure is repeated for anysphere–tessera pair. The computational cost of this step can be reduced, as for the addedsphere generation, if a list of nearby spheres has previously been generated [19].

The first version of the tesserae cutting scheme [23] in GEPOL was based on a simplepartition in sub-tesserae. The resulting tessellation was not differentiable. Because adifferentiable tessellation is essential to use gradient-based automatic geometry optimiza-tion procedures, an analytical calculation of the cut tessera area has been introduced [16].


The geometrical definitions and equations to be used are those of the generalized spher-ical polygon [24], which is the portion of spherical surface delimited by one or moreplanes that pass through the sphere centre. The spherical polygon is generalized if one ormore of the planes do not pass through the sphere centre [13]. In contrast to plane poly-gons, a spherical polygon can have only one or two sides (note that the original uncuttessera is a spherical triangle). Each cutting sphere adds a delimiting plane that does notpass through the centre of the sphere on which the tessera lies. The number of differentcases which can arise from the intersection between a spherical triangle and one or morespheres is very large: details on this topic are beyond the scope of this chapter. The twomost common cases are illustrated in Figure 1.6.

Figure 1.6 Two common cases of intersection. The cutting sphere removes A and B verticesreplaced by D and E (left). The result is a smaller (and irregular) triangle. The cutting sphereremoves vertex B replaced by D and E (right). The result is an irregular quadrilateral polygon(See Colour Plate section).

The final result of the cutting is a generalized spherical polygon, for which the surfacearea of the tesserae can be analytically calculated [23]. The sampling point is taken asthe average of the polygon vertices on the sphere surface. This procedure leads to adifferentiable tessellation but suffers from numerical troubles in some cases [25].

Some Difficult CasesThe various steps of GEPOL we have described above are not fully reliable from anumerical point of view especially when used in a gradient-based geometry optimiza-tion procedure. The contribution of the surface elements to the gradients is calculatedconsidering the variation of shape and area of each tessera with respect to the displace-ment of the intersecting spheres. The primitive spheres are centred on atoms and thusthey follow them in the molecular geometry evolution. The displacement of the addedspheres is related to the atoms that appear in their genealogical trees: also, the radii ofthe added spheres are variable by definition. The evolution of the added spheres duringa geometry optimization can lead to their annihilation when their overlap with otherspheres and/or their radius falls under the selected thresholds. As a result, this variationof the set of added spheres leads to a discontinuity in the description of the solvent reac-tion field [21]. Typical cases in which this discontinuity can occur are those in whichthere is a large variation of the distance between two atoms (dissociations, rearrange-ments, etc.). This kind of numerical instabilities does not alter the final stationary pointreached by the optimization procedure, but it can increase the number of steps needed


to reach it. In extreme cases the optimization procedure may enter an infinite loop,in which the molecular geometry walks around the stationary point but never reachesit. This occurs when the distance between the geometrical place of sphere annihilationand the stationary point is very small. This infinite loop is generally characterized bya pseudo-periodic behaviour of the geometry optimization related parameters (energy,displacement and gradient norms, etc.). This problem can be resolved by a manual restartof the optimization procedure, in particular:

1. Choose a step of the optimization procedure located just before the infinite loop.2. Slightly alter by hand the chosen geometry.3. Restart the procedure.

If this procedure is not successful, a further possibility is to alter the threshold parametersfor the sphere annihilation. In more unlucky cases some patient tuning work is required.

Another possible source of troubles in GEPOL is the presence of ill-defined tesserae,i.e. a very small tessera and/or a tessera with a complex or oblong shape. Some typicalcases are illustrated in Figure 1.7. Ill-defined tesserae can affect the solution of the PCMequations, the convergence of the SCF and the convergence of the geometry optimizationprocedure. A large part of these problems can be solved by using the TsAre option inthe sphere partition procedure and by the usage of group spheres for groups such asCHn� n = 1�2�3. A manual inspection and resolution of problems related to ill-definedtesserae is not possible (the zoomed part of the phenol cavity reported in Figure 1.6 isless than the 1 % of the total surface). Fortunately, many GEPOL versions in use havebuilt-in tests and tricks to intercept and remove these numerical troubles [19, 26]. In thefew cases in which these automatic procedures do not work, a tuning procedure similarto those proposed for the added spheres problem can be used: in this case the parameterto be altered is the TsAre value.

Figure 1.7 A zoomed detail of the phenol cavity near the ring centre. Some typical cuttesserae shapes are shown. A: a tessera with complex cutting but without problematic situ-ations. B: an oblong tessera. C: a very small tessera. D: short edges can cause numericaltroubles. B, C, D are cases of ill-defined tesserae (see Colour Plate section).

Methods Based on Weighted Sets of PointsA completely different approach to solve the possible numerical problems inherent inpartition procedures such as those used in GEPOL is to approximate the tesserae areasby weights calculated using a scale function. The word weight is used instead of area


because the quantity introduced here does not have a well-defined geometrical meaning.In this framework, a tessera has a weight w that is initially equal to the uncut tesserasurface area if a geodesic sampling of the sphere is adopted. Each nearby sphere scalesthe weight by a function of the tessera centre to sphere surface distance:

wi = w0i

∏j

f(∣∣�si −�rj

∣∣−Rj

)(1.52)

where f�x� goes from 1 when the point i is far from the sphere j to 0 when the pointi is far from the sphere j. In the intermediate region of space (the switching region) apolynomial function smoothly interpolates between 0 and 1. Two alternative schemeshave been proposed in the literature to define the polynomial functions. In the first, dueto Karplus [27], the interpolating polynomial is determined by requiring that the valuesof the polynomial’s first and second derivatives are zero at both ends of the switchingregion. The lowest limit of the switching region is located inside the sphere j and theupper limit is located outside. Furthermore, the point charges are replaced by sphericalgaussian distributions of charge so to avoid singularities for very near points and theexponent of the gaussians is chosen to fit the exact values of the Born equation forspherical ions.

In the second approach, the Tessellationless (TsLess) [25], the same conditions at bothends of the switching region apply, supplemented by the requirement that the value of theintegral of the polynomial on the switching region is 1, so to avoid any underestimationof the weights of points lying on the switching region. The lowest limit of the switchingregion is located slightly outside the sphere j and the upper limit at a larger distancefrom the sphere j. The choice of the switching region in TsLess also solves the problemof very near points without altering the physical nature of point charges.

Note that the collocation of a part of the switching region inside the sphere j in theKarplus scheme plays the same role as the polynomial ‘normalization’ in TsLess.

The calculation of the switching function is fast and very similar in both approaches.The product in Equation (6) mimics the geometrical properties of the tesserae-cuttingscheme: the weight of a point is unaffected by far spheres and goes to zero when it iswell buried (Karplus) or very near (TsLess) inside a single sphere. The calculation ofweights is simpler than that of analytical areas using the tesserae cutting procedure, andit is also not affected by the numerical troubles described in the previous section.

1.3.4 Solution of the BEM Equations

In this section we report the most common formulations of the BEM equations forthree different versions of PCM [1], namely IEFPCM (isotropic), CPCM and DPCM.The mathematical and physical significance of these equations are discussed in thecontribution by Cances. Here we are interested only in the computational features.

The most convenient form of the BEM equations for numerical purposes is [18]

Tq = −Gf (1.53)

where T and G are matrices depending on the tessellation and on the solvent dielectricconstant, q are the PCM charges and the f vector contains the molecular electrostatic


potential in the IEFPCM and CPCM formulations and the flux of the electrostatic fieldthrough the corresponding tessera in DPCM. The formulae for the elements of thematrices and vectors introduced here are reported in Table 1.1.

Table 1.1 Definitions of the matrix elements in the BEM equations

BEM equationsformulation F T G

IEFPCM V(

2�

(�−1�+1

)A−1 −D

)S 2�A−1 −D

CPCM V S�−1

�I

DPCM En 2�

(�−1�+1

)A−1 −D∗ I

Matrix elements

Aii = ai Aij = 0 D∗ii = Dii = 1�0694

1Rl

√�

ai

or D∗ij = −

(�si −�sj

)• ni∣∣�si −�sj

∣∣3Sii = 1�0694

√4�

ai

Sij = 1∣∣�si −�sj

∣∣ D�∗�ii = − 1

ai

(2� +∑

i �=jD

�∗�ij aj

)Dij = −

(�si −�sj

)• nj∣∣�si −�sj

∣∣3Two alternative definitions for the diagonal elements of the D and D∗ matrices have

been presented. The first reported in the table is the original one and takes into accountthe curvature of the tesserae (the inverse of the radius Rl of the sphere to which thetessera belongs). The second formulation is based on electrostatic considerations [28].The numerical factor 1.0694 has been empirically adjusted in order to reproduce thevalues given by the exact Born equation for spherical ions [18].

When the attention is focused to the development of the formalism for the calculationof molecular properties and energetic, the most appropriate form of Equation (1.53) is:

q = −Kf (1.54)

where K = T−1G. This form easily connects the charges to the molecular electrostaticpotential (or field) through a linear operator. When attention is focused on the compu-tational aspects, the form with the T and G matrices is more useful, because T and Ghave simple analytical formulations.

In the cases in which the molecular charge partially lies outside the cavity boundary(practically all the cases in which a QM model is used for the description of the molecule)the polarization weights [18]

w = q +q∗

2(1.55)

have to be calculated instead of the charges. The vector q∗ is the solution of the equation

q∗ = −K†f (1.56)


Matrix InversionAs shown above, the straightforward resolution method to obtain the PCM charges issimply to invert the T matrix of Equation (2) and to solve the resulting linear system [29]:

q = −T−1Gf = −Kf (1.57)

If the pairs of tesserae sampling points are not too close in space, the T matrix is strictlydominated by the diagonal elements, i.e.

�Tii� >∑i �=j

∣∣Tij

∣∣ (1.58)

because the diagonal elements of T depend on the tesserae area and solvent parametersbut the off-diagonal elements depends on the inverse of the distance between the pair oftesserae sampling points.

If this condition is fulfilled (this occurs for a well tessellated surface) the chargesobtained are fully reliable, as a strictly diagonal dominated matrix is not singular [30].If there are pairs of very close tesserae (for example tessera i and j), a simple ‘safety’measure is to annihilate the corresponding diagonal elements, Tij and Tji. Note that themethods based on tesserae weights are implicitly not affected by this problem.

Derivatives with respect the molecular geometry can be obtained by differentiatingEquation (1.54):

�q�!

= −�K�!

f −K�f�!

(1.59)

where ! is a molecular coordinate. All the derivatives involved in Equation (1.59) can becalculated analytically. More details on the derivatives of the PCM equations are reportedin the chapter by Cossi and Rega.

Iterative ComputationThis is the formulation originally used in continuum models [31] but it has been exten-sively improved through the years so that it now is the method of choice for calculationsin which the computational cost of the ASC calculation is not negligible or serious storagelimitations are present.

The iterative method uses the Jacobi iterative algorithm [32] to solve the linear setof equations. Jacobi iterations are rapidly convergent if the diagonal term dominatesthe linear system equation: this is the case of PCM-BEM equations. The matrix T ispartitioned in two parts: T0 that contains the diagonal elements and T1 that contains theoff diagonal elements. A 0th cycle guess of the charges is given by:

q0 = −T−10 Gf (1.60)

then it is updated by iterating the equation

qn = − �q0 −T1qn−1� (1.61)


until

�qn −qn−1� = en < (1.62)

where is a threshold value. If this iterative calculation is nested into the SCF cycle then can be safely set to 1–2 degrees of magnitude less than the current SCF error norm.

The convergence of the method can be improved by using a slightly different set ofcharges in Equation (1.61):

q′n−1 = −

(q0 −

n−1∑k=1

�kqk

)�

n−1∑k=1

�k = 1 (1.63)

Two proposals have been given to set ��k . In the DAMP scheme [33] only the n-1 andn-2 coefficients are different from zero:

�n−1 = 1/en−1

1/en−1 +1/en−2

(1.64)

In the DIIS scheme [33] they are determined by minimizing the error function:

s =∣∣∣∣∣n−1∑k=1

�kek

∣∣∣∣∣2

�n−1∑k=1

�k = 1 (1.65)

Both schemes are also used as SCF convergence accelerators. The DIIS scheme isparticularly efficient when used in conjunction with CPCM and IEFPCM schemes, inwhich the diagonal dominancy of T is less prominent than in DPCM. DIIS is veryefficient from the point of view of CPU times, but it requires the storage of several setsof intermediate charges. DAMP is less efficient but requires the storage of two sets ofintermediate charges only.

CPU time can be traded versus storage using conjugate gradient schemes [18], whichrequire longer CPU times than DIIS but do not need to store intermediate ASC sets.Another improvement concerns the fast calculations of the A1qk terms, the only ones thatcontain two nested cycles on the charges and thus scale quadratically with the numberof charges. While the original formulation of the iterative scheme eliminates the needof the storage of T (T1 can be calculated freshly at each iteration), it does not scalelinearly with the number of charges. The linear scaling can be achieved by looking atthe electrostatic nature of the T1qk terms:

�T1qk�l =

⎧⎪⎪⎪⎨⎪⎪⎪⎩∑j �=l

qkj

��sj−�sl� = V��sl"qk� for CPCM∑j �=l

qkj��sj−�sl�•�nl

��sj−�sl�3 = �V��sl"qk�

��nlfor DPCM

T��sl"qk� for IEF

(1.66)

where V��sl"qk� is the electrostatic potential at the tessera l sampling point due to the qk

set of charges. T��sl"qk� has a more complex expression without a electrostatic meaning


but similar to V��sl"qk�. Given these properties, approximated expressions of ��sl" qk�can be obtained using local multipole expansions [34] or the powerful fast multipolemethod (FMM) [35]. For IEFPCM a custom version of FMM has to be used [36]. Analternative approach to IEFPCM involves a partition of the charges into two contributions,one similar to the CPCM one and the other similar to the DPCM one [34]. Thus, twofull iterative procedures have to be performed to calculate the two sets of charges thatsummed give the final IEFPCM charges. When coupled to linear scaling electrostaticengines like FMM, the storage and CPU time of the iterative method are both linear withrespect to the number of tesserae.

The iterative method is very sensitive to the cavity quality, especially for CPCM andIEFPCM in which the interaction between two tesserae depends on the inverse of thedistance. Some unpublished tests performed by the author on slowly convergent iterativecalculations have shown that in the last steps almost all the error norm is due to a fewcharges that still have very large variations with respect the previous iteration cycle,whereas all the other charge variations are several orders of magnitude smaller.

Iterative methods also allow the calculation of derivatives of charges with respect tomolecular geometry. By differentiating Equation (1.53), we obtain:

�T�!

q +T�q�!

= −�G�!

f −G�f�!

(1.67)

All the quantities can be calculated analytically except �q/�!, which can easily becomputed by applying the iterative scheme to a rearranged Equation (1.67):

T�q�!

= −�G�!

f −G�f�!

− �T�!

q (1.68)

The iterative scheme for the derivatives is very similar to that used for the originalcharges, because the matrix to be partitioned is the same in both cases.

A method similar to the iterative, is the partial closure method [37]. It was formulatedoriginally as an approximated extrapolation of the iterative method at infinite number ofiterations. A subsequent more general formulation has shown that it is equivalent to usea truncated Taylor expansion with respect to the nondiagonal part of T instead of T−1 inthe inversion method. An interpolation of two sets of charges obtained at two consecutivelevels of truncations (e.g. to the third and fourth order) accelerates the convergence rateof the power series [38]. This method is no longer in use, because it has shown seriousnumerical problems with CPCM and IEFPCM.

1.3.5 Conclusions

Computational methods have accompanied the development of the Polarizable ContinuumModel theory throughout its history. In the building of the molecular cavity and itssampling together with the resolution of the BEM equations we nowadays have a largechoice of alternative algorithms, suitable for all kinds of molecular calculations. Linearscaling both in time and space is achieved in both fields.

Cavities based on interlocking spheres allow a simple and accurate calculation oftessellation elements, thanks to weight function methods. A question not solved yet is


a full smooth description of solvent-excluded volume with the use of spherical objects.Alternatives could be the development of methods based on more complex geometricalshapes and fully differentiable or the use of isodensity methods. The field of the numericalsolution of BEM equations does not show nowadays problems of this magnitude. Theinversion method is full reliable for small molecular systems and the iterative for largemolecular systems.

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1.4 A Lagrangian Formulation for Continuum Models

Marco Caricato, Giovanni Scalmani and Michael J. Frisch

1.4.1 Introduction

Implicit solvation models have proved themselves very effective in providing a compu-tationally feasible way to simulate the microscopic environment of molecules in solu-tion [1–3]: accurate free energy of solvation can be computed, and the spectroscopicproperties of solutes can be corrected to take into account solvent effects.

While all implicit solvent models share the same advantage with respect to explicitones, i.e. the very significant reduction in complexity achieved through the descriptionof the solvent as a uniform continuum, they can be grouped in various ways accordingto the theoretical framework used to describe the solute, the solvent and the interfacebetween them.

In the Generalized Born model [2–5], the solvent is described in a extremely simpli-fied way and there is no mutual polarization between solute and solvent. The Onsagermodel [6] allows for solute–solvent polarization, but the description of the cavity and ofthe solvent is still very crude.

A more sophisticated description of the solvent is achieved using an Apparent SurfaceCharge (ASC) [1, 3] placed on the surface of a cavity containing the solute. This cavity,usually of molecular shape, is dug into a polarizable continuum medium and the properelectrostatic problem is solved on the cavity boundary, taking into account the mutualpolarization of the solute and solvent. The Polarizable Continuum Model (PCM) [1,3,7]belongs to this class of ASC implicit solvent models.

Finally, other models [8–10] define the dielectric constant as a function of the pointin space around the solute and solve the three-dimensional electrostatic problem, usuallyby a finite differences method.

In recent years many attempts have been made to extend the implicit solvent models tothe description of time-dependent phenomena. One of these phenomena is nonequilibriumsolvation [3] and it can be described effectively in a very simplified way, despite thefact that it actually depends on the details of the full frequency spectrum of the dielectricconstant. Typical examples of nonequilibrium solvation are the absorption of light bythe solute which produces an excited state which is no longer in equilibrium with thesurrounding polarization of the medium [11–13]. Another example is intermolecularcharge transfer within the solute, also leading to a nonequilibrium polarization [14].

In the simplest picture of the nonequilibrium state, only a fraction of the solventdegrees of freedom is able to ‘follow’ the quick change in the electronic structure of thesolute, while the ‘slow’ degrees of freedom take a longer time to equilibrate with thenew state of the solute. More detailed descriptions of the time evolution of the solventpolarization have been reported [15] and similar results have also been recently achievedin the context of the PCM [13, 14].

Aiming to describe any kind of time-dependent phenomena, it would be highly desir-able to couple the standard molecular dynamics (MD) methods, both classical and abinitio, with the implicit solvent model. This can be achieved either by solving the


electrostatic problem at every step of the dynamics or by defining an extended Lagrangianwhich includes the polarization of the medium as a dynamical variable.

In the first scheme, the only significant issue is to ensure that the solvation potentialgiven by the implicit solvent model is a continuous and smooth function of the nuclearcoordinates. There are numerous examples of successful application of this strategy inthe literature. The Generalized Born method has been effectively coupled with MDusing classical force fields and the GB–MD technique is nowadays widely used in clas-sical MD simulations of large molecules and proteins [2, 4, 16]. The Car–ParrinelloLagrangian has been extended by De Angelis and co-workers [17] using an ASC implicitsolvent model, namely the conductor-like flavor of the PCM model (CPCM), to includethe interaction energy between the solute’s electrostatic potential and the polarizationcharges. A similar approach has been proposed by Fattebert and Gygi [8–10], also in thecontext of the Car–Parrinello method. They introduce a dielectric permittivity which is asmooth function of the solute’s density, and solve by finite differences the Poisson equa-tion. The results is the electrostatic potential produced by the polarized medium whichinteracts with the solute’s electronic density. Finally, Rega recently reported [18] thecombination of the Atom-centered Density Matrix Propagation (ADMP) [19] techniquewith CPCM.

All the methods mentioned share two common drawbacks. First, the time dependencyof the medium polarization is lost in the sense that it is assumed to evolve much fasterthan the geometry of the solute. No phenomena involving nonequilibrium solvation canbe described in this way. A partial solution to this problem would be the use of mixedimplicit–explicit solvent models as proposed be Brancato et al. [20, 21]. The seconddrawback is the high computational cost involved in solving the electrostatic problemfor each nuclear configuration. In particular in the case of solutes described at a classicallevel, this added cost is exceedingly large with respect to the cost of running the simulationin vacuo and probably also larger than the use of a box of explicit solvent molecules.

As previously mentioned, an alternative strategy can be used to couple MD methods andimplicit solvent models. The Lagrangian describing the solute can be extended to includethe medium polarization as a dynamical variable. Such an approach has the advantage ofproviding a proper description of the time evolution of the solvent polarization coupledto the evolution of the solute geometry. Also, it is potentially characterized by a lowercomputational cost since the full electrostatic problem is not solved at each nucleargeometry, but rather the medium polarization is propagated in time and allowed tooscillate around the solution of Poisson’s equation.

The main difficulty arising from this scheme is the need for a potential energy functionalwhich is valid, i.e. corresponds to the free energy of the interacting solute–solventsystem, for an arbitrary medium polarization, and not only for the polarization that solvesthe Poisson equation. This functional also needs to be variational with respect to boththe geometrical and the polarization degrees of freedom so that, when minimized, thefree energy of the system at equilibrium polarization is recovered. Other issues are thepotentially strong coupling between the geometrical and polarization variables and theneed to assign a fictitious mass to the polarization degrees of freedom.

In the following sections we will review the possible choices of free energy functionalsfor both dielectric and conductor boundary conditions, focusing on their applicability inthe context of ASC implicit solvent models. Then in Section 1.4.5 we will present our


formulation of a smooth extended Lagrangian for the PCM family of solvation models.Finally, in Sections 1.4.6 and 1.4.7 we report numerical examples and prototypicalapplications of the PCM extended Lagrangian.

Before turning our attention to the free energy functionals, we recall a few funda-mental concepts that will be used throughout in the following. We start from the generalexpression for the electrostatic energy of a charge density �0 in a nonlinear dielectricmedium [22]:

W = 14�

∫d3r

∫ D

0E · D (1.69)

where E is the electrostatic field and D is the electric displacement, defined by:

D = E+4�P (1.70)

and P is the electric dipole polarization of the medium. In the case of a linear response:∫ D

0E · D = 1

2E · D� (1.71)

so that the electrostatic energy is simply:

W = 12

∫�0� d3r (1.72)

where � is the total electrostatic potential, E = −��, and D = �E, where we alsoassumed the dielectric to be isotropic. When the dielectric is fully polarized, the Poissonequation holds:

� · �� = −4��0 (1.73)

1.4.2 Ad Hoc Functionals

In this section we describe some examples of functionals proposed to compute theelectrostatic potential �, which is used in Equation (1.72) to solve for the electrostaticinteraction energy between the charge density �0 and the dielectric medium. This classcontains functionals which are not energy functionals, in the sense that their minimizationdoes not lead to the electrostatic free energy, Equation (1.72). However, at the end ofthe variational process they provide an electrostatic potential (or a polarization) whichsatisfies Equation (1.73) and thus it can be used to compute the electrostatic energy.

Although these functionals can be robust from the numerical point of view, they donot correspond to an energy and this prevents their direct use in MD simulations, as partof an extended Lagrangian, since it would not yield the correct forces.

By using the electrostatic potential as the variational parameter York and Karplus [23]proposed two general functionals. The first one can be expressed in the form:

W��"�0� �� =∫

�0� d3r − 18�

∫�� · � ·�� d3r (1.74)


where �0 and � are considered functional parameters. When the first derivative of thisfunctional with respect to � is nil, the Poisson differential Equation (1.73) is satisfied.

However, for � > 0 this functional happens to be concave with respect to the potential,so it is a maximum at the stationary point, since it can be demonstrated that the secondderivative is negative. This fact makes the above functional not easy to handle, sincenormal minimization algorithms cannot be used.

In the same paper [23] the authors proposed another functional, namely:

2e ��"�0� �� = 1

2

∫�(E− �−1E0

)2d3r (1.75)

in which the unconstrained parameter is still the electrostatic potential. This functionalis analogous to the function that is minimized in least-square fitting procedures. Thestationary point of this functional is equivalent to that of Equation (1.74) but in this casethe functional is convex with respect to �, thus the functional in Equation (1.75) mustbe minimized. The functionals in Equations (1.74) and (1.75) can also be expressed interms of the variations in the polarization potential �pol = �−�0, see ref. [23].

If the solute charge density �0 is completely contained inside a cavity surrounded bythe dielectric medium, which mimics the solvent, both the functionals can be variationallyoptimized constraining the variation of the polarization density to be on the cavity surface.

Another variational approach is proposed by Allen et al. [24]. In that work the authorsdeal with the problem of the ion channels through membranes, in which the roles of thesolvent and the solute are interchanged. However, the functional they proposed can beused in general solvation problems. The form of this functional is:

W �� = 12

∫�� ·�� d3r −

∫�

[4��0 + 1

2� · ��

]d3r (1.76)

where = �−1 is the dielectric susceptibility.The authors demonstrated that the minimum of the functional in Equation (1.76)

corresponds to the solution of the Poisson equation, Equation (1.73). However the valueof the functional in the minimum correspond to minus the electrostatic energy.

The functional (1.76) still depends on the total electrostatic potential, but it can beturned into a functional of the polarization charge density, see ref. [24].

When a well defined separation between the dielectric medium and the charge density�0 is assumed, so that the dielectric susceptibility undergoes a step discontinuity on thesurface boundary with the dielectric, the induced polarization charge reduces to a surfacecharge, and the integrals involving this quantity can be reduced to surface integrals [24].

Even if the functionals presented in this section cannot be directly used in the contextof ASC implicit solvent models to define an extended Lagrangian for MD simulations,the electrostatic potential obtained at the stationary point can then be used to deduce theelectrostatic forces acting on the nuclei. This description of the electrostatic interactionbetween solute and solvent corresponds to a situation in which the dielectric polarizationinstantaneously follows the change in the solute charge distribution. This means that ateach step of the simulation solute and solvent are in equilibrium.


1.4.3 Free Energy Functionals

The theory of electronic polarization in dielectric media [25] provides the framework forthe derivation of a free energy functional that meets the requirements set forth in theIntroduction. In particular, the additional free energy of the system due to a polarizationP(r) can be expressed as [26]:

W�P� = 12

∫P�r� ·−1�r� ·P�r�dr

+ 12

∫ ∫ �� ·P�r�� ′ ·P�r′��r − r′� dr dr′ (1.77)

−∫

�� ·P�r��0�r�dr

where �r� is the dielectric susceptibility of the medium and �0�r� is the potentialproduced by the charge density �0�r�. The above functional is valid for an arbitrary valueof the polarization field and has a stationary point at

W

P�r�= 0 � P�r� = �r� ·E�r� (1.78)

where the electric field E(r) is given by

E�r� = −��0�r�+�∫ �� ′ ·P�r′��

�r − r′� dr′ (1.79)

and D = E+4�P satisfies Poisson’s equation. This stationary point is indeed a minimumas �r� is a positive definite tensor everywhere. Unfortunately, the functional in Equa-tion (1.77) is not easily applicable in the context of ASC implicit solvation models asthe polarization is represented by a vector field.

Marchi and co-workers [27,28] have applied Equation (1.79) in the context of classicalMD by using a Fourier pseudo-spectral approximation of the polarization vector field.This approach provides a convenient way to evaluate the required integrals over allvolume at the price of introducing in the extended Lagrangian a set of polarization fieldvariables all with the same fictitious mass. They also recognized the crucial requirementthat both the atomic charge distribution and the position-dependent dielectric constantbe continuous functions of the atomic positions and they devised suitable expressionsfor both.

A functional even more general than that in Equation (1.77) was given by Marcus [29]in order to describe a system where only a portion of the polarization is in equilibrium.However, also in this case, the functional is in terms of three-dimensional polarizationfields and thus it cannot be readily introduced in an ASC implicit solvation model.

Recently, Attard [30] proposed a different approach which provides a variationalformulation of the electrostatic potential in dielectric continua. His formulation of thefree energy functional starts from Equation (1.77), which he justifies using a maximumentropy argument. He defines a fictitious surface charge, s, located on the cavity boundary.The charge s, which produces an electric field f , contributes together with the solute


density charge � to polarize the dielectric, producing an apparent surface charge � . Whenthe mutual polarization between the solute, the fictitious charge and the dielectric reachesan equilibrium, the fictitious and the induced surface charges are expected to coincides = � . Defining � the electrostatic potential produced by the surface charge � , a freeenergy functional can be written in the form:

W�s�� = 12

∫dr��r��r�+ 1

2

∮dr��r� ��r�−f�r�� (1.80)

where the expression for the potentials are:

f�r� =∫

dr′ ��r′��r − r′� +

∮dr′ s�r′�

�r − r′� (1.81)

��r� =∫

dr′ ��r′��r − r′� +

∮dr′ ��r′�

�r − r′� (1.82)

To the best of our knowledge, this is the only free energy functional that can be readilyintroduced in an ASC implicit solvent model as it involves only surface integrals in termsof the independent polarization variable which is no longer a three-dimensional field, butinstead assumes the form of a surface charge distribution on the dielectric boundary.

1.4.4 Free Energy Functional for the Conductor-like Model

In the case of the conductor-like model a free energy functional that meets the require-ments set forth in the Introduction is readily available. Indeed, in the limit of a conductor,the potential must vanish inside and at the boundary of the medium. Thus the functionalcan be written in the form:

W �� = −∫

dr∮

dr′ ��r��r′�

�r − r′� + 12

∮dr

∮dr′ ��r��r

′��r − r′� (1.83)

The minimization of this functional satisfies the condition at the boundary:

W

�= −

∫dr

��r��r − r′� +

∮dr

��r��r − r′� = 0 (1.84)

This functional is also physically motivated as it expresses the balance of two terms:a favorable (negative) solute–solvent interaction energy and an unfavorable (positive)solvent–solvent interaction. At equilibrium the second term is equal to half of the firstas expected also from basic electrostatic arguments.

Despite the simple form of Equation (1.83), the detailed formulation of an extendedLagrangian for CPCM is not a straightforward matter and its implementation remainschallenging from the technical point of view. Nevertheless, is has been attempted withsome success by Senn and co-workers [31] for the COSMO–ASC model in the frameworkof the Car–Parrinello ab initio MD method. They were able to ensure the continuity ofthe cavity discretization with respect to the atomic positions, but they stopped short ofproviding a truly continuous description of the polarization surface charge as suggested,


for example, by York and Karplus [23]. This led to the need for different time steps foratomic degrees of freedom and polarization charges, and to the use of micropropagationsteps for the latter.

1.4.5 A Smooth Lagrangian Formulation of the PCM Free Energy Functional

The strategy to obtain a Lagrangian formulation of PCM is to consider the PCM apparentcharges as a set of dynamic variables, exactly as the solute nuclear coordinates. Thealgorithm proposed in the present chapter is applied within the MM framework, since itallows a simplified notation and faster calculations. However, we point out that it can bestraightforwardly extended to QM calculations.

It has to be noted that only the values and not the positions of the PCM charges nowbecome independent on the nuclear coordinates. In fact we still keep the PCM cavityas a series of interlocking spheres centered on the nuclei. Thus when a nucleus movesthe sphere centered on it also moves and the surface elements located on that spheremove as well. Here we also point out that, when two intersecting spheres move followingthe motions of the nuclei which they are centered on, the number of surface elementsexposed to the solvent changes, and only the apparent charges exposed to the solventcontribute to the free energy. With the term ‘exposed’ we mean the apparent chargeswhich are not inside the volume of the cavity (i.e. which are in a region of the surfaceof a sphere which is covered by an adjacent intersecting sphere). We stress that, thoughthe term ‘exposed’ is not rigorous, as the charges are not exposed to the solvent but theyare the solvent, we continue to use that term to distinguish the surface elements whichcontribute to the free energy from those that do not contribute.

CPCM FunctionalAs outlined in Section IV, in the conductor-like version of PCM we have a simpleexpression of the energy functional, Equation (1.15). It can be discretized as:

W�r�q� = −qV + �

2��−1�qSq (1.85)

where the matrix S represents the electrostatic potential induced by the apparent chargeson the surface cavity [3]. The last term on the right hand side represents the polarizationof the dielectric medium. In this form the value of the variables q does not explicitlydepend on r, while this dependence is present for the electrostatic potential V and for thePCM matrix S, though we omit it in the equation. When the free energy is minimized (atleast with respect to the PCM apparent charges variables) then q satisfy the PCM systemof equations and the second term in the above equation becomes exactly one half of theqV term.

As said above, minimizing the functional in Equation (1.85) with respect to the chargesq is equivalent to solve the CPCM system of equations:

�

�−1Sq = V (1.86)

However the present strategy also implies new technical difficulties. The first obstacleis represented by the diagonal elements of the matrix S� Sii = fi/ai, as they contain


the area of the surface element i� ai, as the denominator. If the surface element i is inthe region of the intersection between two spheres, the gradient of the energy functionalwith respect to the charge qi can become very large when ai becomes small, leading tonumerical instabilities of the optimization algorithm.

A more important source of instability is that, as the solute geometry changes duringa geometry optimization or an MD trajectory, some charges becomes buried while someothers become exposed to the solvent, following the motion of the spheres where they arelocated. This fact leads to a discontinuity in the energy derivatives (with respect to boththe nuclear and the charges degrees of freedom), as the number of dynamic variableschanges.

Furthermore this appearance and disappearance of the PCM charges can representa more severe source of instability in a MD simulation, because no forces act on thecharges inside the cavity (because there are not terms of the gradients which involvethese charges). This fact means that, when a charge is buried and after a time intervalt it is again exposed to the solvent, its value could be arbitrarily large, leading to anonconservative behavior of the energy.

To overcome both these problems we introduce a new set of variables q, which havethis relation with the PCM charges:

qi = qia1/2i (1.87)

where ai is the area of the ith surface element. Thus the value of the charge qi is nonzerowhen the area ai is not zero, i.e. when the ith surface element is exposed to the solvent.The opposite relation qi = qia

−1/2i is valid only if ai is nonzero. This charges q are a sort

of area-weighted apparent surface charges and their definition is in a way reminiscent ofthat of the mass-weighted nuclear coordinates. During the optimization the value of theq can be nonzero even if the corresponding surface element is inside the cavity: we callthese shadow q �qsh�.

Thus we introduce an energy term involving the shadow q in the energy functionalsand this term has to vanish when the functional is minimized. Still using the CPCMformalism, the functional is now given by:

W�r� q� = −A1/2qV + �

2��−1�qA1/2SA1/2q + �

2��−1�fq2

sh (1.88)

where the last term is a sort of self-interaction of the shadow charges involving thediagonal term of the matrix S� Sii = fi/ai. We note that for these terms the dependenceon the area of the surface elements ai disappears when we pass from q to q. Moreoveras the fi elements are positive the last term is positive and the only way to minimize itis to set to zero all the shadow q.

The form of this self-interaction term for CPCM seems very plausible if we consideran extended CPCM system of equations, analogous to that in Equation (1.86), collectingboth the exposed and the shadow charges q. Starting from the CPCM equation (switchingfrom q to q):

�

�−1A1/2SA1/2q = A1/2V


which can be also written in an extended form:

�

�−1Sextq = Vext (1.89)

where:

Sext =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

/ / /

/ S / 0/ / /

f 00 f

0 f

⎤⎥⎥⎥⎥⎥⎥⎥⎦� Vext =

⎡⎢⎢⎢⎢⎢⎣/

V/

0

⎤⎥⎥⎥⎥⎥⎦ (1.90)

The nonzero block in the upper left corner of the matrix Sext interacts with the chargesexposed to the solvent. When the minimization of the functional (1.88) is complete thevector of the q will look:

qext =

⎡⎢⎢⎣/q/0

⎤⎥⎥⎦ (1.91)

The last technical but essential note is that, as the description of the charges in theregions of the intersection of the spheres represents a critical numerical issue, we foundthat the use of the Karplus smoothing scheme [23], recently extended to the variousPCM versions by Scalmani and Frisch [32], is crucial to allow for a smooth behaviorof the electrostatic potential and thus of the free energy functional in those regions. Wepoint out that, though the expression of the matrix S is a bit more complicated in thePCM formulation which uses the Karplus weights [23], the expressions presented in thepresent contribution are still valid for the purpose of illustrating our approach.

DPCM FunctionalThe case of the dielectric version of PCM (DPCM) is more complicated than CPCM, asthe system of equations which must be solved to compute the apparent surface chargesis [3]: (

2��+1�−1

A−1 −D∗)

q = −E⊥ (1.92)

and so a different approach must be used.Following the work of Attard [30], we discretize the integrals involved in the functional

W�s��, Equation (1.80), which assumes the following form:

W�q� r� = 12

V†q�q�+ 12�q�q�−q�† Sq�q� (1.93)


where q and q are the charges corresponding to the surface charge s and � respec-tively. We omitted the dependence on the nuclear coordinates r, but we emphasized thedependence of q on q. This explicit dependence is:

q = −�−14�

A �E⊥ −D∗q�− �−12

q (1.94)

When the mutual polarization reaches equilibrium then q satisfies the DPCM equationsand q = q, as can be demonstrated looking at Equation (1.94). The value of the functional(1.93) is then:

W�q� r� = 12

V†q�q� (1.95)

which is the expression of the equilibrium free energy of the system.To verify that the constraints, i.e. the DPCM equations, on the functional are satisfied,

the derivative of the functional with respect to the charges must be equal to zero. Byusing the relation between the solute potential and the normal component of the soluteelectrostatic field [3]:

�2�I −DA�V = SAE⊥ (1.96)

which holds only in the case that all the solute charge density is contained inside thecavity, and assuming that:

DAS = SAD∗ (1.97)

which holds exactly for the integral operators but is still very accurate when their matrixrepresentation is used, the final expression for the functional partial derivative is:

�W

�q=

(�−14�

)2

�2�I −DA�SA[(

2��+1�−1

A−1 −D∗)

q +E⊥

](1.98)

This equation can be equal to zero only if the term in the last parentheses is equal tozero, which is equivalent to satisfying the DPCM equations. We note that the term in thesecond parentheses cannot be equal to zero since Equation (1.96) holds.

At the end of the optimization, when both the derivatives of the functional with respectto the nuclear coordinates and the PCM charges are nil, the electrostatic equations forthe dielectric are satisfied and the equilibrium DPCM charges are obtained.

Now we reintroduce the q, which are necessary to take into account the contribution tothe functional from the charges buried inside the cavity. The relation between the barredcharges and the normal ones is:

q = qA1/2 (1.99)

q = ¯qA1/2 (1.100)


The relation between the ¯q and q then becomes:

¯q = −�−14�

A1/2(E⊥ −D∗qA1/2

)− �−12

q (1.101)

When the surface element i moves into the cavity, the corresponding area ai = 0, soEquation (1.101) becomes:

¯qi =�−14�

�2� −gi� qi (1.102)

since the diagonal element of the matrix D∗ is defined as D∗ii = gi/ai.

The contribution of the shadow charges to the energy functional is:

12

∑i��

fi¯qi

( ¯qi − qi

) = 12

(�−14�

)2

fi �2� −gi�2 q2

i + 12�−14�

fi �2� −gi� q2i (1.103)

which is a positive term, since �2� − gi� is positive, as can be demonstrated looking atref. [33]. Also, as the fi elements are positive, the only way to minimize this term is toset qi = 0 and thus, at the end of the optimization the charges inside the cavity do notcontribute to the energy as expected.

1.4.6 Prototypical Application: Simultaneous Optimization of Geometry andReaction Field

A first important application of this new strategy is constituted by the geometry opti-mizations. In fact, the internal energy of the system in the MD methods coincides withthe energy functional which has to be minimized in the geometry optimizations, and thesame derivatives of the energy with respect to the nuclear coordinates are involved. Westress that our interest focuses on the technical issues and not on the specific character-istics of the systems we use as test molecules. The calculation were performed with adevelopment version of Gaussian [34].

We choose three test molecules: formaldehyde, proline and 2-phenylphenoxide. Thestructure of these systems is shown in Figure 1.8. The calculations were performed invacuo and in water solution, with the C and the D versions of PCM with the standard andthe simultaneous approaches. Here we note that we used the same solute-shaped cavityfor all the optimizations of each system. The force field we used for all the calculations,both in vacuo and in solution, is the UFF [35] and the nuclear charges at the initialpoint were estimated with the QEq [36] algorithm. As we are not interested in obtainingresults comparable with experimental data or with other calculations, but only in thePCM results with the different optimization schemes, the choice of the force field is nota critical point. The only requirement is that we performed all the calculations with thesame force field.

In Table 1.2 the energy for the three molecules in vacuo and in solution are reported.The data show that the approach of simultaneously optimizing the geometry and thepolarization succeeds in providing the same minimum geometry found with the standardapproach.


O–

(a) (b) (c)

O

C HN

C OH

O

Figure 1.8 Structure of (a) formaldehyde, (b) proline and (c) 2-phenylphenoxide.

Table 1.2 Energy �kcal mol−1 for the three molecules in Figure 1.8 invacuo and in solution are reported. CPCM and DPCM indicate the calcu-lations performed with the standard version of the models, sCPCM andsDPCM indicate the calculations where the geometry and the polarizationare optimized simultaneously

H2CO Proline Phenoxide

vacuum 0�00 101�79 51�31CPCM −4�86 93�24 4�72sCPCM −4�86 93�23 4�72DPCM −4�84 93�27 5�70sDPCM −4�84 93�27 5�67

Thus now we discuss the features of the CPCM and DPCM free energy functionalspresented in Sections 1.4.5 and 1.4.6 in terms of their computational cost with respect tothe standard approaches. We outline that this comparison is qualitative since it is basedonly on some of the parameters that influence the final computational time and we arealso limiting our discussion on the small molecules presented in this section.

The bottleneck of a calculation in solution is the evaluation of the polarization which,in the case of PCM, corresponds to the evaluation of the apparent surface charges. Inparticular, the bottleneck is represented by the evaluation of the products between theintegral matrices of the electrostatic potential (matrix S in Equation (1.8.6)) or of thenormal component of the electric field (matrix D∗ in Equation (1.92)) and the apparentcharges vector q. Thus the criterion we use to compare the standard and the simultaneousapproach is based on the number of matrix products (Sq or D∗q) necessary in the wholeoptimization process. We also remind the reader that the dimension of the matrices isequal to the square of the number of the surface elements.

The advantage of the new strategy is that, for each step of the optimization, a smalland constant number of matrix–vector products are necessary (three for CPCM and ninefor DPCM). In contrast, for the standard approach the evaluation of the apparent charges


requires many more matrix–vector products to be solved for the charges using an iterativeapproach [37], plus some others for the evaluation of the gradients. We must point outthat, if the PCM matrices are small enough to be kept in memory during the iterativesolution of the PCM equations, the computational time needed to compute the apparentcharges greatly reduces. However, this is not likely to be possible for large molecules. Weused the conjugate gradient algorithm for the geometry optimization, since it is cheap, soit is a good choice for MM calculations. However, this choice may not be the best onewhen the simultaneous optimization is performed, since this algorithm does not take intoaccount the coupling between the two different sets of variables (the nuclear coordinateand the solvent charges), because the Hessian (or at least an estimation of the Hessian)is not computed.

With all those assumptions and limitations in mind we can analyze the number ofmatrix–vector products necessary to perform the geometry optimization for the threemodel molecules, reported in Table 1.3.

Table 1.3 Estimation of the number of matrix–vector products necessary to optimize thesystems in Figure 1.8. The number in parentheses represents the steps necessary to reach theminimum geometry. The first energy is computed by solving the PCM equations for all theschemes. CPCM and DPCM indicate the calculations performed with the standard versionof the models, sCPCM and sDPCM indicate the calculations where the geometry and thepolarization are optimized simultaneously

H2CO Proline Phenoxide

CPCM ∼ 180 �7 ∼ 1280 �31 ∼ 3700 �92sCPCM ∼ 60 �13 ∼ 910 �290 ∼ 950 �303DPCM ∼ 180 �7 ∼ 10550 �319 ∼ 11100 �336sDPCM ∼ 400 �44 ∼ 25250 �2805 ∼ 30000 �3334

Let us start the analysis from CPCM. From Table 1.3 it is evident that the energyfunctional performs better than the standard scheme, even if a very simple optimizationalgorithm is used and even if the two sets of variables are treated on the same footings.So even if the number of steps necessary to reach the minimum geometry is larger forthe simultaneous scheme than for the usual one, as expected since in the first case thevariables are many more, the total number of matrix–vector products is lower. Thus onecan expect that, with a better choice of the optimization algorithm, the number of stepsshould greatly decrease for the simultaneous approach, especially in areas of the energysurface close to the minimum, and this approach should become more convenient thanthe usual one even for smaller molecules.

The situation is the opposite when we consider the DPCM results. Indeed in this case,even if the ratio between the number of steps for the simultaneous and the standard schemeis comparable to the ratio in the CPCM case (for the two larger molecules) the sDPCMscheme requires a larger computational effort than the DPCM one. This is due to themore complicated expression for the DPCM free energy functional, Equation (1.93), thanfor the CPCM one, Equation (1.88). The functional (1.93) appears difficult to deal withfrom a numerical point of view. Numerical instabilities are probably arising from a strongcoupling between the two different sets of variables which must be better investigated.Moreover, the potential energy surface in the DPCM case looks more complicated than in


the CPCM case, as can be seen by comparing the number of steps necessary to reach theconvergence, even when the standard scheme is used for both methods. Numerical issuesare particularly severe close to the energy minimum, and the optimization algorithmoscillates for many steps around the minimum before reaching it. This behavior preventsthe use of the DPCM free energy functional with molecules larger than those proposedin this section. Furthermore the coupling between the two sets of variables, on the otherhand, makes the separation of the nuclear normal modes from the charges oscillationsdifficult; thus in the next section only dynamics simulation performed with the CPCMfunctional are presented.

The data shown in this section demonstrate that the simultaneous optimization of thesolute geometry and the solvent polarization is possible and it provides the same results asthe normal approach. In the case of CPCM it already performs better than the normal scheme,even with a simple optimization algorithm, and it will probably be the best choice when largemolecules are studied (when the PCM matrices cannot be kept in memory). This functionalcan thus be directly used to perform MD simulations in solution without considering explicitsolvent molecules but still taking into account the dynamics of the solvent. On the other hand,the DPCM functional presents numerical difficulties that must be studied and overcome inorder to allow its use for dynamic simulations in solution.

1.4.7 Prototypical Application: Time Propagation of Geometry and ReactionField

In this section we compare the behavior of the CPCM extended Lagrangian classicaldynamics with a dynamics in which the charges are equilibrated, i.e. the PCM systemof equations is solved at each time step. The main point which differentiates the twodynamics is that, when an extended Lagrangian is introduced, the solvent apparentcharges, or better the area-weighted apparent charges q, have their own time evolution.A kinetic energy term appears, which takes into account the velocity of the changes inthe space of the charges values, and a fictitious mass must also be defined. This masscan be tuned to obtain different responses of the solvent to the changes in the solutegeometry. In the simulations we present in this section we assigned the same mass toall the charges independently of where they are located on the cavity surface. We chosethis mass in such a way that the charges are light enough to rapidly follow the motionof the solute. In this way we managed to run an equilibrium dynamics by using thesame time step used for the dynamics in which the PCM equations are solved at eachstep. The latter can be seen as a dynamics in which the charges are infinitely light, sothey instantaneously equilibrate with the solute charge distribution at each time step. Theadvantage of the new approach is that the number of matrix–vector products is greatlyreduced, as also shown in the previous section, so it is possible to run much longertrajectories.

We studied two of the test molecules used in the previous section (formaldehyde andphenoxide) in water. As far as the formaldehyde dynamics is concerned we will analyzethe energy conservation as well as the oscillations of the potential energy. As for thephenoxide we will examine the solvent shift in the normal mode frequencies.

The formaldehyde dynamics ran for 25 ps, with a time step of 0.1 fs. Figure 1.9 reportsthe results obtained with the charges equilibrated at each step and with the extended


(a)

(b)

Figure 1.9 Total and potential energy (au) of formaldehyde in water, (a) with thePCM charges equilibrated at each time step and (b) with the PCM extended Lagrangianformulation.

Lagrangian, respectively. At first we note that the total energy is conserved in boththe dynamics, with oscillations orders of magnitude smaller than the oscillations of thepotential energy. The latter presents on the other hand a behavior that is quite differentin the two cases. For the case in which the charges are equilibrated at each step, theoscillations are quite large, of the order of 3�5 × 10−3 au, and they last for the wholetrajectory. On the other hand, for the extended Lagrangian approach, after an initial period


of equilibration, the potential energy oscillations are smaller. The initial equilibration isdue to the fact that we started the dynamics with a nil velocity of the charges. The smalleroscillations of the energy are probably due to the mass of the charges, which drags themotion of the nuclei. However, this mass is small enough to prevent an overlap of thenuclear vibrational frequencies with the solvent charges ones.

This example shows that also in the case of MD simulations, the extended Lagrangianapproach is promising, in the sense that it provides a more stable expression for thepotential energy, allowing a better energy conservation. It is also less computationallydemanding, because the charges are propagated with the solute nuclear coordinates, thusno linear system must be solved at each point. We stress that, contrary to the formulationproposed in ref. [31], in our formulation the solvent charges are propagated with thesame time step of the nuclei and no micropropagations are necessary.

In Figure 1.10 the low frequency region of the spectrum of phenoxide is presented.It is obtained by the Fourier transform of the velocity–velocity autocorrelation function,after a trajectory of 20 ps in vacuo and 4 ps in solution with the two approaches. Thetime step is 0.1 fs. We consider the first four vibrational frequencies, which present thelargest solvent shift. The harmonic values of these frequencies, computed analyticallyin vacuo and in solution at the equilibrium geometries, are reported in Table 1.4. Thefirst and the fourth frequencies, which are those with the larger shifts, correspond to thetorsion of the dihedral angle between the two rings and to the motion out of plane of theoxygen, respectively.

Figure 1.10 Vibrational spectra (frequencies in cm−1) of the phenoxide molecule in vacuoand in solution obtained by the MD simulation. The intensities were scaled in order to fit onthe same scale.

The results in Figure 1.3, even if the picks are not completely resolved because thedynamics were probably too short, show that the two approaches in solution match.


Table 1.4 Analytical first four harmonic vibrationalfrequencies �cm−1 of the phenoxide molecule invacuo and in solution

1 2 3 4

vacuum 86�2 103�1 149�9 245�0water 76�7 98�7 143�0 234�9

Moreover the shifts in the frequencies passing from the gas phase to the solutionare qualitatively correct (we did not consider any anharmonicities in the analyticalcalculations). Thus also in the case of a larger test molecule, the extended Lagrangianformulation of CPCM is successful in describing the solvation effect.

1.4.8 Conclusion and Perspectives

The aim of this contribution was to review the efforts that have been made so far in theformulation of a Lagrangian for the implicit solvation model. The goal is to provide asimple and computationally efficient way to describe the very complex phenomenon ofsolvation, which involve a large number of molecules, by using a strongly reduced setof degrees of freedom.

Among the approaches presented in this contribution, those that seem more appealingare based on free energy functionals, since they can be directly used in moleculardynamics simulation. We used this approach to define the functional for CPCM andDPCM in Section 1.4.5. As for the former, its simple expression makes it feasible tobe used with medium sized molecules for simultaneous optimization of geometry andpolarization and also to perform MD simulations. The latter, on the other hand, presentsnumerical difficulties that must be overcome to make it generally useful.

Although much work must yet be done to understand the features and the limitations ofthese functionals, their range of applicability and their accuracy, we consider the resultspresented in this contribution as encouraging.

Acknowledgments

The authors would like to thank Prof. Berny Schlegel for his contribution in the discussionthat led to the idea of the area-weighted apparent surface charges. Also we would liketo thank Prof. Benedetta Mennucci for her continuing interest and her encouragement.

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Chem. Acc., 90 (2004) 111.

1.5 The Quantum Mechanical Formulation of Continuum Models

Roberto Cammi

1.5.1 Introduction

The quantum mechanical (QM) (time-independent) problem for the continuum solvationmethods refers to the solution of the Schrödinger equation for the effective Hamiltonianof a molecular solute embedded in the solvent reaction field [1–5]. In this section wereview the most relevant aspects of such a QM effective problem, comment on thedifferences with respect to the parallel problem for isolated molecules, and describe theextensions of the QM solvation models to the methods of modern quantum chemistry.Such extensions constitute a field of activity of increasing relevance in many of thequantum chemistry programs [6].

In our discussion the usual Born–Oppenheimer (BO) approximation will be employed.This means that we assume a standard partition of the effective Hamiltonian into anelectronic and a nuclear part, as well as the factorization of the solute wavefunctioninto an electronic and a nuclear component. As will be clear soon, the correspondingelectronic problem is the main source of specificities of QM continuum models, due tothe nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclearproblem, whose solution gives information on solvent effects on the nuclear structure(geometry) and properties, has less specific aspects, with respect the case of the isolatedmolecules. In fact, once the proper potential energy surfaces are obtained from thesolution of the electronic problem, such a problem can be solved using the standardmethods and approximations (mechanical harmonicity, and anharmonicity of variousorder) used for isolated molecules. The QM nuclear problem is mainly connected withthe vibrational properties of the nuclei and the corresponding spectroscopic observables,and it will be considered in more detail in the contributions in the book dedicated tothe vibrational spectroscopies (IR/Raman). This contribution will be focused on the QMelectronic problem.

The structure of the contribution is as follows. In Section 1.5.2 we discuss the structureof effective nonlinear Hamiltonians for the solute. In Section 1.5.3 we present a two-stepformulation of the QM problem, with the corresponding Hartree–Fock (HF) equation.In Section 1.5.4 we introduce the fundamental energetic quantity for the QM solvationmodels while in Section 1.5.5 extensions beyond the HF approximation are presentedand discussed.

1.5.2 The Structure of the Effective Hamiltonian

The effective electronic Hamiltonian, Heff , for the solute has already been introduced inthe contribution by Tomasi. It describes the solute under the effect of the interactionswith its environment and determines how these interactions affect the solute electronicwavefunction and properties. The corresponding effective Schrödinger equation reads

Heff �� = E �� (1.104)


Heff is composed by two terms, the Hamiltonian of the solute H0M (i.e., the molecular part

M of the continuum model) and the solute–solvent interaction term Vint:

Heff = H0M + Vint (1.105)

The structure of Vint depends, in general, on the nature of the solute–solvent interactionconsidered by the solvation model. As already noted in the contribution by Tomasi,a good solvation model must describe in a balanced way all the four fundamentalcomponents of the solute–solvent interaction: electrostatic, dispersion, repulsion, chargetransfer. However, we limit our exposition to the electrostatic components, this beingcomponents of central relevance, also for historical reason, for the development of QMcontinuum models. This is not a severe limitation. As a matter of fact, the QM problemassociated with the solute–solvent electrostatic component defines a general frameworkin which all the other solute–solvent interaction components may be easily collocated,without altering the nature of the QM problem [5].

The operatorial form of Vint depends on the method employed to solve the electrostaticproblem which has to be nested into the QM Equation (1.104) to determine the reactionpotential produced by the polarized solvent on the solute. Here we shall consider themore general case of Vint corresponding to the ASC version of the continuum solvationmodels (see the contribution by Cancès). The operator Vint can be divided into four termshaving a similarity to the two-, one-, and zero-electron terms present in the Hamiltonianof the solute. To show it we consider the solute–solvent interaction energy Uint given asthe integral of the reaction potential times the whole charge distribution �M, convenientlydivided into electronic and nuclear components �M�r� = �e

M�r�+�nM�r�.

The reaction potential has, as sources, the two components of �M and thus it iscomposed of two terms, one stemming from the electronic distribution of the solute Mand one from the corresponding nuclear distribution. As a result, Uint is partitioned intofour terms:

Uint= U ee+U en+U ne+U nn (1.106)

where U xy corresponds to the interaction energy between the component of the interactionpotential having as source �x

M�r�, namely V xint, and the charge distribution �

yM�r�.

Following this formalism, three different QM operators appear, namely V nn� V ne

(it may be shown that U ne and U en are formally identical), and V ee. These have acorrespondence, respectively, to zero-, one-, and two-electron terms of H0

M. We note thatthe zero-order term gives rise to an energetic contribution U nn which is analogous to thenuclei–nuclei repulsion energy Vnn and thus it is generally added as a constant energyshift term in H0

M. The conclusion of this analysis is that we may define four operators(reduced in practice to two, plus a constant term) which constitute the operator Vint ofEquation (1.105).

To make the exposition of Vint more explicit we present here the Schrödinger equationwith the introduction of a new formalism:

Heff �� >=[H0

M + #er V

Rr + #e

r VRrr′< � �#e

r′ �� >]�� >= E�� > (1.107)


With the superscript R we indicate that the corresponding operator is related to thesolvent reaction potential, and with the subscripts r and rr′ the one- or two-electronnature of the operator. The convention of summation over repeated indices followed byintegration has been adopted. �e

r is the electron density operator and �er V

Rr is the operator

which describes the two components of the interaction energy we have previously calledU en and U ne. In more advanced formulations of continuum models going beyond theelectrostatic description, other components are collected in this term. V R

r is sometimescalled the solvent permanent potential, to emphasize the fact that in performing aniterative calculation of �� > in the BO approximation this potential remains unchanged.

The �er V

Rrr′ < � ��e

r′ �� > operator corresponds to the energy contribution that we previ-ously called U ee. This operator changes during the iterative solution of the equation.V R

rr′ is said to be the response function of the reaction potential. It is important to notethat this term induces a nonlinear character to Equation (1.107). Once again, in passingfrom the basic electrostatic model to more advanced formulations other contributions arecollected in this term. The constant energy terms corresponding to U nn and to nuclearrepulsion are not reported in Equation (1.107).

Summing up, the structure of the effective Hamiltonian of Equation (1.107) makesexplicit the nonlinear nature of the QM problem, due to the solute–solvent interactionoperator depending on the wavefunction, via the expectation value of the electronicdensity operator. The consequences of the nonlinearity of the QM problem may be essen-tially reduced to two aspects: (i) the necessity of an iterative solution of the SchrödingerEquation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity,not described by the effective molecular Hamiltonian. The contrast with the correspondingQM problem for an isolated molecule is evident.

1.5.3 A Two-Step Formulation of the QM Problem: Polarization Charges andthe Hartree–Fock Equation

As said before, the nonlinear nature of the effective Hamiltonian implies that the Effec-tive Schrödinger Equation (1.107) must be solved by an iterative process. The procedure,which represents the essence of any QM continuum solvation method, terminates whena convergence between the interaction reaction field of the solvent and the charge distri-bution of the solute is reached.

The most naive formulation of these processes, which corresponds to the mutualinteraction between real and apparent charges, is that used in the first version of thePolarizable Continuum Model developed in Pisa, also denoted as DCPM [2]. We recall ithere, as it is helpful in the understanding of the basic aspects of the mutual polarizationprocess. One starts from a given approximation of #e

M (let us call it #0M) that could be

a guess, or the correct description of #eM without the solvent, and obtains a provisional

description of the apparent surface charge density, or better, of a set of apparent pointcharges that we denote here �qo�o

k . These charges are not correct, even for a fixedunpolarized description of the solute charge density because their mutual interaction hasnot been considered in this zero-order description. To get this contribution, called mutualpolarization of the apparent charges, an iterative cycle of the PCM equation (includingthe self-polarization of each qk) must be performed at fixed #0

M (see the contribution byPomelli for more details). The result is a new set of charges �qo�f

k , where f stands for


final. The �qo�fk charges are used to define the first approximation to Vint, and a first QM

cycle is performed to solve Equation (1.104). With the new #1M the inner loop of mutual

ASC polarization is performed again giving rise to a �q1�fk set of charges. The procedure

is continued until self-consistency.We remark that, in this formulation, we have collected into a single set of one-electron

operators all the interaction operators we have defined in the preceding section, and, inparallel, we have put in the �qk set both the apparent charges related to the electrons andnuclei of M. This is an apparent simplification as all the operators are indeed present.It is interesting here to note that this nesting of the electrostatic problem in the QMframework is performed in a similar way in all continuum QM solvation codes.

Following a canonical order to get molecular wavefunctions, we introduce here theHartree–Fock (HF) level of the two-step approach described above. In this frameworkwe have to define the Fock operator for our model. We adopt here an expansion of thisoperator over a finite basis set � and thus all the operators are given in terms of theirmatrices in such a basis. The Fock matrix reads:

F = h0 +G0�P�+hR +XR�P� (1.108)

The first two terms correspond to H0M, the third to #e

r VRr and, the last to #e

r VRrr′ <� �#e

r′ ��>.Assuming the reader’s familiarity with the standard HF procedure and formalism,

we recall that all the square matrices of Equation (1.108) have the dimensions of theexpansion basis set, and that P is the matrix formulation of the one-electron densityfunction over the same basis set. According to the standard conventions P has been placedas a sort of argument to G0 to recall that each element of G0 depends on P. For analogy,we have made explicit a similar dependence for the elements of XR. We also remarkthat the standard HF equation is nonlinear in character and that in the development ofthis method its nonlinearity is properly treated. The new term XR�P� adds an additionalnon-linearity of different origin but of similar formal nature, that has to be treated inan appropriate way. This fact was not immediately recognized in the old versions ofcontinuum QM methods, giving rise to debates about the correct use of the solute–solventinteraction energy. This point will be treated in the next section.

It should be noted that, as in the previous analysis of the Schrödinger Equation (1.104),in the Fock matrix expression (1.108) we have used a single term to describe the one-electron solvent term. We remark, however, that in the original formulation two matrices,jR and yR, were used, namely:

jR$� = ∑

k

V$��sk�qn��sk� (1.109)

yR$� = ∑

k

V n��sk�qe$��sk� (1.110)

In both expressions the summation runs over all the tesserae (each tessera is a singlesite where apparent charges are located), V$��sk� is the potential of the ∗

$� elementarycharge distribution computed at the tessera’s representative point, V n�sk� is the potentialgiven by the nuclear charges, computed again at the same point, qn�sk� is the apparentcharge at position sk deriving from the solute nuclear charge distribution, and qe

$��sk� is


the apparent charge, at the same position, deriving from ∗$�. The two matrices (1.109)

and (1.110) are formally identical, as said before, and thus in Equation (1.108) we havereplaced them with the single matrix:

hR = 12

(jR +yR

)(1.111)

We note that in computational practice, the more computationally effective expression(6) is generally used.

The elements of the second solvent term in the Fock matrix (1.108) can be put in thefollowing form:

XR$� = ∑

k

V$��sk�qe��sk� (1.112)

with

qe��sk� = ∑$�

P$�qe$��sk� (1.113)

In this way, we have rewritten all the solvent interaction elements of the Fock matrix interms of the unknown qe and qn apparent charges (the last, not being modified in theSCF cycle, can be separately computed at the beginning of the calculation).

1.5.4 The Basic Energetic Quantity: the Free Energy Functional

The second, and more far reaching, implication of the nonlinearity of the QM problemin continuum models involves the fundamental energetic quantity for these models. Tounderstand this point better it is convenient to compare the standard variational approachfor an HF calculation on an isolated molecules with the HF approach for molecules insolution.

For an isolated molecule the Fock operator:

F0 = h0 +G0�P� (1.114)

is used to determine the variational approximation to the ground state exact wave function∣∣� 0HF

⟩corresponding to the system specified by H0. This is determined by minimizing

the appropriate energy functional E��, namely

E0HF =

⟨�HF

∣∣H0∣∣�HF

⟩(1.115)

or, in a matrix form:

E0HF = tr Ph0 + 1

2tr PG0�P�+Vnn (1.116)

where we have used same formalism used in the previous section and we have introducedthe trace operator (tr). Obviously, the nuclear repulsion energy, Vnn, in the BO approx-imation is a constant factor. We note that in Equation (1.116) there is a factor 1/2 in


front of the two-electron contribution. This factor is justified in textbooks by the need toavoid a double counting of the interactions, but this double counting has its origin in thenonlinearity of the HF equation.

Let us now pass to continuum models. As for the isolated molecule, also here thenew Fock operator defined in Equation (1.108) and determining the new solution

∣∣� SHF

⟩is obtained by minimizing an appropriate functional. However, now the kernel of thisfunctional is not the Hamiltonian Heff given in Equation (1.105) but rather Heff − Vint/2and thus the energy of the system is given by

GSHF =

⟨�HF

∣∣∣∣Heff − 12Vint

∣∣∣∣�HF

⟩(1.117)

which, expressed in a matrix form similar to that used for E0HF, reads:

GSHF = tr Ph0 + 1

2tr PG0�P�+ 1

2tr P�j+y�+ 1

2tr PX�P�+Vnn + 1

2Unn (1.118)

where the solvent matrices, j, y and X are those defined in Equations (1.109), (1.110)and (1.112) (here we have only dropped the ‘R’ superscript). We have also added onehalf of the solute–solvent interaction term related exclusively to nuclei, which in the BOapproximation is constant.

Similar expressions and properties of the free energy functional (1.118) hold for allother levels of the QM molecular theory: the factor 1

2 is present in all cases of lineardielectric responses. More generally, the wavefunctions that make the free energy func-tional (1.117) stationary are also solutions of the effective Schrödinger Equation (1.107).

The change of the basic energy functional arises from the nonlinear nature of theeffective Hamiltonian. This Hamiltonian has in fact an explicit dependence on the chargedistribution of the solute, expressed in terms of �e

M, which is the one-body contraction of∣∣�SHF

⟩ ⟨�S

HF

∣∣, and thus it is nonlinear. It must be added that this nonlinearity is of the firstorder, in the sense that the interaction operator depends only on the first power of �e

M.Some comments about nonlinearities in the Hamiltonian may be added here. The case

we are considering here is called scalar nonlinearity (in the mathematical literature itis also called ‘nonlocal nonlinearity’) [7]: this means that the operators are of the formP�u� = �Au�u�Bu where A, B are linear operators and <� ��> is the inner product ina Hilbert space. The literature on scalar nonlinearities applied to chemical problems isquite scarce (we cite here a few papers [2, 8]) but the results justified by this approachare of universal use in solvation methods.

The symbol G used for the energy functional emphasizes the fact that this energyhas the status of a free energy. The explicit identification of the functional (1.117)with the free energy of the solute–solvent system was first done by Yomosa [2], onthe basis of electrostatic arguments. In the Tomasi–Persico 1994 review [4a] alternativejustifications for the factor 1

2 in the expression of the energy were given starting fromperturbation theory, statistical thermodynamics, and classical electrostatics, all valid fora linear response of the dielectric.

We report here only a consideration based on classical electrostatics. Half of the workrequired to insert a charge distribution (i.e. a molecule) into a cavity within a dielectric


corresponds to the polarization of the dielectric itself and it cannot be recovered bytaking the molecule away and restoring it to its initial position. This one half of the workexpended is irreversible, and it has to be subtracted from the energy of the insertionprocess to obtain the free energy (or the chemical potential). Let us now return to theHF level to illustrate some properties which follow from the variational formulation interms of the free energy.

1.5.5 QM Descriptions Beyond the HF Approximation

In the past few years, a great effort has been devoted to the extensions of solvation modelsto QM techniques of increasing accuracy. All these computational extensions have beenbased on a reformulation of the various QM theories describing electron correlation soas to include in a proper way the effects of the nonlinearity of the solvation model byassuming the free-energy functional as the basic energetic quantity.

Most of these extensions have involved electron correlation methods based on varia-tional approaches (DFT, MCSCF, CI,VB). These methods can be easily formulated byoptimizing the free energy functional (1.117), expressed as a function of the appropriatevariational parameters, as in the case of the HF approximation. In contrast, for nonva-riational methods such as the Moller–Plesset theory or Coupled-Cluster, the parallelextension to solvation model is less straightforward.

DFTDensity Functional Theory does not require specific modifications, in relation to thesolvation terms [9], with respect to the Hartree–Fock formalism presented in the previoussection. DFT also absorbs all the properties of the HF approach concerning the analyticalderivatives of the free energy functional (see also the contribution by Cossi and Rega),and as a matter of fact continuum solvation methods coupled to DFT are becoming theroutine approach for studies of solvated systems.

MCSCFApplications of continuum solvation approaches to MCSCF wavefunctions have requireda more developed formulation with respect to the HF or DFT level. Even for an isolatedmolecule, the optimization of MCSFCF wavefunctions represents a difficult computa-tional problem, owing to the marked nonlinearity of the MCSCF energy with respectto the orbital and configurational variational parameters. Only with the introduction ofsecond-order optimization methods and of the variational parameters expressed in anexponential form, has the calculation of MCSCF wavefunction became routine. Thus,the requirements of the development of a second-order optimization method has beenmandatory for any successful extension of the MCSCF approach to continuum solva-tion methods. In 1988 Mikkelsen et al. [10] pioneered the second-order MCSCF within amultipole continuum model approach in a spherical cavity. Aguilar et al. [11] proposedthe first implementation of the MCSCF method for the DPCM solvation model in 1991,and their PCM–MCSCF method has been the basis of many extensions to more robustsecond-order MCSCF optimization algorithms [12].

It is worth recalling here that the building blocks of a second-order MCSCFoptimization scheme, the electronic gradient and Hessian, are also the key elementsin the development of MCSCF response methods (see the contribution by Ågren and


Mikkelsen). Linear and nonlinear response functions have been implemented at theMCSCF level by Mikkelsen et al. [13], for a spherical cavity, and by Cammi et al. [14]and by Frediani et al. [15] for the PCM solvation models.

CIThe conceptual simplicity of the configuration interaction (CI) approaches has attractedthe interest of researchers working in the field of solvation methods [2,16,17] to introduceelectron correlation effects. However, despite this apparent simplicity, the application ofthe CI scheme to solvation models raises some delicate issues, not present for isolatedmolecules.

The nonlinear nature of the Hamiltonian implies a nonlinear character of the CI equa-tions which must be solved through an iteration procedure, usually based on the two-stepprocedure described above. At each step of the iteration, the solvent-induced componentof the effective Hamiltonian is computed by exploiting the first-order density matrix (i.e.the expansion CI coefficients) of the preceding step. In addition, the dependence of thesolvent reaction field on the solute wavefunction requires, for a correct application ofthis scheme, a separate calculation involving an iteration optimized on the specific state(ground or excited) of interest. This procedure has been adopted by several authors [17](see also the contribution by Mennucci).

A further issue arises in the CI solvation models, because CI wavefunction is notcompletely variational (the orbital variational parameter have a fixed value during the CIcoefficient optimization). In contrast with completely variational methods (HF/MFSCF),the CI approach presents two nonequivalent ways of evaluating the value of a first-orderobservable, such as the electronic density of the nonlinear term of the effective Hamil-tonian (Equation 1.107). The first approach (the so called unrelaxed density method)evaluates the electronic density as an expectation value using the CI wavefunction coeffi-cients. In contrast, the second approach, the so-called ‘relaxed density’ method, evaluatesthe electronic density as a derivative of the free-energy functional [18]. As a conse-quence, there should be two nonequivalent approaches to the calculation of the solventreaction field induced by the molecular solute. The ‘unrelaxed’ density approach is byfar the simplest to implement and all the CI solvation models described above have beenbased on this method.

The CI ‘relaxed’ density approach [18] should give a more accurate evaluation ofthe reaction field, but because of its more involved computational character it has beenrarely applied in CI solvation models. The only notably exception is the CI methodsproposed by Wiberg at al. in 1991 [19] within the framework of the Onsager reactionfield model. In their approach, the electric dipole moment of the solute determining thesolvent reaction field is not given by an expectation value but instead it is computed asa derivative of the solute energy with respect to a uniform electric field.

VB MethodsThe powerful interpretative framework of the Valence Bond (VB) theory has beenexploited in several couplings and extensions with continuum models. We mention herethe most relevant in the present context.

Amovilli et al. [20] presented a method to carry out VB analysis of completeactive space-self consistent field wave functions in aqueous solution by using theDPCM approach [3]. A Generalized Valence Bond perfect pairing (GVB–PP) level


combined with a continuum description of the solvent using the DelPhi code [21] toobtain a numerical solution of the electrostatic problem as been developed by Honiget al. [22].

Bianco et al. [23] proposed a direct VB wavefunction method combined with a PCMapproach to study chemical reactions in solution. Their approach is based on a CIexpansion of the wavefunction in terms of VB resonance structures, treated as diabaticelectronic states. Each diabatic component is assumed to be unchanged by the interactionwith the solvent: the solvent effects are exclusively reflected by the variation of thecoefficients of the VB expansion. The advantage of this choice is related to its easyinterpretability. The method has been applied to the study of the several SN1/2 reactions.

Another method from the same PCM family of solvation methods, namely the IEF–PCM [24] (see also the contribution by Cances), has recently been used to develop anab initio VB solvation method [25]. According to this approach, in order to incorporatesolvent effect into the VB scheme, the state wavefunction is expressed in the usual termsas a linear combination of VB structures, but now these VB structures are optimizedand interact with one another in the presence of a polarizing field of the solvent. TheSchrödinger equation for the VB structures is then solved directly by a self-consistentprocedure.

MPn methodsThe quest for methods able to account for the effects of dynamical correlation incontinuum solvation models has lead to several proposals of Møller–Plesset methodsfor the descriptions of the solute. The question of the electron correlation in solvationmodels deserves a few words of comment. The introduction of correlation modifies thetotal electronic charge distribution, with respect to the HF reference, and as a conse-quence the solvent reaction potential is also changed. On the other hand, the polariza-tion induced by the solvent through the reaction field modifies the electron correlationeffects. The decoupling of these effects may give useful information about the solventeffects on the molecular properties of the solute. In this regard, the correlation methodsbased on the perturbation theory give both a conceptual and a computational framework.However, their extension to solvation models involves several difficulties and has beensomewhat controversial. This is reflected in the numerous variant of the MPn methodsfor continuum solvation models. Perturbation theory within solvation schemes has beenoriginally considered by Tapia and Goscinski [1b] at the CNDO level.

An ab initio version of the Møller–Plesset perturbation theory within the DPCMsolvation approach was introduced years ago by Olivares et al. [26] following the aboveintuitive considerations based on the fact that the electron correlation which modifiesboth the HF solute charge distribution and the solvent reaction potential depending onit can be back-modified by the solvent. To decouple these combined effects the authorsintroduced three alternative schemes:

(1) MPn–PTE: the noniterative ‘energy-only’ scheme (PTE), where the solvated HF orbitals areused to calculate MPn correlation correction;

(2) the density-only scheme (PTD) where the vacuum MPn correlated density matrix is used toevaluate the reaction field;

(3) the iterative (PTED) scheme, where the correlated electronic density is used to make thereaction field self-consistent.


PTE and PTD describe, respectively, the effects of the solvation on the electron correlationon the solvent polarization and vice versa; the PTED scheme leads instead to a compre-hensive description of these two separate effects, revealing coupling between them.However, the PTDE scheme is not suitable for the calculation of analytical derivatives,even at the lowest order of the MP perturbation theory.

All the alternative variants of the MPn may be implemented using a ‘relaxed’ densitymatrix or a ‘unrelaxed’ density matrix, in analogy with the CI solvation methods. In thefirst case the correlated electronic density is computed as a first derivatives of the freeenergy, while in the second case only the MPn perturbative wavefunction amplitudes arenecessary.

An analysis of the ‘unrelaxed’ MPn methods in continuum solvation models has beenperformed by Angyan [27]. By rigorous application of the perturbation theory for a nonlinearHamiltonian, as is the case for continuum models, it has been shown that the nth-ordercorrection to the free energy is based on the (n-1)th-order ‘unrelaxed’ density. This meansthat the correct MP2 solute–solvent energy has to be calculated with the solvent reac-tion field due to the Hartree–Fock electron density, as is the case of the PTE scheme.Following this analysis the PTED scheme at the MPn level is not analogous to standardvacuum Mller–Plesset perturbation theory as terms higher than the nth order are included.

Other MP2 based solvent methods consist of the Onsager MP2–SCRF [19], withina ‘relaxed’ density scheme analogous to the PTDE scheme, and a multipole MP2-SCRF model [28], based on a iterative ‘unrelaxed’ approach. The analytical gradientsand Hessian of the free energy at MP2–PTE level, has been developed within thePCM framework [29].

Coupled-cluster MethodsAlthough the correlative methods based on the coupled-cluster (CC) ansatz are amongthe most accurate approaches for molecules in vacuum, their extension to introducethe interactions between a molecule and a surrounding solvent have not yet reached asatisfactory stage. The main complexity in coupling CC to solvation methods comes fromthe evaluation of the electronic density, or of the related observables, needed for thecalculation of the reaction field. Within the CC scheme the electronic density can onlybe evaluated by a ‘relaxed’ approach, which implies the evaluation of the first derivativeof the free energy functional. As discussed previously for the cases of the CI and MPnapproaches, this leads to a more involved formalism.

The only example of a CC solvation model appearing so far in the literature is theCC/SCRF method developed by Christiansen and Mikkelsen [30] using the multipolesolvation approach; the same scheme has also been extended to the CC response methodincluding both equilibrium and nonequilibrium solvation [31]. The CC/SCRF method,exploiting the general concept of variational Lagrangian commonly used in quantumchemistry, defines a coupled-clusters Lagrangian in terms of the free energy functional(14) which leads to a set self-consistent equations. However, the need to evaluate theelectric dipole moments of the solute as a first derivative of the Lagragian requires theintroduction of set of auxiliary CC parameters, which have to be determined in addition tothe CC amplitude. A systematic coupling of CC theory to other continuum methods, likethe ASC based methods is still an open problem, and thus great advances are expectedin the near future.


1.5.6 Conclusion

Molecular solutes described within QM continuum solvation models are characterizedby an effective Hamiltonian which depends on the wavefunction of the solute itself.This makes the determination of the wavefunction a nonlinear QM problem. We haveshown how the standard methods of modern quantum chemistry, developed for isolatedmolecules, have been extended to these solvation models. The development of QMcontinuum methods has reached a satisfactory stage for completely variational approaches(HF/DFT/MCSFC/VB). More progress is expected for continuous solvation model basedon MPn or CC wavefunction approaches.

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1.6 Nonlocal Solvation Theories

Michail V. Basilevsky and Gennady N. Chuev

1.6.1 Introduction

In this chapter we consider the extension of continuum solvent models to nonlocal theoriesin the framework of the linear response approximation (LRA). Such an approximationis mainly applicable to electrostatic solute–solvent interactions, which usually obey theLRA with reasonable accuracy. The presentation is confined to this case.

The medium effects are introduced in terms of � (the dielectric permittivity) or (thesusceptibility). At space point �r conventional electrostatic expressions relate the electricfield strength E��r�, the dielectric displacement D��r� and the polarization field P��r� as

D= �E" P= E" � = 1+4� (1.119)

Generally, � and are tensorial quantities. They reduce to scalars in the case of isotropicmedia, and then describe the longitudinal polarization effects. Our presentation is devotedto this simple transparent case. Complications introduced by anisotropic phenomena arenot considered; they do not change the main idea of nonlocal theory only making thenotation cumbersome.

According to the nonlocal theory the vector fields E��r�� D��r� and P��r� in Equa-tions (1.119) can be treated as time dependent and they obey the Maxwell equations [1].Within the LRA, most general expressions are valid:

D��r� t� =∫

d3r ′dt′��r��r ′� t� t′�E��r ′� t′�

P��r� t� =∫

d3r ′dt′��r�r ′� t� t′�E��r ′� t′�(1.120)

By introducing the integral operators ��r� t�" ��r� t�, we reformulate Equations (1.120)in the contracted form

D = ∧�E" P = ∧

E (1.121)

complemented by the relation between susceptibility operators:∧� = I +4�

∧, where I is

the identity operator. In the most common uniform case (both temporal and spatial) theintegral kernels depend only on differences of their arguments:

��r� �r ′� t� t′� =��r −�r ′� t− t′�"

��r� �r ′� t� t′� =��r −�r ′� t− t′�"

��r� �r ′� t� t′� = ��r −�r ′� �t− t′�+4��r −�r ′� t− t′�

(1.122)


Within this additional constraint the Fourier transforms are useful:

��k�� =∫

d3�r�d�t� exp�i��k�r�+ i�t��r�t�

��k�� =∫

d3��r�d�t� exp�i��k�r�+ i�t��r�t�

(1.123)

where �r = ⇀r − �r ′ and t = t − t′ (�k and � are wavevector and frequency variables).

Scalar products of vectors �k and �r are denoted as ��k�r�. Correspondingly

E��k�� =∫

exp�i��k�r�+ i�t�E��r� t�d3rdt

D��k�� =∫

exp�i��k�+i�t�D��k�r� t�d3rdt

P��k�� =∫

exp�i��k�r�+i�t�P��rr� t�d3rdt

(1.124)

In the ��k�� domain Equations (1.120)–(1.122) reduce to

D��k�� = ��k��E��k��

P��k��= ��k��E��k��(1.125)

This looks quite similar to the conventional electrostatic Equations (1.119) with theinevitable complication that the susceptibility functions ��k�� and ��k�� becomecomplex valued. Consequently, although the applied electric field E��k�� can be alwaystreated as a real quantity, the response fields D��k�� and P��k�� are complex. Undercertain constraints on �k and �, Equations (1.120), (1.121) and (1.125) can be consideredas solutions to time-dependent electrodynamic (Maxwell) equations. This is a legitimateapproximation provided relativistic (i.e. magnetic) effects are negligible. We follow thisapproach, which will be called the ‘quasilectrostatic approximation’ in the forthcomingtext. It becomes exact in the true electrostatic limit � = 0. Then ��k�0� = ��k� and��k�0� = ��k� represent pure effects of spatial dispersion. In practical implementationstemporal (or frequency) dispersion and spatial dispersion effects are often treated sepa-rately, sometimes being combined within simple models. We follow this strategy in thepresent contribution.

The technique of complex-valued dielectric functions was originally applied to solva-tion problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfertheory. They reformulated in terms of ��k�� the familiar golden rule rate expressionfor electron transfer [3]. This idea, thoroughly elaborated and extended by Dogonadze,Kuznetsov and their associates [4–7], constitutes a background for subsequent nonlocalsolvation theories.


1.6.2 Temporal (or Frequency) Dispersion

We consider as an example the second relation of Equations (1.120) and (1.125) with-drawing from them both k-dependence and anisotropic effects:

P�t� =+�∫

−�d��t�E�t− ��" P��= ��E�� (1.126)

Note that the identity

+�∫−�

�t�E��d� =+�∫

−��t− ��E��d�

is valid and its first version accepted in Equation (1.126) is more convenient. Fields P andE depend on space points whereas susceptibility is r-independent. Equation (1.126) isnonlocal in the time domain which means that the response P�t� is determined by thewhole evolution of E over the period �t − ��. The causality principle requires that theresponse cannot precede the input signal. This implies the condition �� = 0�� < 0�, i.e.the susceptibility must be a step function. With positive �, the response P�t� lags behindthe driving force E�t− �� [8].

Another condition arises because P and E are real in the time domain. Combined withthe causality this establishes the following form of the complex susceptibility function�� [9]

�� =�∫

0

�t�exp�i�t�dt= 1��+i2��" 1�−�� = 1��2�−��= −2��

(1.127)

Hence, 1 and 2 are real and, respectively, even and odd functions of frequency. As aconsequence of this property, the important Kramers–Kronig relation arises [9, 10]:

1�� = 2�

P

�∫0

�′2��′�

�′2 −�2d�′ (1.128)

here symbol P denotes the principal value of the integral. Note that the real part ofthe susceptibility, i.e. 1�� is responsible for dielectric screening effects whereas theimaginary component 2�� accounts for the absorption of the radiation field.

Frequency regions where 2�� vanishes and �� = 1�� are called transparencyregions. No energy is absorbed here. Provided � is located in a transparency region, theKrames–Kronig relation holds for �� as well as for 1��. This is always true for� = 0, so the static permittivity can be expressed as

0= �0� =�∫

0

2��′�

�′ d�′ (1.129)

We assume here that the integrand behaves properly when �′ → 0.


Typically, another transparency region exists: �a < � < �� where ��, called theoptical frequency, denotes the lower bound of the electronic (optical) absorption spectrum.Provided this transparency band is wide (say, �0/�� < 102; typically �� ≥ 1016 s−1 and�a ≈ 1013–1014 s−1), one can define the optical dielectric permittivity, �� = 1 + 4��.The real quantity � is defined for � < �a

2�

�∫��

�′2��′�

�′2 −�2d�′ � 2

�

�∫��

2��′�

�′ d�′ = �

We obtain as a consequence:

1�� 2�

P

�a∫0

�′2��′�

�′2 −�2d�′+� (1.130)

The following interpretation can be suggested [3–5] for Equation (1.129), which is exact,and Equation (1.130), which is approximate but becomes exact when � = 0. The staticdielectric screening effects arise due to the accumulation of the radiation absorptionover the whole frequency range. Within the LRA, solvent behaves as an ensemble ofharmonic oscillatory modes with frequency �′ which is much higher than the frequency� of the applied field �� < �a�. Thereby �� is a real constant, �� is local andthe corresponding electronic oscillators ��′ > �� are not involved in the observablemedium dynamics, being responsible only for screening effects, measurable in termsof the dielectric constant ��. This is a formulation of the adiabatic approximation forelectronic modes. On the other hand, the oscillators which are sluggish ��<�a� behave asdynamically active ones and produce retardation effects as expressed by Equation (1.126).They govern the solvent relaxation on time scales � >> ��−1.

1.6.3 Time-Dependent Polarizable Continuum Model

In the solvation theory a reformulation of electrostatic Equations (1.119) is expedient.The solute charge density #�r� serves as an input variable, i.e. the driving force. Thetarget of a computation is the scalar solvent response potential &�r�. In the frameworkof LRA the basic relation

&��r� = K��r� =∫

d3rK��r� �r ′��r ′� (1.131)

is valid, where K is the integral response operator. Its symmetric kernel K��r� �r ′�=K��r ′� �r�is called the electrostatic Green function [11]. The expression for K��r� �r ′� depends onthe explicit formulation of a specific problem.

Within this framework the input quantities are the dielectric permittivity �, the solutecharge ��r� and the excluded volume cavity occupied by a solute. The response field iscreated by the surface charge density ��r� (the apparent charge) arising, as a result ofmedium polarization, on the cavity surface S:

&��r� =∫S

d2r ′ ��r ′��r −�r ′� (1.132)


A connection to vector fields (1.119) is established by the notion that � is equal tothe normal component of the polarization vector P��r� located on the external side of S.Polarization vanishes in the bulk of the medium provided the dielectric constant doesnot change there. The apparent charge ��r� found in terms of numerical algorithms [12]is, in turn, a linear functional of ��r�. Its computation is equivalent to a solution ofthe Poisson equation with proper matching conditions for &��r� on the boundary of thecavity, i.e. on surface S.

This solution, formally expressed as Equation (1.131), is essentially nonlocal in space,although the problem is originally formulated in terms of local Equations (1.119). Thespatial nonlocality arises from boundary conditions on S. Simple solutions are avail-able only for spherically symmetrical cases (Born ion or Onsager point dipole). Theequilibrium solvation energy is expressed as

Usolv = 12�&��r�·��r�� = 1

2∫∫d3rd3r ′ ��r�K��r� �r ′��r ′� (1.133)

where scalar product �&��r�·��r�� denotes the volume integral.Let us now consider time-dependent phenomena which can be described in terms of a

quasielectrostatic extension of Equation (1.131) based on Equation (1.126):

&��r�� = K��r�� (1.134)

It is assumed that the time-dependent charge ��r� t� and response &��r� t� are connectedby the linear integral operator K with the time-dependent kernel K��r� �r ′� t�; the quantitiesin Equation (1.134) are the relevant Fourier transforms. The solution can be found [13]for the special case

��r� t� = ��r�'�t� (1.135)

where '�t� is an arbitrary function of time. We consider the Poisson-like equation&��r�� = K��r�, with a solution similar to Equation (1.132):

&��r�� =∫S

d2r ′ ��r ′��

��r −�r ′�

For given � value the apparent charge density ��r�� is available in terms of theextended PCM procedure with a complex-valued dielectric function �, namely, �� =�1��+ i�2�� where �1�� = 1+4�1�� and �2�� = 4�2�� with complex-valuedsusceptibilities defined in Equation (1.127). The complication that both ��r�� and&��r�� become complex is inevitable. However, after applying the inverse Fourier trans-form, they become real in the time domain. This is warranted by the symmetry properties,


the consequence of the causality principle: &��r�� = &1��r��+ i&2��r�� &1��r�−�� =&1��r�� &2��r�−�� = −&2��r��. All derivations follow those for in Equa-tions (1.127)–(1.129). By combining the inverse Fourier transform with the Kramers–Kronig relation (similar to Equation (1.128)) one obtains the real causal function:

&��r� t� = 12�

+�∫−�

&��r�� exp�−i�t�d�"

&��r� t > 0� = 2�

�∫0

&2��r�� sin��t�d�� &��r� t < 0� = 0

(1.136)

The transformation for ��r�� is quite similar. The final solution for the case (1.135)is straightforward because the procedure implemented for computing &��r��, beinglinear, can be extended for &��r�� as well: &��r�� = &��r��(��, where (��is the Fourier transform of (�t�. The inverse Fourier transform gives &��r� t� =+�∫−�

&��r� ��(�t − ��d�. A common selection for (�t� is the step function �(�t > 0� =1�(�t < 0� = 0�. This implies that the solute charge ��r� t� is created instan-taneously at t = 0 and then remains constant, a situation typical for spectro-scopic applications. By taking Equation (1.136) into account we find the basicresult:

&��r� t > 0� =t∫

0

)��r� ��d� (1.137)

This approach, based on a complex-valued realization of the PCM algorithm, reducesto a pair of coupled integral equations for real and imaginary parts of apparent chargedensity for ��r��) [13]. An alternative technique avoiding explicit treatment of thecomplex permittivity has been also derived [14, 15]. The kernel K��r� �r ′� t� of operatorK does not appear explicitly. However, its matrix elements can be computed for anypair of basis charge densities �1��r� and �2��r� �

⟨�1�K��2

⟩= ∫

�1��r�&��r� t�d3r, where

&��r� t�, given by Equation (1.137), corresponds to ��r� = �2��r�.

1.6.4 Formulation of the Spatial Dispersion Theory

Spatial dispersion effects are usually considered separately from time dependences andcorrespond to static limit �= 0. Consequently ��k�0�= ��k� and ��k�0�= ��k� are basicsusceptibility functions. Within the LRA the relation similar to Equation (1.131) is valid.It formally represents a solution to the nonlocal Poisson equation with a k-dependentsusceptibility.

In computational practice, such solutions are restricted by the approximation that thesolvent is uniform and isotropic. It defines in the real space the susceptibility kernel as��r� �r ′� = ��r −�r ′��. The counterpart in the k-domain obtained via Fourier transform,reads ��k� = �k�, where k = ��k�. The representation for � is similar. Parameterization


of such functions is a question of practical importance. It is formulated in the k-domain,usually, as a Lorentzian function

�k� = � + 0 −�1+�2k2

(1.138)

with static and optic values 0 and � (Section 1.6.2). The transformation to the realspace yields

��r −�r ′�� = � ��r −�r ′��+ 0 −�4��2

exp�−��r −�r ′�/��r −�r ′� (1.139)

It is seen that � serves as a screening length, reflecting a correlation between theneighbouring solvent particles; the local uncorrelated model corresponds to � = 0 and��r −�r ′�� ∝ ��r −�r ′��. This notion explains the usually applicable term ‘the correlationlength’ [6]. Equation (1.139) implies that the electronic polarization is local, i.e. nocorrelation exists inside solvent particles, which is an approximation.

Originally, the representation similar to Equation (1.138) was applied to another dielec-tric function [4–6]:(

1− 1��k�

)=

(1− 1

��

)+

(1�0

− 1��

)1

1+*2k2(1.140)

This quantity proves to be proportional to the correlation function of the medium polar-ization (see Section 1.6.7) and Equation (1.140) has the advantage that its parameterscan be extracted from the direct experimental measurements of this correlation function,or from its simulations. Formally Equations (1.138) and (1.140) are equivalent provided� = √

�0/��*, where �0 is the static dielectric constant (see Section 1.6.7).A more refined parameterization allows for the several Lorentzian terms in equations

similar to Equations (1.138) and (1.140) [5, 6, 16]. They contain a number of correlationparameters �i or *i�i= 1�2� � � � �; the interrelations between parameters �i and *i dependon this number.

Representation of the static susceptibility as Equation (1.138) or its multi-term coun-terpart returns us to the frequency dispersion theory (Section 1.6.2). Similar to Equa-tion (1.129), it states that for the static case �k� accumulates additively the contributionsfrom medium polarization modes over the whole frequency absorption spectrum, whichis represented by the imaginary part of the complex susceptibility, i.e. the function2��, or 2�k�� in the present case. As in Equation (1.130), the electronic (inertia-less) modes are separated and assumed to be local. The nonlocality of inertial modes isintroduced by means of correlation lengths �i or *i, which correspond to medium oscil-lators confined within a lower frequency ranges and separated from electronic modesby a transparency region. For instance, an appropriate parameterization of water [6, 16]suggests two Lorentzian terms, associated with infrared (vibrational � = 1013–1014 s−1)and Debye (orientational � = 1011–1012 s−1) absorption. Correlation lengths *i (but not�i) are, roughly speaking, comparable in magnitude with the size of solvent particles.The importance of nonlocal effects is measured by the ratio */Rsol, where Rsol denotes


the characteristic radius which measures the size of the solute (i.e. of its cavity). Thelimit when this ratio vanishes corresponds to the local continuum medium model:

size of solvent particles

size of a solute particle<< 1 (1.141)

By introducing k-dependent susceptibilities one can, at a phenomenological level, imitatethe molecular structure of solvent around the solute with any desired degree of accuracy.Invoking isotropic and uniform approximations such as Equations (1.138) or (1.140)constrains the ability of such an approach to a certain degree. In any case, this is anessential extension of structureless local models of solvent.

1.6.5 Spatial Nonlocal Equations

We consider the formulation which accounts for the excluded volume of a solute particle.This nonlocal extension of the PCM deals with the stepwise dielectric functions ��k�and �k�. Their inverse Fourier transforms change on the boundary of the cavity surface:� = 1� = 0 inside the cavity and � = ��r −�r ′�� = ��r −�r ′�� outside. The starting

point is Equation (1.121) where time variable t is suppressed in operators∧� and

∧, and

Equation (1.125) where frequency is suppressed. By replacing vector field E =−� + bypotential + , the Poisson equation appears and it changes its standard form �2+ = −4��,valid only inside the cavity, to ��

∧��+� = 0 outside.

The boundary conditions require that � remains continuous at the cavity surface, butits normal derivative displays a step. Compared to the PCM matching condition, thematching expression is more complicated because it includes operator

∧� and is nonlocal

in space.General solutions to this problem have been suggested [17–21]. The algorithm is

complicated, requires cumbersome notation and has been actively performed only forsimple spatially symmetric cases. We consider below the spherical case as an illustration.The solution is represented [19] as

+ ��r� = ,��r�+-��r�+&��r� (1.142)

,��r� =∫Vi

d2r ′ ��r ′��r −�r ′� � -��r� =

∫Vl

d2r ′ g��r ′��r −�r ′� � &��r� =

∫S

d2r ′ ��r ′��r −�r ′� (1.143)

The vacuum potential ,��r� is a solution to the ordinary Poisson equation with � = 1in the whole space. The induced potential consists of two components - and & createdby the external g��r� and surface ��r� charge distributions. The normal derivatives �/�nof the volume potentials , and - are continuous; moreover �-/�n = 0 on the surfaceS. The single layer potential &��r�, however, obeys a singular matching condition onS � ��&/�n�i − ��&/�n�e = 4�� r ∈ S�, where subscripts i and e denote internaland external sides of the surface. Its presence allows for a step in �&/�n. With thiscondition Equations (1.142) and (1.143) describe the solutions valid for the general case,without symmetry restrictions. The equations to be solved comprise a procedure for


simultaneously finding unknown functions g��r� and ��r�. As a supplement to PCM, thevolume charge g��r� and its field -��r� appear in the nonlocal theory.

In the spherically symmetrical Born case we consider the charge ��r� = Q ��r� locatedat the centre �r = 0 of the sphere with radius a. The problem reduces to a single dimen-sion: ,��r� = ,�R�� g��r� = g�R�� -��r� = -�R�� &��r� = &�R�� = const, and also,(when R = a) �,/�R = −Q/a2� �-/�R = ��)/�n�i = 0� ��)/�R�e = −4�� . Sphericalcoordinates �r = �R�,�� and �r ′ = �R′�,′� �′� are used here. With this notation, g�R� and� are determined by equations

g�R�+ 4��

�∫a

.�R�R′�g�R′�dR′ = − 1��

(Q

a2+4��

).�R�a� (1.144)

� =�� + 1��

�∫a

dR′ ��R′ 0�a�R′�� = − Q

4�a2

(1− 1

��

)(1.145)

The integral susceptibility kernel is expressed in the form ��r −�r ′�� = � ��r −�r ′��+��r −�r ′��, where is nonlocal and one-dimensional kernels in Equation (1.144) appearas a result of its averaging over angular variables:

.�R�R′� = R′2∫

d,′ sin �′d�′��r −�r ′��

0�R�R′�cos� = R′2∫

d,′ sin �′d�′ ��r −�r ′��(1.146)

As a result of the spherical symmetry the right-hand parts of Equations (1.146) proveto be angle independent; therefore their calculations can be performed with � = , = 0.An analytical solution is available [18, 20] with simple Lorentzian form for the Fouriertransform of susceptibility (see Equation (1.138)) with single correlation length �):

�k� = 14�

[�� −1+ �0 −��

1+�2k2

](1.147)

The corresponding potentials are:

-�R < a� = 4�*g0

-�R > a� = 4�*2g0�1+a/*− exp��a−R�/*��/R

&�R < a� = 4�a�

&�R > a� = 4�a2�/R

(1.148)

with the notation

g0 = − a

4�*

(Q

a2+4�a�

)(1��

− 1�0

)��

1+√��/�0 coth�a/��

� = �� − Q

4�a2

(1��

− 1�0

) √�0/��coth�a/��−�/a�√

�0/��coth�a/��−�/a�+�/a√��/�0 +1

(1.149)


The solvation energy is generally expressed as [20] Usolv = 0�5∫ d3r ��r��-��r�+&��r��.For case (1.147) this reduces to the result

Usolv = −Q2

2a

(1− 1

��

)− Q2

2a

(1��

− 1�0

)1+√

�0/��coth�a/��−�/a�√�0/��coth�a/��−�/a�+�/a

√��/�0 +1

(1.150)

This example shows the degree of complication inherent in the nonlocal extension of thecontinuum theory even for the simplest Born-like case. In accord with Equation (1.141),the dimensionless parameter �/a measures the importance of nonlocality effects; thelocal Born limit is recovered when �/a → 0. The opposite strongly nonlocal limita/� → 0 corresponds to the unscreened solvation: Usolv = −Q2�1 − 1/��/2a. For thegeneral form of the dielectric function !�k� a numerical solution for one-dimensionalEquation (1.144) is straightforward [19]. However, there exists a principal difficultyhindering such solution when !�k� has poles on the real k-axis (see Sections 1.6.7 and1.6.8). This creates oscillating kernels !��r −�r ′�� in the real space.

1.6.6 Uniform Approximation

Let us consider the nonlocal Poisson equation ��+� = −4�� in the uniform space.The singular boundary condition on the surface of the solute cavity is neglected. Notethat this condition furnishes the mechanisms of the excluded volume effect. The solute ischarged and spherical, i.e. ��r� = ��R�. The solution ��R� is obtained by using Fouriertransform [6, 16]; it is valid outside the cavity �R > a�,

+ �R� = 2�

�∫0

dk��k�

sin�kR�

kR��k� (1.151)

Here ��k� is the Fourier transform of ��R�. This Born ion is considered as a conductingsphere with its charge Q being smeared over the surface of its cavity: ��R� =�Q/4�a2� �R − a�� k� = Q sin�ka�/ka. Outside the cavity the electrostatic fieldcreated by this charge is fully equivalent to the field due to the point charge Q consideredearlier. By this means for R > a

+ �R� =2Q�

�∫0

dk��k�

sin�kR�

kR

sin�ka�ka

(1.152)

The solvation energy is obtained from Usolv = 0�5��·�+ −,�� where ,�R� = Q/R is thevacuum potential. This produces the final result [4, 22]:

Usolv = −Q2

�

�∫0

dk(

1− 1��k�

)(sin�ka�

ka

)2

(1.153)


In the same manner [6,16] the interaction energy between a pair of spherical ions (chargesQ1 and Q2 with radii a1 and a2 can be derived:

Usolv�R > a1 +a2� = 2Q1Q2

�

�∫0

dk��k�

sin�ka1�

ka1

sin�ka2�

ka2

sin�kR�

kR(1.154)

Here R denotes the distance between the ion centres. The important condition is that thetwo spheres do not overlap. Equations (1.152)–(1.154) are approximate because of theimplicit assumption that uniform potential (1.151) represents the true potential actuallyexisting in the vicinity of the ion. In fact, this expression is perturbed by matchingconditions on the boundary, which are neglected.

The validity of the uniform approach is illustrated in Figure 1.11 where two solvationenergies are compared: that given by Equation (1.153) and another obtained by the exacttreatment of Equation (1.150). The dielectric function is ��k�= �� +��0 −��/�1+�2k2�and uniform result proves to be the excellent fit for this particular case [20].

Figure 1.11 Solvation energy Usolv versus cavity radius a: solid line corresponds to Equa-tion (1.150) [20]; circles to Equation (1.153) [6]; dashed line to the Born theory (�0 =78�39� �� = 1�7756, (a) = 4�83 Å, (b) � = 0�72 Å).

The approach described can be extended to a more complicated nonspherical case.Similar to Equation (1.154), we consider a neutral system composed of two Born sphereswith Q1 = Q and Q2 = −Q. It is usually called ‘the dumbbell’. For the isolated sphereswe denote their charge densities as �1 and �2, their response fields as &1 = �1 −,1

and &2 = +2 −,2, where +i and ,i �i = 1�2� are defined similar to the single spherecase. The solvation energy for such system equals to Usolv = 0�5��&1·�1�+ �&1·�2�+�&2·�1�+ �&2·�2��. The scalar products mean volume integrals. The reasonable estimatefor separate terms in will be Ui = 0�5 �&i·�i�� i = 1�2�� Uint�R� = �&1·�2� = �&2·�1�,where U1 and U2 are solvation energies obtained in terms of Equation (1.153) whereas theinteraction energy is identified with Equation (1.154). In this result we assume that the


electrostatic energy contributions for each ion can be computed neglecting the presenceof the neighbouring ion. This assumption is acceptable when the spheres do not overlap.Bearing in mind how complicated are accurate nonlocal solutions, the uniform modelcomprises a useful practical tool for estimates of nonlocal solvation effects [6, 16].

1.6.7 Modelling Dielectric Functions

The nonlocal theory was originally based on the approximation of ��k� in the form ofEquation (1.140) [6, 16], but much effort has been focused on calculations of dielectricfunction ��k�. Earlier studies have been based on the integral equations theory (IET) [23]and used the mean spherical approximation (MSA) [24] or the hypernetted chain (HNC)model [25]. Using a few fitting parameters (hard sphere radius in the MSA or Lennard-Jones parameters in the HNC and diffusion coefficients), researchers are able to calculatethe frequency and spatial dispersions. Concerning the frequency dependence the modelsare satisfactory to predict accurately the behaviour at low frequencies, while they provideonly qualitative effects at optical frequencies [26]. The static dielectric properties ofmolecular liquids have been studied more intensively on the basis of the IET [27, 28] ormolecular dynamics (MD) simulations [29–34]. Figure 1.12 shows the static dielectricfunction ��k� of water under normal conditions, which is obtained by the MD and by theIET with the employment of the reference interaction site model [35]. As can be seenthe IET reproduces the qualitative behaviour of ��k�, although the description of detailsis less satisfactory due to application of the rigid model of water molecule.

0

–10

–20

–30

–40

0 2 4 6 8 10 12

k [A–1]

1ε (k)

Figure 1.12 Dielectric function �−1�k for bulk water calculated with the RISM method(dashed line) and for MD simulations (solid line) [35].

The IET as well as the simulations indicate that the dielectric constant increases fromthe macroscopic dielectric value to infinity and then becomes negative at some valueof k. Such exotic pole-like behaviour is not unique and has been reported for the one-component plasma and the degenerate electron gas [36]. This overscreening effect leads to


repulsion between two unlike charges and attraction between two similar charges at shortdistances. The overscreening effect is found to have a multi-scale origin. The first reasonis trivial and is caused by the discreteness of molecular liquids, when discrete dipolesoriented around an intruding charge provide an overscreening at a submolecular scale.However, the dielectric overscreening may also be due to intermolecular correlations andcoupling between polarization and density fluctuations [37].

The profiles of dielectric functions in Figure 1.12 obviously disagree with theirLorentzian models considered in Section 1.6.4, which suggest they have a peak at k = 0.It is expected that Lorentzian peaks can survive in the range of small k��k � 1� wherethe accuracy of molecular simulations is insufficient to reveal quite definitely their exis-tence [31]. The question of justifying phenomenological models of ��k� at a microscopiclevel remains open. The pole structure of ��k� leads to an oscillatory behaviour of thenonlocal kernel ��r −�r ′�� and such an oscillating kernel results in an irregular behaviourof the solvation energy as a function of the solute radius, complicating computationsof the solvation energy with the use of non-Lorentzian ��k�. On the other hand, theexotic behaviour of ��k� also leads to several interesting and unexpected consequenceswith important implications. For example, the overscreening effect is believed to berevealed as charge inversion in chemical and biological systems [38] observed as anaggregation of biomolecules. Another example of the exotic behaviour is the insulator–metal transition in metal–ammonia solutions and the associated phase separation. Atlow metal concentrations, the solutions are nonmetallic and have an intense blue colour,characteristic of the formation of solvated electrons. At intermediate concentrations andbelow a critical temperature, the two different phases separate within a miscibility gap.At high enough concentrations of metals, the solutions are metallic with a character-istic bronze coloration. As indicated in ref. [39], these phenomena are strongly relatedto the frequency-dependent dielectric function of the solution. At a finite concentra-tions, owing to the large frequency-dependent polarizability the solvated electrons inducea polarization catastrophe leading to a markedly increased dielectric constant and theinsulator–metal transition.

1.6.8 Applications

Among most familiar applications the time-dependent Stokes shift in absorption–emissionspectra is essentially an effect governed by the nolocal time evolution of solvationshells surrounding electronically excited states. This phenomenon is discussed in thecontribution of Ladanyi to this volume. We only comment here that Sections 1.6.2and 1.6.3 of the present contribution provide a methodological background for this theme.In such a context, spatial nonlocality is usually ignored, although microscopic solventmodels, even the most simplified ones [40–43], actually account for the nonlocal effects.Explicit functions ��k�� have been considered in only few publications [44,45] whereasinvoking �� is a standard way to treat the Stokes shift. To get a satisfactory descriptionof the experiment rather sophisticated functions are required [21,46–49]; simple Debye-like models of �� are hardly efficient.

Applications of spatially nonlocal electrostatic theory are not so numerous. Limitedby simple models reducible to a one-dimensional problem, they only include systemsobeying spherical or planar symmetry. A traditional treatment of hydration free energies


of small spherical ions within a uniform approximation as considered in Section 1.6.6 issuccessful. Fitting the experimental data with the refined multi-term Lorentzian spectralfunctions is surveyed in refs [6,16]. By tuning ion radii and correlation lengths reasonableaccuracy is gained. Three-dimensional computations for small ions are also mentionedin ref. [50]. The interfacial solvation effects accompanying electrochemical processes inthe vicinity of a planar surface have been studied [6, 16]. Nonlocality is significant atrather small distances between the ‘solute’ (ion or electrode surface) in comparison withthe solvent correlation length. The formation of a dynamically ordered water shell is animportant factor determining hydration in biological systems.

Non-Lorentzian dielectric functions discussed in Section 1.6.7 cannot be directlyapplied to treat solvation energies. The poles of ��k� promote numerical instabilities incalculations. They have deep physical roots originating from the interference betweenpolarization and density fluctuations in the vicinity of the solute [37]. Attempts to suppressthis complication in terms of unusually sophisticated methods have been reported [51,52].However, simple traditional solutions look more expedient and efficient. Restricting thetreatment by purely Lorentzian functions ��k� resolves the problem and provide a consis-tent and satisfactory semi-empirical theory for ordinary practical implementations.

Lorentzian dielectric functions have also been used to treat solvent reorganization ener-gies in electron transfer reactions [53, 54] within the framework of the uniform approx-imation. Nonlocal effects reduce their values compared with conventional estimates interms of the Marcus theory. The role of overscreening has been discussed [55]. However,it is not so obvious how to reveal deviations of this type in experimental data, sincenonlinear effects, short rage forces, etc. provide alternative sources of possible compli-cations masking the real physical consequence of spatial dispersion. Still, at least oneconsequence is certain. This is the nonzero values of reorganization energies in nonpolarsolvents (benzene, dioxane, etc) with vanishing permanent dipoles and �� = �0 [55–57].Local electrostatic models predict that the solvent reorganization energy must disappearin such solvent but the values of 0.1–0.3 eV have been observed [55–57] and reproducedin molecular level computations [58, 59]. This effect would arise immediately in termsof the nonlocal theory by invoking the simplest Lorentzian models, although no suchstudies have so far been published.

1.6.9 Conclusions

Discreteness of molecular liquids is a source of microscopic inhomogeneity of a solventrevealed as the formation of a structured shell around the solute. Because of the long-range nature of electrostatic interactions, modelling the electrostatic response by molec-ular simulations taking into account detailed solvent structure requires cumbersomecomputations. The nonlocal theory can in principle provide a computationally tractableapproach and it is therefore a serious candidate for a realistic description of solventeffects. Unfortunately, at its present technical level, the nonlocal approach is consider-ably more demanding than local continuum schemes such as PCM. A numerical solutionof coupled three-dimensional integro-differential equations becomes a formidable taskfor really interesting large solutes. The absence of available universal computer packagesrestricts practical implementations of the method. This is why it has been applied mainly


to analyse idealized one-dimensional models and to reveal common trends in experimentwith the use of additional approximations leading to analytical results.

Nevertheless, the concept of spatial dispersion provides a general background fora qualitative understanding of those solvation effects which are beyond the scope oflocal continuum models. The nonlocal theory creates a bridge between conventional andwell developed local approaches and explicit molecular level treatments such as integralequation theory, MC or MD simulations. The future will reveal whether it can surviveas a computational tool competitive with these popular and more familiar computationalschemes.

Acknowledgement

MVB and GNCh thank the RFBR (grant 04-03-32445).

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1.7 Continuum Models for Excited States

Benedetta Mennucci

1.7.1 Introduction

For long time it has been well known that solvents strongly influence the electronicspectral bands of individual species measured by various spectrometric techniques (UVvisible spectrophotometries, fluorescence spectroscopy, etc.). Broadening of the absorp-tion and fluorescence bands results from fluctuations in the structure of the solvation shellaround the solute (this effect, called inhomogeneous broadening, superimposes homoge-neous broadening because of the existence of continuous set of vibrational sublevels) [1].Moreover, shifts in absorption and emission bands can be induced by a change insolvent nature or composition; these shifts,called solvatochromic shifts, are experimentalevidence of changes in solvation energy. In other words, when a solute is surroundedby solvent molecules, its ground state and its excited state are differently stabilized bysolute–solvent interactions, depending on the chemical nature of both solute and solventmolecules [2, 3].

The accurate modelling of excited state formation and relaxation of molecules in solu-tion is a very important problem. Despite this recognized importance and the numerousapplications that such a modelling might have not only in photochemical or spectro-scopic studies but also in material science and biology, the progress achieved so far isnot as great as that achieved for ground state phenomena. This delay in the developmentof accurate but still computationally feasible strategies to study excited states in solutionis due to the complexity of the problem.

The modelling of electronically excited molecules when interacting with an externalmedium, in fact requires the introduction of the concept of time progress, a concept whichcan be safely neglected in treating most of the properties and processes of solutes in theirground states. In fact, in these cases, and also when introducing reaction processes, onecan always reduce the analysis to a completely equilibrated solute–solvent system. Incontrast, when attention is shifted towards dynamic phenomena such as those involvedin electronic transitions (absorptions and/or emissions), or towards relaxation phenomenasuch as those which describe the time evolution of the excited state, one has to introducenew models, in which solute and solvent have proper response times which must not becoherent or at least not before very long times.

In the previous contributions of this book, an extensive description of continuumsolvation models has been given for equilibrated solute–solvent systems. Here, in contrast,an extension of these models will be given in order to describe solvent effects onelectronic excitation/de-excitation processes.

Different semiclassical schemes [4] have been proposed to evaluate solvatochromicshifts (i.e. the excitation energy difference between gas phase and solution for a givensolute) from the properties of the gas phase molecule. These different schemes usuallyexploit Onsager’s solvation model [5], enclosing the solute in a spherical cavity builtin the continuous dielectric representing the solvent and considering the solute as apolarizable dipole. The solvatochromic shifts are finally given in terms of the ground and


excited state dipoles and polarizabilities of the solute considered in the gas phase, and ofthe static and optical dielectric constant of the solvent.

As shown in the other contributions, continuum models have been significantly modi-fied and improved with respect to the older versions; the same improvements havealso been achieved for their extensions to the study of vertical excitation/de-excitationprocesses. These extensions will be reviewed here but before that, a brief overview willbe given on the main physical aspects to be accounted for in any theoretical model aimedat reliably reproducing solvatochromic shifts.

1.7.2 Physical Aspects

It was mentioned in the Introduction that shifts in absorption and emission bands can beinduced by a change in solvent nature or composition. These shifts, called solvatochromicshifts, are experimental evidence of changes in solvation energy and they have beenwidely used to construct empirical polarity scales for the different solvents.

It is worth mentioning here the use of solvatochromism of betaine dyes proposed byReichardt [6] as a probe of solvent polarity. The exceptionally strong solvatochromismshown by these compounds can be explained by considering that in their ground state theyare zwitterions while, upon excitation, electron transfer occurs exactly in the direction ofcancelling this charge separation. As a result, the dipole moment which is about 15 D inthe ground state becomes almost zero in the excited state and thus solvent interactionschange markedly leading to the observed negative solvatochromism.

An alternative approach to quantify polarity effects was proposed by Kamlet et al.[7]. According to this approach the positions of the bands in UV–visible absorption andfluorescence spectra can be determined as

� = �0 + s�∗ +a!+b0 (1.155)

where � and �0 are the wavenumbers of the band maxima in the solvent consideredand in the reference solvent (generally cyclohexane), respectively, �∗ is a measureof the polarity/polarizability effects of the solvent, ! is an index of solvent hydrogenbond donor acidity and 0 is an index of solvent hydrogen bond acceptor basicity. Thecoefficients s� a and b describe the sensitivity of a process to each of the individualcontributions.

The �∗ scale of Kamlet and Taft deserves special recognition not only because it hasbeen successfully applied in many studies (not limited to UV or fluorescence spectra, andincluding many other physical or chemical parameters such as reaction rate, equilibriumconstant, etc.) but also because it gives a very clear introduction of the problem. Namely,Equation (1.155) indicates that the two main aspect to consider when modelling solventeffects on transition energies are polarity/polarizability effects and hydrogen bonding.Let us briefly analyse these two aspects separately starting from the latter one.

Specific InteractionsSeveral examples have shown that specific interactions such as hydrogen bonding interac-tions should be considered as one of the intrinsic aspects of solvent effects on absorptionor fluorescence spectra.


A well-known example is the case of n → �∗ transitions in solutes with carbonylor amide chromophores in protic solvents. In such transitions, the electronic density onthe heteroatom (either oxygen of nitrogen) decreases upon excitation. This results in adecrease in the capability of this heteroatom to form hydrogen bonds. The effect onabsorption should then be similar to that resulting from a decrease in dipole moment uponexcitation, and a blue shift of the absorption spectrum is expected; the higher the strengthof hydrogen bonding, the larger the shift. This criterion is convenient for assigning ann�∗ band while the spectral shift can be used to determine the energy of the hydrogenbond. It is easy to predict that the fluorescence emitted from a singlet state n�∗ will bealways less sensitive to the ability of the solvent to form hydrogen bonds than absorption.In fact, if n → �∗ excitation causes hydrogen bond breaking, the fluorescence spectrumwill only be slightly affected by the ability of the solvent to form hydrogen bonds becauseemission arises from an n�∗ state without hydrogen bonds.

Another case in which hydrogen bonding can play a role is represented by the � → �∗

transitions. In these cases, it is often observed that the heteroatom of a heterocycle (e.g.N) is more basic in the excited state than in the ground state. The resulting excitedmolecule can thus be hydrogen bonded more strongly than the ground state. As a result,� → �∗ fluorescence is generally more sensitive to hydrogen bonding than � → �∗

absorption.These simple observations clearly show that a change in the ability of a solvent to

form hydrogen bonds can affect the nature �n�∗ versus ��∗� of the lowest singletstate. Some aromatic carbonyl compounds often have low-lying, closely spaced ��∗

and n�∗ states. Inversion of these two states can be observed when the polarity andthe hydrogen-bonding power of the solvent increases, because the n�∗ state shifts tohigher energy whereas the ��∗ state shifts to lower energy. This results in an increase influorescence quantum yield because radiative emission from n�∗ states is known to beless efficient than from ��∗ states. The other consequence is a red shift of the fluorescencespectrum.

From these few examples it is apparent that the shifts occurring in hydrogen-bondingsolvents are complex and may occur in either direction, but the take-home message is thatspecific first solvation-shell effects cannot be ignored. On the basis of this picture, onemight guess that a good computational prediction of the excitation energies of hydrogen-bonding solute–solvent systems is obtained in terms of clusters of solute plus few solventmolecules, namely those interacting with H-bond accepting and donating sites in thesolute. In contrast, from many studies it follows that this picture is not completelyright, or at least it is incomplete. These analyses in fact show that the supermoleculeapproach is surely needed to predict the blue or red character of the solvent-inducedshifts. However, a better agreement with experimental observation is found when acontinuum model is added on top of the aggregates containing the solute and some explicitsolvent molecules [8]. This result can be explained by considering that continuum modelsrepresent an effective way to include the electrostatic long range effects missing in thecluster-only description. An alternative approach found in the literature producing similarresults considers explicitly solvent molecules belonging to the second and outer solvationshells. It is easy to understand that, because of the disordered nature of the solvent, a largenumber of calculations on different clusters are needed in this type of model to achieveconvergency in the statistical sampling. By contrast, the use of a continuum description


allows the consideration of many different solute–solvent configurations to be avoided,as by definition it accounts for an implicit average.

Polarity Effects: the Nonequilibrium SolvationIn order to analyse bulk polarity effects it is common to represent the electrostaticresponse of the solvent in terms of the polarization function P. This vectorial functionin fact can be directly connected to any electric field (here that produced by the solute)through a single quantity, the susceptibility , or equivalently the permittivity � [9].

To apply this picture to solvatochromism we have to consider that the responses of themicroscopic constituents of the solvent (molecules, atoms, electrons) required to reacha certain equilibrium value of the polarization have specific characteristic times (CT).When the solute charge distribution varies appreciably within a period of the same orderas these CTs, the responses of these constituents will not be sufficiently rapid to build upa new equilibrium polarization, and the actual value of the polarization will lag behindthe changing charge distribution.

To understand this point better, it is convenient to introduce a partition of the sourcesof the dynamical behaviour of the medium into two main components. One is representedby the molecular motions inside the solvent due to changes in the charge distribution,and/or in the geometry, of the solute system. The solute when immersed in the solventproduces an electric field inside the bulk of the medium which can modify its structure,for example inducing phenomena of alignment and/or preferential orientation of thesolvent molecules around the cavity embedding the solute. These molecular motions arecharacterized by specific time scales of the order of the rotational and translational timesappropriate to the condensed phases. In a analogous way, we can assume that the singlesolvent molecules are subjected to internal geometrical variations, i.e. vibrations, due tothe changes in the solute field; once again these will be described by specific shortertime scales. The translational, the rotational and/or the vibrational motions all involvenuclear displacements and therefore, in the following, they will be collectively indicatedas ‘nuclear motions’. The other important component of the dynamical nature of themedium, complementary to the nuclear one, is that induced by motions of the electronsinside each solvent molecule; these motions are extremely fast and they represent theelectronic polarization of the solvent.

These nuclear and electronic components, owing to their different dynamic behaviour,will give rise to different effects. In particular, the electronic motions can be considered asinstantaneous and thus the part of the solvent response they cause is always equilibratedto any change, even if fast, in the charge distribution of the solute. In contrast, solventnuclear motions, markedly slower, can be delayed with respect to fast changes, andthus they can give rise to solute–solvent systems not completely equilibrated in the timeinterval of interest in the phenomenon under study. This condition of nonequilibrium willsuccessively evolve towards a more stable and completely equilibrated state in a timeinterval which will depend on the specific system under scrutiny.

If we limit our description to the initial step of the whole process, i.e. the verticalelectronic transition (absorption and emission), we can safely assume a Franck–Condonlike response of the solvent, exactly as for the solute molecule; the nuclear motionsinside and among the solvent molecules will not be able to follow immediately the fastchanges in the solute electronic charge distribution and thus the corresponding part of the


response (also indicated as inertial) will remain frozen in the state immediately prior tothe transition. Within this framework, the polarization can be split into two components(see also the contribution by Tomasi):

P � Pfast +Pslow (1.156)

where fast indicates the part of the solvent response that always follows the dynamicsof the process and slow refers to the remaining slow term. Such splitting in the mediumresponse gives rise to the so called ‘nonequilibrium’ regime. Obviously, what is fast andwhat is slow depends on the specific dynamic phenomenon under study. In a very fastprocess such as the vertical transition leading to a change of the solute electronic state viaphoton absorption or emission, Pfast can be reduced to the term related to the response ofthe solvent electrons, whereas Pslow collects all of the other terms related to the variousnuclear degrees of freedom of the solvent.

This analysis shows that in order to account properly for solvent polarity effects, asolvation model has to be characterized by a larger flexibility with respect to the samemodel for ground state phenomena. In particular, it should be possible to shift easily froman equilibrium to a nonequilibrium regime according to the specific phenomenon underscrutiny. In the following section, we will show that such a flexibility can be obtained incontinuum models and generalized to QM descriptions of the electronic excitations.

1.7.3 Quantum Mechanical Aspects

Within the QM continuum solvation framework, as in the case of isolated molecules,it is practice to compute the excitation energies with two different approaches: thestate-specific (SS) method and the linear-response (LR) method. The former has a longtradition [10–24], starting from the pioneering paper by Yomosa in 1974 [10], and it isrelated to the classical theory of solvatochromic effects; the latter has been introduced fewyears ago in connection with the development of the LR theory for continuum solvationmodels [25–31].

The state-specific method solves the nonlinear Schrödinger equation for the state ofinterest (ground and excited state) usually within a multirefence approach (CI, MCSCFor CASSCF descriptions), and it postulates that the transition energies are differencesbetween the corresponding values of the free energy functional, the basic energeticquantity of the QM continuum models. The nonlinear character of the reaction potentialrequires the introduction in the SS approaches of an iteration procedure not present inparallel calculations on isolated systems.

A different analysis applies to the LR approach (in either Tamm–Dancoff, RandomPhase Approximation, or Time-dependent DFT version) where the excitation energies aredirectly determined as singularities of the frequency-dependent linear response functionsof the solvated molecule in the ground state, and thus avoiding explicit calculation ofthe excited state wave function. In this case, the iterative scheme of the SS approachesis no longer necessary, and the whole spectrum of excitation energies can be obtained ina single run as for isolated systems.

Although it has been demonstrated that for an isolated molecule the SS and LRmethods are equivalent (in the limit of the exact solution of the corresponding equations),


a formal comparison for molecules described by QM continuum models shows that thisequivalence is no longer valid.

The origin of the LR–SS difference was imputed to the incapability of the nonlineareffective solute Hamiltonian used in these solvation models to correctly describe energyexpectation values of mixed solute states, i.e., states that are not stationary. Since in aperturbation approach such as the LR treatment the perturbed state can be seen as a linearcombination of zeroth-order states, the inability of the effective Hamiltonian approachto treat mixed states causes an incorrect redistribution of the solvent terms among thevarious perturbation orders [32].

A simple but effective strategy (‘corrected’ LR, or cLR) aimed at overcomingthis intrinsic limit of the nonlinear effective solute Hamiltonian when applied to LRapproaches has been first proposed by Caricato et al. [33]. With such a strategy, the state-specific solvent response is recovered within the linear response approach. As a result,the LR–SS differences in vertical excitation energies are greatly reduced (still keepingthe computational feasibility of LR schemes).

Operative EquationsIn the previous contributions to this book, it has been shown that by adopting a polar-izable continuum description of the solvent, the solute–solvent electrostatic interactionscan be described in terms of a solvent reaction potential, V� expressed as the electrostaticinteraction between an apparent surface charge (ASC) density � on the cavity surfacewhich describes the solvent polarization in the presence of the solute nuclei and elec-trons. In the computational practice a boundary-element method (BEM) is applied bypartitioning the cavity surface into Nts discrete elements and by replacing the apparentsurface charge density � by a collection of point charges qk, placed at the centre of eachelement sk. We thus obtain:

V��r� = ∑k

1

�r − sk�q�sk" ��GS� (1.157)

where r is the electronic coordinate and we have indicated the explicit dependence ofthe apparent charges q on the solvent dielectric constant � and the solute ground statedensity �GS (including the nuclear contribution).

The corresponding energetic functional to be minimized becomes:

G = �� ∣∣H0 + V�

∣∣��− 12�� ∣∣V�

∣∣�� (1.158)

and its minimization for the ground state gives the equation:

Heff �� =[H0 + V�

]�� = EGS �� (1.159)

This approach allows us to rewrite Equation (1.158) as

GGS = EGS − 12

∑i

VGS�si�qGS�si� (1.160)


where VGS�si� is the electrostatic potential produced by the solute in its electronic groundstate on the cavity.

The free energy expression given in Equation (1.160) for a ground state can begeneralized to both an equilibrium and a nonequilibrium excited state K.

By rewriting the solute electronic density (in terms of the one-particle density matrixon a given basis set) corresponding to the excited state K as a sum of the GS and arelaxation term P, and by assuming a complete equilibration between the solute in theexcited state K and the solvent, we obtain

G eqK = EK

GS − 12

∑i

VGS�si�qGS�si�+ 12

∑i

V�si"P�q�si"P� (1.161)

where we have defined:

EKGS =

⟨�

eqK

∣∣H0 + V��GS�∣∣� eq

K

⟩(1.162)

=⟨�

eqK

∣∣H0∣∣� eq

K

⟩+∑

i

VK�si�qGS�si�

as the excited state energy in the presence of the fixed reaction field of the ground state�V��GS��

In the above equations we have exploited the linear dependence of the solvent chargesand the corresponding reaction potential on P, namely:

VK�si� = VGS�si�+V�si"P�

qK�si� = qGS�si�+q�si"P�

The nonequilibrium equivalent of Equation (1.161) can be obtained using two alternativebut equivalent schemes (often associated to the names of Pekar and Marcus). The twoschemes are characterized by a different partition of the low and fast contributions of theapparent charges, namely we have [34]:

Partition I qK = qorGS +qel

K

Partition II qK = qinGS +q

dynK (1.163)

In PI, the slow and fast indices are replaced by the subscripts or and el referring to‘orientational’ and ‘electronic’ response of the solvent, respectively, while in PII thesubscripts in and dyn refer now to an ‘inertial’ and a ‘dynamic’ polarization response ofthe solvent, respectively.

The differences between the two schemes are related to the fact that, in partition I,the division into slow and fast contributions is done in terms of physical degrees offreedom (namely, those of the solvent nuclei and those of the solvent electrons), whereasin partition II, the concept of dynamic and inertial response is exploited. This formaldifference is reflected in the operative equations determining the two contributions toq as, in II, the slow term �qin� includes not only the contributions due to the slow


degrees of freedom but also the part of the fast component that is in equilibrium with theslow polarization, whereas, in I, the latter component is contained in the fast term qel.This difference is made evident by the electrostatic equations defining the correspondingapparent surface charge densities

Partition I(�V

�n

)in

− ��

(�V

�n

)out

= 4��orGS

Partition II(�V

�n

)in

− ��

(�V

�n

)out

= 0

As two different partitions of the solvent charges are introduced, in order to obtainequivalent results, we have to use two different expressions for the nonequilibrium freeenergy, namely:

Partition I

⎧⎪⎨⎪⎩Gneq

K = Gel +Gor − 12

∑i V

orGS�si�

(qor

GS�si�−qelK�si�

)Gor = ∑

i VK�si�qorGS�si�− 1

2

∑i VGS�si�q

orGS�si�

Gel = E0K + 1

2

∑i VK�si�q

elK�si�

Partition II

⎧⎪⎨⎪⎩Gneq

K = Gdyn +Gin

Gin = ∑i VK�si�q

inGS�si�− 1

2

∑i VGS�si�q

inGS�si�

Gdyn = E0K + 1

2

∑i VK�si�q

dynK �si�

In order to obtain a more compact formalism, from now on the partition II will be used.By introducing the following partitioning of the charges:

qdynK = qdyn

GS +qdyn (1.164)

qinK = qin

GS

after some algebra, we get

GneqK = E

K�neqGS − 1

2

∑i

VGS�si�qGS�si�+ 12

∑i

V�si"Pneq �qel

�si"Pneq � (1.165)

which is parallel to that obtained for the equilibrium case but this time the last term iscalculated using the dynamic charges q

dyn .

The vertical transition (free) energy to the excited state K is finally obtained bysubtracting the ground state free energy GGS of Equation (1.160) to Gneq

K of Equa-tion (1.165):

�neqK = E

K0�neqGS + 1

2

∑i

V�si"Pneq �q

dyn �si"Pneq

�

This equation shows that vertical excitations in solvated systems are obtained as a sumof two terms, the difference in the excited and ground state energies in the presence of afrozen ground state solvent and a relaxation term determined by the mutual polarization


of the solute and the solvent after excitation. The latter term is obtained taking intoaccount the fast and slow partition of the solvent response. In the following section weshall show that it is this relaxation term that leads to differences in the two alternativeSS and LR approaches

State Specific vs. Linear ResponseThe requirement needed to incorporate the solvent effects into a state-specific (multirefer-ence) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159).The only specificity to take into account is that in order to calculate V� we have to knowthe density matrix of the electronic state of interest (see the contribution by Cammi formore details). Such nonlinear character of V� is generally solved through an iterativeprocedure [35]: at each iteration the solvent-induced component of the effective Hamil-tonian is computed by exploiting Equation (1.157) with the apparent charges determinedfrom the standard ASC equation with the first order density matrix of the preceding step.At each iteration n the free energy of each state K is obtained as

GnK = ��n

K�H0 ��nK�+ 1

2

∑i

��nK� V��

n−1K � ��n

K� (1.166)

where the solvent term V��n−1K � has been obtained using the solute electronic density

calculated with the wavefunction of the previous iteration.At convergence �n and �n−1

must be the same and Equation (1.166) gives the correct free energy of the state K.We note that this procedure is valid for states fully equilibrated with the solvent; the

inclusion of the nonequilibrium effects needs in fact some further refinements.In particular, the inclusion of nonequilibrium effects requires a two-step calculation: (i)

an equilibrium calculation for the initial electronic state (either ground or excited) fromwhich the slow apparent charges, qs, are obtained and stored for the successive calculationon the final state; (ii) a nonequilibrium calculation performed with the interaction potentialV� composed by two components:

V� = Vfixed +Vchang

Vfixed is constant as a result of the fixed slow charges qs of the previous calculation, whileVchang changes during the iteration procedure. It is defined in terms of the fast charges qf

as obtained from the charge distribution of the solute final state.In order to derive the alternative LR equations, the effective Hamiltonian defined in

Equation (1.159) has to be generalized as

Heff �t� = H0 +V��t�+W�t� (1.167)

where W�t� is a general time-dependent perturbation term that drives the system andinduces a time dependence in the solute–solvent interaction term V� . This time depen-dence originates from dynamic processes involving inertial degrees of freedom of thesolvent. The time scale of these processes is orders of magnitude higher than the timescale of the electron dynamics of the solute, and an adiabatic approximation can be usedto follow the electronic state of the solute, which can be obtained as an eigenstate of thetime-dependent effective Hamiltonian (Equation (1.167)).


As for isolated systems, also for solvated ones, we can express the TD variational wavefunction ��t� in terms of the time-independent unperturbed variational wave function��t� = �0 +��0d+· · · and limit the time-dependent parameter d to its linear term [36].Instead of working in terms of time, we then consider an oscillatory perturbation andexpress W�t� by its Fourier component. In this framework, the linear term in the param-eter assumes the form d = �X exp�−i�t�+ Y exp�i�t��/2 where the (X, Y) vector isdetermined by solving the following system:(

�WW�

)+ �1−��

(XY

)= 0 (1.168)

where

�1−�� =(

A BB∗ A∗

)−�

(1 00 −1

)(1.169)

is the inverse of the linear response matrix for the molecular solute. In Equation (1.169)A and B collect the Hessian components of the free energy functional G with respect tothe wave function variational parameters.

The response matrix depends only on intrinsic characteristics of the solute–solventsystem, and it permits one to obtain linear response properties of a solute with respect toany applied perturbation in a unifying and general way. The poles �±�n� of the responsefunction give an approximation of the transition energies of the molecules in solution;these are obtained as eigenvalues of the system

�1−�n�

(Xn

Yn

)= 0 (1.170)

where �XnYn� are the corresponding transition eigenvectors.This general theory can be made more specific by introducing the explicit form of

the wavefunction; in such a way, by using an HF description, we obtain the randomphase approximation (RPA) (or TDHF). Within this formalism, the free energy Hessianterms yield

Bmi�nj = �mn � ij�+Bmi�nj (1.171)

Ami�nj = mn ij��m −�i�+�mj � in�+Bmi�nj (1.172)

where �mn � ij� indicates two-electron repulsion integrals and �r orbital energies. Herewe have used the standard convention in the labelling of molecular orbitals, that is,�i� j� � � � � for occupied and �m�n� � � � � for virtual orbitals, respectively.

In the definitions (1.171) and (1.172) the effect of the solvent acts in two ways,indirectly by modifying the molecular orbitals and the corresponding orbital energies (theyare in fact solutions of the Fock equations including solvent reaction terms) and explicitlythrough the perturbation term Bmi�nj [26]. This term can be described as the electrostaticinteraction between the charge distribution 2∗

m2i and the dynamic contribution to thesolvent reaction potential induced by the charge distribution 2∗

n2j and it can be written in


terms of the vector product between the electrostatic potential and the induced apparentfast charges, determined by the corresponding transition density charge, namely:

Bmi�nj = ∑k

⟨2m

∣∣∣∣ 1

�r − sk�∣∣∣∣2i

⟩qdyn�sk"2

∗n2j� (1.173)

where the charges qdyn are calculated according to the partition II (Equation (1.163))described in the section Operative Equations.

A parallel theory can be presented for a DFT description; in this case the term TDDFTis generally used. Within this formalism an analogue of Equation (1.170) is obtainedbut now the orbitals to be considered are the occupied and virtual Kohn–Sham orbitalsand the two-electron repulsion integrals have been replaced by the coupling matrixKmi�nj containing the Coulomb integrals and the appropriate exchange repulsion integralsdetermined by the functional used. We note, however, that the explicit solvent term hasexactly the same meaning (and the same form) as the Bmi�nj defined in the HF method(see Equation (1.173)).

A Linear Response Approach to a State-specific Solvent ResponseIn Equation (1.161) (or equivalently in Equation (1.165) for the nonequilibrium case) wehave shown that excited state free energies can be obtained by calculating the frozen-PCM energy EK

GS and the relaxation term of the density matrix, P (or Pneq ) where the

calculation of the relaxed density matrices requires the solution of a nonlinear problemin which the solvent reaction field is dependent on such densities.

If we introduce a perturbative scheme and we limit ourselves to the first order, anapproximate but effective way to obtain such quantities is represented by the LR schemeas shown in the following equations.

Using an LR scheme, in fact, we can obtain an estimate of EK0GS = EK

GS −EGS whichrepresents the difference in the excited and ground state energies in the presence of afrozen ground state solvent as the eigenvalue of the following non-Hermitian eigensystem(1.170) where the orbitals and the corresponding orbital energies used to build A andB matrices have been obtained by solving the SCF problem for the effective Fock (orKS operator), i.e. in the presence of a ground state solvent. The resulting eigenvalue �0

K

is a good approximation of EK0GS in the sense that it correctly represents an excitation

energy obtained in the presence of a PCM reaction field kept frozen in its GS situation.By using this approximation, the equilibrium and nonequilibrium free energies for theexcited state K become:

GeqK = GGS +�0

K + 12

∑i

V�si"P�q�si"P� (1.174)

GneqK = GGS +�0

K + 12

∑i

V�si"Pneq �q

dyn �si"Pneq

� (1.175)

The only unknown term of Equations (1.174) and (1.175) remains the relaxation partof the density matrix, P (or Pneq

) (and the corresponding apparent charges q or qdyn ).

These quantities can be obtained through the extension of LR approaches to analyt-ical energy gradients; here in particular it is worth mentioning the recent formulation


of TDDFT-PCM gradients [37]. In these extensions the so called Z-vector [38] (orrelaxed-density) approach is used. The solution of the Z-vector equation as well as theknowledge of eigenvectors �XK�YK� of the linear response system allow one to calculateP for each state K as:

P = TK +ZK (1.176)

where TK is the unrelaxed density matrix with elements given in terms of the vectors�XK�YK� whereas the Z-vector contribution ZK accounts for orbital relaxation effects.

Once P is known we can straightforwardly calculate the corresponding apparentcharges qx

= q��x�Px� where⎧⎪⎨⎪⎩

�x = �

Px = P if an equilibrium regime is assumed

qx = q⎧⎪⎨⎪⎩

�x = ��Px

= Pneq if a nonequilibrium regime is assumed

qx = qdyn

By introducing the relaxed density P and the corresponding charges into Equa-tions (1.161) (or (1.165)) we obtain the first-order approximation to the ‘exact’ freeenergy of the excited state by using a linear response scheme. This is exactly what wehave called the ‘corrected’ Linear Response approach (cLR) [33]. The same scheme hasbeen successively generalized to include higher order effects [39].

1.7.4 Conclusions

In this contribution we have presented some specific aspects of the quantum mechanicalmodelling of electronic transitions in solvated systems. In particular, attention has beenfocused on the ASC continuum models as in the last years they have become the mostpopular approach to include solvent effects in QM studies of absorption and emissionphenomena. The main issues concerning these kinds of calculations, namely nonequilib-rium effects and state-specific versus linear response formulations, have been presentedand discussed within the most recent developments of modern continuum models.

In these concluding paragraphs it is useful to add that, besides vertical processes,polarizable continuum models can be (and have been) generalized to treat also morecomplex aspects of the relaxation of the excited state following the vertical excitation,or inversely that of the ground state after emission. These are more general dynamicprocesses in which solute and solvent dynamic behaviours mutually interact. In othercontributions to the book some of these processes (such as excitation energy transfers andexcitation-induced electron and proton transfers) are analysed in terms of the availablemodels. Here, however, it is important to stress that in order to account accurately forthe time dependence of the solvent response in many dynamic processes new ideas andnew computational strategies are still required. A possible direction has recently beenproposed in terms of solvent apparent charges continuously depending on time [33, 40].


These are obtained by introducing an explicit time dependence of the permittivity. Thisdependence, which is specific to each solvent is of a complex nature, cannot in generalbe represented through an analytic function. What we can do is to derive semiempiricalformulae either by applying theoretical models based on measurements of relaxationtimes (such as that formulated by Debye) or by determining through experiments thebehaviour of the permittivity with respect to the frequency of an external applied field.

It is evident that these ideas represent only a preliminary indication of a possibledirection to follow which is certainly not the only one or maybe not even the best one,but the good news is that something is moving. We are thus quite confident that now itis time for continuum models to take a new important step further and to extend theirapplication to real time-dependent phenomena. However, this extension should not bedone independently of the experience achieved in past years on more standard applicationsof the models to study energy/geometries and properties of solvated systems. From thesestudies in fact it appears evident that continuum only approaches are often too simplisticand their combinations or couplings with discrete approaches are not only beneficialbut in some cases essential. It seems thus necessary to accept from the very beginningthat hybrid or combined approaches, mixing not only different levels of calculation (asfor example in QM/MM or other similar methods nowadays largely diffused) but alsodifferent ‘philosophies’ (as for example continuum and discrete descriptions but alsoelectronic calculations and statistical analyses), represent very promising strategies.

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