From Force Fields to Dynamics: Classical and Quantal Paths

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  • From Force Fields to Dynamics:Classical and Quantal PathsDONALD G. TRUHLAR AND MARK S. GoRDON

    Reaction path methods provide a powerful tool for bridg-ing the gap between electronic structure and chemicaldynamics. Classical mechanical reaction paths may usuallybe understood in terms of the force field in the vicinity ofa minimum energy path (MEP). When there is a signifi-cant component of hydrogenic motion along the MEPand a barrier much higher than the average energy ofreactants, quantal tunneling paths must be considered,and these tend to be located on the corner-cutting side ofthe MEP. As the curvature of the MEP in mass-scaledcoordinates is increased, the quantal reaction paths maydeviate considerably from the classical ones, and the forcefield must be mapped out over a wider region, called thereaction swath. The required force fields may be repre-sented by global or semiglobal analytic functions, or thedynamics may be computed "directly" from the electronicstructure results without the intermediacy of potentialenergy functions. Applications to atom and diatom reac-tions in the gas phase and at gas-solid interfaces and toreactions of polyatomic molecules in the gas phase, inclusters, and in aqueous solution are discussed as exam-ples.

    THHE DEVELOPMENT OF PRACTICAL AND ACCURATE METHODSto treat the dynamics of chemical reactions is a criticalchallenge to theoretical chemistry. Most approaches involve

    two steps-the standard approach is first to use some means todetermine or approximate as much as possible of the potentialenergy function (PEF) (1) and then to use that PEF as a startingpoint for investigations of the dynamics. In principle, it is desirableto know the dependence of the PEF on a wide range of all variables,but this becomes increasingly impractical as the complexity of thereaction increases, so computational efforts must be focused onessential features.Many approaches have been used for mapping PEFs for chemical

    reactions. One method is to fit an analytic PEF so that dynamicscalculations agree with available experimental rate data. The diato-mics-in-molecules procedures developed from the early work ofLondon, Eyring, Polyanyi, and Sato (2) (often referred to as theLEPS or generalized LEPS method) have been very useful for suchsemiempirical functions. This approach has been used for manyyears, for example, for the H + H2 and Cl + H2 reactions (3).D. G. Trhiar is in the Department of Chemistry and the Supercomputer Institute,University ofMinnesota, Minneapolis, MN 55455. M. S. Gordon is in the Departmentof Chemistry, North Dakota State University, Fargo, ND 58105.

    3 AUGUST 1990

    In recent years the capability to perform accurate electronicstructure calculations ofPEFs has improved dramatically (4, 5), andit is now possible to improve upon the purely semiempiricalapproach by fitting ab initio or semiempirical electronic structurecalculations at important points on the PEF or by combiningelectronic structure calculations for certain features of the PEF withadditional refinement based on comparing dynamics calculations toexperiment. For reactions with more than four atoms, for example,the reaction ofCH3 with H2 (6, 7), the LEPS approach is sometimesapplied to atoms involved in bond breaking and making andcombined with "molecular mechanics"-type models for nonbondedinteractions. Other approaches are reviewed elsewhere (8). Determi-nation of the critical geometries, interpolation of the electronicstructure results, and analytic representation of the whole PEF frominformation at selected points is a stimulating challenge to thedynamicist's skill and experience, and it allows for the artfulinterplay of theory and experiment. Nonetheless, it is often prefera-ble for such investigations to be more systematic in order to allow awider range of applications.

    In order to capture most of the physics while keeping ourtreatment computationally tractable, we need to ignore large partsof the PEF. The first step in such a development is the recognitionofkey topological features ofPEFs that must be treated as accuratelyas possible for reactive systems. The most significant of these are thepoints that have a zero gradient ofthe energy with respect to atomiccoordinates and therefore correspond to stationary points. If thematrix of second derivatives of the potential energy with respect toatomic coordinates (hessian) at such a point is positive semidefinite,then the point is a local minimum. If the hessian has one (and onlyone) negative eigenvalue, then the associated eigenvector gives thedirection of (downhill) motion to reactants and products, and thepoint is a saddle point. In conventional transition state theory (TST)(9), rate constants for chemical reactions are approximated entirelyon the basis of the geometries, energies, and energy derivatives atstationary points (by the calculation of the appropriate partitionfunctions from molecular structures and vibrational frequencies).This approach provides a simple model for the prediction ofBoltzmann-averaged rate coefficients. Modem techniques in elec-tronic structure theory now make it relatively straightforward todetermine the gradient and hessian at enough geometries to find andcharacterize the stationary points even for many-atom systems (10).

    In many cases we are not satisfied with a description based onlyon stationary points or with only calculating thermal rate coeffi-cients, and we seek to map out and utilize a larger portion of theinteraction space, including the minimum-energy valleys connectingreactants and products to saddle points and the ridges throughwhich reactions may proceed by tunneling. These features certainlyaffect rate constants significantly in many cases, and in addition they

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  • are crucial for determining state-to-state cross sections, especiallywhen the product energy distribution is nonstatistical. We find thatgeneralized transition state theory and semiclassical tunneling meth-ods allow us to base calculations of rate constants, thresholdenergies, and tunneling probabilities on the most critical parts ofthePEF without having a global representation of this function. Themost systematic way to calculate these critical PEF features is tobegin with the steepest descents paths that connect saddle points tominima. Such a path is referred to as a minimum energy path(MEP). Physically intuitive approximations to the dynamics in thevicinity of an MEP [such as one-dimensional (1-D) tunneling,internal centrifugal effects, vibrational adiabaticity, and infinitelydamped trajectories] are facilitated when the path is calculated inisoinertial coordinates, that is, in any coordinate system that is massscaled to make the effective reduced mass the same for all atomicdisplacements [a familiar example is the mass-weighted system (11)of infrared spectroscopy in which the effective reduced mass ischosen as 1 amu]. The MEP in an isoinertial coordinate system (12-16) is sometimes referred to as the intrinsic reaction coordinate(IRC) (15), and we use the two terminologies (MEP and IRC) assynonyms.When potential energy contours are plotted in isoinertial coordi-

    nates, the angle (commonly called the skew angle) between thereactant and product valleys depends on the mass combination, andhence so does the curvature of the MEP. This is illustrated in Fig. 1,which shows four examples ofMEPs: that for the H + D2 reactionhas small curvature because the skew angle is 65.90; those forO + H2 and Cl + HD have medium curvature and skew angles of46.7 and 36.40, respectively; and that for Cl + HCI' (where Cldenotes MC(l and Cl' denotes 37C1) has large curvature because theskew angle is only 13.40. The physical meaning of the skew anglefollows from the fact that mass scaling converts mass effects intorelative distances. Thus, for example, in a light-atom transfer, themotion from the reactant valley to the product valley corresponds toa small reduced mass and hence a fast motion-thus in mass-scaledcoordinates the two valleys must be very close, resulting in a smallskew angle.

    a0- cx

    RH,D2

    0I

    Knowledge of the sequence of geometries along the MEP allowsone to calculate the curvature of the path and hence the reactionpath kinetic energy (13, 17-20), and knowledge of the potentialalong the path and the force constants for motion transverse to itallows one to write a convenient reaction-valley potential (13, 14, 17,18, 21) for low-energy dynamical motions connecting reactants andproducts. One method based on such information is variationaltransition state theory (VTST). VTST uses a criterion ofminimumflux (22), maximum vibrationally adiabatic energy (23, 24), orminimum free energy of activation (20, 21, 24-26) to determine adynamical bottleneck for overbarrier processes, which tend toinvolve only motions in the vicinity of the MEP. When thecurvature of the MEP is small, realistic tunneling probabilities canalso be calculated entirely from the reaction path, its curvature, andthe reaction-valley potential (20, 27, 28). When, however, the MEPis highly curved, tunneling may proceed by shortcuts throughregions beyond the radius of curvature of the MEP (29, 30) asillustrated in Fig. IC; such regions are wider than the valley that canbe described in curvilinear coordinates based on the path itself. Thenthe reaction path, its curvature, and the reaction-valley potential donot provide enough information for an adequate treatment ofreaction dynamics, and one needs to explore a region adjacent to thereaction path called the reaction swath. In either the small- or large-curvature cases, the effects of tunneling may be incorporated intoVTST by a ground-state transmission coefficient (20, 28, 30, 31).

    In the present article, we discuss the use of minimum energypaths, reaction-valley potentials, and quantal tunneling paths forreaction dynamics. First we discuss the essential features of varia-tional transition state theory and semiclassical treatments of tunnel-ing. This is followed by a discussion ofreaction paths, reaction-pathforce fields, dynamics, and recent applications.

    Variational Transition State TheoryA generalized transition state (GTS) is a hypersurface in phase

    space (which is the space of all position and momentum coordi-

    Fig. 1. Potential energy contours (black), transi-tion states (red), minimum energy paths (green),and tunneling paths (blue) for four representativereactions A + BC -* AB + C in isoinertial coor-dinates. RA,BC is the distance from A to BC; YBCis the mass-scaled distance between B and C; andonly collinear geometries are illustrated. The tran-sition states are surfaces dividing reactants fromproducts; they are located at the saddle point and

    e---- _ have zero-point vibrational amplitude for theR vibration transverse to the minimum energy path.CI,HD The reaction proceeds from lower right to the top

    of the figure in each case. (A) H + D2 -* HD +D. The tunneling path is a Marcus-Coltrin path(24, 27), which is appropriate for small-curvaturesystems. (B) 0 + H2--OH + H. The tunnelingpath is the optimum LAG path (39, 69) for a(1\ C_typical tunneling energy at room temperature.(C) Cl + HD HCI + D. The tunneling pathsare three trial paths for a least-action tunnelingcalculation (39) at a typical tunneling energy. The

    R optimum tunneling path in this case is betweenCI,HCI' the two blue curves lying closest to the greencurve. (D) 35Cl + H37CI- H35CI + 37CI. Thetunneling paths are large-curvature paths (30) fortwo different energies: the one farther from thegreen curve for a typical tunneling energy and theone closer to the green curve for a higher energy.

    SCIENCE, VOL. 2494.92

  • nates) or coordinate space that divides reactants from products (21,22, 31). The conventional transition state is a special case dependingonly on coordinates (not momenta) and passing through the saddlepoint; it is constrained to be a hyperplane in isoinertial coordinatesand to be perpendicular to the imaginary frequency normal mode. Itcan be shown (22) that in classical mechanics under thermalequilibrium conditions of the reactant [which conditions are usuallyassumed to hold (32)] the rate coefficient k(T) is bounded fromabove by the one-way flux coefficient kGT( T) for passage throughany GTS, and if the GTS is varied without constraints, then minkGT(TT) = k(T). This is the basis for VTST: we vary the GTS tominimize the calculated flux coefficient, and the minimum value isaccepted as an approximation to k(T). The GTS corresponding tothe minimum flux constitutes a "dynamical bottleneck" to thereaction.The theory as presented above is not useful for two reasons: (i) for

    an N-atom system, phase space has 6N dimensions, and uncon-strained variations of the GTS are impractical; and (ii) quantumeffects, especially zero-point motions and tunneling, are very impor-tant, and classical mechanics is inadequate. Current practical VTSTand tunneling methods (20, 24, 26, 31, 33) have evolved as a way toovercome these limitations.The signed distance along the MEP from the saddle point,

    measured in the isoinertial coordinate system, is called s. We define aone-parameter sequence of GTSs, where the parameter is the valueof s at which the GTS crosses the MEP. Each GTS is, at least in thevicinity of the MEP, a hyperplane in isoinertial coordinates andorthogonal to the MEP at the point of intersection.

    Practical VTST calculations are carried out as follows: First theMEP is calculated on a grid with stepsize bs; this requires at least onegradient evaluation for each step. Then, at typically larger intervalsAs, one calculates the curvature components and a hessian andperhaps higher derivatives depending on whether anharmonicity(33, 34) is to be included in the partition functions. At these hessianpoints one also calculates the GTS standard-state molar-free energyof activation AGGT,O(T,s) for each temperature of interest. Thecurve of AGGT(T, s) as a function of s is called a free energy ofactivation profile. The VTST rate constant for a canonical ensembleat temperature T is given by (21, 24, 26)

    kCvT(T) = min (kBTIh)K0 exp[-AGGTO(T,s)IRT] (1)s

    where CVT denotes canonical variational theory, kB is Boltzmann'sconstant, h is Planck's constant, K is the reciprocal of the standardstate concentration for bimolecular reactions and is unity forunimolecular reactions, and R is the gas constant. Thus the varia-tional transition states are the maxima with respect to s ofAGGT,O( Ts).At low temperature, th...

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