reference interaction site model polaron theory of the

Reference interaction site model polaron theory of the hydrated electron Daniel Laria,“) David Wu, and David Chandler Department of Chemistry, University of Cal$ornia, Berkeley, California 94720

(Received 16 April 199 1; accepted 4 June 199 1)

We have extended the reference interaction site model (RISM)-polaron theory of Chandler er al. [J. Chem. Phys. 81, 1975 ( 1984) ] to treat self-trapping and localized states of excess electrons in polar fluids. The extension is based on a new closure of the RISM equation presented herein. The theory is applied to the hydrated electron employing a simple class of electron-water pseudopotentials. Included in this class are models coinciding with those already examined by others using computer simulations. In those cases, the results for both structural and energetic properties compare well with those of simulation. The work function, or equivalently, the excess chemical potential of the hydrated electron are also computed; the theoretical result agrees with experiment to about 1%. Most interesting, however, is that as the parameter characterizing the pseudopotentials is varied, a critical parameter is found where the electron behavior changes essentially discontinuously from a trapped state to a “super”- trapped state. This transition may have a direct bearing on theoretical efforts to explain the properties of solvated electrons.

I. INTRODUCTION

Theoretical work employing the path integral formula- tion of quantum mechanics”2 has spurred interest in the phenomena of solvated electrons. By combining the path integral approaches with modern numerical methods, it is now possible to treat exactly pseudopotential models for electrons in random fluctuating environments.3 In addition, the path integral approach has been combined with approximate theories of liquid structure-specifically the reference interaction site model (RISM)4-to derive computational- ly convenient integral equation theories for the equilibrium properties of solvated electrons- The approximate theories, representing an extension of Feynman’s polaron theory,’ have been applied with reasonable success to the primi- tive model of an electron in a fluid of hard sphere scatterers.3’b”8 The theoretical results provided guidance to subsequent simulation studies examining localization or self-trapping of electrons in simple fluids.8*9 The RISM-polaron theory has also been used to interpret the anomalous diffusion of excess electrons in liquids such as argon and xenon.6(d) The purpose of the present paper is to further extend this polaron theory to treat the equilibrium properties of electrons in polar fluids such as water.

The response of the fluid density to excluded volume forces leads to the localization of the electron in a nonpolar fluid. A somewhat different mechanism, however, is opera- tive in a high dielectric solvent. Here, it is the coupling of the polarization fluctuations to the charge distribution of the electron that is most important. Despite the differences, however, the general framework of the RISM-polaron theo- ry5 is capable of treating both situations. In fact, by employing the RISM-polaron theory, Malescio and Parrinello” were able to compute the self-correlation function for the trapped e - particle in molten K + Cl -. The results they obtained were in reasonable agreement with those obtained from a path integral molecular dynamics simulation of the

“Presently at CECAM, Universitt Paris-Sud, 91405 ORSAY Cedex PRANCE.

same system. 3(a) Unfortunately, we have found that the Malescio and Parrinello version of the RISM-polaron theory fails to converge for an electron in water at room temperature. Therefore, certain aspects of their approach must be improved upon. Nevertheless, their work with this theory is significant, and their success for the molten salt system moti- vated the extension of the polaron theory presented herein.

In Sec. II, after reviewing the basic elements of the RISM-polaron theory, we discuss the new feature of the theory we have developed. This development, a particular closure relation, is contrasted with earlier extensions of RISM, including that of Malescio and Parrinello. Numerical results obtained from our theory for the electron in liquid water are presented in Sec. III. We conclude with a brief discussion in Sec. IV.

II. THEORY A. Basic elements (Ref. 5)

The general class of many-body systems we are considering is a single electron in classical polar fluid. The pseudopotential between the electron at position r and solvent molecule is of the form

C G(Ir-r~~)l), s where ri(‘) is the position of the sth interaction site in the ith molecule. In the absence of the electron, the response functions or densitydensity pair correlation functions of the flu- idareX,(]r-r’I>.

The ultimate property one wishes to determine is the electron self-correlation function (or pair correlation function of the isomorphic electron polymet ), w,(]r-r’];t-1’) = (6[r-r’-r(t) +r(t’)]). (2.1) Its zero frequency component is

cm w, ( Ir - r’] ) = (/3fi) - ’

s d(t-t’)~,(Ir-r’l;t-t’).

0

(2.2)

4444 J. Chem. Phys. 95 (6). 15 September 1991 0021-9606/91/184444-10$03.00 0 1991 American Institute of Physics

Here, r(t) is a point on the electron path in imaginary or Euclidean time, and the angular brackets denote the equilibrium ensemble average over positions of the solvent molecules and configurations of the electron paths. As usual, 2&i is Planck’s constant and p - ’ is Boltzmann’s constant times temperature. Integration over the solvent coordinates leads to an influence action functional for the electron. In the RISM-polaron theory, the weight functional for electron paths in imaginary time is governed by the action

s=s,[r(t)l +&[Wl, (2.3) where

so = -fi-’ fmdr+mli(t)12

and JO L

Here, m is the electron mass, and the solvent induced interaction has the diagrammatic form illustrated in Fig. 1; i.e.,

c(k) = c %s(k).$‘, (k)?,,(k), Sd

where the overcarets denote Fourier transforms. The quanti- ty E, (k) is the Fourier transform of the coupling function or direct correlation function between the electron and the solvent site s.

‘(0 - r(t’)jl.

(2.4)

dt’ v[ Ir (2.5)

(2.6)

Here, our terminology is generally consistent with earlier works, e.g., Ref. 4 as well as Ref. 5, in our usage of Roman letters to label solvent interaction sites. The solvent density, previously ps in Ref. 5, is now denoted by p.

The coupling functional c, ( r) is the site-site direct correlation function4 It must be determined self-consistently. In particular, for a given average pair structure of the solvated electron, c, (r) is the solution to a RISM equation [see

Laria, Wu, and Chandler: Reference interaction site model polaron theory 4445

Eq. (5.2) of Ref. 5 and also Fig. 1 herein],

pfi, (k) = 2 G;, (k)L (k)2,, (k), ti

(2.7)

augmented with an additional closure condition relating c,, (r) to the electron-solvent pair correlation functions, h, (T), and the e - -solvent pair potentials, U, (r) . Further, the electron path integral must be performed to determine the zero frequency electron correlation function, w, ( jr - r’] ), and the weight functional for this average over quantum paths depends upon u( (r - r’j). Hence, w, ( ]r - r’( ) is a functional of c, ( r), yet c, (r) is a functional of w, ( (r - r’] ) through the RISM equation.

This self-consistent link between the solution of the RISM equation and the electronic path integration is the result of a type of mean field theory, or more precisely, a type of mean pair interaction theory. Without this additional averaging of disorder, which is of course an approximation, the RISM equation (2.7) would depend upon the instanta- neous electron path rather than its average behavior. It is only after averaging that all pairs of points on the path are equivalent. Hence, without this averaging, Eq. (2.7) would become a formidable time-dependent RISM equation, and the influence functional would not be pair decomposable.

Even with this averaging approximation, discussed in more detail in Sec. V of Ref. 5, the resulting influence functional pair interaction, v( jr - r’] ), is quite complicated and not harmonic. To perform the electronic path integral, some additional approximation is required. The method we use is that of Feynman:’ the influence functional, S,, is mimicked by a quadratic form, which is to within an additive constant,

P Sref= +W2 dt

s I

m dt’ r(t - t’)

0 0

X h-(t) - r(t’) 1’. (2.8) The function I( t - f ‘) is determined variationally in accord with the standard bound Z)Z, , where Z is the electronic partition function and Z, [I ( t - t ‘> ] is the first-order perturbation theory estimate for Z when the reference action is So + Sref. The specifics of this variational calculation are described in Sec. VI of Ref. 5.

v(lr-r’l) = r”--0, = 2---J c-h) e7 w-’

ph,(lr-r’l) = eT w

(a)

(c)

B. New closure The remaining feature that must be addressed concerns

the closure of Eq. (2.7). To develop one that is sensible, perturbation theory provides useful guidance. First, consid- er the limit of low solvent density, p -* 0. In that case, h, (r) is determined in principle by solving the scattering problem of the electron interacting with a single solvent molecule. After carrying out the appropriate thermal averages, the result can be expressed diagrammatically as a series of ladder diagrams. For pseudopotentials, U, (r), which vary suffi- ciently slowly in space, this series is well approximated by

FIG. 1. Influence functional interactions and pair correlations. (a) Sche- matic picture illustrating that u( )r - r’l ) bonds couple all pairs of points on the cyclic electron path. (b) The diagrammatic meaning of Eq. (2.6); i.e., the c bonds stand for the c,, (r) functions, and the hatched rectangle represents the solvent response functions, xSf (Ir - r’l). (c) The diagrammatic meaning of Eq. (2.7) for the electron-solvent pair correlation function, with the hatched circle representing the electron response function.

~yh,(r)zexp[ -fi,(r)] - 1,

where

S,(r) =~o,*u,~*w,~,(r). s’

(2.9)

(2.10)

J. Chem. Phys., vol. 96, No. 6, 15 September 1991

4446 Laria, Wu, and Chandler: Reference interaction site model polaron theory

Here, the asterisks indicate convolution integrals and cod ( Ir - r’ 1) = (6 [r - r’ - ris) + ri”’ ] ) (2.11)

is the intramolecular pair distribution function for the solvent.

At finitep, we correct Eq. (2.9) by accounting for the effects of surrounding solvent particles. For pseudopotentials with Coulombic interactions, it seems most important to incorporate the dielectric screening induced by these surrounding particles. This can be done by adding a reaction field correction to - flu, (r) . Such a correction is derived by considering diagrams through second order in perturbation theory, or equivalently, by adopting the model of Gaus- sian density fields linearly coupled to the electron. An alter- nate derivation baaed on an optimized perturbation theory is given in Appendix A. The result is

h[ 1 + h,(r)] z -BL(r) -PAwa(r), (2.12a) where

If II, (r) is purely Coulombic for r greater than some distance, R, then because of Gauss’ law, & u,~ *wj, (r) = 0 for r>R + I&, where Id is the distance to the farthest fixed site s’ from s. In other words, the effective pair potential ii, (r) is that of a charge to concentric shells of charge. This spherical averaging of the pair potential blurs some orientational in- formation, such as the dipolar correlations of the polar molecules to the electron. However, the electron-site interaction strength, and in particular, the reaction field energy, should still be reasonably approximated.

C. Dielectric response and long-ranged behavior of cem

- D Aw, 09 = C w, wed *xdse *c,., *wFs (r)

= C a, *ted *phi, ( r> . d

(2.12b)

In the last equality, we introduced the intermolecular pair correlation function of the pure solvent, h,, (r). The diagrammatic meaning of Eqs. (2.12) is illustrated in Fig. 2.

Equations (2.12) represent the new closure to the RISM equation (2.7). This closure is reminiscent of the HNC closure

At large length scales, it is reasonable to expect that the dielectric screening of a charge q by the surrounding polar fluid becomes that of a continuum dielectric. Specifically, the integrated charge of the screening polarization cloud should approach 4% = - ( 1 - l/e)q. (The subscript “SC” denotes “screening.“) To conserve charge, there is a corre- sponding charge density accumulated at the outer macro- scopic surface of the fluid. However, as the volume tends to infinity, the pair correlation functions integrated up to any finite radius surrounding the test charge will show a net ac- cumulation of integrated charge density within that radius. Thus the usual stoichiometric condition, that lim,+, &, (k) be independent of the site label s, must be generalized to

FII,I, -&d,(k) = qsc, - s

(2.14)

141 + LWIHNC z -Bu,(r) -I- [h,(r) --c,,(r)] (2.13)

adopted by Malescio and Parrinello. When considering simple classical fluids, the same reasoning we have adopted here would lead directly to Eq. (2.13). For polyatomic fluids and for quantum or isomorphic’ polymeric systems, however, Rq. (2.13) does not have the structure of an interacting tagged pair of particles dressed by a reaction field correction-the latter arising from particles other than the tagged pair. As a result, Eq. (2.13) appears to be inappropriate for the class of systems considered herein. The same criticism could be leveled at the straightforward HNC closure of RISM-the so-called “extended” RISM theory.” Improve- ment of extended RISM, however, for classical molecular fluid structure might not be as simple as Eq. (2.12).

where z, is the charge on solvent site s. Therefore, h, (0 + > can differ, up to a constant, for different sites.

This behavior in dielectric fluids is to be contrasted with the perfect shielding found in electrolyte solutions, which can conduct current, i.e., E(k-+O) = 00. The sum rule (2.14) is analogous to the local electroneutrality conditions for electrolyte solutions.‘2

The relationship between Eq. (2.14) and the coupling functional, 2, (k), can be, elucidated by substituting the RISM equation (2.7) for h, (k) and also expanding up to the first two orders in k 2,

CL =p ~zJ,(o+ 1 s

One consequence of the closure is that the bare potential enters only in the combination Ii, (r) = 8, w, *II,,, *ads (r) .

u Infl+hea(lr-r’l)] = M w 87’ e,r 8,r ’

FIG. 2. Closure relation, Eq. (2.12). The u bond represents the pseudopotentials in units of - ka Z’, i.e., - flu, (r) . The left and right c bonds are the direct correlation functions c,$ (r) and c,.~ (r), respectively. The latter depend upon the pure homogeneous solvent only. The former are associated with the solvated electron. The hatched rectangle represents the pure solvent response function, xTS. (jr - r’ I). The right hatched circle is the intramolecular part of that function, i.e., oil (Ir - r’j). The left hatched circle represents the electron response function.

=~z$,(O+) pan* s

=p + #)k 2) (

&- 2) + + @ >

(x +x2’k2)

-I- O(k 2,

=z zsczd- 2) (2) XYS * (2.15)

The integer superscripts indicate the coefficients of powers of k. We have used Z,z, = 0 to eliminate terms. For a nonpolar fluid, I$- 2, = 0, and the usual stochiometric condition holds. However, for long-ranged c, (r) functions, as is the case for polar fluids, c:s-2’(l/k2) = -y&s(k) = - y(4?rgz,q/k*), (2.16)

J. Chem. Phys., Vol. 95, No. 6, 15 September 1991

where q = - e is the charge of a “test” electron, and where y is possibly # 1 if the bare potentials are renormalized. Then

qx = - yC z,z,,y~~‘4?&!3q = - ( >

1 - J- hd E qy

(2.17)

and we see that y also represents the fraction of the continuum dielectric response charge that is predicted to accumu- late around a test charge,

If we confine our attention to long distance behavior only, it is probably true that the screening fraction y should equal unity. The reason is that the response to a full charge - e can be written in a diagrammatic series,i3 where the

longest-ranged term is - L&k *xds (r)zdzs - ( 1 - l/E)pe/r. We find, however, in our new closure that y is not exactly equal to one. This fact is understood by re- writing the RISM equations (2.12) and (2.7) as

1 +h,(r)mxp[ -EL(r) +h,(r) ---L(r)] (2.18a)

and

- h, (4 = L (r) +p d2 Zed*ods. ‘*bS (r), (2.18b)

respectively. Here w, ’ ( Ir - r’] ) is the matrix-functional in- verseofo,(Ir-r’j),and

G(r) = Cw,*cey*aW= --pii, + AC,(r). s’

(2.18~)

Thus, we have

Z,(k) = --P&(k) +C~,‘(k)AE,(k)~;,*(k). d

(2.19) As k-+0, - pii, (k) is asymptotic to 4?r@ez,/k ‘, and 6); ’ (k) tends to 1. Furthermore, AC, (r) is positive and short ranged, as is quickly verified from Eq. (2.18a). Hence, AE, (k) is not singular at small k. The matrix of functions $2 ’ (k) is singular, however, being proportional to k - 2. The constants of proportionality obey a neutrality-like condition, namely, 2, k ‘$2 ’ (k) -0 as k-+0, but in that limit AE, (k) tends to a finite number, which depends on the label s. Therefore, since 4rr/k ’ behavior at small k is equivalent to r - ’ at large r, we arrive at the asymptotic result

c, 09 -P%h, (2.20) where the renormalized charge, Z,, obeys neutrality,

-ps = 0, I

and specifically

(2.21)

% = zs +; F AL (O)qbs, (2.22)

with

t,b,+ = F”, (k 2/4,rr)& l(k). (2.23)

It is the aforementioned Gauss’ law effect that allows c, ( r) to have a renormalized charge, since the requirement that AC,(~) be short ranged is fulfilled, regardless of the

coefficient of l/r in c, (r) . The criterion for the choice of ‘y, then, comes from the small I behavior in ii, (r). Although it has been demonstrated that the asymptotic behavior of bulk fluid correlation functions is only weakly coupled to short- ranged structure,14 in our new closure, the value of y has a direct effect on the global area of the short-range part of h, (r) , since most of the screening cloud is close to the electron. However, in the calculations we performed for the pseudopotentials used in simulation, we find that the z,‘s are reduced in magnitude by only 2-8%.

An anomaly in the sign of the z,‘s can occur, however, if the electron to positive site interactions are not Coulombic until very large distances. The resulting shallow attraction to the positive site is then overwhelmed by the spherically averaged repulsion to another site. This situation will be discussed further in Sec. III E.

III. THE HYDRATED ELECTRON A. Models

Of the several groups that have studied the hydrated electron by simulation, Sprik ef aL3(=) computed the equilibrium properties of the hydrated electron by performing stag- ing path integral Monte Carlo. In their simulations, the water solvent was modeled with the SPC pair potential, Is and two different electron-water pseudopotentials were examined. In both, the pseudopotentials can be written as

u,(r) = - ez,/R,, r-=cR,, =- ez,/r, r> R,. (3.1)

Here z, is the partial charge on the sth site of the SPC molecule (there are three such sites; one for each atom). The cutoff distance, R,, is zero for the oxygen site, and can be some finite value for the proton sites. In one model examined by Sprik et a1.,3(e) R, = = 1 A (model II).

0 A (model I); in the other, R,

Schnitker and Rossky3’d’ considered a somewhat different electron-water pseudopotential’ when they performed a path integral molecular dynamics simulation of the hydrated electron. Numerically, the Schnitker-Rossky pseudopotential is very close to model II of Sprik et al. The simulations of both these groups are useful benchmarks for testing the accuracy of the RISM-polaron theory. Therefore, for model I and also for model II, we have carried out RISM- polaron calculations employing the closure, Eq. (2.12). In addition, we have studied several other choices of R,.

B. Solvent structure factors

The calculations required as input the partial structure factors, i.e., the tti (k) functions, for pure SPC water. We determined these quantities by a simulation’7 of the pure solvent. As discussed below, the theory is sensitive to the structure factors, particularly in the small wavelength regime, k < 1 A - ‘. In our determination of these quantities, we Fourier transformed the pair correlation functions obtained from the simulations. For ka0.2 A- *, the Fourier transforms were smoothed with spline fits, and we employed an iterative inverse Fourier transform and Fourier transform scheme that ensured consistency between the r-space and k-


J. Chem. Phys., Vol. 95, No. 6,15 September 1991

space representation of the pair correlation functions. We also ensured that the quadratic extrapolation to the small wave vector regime, k < 0.2 A - I, was consistent with the compressibility theorem and correct dielectric behavior with a dielectric constant of 70.‘3*‘8 (The dielectric constant of SPC water is 72 f 7.) I9 The partial structure factors obtained in this way are tabulated and presented elsewhere.”

trix is not invertible at k = 0. is,* * The divergence in 2~ ’ ( k) is controlled, however, by a sum rule akin to a neutrality condition, and the product X,.x; ‘(k&., (k) is well be- haved as k+O. Although the behavior of the solvent pair correlation functions at small k is magnified upon taking the inverse oftse (k), we find that the features of the electronic structure are only slightly affected by sensible changes in j& (k) at small k.

C. Numerical procedure

Armed with these functions, the following numerical D. Results

method was used to solve the RISM equations (2.12) and (2.7) : The equations are rewritten as

h,(r)zexp[ --EL(~) + L(r)] - 1 (3.2a) and

In Fig. 3, we show our converged results for S?(t-t’) = (Ir(t) -r(f’)(*)“‘,

t,,(r) = h, (4 - 1 ph, *xG’l*qQ (4, d,s”

(3.2b)

where Z,,X~;.‘*W,~, (r) is a function of the solvent only, and needs to be computed only once prior to any iterations.

For a given electronic 2, (k), Eqs. (3.2) represent a pair of self-consistent equations for 3, (k), or fi, (k), which we solve using a Picard procedure with a mixing ratio. At each step, the new trial function for tes (k) is diluted with a fraction of the previous trial ;,, (k) before inserting into the next iteration. For model I we use a mixing ratio of 5%, i.e., a mixture of 5% new trial and 95% old. For R, = 1 A (model II) or greater, a mixing ratio of 0.5% is required for convergence.

as we compare them with those of simulations. The behavior of this root mean square displacement function illustrates the ground state dominance of the trapped electron6 The theory is in close agreement with the simulation. Radial distribution functions obtained from the theory are compared with those of simulation for model I in Fig. 4 and for model II in Fig. 5. Here too, there is reasonable qualitative agreement, perhaps as close as one might expect for such a theory. Although this theory gives an accurate prediction for the size of the electron cloud, the surrounding water molecules are kept at a slightly farther distance from the electron than found in the simulation. Nevertheless, the characteristic broad features of the hydrated electron are reproduced, with the expected screening cloud. The integrated solvent screening charge,

The RISM-polaron theory also requires the self-consistent determination of w, (r) . To perform such calculations, we begin with o, (r) set equal to that of a free thermally equilibrated electron. The Fourier transform of that function is

s

R

4% CR) = 4m- 2 drp C z,g,, ( r), 0 *

is shown in Fig. 6. We have also computed the average energetics. Our re-

sults are shown in Fig. 7, where we compare our predictions with available simulation and experimental data for the hydrated electron. As noted by Malescio and Parrinello,‘” the average kinetic energy of the electron is given by

&dk) = Wd - ‘lmdtexp( - k’B;r,, (0

6 ), (3.3)

with 9fr, (f> = 3Aftu3fi - t)/(mz, oct<Pi, (3.4)

where /2, = (@/m ) “* is the thermal “wavelength” of the electron. With this starting point, the RISM equation can be solved, as described above, to determine ?es (k), from which we can calculate c, (r) and therefore the influence functional interaction, v(r). With v(r) set, the polaron variational calculation can be performed to find a new o, (r). How- ever, rather than iterating either Eqs. (3.2) or the polaron equations alone to full self-consistency before proceeding with the other, we perform a Picard iteration on both equations simultaneously. A single cycle consists of five iterations of the polaron equations for model I, or two iterations for model II, followed by up to 200jterations of the RISM equations. Less than 200 are used if t,, (k) is converged LO within 0.000 1. About 20 cycles are required to converge t, (k) to this accuracy, and $, (k) to one part in 106.

Equation (3.2b) makes explicit the reason why the new closure, Eq. (2.12)) depends sensitively upon the small wave vector portion of the pure solvent. These quantities appear always in the combination 2,“x~ ‘*a,.,. (r) or, equivalently, 8,-2s; ‘( k)$,., (k). Due to compressibility, thej& (k) ma-


(3.5)

(ke) =t,.,(, + C ‘,a w Pm% -t- yn

), (3.7)

where Y,, is (essentially) the Fourier component of

FIG. 3. Root mean square correlation function for an electron in water. The lines refer to the theoretical results. The circles and squares are simulation results [Ref. 3 (e) ] for the same models, models I and II, respectively. The . . solid parabolic hne IS I, (t - f ‘) .

J. Chem. Phys., Vol. 95, No. 6, 15 September 1991

I’ I ’ 1 ’ 1 ’ 1’1 I I

Model I

0 2 4 6 a r (4

FIG. 4. Electron-solvent radial distribution functions for pseudopotential model I ofan electron in water. The solid lines are our theoretical results for the electron-hydrogen and electron-oxygen distributions. The circles are the simulation results [Ref. 3(e)] for the same distributions.

6

I , I 1 1 I I I 1

I I I 1 I I I I I

1 , I , I , I I I

2 4 6 8 10 f (4

FIG. 5. Electron-solvent radial distribution functions for pseudopotential model II of an electron in water. The solid lines are our theoretical results for the electron-hydrogen and electron-oxygen distributions. The circles are the Monte Carlo simulation results [Ref. 3 (e) 1, and the squares are the molecular dynamics simulation results [Ref. 3(d)] for the same distributions.

FIG. 6. Integrated solvent screening charge q, (r) surrounding the electron up to a radius r, shown as the dotted line. The model I pseudopotential was used. The electron-solvent radial distribution functions are shown as the solid lines to orient the viewer. The dashed line is the response charge of a dielectric continuum with E = 70.


2I(t- t’) [see Eq. (6.6) of Ref. 51, and depends on the frequencies Q,, = 2rrn/@i, - 00 < n < 00. The average potential energy of the electron is given by

tie) =pzJ drg,W~,(r), (3.8) s

where g,, (r) = h, (r) + 1 is the electron-solvent site radial distribution function computed from the RISM equation. Finally, the excess chemical potential, A,u, also shown in Fig. 7, is given by

PAP= -pC&JO)-3 C Yn s n>l (PmQ’, + yn 1

(3.9)

which is a very slight generalization of Eq. ( 12) of Ref. 6(a). The experimental value of A,u has been estimated through thermodynamic cycle analysis of electrochemical measure- ments.**

Our results for the kinetic and potential energies are comparable to those found from simulation, but do not re- produce the Coulomb virial relation (pe) = - 2 (ke) found in the model I simulation. It might seem that the (ke)/(pe) ratio is spuriously affected by the fact that the polaron theory uses a harmonic reference where (pe) = (ke). However, a simple variational perturbation theory calculation of a particle in a single Coulomb potential, using a harmonic reference, shows that the exact Coulomb virial theorem can be recovered, and is, in fact, equivalent to requiring a variationally optimum reference. This fact is demonstrated in Appen- dix B. Furthermore, the quantitative accuracy of the polar- onic approximation to the configuration integral using the self-consistent influence functional of this theory has been verified by a separate Monte Carlo simulation.23 Thus, the primary sources of error in our theory are the approximations associated with the closure to the RISM equation.


Laria, Wu, and Chandler: Reference interaction site model polaron theory

4 - 1 > s TiO 1

I i

/

?------ n 1

-41 1 6 -1.0'

0 J

2.0 4.0 6.0 8.0 1.0

r A 8-

6 3 A- KE

6 cl O- ,----- -

l- 1 ---- 1: -___ ---

b -8: ______c 0.5 ----- -8

I I 1.5 2.0

FIG. 8. Electron-solvent pair correlation functions for the super-collapsed R, = 1 A state. The solid and dashed lines are the electron-oxygen and electron-hydrogen distribution functions, respectively.

FIG. 7. Energetics of the hydrated electron as a function of R,. The solid and short-dashed lines are the theoretical results for the kinetic and potential energies, respectively. The long-dashed line is the theoretical result for the chemical potential. Model I energetics (triangles) were computed using the Monte Carlo method [Ref. 3 (e) 1. Model II energetics (squares) were computed using molecular dynamics by Schnitker and Rossky [Ref. 3 (d) 1. Their pseudopotential, while not precisely the same as model II, is very close to it numerically. The value shown for the model II chemical potential (6lled square), however, is an approximate estimate based upon computed average energies, rather than reversible work functions. The diamond is the experimental work function value reported in Ref. 22.

range of 0.95 8, CR, < 0.97 A. Moreover, the value of the diameter when R, is just less than 1 A is very close to the simulation value. As R, is varied from 0 to just under 1 A, the structure changes qualitatively only in that the value of the intercept of h,, (r = 0) gradually decreases, with the best match of the intercept to the model II simulation data occurring around R, = 0.5 to 1 A. From Fig. 5 it can be seen that the variation in h,, (r) between the simulation structures is consistent with the qualitative changes we describe above for pseudopotentials that ditfer only slightly numerically.

E. Sensitivity of correlation functions to the cutoff radius

In our comparisons with the model II simulations, we have chosen a value of RH just below I A. While the theory gives qualitative agreement with the h, (r)‘s for model I and for model II when R, = 0.95 A, the actual calculated structure for R, = 1 A seems to be completely different (see Fig. 8). In comparison with the simulation structure, for which the electron cloud diameter, 8 (@/2), is about 3 A in diameter, the integral equation predicts the electron is further collapsed to about 1.3 .& in diameter. Interestingly, we ob- serve that during the course of the Picard iterations, the progress toward convergence seems to pause at a near fixed point, with a structure very similar to that found in simulation. Further iterations then cause the structure to collapse to the smaller hnal state described above for R, = 1 A. This behavior is suggestive of a sudden transition with respect to changes in the parameter(s) that characterize the model.

Since model I is simply model II with the cutoff radius set to zero, we investigated the transition of the converged structures from the 3 A diam to the 1.3 A diam as R, is varied continuously from 0 to I A. We discovered that the electronic structure is mostly insensitive to the exact value of the cutoff, except for a very precipitous transition in the

The collapse occurs when the cutoff R, is extended so far that the spherically averaged pair potential E, (r) becomes ineffective in attracting the hydrogens toward the electron. The electron then feels a greater overall repulsion from the water molecules, and becomes further compressed. One peculiarity of the theory is that the effective charges for c,, (r) can become opposite in sign to - JTu,, (r), that is, y is found to be around - 0.2 for the super-collapsed state. This erroneous implication of a negative charge cloud surrounding the electron arises because the spherical averaging effect creates a slightly more favorable interaction between electron and oxygen over electron and hydrogen. However, the difference in the pair correlation function is small enough, as is also evidenced by the low electron-water potential energy, that only 20% of an electron charge accumulates after inte- grating over the bulk. Although the preference of the electron for oxygen or hydrogen is reversed in this case by the theory, we expect that the dominant effect is one of overall repulsion from the water, and so expect the prediction of a collapse to be physical. The quantities &’ (@/2) and y are plotted in Fig. 9 as a function of the cutoff R, in the region where the electron undergoes collapse.

These observations suggest that the integral equation theory is qualitatively correct in the description of the states prior to the collapse, and predicts a collapse transition that is sensitive to the precise value of the cutoff, which by coinci- dence is within a few percent of that chosen for use in the



V.”

I I 1 I I

0.2LA . .

0.1 -

0 I I I 0 0.5 l;O(A) 1.5 2.0

FIG. 9. Size 54(@i/2)/.%‘, (@/2) and charge renormalization y of the electron through the collapse transition. The solid lines are the RISM-polaron results, while the two circles are the simulation results of Ref. 3(e). The dotted line is the “metastable” solution to the RISM-polaron equations prior to further collapse.

model II simulation. With more exploration, the collapse should be observed by simulation, though the precise parameter values for the transition will no doubt differ somewhat from those predicted by our theory.

IV. CONCLUSIONS The good agreement between theory and experiment for

the work function or chemical potential is an important out- come of the theory we have presented here. Our approximate treatment of the solvated electron is unique, in the sense that it is a Hamiltonian based theory. The agreement with simulation for the several structural properties further empha- sizes the apparent correctness of the picture we have con- structed with the RISM-polaron theory. We know of no other theory capable of such performance.

Our theory also suggests that there are subtle dependen- cies of the structure of solvated electrons upon changes in the pseudopotential parameters. We find that although the structure of the hydrated electron varies only slightly within a certain range of values for the pseudopotential cutoff, a dramatic transition can occur with further small changes in the pseudopotential that cause the balance in solvation effects to shift from attractive to repulsive. Our theory de- scribes these predicted structures, except for errors in the orientational correlations of water molecules in the case of the super-collapsed electron, due to the Gauss’ law effect of

orientational averaging of the pair potentials. Nonetheless, it appears that this averaging approximation is a good estimate of the total strength of the interaction at close distances.

Possible extensions of the calculations presented here are ( 1) the determination of the density of states and (2) the calculation of the Franck-Condon factors and the associated excitation spectrum of the hydrated electron. Both seem worthy of future attention. Examination of temperature dependence as well as solvent dependence may also be of interest. This could require further pseudopotential modeling and the fitting of partial structure factors for different temperatures and for different solvents. Finally, the generalization of the polaron theory to the case of two electrons and predictions on formation and stability of the bipolaron would seem well within the grasp of this theory since within the context of the polaron approximations, the problem of alternating weights and fermionic exchange can be solved analytically. This possibility may be particularly interesting for the case of liquid ammonia, where the bipolarons might play a significant role in metal insulator transitions.

ACKNOWLEDGMENTS We are grateful to Toshiko Ichiye and Robert Kuharski

for their help in the fitting of the SPC water partial structure factors. We are grateful to M. L. Klein and co-workers for reporting to us the kinetic and potential energies obtained from their simulations [Ref. 3 (e) 1. This research was sup- ported in part by the National Science Foundation.

APPENDIX A: DERIVATION OF THE CLOSURE FROM AN OPTIMIZED PERTURBATION THEORY

The RISM closure used in this paper can be derived from a perturbative estimate of the following ansatz for the pair correlation function of site a on solute 1 and site y on solute 2 in a solvent,

g,,((r--‘I) = I dld2S(r-r~“))S(r’-r~Y))

X8( l)sW exp CfvA (lr, VA

(rl) - $)I)),

(Al) where the fvA functions account for the direct potential and the reaction field, fa7(Ir-r’l) = -Buay(lr-r’l)

+ c ca9 *xvA*cn7(lr - r’l), ‘1J

L42)

and s( 1) is the intramolecular distribution function of all the coordinates of molecule 1, {rlq’), and s(2) is similarly defined for solute molecule 2. In the end, solute 1 will be the electron, and solute 2 will be a tagged water molecule in the solvent.

Now let us use as a reference distribution

g$(lr-r’l) = s

dld2S(r-ri”))6(r’-r:Y))

XdlW) exp[~~,!Jr~“‘-r:Y’I)], (A3)



where?=,, (r) will be chosen to mimic the combined effect of all the f,, functions. In particular, we can use the fact that

For a particle in a Coulomb potential, the Hamiltonian is

H=p2/2m-e’/r=T+V

and the partition function is

Q=TreFBH= g[r(t)]ewtr(r)i, s

ga,,(lr-r’l) = dld2p,,(l)pYf(2)

X ew (, Cf$ ( Ir,

= (-(

(7) - riy) I ))

w )) where

ar.yf w[r(t)l = -P

dNr-r’l) exP((~f,, -7Ly)a,,)p 9.v By writing (A4) 2lrn r(t) =Cr,e-‘Onf, s1, =-,

n Pfi where the average indicated by ( * * .)a,.yr’ is taken with the distribution

p,,( l&,, (2) =S(I)s(2)6(ria) - r)G(riY’ -r’). (A5)

The first-order perturbation correction to

gay(r)zg2G(r) =exp[fay(r)]

can be set to zero by choosing

Xy = (Cf9v ( Ir19) - riv)l 1) , 9.v ar.yf

which yields

C-46)

(A7)

L,(r) = p%9*fvA*%y(d. (A81 9.A

As a result, (A6) becomes

gayW- C~,,*fvA*~Ay(lr-r’l> (A9) 94 >

and the closure ( 2.12 ) follows from going to the continuum limit P-+ CO for the electron,

w,(r) =$Co,,(I) (A101 9

and

LA (r) = PfvA (r). (All)

APPENDIX 6: COULOMB VIRIAL THEOREM FROM A HARMONIC REFERENCE

Here we show that the reference average of the Coulomb potential (V), as calculated by a variational perturbation theory using a generalized harmonic reference satisfies the Coulomb virial theorem,

(V, = - 2(0,, 031) even while the reference system itself satisfies its own harmonic virial theorem,

(v,), = <no. 032) Here T and V are the kinetic and potential energies, respectively, and will be defined more explicitly below.

(B3)

(B4)

(B6)

and noting r, = r*_ n since r(t) is real, we have

+,I = --+C WmfX)Ir,12+wI[r,], n

where

037)

.

The variational condition is

Q = s Q dr, e”[‘“l>Qoe’“- w“)“, (J39)

where we choose the reference w, to be the most generalized harmonic reference,

w,[r(Ol= -B ~~~(~mli(i)l’+fklr(t)li) 1 m mdt dt’ -- 2

IS -- 0 0 Bfi Pfi

Xl?(t- t’)lr(t) -r(t’)j=. In Fourier components,

(BlOa)

wo[bJ =

wherey,, =2(I’, -IO) [seeEq. (6.6) ofRef.51. We solve the variational equations by varying the pa-

rameters f, = /?k + yn ,

-& (In Q, f (w,), - (w,,>,) = 0, (Bll) n

W I.0 =

A straightforward calculation gives

where

U312)

T @ma”, +fn)w1)-3’2. (B13)

Since the right-hand side is independent of n, f, = fng = f. =/Sk. Thus, all the fs must be zero, and there is only one nontrivial variational parameter,

k= c (firnnz + flk) - I) - 3’2. (B14) n



In terms of the oscillator frequency w = fi, Eq. (B14) yields the self-consistent equation

o = ( 16me4/9d3) tanh3(/3ti/2). (B15)

Here we have used

2 (n’+.*)--=$coth(?ra). n= --m

(B16)

Now we calculate (V), and (T),,

(V>, = -B -1(wI)o

= - 2e2[ (mw/d) tanh(/?fiw/2)]“*, (B17a) as computed from Eq. (B8 ) and again using Eq. (B 16).

Substituting for the variational solution of o [Eq. (BIS)], weobtain

(V), = - 2(3m02e4/4n)“3. (Bl7b)

The kinetic energy can be written from Eq. (3.7) as

(T), d-2 ‘fk+gk =+&cothF, n n

818) .

(B and substituting again for the variational solution gives

(T), = (3rno~~e~/4~)“~, (B 19)

and therefore we recover the Coulomb virial theorem, Eq. (Bl ), as an equivalent condition to the polaron variational condition.

It should be noted that although ( V ), satisfies the Cou- lomb virial theorem, the reference system itself is harmonic and must satisfy the harmonic virial theorem

(V,),=(~kr’) =$k~[flmR2,+/?k]-1=(T),. 0 R

(B20) Finally, it is interesting to note that the absolute energy predicted from the variational treatment is reasonably accurate. Forexample,from (Bl5), (Bl7),and (BlB),weseethatthe ground state energy, E(B) as p-+ CO, is given by - ( me4/W) (4/3~), whereas the exact solution to the Cou-

lomb problem gives - (me”@) (l/2). This approximate result for the ground state energy can be obtained alterna-

tively and straightforwardly by variationally optimizing a Gaussian wave function for the hydrogen atom.

’ R. P. Feynman and A. R. Hibbs, Quantum Mechanicsand Path Integrals (McGraw-Hill, New York, 1965).

‘D. Chandler and P. G. Wolynes, J. Chem. Phys. 74,4079 ( 1981). 3 Included among the many examples are (a) M. Parrinello and A. Rah-

man, J. Chem. Phys. 80,860 (1984); (b) M. Sprik, M. L. Klein, and D. Chandler, Phys. Rev. B 32,545 (1985); (c) M. Sprik, R. W. Impey, and M. L. Klein, Phys. Rev. Lett. 56,2326 ( 1986); (d) J. Schnitker and P. J. Rossky, J. Chem. Phys. 86,347l ( 1987); (e) M. Sprik, R. W. Impey, and M. L. Klein, J. Stat. Phys. 43,949 (1986); (f) D. Coker, B. J. Beme, and D. Thirumalai, J. Chem. Phys. 86,5689 ( 1987); (g) M. Marchi, M. Sprik, and M. L. Klein, J. Phys. 2, 5833 (1990); (h) A. Wallqvist, D. Thiruma- lai, and B. J. Berne, J. Chem. Phys. 86,6404 (1987).

4For reviews see D. Chandler, in Studies in Statistical Mechanics, Vol. VZIZ, edited by E. W. Montroll and J. L. Lebowitz (North-Holland, Am- sterdam, 1981), p. 275; and, more recently, D. Chandler in The Liquid State and Its Electrical Properties, Proceedings NATO ASI, edited by E. B. Kunhardt, L. G. Christophorou, and L. H. Luessen (Plenum, New York, 1988), p. 1.

‘D. Chandler, Y. Singh, and D. M. Richardson, J. Chem. Phys. 81, 1975 (1984).

’ (a) A. L. Nichols III, D. Chandler, Y. Singh, and D. M. Richardson, J. Chem. Phys. 81,5109 (1984); (b) A. L. Nichols, III, and D. Chandler, ibid. 84,398 (1986); (c) ibid. 87,6671 (1987); (d) D. Hsu and D. Chan- dler, ibid. 93, 5075 ( 1990).

‘R. P. Feynman, Phys. Rev. 97,660 (1955). * M. Sprik, M. L. Klein, and D. Chandler, J. Chem. Phys. 83,3042 ( 1985). 9 D. Laria and D. Chandler, J. Chem. Phys. 87,408s ( 1987). “‘G. Malescio and M. Parrinello, Phys. Rev. A 35, 897 ( 1987). ” B. M. Pettitt and P. J. Rossky, J. Chem. Phys. 77, 1451 ( 1982). ‘*F. Stillinger, Jr., and R. Lovett, J. Chem. Phys. 49, 1991 (1968). “D. Chandler, J. Chem. Phys. 67,1113 (1977). I4 P. J. Rossky, M. Pettitt, and G. Stell, Mol. Phys. 50, 1263 (1983). “H. J. C. Berendsen, J. P. M. Postma, W. F. Van Gunstereu, and J. Her-

mans, in Intramolecular Forces, edited by B. Pullman (Reidel, Dor- drecht, 1981), p. 331.

I6 J. Schnitker and P. J. Rossky, J. Chem. Phys. 86,3462 ( 1987). “The simulations were carried out on the University of California, Berke-

ley, CRAY-XMP. The code was a modified version of the molecular dynamics program written by R. W. Impey, 1986. The simulated system contained 230 SPC water particles, and Ewald sums were performed. The temperature was 300 K, and the mass density was 1 g/cm3. The simulations of the hydrated electron in Refs. 3 (d) and 3 (e) were performed at slightly different temperatures, but for the purposes of this work we be- lieve these temperature differences are insignificant.

‘*J. HQye and G. Stell, J. Chem. Phys. 65, 18 (1976). I9 K. Watanabe and M. L. Klein, Chem. Phys. 131, 157 ( 1989). “D. Laria, Ph.D. thesis, University of California, Berkeley, 1987. ” P. T. Cummingsand G. Stell, Mol. Phys. 46,383 ( 1982); P. T. Cummings

and D. E. Sullivan, ibid. 46,665 (1982). **J. Jortner and R. M. Noyes, J. Phys. Chem. 70,770 (1966). 23 D. Wu (unpublished work).


reference interaction site model polaron theory of the

Documents