Symmetry of a free energy curve in polar solution

5 August 1994

ELSEVIER

CHEMICAL PHYSICS LElTERS

Chemical Physics Letters 225 (1994) 494-498

Symmetry of a free energy curve in polar solution A. Yoshimori DepartmentofPhysics,Nagoya University,Nagoya464-01, Japan kceived

19 February 1994

AhStTlWt Two kinds of symmetry causing a symmetric shape of the free energy curve are presented in a Hamiltonian of a neutral reactant state for polar solution. They are the symmetry for inversion of solvent molecules and a reactant molecule. In addition, the deviation from the symmetry is formulated using a perturbation method. The antisymmetric part of the free energy curve is expressed by the symmetric part using a thermal average of the deviation in the Hamiltonian and the reaction coordinate.

1. Introduction A free energy curve [ 1,2] for polar solution is important in electron transfer reactions and solvation dynamics. The free energy gap dependence of an electron transfer rate can directly be obtained by the free energy curve in equilibrium theories [ 3-5 1. In nonequilibrium theories, many authors have calculated probability distribution functions on the free energy curve [ 1,2,6-8 1. In addition, some theories of solvation dynamics include dynamics along the free energy curve [ 9- 13 1. Particularly, symmetry of the free energy curve has long been discussed in the free energy gap dependence of an electron transfer rate. A symmetric shape of the free energy gap dependence predicted by Marcus [ 14,151 has been experimentally observed for some reactions [ 16- 18 1. However, a very asymmetric shape of the free energy gap dependence has been obtained for photoinduced charge separation reactions [ 19,201. This asymmetric shape arises from asymmetry of the free energy curve in neutral reactant systems, if the free energy curve directly reflects the free energy gap dependence [ 31. On the other hand, we have recently shown that the free energy gap

dependence is asymmetric when distances of acceptors and donors have a distribution [ 2 1,221. Even then, we should investigate symmetry of the free energy curve in neutral reactant states, since it has large effects on the free energy gap dependence. The symmetry of the free energy curve also has large effects on a change in number density of a solvent, such as electrostriction [ 231. If number density changes owing to the electric field from reactant charges, the symmetry of the free energy curve plays an important role in electrostriction. When the free energy curve has a symmetric shape, nonlinear effects alone induce electrostriction. The symmetry of the free energy curve directly affects the coupling between the number density and the electric field. While a shape of the free energy curve is important in these chemical phenomena in polar solution, many authors have considered only a symmetric shape. Some molecular dynamics simulations have given symmetrical shapes in free energy curves of polar solution [ 11,12,24-261. The expressions are only the free energy curves for solvents with electric polarization [ 27 ] and for the twisted intramolecular electron transfer [ 28 1. Hence, most theories have included a symmetric shape of the free energy curve in the neu-

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A. Yoshimori/Chemical PhysicsLetters225 (1994) 494-498

tral reactant state [6-8,10,14,15,21-231. A symmetric shape, however, is not obvious for the free energy curve of polar solution. Neutral reactant systems do not possess translational invariance because the reactant exists. In addition, some shapes of the reactant often break rotational invariance. The interaction between a solvent and a reactant does not possess translational or rotational invariance either, since the electric field from reactant charges varies spatially. A symmetric shape of the free energy curve requires symmetry of the Hamiltonian in the neutral system and antisymmetry of the interaction between a solvent and a reactant. The purpose of the present study is to investigate the symmetry of a Hamiltonian causing a symmetrical shape of the free energy curve and the deviation from the symmetry. Two hinds of the symmetry are presented in solvent and reactant molecules. Then, the deviation from the symmetry is formulated using a perturbation method.

short-range interaction between two solvent molecules i and i and between solvent molecule i and the reactant molecule. In addition, VP(ri- rj, s)i, Qj) and vf (ri, Qj, @) denote the long-range interaction given by Vp(ri-rj,

Qi,

=

=

Qj)

ss ss P(rc

The reaction considered in the present study is charge separation where a charge of the reactant changes from 0 to a value. Then, the initial Hamiltonian H’ and the final Hamiltonian HF are expressed by H’({ri, Qi})=Hss+

F vY(ri, Qi, Qr) 3

HF({ri, Qi})=Hss+

C C(ri, i

(1)

Qi, Q)

with

Qi})EH’({ri,

e--BG(x)=j3-1

ij

f2i, Qj)

(5)

fii))-HF({ri,

QI)

(3)

where ri denotes the position of solvent molecule i and Qi and Qr denote the orientation of solvent molecule i and the reactant molecule. For d)i and 9, the direction is the same as the axis fixed to molecules and the magnitude is the angle of rotation. Furthermore, U: (ri- rj, Qj, 4) and of (ri, Qi, Q) denote the

(6)

JGcf({ri, Oi})-X) (7)

where fi= (kJ)-’ and r is the normalized configuration space of solvents. For the Hamiltonians, symmetry in solvent molecules is introduced to obtain a symmetrical shape of the free energy curve. It is the symmetry for inversion of solvent molecules [ 3 1,

Uf(ri,

Y

(4)

xexP[-BH’((ri,~i})lcir,

VP(ri-rj,

Qi, 4)

Irei-reil

Arei, ri, Qi)Pc(rer, fir) dr dr Ir,i-r,,I ‘I er’

Arei, ri, -ai)

+ C vl”(ri-rj9

dr

CI CJ)

= - F vf(ri, Oi, Q) 3

2. Symmetry of Hamiltonian

C VP(ri-rj, ij

rj, Qj) dr

ri, Q)P(r.i,

Here, p( r,i, ri, Pi) and pC(r,, Q) denote the charge distribution of solvent molecule i and the reactant molecule, where rOiand r, denote the positions of the electrons. In the Hamiltonian, the reaction coordinate and the free energy curve are defined by [ 3,5 ] fl{ri,

H-Z

495

= -P(rb,

-Sri,

-s)i,

ri, Qi) 9

(8)

-5))=@(ri-rj,

Q)

=Vf(ri,

Oi,

b)i,

Qr)

.

Qj)

,

(9)

(10)

The above symmetry of the Hamiltonian gives a symmetric shape of the free energy curve: G ( -x) = G(x) . For the initial Hamiltonian and the reaction coordinate, H’({rj, -Qi])=H’({ri, AIri,

Thus,

-Qi))=-f({ri,

Sri}) 3 Qi)) .

(11)

(12)

496

A. Yoshimori / Chemical Physics Letters 225 (I 994) 494-498

e-BG(-x)=jl-l

s

These terms include the deviation in two-particle interaction of solvent molecules. On the other hand, the deviations from the symmetry of the reactant molecule are also expressed by

Scf({ri, Q})+x)

X exP[-BH1((ri9Qi))l~ =B-’ J X

Wt{ri,

-Qi))-Xl

GH~~‘((ri,~i})-H*({-rj,

-Qi})lw

exP[-Bff’((ri,

&f({ri,

=e-/Wx).

(13)

Symmetry in the reactant molecule is also introduced to obtain a symmetric shape of the free energy curve. The reactant molecule is assumed to have the symmetry for inversion as follows: Pc(&n -Q)

= -P&r,

Qr) 9

(14)

Ur(ri, Qj, -Qr) =Uf(ri, Qi, Qr) *

(15)

Qi})+f<{-ri3

-Qi}) -pi)>

,

.

(21) (22)

These include the deviation alone in one-particle interaction of solvent molecules, For the deviations, perturbation to the free energy curve can be estimated using the inhomogeneous free energy F( 2) defined by e--BFCA)z

I

eXp[ -jW({ri,

Qi})

+Q!ft{%Qi})W.

(23)

The inversion of the reactant molecule is equivalent to inversion of the whole solvent system for the fixed reactant molecule. In addition, this inversion of the solvent system does not change H”. As a result, Eqs. (14)and(l5)read

If the perturbed terms Sfand 6H are small, in the first order, we obtain

ff’({-ri,

where ( )1 denotes the average under the inhomogeneous Hamiltonian H’ ( { ri, Qi} ) - Af( { ri, Qi} ) . The antisymmetric part of the inhomogeneous free energy @(A) gives the antisymmetric part of the free energy curve gG( x) by the relation shown in the Ap pendix. Using Eq. (A.5 ) ,

-Qi})=ff’({~i,Qi>)~

_!I{-ri,

(16)

(17)

-Qi})=-f({ri9s)i})m

Thus, e-fl(-X)=j3-’

I

Gcf({ri, s),})+X)

X exP[-BH’({ri, =b-’ X

eXp[

=e

J Gcf({-ri, -/W((-ri,

lfl

WI

=F(l(x)) +n(x)x+n(

-Qi})]W

-&G(x)

(18)

3. Perturbationto symmetricHamiltonian

Qi})+f({G

-Qi})

(24)

-Bi}) .

,

(19) (20)

-F(A(

-x) -x))

-x)x.

In the first order of 6Adefined by J(x) +A( -x), obtain

= @0(x))

Though few real molecules possess the exact symmetry introduced in the previous section, the deviation is small for many systems of polar solution. In the initial Hamiltonian and the reaction coordinate, the deviations from the symmetry of solvent molecules are expressed by

W~_ft{&

= ( 6H-@i)A,

6G(x) = G(x) -G(

-Qi})-X)

6HEH’({ri, Qi})-ZP({ri,

W(n) =F(A) -F( -A)

>

(25)

we

(26)

where the dash indicates the differential and Eq. (A.8) is employed. From Eq. (A.6), in the zeroth order of M, n(x)=Gb(x)=i[G’(x)-G’(-x)].

(27)

Eqs. (26) and (27) show that the antisymmetric part of the free energy curve is expressed by the symmetric part G,,(x) within the small deviation from the symmetry.

A. Yoshimori /Chemical Physics Letters 225 (1994) 494-498

When the antisymmetric part of the inhomogeneous free energy W(n) can be expanded in terms of 2, we can calculate the free energy curve G(x) =G,(x)+ 16G(x). If terms higher than the second order of 1 are neglected in Eq. (24)) then W(n) =&a )

(28)

with a=
Qi}).))6H-&f)0.

(29)

For this expansion, the asymmetric part of the free energy curve is obtained by 6G(x) =aGb(x)

.

(30)

To show an explicit example of Eq. (30), the symmetric part of the free energy curve Go(x) is assumed to be G,(x)=C,x4+CZxz.

(31)

Then, from Eq. ( 30 ) , we obtain 6G(x) =4aC4x3+2aCzx.

(32)

4. Discussion In the present study, two kinds of symmetry were presented for the Hamiltonian giving a symmetric shape of the free energy curve. Few real systems of polar solution, however, possess the symmetry in solvent or reactant molecules. For example, in solvents, water molecules have an asymmetric charge distribution, though having spherical symmetry in the short-range interaction. Acetonitrile, which is a solvent employed in many experiments, is asymmetric in the charge distribution and the short-range interaction. Particularly, in the Edwards’ model of acetonitrile, the Lennard-Jones parameter e for methyl is about four times as large as that for nitrogen [ 29 1. Methanol molecules also have asymmetry, which is larger for the charge distribution than for the shortrange interaction. On the other hand, the reactant molecules possess the symmetry only in a few systems, such as an ion pair of the same size [ 11,25 ] or a spherical dipolar molecule [ 30,311. Many real reactants exhibit a more complex structure for the charge distribution and the short-range interaction [ 12,28 1. Though these molecules do not possess the

491

present symmetry, the deviation in the Hamiltonian from the symmetry is too small to observe large asymmetry of the free energy curve. The deviation from the symmetry was formulated using a perturbation method in the present study. The obtained perturbation has different dependence on solvent density for the symmetry of solvent molecules from that of the reactant molecule. In the first order of I, the perturbed terms (29) include terms of pz for the symmetry of solvent molecules, where pSis solvent density. On the other hand, they include terms ofp: for the symmetry of the reactant molecule. This shows that the perturbation is affected by correlation of more particles on the symmetry of solvent molecules than that of the reactant molecule. Even if a system of polar solution does not possess the present symmetry, the linear response approximation gives a parabolic shape of the free energy curve [ 14,151. Then, the deviation from the symmetry causes only a shift of the minimum in the free energy curve. On changes in number density such as electrostriction, however, the deviation from the symmetry has large effects because it gives the linear coupling between the number density and the electric field. On the other hand, when the linear response is not applicable, the present symmetry is important. Particularly, in the nonequilibrium solvation dynamics, where nonlinear effects are strong [ 11,26,29,32], asymmetric dynamics is expected if the system does not possess the present symmetry. The present study did not include quantum effects. Quantum effects induce an asymmetric shape of the free energy gap dependence in an electron transfer rate, even if the free energy curve has a symmetric shape. Since the quantum effects have been investigated only in the displaced harmonic potentials model [ 33 1, it should be studied for the nonlinear Hamiltonian in future.

Acknowledgement

The author would like to thank Professor T. Kakitani for his great help in studying the present problem, again.

498

A. Yoshimori /Chemical Physics Letters 225 (I 994) 494-498

Appdix

A

In the Appendix, the free energy curve (7) is associated with the inhomogeneous free energy defined by Eq. (23). To do this, GA(x) is introduced by

(A-1) =G(x)

64.2)

-Lx.

We can associate GA(x) with the inhomogeneous free energy F(1) as follows: fij

exp[ -BGA(x)]dx=e-p(‘).

(A-3)

The method of steepest descent permits the integral in the left-hand side of Eq. (A.3): F(I) rGJx*)

,

(A.4)

=G(x*)-LX*

(A.5)

with G’(x*) ~13..

(‘4.6)

In Eq. (AS), Eq. (A.2) was employed. By differentiating Eq. (AS) with respect to 1 so that Eq. (A.6) is satisfied, we find F’(L) = [ G’(x*) -11 d$ =-x

l

.

-x* ,

(A-7) (A-8)

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