Quantum-chemical evaluation of energy quantities governing electron transfer kinetics: applications to intramolecular processes

THEO CHEM ELSEVIER

Journal of Molecular

Structure (Theochem) 371 (1996) 191-203

Quantum-chemical evaluation of energy quantities governing electron transfer kinetics: applications to intramolecular processes M.V. Basilevskya’*,

G.E. Chudinovla,

I.V. Rostova, Yi-Ping Liub, M.D. Newtonb

“1.V. Rostov, Karpov Institute of Physical Chemistry, Vorontsovo Pole 10, 103064 Moscow, Russia hDepartment of Chemistry, Brookhaven National Laboratory, Box 5000, Upton, NY 11973-5000, USA Received 8 January

1996; revised 24 April 1996; accepted 23 May 1996

Abstract Important energy quantities governing electron transfer (ET) kinetics in polar solutions (reorganization energy, E,, and net free energy change, AU) are evaluated on the basis of quantum-chemical self-consistent reaction-field (SCRF) models. Either self-consistent field (SCF) or configuration interaction (CI) wavefunctions are used for the solute, which occupies a molecular cavity of realistic shape in a dielectric continuum. A classical SCRF model together with unrestricted Hartree-Fock SCF wavefunctions based on the semiempirical PM3 Hamiltonian is applied to the calculation of the solvent portion of E, (denoted E,) for two different series of radical ion ET systems: radical cations and anions of biphenylyl/naphthyl donor/acceptor (D/A) pairs linked by cyclohexane-based spacer groups and trans-staggered radical anions of the type (CH,),, m = 2-5. Results for E, based on two-configurational CI wavefunctions and an alternative reaction field (the so-called Born-Oppenheimer model, which recognizes the fast timescales of solvent electrons relative to those involved in ET) are also noted. Results for innersphere (i.e. intra-solute) reorganization, Ei, and for AU are also reported. The semiempirical E, results are quite similar to corresponding ab initio results and display the form of the two-sphere Marcus model for E, as a function of D/A separation. Nevertheless, in the one case where direct comparison is possible, the calculated E, result is more than twice the magnitude of the estimate based on experimental ET kinetic data. To reconcile this situation, a generalized SCRF model is proposed, which assigns different effective solute cavity sizes to the optical and inertial components of the solvent response, using ideas based on non-local solvation models. Keywords:

Electron transfer; Reorganization

energy

1. Introduction The mechanistic elucidation of electron transfer (ET) kinetics poses a major challenge in contemporary physical chemical research [l-6]. The activation energy, AU ’ for thermal ET in a polar medium * Corresponding ’ Deceased.

authors, M.V. Basilvsky

and M.D. Newton.

linearly coupled to the solute in the limit of weakly interacting donor (D) and acceptor (A) sites may be expressed as [2,3] AU* = (E, + AU)2/4E, where AU is given to the

0166-1280/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved PII SO166-1280(96)04735-5

(1.1)

E, is the so-called reorganization energy and the net free energy change. (In the earlier report in Ref. [7], AU and Q correspond, respectively, current AU ’ and AU.) The quantity E,, which

192

M.V. Basilevsky et aLlJournal of Molecular Structure (Theochem) 371 (1996) 191-203

for a linear model may be equated to half of the Stokes shift for the optical (vertical) ET process [8], is a manifestation of the change in equilibrium coordinates of the system (both the solute and the surrounding medium) on passing from the initial or precursor (P) state (where the transferring charge is localized at the D site) to the final or successor(S) state (where the transferring charge has moved to the A site) [l-4]. If we represent these states, respectively, as $’ and $‘, with corresponding equilibrium coordinates {Xkp}and {Xk}, then E, for a linear system is given as the following difference of free energies U{X,}: E, = UP({X,s]) - UP({X,P])

(1.2a)

mm) - mw3)

(1.2b)

=

tally equivalent [8]. In practice, such effects appear to be quite minor. An additional simplification of the linear coupling model is the partitioning of E, into solvent (Es) and solute or “inner-sphere” (Ei) contributions [2,3]: E,=E,

+Ei

(1.3)

The primary aim of the present study is to report on the quantum-chemical evaluation of E,, Ei, and AU

for a sequence of ET processes of major recent interest - namely, the intramolecular ET in radical anions or cations comprising a biphenylyl donor and naphthyl acceptor, covalently linked by cyclohexane-based spacer groups [9]. We also present E, results for a series of radical anions comprising terminal methylene (CH,) D and A groups linked by a trans-staggered alkane bridge containing two, four, six or eight CH2 units (see also Refs. [7] and [S]). Further details concerning these two series of ET systems, denoted, respectively, as l(n) and 2(n), are presented in Table 1, where the index n is defined. Evaluation of Ei is relatively straightforward, using standard gradient techniques to evaluate those subsets

where U’({Xi}) is the free energy associated with $‘, subject to the constraint X, = Xl, etc. While the equality between Eqs. (1.2a) and (1.2b) holds strictly for fixed $’ and @, slight departures from equivalence may arise if 4’ and $’ are allowed to vary with {X,} in cases where the D and A groups are not symmetriTable 1 Electron transfer systems System”

Db

A’

(A) Cycloherane-based l(n); II - 2-5, 8

bridge systems 4-Biphenylyl

2-Naphthyl

l(2) l(3) l(4) W) l(g) (B) Trans-staggered (CHd a+3radical anions 2(n); n = 1, 3, 5, 7 CHr-

CH2-

Bd

bA

(&’

Active space

*

Radical anions: le/homo(D),lumo(A) Radical cations: 3e/homo(D),lumo(A) ck-1,3-Cyclohexane trans-1,4Cyclohexane 2,7-Decalin 2,6-Decalin 3,16-Androstane

1.00 11.8 12.5 14.0 17.4

-(CHz)n+r-

2.49 + 1.24n

3e/homo(D),lumo(A)

’ The notation l(n) and 2(n) denotes the number (n) of CC bonds in the shortest bonded path between the indicated points of attachment of D and A to the bridge (B); see footnotes b and c. b Local donor group, site of transferring charge in precursor (P) state. ’ Local acceptor group, site of transferred charge in successor (S) state. d The acyclic, cyclic or polycyclic bridge group, attached to D and A at the indicated positions. For the l(n) systems, D and A are attached in the equatorial conformation. ’ The distance between the midpoints of the D and A groups: for l(n) see Ref. [9]; for 2(n), r nA is the separation distance of the terminal C atoms. f As indicated, the radical anion and cation systems involve either one or three electrons distributed in two active MOs, either the highest-lying occupied (homo) or lowest-lying unoccupied (lumo) pair. The D,A labels denote spatially localized active-space MOs (for the (CH2)2.+s series; equivalent results are obtained using the symmetrically-delocalized counterparts of the D,A MO pairs [7]).

M.V. Bade&y

193

et aLlJournal ofMolecular Structure (Theochem) 371 (1996) 191-203

of {Xi} and {XE} which pertain to the discrete modes of the solute. The evaluation of the solvent-dependent quantities, E, and AU, is conveniently implemented by a self-consistent reaction-field (SCRF) procedure, which combines a quantum-chemical treatment of the solute self-consistently with a dielectric continuum treatment of the surrounding polar medium [7,8,10141. Recently-developed numerical techniques allow the solute to be placed in a cavity of realistic molecular shape [7,8,10-141. The evaluation of AU involves a conventional equilibrium SCRF calculation, whereas E, is a non-equilibrium free energy (Eqs. (1.2a) and (1.2b)); i.e. in an activated or reorganized state the {X,} corresponding to the inertial polarization modes of the medium are not in equilibrium with the solute charge distribution. Nevertheless, it has been shown [8] that non-equilibrium quantities like E, may be expressed as a sum of suitably-defined equilibrium free energies, and thus their calculation is relatively straightforward using standard SCRF computer codes. To maintain continuity with previous usage, we employ the designations “inertial” and “inertialess” to distinguish, respectively, the response involving the “light” (electrons) and “heavy” (nuclear) particles of the system; these labels are not to be taken literally, since in fact much of the latter response may be diffusive in character. We also note that the “inertialess” component of the solvent polarization is given by the constitutive relation P, = (Ed - 1)6/47r, in contrast to the more conventional “electronic” _polariz_tion given by ?,t =(ea - l)E/47~, where D and E are, respectively, the displacement field and the electric field [4,7]. In examining either of the two series, l(n) and 2(n), it is of great interest to assess the dependence of E, on the basic structural and dielectric properties of the given systems. An especially important reference point in this regard is provided by the Marcus twosphere model [2,3,8]:

radii of the spherical donor and acceptor site cavities, and the separation between these sites. In Section 4, below, we will compare E, values calculated for the actual systems of interest with those based on effective two-sphere models (Eqs. (1.4) and (1.5)), and also with estimates of E, (as well as Ei and AU) based on experimental kinetic data [9]. Systematic deviations of calculated and experimental estimates for E, will lead us to propose an extension of the SCRF model suitable for calculations in which both equilibrium and non-equilibrium solvation free energies are required.

E, = C(Aq)2(1/2r,

This can be written more explicitly

+ 1/2r,

- l/r&

(1.4)

2. Formulation of the ET phenomenon in terms of

SCRF schemes 2.1. The equilibrium

SCRF model

A quantum-chemical SCRF calculation generally consists of two steps. The quantum-chemical step solves a Schrodinger equation for the solute wavefunction: (Ho - +)$ = wlc/

(2.1)

where Ho is the solute gas-phase Hamiltonian and $ is the solvent reaction-field potential, denoted below as simply the “reaction field” (the minus sign preceding 4 reflects the fact that 6 in the electronic Schrodinger Hamiltonian has been multiplied by the electronic charge, e = - 1 in atomic units). The solute charge density p is then constructed in terms of 4. Thereby, we can write p as a functional of f#~: (2.2)

P = P[41

The second (electrostatic) tion of 4 as a functional response model:

step involves the computaof p in terms of a linear

d = dJ[Pl

(2.3) as [4]:

A

where

$(r) = [Kp](r)

=

d3r’K(r, r’)p(r’) s

C=(l/.%-l/&a)

(1.5)

is the Pekar factor, involving the optical (Ed) and static (Ed) dielectric constants, and where Aq, To, rA, and IDA are, respectively, the amount of charge transferred (one electron in the present cases), the effective

(2.4)

Here ~,r’ represent spatial point vectors, whereas l? is a linear integral operator (Green’s operator) with kernel K(r,r’) (Green’s function). The form of operator I? depends on the specific procedure adopted to perform the electrostatic calculation [7].

194

M. V. Basilevsky et al.lJournal of Molecular Structure (Theochem) 371 (1996) 191-203

A commonly applied methodology invokes a classical uniform medium model with a cavity, henceforth called the “Born-Kirkwood-Onsager (BKO) model” [10,14]. According to this approach, the solute charge p is confined within a cavity with dielectric constant E = 1 surrounded by a structureless solvent characterized by a single parameter, the static dielectric constant E = eo. Recent computer codes allow efficient solvation of the corresponding Poisson equation for cavities with arbitrary shape [lo-131. The reaction field $ is then expressed in terms of a surface charge distribution (Tgenerated on the boundary of the cavity: 4(r) = / d2rr

&

2.2. Adaptation

(2.5)

to ET

In the context of the ET theory, the above treatment must be extended by introducing a pair of linear operators: R=R(e,) k, =R(e,)

(2.6)

The notation of Eq. (2.6) implies that the corresponding BKO procedure is performed twice using, respectively, the values of external dielectric constant E indicated in parentheses. By this means, the total (4) and intertialess (&) reaction fields are obtained, the latter being associated with the electronic (inertialess) polarization: & =R,p

(2.7)

The definition of the inertial reaction field, @in, associated with the orientational (inertial) medium polarization then reads: +in = d - +rn = (i -fico)Pin

(2.8)

For full electrostatic equilibrium, Pin is the same as the actual charge density p. In more general (non-equilibrium) situations (see below), Pin f p [4,7,14]. These simple expressions follow from the linear response treatment. We emphasize that although the numerical tests reported below correspond to the above BKO treatment, the formulation in terms of Eqs. (2.1)-(2.8) is much more general. It can be extended to a more

sophisticated model allowing for the internal structure of the medium. All necessary changes will be included in a refined formulation of the Green’s operators I?, IL. In terms of the general SCRF procedure given by Eqs. (2.1)-(2.5) it is expected that inserting Eq. (2.3) into Eq. (2.2) or vice versa must give an identity. In order to obtain this desired ultimate result an iterative procedure is adopted, starting from some initial approximation for p in Eq. (2.3), until self-consistency between Eqs. (2.2) and (2.3) is achieved. The most important consequence of this iterative scheme is that the quantum-chemical calculation (Eqs. (2.1) and (2.2)) and the electrostatic calculation (Eqs. (2.3) and (2.4)) become conjugated in a non-linear fashion. This is of fundamental importance to the ET problem. A non-linear algorithm can generate multi-valued solutions and this is what we expect to find. More specifically, we expect the calculation to yield at least two solutions, namely p’(r), 4’(r) and p’(r), 4’(r). They represent different solute charge densities equilibrated with different solvent fields under a fixed geometry of the solute system, corresponding to the P and S states defined in Section 1. This effect of multivalued solutions has been firmly established in several recent calculations [7,8,15]. In the following, we discuss some of its kinetic consequences by considering transitions between the two ET states in terms of Marcus theory [l-3]. Related instability in quantum-chemical SCF calculations carried out in vacuum has been reported in earlier work and is referred to as a “sudden polarization” [16-181. In these cases, the non-linearity generating multi-valued solutions results from mutual polarization effects within the solute electronic manifold. More recently, this type of behaviour has been observed specifically for ET systems [5,19], once again in calculations which did not explicitly incorporate solvation effects. For a physical discussion of ET in terms of a medium reorganization we have to return to Eqs. (2.1)-(2.8).

3. Free energy functionah coordinates

and medium

We have already introduced (Eqs. (l.l)-(1.3)) the free energy quantities which arise in the context of

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M.V. Basilevsky et al./Journal of Molecular Structure (Theochem) 371 (1996) 191-203

Marcus ET theory [l-3]. We now proceed to formulate these free energies in terms of SCRF models. Generally, in order to treat non-equilibrium solvation effects we employ the inertial field $in (Eq. (2.8)) as an independent functional variable and define the free energy functional (FEF) CJ[+i”] such that AU’ = Up[4z]-

uP[+in(r)]

(3.1)

where +i,’ and $in(T) correspond, respectively, to the saddle point and precursor minimum on this FEF (the FEFs U[$in] are equivalent to the quantities Cr({X,}) introduced in Section 1). Several techniques exist for non-equilibrium SCRF calculations of U[+in], depending on the type of solute wavefunction and the approximation used in the treatment of the inertialess field & (Eq. (2.7)). It is not at all necessary for the FEFs UP($i”) and us(+in) to have the twoparaboloid form of the Marcus diabatic model, which underlies the limiting case of Eq. (3.1) given by Eq. (1.1). The solute wavefunctions may be found either by Hartree-Fock or configuration interaction (CI) quantum-chemical procedures. The inertialess field may be treated either classically (the so-called iterative selfconsistent (SC) method) or quantum-mechanically (the Born-Oppenheimer (BO) method) [6,7]. A discussion of these approximate procedures and their interrelation may be found elsewhere [6,7,20,21]. 3.1. The SC method: classical treatment of inertness polarization When approximate wavefunctions are employed (e.g. the unrestricted Hartree-Fock (UHF) model used in the applications of Section 4) the eigenfunction equation, Eq. (2.1) may be replaced by the expression for the expectation value W (see Eq. (2.1)): w=

< $IH,-95l$>

(3.2)

The FEF, u[$i”], has the form [21]: u[tii,l

=Si, + Sm + w[4,i,l

(3.3)

where Sin and S, are self-energies corresponding, respectively, to inertial and inertialess fields:

S, = - i

d3r p(r)&(r)

(3.4)

and where the expectation value W may be expressed as a functional of the inertial part of the reaction-field potential, since the total energy is always minimized with respect to the inertialess part [7,14]. The inertial self-charge pin is equilibrated to a given din according to Eq. (2.8). However, p represents the solute charge distribution associated with the wavefunction $ in Eq. (3.2) and dZ is related to p according to Eq. (2.7). We recall that p # Pin for a non-equilibrium situation. Altogether, given the independent variable 4in, generally non-equilibrium, 4% is calculated and then the total field (obtained by rearranging the first equality of Eq. (2.8)): d = +in + +,r

(3.5)

inserted into the solute Hamiltonian (see Eq. (3.2)). This treatment gives rise to a generalized SCRF algorithm which mutually adjusts the fields p and $p in an iterative non-linear procedure (this procedure was implemented at the CI level in [7] and will be applied below (Section 4) using UHF wavefunctions). For an equilibrium case it reduces to the standard SCRF scheme. We now return to the pair of equilibrium fields pP(r), r&(r) and p’(r), &.,(r) pertaining to the generic ET system described above. We can define

(Pin =+f;,=4: +h[4~- +LII Owing to the linearity, Pin ‘pk ‘pp + h(ps -pp)

05X51

we deduce from Eq. (3.6) that OS-X51

(3.7)

Proceeding with this notation, we define for a given X a pair of functions p(“) and +$‘) which provide solutions to the above-mentioned generalized SCRF procedure. The corresponding wavefunctions and energies are $‘(“), W’(“) and J/““‘, W”“‘. The quantity h may be considered as a medium coordinate [l-4]. It generates the free energy profiles uPCX)= U’[&]

(3.8)

and us@’ I US[$&]

Sin = - i 1 d3r pi”(r)&(r)

(3.6)

for which equilibrium

(3.9) is given, respectively,

by Up(‘)

M. V. Basilevsky et aLlJournal of Molecular Structure (Theochem) 371 (I 996) 191-203

196

and Us(‘). Up(‘) and Us(‘) are illustrated in Fig. 1 for the radical anion states of species l(2) (see Table 1). The details of the calculations are given below. The medium reorganization energy (E,) is then evaluated as (see Eqs. (1.2a) and (1.2b)) E;=U P(l) _ uw3

(3.10)

E;=U S(O)_ @(l)

(3.11)

For the case reported in Fig. 1, Eqs. (3.10) and (3.11) yield quite similar results. These results are essentially purely electrostatic in nature since the gas-phase contributions (i.e. those involving Ho (see Eq. (2.1)) are found to be quite small in magnitude. In the general case, where EF may differ appreciably from Ef, the Stokes shift is given by E! + E”, and half this quantity serves as a useful estimate of the mean solvent reorganization energy.

0.8

0.6

3.2. The U-B0 method: quantum-mechanical treatment of inertialess polarization We now turn to the BO reaction-field model [6,7], which is appropriate when the timescale of the transferring solute electron is slow (as in many ET processes) relative to that of the medium electrons. We develop a generic two-state CI model based on two electronic configurations (single determinants), Di a common electronic and D2, which comprise “core”, and two molecular orbitals (MOs), x1 and x2, of variable occupancy (the “active space”, defined in more detail in Table 1). These MOs are linear combinations of all the basis orbitals of the ET system, but are dominated by contributions from the local (weakly-coupled) D and A orbitals. The results reported below (including the specific applications in Section 4) are invariant under a unitary transformation of the two active-space MOs [7], so that MOs may be employed which are either localized or delocalized with respect to the D and A sites. For symmetry-inequivalent D/A sites, a localized pair of MOs arises most naturally, while for symmetryequivalent D/A sites there is often the choice of a symmetry-equivalent pair of MOs (D and A type) or a symmetric and antisymmetric pair of delocalized MOs [5,19]. The Cl Hamiltonian matrix has the form H=

5: Q‘ 2

(

HII

ff12

H21

H22

)

where

0.4

Hii =hii +

ff22

=A22

J

Pii&nd3r

P22hnd3r

+ s

H12

-0.2 0.0

0.2

0.4

0.6

0.8

1.0

h Fig. 1. FEF profiles with respect to the X coordinate (Eqs. (3.8) and (3.9)) for the radical anion of species l(2) (see Table l), calculated with the classical UHF SCRF model (Section 3.1). The results are based on the geometry of the neutral parent and include no “innershell” reorganization (Ei, Eq. (1.3)).

=h2

+

P124ind3r

(3.12)

Here, hnb (a,b = 1 or 2) represent gas-phase matrix elements (Ho, Eq. (2.1)), modified by an equilibrium solvation due to the inertialess solvent polarization. According to the BO scheme [6,7], the inertialess contribution is different in each CI matrix element, so that the screening potential created by solvent electrons forms a 2 x 2 matrix. The solute charge

197

M.V Basilevsky et al.iJournal of Molecular Structure (Theochem) 371 (1996) 191-203

density matrix p consists of electronic (pJ and nuclear (p,) components: Pab = (&&b

- h

The electronic

(3.13)

jab

component

is computed

coordinates Xl =

as X2=

(dab

a,b=l,2

= (Da 16, IDb)

(3.14)

where j, is the electronic density operator. The negative sign is consistent with the negative sign of the field contribution ( - 4) in the Schr6dinger equation (Eq. (2.1)). A preliminary, purely electrostatic calculation prepares the matrices of the inertialess (($m)&) and inertial ((4j,),b) reaction fields equilibrated to the charge distribution pa& a,b=l,2

(4&b

=k&b

(&n)ab

=(R-R=)Pab

(3.15)

The inertialess screening potential matrix, a component of hab in Eq. (3.12), is then calculated as [7] VQb=; c &

J

(3.16)

d3&&‘&b

The limited summation in Eq. (3.16) represents a particular implementation of an approximation to the BO matrix elements based on matrix multiplication [7]. The matrix Vob was denoted Uob in Ref. [7]. We now introduce the notation Yab =

.r

(3.17)

Pnb4ind3r

and consider the quantities Y& as three medium coordinates [7,14]. Their number can be reduced by slightly changing the CI matrix as

fi=

( :,Z

&ZL1,)

a=h

22 -hll

+

P=&+

.I

s

(~22

= (;+Xz

-Pll)(4dlld3r

J

1:;)

(3.18)

Pn(4dlld3r

where the notation (+in)q was introduced in Eq. (3.15). This matrix yields the same CI expansion as the original Hamiltonian (Eq. (3.12)). The two new

J

X1, X2 are defined as (3.19)

d3r(P*2-P11)(4in-(4in)11)

d3rh)(4i”

-4in)ll)

The states of this 2 x 2 CI-BO energies [7]:

system

have the

(3.20) Here, W1, W2 are eigenvalues of the matrix fi (Eq. (3.18)) and the elements of the reorganization matrix T are defined in Eq. (3.21). The ground state energy surface U, represents the FEF for an ET reaction as a function of two collective medium coordinates X1, XZ. The first term in Eq. (3.20) is the inertial field self-energy (common to both states), expressed in terms of the inverse of the symmetric T matrix: T,,=

1

T12 = -

s

T,, = s

d3r(P22-P11)((4in)22-4in)11)

d3G22

-Pll)(4in)l2

(3.21)

d3rpd4in)n

(note that a minus sign should be included on the right-hand-side of Eq. (5.9) in Ref. [7]; conversely, the minus signs appearing in footnote b of Table 1 in Ref. [7] should be omitted, and T’ 11,ooreplaced by T ll,ll, * matrix T’ in Ref. [7] is the same as the present matrix T). For cases of ET between weakly-coupled D and A sites, the function U,(X,,X,) is a two-dimensional double-well potential. When a straight path in the two-component inertial space (X1,X,) exists such that the FEF profile along this line contains both of the minima and the saddle-point, then the barrier formula given by Eq. (1 .l) will be a good approximation, at least in cases of relatively weak D/A coupling. The essential coordinate is either X1 (if the original MOs are well localized) or XZ (if they are delocalized). The latter case is illustrated in Fig. 2 (details of the calculation are given in Section 4).

198

h4.V. Bade&y

et aLlJournal of Molecular Structure (Theochem) 371 (1996) 191-203

When (p’Oc)r2is small, as in weak D/A coupling, the ET problem becomes one-dimensional. It can be proved that in the limit of vanishing (p10c)r2,the potential Ur(Xr,X~) (Eq. (3)) generates an exact one-dimensional profile, for which the Marcus reorganization energy, E,, equals

2.5

Es=

2.0 z 2

ITll 2 i 2T22

(localized

x1, x2, asymmetrical

(symmetrically

delocalized

solute)

x1, x2) (3.22)

1.5

1.0

0.5

0.0 -1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X2 WI Fig. 2. FEF profiles (Eq. (3.20)) for the radical anion of the species 2(l) (see Table l), calculated at the two-configuration BO level (Section 3.2), using delocalized active space MOs ,yl and ~2 obtained from an SCF calculation for the dianion in vacuum. The coordinate X2 is defined in Eq. (3.19).

To gain further perspective on the two-coordinate space, let us introduce the exchange charge density to some optimally-defined (P’Y 12, corresponding localized orbital description, in contrast with the “arbitrary” density matrix Pab (i.e. arbitrary under a 2 x 2 unitary transformation within the D and A orbital space spanned by x1 and ~2). In the limiting cases where Pob approaches a fully localized or fully delocalized representation of the active (p1”)t2 = p12 and space, we have, respectively, Since it is not affected by the (Ph2 = (p22 -ptr)/2. choice of orbital [7], the CI-BO scheme does not explicitly require the specification of (p’Oc)t2, a formal advantage which avoids the problem of defining optimal localization, a particular drawback in cases of symmetry-inequivalent D and A sites. This flexibility is gained at the expense’ of introducing two coordinates.

The first case corresponds to a conventional picture of ET, with coordinate X1 being an obvious generalization of a commonly accepted collective medium mode [22-281, with allowance made for boundary effects on the cavity surface [14]. Generally, if (p10f)r2 # 0, the 2 x 2 CI-ET problem becomes essentially two-dimensional. The true reaction coordinate is a combination of Xi and X2. Moreover, the saddle point may occur in the X1 and XZ plane off the straight line connecting the precursor and successor minima. In the latter case, the conventional barrier formula (Eq. (1.1)) fails to be valid. Finally, a multi-configurational CI wavefunction generating even greater numbers of coordinates may be needed to describe ET in cases where precursor and successor charge distributions overlap significantly. All these possibilities were numerically tested in the present work and were found to be of little importance for the systems under consideration. This allows us to continue our discussion in terms of a simple onedimensional description of an ET process.

4. Results and discussion Calculated values of E, were obtained for the series l(n) and 2(n), implementing the procedures developed in Section 3, together with the PM3 semiempirical electronic structural method [29,30] adapted [12,15] so as to include an SCRF. The E, values are presented in Tables 2 and 3, supplemented for purposes of comparision with available results based on ab initio calculations and experiment. In the case of l(n), estimates of AU are also provided (Table 3). Finally, calculated values of Ei (Eq. (1.3)) are presented on Table 4.

M. V. Bade&y

Table 2 Calculated

et al./Journal of Molecular Structure (Theochem) 371 (I 996) 191-203

Table 3 Calculated

E, values (eV) for 2(r1)~ in waterb

199

E, (ev) and AU (eV) values for l(n) in 1,2-dichloro-

ethane a ?I

Present semiempirical

Ab initio resultsd resultsC

1 3 5 7

1.28 2.10 2.53 2.76

1.45 2.18 2.60 2.83

A& (e) ~D,A(A)

1.04 2.4

1.08 2.6

a See Table 1. Standard geometries were used [8]. bE, is the solvent contribution to E, (Eq. (1.3)). The aqueous dielectric continuum was represented by ea = 80.0 and em = 1.8 [8]. ’ Calculated at the UHF level (Eq. (3.10) or (3.11)) using the PM3 semiempirical Hamiltonian [29], as implemented in the MOPAC code [30], and modified as described in Refs. [12] and [15]. The solute cavity was defined in terms of the effective atomic radii determined in Ref. [12]. d ROHF results obtained with a 6-31G” orbital basis, as described in Ref. [8], where the relevant E, is defined in Eq. (10). ’ The effective amount of charge transferred (Ages) and effective radius of the D and A sites (foiA), obtained in terms of Eq. (1.4) by linear least-squares fits of calculated E, values vs. l/rnA, using the roA values given in Table 1 (linear regression coefficient r 2 0.999).

4.1. E, results for 2(n) The results in Table 2 for aqueous solvent, based on a SCF model with a classical SCRF (Section 3.1), show good agreement between the semiempirical and ab initio [8] approaches. Also, in spite of the chemical complexity of the organic species, the variation of E, with l/rDA (for n = 1, 3, 5, 7) follows essentially the exact form of the simple two-sphere Marcus model (Eq. (l)), as inferred from leastsquares fitting, which yielded reasonable values of the effective transiered charge ( = le) and local D/A radii ( = 2SA, close to the expected effective radius for a terminal CH2 group). This result, also found for the l(n) series (Section 4.2), indicates that the relatively small volume occupied by the lowdielectric bridge has little influence on the overall magnitude of E,. Fig. 2 displays the FEFs obtained for the low-lying adiabatic states of 2(l), using CI with a SCRF based on the BO approach (Section 3.2). The quantity E, may be taken as the vertical energy gap at either of the local minima (the somewhat larger value compared with Table 2 reflects the modest difference

n

Radical cations

2

1.02

Radical anions 1.07

[ 1.00 (ab initio)] b 3

1.09

1.16

[ 1.Ol (ab initio)]’ 4 5 8

1.14 1.18 1.27

4%

1.04 3.7 - 0.23 2 0.02’ [ - 0.27 (ab initio)]’ [ - 0.04 (exp)lg

FD/A 6)"

AU

1.20 1.25 1.34 [0.62 (exp)]’ 1.07 3.7 - 0.16 ? 0.02’ [ - 0.16 (ab initio)]’ [ - 0.05 (exp)lp

a Except as noted otherwise, the listed data are obtained at the semiempirical PM3 UHF level (see footnote c of Table 2). The solvent was represented at the continuum level by ea = 10.37 and E, = 2.078 [42]. The l(n) geometries are those optimized in vacuum at the PM3 level for the neutral parent systems. The calculated E, values are based on the Us FEF (see Eq. 3.11). b Ab initio results using a minimal STO-3G-basis [43] and the procedure described in Ref. [8], neglecting the small promotion energy term in Eq. (10) of Ref. [8]. ’ [31,32]. d See footnote e, Table 2 (here we find r 2 0.995). ’ A constrained value of ALI obtained at the PM3 UHF level, with solute geometry optimized for the neutral parent system in vacuum (see also footnote a). f Estimates based on the isolated D and A species: Bph ’ + N --t Bph + N ’ The geometries were separately optimized for the reduced and oxidized states of each species. The corresponding estimates of AU based on reduction potentials measured in acetonitrile are - 0.17 eV (radical cations) and - 0.07 eV (radical anions) ]441. g 191.

between results based on UHF wavefunctions and CI wavefunctions based on fixed MOs obtained from vacuum calculations). An estimate of the effective D/A electronic coupling element ( = 0.25 eV) is obtained as half the vertical gap at the transition state (YQ = 0.0, where a delocalized MO set is employed). 4.2. E, and AU results for l(n) The solvent-dependent quantities E, and AU are presented in Table 3 and Fig. 1. The results were obtained at the SCF level (Section 3.1), with the solvent 1,2-dichloroethane (chosen for comparision

200 Table 4 Calculated

M. V. Basilevsky et aLlJournal of Molecular Structure (Theochem) 371 (1996) 191-203

Ei values for

l(n) systems” Ei @‘)

(A) Semiempirical resultsb for radical cations of l(4) (B) Ab initio resultsC based on separate biphenyl (Bph) and naphthalene

ET kinetics for the radical anion of the

Radical anion

0.27

_

0.37 (0.15) 0.23 0.60

0.52 (0.20) 0.27 0.79

0.63 (0.13 2 .03)

0.58 (0.13 -c .03)

(N) molecules

Bph Total d (Torsion only)’ N” TotaId (Bph + N) (C) Estimate from experimental Total (Torsion only)g

Radical cation

l(n) systems’

a The solute contribution to E, (see Eqs. 1.2 and 1.3). h Based on PM3 SCF results for the equilibrium geometriesin vacuum, using RHF and UHF for the neutral and ionic states (see footnote b, Table 2). Uncertainty in Ei due to anharmonicity was < 0.05 eV. ’ Calculated with a 6-31G” orbital basis using the PS-GVB and Delphi computer codes [45,46]. Neutral and ionic states were treated, respectively, at the RHF and ROHF levels. The calculated Ei values were essentially the same for calculations with and without solvent (using the vacuum-based equilibrium in both cases). d Uncertainty in E, due to anharmonicity was 5 0.06 eV. ’ Obtained by varying only the torsional angle of Bph, keeping other coordinates fixed (the equilibrium conformation is almost planar for the ionic states and about 45” for the neutral state; see also Ref. [32]). Owing to coupling between the torsional mode and the other coordinates, the E, contribution is rather sensitive to the precise choice of these fixed values. For cations and anions, the resulting uncertainties in the torsional E, contributions are, respectively, about 0.05 and 0.10 eV. ‘See Refs. [9], [31] and [32]. g Based on a combination of experimental and computational results [31,32].

with the results given in Ref. [9]) represented by a classical SCRF (comparable results were obtained at the BO level). The listed values are those defined in terms of the Us FEF (Eq. (3)). Owing to the lack of symmetry-equivalence between the D and A sites, there is no requirement that Up yield the same value of E,. As revealed by Fig. 1 for the radical anion of l(2), the value based on the precursor (P) state is approximately 10% smaller. As for 2(n), we find the variation of E, with l/rDA to conform very well to an effective two-sphere model (Eq. (1.4)). In the few cases where comparision is possible, the semiempirical and ab initio results are quite similar for both E, and AU. The AU values are somewhat more exothermic than the experimental estimates [9]. The primary conflict with experiment involves the E, values, where the calculated result for the radical anion of l(8) exceeds the experimental estimate by more than a factor of 2. A similar discrepancy is found for the lower members of the Series if it is assumed that the experimental values vary in conformity with

the two-sphere model. If Aq is set to unity, the experimental value of 0.62 eV in conjunction with Eq. (1.4) yields an effe$tive mean radius for the D and A groups (TniA) of 5.9 A, a 50% increase in the value (3.9 A) inferred [9] from structural data. A possible extension of the reaction-field model taking into account this effect (which appears to be a common feature of the semiempirical and ab initio results) is discussed below in Section 5. An alternative to empirically adjusting ?n/A has been proposed [31,32], which involves maintaining the “structural” value of pn,* (3.9 A) and adjusting the prefactor C(Aq)’ (Eq. (1.4)). Accommodating the experimental E, values leads to a 40% reduction of the prefactor [31,32]. 4.3. Ei results for l(n) Table 4 summarizes Ei estimates obtained from calculations and experimental data. The semiempirical (PM3) estimate, seen to be about 50% of the other estimates, is a UHF result based on the l(2) radical

M.V. Bade&y

et alSJourna1 of Molecular Structure (Theochem) 371 (1996) 191-203

cation states. The ab initio SCF results are based on the charged and neutral states of the separate biphenyl (Bph) and naphthalene (N) molecules. The experimental estimate is based on a semi-classical analysis of the kinetic data [9]. The values of Ei are similar for radical cation and anion systems, and depend only weakly on n and the solvent. The torsional mode of Bph is found to make a sizable contribution to the total E, magnitude, an effect arising from the difference in conformation for the neutral (a dihedral angle of about 45”) and ionic (almost planar) states. Finally, we note a recent ab initio study of Ei for the N/IV pair [33], which suggests that the inclusion of electron correlation may appreciably reduce the magnitude of Ei relative to the SCF value.

terms of the static dielectric constant, E = co. For spherical cavities, such a solvent model was introduced earlier [37,38] in a computation of equilibrium solvation effects, and we are presently implementing such an approach in the framework of a SCRF quantum-chemical procedure by adapting the semiempirical computer codes employed in the current study. The extended treatment will first be calibrated for equilibrium free energies and then applied to the calculation of E,. The essence of the approach may be illustrated for the case of equilibrium solvation involving a single spherical cavity: u solv=

-[(1-1/&r)/~r+(1/E~-1/EO)lri,142/2 rr 5 rin

5. Generalized

SCRI? model

The results for E, collected in Table 3, including the semiempirical and ab initio SCF results and also those obtained from the simple Marcus model (Eq. (1.4)), suggest that dielectric continuum models based on cavity sizes fitted to accommodate equilibrium solvation free energies may be grossly inadequate for estimating non-equilibrium free energy quantities associated with the inertial or low-frequency part of the solvent response. Specifically, in the case of ET processes modelled using dielectric cavities defined in terms of effective radii of the constituent solute atoms, we suggest that the constraint of using the same fixed atomic radii must be relaxed for different types of polarization response. We thus propose an explicit distinction between spatial properties of inertialess (electronic) and inertial (nuclear, mainly orientational) components of the medium polarization, introduced by allowing for different boundaries of the corresponding cavities associated with two types of response. Support for this idea is found in studies of frequency-dependent solvation based on the mean spherical approximation [34,35] and non-local dielectric continuum theory [36]. We consider a solute surrounded by a medium with a dielectric constant E that changes in stepwise fashion, so that around the cavity with E = 1 there exists an intermediate layer with e = em, approximately spanning the first solvation shell of the solute. The part of the medium more remote from the solute is treated in

20 I

(5.1 I

where rz and Tin are associated, respectively, with E, and aa (i.e. beyond the larger radius, rin, the inertial as well as the inertialess response is operative). In the usual Born model, rr = Tin. According to the non-local theory given in Ref. [36], based on an assumed model for the correlation of polarization fluctuations, rx and Tin are related by the expression I,/rin

= 1 - (1 - exp( -X))/X

(5.2)

where x=2r,/h,

(5.3)

and X, is the correlation length associated with the solvent inertial response, expected to be of the order of size of a solvent molecule. It is straightforward to extend the above results for equilibrium solvation (Eqs. (5.1)-(5.3)) to obtain the counterpart of the two-sphere BKO expression for E, (Eq. (1.4)) in terms of the refined model based on distinct cavity sizes for low- and high-frequency polarization response. The result proves to have the same form as Eq. (1.4) with rn and rA being given values (ri,,) pertaining to the cavities associated with the low-frequency (inertial) polarization response. Given the results reported in Section 4.2 (where an increase of about 50% in the effective mean radius ?n,A was required to accommodate the experimental estimate of E, for l(8) in 1,2-dichloroethane in terms of the two-sphere model (Eq. (1.4))) the present model implies Tin/r=-‘I 2, or X = 1.3r,, a physically plausible result. The model described by Eqs. (5.1)-(5.3) may be

202

M.V. Bade&y

et al./Journal of Molecular Structure (Theochem) 371 (1996) 191-203

viewed as a special limiting case of a more general non-local solvation theory, in which each of several frequency ranges of the polarization response is associated with a characteristic correlation length. Such as a model was originally introduced specifically in terms of an exponential form for the decay of the spatial correlation functions governing the solvent fluctuations [36]. The approach may, however, equally well be extended to include oscillating spatial correlations [39], consistent with the results of recent microscopic studies of polar fluids [40]. Future efforts to determine distinct values for effective optical and inertial radii will exploit experimental estimates of E, for other ET processes, such as those occurring in norbornyl-based bridge systems [41], for which calculations of the type reported here also yield E, values exaggerated by a factor of approximately 2 in comparison with the experimentally-based estimates (unpublished work).

Acknowledgements This research was carried out at Brookhaven National Laboratory under contract DE-ACOZ 76CHOOO16with the US Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences. M.V.B. and I.V.R. thank the Russian Foundation of Fundamental Research and the International Science Foundation for financial support. Most of the calculations reported here were carried out during a visit by G.E.C. to BNL in 1993. One of us (M.D.N.) benefited from discussions with Dr. John R. Miller.

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[45] M.N. Ringnalda, J. Langlois, B.H. Greeley, T.V. Russo, R.P. Miiller, B. Marten, Y. Wong, R.E. Donnelly, Jr., T. Pollard, G.H.G. Miller, W.A. Goddard III and R.A. Friesner, PS-GVB ~2.0, Schriidinger, Inc., Pasadena, CA, 1994. [46] K.A. Sharp and A. Nicholls, Delphi ~3.0, Columbia University, New York, 1989.