99
J . Phys. Chem. 1993,97,99-106
Direct Determination of the Effective Electronic Coupling in Electron-Transfer Problems Javier Fernhdez Sanz' Departamento de Quimica F'isica, Facultad de Quimica, Universidad de Sevilla, E-41012, Sevilla, Spain
Jean Paul Malrieu Laboratoire de Physique Quantique (URA 505 du CNRS). Uniuersiti Paul Sabatier, 31 062 Toulouse Cedex, France Received: July 14,1992;In Final Form: October 15, 1992
The theory of effective Hamiltonians is used to calculate the effective electronic coupling between the two nearly degenerate A+-L-B and A-L-B+ structures in electron-transfer problems. A procedure is proposed to include the leading contributions of electron correlation to the effective coupling, either in a second-order approach or through a limited C1 (of the type two holes-one particle), giving results of higher accuracy. The method is applied to a series of (CH*=CH-L-CH=CH*)+ systems, where bridge L is a ligand of various lengths. The results are stable under the choice of starting MOs, and the contributions of through-space versus through-bond are analyzed.
Introduction
'otential Energy
Electron-transfer (ET) reactions are ubiquitous in both chemical and biochemical processes and have been the subject of considerable research as reflected in a large body of literature. In order to understand the mechanisms of ET reactions, a wide variety of theoretical models have been proposed and summarized in excellent reviews and monographs.'-8 When the cla~sical,~ semiclassicali0 or quantum mechanical] natures of these approaches are disregarded, the key role in them is played by the electron-transfer matrix element Vir, which in a two-state model is conventionallydefined as half the splitting between the adiabatic potential energy surfaces at the crossing seam (Figure 1):
2Ff= E + - E
(1) Within the two-state framework, E+ and E- are solutions for the secular equation
where Hii = (+iIfNi), H r r = ($#$h), Hif = ($iIqh), Sif = ($il$f), H being the electronic Hamiltonian, and the eigenvalue E is energy. At the crossing point Hii = Hffand if Sir is small, V,r almost equals the transfer integral coupling the states +i and Hir. Equation 2 should be valid out of the crossing $f: seam (Hii = Hfr), and it is expected that, according to the Condon principle,8 Hir does not vary significantly under the geometry changes of the partners of the charge transfer. Ab initio SCF techniques have been successfully used in transfer integral calculations by exploiting the properties of symmetrybroken SCF solutions for weakly coupled systems.l-14 Moreover, sincegenerally the interest is in the transfer integral at the crossing seam, gradient-based techniques for locating this point have been developed by Toga and MorokumalSaand later generalized and implemented in an all purpose program by Farazdel and Dupuis.15b In spite of these theoretical advances, the problem arises when dealing with transfer integral calculation, as pointed out by Newton,' from the reliability of computed values. Effectively, the transfer integral is an extremely small quantity for which typical values range between 100 and 1000 cm-1, and if the coupling is weak, Hifbecomes I 1 0 cm-1. If the inaccuracies due to numerical precision are disregarded, it appears clear that, in a proper description, effects of electron correlation should be
Nuclear Configuration Figure 1. Schematic potential energy profile for an electron-transfer reaction.
taken into account.I6 However, whatever the method of calculation for Hif is (via energy splitting or from symmetry-broken wave functions), usual techniques of incorporating electron correlation would lead to a huge computational effort for mediumand large-sized systems. In the present work we present an efficient strategy for the calculation of the transfer integral Hif. The method is based on the concept of effective Hamiltonian. The two states i and f define a model space on which the exact effective Hamiltonian is approached. Effective Hamiltonians are usually built up from the quasi degenerate perturbation theory. This idea will be used along two directions: (i) direct perturbative calculation of the matrix element Hif through a second-order development (this calculation involves a minimal set of determinants interacting with both $i and +f) and (ii) variational calculation by means of diagonalization of the CI matrix restricted to the above set of determinants (the use of such a CI procedure incorporates higher order effects and improves the reliability of the method). Notice
0022-365415812097-0099%04.00/0 0 1993 American Chemical Society
100 The Journal of Physical Chemistry, Vol. 97, No. 1 , 1993
that our computational strategy does not involve either explicitly correlated wave functions or correlated adiabatic states but it only accounts for correlation effects which contribute to the transfer integral. In the next section a detailed description of the methodology proposed is given. This is followed by some applications to the calculation of the transfer integral for simple models, namely the ET between two ethylene subunits separated by a variable bridge.
Theoretical Method Let us consider a monopositive ion (A-L-B)+ where A and B are the donor and acceptor subunits between which the electronic jump takes place. In our model A and B are two ethylenes: in reactants, B bears the positive charge while, in products, the charge is in A. L accounts for a symmetric bridge in such a way that a t the crossing seam A and B are mirror images. Then the exact solutions of the electronic Hamiltonian H are symmetryadapted; i.e. the charge is symmetrically delocalized. The HartreeFock wave function for the ground state can be written
Sanz and Malrieu The rotation angle y may be determined by projection techniques or by usual localization procedures, for instance that of Boys. The two states 40aand 4 O b are orthogonal since (a(b) = 0 by definition and eq 2 reduces to
(3) i.e. Hif = Hab, Sir = 0. Calculation of Hab at the correlated level, without using explicitly correlated wave functions, can be carried out from the quasi degenerate perturbation theory (QDPT).'7-23 The main idea is to bring partial exact information into a limited model space So. Briefly, the non-Hermitian formulationl8 of the QDPT allows us to obtain exact eigenvalues E, and eigenfunctions +,, of an exact Hamiltonian H
W m ) = EmI$,
)
from model functions q, which are the projection of model space
+,
into the
40, = 11 i 2 2 ... g g U I where 1,2, ...are innermolecular orbitals and g and u are, mainly, plus and minus linear combinations of ethylene ?r orbitals and therefore have equal amplitudes on A and B (labels g and u refer to gerade and ungerade MOs and are not strictly applied here). For the upper state, the wave function 4 O 2 is
In the model space the action of H is replaced by an effective operator HCff with eigenfunctions qmand exact eigenvalues E,:
i.e.
4 0 =~11 i 2 2 ... g u UI m
Let us now define
+
and a = (g u)/& b = (g - u)/& These two MOs are localized on A and B, respectively, and can be used to build two determinants:
In the present case the model space is spanned by the determinants doaand and the expression for Herris the 2 X 2 matrix
40, = 11 12 2 ... a a b(
(4)
= 11 1 2 2 ... a b 61
Again, the off-diagonal element of this matrix has to be identified with the transfer integral Hif of eq 2.24 Herrcan be obtained using perturbational approaches and, in particular, at the second order, we have
+Ob
These two determinants have the hole on B and on A, respectively, and are degenerate at theseam and quasi degenerate in its vicinity. Notice that a switch can be made from localized solutions 40a and 4 O b to delocalized ones:
Le., symmetry-adapted determinants I and 4 O 2 incorporate the resonance between 40aand described with the unique set of MOs a and b. Also, it should be noted that these orbitals are a compromise: a is halfway between the best orbital for A in the presence of B+ and the best orbital for A+ in the presence of B. Out of crossing seam, the ground-state RHF eigenfunction becomes where i and j, the two highest occupied MOs, may be of the same symmetry. These two orbitals may be transformed into localized MOs a and b by using an appropriate rotation U:
a = (cos y)i + (sin y ) j
b = -(sin y)i + (cos y ) j
where ( 4 O U ]are excited determinants which do not belong to the model space and €0and ea are respectively the zeroth-order energies of supposedly degenerate model functions
and of the
determinants
The degeneracy of the model space (eo = (Haa 4- &)/2) is introduced to maintain HeffHermitian. The crucial points are the following: (i) When Ha, = Hbb (on the seam), the energy splitting is given by HCfr& only, the calculation of which is sufficient. It is worth
Electronic Coupling in Electron-Transfer Problems noting that the determinants $Painteracting with both 40a and 4'b are much less numerous than those interacting with either or 4'b. The expression of Heff'(')abis given in the Appendix according to a diagrammatic representation of the many body perturbation theory, together with the set of useful determinants. Their number is proportional to no2nvwhere no and n, are the number of occupied and virtual MOs, respectively, while the number of determinants interacting with 40a is proportional to nO2nv2. (ii) For nonsymmetrical out of the seam situations, research of the adiabatic energy differences requires the calculation of Herfa,- Hcffbb. It is easy to show that the determinants which contribute to this energy difference are exactly those which interact with both 40a and 4 O b . Instead of using the second-order perturbative development, a diagonalization of the CI matrix restricted to 40a,f$Ob. and the above set of determinants, i.e., the space (Pa + 4'b + (4"&, can be performed. These determinants may be called two hole-one particle (2h-lp). Thecorresponding CI, labeled (2h-lp)CI, will give the correct energy splitting for both symmetrical and nonsymmetrical situations. This approach introduces higher order contributions, and a detailed formulation is given in the Appendix. Since in such variational procedures instabilities due to the possible smallness of denominators are avoided, higher reliability is expected. (Hereafter the notation H e d and Heffvar will be used for the off-diagonal effective element.)
Results and Discussion The methodology developed above has been applied to the calculation of the transfer integral H;f for the electronic jump between two ethylenes connected by a bridge. The bridge is an aliphatic chain -(CH2),,- with n = 1-3. Among all possible conformational isomers for these compounds, only the following structures, in which all carbon atoms are in the same plane, have been selected as models
2
1
CH2-CH'
3
Although, as stated above, computational schemes for locating geometries a t the crossing seam have been proposed, in this work and because of their simplicity, they have been estimated by interpolating between the initial and final equilibrium structures, keeping the bridge fixed. One of our main concerns in the present study was the choice of the molecular orbitals basis set used in the perturbational or variational calculation of the transfer integral Hif. In principle any delocalized set of MOs is valid: restricted open-shell HartreeFock (ROHF) orbitals, unrestricted Hartree-Fock natural orbitals (UHFNOs), or multiconfigurational self-consistent field (MCSCF) orbitals arising from a state-averaged calculation where the states 4; and &are mixed. All of these, if delocalized, satisfy the criteria of being 'democratic" at the seam; i.e. they do not polarize the bridge toward the left or toward the right. All of them are available when the initial and final structures are related by a symmetry transformation, i.e. for a process as the E T between A and B:
+
Areact-L-Breact Aprd+-L-Bprd where A,,,, = Bprd,B,,, = Aprd, and the bridge is symmetric at the seam. However, when the ET reaction is not symmetric because either the donor and acceptor sites are different (for +
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 101
TABLE I: Transfer Int a19 (cm-1) Determined from Zeroth-Order (Heno)and%ond-Order Perturbations (W) and Variational (H."") Calculations Using Both CSMOs and SAM&'
1 2 3 4 2' 5 6 b
-2754 -3226 -2911 2.5396 -3031 -3185 -3022 3.8990 -134 -24 -157 -313 -77 -110 557 564 5.1130 489 498 564 564 2561 2307 3.0309 2359 2096 2528 2405 2002 1854 3.0309 1784 1613 1964 1942 -9863 -9246 -7833 2.4098 -1413 -9727 -9385 2.1249 -11037 -12133 -9424 -11105 -12477 -9519
0 In all calculations, the two ?r orbitals a and b have the same phase. Ethylene4hylene distance in A.
instance two different transition metals) or the bridge is disymmetric (as for instance-CH2-O-CH2-CH2-), both R O H F and U H F orbitals are localized and, therefore, the bridge is polarized. Such orbitals are no longer *democratic" for the calculation of the transfer integral. In these cases, only the stateaveraged MCSCF molecular orbitals (SAMOs) preserve this condition, since they minimize the energies of the two lowest states of the ion. Although SAMOs are the best rational choice in the calculation of Hif, their availability in practice is limited to medium-sized systems because of well-known computational restrictions. That is why we have examined a fourth possibility, namely the use of closed-shell molecular orbitals (CSMOs) arising from a restricted Hartree-Fockcalculation on the neutral species. Because of the Koopman's theorem, positive ions can easily be built up in the CSMOs basis set, provided electronic reorganization effects are incorporated in the perturbational or variational process.25 In Table I, results of H;r calculated for compounds 1-3 using both CSMOs and SAMOs are reported. For 1, states i and f a r e strongly coupled, the transfer integral being-303 1 cm-I (CSMOs) and -2754 cm-' (SAMOs) a t zeroth order. Second-order perturbation corrections moderately increase the coupling (7 and 14%, respectively), whilevariational approach introduces smaller changes. It is worth noting that although the difference between He#, with respect to the type of molecular orbitals, is noticeable, the agreement in Herfaris better than 4%. In 2, the coupling is small and although at zeroth order Hif is under- or overestimated with respect tovariational results,again the latter arein reasonable agreement (in absolute value). For compound 3, Hif has a moderate value and, as can be seen, whatever the MOs are used, second-order and variational results are in fairly close agreement and are 60-70 cm-I higher than those calculated a t zeroth order. Rationalization of the evolution of the transfer integral in these compounds is not obvious since they arise from an overlap of contributions, among which two mechanisms have been classically assumed: through-bond (TB) and through-space (TS). TB and TS mechanisms clearly depend on the type of bridge and on the distance between donor A and acceptor B which, as can be seen from Table I, changes dramatically, going from 1 to 3. In order to get a qualitative description of the TS contribution, a series of calculations of Hifina system formed by twocollinear ethylenes without a bridge has been carried out. From Figure 2 it turns out that (i) The TS contribution increases when the intergroup distance lowers, (ii) second-order Hif is always lower than that computed at zeroth order, and (iii) the matrix element coupling states i and f are always positive. On the other hand, the straight line found in the logarithmic plot shows unambiguously the exponential behavior of the TS contribution with respect to the donor-acceptor distance in agreement with previous work.' Disregarding electronic screening due to the bridge, TS contribution for compound 3 is 13 cm-1, thus 2.2%. For 1 and 2 they are 128 and 2282 cm-I, respectively, and since, for these compounds, H i fvalues appear to be negative, it turns out that
Sanz and Malrieu
6000
Whether the contribution of the bridge to Hif is additive or not can be analyzed by considering a model with two bridges like compound 4:
-
5000 -
/ CHz-CHz \
CHz=CH
4000
3000 -
2oool
4
1000
0 0
2
1
3
,-
4
- --
-
-
6
6
7
5
6
7
r(et-et) / A +
1.OE-01
CH=CH2
i
order 0
* order
2
‘ 0
1
2
3
4
det-et) / A +-
order 0
-a order 2
2’
In order to make reliable comparisons, compound 2 is not a suitable partner, as both intergroup distances and electronic screening are different. That is why model 2’, in which the ethylene-ethylene distance has been fixed to the same value as in 4, has also been considered. Since, according to Figure 2, the TS contribution to Her? in 2’ is about 900 cm-I, a simple analysis suggests the bridge contribution to be about 1000 cm-1. These values would lead, for compound 4, to an Hifcloseto 2900 cm-I, which actually is found to be 2528 cm-I. This result shows that although the bridge effect is not strictly additive, at least it is constructive.30 On the other hand, when the values of Hir for compounds 2 and 2’ are compared, they appear to be considerably different, even though the number of links in the bridge is the same, showing, thus, the extreme dependence of the transfer integral on the structure of the bridge. Up to the present only models in which the bridge is a saturated hydrocarbon have been explored. For these compounds, the zeroth-order Hefrois a rough approximation of the transfer integral. Attention will be focused now on compounds in which the bridge is clearly more (model 5) or less electronegative (model 6 ) than donor and acceptor.
Figure2. Variation of the transfer integral H,f? for thesystem (Et-Et)+ versus inter-ethylene distance. (Bottom: logarithmic representation).
they are alsodominated by contributions other than TS. A further aspect to be considered is the angle between the two ethylenes. Calculations of Hif for the same system at different angles are reported in Figure 3. (In the supermolecule C A = C ~ C ~ = C D , values on the x axis are the angles between the vectors ~ B Aand rCD, i.e., 180 and 0’ when ethylenes are collinear and parallel, respectively, under a CZ,arrangement.) As can be seen from Figure 3, whatever the inter-ethylene distance is, variation of Hif isquite moderateup to angles larger than 60’. Since in compounds 1-3, the ethylene subunits are almost collinear, one may conclude that the noticeable difference found in Hif must be accounted for on the grounds of the bridge itself.
6
5
These compounds are not of the same kind as the preceding ones in the sense that both oxygen and sulfur bear an electron pair delocalized on the A system. However, they provide two simple examples for which zeroth-order couplings are a bad guess. Thus the variational coupling for compound 5 is about 14001500 cm-I higher than the zeroth-order solutions, while for compound 6 the opposite trend is observed and the zeroth-order values are found to be overestimated by about 1600 cm-I. In these cases, second-order values must be discarded because of the
Hab / cm
300 2500
looo~
i
500 0‘
0
120
60
Anglo. degree
-
order 0
+order 2
I mo
0
0
60
120
Angl..
-
order 0
110
degN0
+order 3
( E t f t b , 3.0 A
Figure 3. Variation of the transfer integral He($ for the system (Et.-Et)+ versus inter-ethylene angle (see text for explanation). The Et-Et intergroup distances are 3.0 A (left) and 5.1 A (right).
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 103
Electronic Coupling in Electron-Transfer Problems
1 40
2
(kcal/mol)
26
(kcal/mol)
20
so
16
20 10
10
6
0
-2
0
-1
-adlab
0 -2
2
1
0
-1
-adlab
dlrb
1
2
dlab
3
-2
0
-1
1
2
alsp
-adlab
dlab
Figure 4. Diabatic (hashed curves) and adiabatic (solid curves) reaction profiles of compounds 1-3 at the second-order perturbational level. The reaction coordinate is defined by a step as follows: step 0 refers to the system at the seam while steps -1 and +1 correspond to nuclear configurations of the system fully relaxed with the charge localized on the right and on the left, respectively (Le., the minima of the i and f diabatic states).
low-lying states introduced by the bridge (mainly sulfur derivative), whose contribution to the coupling is poorly described in the perturbational expansion. A proper description of this problem could be undertaken using an approach based on intermediate Hamiltonians. On the other hand, application of methodology developed in the theoretical section is not limited to the calculation of Hir at the crossing seam but it is also useful for studying the features of both diabatic and adiabatic potential energy surfaces (PES) around the seam. This procedure involves calculation of both diagonal (diabatic description) and off-diagonal elements of the effective Hamiltonian which is then diagonalized (adiabatic description). However, it should be noted a t this point that, as pointed out in the previous section, the process for obtaining localized molecular orbitals as plus and minus combinations of pseudo-g and pseudo-u delocalizated orbitals is no longer valid and either a projection scheme or a Boys localization are in order. Using the later, adiabatic PES cross sections along a deformation coordinate Q have been computed for compounds 1-3. Results
obtained by using a second-order perturbation effective Hamiltonian and a SAMOs basis set are reported in Figure 4. As can be seen, there is smooth behavior around the seam, proving the usefulness of our approach. It is also worth noting that all along the considered relaxation geometry the transfer integral H,d was almost constant, its variation never exceeding 0.5% in agreement with the Condon approximation.* When parts a-c (for compounds 1-3) of Figure 4 are compared, it turns out that while compounds 2 and 3 show two well-differentiated minima, only a minimum is observed for compound 1. The existence of one or two minima on the PES arises from a compromise between the size of the transfer integral and relaxation energy. As stated by one of us in a previous work,” two minima are expected only if the transfer integral is lower (in absolute value) than twice the relaxation energy AE defined as the gap between the seam and the minimum of the diabatic PES cross section. For 2 and 3 values of AE are 0.0043 and 0.004 1 hartree whileH,$ is 0.000 35 and 0.0025 hartree, respectively, and therefore, two minima must be found. For 1, the relaxation energy is found to be 0.0040
Sanz and Malrieu
104 The Journal of Physical Chemistry, Vol. 97, No. 1, 1993
hartree and, since He$ is 0.0145 hartree, only one minimum is predicted in agreement with Figure 4a.
a
Conclusions
i
As expected for nearly degenerate situations, the theory of effective Hamiltonians is the natural and rational guideline to go beyond monoelectronic evaluation of energy splittings or electronic couplings. The transfer integral Hir present in electron-transfer reactions falls into the domain of relevance of that approach. Second-order corrections already incorporate the major contributions to the electronic coupling between A+-L-B and A-LB+. It simply requires the calculation and storage of n3 (exactly no%") molecular bielectronic integrals (iajp) and therefore is applicable to rather large systems. The method also provides a direct diabatic picture. Better accuracy (and better stability with respect to the choice of the starting MOs) is obtained through diagonalization of the determinants contributing to Hefr,(2), i.e. the two hole-one particle CI. The sizeof such a CI is n,2n, and all (ij,kl) molecular integrals must be calculated. Recent progress on direct CI algorithms for arbitrarily selected spaces32 should make this procedure rather easy. For very large systems, observable-dedicated MOs (i.e. MOs most contributing to the effective coupling) can also be defined,33allowing a reduction of the variational CI to the most involved MOs, the remainder being treated perturbatively. These are the methodological proposals of the present work. The applications concerning the electron transfer between two ethylene subunits connected through a saturated hydrocarbon chain of various lengths are quite satisfactory. Whatever the set of starting MOs is (neutral ground state (ALB) MOs or stateaveraged MOs for the two states of the cation (ALB)+), the values of the effective electronic coupling HcfrVar are the same within 100 cm-l. The leading role of through-bond mechanism is shown by comparison with nonbridged ethylenes in the same relative positions.
,I
._- - - - F I,
- - - - _ _F b
11
a t
bl ifi=a
z z j
bt ori=b
Acknowledgment. This work was supported by the Spanish DirecciQ General Cientifica y Tknica, Project NO.PB89-056 1. J.F.S. thanks the Junta de AndalucIa for a grant during his visit to the Laboratoire de Physique Quantique. The authors also thank Dr. F. Spiegelmann for a copy of his determinant-based CI program. Appendix PerturbationalCalculationof the Off-DiagonalTerm He$. Let (1, 2 , ..., i, j, ..., a, b, ..., p, ...) be molecular orbitals of a model system (A-L-B)+ where a and b accounts for localized MOs. The reference is the closed-shell determinant: 11 1 2 2 ... a a b
61
F denotes the Fock operator corresponding to this reference and J and K are Coulomb and exchange operators. Indices i j run over occupied orbitals and p over virtuals. co is taken as 0'
= (Faa + Fbb)/2
and the molecular integrals (ij,kl) are defined as
The second-order off-diagonal effective Hamiltonian Her+is then written as a summation of the following contributions: -zeroth
order
-Fab
-one hole, first order
bl ifi=a
at
P
(
- cjlFIp>)(ba,pj) - PP
Electronic Coupling in Electron-Transfer Problems
The Journal of Physical Chemistry, Vol. 97, No. I, 1993 105
P being the projection operator
ori=b
= b o a ) ( 4 ' a 1 + Id'b) ( d o b l In the non-Hermitian (Bloch) formulation,'* the effective Hamiltonian Herf is written
bt \
with (P+i)l= S - ' ( P $ i } , S being the overlap matrix of the model eigenfunctions tP+i}. The nondiagonal term Herfarcan then be obtained as
\
=
(#alflfrbb)
(#'alp$])
(p$IL140b) E2(4°a/P$,)
+ ( p$2L140b)
For the problem wearedealing with here, theuseofa Hermitian effective Hamiltonian is more convenient. In the Hermitian (des Cloizeaux) formulation,19 a set of eigenfunctions VI' and (p2' of the effective Hamiltonian HcffdC is first obtained as
{'Pi)= s l ' 2 ( P $ i ) i.e.
= E,I~;)
H"ffdclqi')
bt [(plqb) - (blJab)
-c
The effective Hamiltonian is then written
+ ( b l K a l p ) l [ ( a l q p ) - ( a l J b b ) + (alKblp)l eo- t p
P
In order to get the right splitting when 40a and 4'b are no longer degenerate, contributions having a differential effect on the effective energy difference Hcfraa - Hcrrbb must be considered. These differential second-order effects necessarily involve the propagation lines a i or bl. They are of the type
"I "t and therefore, they concern the 2h-lp determinants already considered in the off-diagonal calculation. Variational Determination of the Off-Diagonal Term H e p . Let El and E 2 be eigenvalues of the CI matrix and + I and $2 the corresponding eigenfunctions: q$i)
qi = cia40a+ Cib4'b
= E,l$i)
+ C c i L I ~ o a i = 1,2 LI
40a and 4 O b being the zeroth-order localized determinants' spanning the model space. The effective Hamiltonian Herfis defined in such a way that
Hefflcpi) = E J ~ ) i = 1 , 2 where
c p ~and cp2
are the model eigenfunctions defined as
[vi)=
i = 1,2
i = 1, 2
= EIIql')
IieffdC
((01'1
+
('Pi1
and the off-diagonal term Herrvar
=
(4a1Heffd&b)
= El
E2(dJ0al'Pi)
(#'aIql')
('Pibob)
+ = E I C ~ ~ C+~E~cZ;C~< < ('Pl'l4'b)
and since cl; = cad and Clb' = -cZaf
HeffdC= Cl'C;(E1 - E2) Notice that a t the seam, cI' = c; = 1 / 4 2 . As an additional remark, notice that the set of 2h-lp determinants is invariant under the unitary transformation of the holes among themselves and/or of the particles among themselves. That space being invariant, the result of the (2h1p)CI is invariant. Therefore, localized and delocalized unitarilytransformed MOs may be indifferently used in the variational calculation. This invariance is, of course, no longer guaranteed when occupied and virtual MOs are mixed. Also, it is easy to show that the (2h-1p)CI is invariant under any rotation of the occupied MOs among themselves and of the virtual MOs among themselves.
References and Notes (1) For a recent review see: Newton, M. D. Chem. Reu. 1991, 91,767. ( 2 ) Dogonadze, R. R.;Kuznetsov, A. M.; Maragishvili, T. A. ECctrochim. Acia 1980, 25, 1. (3) Sutin, N . Prog. Inorg. Chem. 1983, 30, 441. (4) Newton, M. D.; Sutin, N . Annu. Reu. Phys. Chem. 1984, 35, 437. (5) Mikkelsen, K. V.; Ratner, M. A. Chem. Reu. 1987, 87, 113. (6) Cannon, R. D. Electron Transfer Reactions; Butterworth: Stoneham, MA, 1980. (7) DeVault, D. Quantum-Mechanical TunnelinginBiologicalSystems; Cambridge University Press: Cambridge, U.K., 1984. (8) Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer-Verlag: New York, 1979. (9) Marcus, R. A. J . Chem. Phys. 1956, 24, 979. Marcus, R. A. J . Chem. Phys. 1965,43,679. Hush, N . S.Trans. Faraday Soc. 1%1,57,155. Hush, N . S. Electrochim. Acta 1968, 13, 1005. (10) Brunschwig, B.S.;Logan,J.;Newton,M.D.;Sutin,N.J.Am.Chem. Soc. 1980, 102, 5798. (1 I ) Do1in.S. P.;German, E. D.;Dogonadze, R. R. J . Chem. Soc., Faraday Trans. 2 1977, 73, 648. (12) Kestner, N . R.; Logan, J.;Jortner, J. J . Phys. Chem. 1974,78,2148.
I06 The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 (13) Newton, M. D. Inr. J. Quantum Chem., Quantum Chem. Symp. 1980, 14, 363. (14) For a recent formulation see: Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J. Am. Chem. SOC.1990, 112, 4206. (15) (a) Koga, N.; Morokuma, K. Chem.Phys.Lett. 1985,119,371. (b) Farazdel, A,; Dupuis, M. J. Compur. Chem. 1991, 12, 276. (16) Braga, M.; Bros, A.; Larsson, S.Chem. Phys. 1991, 156, I . ( I 7) Van Vleck, J. H. Phys. Rev. 1929, 33, 467. (18) Bloch, C. Nucl. Phys. 1958, 6, 329. (19) des Cloizeaux, J. Nucl. Phys. 1960, 20, 321. (20) Brandow, B. H. Rev. Mod. Phys. 1967, 39, 771. (21) Lindgren, I. J. Phys. E A f . Mol. Phys. 1974, 7, 2441. Lindgren, 1.; Morrison, J. Atomic Many Body Theory; Springer: Berlin, 1982. (22) Shavitt, 1.; Redmon, L. T. J. Chem. Phys. 1980, 73, 5711. (23) A recent monograph on effective Hamiltonians is given in: Durand, Ph.; Malrieu, J. P. In Advances in Chemistry and Physics; Ab Initio Methods in Quantum Chemistry. Part I ; Lawley, K. P., Ed.; Wiley: Chichester, U.K., 1987. (24) Schemes based on effectiveenergy-dependent Hamiltonians have been previously used although in a rather different approach. See for instance: Larsson, S. J. Am. Chem. SOC.1981, 103, 4034.
Sanz and Malrieu (25) Calculations were carried out using the pseudopotential approximation to describe core electrons of carbon atoms, while for the valence, a double-{ basis set was e m p l ~ y e d . For ~ ~ .hydrogen ~~ atoms, the standard (4s)/ [Zs] basis set was used.28 Both RHF-SCF and MCSCF calculations were undertaken using the system of programs HONDO-7.29 (26) Durand, Ph.; Barthelat, J. C. Theor. Chim. Acta 1975, 38, 283. Barthelat, J . C.; Durand, Ph.; Serafini, A. Mol. Phys. 1977, 33, 159. (27) Molecular ab initio Calculations using Pseudopotentials. Technical Report: Laboratoire de Physique Quantique: Toulouse, 198 1 . (28) Dunning, T. H. In Modern Theoretical Chemistry; Schaefer, H. F., 111, Ed.; Plenum Press: New York. 1977; Vol. 2. Huzinaga, S.J . Chem. Phys. 1965, 42, 1293. (29) Dupuis, M.; Wats, J. D.; W a r , H. 0.; Hurst, G.J. B. HONDO-7. IBM Technical Report KGN-169; IBM: Kingston, NY, 1988. (30) Liang, C.; Newton, M. D. J. Phys. Chem. 1992, 96, 2855. (31) Durand, G.;Kabbaj, 0. K.;Lepetit, M. B.; Malrieu, J. P.; Marti, J. J . Phys. Chem. 1992, 96, 2162. (32) ,Caballol, R.; Malrieu, J. P. Chem. Phys. Lett. 1992,188,543. Povill, A.; Rubio, J.; Illas, F. Theor. Chim. Acta 1992, 82, 229. (33) Miralles, J.; Caballol, R.; Malrieu, J. P. Chem. Phys. 1991, 153.25.
