J. Phys. Chem. 1992, 96, 2855-2866 sites i and j , respectively.8 Using eq 5 , the location of hot and cold spots can be probed using improved electronic structure methods.
We thank A‘ and B’ Gray for helpful discussions. This work was peformed in part at the Jet Propulsion Laboratory, California Institute of Technology, and was sponsored in part by the Department of Energy’s Catalysis/Biocatalysis Program (Advanced Industrial Concepts
2855
Division), through an agreement with the National Aeronautics and Space Administration. J.N.O. thanks the National Science Foundation (Grant DMB-9018768) and the Department of Energy’s Catalysis/Biocatalysis Program for a research contract administered by JPL. J.N.O. is in residence at the Instituto de Fisicae Quimica de Sb Carlos, Universidadede s.o paulo, 13560, sgo Carlos, sp, Brazil, during the summers. Registry No. Cytochrome c, 9007-43-6.
Ab Initio Studies of Electron Transfer: Pathway Analysis of Effective Transfer Integrals Congxin Liang’ and Marshall D. Newton* Department of Chemistry, Brookhaven National Laboratory, Upton, New York 1 1 973 (Received: October 28, 1991; In Final Form: December 16, 1991)
Effective transfer integrals ( r ) have been evaluated for u- and n-type electron and hole transfer in radical ion systems comprising methylene donor/acceptor groups linked by various saturated organic spacer groups. The T values have been calculated on the basis of ab initio self-consistent-fieldwave functions for the radical ion states, obtained either directly for the system of interest ( TwF)or from the associated neutral state via Koopmans’ theorem (TKT), and employing either minimal (STO-3G) or split-valence(3-21G) basis sets. The TKT values have been decomposed into additive contributions from individual pathways, both through-space (Ti;) and through-bond (Ti:), using perturbation theory as formulated by Ratner together with a localized orbital basis represented by natural bond orbitals (NBO’s) as defined by Weinhold et al. The overall coupling has been shown to arise from interference among a large number of competing pathways, none of which is strongly dominant. Nearest-neighborpathways of the McConnell type are significant for transfer in some radical cation systems, but are frequently of very minor significance in comparison with lower-order superexchange pathways, often with contributions differing in sign from the overall TKTvalues. These latter conclusions are generally consistent with those based on studies involving different saturated spacer groups by Naleway et al. (for radical anions) and Jordan and Paddon-Row (for radical anions and cations). We find that transfer in radical anions and cations is generally dominated, respectively, by “electron” and ”hole” pathways, but both mechanisms are found to be significant in both types of transfer. A number of transferability relationships have been identified for generic pathway types, and the important influence of stereochemistry on coupling has been illustrated, with regard to both orientation of donor/acceptor groups relative to the spacer and internal conformation of the spacer, showing the competition between coupling via the carbon framework and via CH bonds (i.e., hyperconjugation).
I. Introduction Electron transfer over long distances (up to tens of angstroms) has recently attracted much attention in the chemical communitye2S3 In particular, there is rapidly increasing interest in the details of indirect donor/acceptor (D/A) coupling mediated by intervening materials (“spacers”) which in many cases are completely saturated electronically. This sort of indirect or superexchange4 interaction, termed through-bond (TB) coupling5 in cases where the local D and A sites are linked by a sequence of chemical bonds, is crucial not only for efficient long distance electron (or hole) transfer:,’ but also for long-range triplet energy transfer8p9and for static molecular properties such as splittings ( I ) Present address: BIOSYM Technologies, Inc., 10065 Barnes Canyon Road, San Diego, CA 92121. (2) Newton, M. D. Chem. Reo. 1991, 91, 767. (3) (a) Kuki, A. Siruciure and Bonding, Springer-Verlag: Berlin, 1991; pp 50. (b) Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985,811,265. (4) (a) George and Griffith. The Enzymes; Academic Press: New York, 1959; Vol. I , p 347. (b) Halpern, J.; Orgel, L. E. Discuss. Faraday Soc. 1960, 29, 32. (c) Kosloff, R.; Ratner, M. A. Isr. J . Chem. 1990, 30, 45. ( 5 ) (a) Hoffmann, R.; Imamura, A,; Hehre, W. J. J. Am. Chem. SOC. 1967,90, 1499. (b) Hoffmann, R. Acc. Chem. Res. 1971,4, I . (c) Brunck, T.K.; Weinhold, F. J . Am. Chem. Soc. 1976, 98,4392. Weinhold, F.; Brunck, T.K. J . Am. Chem. SOC.1976, 98, 3745. (d) Imamura, A,; Ohsaku, M. Teirahedron 1981, 37, 2191. (6) Miller, J. R.; Beitz, J. V. J. Chem. Phys. 1981, 74, 6746. (7) (a) Miller, J. R.; Beitz, J. V.;Huddleston, R. K. J. Am. Chem. SOC. 1984, 106,5057. (b) Closs, G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. W. J . Phys. Chem. 1986, 90, 3763. (c) Closs, G. L.; Miller, J. R. Science 1988, 240, 440. (d) Johnson, M. D.; Miller, J. R.; Green, N. S.;Closs, G. L.J . Phys. Chem. 1989, 93, 1173. (8) Closs, G. L.; Johnson, M. D.; Miller, J. R.; Piotrowiak, P. J . Am. Chem. SOC.1989, 1 1 1 , 3751. (9) Closs, G. L.; Piotrowiak, P.; MacInnis, J. M.; Fleming, G. R. J . Am. Chem. SOC.1988, 110, 2652.
0022-3654/92/2096-2855$03.00/0
of ionization potentials (IP) and electron affinities (EA)I0,l1and magnitudes of magnetic exchange coupling.I2 Several groups have carried out experimental studies with homologous spacer groups to investigate the attenuation of TB interaction with effective D/A separation distance or with number of intervening chemical bonds. Closs and Miller et al.7-9 have studied intramolecular electron transfer, hole transfer, and triplet energy transfer through spacers such as cyclohexane, decalin, and steroids. Paddon-Row and Jordan et a1.lo5” have focused primarily on electron transfer and IP and EA splittings associated with polynorbornyl spacer groups. Systematic studies of long distance electron transfer through olig~bicyclooctane,~~ ~piroalkane,’~ and proliieI5 spacers and through rigid g l a s ~ ~have , ’ ~ also been reported. As far as experimental uncertainty permits, these experiments (IO) Paddon-Row,M. N. Acc. Chem. Res. 1982, IS, 245. (1 I ) (a) Paddon-Row, M. N.; Jordan, K. D. In Modern Models of Bonding and Delocalization; Liebman, J. F., Greenberg, A., Eds.; VCH Publishers: New York, 1988; p 115. (b) Balaji, V.; Ng, L.; Jordan, K. D.; Paddon-Row, M. N.; Patney, H. K. J . Am. Chem. SOC.1987, 109, 6957. (12) (a) Kramers, H. A. Physica 1934, I , 182. (b) Anderson, P. W. Phys. Reo. 1950, 79, 350. (c) Anderson, P. W. Phys. Rev. 1959, 115, 2. (13) (a) Leland, B. A,; Joran, A. D.; Felker, P. M.; Hopfield, J. J.; Zewail, A. H.; Dervan, P. B. J . Phys. Chem. 1985,89,5571. (b) Joran, A. D.; Leland, B. A.; Geller, G. G.; Hopfield, J. J.; Dervan, P. B. J . Am. Chem. SOC.1984, 106, 6090. (14) (a) Stein, C. A,; Lewis, N. A,; Seitz, G. J. J . Am. Chem. SOC.1982, 104,2596. (b) Knapp, S.; Murali Dahr, T. G.; Albaneze, J.; Gentemann, S.; Potenza, J. A.; Holten, D.; Schugar, H. J. J . Am. Chem. Soc. 1991, 113,4010. (15) (a) Vassilian, A,; Wishart, J. F.; van Hemelryck, B.; Schwarz, H.; Isied, S.S. J . Am. Chem. SOC.1990, 112, 7278. (b) Isied, S. S.; Vassilian, A.; Wishart, J . F.; Creutz, C.; Schwarz, H. A,; Sutin, N. J . Am. Chem. SOC. 1988, 110, 635. (c) DeFelippis, M. R.; Faraggi, M.; Klapper, M. H. J . Am. Chem. SOC.1990, 112, 5640. (d) Schanze, K. S.;Sauer, K. J . Am. Chem. SOC.1988, 110, 1180.
0 1992 American Chemical Society
2856 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
indicate that the through-bond interaction decreases exponentially with increasing distance or with increasing number of chemical bonds or spacer groups. Numerous theoretical attempts have been made to understand the mechanism of long distance electron t r a n ~ f e r . ~ , l ~In -~~ particular, several methods have been proposed to correlate the electronic structure of the bridge linking the donor and acceptor with the magnitude of the effective transfer integral, TDA, controlling the D/A interaction. Three decades ago McConnellI6 proposed a superexchange model which predicted exponential decay of TDA with number ( m ) of spacer units, S, in a linear, homologous spacer of type (S),,,. Several generalizations of McConnell's superexchange model have been proposed.2 Recently, Ratner20 derived and extended McConnell's equation using an iterative version of the Lippmann-Schwinger equation.30 This method gives TDA as the sum of contributions from all possible pathways linking the donor and acceptor. This superposition exposes a variety of interesting possibilities for constructive or destructive interference. Other studies of interference effects have been undertaken by BeratanI9 et al. and by Gruschus and K ~ k i . * ~ Larssonlsa has employed Lowdin's partitioning method3' to define an effective two-state Hamiltonian for the donor and acceptor in which the influence of the bridge orbitals (states) is treated as a perturbation. A number of studies have analyzed TB interactions in terms of localized bonding (or antibonding) bridge orbita l ~ . ~ ~ J Paddon-Row ~ , ~ ~ * and ~ ~Jordan28b,cie ~ - ~ ~ and ~ , Naleway, ~ ~ Curtiss, and Miller29 have attempted to assess the relative importance of individual local bridge orbitals to the overall coupling, using ab initio molecular orbital techniques. In the present paper we pursue the goal of achieving a useful additive decomposition of TB coupling based on individual contributions expressed in terms of suitably-defined bridge orbitals. Such detailed information should help in enhancing the ability ultimately to design molecular bridges with specified charge or energy-transfer characteristics. The viability of a decomposition scheme requires the consideration of a number of issues. The desired additivity of contributions leads to the adoption of perturbation-theoretic models, and thus the convergence behavior of the perturbation series is an important issue. Another issue arises from the fact that, for a decomposition to be optimally (16) McConnell, H. M. J . Chem. Phys. 1961, 35, 508. (17) Heilbronner, E.; Schmelzer, A. Helu. Chim. Acta 1975, 58, 936. (18) (a) Larsson, S. J . Am. Chem. SOC.1981, 103, 4034. (b) Siddarth, P.; Marcus, R. A. J . Phys. Chem. 1990, 94, 2985. (19) (a) Beratan, D. N.; Hopfield, J. J . J . Am. Chem. SOC.1984, 106, 1584. (b) Beratan, D. N. J . Am. Chew. SOC.1986, 108,4321. (c) Onuchic, J. N.; Beratan, D. N. J . Am. Chem. SOC.1987, 109, 6771. (d) Onuchic, J. N.; Beratan, D. N. J. Chem. Phys. 1990, 92, 722. (20) Ratner, M. A. J . Phys. Chem. 1990, 94, 4877. (21) (a) Kuki, A.; Wolynes, P. B. Science 1987, 236, 1647. (b) Marchi, M.; Chandler, D. J . Chem. Phys. 1991, 95, 889. (22) Gruschus, J . M.; Kuki, A. J . Am. Chem. SOC.,submitted for publication. (23) (a) Newton, M. D. J . Phys. Chem. 1988, 92, 3049. (b) Newton, M. D. J . Phys. Chem. 1991, 95, 30. (c) Newton, M. D.; Ohta, K.; Zhong, E. J . Phys. Chem. 1991, 95, 2317. (24) Ohta, K.; Closs, G. L.; Morokuma, K.; Green, N. J . Am. Chem. SOC. 1986. /08. 1319. . . - ., . .., . .. (25) Cave, R. J.; Baxter, D. V.; Goddard, W. A,, 111; Baldschweiler, J. D. J . Chem. Phys. 1987.87, 926. (26) Farazdel, A.; Dupuis, M.; Clementi, E.; Aviram, A. J . Am. Chem. SOC.1990, 112, 4206. (27) Broo, A.; Larsson, S. Chem. Phys. 1990, 148, 103. (28) (a) Paddon-Row, M. N.; Wong, S. S. Chem. Phys. Left. 1990, 167, 432. (b) Paddon-Row, M. N.; Wong, S. S.; Jordan, K. D. J. Am. Chem. SOC. 1990, 112, 1710. (c) Paddon-Row, M. N.; Wong, S. S.; Jordan, K. D. J . Chem. SOC.,Perkin Tram. 2 1990,425. (d) Jordan, K. D.; Paddon-Row, M. N. J . Phys. Chem., in press. ( e ) Jordan, K. D.; Paddon-Row, M. N. Chem. Reu., in press. (29) Naleway, A. C.; Curtiss, L. A.; Miller, J . R. J . Phys. Chem. 1991, 95, 8434. (30) (a) Lippmann, B.; Schwinger, J . Phys. Rev. 1949, 79,469. (b) The definition of effective coupling based on ref 30a arises in a dynamical context. Of course, the coupling may also be obtained directly in terms of stationary states based on time-independent perturbation theory (e&, see ref 52). (31) Lowdin, P. 0. J . Mol. Spectrosc. 1963, 10, 12; Phys. Rec. 1965, 139, A351.
Liang and Newton
Figure 1. Molecules employed as models for r (1-5) and u (6-10) electron transfer. T h e donor (D) and acceptor (A) orbitals are indicated schematically. All molecules have Czusymmetry except for 6 and 8, which belong to D3*. The symmetry-distinct bridging units are denoted by a and b.
useful, it should be as compact as possible. Thus in implementing the above approach in an a b initio molecular orbital framework we desire to keep the orbital basis set as small as possible, while retaining enough flexibility to reproduce with reasonable accuracy the correct TDAvalues (as defined by the results obtained with more flexible basis sets). We apply the above techniques to models for T - and u-electron transfer in radical anion and cation systems involving five different saturated spacer groups. The relative importance of various tight-binding (nearest-neighbor) and longer-range pathways is examined, and topics such as additivity and interference, and stereoelectronic features are addressed. 11. Theoretical and Computational Details A. Molecular Systems. The molecular systems employed in
the present study are displayed in Figure 1, and their structures were obtained as follows. The structures of bicyclo[ 1.1.l]pentane, bicyclo [2.2.1 ]heptane (norborane), bicyclo [2.2.21octane, cyclohexane (boat conformation), and butane (cis conformation) were optimized at the self-consistent-field molecular orbital level (SCF MO), using the minimal STO-3G basis set.32 Geometry optimization for bicyclooctane was performed for convenience with the constraint of D3hsymmetry (the D3hstructure is just slightly higher in energy than the energy minimum which has D3 symm e t r ~ ~ ~For ) . systems 1-5, a hydrogen atom at each bridgehead or terminal carbon atom (Cl, el,)is replaced by a methylene group, thereby yielding local D and A groups of the Ir-type @e., (32) Hehre, W. J.; Stewart, R. F.; Pople, J. A. J . Chem. Phys. 1969, 51, 2657. (33) Schmitz, L. R.; Allinger, N. L.; Flurchick, K. M. J . Compur. Chem. 1988, 9, 28 1 ,
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2857
Ab Initio Studies of Electron Transfer the methylene nonbonded orbitals (indicated in Figure 1) have a a-type orientation with respect to the C-C vectors joining the methylene groups to the spacer).34 Correspondingly, in systems 6 through 10 the “dangling” hybrids generated by the removal of bridgehead hydrogen atoms serve as a-type D/A orbitals and define local sites for u-type D/A groups. As in earlier studies,24 the CH2 D/A groups in 1-5 can serve as models for more complex D/A groups, such as olefinic or aromatic species. The main structural difference in bridges of 3,4, and 5 lies in the LCIC2C2’ angle which increases from 3 (109.7’) to 4 (1 12.6’) to 5 (1 16.5’). Consequently, the distance between bridgehead atoms increases in the same direction (2.599 A for 3, 2.744 A for 4, and 2.937 A for 5). The methylene groups in 1-5 are constrained to be planar, with C-H distance of 1.082 A and LHCH angle of 120’. The C-C bond linking methylene groups to bridges is fixed at a length of 1.525 A in all cases. All 10 structures correspond to the C2, point group except for 6 and 8, which belong to D3h. Systems 1-10 are chosen for several reasons. First, they are chemically significant systems. The spacer in molecules 3 and 8, namely the bicyclooctane framework, has been dealt with in previous experimental” and theoreticalig studies of a-electron transfer, and questions remain concerning its efficiency in facilitating a-electron transfer. It also offers a classic example of TB u interaction^.^ The spacer in 1 and 6 (bicyclopentane) contains two-bond bridges between the bridgehead atoms, in contrast to the three-bond bridges of 3 and 8, and it is of interest to compare the properties of the different spacers in transmitting u and a electrons. The bridgehead atoms in the n~rbornane’~ spacers in 2 and 7 are linked by both two-bond and three-bond bridges, thus enabling one to study the possibility of interference among the different bridge types. Compounds 4 and 9, and 5 and 10, have the same kind of threebond bridges as 3 and 8, but fewer of them. Note that the combined set of bridges in 4 and 5 (9 and 10) yields the set in 3 (8), and thus the possibility of additivity relationships in TB coupling may be studied. For each of the five chosen spacer groups, the comparisons of a and u coupling in both radical anion and cation states (seebelow) provide different probes of the various components of the spacer electronic manifold. Stereoelectronic aspects of TB coupling are of considerable interest,2~11~’8a~z716~37 both with regard to internal spacer geometry, and the geometry of the linkage between the D and A groups and the spacer. All of the three-bond bridges in 1-10 have cis conformations, and in 1-5 the two D/A methylene groups lie in a common plane, perpendicular to the reflection plane containing C1 and CIt(see Figure 1). After discussing the calculated results for 1-10, we shall focus on stereoelectronic features by comparing the coupling in 3-5 and some of their conformational variants. B. Electrooic States and Energies. The radical ions of molecular species 1-10 (with net charge f 1) are employed in chargetransfer processes as specified by the following electronic states (with standard notation for single determinant wavefunctions comprised of spin orbitals): I(core)DDAl I(core)DI
T&n)
I(core)Dul
TM(wt)
I(core)Al
(anion) (cation)
(la) (lb)
where “(core)” refers to the portion of the occupied manifold whose orbitals are not explicitly displayed, and D and A as orbital designations refer to the effective donor and acceptor orbitals in the process. The decomposition of D and A into local contributions and “tails” arising from delocalization onto the spacer will be (34) In the case of a single alkyl bridge (as in 5). the *-type. D / A groups (CH,) emplo ed here are analogous to the sp3 NH2 group studied by Broo and LarssonJ Their sp2 NH2 groups are twisted so as to correspond to the hyperconjugative coupling discussed in section IVC of the present paper. (35) Due to the manner in which it is coupled to the D / A groups, the norbornyl spacer in 2 and 7 plays a role distinct from that in the spacers designed by Paddon-Row.’OJ’ (36) (a) Woitellier, S.;Launay, J. P.; Joachim, C. Chem. Phys. 1989, 131, 481. (b) Larsson, S.; Broo, A.; Kallebring, B.;Volosov, A. In?. J . Quanr. Chem., Quant. Bioi. Symp. 1988, IS, 1. (37) Cave, R. J.; Siders, P.; Marcus, R. A. J . Phys. Chem. 1986,90, 1436.
discussed below. The processes are controlled by the transfer integral, TDA, which couples the initial and final states. The schemes ((la) and (lb)) imply a core and D/A orbital pair common to both the initial and final states in each process. For the implementations reported below, this constraint (implying in effect a strict one-particle model) is either obeyed exactly or is found to be a good a p p r o x i m a t i ~ n .However, ~ ~ ~ ~ the orbitals for the anion and cation processes are obtained separately and are distinct from each other. These distinctions will be made explicit only when necessary for clarity. Processes l a and l b are sometimes referred to, respectively, as “electron transfer” and “hole t r a n ~ f e r ” . ~ J ) However, ~~J~~ “electron” and “hole” mechanisms (involving, respectively, the participation of empty and filled spacer orbitals) may in general each contribute to both types of process ((la) and (lb)), as discussed below. Accordingly, in the remainder of the paper we simply refer to processes l a and l b as radical anion charge transfer and radical cation charge transfer. As an alternative to the above dynamical picture one may focus on the stationary states formed from the neutral triplet diradical by ionization or electron attachment. If the D/A sites are symmetry-equivalent as in the present case, then the “natural” representation would involve the sum (+) and difference (-) of the members of each of the initial-state/final-state pairs in $*(an) = (I(core)DDAl f I(core)DAA)I/fi #*(cat) = (I(core)DI f I(core)Al)/fi
(cation)
(anion) (2a) (2b)
These states can be rewritten in terms of symmetrically delocalized D/A orbitals, @*= (D f A ) / f i
(3)
$*(an) = -I(core)r$+b*#-l
(4a)
$*(cat) = I(core)@l
(4b)
as follows:
+
Note that the labels and - correspond respectively to wave functions which are symmetric (s) and antisymmetric (a) with respect to interchange of D and A in the case of \l+(cat), whereas for $*(an), and - are associated respectively with a and s symmetry. As has been discussed extensively in the literat ~ r e , Z J the ~ , ~magnitude ~ * ~ ~ of the transfer integrals, TDA(an)and TDA(cat),can often be equated approximately to one-half the magnitude of the energy splittings of the delocalized ion states (the relationship is exact to the extent that a strict one-particle model is ~ a l i d ) . ~We J ~ employ the following sign conventions for the splitting energy:
+
A
(E- - E+)
where E* are the total energies associated with $*. Equation 5 can be applied to both anion and cation states. As defined, A is positive when the energies of the ion states have the “normal” order, Le., when the occupancy of r$+ is energetically preferable to occupancy of d-. A negative sign implies “inverted” ordering.5910$l I To the extent that Koopmans’ theorem (KT)38 is valid, the splitting parameter A may be approximated by the corresponding splitting of the orbital energies, t+ and e-, associated, respectively, with d+ and 4- as obtained from the neutral triplet diradical species, $(triplet) = I(core)d+@-l
(6)
We define (7)
(once again, the definition yields a positive sign for the “normal” (38) Koopmans, T. Physica (Ufrechr) 1933, 1 , 104.
2858 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
orbital energy ordering). This approach has been exploited by Paddon-Row and and by Broo and Larswn,2’ and is also employed in the present study, thereby allowing the more complicated many-electron splittings and transfer integrals (which at the SCF level are denoted below, respectively, as A,, and TSCF) to be approximated by simpler one-electron quantities. The AKT values reported below for anion and cation diradicals are based on the orbital energy splittings, respectively, of the lowest unoccupied and highest occupied pair of MO’s of the neutral triplet diradical SCF wave function (eq 6) obtained at the unrestricted Hartree-Fock level (UHF).39 Since the effective transfer integral is related to the energy splitting as2
TDA = (E’ - E ) / 2
(8)
we see that the relationship between TDA and the splitting parameter A is given by
TDA
-A/2
(9)
In the following, the symbol Tis understood to refer to TDA, and any subscripts applied to T denote the particular method or level of evaluation, while superscripts will be employed on occasion to distinguish TS and TB or to indicate order of perturbation theory. The calculated transfer integrals will be presented below as -TDA so as to conform to the adopted sign convention according to which positive (negative) quantities correspond to normal (inverted) energy ordering. C. Decompition Schemes. In order to facilitate the decomposition of A (or equivalently, TDA) into TS and TB contributions, we adopt as a convenient local orbital basis set, the natural bond orbitals (NBO) defined by Weinhold et a1.@ The NBO basis has been employed in previous analyses of electronic coupling in terms of local bonding c o n t r i b ~ t i o n s . ~The , ~ ~NBOs are optimally localized, subject to the constraint that they remain orthonormal and are obtained by suitable unitary transformation of canonical SCF MO’s (CMO’s). Unlike most other candidates for local orbitals, the NBO’s allow mixing between occupied and unoccupied CMOS. In using NBO’s as a local basis one must, of course, ascertain the significance of delocalized tails arising from the imposed orthonormality, a topic considered in section 111. Adopting the KT model (eq 7), we seek to decompose AKT additively in terms of the elements of the Fock matrix (F) in the NBO basis. The NBO basis becomes a zeroth-order set of MO’s, and the associated diagonal and off-diagonal Fock elements serve respectively as zeroth-order eigenvalues ((e!’)), and first-order coupling elements. In the NBO context, D and A will denote the donor and acceptor NBOs, which correspond closely to the orbitals depicted in Figure 1, with energy FDD = FAA, where the degeneracy of D and A corresponds to the transition state for resonant charge transfer. A convenient definition of TS coupling is provided by the first-order splitting, (,+)TS E e b0) A
FDA
(10)
The splitting is then
=-~FDA
(1 la)
- A E t / 2 = FDA
(1 lb)
Ai:
and hence -.
@T
G:
TT,s, =
-
TKT
groups. Accordingly +*(an) correspond respectively to states of 2B2and 2AI symmetry, while $*(cat) transform as 2Al and 2B2. In the following, we employ the notation of the C , group (which is either the full point group or a subgroup (for 6 and 8)) for the 10 systems under study. We now consider two different perturbative approaches to the decomposition of A:!. 1. Partitioning Method. Using the partitioning method,3’ we definelBaan effective, energy-dependent two-state Fock matrix (F’). We partition the full NBO orbital space into the D,A pair and the remainder, Ix:},which constitutes the “spacer” set.41 The Fock operator is then diagonalized with respect to (x:),yielding spacer eigenvalues ((e:)) and eigenvectors (I&}). The elements of F’ are then expressable as
+ C ( F J 2 / ( e - e;),
x = D,A
(12a)
F’DA(c)= F ’ d e ) = FDA+ E(FDjFiA)/(e - e;)
(12b)
F:,(e) = F,,
i
i
where the sum is over all 4:. The lowest-order approximation for the effective transfer integral, F’DA,is to set e = cgIA in eq 12b. A higher-order approximation, which may be considered as next step in an iterative procedure,18ais to replace e in eq 12b with the diagonal element F h , obtained from eq 12a by setting e = e#,. This latter approach has the effect of increasing the magnitude of the energy denominators and thus improving convergence, as shown below. While eq 12b provides an additive partitioning of the transfer integral, TKT
N
Tprt
E
F’DA
(13)
the price of the low-order of perturbation theory (i.e., second) is that the spacer contributions are cast in terms of the delocalized spacer M O s 4:. One could, of course, attempt a second-order treatment in terms of localized bond orbitals, but in general this would not be expected to yield a reliable model. We thus turn to higher-order alternatives involving the original N B O s so as to provide more insight into local contributions. 2. NBO-Based Pathway Analysis. As Ratner has recently shown,20 higher-order perturbative expressions for TDA may be straightforwardly generated from an iterative version of the Lippman-Schwinger equatiodOfor the effective transfer integral. We illustrate the result through third-order (i.e., with pathways involving up to two intermediate spacer states:
i#J
where i j refer to the spacer N B O S . ~ ’In eq 14, the more general expression, TDA(c),has been evaluated for e =: tgjA,the energy of the zeroth-order D/A level. Naleway et al.29 have recently implemented this type of approach in an ab initio framework, with applications to radical anion systems based on ethyl and cyclohexyl spacer groups. Equation 14 is a generalization of the McConnell tight-binding modelI6 in which only Fij factors involving nearest-neighbor spacer orbitals are retained. For a spacer with a single bridge consisting of a linear sequence of m orbital units, xi, the McConnell model yields,
- FDA
(1 IC) The sign of FDA clearly depends on the relative phases of D and A. We define these by the convention that D and A are converted point into each other by symmetry operations of the C2, (or D3*) TKT
Liang and Newton
(39) (a) Pople, J. A.; Nesbet, R. K. J. Chem. Phys. 1959,22, 571. (b) The use of Koppmans’ theorem here and in refs 27 and 28 is distinct from that in ref 29. As a result, the relative importance of electron and hole pathways observed in the present study differs from that reported in ref 29. (40) (a) Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem. Reu. 1988, 88, 899 and references therein. (b) Reed, A. E.; Weinhold, F. J . Chem. Phys. 1985, 83, 1736.
If the xi from a homologous sequence, then eq 15 displays an m dependence which is exponential in magnitude and which may alternate in sign (the so-called parity rule).2*10742One of the main findings of Naleway et and also Jordan and Paddon-Row,28e is the great importance of pathways beyond the scope of eq 15, (41) For the a-type D/A cases the “spacer” orbital space includes all the NBO’s associated with the CHI groups other than the D and A orbitals. However, these NBO’s have a negligible role in the TB coupling. (42) Verhoeven, J. W.; Pasman, P. Tetrahedron 1981, 37, 943.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2859
Ab Initio Studies of Electron Transfer
b) 3-21G
a) STO-3G
P
z 0.5
CC*ICH*
0.33
0.32
0.34
n w CC*/CH*
0.28
0.0
-0.5
0.30
1-5
6-10
(K)
(0)
Figure 2. Diagonal elements (in hartrees) of the Fock matrices for the neutral triplet diradical species in the N B O basis. The levels for the D / A pairs of the anion (an) and cation (cat) are based respectively on the minority (8) and majority (a)spin Fock matrices obtained from the U H F calculations.
For the other levels (bonding and antibonding), the a and 6 Fock matrices yield very similar results (for which mean values are given in the figure). The vertical width of the rectangular boxes denotes the range spanned by the indicated set of molecules (1-5 or 6-10). For the 3-21G basis, the Rydberg N B O levels lie above the CC*/CH* N B O levels.
involving coupling between nonadjacent spacer orbitals. D. Gmputaliom and Red&. We have implemented the above models in conjunction with the GAUSSIAN 136 program43and the NBO suite of prognuns.@ The calculations employed the minimal STO-3G basis,32the split-valence 3-21G basi~,4~ and in some cases, the extended 6-3 lG* basis& containing polarization functions. The diagonal energies of the N B O s (tjo) F,,)are depicted in Figure 2. In the perturbative treatments we see that the smallest energy gaps (Le., the separation of the D/A and lowest antibonding levels for radical anions, and of the D/A and highest bonding levels for radical cations) are -0.3 h, except in the case of u-transfer in the cations, where the minimal gaps are $0.2 h. The validity of the perturbation theory underlying eq 14 depends, of course, on the magnitude of ratios of the type, f
With the STO-3Gbasis, the magnitude of the largest coupling elements (F,.) involving orbitals in the valence space (Le., D, A and the bonding and antibonding NBOs) is -0.1 h, with those for the radical anion systematically smaller than those for the radical cations in the case of nearest-neighbor interactions, as discussed in detail in section IVB. At the 3-21G level, the most notable new ingredient is the presence of "Rydberg"-type orbitals, Le., those lying outside of the occupied core space and the "valence space"!' While the gaps involving Rydberg orbitals are large (43) Frisch. M. J.; Binkley, J. S.;Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; DeFrets, D. J.; Seeger, R.; Whiteside, R.A.; Fox, D. J.; Fleuder, E.M.; Pople, J. A. GAUSSIAN86; Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1984. (44)Glendening. E. D.; R e d , A. E.; Carpenter, J. E.; Weinhold, F. QCPE 1990, No.504. (45) Pietro, W. J.; Francl, M.M.;Hehre, W. J.; DeFrees, D. J.; Pople, J. A.; Binkley, J. S . J . Am. Chem. Soc. 1982, 104, 5039. (46) Hariharan. P. C.; Pople, J. A. Theor. Chim. Aero 1973, 28, 213.
TABLE I: Comparison of Calculated Transfer IntegaB (T)" STO-3G 3-21G molecule
1 2 3 4 5 6
I 8 9 10 1 2 3 4 5
-T&
-TKTb
-TSCFc
18.9 16.8 -6.3 4.3 -4.3 92.5 19.2 -15.8 -3.7 9.9
19.3 17.5 -5.7 3.6 -3.8 98.7 21.1 -20.1 -6.8 7.4
(B) Radical Cation 16.3 16.7 20.6 16.7 17.5 18.3 0.3 4.2 2.0 2.6 2.0 2.6 1.4 -0.3 0.2
22.1 20.3 -1.7 2.2 -1.9
-TSCFC
(A) Radical Anion 8.5 9.5 13.8 13.8 0.2 -0.1 2.6 2.3 -0.3 -0.2 95.8 104.4 32.3 32.3 -14.5 -17.2 -6.4 -8.1 3.4 2.3
'In millihartrees. The effective transfer integral (T) is half the negative of the splitting parameter (eq 9). Results are presented as -T to conform to the adopted sign convention (positive (negative) quantities correspond to "normal" ("inverted") energy ordering of +* and &* (see section II)). TKT = - A K T / ~(see eq 7). TSCF = -AscF/~ (see eq 5).
( S l h), the off-diagonal elements Fu involving them can also be substantial (as large as -0.5 h), and thus the convergence behavior of the perturbation series (eq 14) at the 3-21G level is potentially (47) (a) The 'Rydbeg" terminology for extra-valence orbitals has been employed previously.28b, It is adapted here merely as a convenient descriptive label and is not intended to imply any formal spectroscopic designation. (b) The core-level N B O s (Le., 1s type) are generally of minor significance, although they do make modest contributions to T coupling in some of the radical cation systems.
Liang and Newton
2860 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 a1 STO-3G
n-transfer
L
,
....... ~
a-transfer
\
I
I
't .......:
I
5
I
.......,,,..
........
(G)
TABLE II: Through-Space and Through-Bond (p&)Coupling (mhartrees)" STO-3G 3-21G molecule -Ti$ - T;; -7% - T;; (A) Radical Anion 1 -0.7 9.2 0.4 18.5 2 -0.6 0.5 16.3 14.4 3 0.3 -0.I -0.2 -6.0 4 0.1 2.5 0.2 4.1 5 0.2 -0.4 0.0 -4.3 6 81.6 14.2 85.2 1.3 7 49.8 -17.5 53.7 -34.5 -52.0 37.4 -53.2 8 37.5 9 30.4 -36.8 34.2 -37.9 10 22.8 -19.4 29.7 -19.8
(B) Radical Cation 3
q... ........ ,
. / - - *. -\. I
,
e ,
-0.3 -0.6 0.2 0.1 0.1
1
I)
2 3 4 5
-g
..
-
........
a
0.2 -0.2 0.1 0.2 0.2
20.4 18.5 0.2 2.4 -0.5
See footnote a in Table I and eq 1 1.
-
n-trdnsfer
%.
16.6 17.3 4.0 2.5 1.3
a-tran*ier Bonding Aniitmnding
-4 9
-0 & a
.......
=#*b b
u 17 07
-0 x
44
25 8
b1 3-21G
n-transfer
4
n-transier
04
.22.3
-0 7
-23.7
5
Figure 3. Orbital diagrams indicating the most important pathways for electron transfer in radical anion systems, based on (a) STO-3G results, and (b) 3-21G results in the two cases where the most important pathway differs from that found with the STO-3G basis. The orbital lobes are drawn approximately to scale on the basis of the calculated NBOs. The bonds (or antibonds) involved in the pathway are denoted by solid lines. For molecules 2 and 6-9, solid lines are also used to denote pathways symmetry-equivalent to those explicitly depicted by orbitals. Other bonds are either not shown ( C H bonds, except for those on the terminal C H 2 groups of 1-5) or indicated by dashed lines. n 7
more problematical than for the STO-3G level. With either basis, perturbation theory is not expected to be applicable for u transfer in the radical cation systems, and in fact, the D/A coupling in these cases is so large that they are not considered further, lying outside the scope of the present study which seeks to analyze relatively weak interactions. TDAvalues based on splittings calculated directly for the ionic species of interest ( T S C Fand ) from the neutral parent via Koopmans' theorem (TKT) at the STO-3G and 3-21G levels are presented in Table I, while decomposition of TKT into TS and TB terms is displayed in Table 11. The results of perturbation theory are summarized in Table 111, and various aspects of pathway decomposition are dealt with in Figures 3-5 and Tables IV, V, VII, and VIII. Table VI displays the influence of stereoelectronic factors on TKT values. 111. Assessment of Results
In this section we assess the performance of the different ap-
-7.5
Figure 4. Summed C-C and C C * pathway contributions to -TPth for x and u transfer in radical anions from individual bridge units (a or b), including one- and two-bond pathways for two-bond bridges and one-, two-, and three-bond pathways for three-bond bridges (pathways involving more than one bridge are not included, nor are those involving backscattering). The top and bottom entry in each pair of numbers refers to the STO-3G contribution involving, respectively, bonding (CC) and antibonding (CC*) orbitals. Contributions from hybrid pathways involving participation of both C C and CC* NBO's are of very minor significance. When bridge a or b denotes a member of a symmetryequivalent set, the appropriate factor of 2 or 3 is included in the displayed results.
proaches, deferring until the next section a detailed discussion of the pathway decomposition. A. Koopmans' Theorem Approximation. Table I shows that
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2861
Ab Initio Studies of Electron Transfer
TABLE III: Comparison of T , Values with Perturbational Results (mhartrees)" STO-3G
3-2 1G
molecule
9 10
8.5 13.8 0.2 2.6 -0.2 95.8 32.3 -14.5 -6.5 3.4
1 2 3 4 5
16.3 16.7 4.2 2.6 1.4
1 2 3 4 5 6
7 8
9.8 (9.5) 15.3 (15.9) 0.3 (0.2) 2.8 (2.9) 0.2 (0.1) 124.1 (125.6) 39.5 (45.8) -18.8 (-17.3) -7.2 (-7.4) 3.9 (3.9) 20.7 20.1 3.0 3.0 1.5
(29.9) (34.8) (18.3) (4.5) (3.1)
(A) Radical Anion 9.6 (9.6) 15.1 (15.8) 0.0 (0.3) 2.7 (2.8) -0.3 (-0.1) 123.2 (129.7) 44.0 (46.6) -18.7 (-17.6) -7.8 (-7.6) 3.8 (3.8)
18.9 16.8 -6.3 4.3 -4.3 92.5 19.2 -15.8 -3.7 9.9
22.1 19.4 -7.0 4.4 -4.9 103.6 24.5 -24.9 -6.8 11.3
(32.5) (17.2) (-22.6) (10.0) (-20.0) (103.2) (27.3) (18.9) (40.6) (32.4)
29.2 (27.2) 26.1 (28.1) -10.8 (-8.oj 2.2 (7.8) -6.0 (-5.0) 98.2 (110.5) 21.2 (31.8) -43.0 (-15.6) -20.4 (-1.7) 4.8 (11.6)
(B) Radical Cation 17.8 (23.8) 16.8 (23.5) -2.4 (1.7) 1.9 (3.2) -1.0 (1.1)
20.6 18.3 0.3 2.6 -0.3
26.7 23.2 -0.4 3.1 -0.5
(30.0) (28.7) (3.7) (3.8) (0.4)
20.6 (23.2) 18.0 (20.3) -5.2 (-3.8) 1.9 (3.0) -2.7 (-1.6)
'See footnotes a and b of Table I. bResults of partitioning theory (eq 13). The results in parentheses employ the zeroth-order donor/acceptor energies ( E t ) A = FDD = FAA)in the energy denominators, while the other results include a second-order correction. CFromthe pathway summation method (eq i4)Tncluding contributions through fourth (fifth, in parentheses) order; Le., pathways with up to three (four) intermediate bridge states; -fiL{ is the total ith-order contribution, as described by eq 14 (e.g., -?$,) = -FDA). TABLE I V Contributions (mbartrees) to Tptbfrom Second-, Tbird-, and Fourth-Order Terms (STO-3G Results)" one bond two bond molecule -7(;ix(-7$i,)) description -7!& (-fi:)) description -fi% (A) Radical Anion CIC2*; C2CI,* ( M ) b -0.45 (-1.67) C&2* -5.73 (-1.60) 1 3.35 (13.66) ClC2*; C2C1,* ( M ) b -0.19 (0.11) -2.27 (4.51) CIC2* 2 4.72(11.16) 3 -2.01 (-4.41) C2C2!* 2.37(3.81) C,C2*;CyCI,* -0.10 (0.35) 4 -0.51 (-0.72) C2C2'* 0.86(3.29) ClH*; CIIH* e (0.07) CIC2*; CyCIc* -0.08 (0.58) 5 -2.29 (-3.08) C2C2'* 2.33 (2.09) 0.69 (1.76) C,C,*; C2C1, (M)* C1C2* -10.52 (-32.46) 6 13.34 (72.21) C,Cj*; C3,Clt* 0.18 (-3.98) CIC2* 3.05 (17.26) 7 13.96 (-19.08) 8 -16.67 (-72.90) C2C2t* 1.94 (15.37) ClC2*; C2rCI,* -0.1 1 (1.25) 9 -16.74 (-46.60) C2C2,* 1.49 (6.97) C,C2*; C2CI,* -0.11 (1.37) 10 -17.63 (-22.08) C2C2,* 1.19 (2.31) CIC2*; C2CI** -0.1 1 (0.79)
c-m
1 2
3 4 5
1.91 (9.67) 2.48 (8.30) -0.79 (-2.99) -0.18 (-0.38) -0.82 (-2.03)
ClC2 CIC2
c2c2, C2C2'
c2c2t* '
(B) Radical Cation* ClC2;C2Clt(M)b 6.44 (7.34) C1C2;C2Clr ( M ) b 5.54 (7.85) c,c2; C2'C1' 1.66(-0.33) 0.84 (1.30) CIH; C,PH 1.20 (-0.37) c,c2;C 2 E , ?
-1.54 1.11 -1.59 -0.40
(1.08) (1.24) (0.76) (0.96) -1.27 (1.25)
three bond description CIC3*; CjCI,*; C2CI,*' C1C2*; C&*; C2CI,*' CIC2; C2C,; C2C1, (M)b*d CIC2; C2CZt; C2rC1, ClC2*; C,C,,*; ClC2*f ClC3*; CIC3n'; CyCIr*c C2C2.;C2H; C2C2,"" C2C2.*; C,C,*; C2C,,*/ C2C2,; C2H; C2Cyd'8
(M)bld
CIC2; C2H; C2Clr C,C,; C2Cli C3C,,c CIC2; C2C2,; C2C1, (M)' C1C2;C2C2,; C2ClI (M)b ClC2;C2C20; C2,CIr( M ) b
4Corresponding, respectively, to pathways involving one, two, and three intermediate states of the spacer group (see eq 14). For each order (n = 2, 3, 4) the pathway yielding the contribution of largest magnitude (fizjx)is indicated, and the total nth-order contribution (fi,$)is given in parentheses. The individual pathway yielding the largest fi!ixvalue (n 5 4) is indicated in boldface italics. The indicated pathway may be a member of a symmetry-equivalent set. Bonds and antibonds are denoted respectively by CC, C H and CC*, CH*, and the atoms are identified by subscripts based on the numbering system given by Figure 1. See also footnotes a and b in Table I. bNearest-neighbor pathways of the McConnell type. cPathways involving more than one bridge of the spacer. "Hole contribution (Le., involvement of occupied bridge orbitals) to coupling in a radical anion. elfi$xl < 0.05 mhartree. 'Involves a backward hop. 8Involves a given bond (or antibond) more than once. *The path analysis for the radical cation should be considered of limited significance since the perturbation expansion seems to diverge gradually for orders of up to -15. 'Electron contribution (Le., involvement of unoccupied bridge orbitals) to coupling in a radical cation.
a
h
Figure 5. Schematic orbital diagram (roughly according to scale, based on STO-3Gresults), depicting the nearest-neighbor interaction of a pair of (a) bonding (CC) and (b) antibonding (CC*) NBO's.
of & to - 3 mhartrees are observed).-The signs of'TfpF a'nd TvT are ihe same except for two cases fo; structure 3, wh&yrelativai magnitudes are involved. For 6-10, a nearly perfect linear relationship ($ = 1.00) exists between TsCFand TKv We conclude
that Koopmans' approximation offers a useful (but not uniformly reliable) model for analyzing the coupling in the ions of l-10.48 B. Comparison of STO-3G and 3-21C Results. For the cases of u transfer in the radical anions (&lo), the two basis sets give generally similar results, and the differences are systematic, as reflected by the good linear relationship which exists between the two sets both for TsCFand TKT (r2 = 0.98). For ?r transfer in both radical anion and cation systems, the results from the two accord, whereas disbasis sets for 2 and 4 are in fairly god crepancies of sign as well as significant deviations in magnitude
(48) The symmetry-restricted results for TsCFgiven in Table I are generally within 30% of values obtained from charge-localizedbroken-symmetry calcu~ations.*~2~
2862 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 TABLE V: Variation in Magnitude of Off-Diagonal NBO Fock Matrix Elements for Systems with Butane-Based Spacer (in hartrees)“ H transfer u transfer orbital oairb transC cisd transC cis‘ Nearest Neighbor D;CiC2 0.089 0.089 0.002 0.003 D;ClC2* 0.071 0.070 0.061 0.057 C,C2; C2C2, 0.707 (0.106) 0.101 (0,100) 0.103 0.098 C,C2*; C2C2,* 0.001 (0.001) 0.013 (0.013) 0.008 0.020 ~~~~
~~~~
D;C2Cy D;C*Cy* C,C2; C2CI, C,C2*; C2Cl,*
~
~
~
Next-Nearest Neighbor 0.020 0.021 0.027 0.027 0.037 (0.036) 0.029 (0.032) 0.042 (0.039) 0.057 (0.055)
0.087 0.073 0.036 0.041
0.085 0.077 0.028 0.053
“Based on STO-3Gresults. The 3-21G basis yields similar results. The bonding and antibonding orbital notation is the same as that employed in Table IV, and D denotes the local donor orbital. The quantities listed pertain to the radical anion systems and are taken from the minority ( p ) spin Fock matrix, except for those in parentheses, which refer to the corresponding radical cation systems and the majority (a) spin Fock matrices. It is clear that for coupling within the spacer, the magnitudes are essentially independent of radical type. The numbering system for the butane bridge is as given in Figure 1 for species 5 and 10. ‘Based on trans variant of 5. dSpecies 5. ‘Based on trans variant of 10. /Species 10.
the STO-3G results reproduce many trends successfully and are of interest in model studies of pathway analysis as discussed in section IV, the results from the more flexible 3-21G basis may be considered as providing more reliable values in cases where discrepancies do occur (see also related comments about basis set sensitivity in ref 28d). In support of this assertion we note that for a limited comparison sample (1,5,6, and lo), the 3-21G TsCF values are in good quantitative accord with values obtained with the more flexible 6-3 lG* basis. The following linear correlation is found: TscF(6-31G*) = -0.630 0.944TscF(3-21G) (17)
+
with regression coefficient = 1.00. As Jordan et al. have emp h a ~ i z e dwhen , ~ ~ the anion states are unbound, as is the case in model studies of the present type,50the use of too flexible a basis set can yield undesired artifactual continuum states. The close agreement between 3-21G and 6-31G* results suggests that this phenomenon is probably not a significant factor in the present calculations, since otherwise a greater sensitivity to increased flexibility of the basis would be expected. C. Degree of Localization of NBOs. A pathway analysis (eq 14) is employed below to give insight into TB coupling on the basis of localized orbitals represented by NBO’s. Thus it is important to assess the precise degree of localization of the NBO’s. As a convenient measure, we evaluate the overlap, SNB0, of an NBO with its “ideally localized” counterpart, obtained simply by zeroing all A 0 coefficients which do not belong to the principal one or two atoms of the NBO and then renormalizing. In terms of the percent deviation of SNmfrom unity, ASNm, we find mean values in the range 0.45480% for species 1-10 when an STO-3G basis is employed. A maximum value of 1.9% was found for the CICz NBO in 1. A somewhat smaller degree of localization is found for bonding and antibonding N B O s in the 3-21G basis (averages in the range 1.6-3.0%, with a maximum value of 8.4% for the CIC, bond in l), and the Rydberg ASNBo values are generally >20%, with a maximum of -75%. Thus a local pathway analysis using (49) (a) Falcetta, M. F.; Jordan, K. D. J . Phys. Chem. 1990, 94, 5666-5669. (b) Chao, J. S.-Y.;Falcetta, M. F.; Jordan, K. D. J . Chem. Phys. 1990, 93, 1125. (c) Falcetta, M. F.; Jordan, K. D. J . Am. Chem. SOC.1991, 113, 2903-2909. (50) The discussion of electron attachment in ref 49 refers to intrinsically
unbound states ( i t . , resonances). The radicals here are unbound but are employed as models for condensed-phase radical anions which do have bound electronic states. In cases where the model unbound anions were to lead to serious artifactual behavior, one could employ a reaction field to help stabilize the excess electron.
Liang and Newton NBO’s at the 3-21G level is possible only to the extent that the Rydberg orbitals are not a major factor in the coupling. While the valence level N B O s are found to be quite localized with both basis sets, it is still possible for their small delocalized tails to have significant energetic consequences, as discussed below in connection with TS c o ~ p l i n g . ~ ’ D. Performanceof Perturbathe Methods. 1. STO-3G Results. The second-order partitioning approach (eqs 12 and 13), which is based on spacer-group eigenfunctions, is seen (Table 111) to be generally reliable provided that the second-order correction is included in the energy denominators. The correction is especially important for A transfer in the radical cations (1-5), where it has the effect of reducing the upper limit of perturbation ratio (R;) magnitudes from -1.0 to the range of 0.3-0.5. For the radical anions, lRsjl 5 0.3 for 7 transfer (1-S), and accordingly, good agreement between TKT and T values is observed. For the corresponding cases of u trans!: (&lo), lRbl ;5 0.4, and the agreement is somewhat less good. Here, Rb, is defined analogously to R , in eq 16 with i and j referring to orbital set D, A, and {$:). Turning to the NBO representation in conjunction with eq 14, we consider the results (T,,,,) through fourth and fifth order (Table 111). These orders correspond, respectively, to pathways involving three and four intermediate spacer states and are high enough to provide useful indications about convergence and also to provide perspective on the role of tight-binding pathways of the McConnell type. For molecules 1-10 such pathways yield third (2-bond)- and fourth (three-bond)-order pathways. Note that in this classification we do not include the single bonds connecting the D/A methylene groups to the spacers in 1-5 since the Fij elements coupling these bonds to D and A are of negligible magnitude. For A transfer in the radical cations (1-5), lRiil 5 0.5 (see eq 16), and higher-order calculations (not shown) reveal slow divergence of the pathway summation (eq 14). For the corresponding radical anions (H), the convergence is satisfactory (IRiil 5 0.3). While the fourth- and fifth-order results for u transfer in the radical anions of 6-10 appear to represent near convergence of the expansion in eq 14 (confirmed by higher-order calculations), it is clear that the Tpathvalues still differ significantly from TKT values (especially for 6 and 7). It must be emphasized that eq 14, by virtue of the approximate evaluation of the energy-dependent denominators (i.e., c = eg)),) and the omission of additional terms involving return “visits” to D and multiple “visits” to A, et^.,^* is not required to converge to TKT.These factors may help to explain the observed convergence behavior. 2. 3-21GResults. The agreement of Tprtand T@,values with TKT for 7 transfer in the radical cations and u transfer in the radical anions is comparable to that exhibited by the STO results. However, the results for A transfer in the radical anions (1-5) are less satisfactory than the STO-3G results, a situation which may reflect the smaller degree of localization in the valence NBOs in comparison with the STO-3G case (see section 1II.C) and the presence of Rydberg orbitals. The sensitivity of Tpthto Rydberg contributions is explored in Appendix A (Table VII), where comparison is made with results in which the Rydberg orbitals in the NBO basis are omitted entirely or are allowed to influence the valence-level NBO Fock matrix perturbatively through a generalized implementation of the partitioning method.53 The Rydberg orbitals clearly make significant (though not dominant) (5 1) (a) Since NBO’s are orthogonal, some of the delocalization must come from this orthogonalization (we term this orthogonalization delocalization). Such “orthogonalization delocalization” may be estimated by calculating AS” for the symmetrically-orthogonalizedideally-localized NBOs described above. For the STO-3G basis we obtain averages of 0.24-0.43%. In other words, by this measure, the orthogonalization delocalization accounts for about 50% of the total delocalization of NBOs. However, a similar analysis of 3-21G results shows that ASNBovalues based on the symmetrically-orthogonalized orbitals may be as large or larger than those based on the original NBO’s. (b) Lowdin, P.-0. J . Chem. Phys. 1950, 18, 365. (52) Messiah, A. Quonium Mechanics; John Wiley: New York, 1962; Chapter XVI. (53) Alternatively, Paddon-Row et al.**bhave attempted to include the influence of the Rydberg NBOs on the valence NBO space by diagonalizing selected blocks of the NBO Fork matrix.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2863
Ab Initio Studies of Electron Transfer TABLE VI: Conformational Dependence of
Tn for I Transfer (3-21G Results in mhartrees)’
conformation D,A/spacer linkage‘ 4 xcos2 e,
spiesb 4 (chair) 4 (boat)
30, 150 30, 150 0, 120, 120
3f 5 (trans) 5 (gauche) 5 (cis) 4 (chair) 4 (boat) 5 (trans) 5 (gauche) 5 (cis)
-TKT
spacer bridgesd gauche, gauche cis, cis cis, cis, cis trans gauche cis gauche, gauche cis, cis trans gauche cis
1.5 1.5 1.5
1.O 1.o 1.o
0 0 0 120, 120 120, 120
0.5 0.5
0.0 0.0
90 90 90
0.0
conformational energy
anion
cation
-11.5 -9.1 -6.3
-3.9 -0.7 +0.3
-8.4 -4.5 -4.3
-11.0 -2.4 -0.3
0.0
-1.7 +4.3
-0.6 +2.6
0.7 13.4
+0.3 +2.0 +9.9
-1.1 +3.6 +7.9
0.4 2.1 13.4
0.0 12.2
1.7 9.6
“See footnote u in
Table I. bThe sample includes structures 3, 4 (boat conformer),and 5 (cis conformer) given in Figure 1, together with the chair conformationalvariant of 4 and the gauche and trans variants of 5. For each framework structure of type 4 or 5 (or their variants) two different conformations of the terminal CH2 groups are considered (see footnote c). CTheangle Bi is the dihedral angle between the plane perpendicular to the donor or acceptor CH2 group and the plane containing the CH2 carbon atom and the closest CC bond of the ith bridge of the spacer (Le., C,C2or CIC3for the donor group and C,C2,or CIE3,for the acceptor group). Thus cos Bi is a convenient measure of the relative degree of overlap of the donor or acceptor w orbital with the carbon framework of the ith bridge unit of the spacer group. Since all structures employed the same 8, value for the donor and acceptor group, the quantity xcos2 e,, summed over all the bridge units, gives an approximate index of the expected D/A splitting for each structure. The idealized 8, values listed here, based on tetrahedral spacer carbon atoms, are very close to the actual values corresponding to the optimized spacer geometries. dThe dihedral angle of each C4bridge unit in the various structures. For those species based on a common spacer type (cyclohexane or butane) the relative conformational energies (3-21G) are given in mhartrees. /The coupling and conformational energy of 3 is essentially independent of 8,. TABLE VII: Sensitivity of -Tp* (mhartrees)to Rydberg Orbitals (3-216 BasirP*b
involvement of Rydberg orbitals via partitioning fully included deleted method through order through order through order 4
5
4
5
4
5 ~
(A) Radical Anion 1 2 3 4 5 6
1 8 9 10
21.4 24.4 -4.0 2.8 -3.6 92.9 28.6 -25.0 -8.5 7.4
21.3 26.0 -2.3 3.8 -3.2 120.5 32.8 -22.8 -8.1 7.4
20.2 18.1 -4.7 2.4 -2.3
23.0 20.6 -3.6 3.1 -1.4
26.1 19.6 -11.0 3.6 -6.4 86.2 20.7 -26.7 -6.3 11.9
26.2 22.0 -8.8 9.4 -5.8 120.6 26.6 -22.3 -4.2 12.8
29.3 26.1 -10.8 2.1 -6.0 98.2 21.1 -43.0 -20.4 4.8
27.2 28.1 -8.0 7.8 -5.0 110.5 31.8 -15.6 -1.7 11.6
20.6 18.0 -5.2 1.9 -2.7
23.2 20.3 -3.8 3.0 -1.6
(B) Radical Cation 1 2
3 4 5
20.5 17.7 -5.0 2.2 -2.6
23.4 20.0 -4.4 2.1 -1.8
“Rydberg denotes the diffuse NBOs which lie outside of the NBO core and valence space.47abEntries give contribution to -Tpaththrough nth order (where n - 1 equals the number of intermediate bridge state (see eq 14)). See footnote c of Table 111. Results are presented as -Tpth to conform to the adopted sign convention (positive (negative) and quantities correspond to normal (inverted) energy ordering of J.’ b* (see Section 111)). contributions to the overall TPthvalues. In summary, we conclude that the advantages of the increased flexibility of the 3-21G basis in comparison with the minimal STO-3G basis are somewhat offset by the disadvantage of being less amenable to a compact perturbational decomposition. IV. Discussion A. Pathway Analysis of TKp Defining a pathway as a sequence of local states corresponding to a single term in the summation in eq 14, we consider how various types of pathways contribute additively to overall T K T values. 1. TS vs TB Terms. Table I1 displays Ti; and Ti: values, obtained as defined in eqs 1l b and 1IC. For the ?r-transfer cases
(1-5), Ti: magnitudes are of minor si nificance relative to those for Ti:. The negative values of - T K!TT at the STO-3G level for 1 and 2 (both for the radical anions and cations) apparently reflect the consequences of delocalized NBO tails (see section 1II.C and ref 28c). In fact, when -Ti: is evaluated by an alternative route using a (CH,), dimer, with the CHI units having the same relative geometry as the terminal CH,’s in 1 and 2, positive values are obtained (in the range 0.3-0.5 mhartree). Two additional cases of negative -Ti:, but with very small magnitude, are found at the 3-21G level. As expected from the shorter D/A separations (see Figure l), considerably larger Ti: values are found for 6-10 in comparison with those for 1-5, constituting the dominant coupling pathway for 6 and 7, and exhibiting substantial “destructive” interference with the TB contributions in 8-10. Inspection of the total -Ti! contributions (Table 11) reveals clear trends in some cases (positive values for both K and u transfer through spacers with exclusively two-bond bridges (1 and 6),and negative values for u transfer through spacers with exclusively three-bond bridges (8-10)). However, patterns in other cases are more complex, indicating the importance of competitive interference among different pathways. Accordingly, we consider in more detail the role of these pathways. 2. TB Pathways. The most important contributions to Ti! in systems 1-10 involve pathways containing one, two, and three intermediate spacer states. The total contributions of each type (fi:)),along with the largest individual contributions ( are presented in Table IV a t the STO-3G level. The corresponding 3-21G results are presented in Appendix B (Table VIII). The most important individual pathway for each radical anion state (1-10) is represented as a schematic orbital diagram in Figure 3. In all but two cases (i.e., 4 and 5 see Figure 3b), these pathways are the same at the STO-3G and 3-21G levels. In the following detailed pathway analysis, we focus primarily on the radical anion systems at the STO-3G level, where reasonable convergence was achieved in most cases and where the complications of diffuse extravalence Rydberg-type orbitals are absent. However, the results for the radical cations at the STO-3G level and the 3-21G results (Table VI11 and Figure 3b) are also of interest in providing various points of comparison. The most important individual pathways can be characterized as follows. They generally involve CC or CC* NBO’s, although CH and CH* NBO’s also have significant roles (e.g., the several examples indicated in Tables IV and VI11 and in Figure 3), as
2864 The Journal of Physical Chemistry, Vo1. 96, No. 7, 1992
Liang and Newton
a homologous sequence of bonding or antibonding states, this is a manifestation of the so-called parity r ~ l e , * J whereby ~ 9 ~ ~ the sign of -T alternates with increasing number of states and is expected to be negative and positive, respectively, for odd and even numbers. The McConnell-type pathways listed in Tables IV and VIII, which involve bonding C-C orbitals, are seen to conform with the above expectations. Furthermore, the radical cation of system 1 offers an example in which a two-bond tight-binding pathway interferes destructively with the related three-bond pathway involving an excursion onto a C H bond. For some of the cases involving antibonding C-C orbitals (Le., for the two-bond pathways in 1, 2, and 6), the sign of the contribution to -T,,, is seen to be “anomalous”. The reason for this may be found by considering the details of the signs of the underlying Fij elements, a topic to which we return in section IV.B, in connection with the discussion of transferability relationships. In some cases, interference patterns are straightforwardly determined by symmetry, as in the cases of pathways involving symmetry-equivalent (or nearly equivalent) spacer bridges (as in 1-4 and 6-9). The interference is, of course, constructive in the case of u transfer (in view of isotropy of D and A with respect to the D/A axis). The consequences of axial anisotropy when D and A are ?r orbitals is dealt with below, where stereoelectronic factors are considered. 3. Coahibutiom Based on Spacer Bridge Type. In general, the data of Tables IV and VI11 reveal a complex pattern of interference effects, both for the individual (-TmaX)and the summed (-TtOt)contributions, and simple comprehensive rationalizations do not seem possible. Here we consider some alternative partial sums. Since collectively the greatest contributions to -TKTcome from pathways involving C C and CC* NBO’s which are confined to a single spacer bridge, it is of interest to consider the cumulative contributions of this type for each symmetry-unique bridge. The results for the radical anions at the STO-3G level are summarized for 1-10 in Figure 4, indicating the distinct role of the bonding (CC) and antibonding (CC*) manifolds. Even though the largest individual pathway contributions conform to the expected pattern of electron transfer in the radical anions and hole transfer in the radical cations (Table IV), we see in Figure 4 that both (electron and hole) mechanisms contribute significantly to the radical anion coupling, and in the case of 1 and 3, the hole contribution exceeds that from the electron mechanism. In several cases the hole and electron terms within a given bridge interfere destructively. The 11 two-bond and three-bond bridges for the a-transfer system 7 contribute comparable magnitudes and interfere destructively, while in the analogous ?r system (2), the magnitude of the twobond coupling is substantially larger than that from the three-bond bridges, and both bridge types conform to the “normal” orbital ordering (Le., -TDA> 0). The bridges of a given length which 12 are rendered distinct because of anisotropy of the *-type D and A orbitals (as in 1 and 3) contribute ’in phase”, with magnitudes It is to be emphasized that the preference for non-tight-binding roughly consistent with the relative orientations of the bridges and pathways is characteristic of NBO’s based on the relatively the D/A orbitals defined with respect to the D/A axis (see section compact STO-3G basis as well as those obtained from the less 1V.C). localized 3-21G NBOs. The departures from the tight-binding The bridge contributions from C C and CC* orbitals (Figure limit do not preclude overall exponential decay of T magnitudes 4)account reasonably well for the overall coupling: Le., their sums with increasing number of homologous units in a spacer but do (over all bridge types) are within -25% or 1 mhartree of the help to explain the occurrence of deviations from such behavTpath values, with the correct sign in all cases. i0r.2*28c*f*55 This topic will be pursued in a future paper which B. Transferability. It is expected that a pathway associated considers alternatives to NBO’s as the “units” for defining variation with a particular sequence of spacer orbitals will give comparable of I7l with increasing “size” of the spacer.55 contributions to D/A coupling in different molecular systems, Simple orbital models indicate that pathways differing by an provided that the relevant spacer units in these systems have odd and an even number of spacer states will interfere, respectively, common structural features. Examples of such approximate in a destructive and constructive f a ~ h i o n . ~ ~In”the ~ ’case ~ ~ of ~ ~ ~ transferability relations may be found in the data of Table IV; e.g., the one-bond pathways for A transfer in the radical anions of 3 and 5 (in the three-bond bridge a), and for u transfer in the (54) Hybrid pathways involving creation or annihilation of electron/hole radical anions of 6 and 7 (in the two-bond bridge a) and also t h a e pairs are expected to be important in cases where D/A levels are near the middle of the gap created by the occupied and empty bridge manifold^,^,'^^ of 8-10 (in the three-bond bridges). In addition, we find that in contrast to the situation depicted in Figure 2. next-nearest-neighbor two-bond pathways yield similar contri( 5 5 ) Liang, C.; Newton, M. D., to be published. butions ( 1-2 mhartree) in a wide range of situations: Le., for ( 5 6 ) For system 12, the central bond can be seen on the basis of symmetry arguments not to contribute to TB coupling. both A (3-5) and u (8-10) transfer in both radical anions and do Rydberg NBO’s at the 3-21G level. For the most part, the indicated pathways for the radical anions and cations correspond, respectively, to electron and hole transfer (Le., involving empty antibonding and filled bonding NBO’s), although a number of significant exceptions occur. Hybrid pathways with both electron and hole components are of minor importance, with Tpalhmagnitudes l$i!l, and in most cases, 1fi;:l > 1fi$1. The especially important role of the central onebond pathways in establishing the inverted order for u coupling in the radical anions of 8-10 is consistent with the earlier analysis of Hoffmann et alaSa The role of non-nearest-neighbor pathways is even more pronounced for larger spacer groups. Thus considering, for example, extensions of 5 and 1, we find that the maximum values of l$:jl for the radical anions occur at n = 3 for both 11 and 12, in contrast to the tight-binding limits, respectively, of n = 7 and n =
Che
$:
+f---/Jw 5 . 5 5 3 5 6
-
N
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 2865
Ab Initio Studies of Electron Transfer
TABLE VIII: Contributions (mhrrtrees) to Tph from Second-, Third-, and Fourth-Order Terms (3-216 Results)'
one bond m o1ecu1e
-.r%(-C3
1 2 3 4 5 6
6.92 (24.41) 9.10(18.07) -5.78 (-10.01) -1.29 (-1.87) -5.67(-7.31) 4.48(31.02) -11.78 (-20.69) -16.19 (-70.40) -18.45 (-50.05) -20.69(-25.69)
7 8 9 10
2.30 (13.05) 3.15 (10.64) -1.75(-5.56) -0.40 (-0.92) -1 39 (-3.96)
description CIC2* CIC2*
C2Czt*
c2c2,* C2C2?* CIC2* C2C2'*
c2c2,* c2c2.* C2C2'*
CIC2 CIC2*C
c2c2,*
C2C2'*
c2c2.*
two bond description (A) Radical Anion
three bond
-.r&c-fia,
-8.88(-3.12)' -2.72 (3.69) 6.27(6.52) 1.37(5.73) 4.32 (3.56) 4.40 (-10.45) 2.44 (1.27) 3.60 (20.27) 5.46 (12.32) 5.19 (5.80)
-4.17 (7.59) -2.92 (3.84) -0.86 (-7.09) 0.41 (-1.93) 0.77(-2.30) -0.86 (-7.58) -0.95 (-13.15) -1.54 (-30.27) -1.49 (-16.93) -2.69 (-5.04)
(B) Radical Cation 3.75(3.52) C2C2;C2CIt(M)b 3.26(4.21) C2C2; CzCI, (MIb -0.94 (0.91) C2C2; C2Cy 0.71 (2.37) CIH; CItH -0.81 (0.87) C$,; C2J21,
-0.90 (3.88) -1.04 (3.29) -0.95 (-0.64) -0.24 (0.16) -0.77 (0.18)
Osee footnote a of Table IV. In addition to valence orbital types arising in Table IV, the 3-21G basis includes diffuse unoccupied orbitals designated as Rydbcrg (Ry) orbitals4'" and identified by the atom to which they belong: D,A denote D/A methylene carbon atoms, and the hydrogen atom (H,) is given the label of its bonded carbon partner. bSee footnote b of Table IV. cSee footnote i of Table IV. "See footnotefof Table IV. CSeefootnote d of Table IV. /See footnote c of Table IV. gSee footnote g of Table IV. cations, and including cases involving both CC (or CC*) bonds in three-bond bridges and analogously disposed pairs of bridgehead CH (or CH*) bonds (in 4). Regularities in contributions to D/A coupling in turn reflect the corresponding properties of the underlying Fij elements and the NBOs on which they depend. Thus we proceed to examine the magnitudes of some representative FiP Results for nearestneighbor and next-nearest-neighbor interactions are displayed in Table V for four different butane-based spacers allowing comparison of different D/A orbital types ( A and u) and different spacer conformation (cis and trans). We focus on the magnitudes of the Fij elements (their signs can be understood in terms of the relative phases of the NBO lobes). The Fy magnitudes indicate a strong degree of transferability of NBO s among the four different systems. In each of the eight cases displayed (i.e., rows), trans and cis conformers for a given transfer type ( A or a) in general yield very similar magnitudes. Interactions including the donor orbital yield, as expected, results which depend on orbital type ( A or u), but pairs within the spacer give similar results for all four cases considered ( A and u, trans and cis). Perhaps most interesting is the roughly order-of-magnitude difference between Fij magnitudes for the bonding and antibonding nearest-neighbor orbital pairs, and in fact the next-nearest-neighbor coupling between CC* NBO's is stronger than for nearest-neighbors (the STO-3G results listed in Table V are quite similar to those obtained with the more flexible 3-21Gbasis). Thus in nearestneighbor pathways (as in the McConnell model)I6 the hole mechanism may be expected to give more effective coupling than the corresponding electron mechanism. Similar results have been found for radical ions involving polynorbornyl spacersazse To understand qualitatively the source of this effect, we consider the orbital diagram in Figure 5. Since the NBOs correspond to sp" hybrids (where n 1 for the CC and 1.5 for the CC* case), we concentrate on the four major lobes of the interacting NBOs and recognize that the overall coupling involves a superposition of contribution from four lobe pairs. In the case of the bonding NBO's (a), the superposition is a constructive one, whereas for the antibonding NBOs (b), destructive interference is unavoidable between the lobe pairs distinguished by solid and dashed lines. Apparently for CCC angles near tetrahedral (as employed for the butane spacers) the competing terms are closely balanced, resulting in the small Fij values. For smaller CCC angles, as in the case of the strained two-bond bridges in 1,2,6, and 7, the dashed-line interactions become relatively more effective and dominate the other terms. Thus the resulting Fij magnitudes are somewhat larger than those in Table V (-0.05-0.08 hartree), but still appreciably smaller than the corresponding quantities for the
-
-
bonding NBO pairs (0.13-0.1 5 hartree). Furthermore, the signs are consistent with the "anomalous" signs of the -T values for the two-bond TB pathways discussed in section IV.A.2 (Le., for coupling of nearest-neighbor CC* NBO's, the sign is negatiue when the relative phases are as given in Figure 5). We also note that for u transfer the Fij elements coupling D with next-nearestneighbor spacer orbitals are larger in magnitude than for the corresponding nearest-neighbor coupling. C. Stereoelectronic Features. The strength of D/A coupling is sensitive to a number of stereoelectronic factors, including the orientation of the D/A orbitals with respect to the spacer and the conformation of the bridges within the ~ p a ~ e r . ~ v Ac~~J~*~~*~ cordingly, we supplement the foregoing analysis of coupling in 1-10 by considering the conformational dependence of coupling strength in cases of A transfer in radical anions with spacers possessing three-bond bridges; i.e., we compare coupling in 3 with that in 4 and 5 and their conformational variants, as displayed in Table VI. Table VI also includes the relative energies of the different conformers. As a single approximate index of the coupling of the D/A orbitals to the spacer bridges we employ Ccos20i, where Oi is the angle between the D (or A) orbital and the local plane formed by the carbon atoms of the D (or A) CH2 group and the first (or last) CC bond of the ith bridge (see footnote c of Table VI). In all cases D and A share a common Oi value. The magnitude of coupling involving spacer CC and CC* bridge bonds is expected to increase with increasing Ems2Oi. Conversely, in the case of a single bridge (i.e., 5), decreasing Ccos2 Oi is expected to favor hyperconjugative coupling of D and A to the spacer CH bonds. Several pronounced trends emerge from Table VI. The -TKT values are seen to increase algebraically both with variation in spacer bridge conformation (trans gauche cis) for fixed Ccos2 8 , and with decrease in Ccos2 Bi, for fixed spacer bridge conformation. For the simple case of a single alkyl bridge, comparable magnitudes (- 10 mhartrees) are obtained in two limiting situations for both radical anion and cation systems: "inverted" coupling (-TKT < 0) for D and A overlapping optimally (cos2 Oi = 1.0) with the carbon skeleton of a trans bridge (a manifestation of the "trans r ~ l e " ) , ~ ~and - l ~"normal" , ~ l ~ coupling (-TKT > 0) for D and A orthogonal to the carbon framework of a cis alkyl bridge (cos2 Bi = O).57 The most important pathway in the hyperconjugative coupling in the latter case is the two-bond route involving the terminal CH2 groups of the bridge (Le., those separated by
- -
(57) A similar pattern for radical cations involving the butane spacer has been obtained by Broo and Larsson2' using the NH2 donor groug, where their sp3 and sp2 configurations are defined, respectively, by 8, = 0' and Oi = 90'.
2866
J. Phys. Chem. 1992, 96, 2866-2868
three CC bonds), a non-nearest-neighbor interaction easily seen to be considerably more effective in the cis conformation than in the gauche or trans conformation. This type of through-space interaction has previously been labeled laticyclic hyperconjugation by Paddon-Row and Jordan in the context of diene D/A systems.! la.28e The relative strength of ?r coupling afforded by the bicyclooctyl (3) and other alkyl spacer groups has been the subject of conFrom Table VI we flicting reports in the see that the coupling in 3 is weaker (Le., the magnitude of TKT is smaller) than in the cyclohexyl system in its lowest-energy conformation (chair, with xcos2 Bi = 1.5) and similarly in the single alkyl chain (5), for both anion and cation cases. However, the coupling via the cyclohexyl and alkyl spacers is very sensitive to conformation, and the entire range of -TKT values for different conformations is seen to bracket the bicyclooctyl values.59 Finally, we consider the bearing of stereochemistry on the relative coupling strength in radical anion and cation systems. For the lowest energy single-chain spacer (trans, with cos2 B = l), somewhat greater magnitude of coupling is found for the cation, whereas the order switches for the case of spacer bridges with gauche and cis conformations, both cyclic (cyclohexane and bicyclooctane) and acyclic (butane), provided that the D/A groups maintain the lowest energy conformation (Le., Ccos2 Bi = 1.O or 1.5). In kinetic studies using spacers based on the chair cyclohexyl no difference in falloff of coupling with D/A separation could be detected in comparisons of anion vs the corresponding cation systems. However, in theoretical studies based on dienes coupled by norbomyl spacers, falloff for radical anions was found to be faster than for corresponding radical cations.28e,f
and through-bond (Tz:), using perturbation theory as formulated by RatnerZotogether with a localized orbital basis represented by natural bond orbitals (NBO’s) as defined by Weinhold et a1.40 The overall coupling has been shown to arise from interference among a large number of competing pathways, none of which is strongly dominant. Nearest-neighbor pathways of the McConne11I6 type are significant for transfer in some radical cation systems, but are frequently of very minor significance in comparison with lower-order superexchange pathways, often with contributions differing in sign from the overall TKT values. These latter conclusions are generally consistent with those based on studies involving different saturated spacer groups by Naleway et alaz9(for radical anions) and Jordan and Paddon-Row28e(for radical anions and cations). We find that transfer in radical anions and cations is generally dominated respectively by electron and hole pathways, but both mechanisms are found to be significant in both types of transfer. A number of transferability relationships have been identified for generic pathway types, and the important influence of stereochemistry on coupling has been illustrated, with regard to both orientation of donor/acceptor groups relative to the spacer and internal conformation of the spacer, showing the competition between coupling via the carbon framework and via C H bonds (i .e., hyperconjugation).
Acknowledgment. This research was carried out at Brookhaven National Laboratory under contract DE-AC02-76CH00016 with the US. Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences. We thank Dr. John Miller for communicating some of his recent results to us during the course of the present research, and for supplying us with a preprint of ref 29. We also benefited from several discussions with Professor K. D. Jordan, who supplied preprints of refs 28d and 28e prior to publication.
V. Summary Effective transfer integrals ( T ) have been evaluated for u- and *-type electron and hole transfer in radical ion systems comprising methylene donor/acceptor groups linked by various saturated organic spacer groups. The T values have been calculated on the basis of ab initio self-consistent-fieldwave functions for the radical ion states, obtained either directly for the system of interest ( T E ~ ) or from the associated neutral state via Koopmans’ theorem ( T K ~ ) , and employing either minimal (STO-3G) or split-valence (3-2 1G) basis sets. The TKTvalues have been decomposed into additive contributions from individual pathways, both through-space (T:?)
Appendix A Sensitivity of Trtb to Rydberg Contributions (3-2lC Basis). Table VI1 exhibits the sensitivity of Tpathto Rydberg contrib u t i o n ~ ~by’ ~showing the results (through fourth and fifth order) in which all pathways involving Rydberg orbitals are excluded or in which the influence of the Rydberg orbitals on the valence NBO Fock matrix is included via the partitioning methods3]
(58) Paddon-Row, M. N. New. J . Chem. 1991, I S , 107-116. (59) It should also be noted (see Table I), that the coupling in 3 displays a significant sensitivity to basis set.
Appendix B Table VI11 presents low-order contributions to Tpth for the 3-21G basis.
Photoirradiation Effect on Colloidal Cadmium Sulfide Formation Processes Toyoharu Hayashi,* Hiroshi Yao, Shigeru Takahara, and Koichi Mizuma Central Research Institute, Mitsui Toatsu Chemicals. Inc., Yokohama 247, Japan (Received: September 5, 1991; In Final Form: January 9, 1992) Cadmium sulfide ultrafine particles of about 3 nm in diameter were prepared by introducing a H,S/He gas mixture into an acetonitrile solution of cadmium perchlorate, styrene monomer, and o-dicyanobenzene while irradiating the solution with light of wavelengths longer than 440 nm. The UV-vis spectral absorption onset wavelengths of the CdS colloidal solutions were blue-shifted in comparison with those of solutions not irradiated. The UV-vis spectra also showed the appearance of a shoulder due to the exciton absorption of quantum-confined CdS particles. These results demonstrate a new method for preparing size-regulated semiconductor ultrafine particles in the size range where quantum confinement of carriers occurs.
Introduction In the past decade, semiconductor ultrafine particles, wellknown as Q-particles,] have been studied extensively. These studies have covered topics such as photogenerated electron-hole carrier ( I ) See review articles such as: (a) Henglein, A. Chem. Rev. 1989, 89, 1861. (b) Steigerwald, M. L.; Brus, L. E. Acc. Chem. Res. 1990, 23, 183.
0022-3654/92/2096-2866$03.00/0
confinement effects on the electronic states of particles: hot camer effects in photo catalyst^,^ and so on. Semiconductor ultrafine particles have also been studied as third-order nonlinear optical materials with large nonlinearity and high operation speed. Here (2) Brus, L. E. J . Chem. Phys. 1983, 79, 5566; 1984, 80, 4403. (3) Williams, F.; Nozik, A . J. Nature 1984, 312, 21.
0 1992 American Chemical Society
