J . Phys. Chem. 1992,96, 1188-1 196
1188
-
-
mechanism consists of an indirect electron density flux C2H4-, HF Hz replacing the direct transfer CzH4 H2. According to this, the catalyst would play the role of an “electronic bridge” connecting reactants in such a way that the electron redistribution along the reaction coordinate can be achieved in a way much more favorable energetically. The only important monoexcitation now is the one in the CzH4 molecule. Table IX presents the A 0 overlap population between 2p, A 0 on C1 of ethylene and 1s A 0 on H1, and between 2p, A 0 on C2 of ethylene and 1s A 0 on H 3 (see Figure IC). These figurea clearly show that when usingthe catalyst pseudoexcitation becomes less important. In effect, the C2-H3 bonding can be achieved through HOMO(CzH4)-LUMO( HF) interaction, and although pseudoexcitation is still important in the C1-H1 bonding the major role is played by the LUMO(CzH4)-HOMO(Hz) interaction.
Conclusions In the present work we have analyzed the bifunctional catalysis of ethylene hydrogenation by HF. The presence of the catalyst makes possible an HOMO-LUMO interaction leading to the formation of the two new C-H bonds, reducing the importance of pseudoexcitation. The bifunctional catalyst provides an alternative path to the direct charge transfer from CzH4to Hz by acting as an electronic bridge. Thus, the electron density rearrangement takes place along the reaction coordinate in a much more energetically economical way.
Acknowledgment. We are greatly indebted to Professors J. BertrPn and A. LLedds for providing us with useful information on some of the systems studied by us in this work. Regism NO. H2, 1333-74-0; H2CeH2, 74-85-1; HF, 7664-39-3.
Long-Range Interactions in a Series of Rigid Nonconjugated Dienes. 1. Distance Dependence of the ?T+,?T-and r+*,?r-* Spllttings Determined by ab Initio Calculations Kenneth D. Jordan* Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
and Michael N. Paddon-Row* School of Chemistry, University of New South Wales, P.O. Box 1 , Kensington, NSW, Australia 2033 (Received: September 20, 1991)
The splittings between the two lowest energy cation states and the two lowest energy anion states are calculated in both the Koopmans’ theorem and ASCF approximations for a series of rigid nonconjugated dienes, with the two double bonds separated by 4-12 C-C u bonds. To determine the role of long-range through-bond coupling on the splittings, the calculations are carried out using several basis sets-STO-3G, 3-21G, 6-311G, 6-31+G, and D95v-which differ in the radial extent of the outermost basis functions. The dependence of the various splittings on n, the number of C-C u bonds separating the ethylenic groups, is examined. The u+,u- splittings for the dienes with bridges containing 8, 10, and 12 C-C bonds are consistent with an exponential dependence on n, independent of the basis set employed. Similarly, when the STO-3G basis set is used, the x+*,T-* splittings for these compounds are also consistent with an exponential n dependence. However, with more flexible basis sets, the u+*,u-* splittings for the longer dienes show small, but significant,deviationsfrom an exponential n dependence. Moreover, when the results for the dienes with the four- and six-bond bridges are included, single exponentials no longer provide good fits to the *+,A- or the a+*,*-*splittings for the entire series of compounds. The deviations from exponential behavior are greater with the nonminimal basis sets and at the Koopmans’ theorem than at the ASCF level of theory. It is suggested that the nonexponential distance dependence is due to the importance of multiple through-bond pathways.
I. Introduction In recent years much experimental data have become available demonstrating that electron transfer (ET) and hole transfer (HT) can occur between redox centers separated by distances significantly exceeding the sum of the donor and acceptor van der Waals
(1) Calcatera, L. T.; Closs, G. L.; Miller, J. R. J. Am. Chem. Soc. 1988, 105,670. Closs, G. L.; Calcatera, L. T.; Green, N. J.; Penfield, K. W.; Miller, J. R. J . Phys. Chem. 1986, 90,3673. (2) (a) Oevering, H.; Paddon-Row, M. N.; Heppener, M.; Oliver, A. M.; Cotsaris. E.; Verhoeven, J. W.; Hush, N. S.J . Am. Chem. Soc. 1987, 109, 3258. (b) Penfield, K. W.; Miller, J. R.; Paddon-Row, M. N.; Cotsaris, E.; Oliver, A. M.;Hush, N. S. J. Am. Chem. Soc. 1987,109, 5061. (c) Wannan,
J. M.; de Haas, M. P.; Paddon-Row, M. N.; Cotsaris, E.; Hush, N. S.; Oevering, H.; Verhoeven, J. W. Notum 1986, 320,615. (d) Oliver, A. M.; Craig, D. C.; Paddon-Row, M. N.; Kroon, J.; Verhoeven, J. W. Chem. Phys. Letr. 1988, /SO, 366. (e) Wannan, J. M.; Smit, K. J.; de Haas, M. P.; Jonker, S.A.; Paddon-Row, M.N.; Oliver, A. M.; Kraon, J.; h e r i n g , H.; Verhoeven, J. W. J . Phys. Chem. 1991,95, 1979. ( f ) Antolovich, M.; Keyte, P. J.; Oliver, A. M.; Paddon-Row, M. N.; Kroon, J.; Verhoeven, J. W.; Jonker, S. A.; Warman, J. M. J . Phys. Chem. 1991, 95, 1933. (8) Paddon-Row, M. N.; Verhoeven, J. W. New J . Chem. 1991, 15, 107.
0022-365419212096-1188%03.00/0
In the nonadiabatic limit, in which the coupling between the donor and acceptor groups is weak, the application of the golden rule gives the following expressions for the ET and HT transfer rates” k,, (2*/h)HbZFCWD (1) (3) (a) Connolly, J. S.;Bolton, J. R. In Photoinduced Electron Transfer, Fox, M. A., Chanon, M., Eds.; Elsevier: Amsterdam, 1988; Part D, p 303. (b) Wasielewski, M. R. In Photoinduced Electron Transfer, Fox, M.A,, Chanon, M.,Us.; Elsevier: Amsterdam, 1988; Part A, p 161. (c) Stein, C. A.; Lewis, N. A.; Seitz, G. J . Am. Chem. Soc. 1982, 104, 2596. (d) Gust, D.; Moore, T. A. Science 1989, 244, 35. (e) Joran, A. D.; Leland, B. A.; Geller, G. G.; Hopfield, J. J.; Dervan, P. B. J . Am. Chem. Soc. 1984, 106, 6090. (f) Leland, B. A.; Joran, A. D.; Felker, P. M.; Hopfield, J. J.; Zewail, A. H.; Dervan, P. B. J. Phys. Chem. 1985,89,5571. (g) Joran, A. D.; Leland, B. A.; Felker, P. M.; Zewail, A. H.; Hopfield, J. J.; Dtrvan, P. B. Noture (London) 1987,327,508. (h) Gust, D., Moore, T. A., Eds. Covalently Linked Donor-Acceptor Species for Mimicry of Photosynthetic Electron and Energy Transfer, Tetrahedron Symposium-in-Print No. 39; Tefrohedron1989, 45. (4) Levich, V. G. Electrochem. Electrochem. Eng. 1966, 4, 249. Dogonadze, R. R.; Kuznetsov, A. M. Elekfrokhimiya 1967, 2, 1324. (5) Marcus, R.; Sutin, N. Biochim. Biophys. Acto 1985,811,265. Newton, M. D.; Sutin, N. Annu. Rev. Phys. Chem. 1984,35,437 and references therein.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1189
Long-Range Interactions in Nonconjugated Dienes and kht
(2*/h)Hb2FCWD
(2)
where Hb is the electronic coupling between the donor and acceptor groups and FCWD refers to a Franck-Condon weighted density of states, which includes the contributions due to the nuclear distortions. Of course, both Hdaand the Franckxondon contributions are different in the ET and HT cases. Particular emphasis has been placed on understanding the role of the electronic coupling, particularly through saturated bridges, on the rate of electron and hole transfer. Much of the progress in this area has come from experimental studies of a series of donor-bridgeacceptor (D-B-A) compounds with the same donor and acceptor groups,but with different length bridges,'" illustrative of which are compounds 1-!k2 Along such series of compounds the differences in the ET (and HT) rates should be determined primarily by the variations in the electronic couplings.68 Studies of photoinduced ET in 1-5 and other D-B-A compounds, as well as of thermal ET in the associated radical anions, strongly euggest that through-bond (TB)9 interactions are largely responsible for the high rates of ET.
etries of the neutral molecules between the two u cation states (AIP) and between the two ** anion states (AEA), respectively. Specifically, to a good approximation, AIP = 2Hbh and AEA = 2Hbc, where the superscripts h and e refer to the hole- and electron-transfer cases, respectively.
6
7
8
CN
9
CN Med
2
10
CN CN
CN CN
CN CN
One particularly fruitful approach to obtaining a detailed understanding of how the characteristics (length, bond angles, cross linkages, etc.) of a particular bridge influence the electronic coupling has been to examine model compounds with the same (or similar) bridges, but with simpler chromophores.8.10J1For example, the dienes 6-10 serve as model compounds for understanding the electronic coupling in 1-5. In symmetrical dienes, such as 6-10. the electronic couplings relevant for HT and ET can be approximatelyassociated with the splittings at the geom(6) Mikkelsen, K. V.; Ratner, M. D. Chem. Rev. 1987,87, 113 and references therein. (7) Newton, M. D. Chem. Reu., in press. (8) 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. (9) (a) Hoffmann, R. Acc. Chem. Res. 1971,4, 1. (b) Gleiter, R. Angew. Chem., In;. Ed. E n d . 1974,13,696. (c) Paddon-Row, M. N. Acc. Chem. Res. 1982, 15, 245(10) Paddon-Row, M. N.; Patney, H. K.; Brown, R. S.; Houk, K. N. J . Am. Chem. Soc. 1981,103,5575. (b) Paddon-Row, M. N.; Jorgensen, F. S.; Patney, H. K. J . Chem. Soc., Chem. Commun. 1983, 573. (c) Jorgensen, F. S.: Paddon-Row. M. N. Teirahedron Lett. 1983.5415. (d) Paddon-Row. M. N:;Patney, H. K.;Peel, J. B.; Willett, G. D. J. Chem. S&:, Chem. Commun. 1984,564. (e) Balaji, V.; Ng, L.; Jordan, K. D.; Paddon-Row, M. N.; Patney, H. K. J . Am. Chem. Soc. 1987, 109, 6951. ( 1 1 ) Paddon-Row, M. N.; Wong, S. S.Chem. Phys. Lett. 1990,167,432.
Two advantages of studying the model dienes rather than the more complicated D-B-A compounds are that they are more amenable to theoretical calculations and that the splittings between the anion and between the cation states can be directly measured (at least for 6 and 7). Indeed, the vertical u ionization potentials and u* electron affinities of 6 and 7, and several related compounds, have been determined by means of photoelectron spectroscopy (PES)8J08" and electron transmission spectroscopy (ETS),'OCrespectively. These measurements give AIP and AEA values much larger than can be accounted for by direct, through-space (TS) interactions between the ethylenic groups, thereby confirming the importance of TB coupling via the u and u* orbitals associated with the bridges. The ET and PE results are exceedingly valuable since they provide a check on various theoretical methods for calculating PIP and AEA. Of particular interest is the finding that even MO calculations using minimal basis sets and invoking Koopman's theorem (KT)'* are able to account in a semiquantitative manner for the observed splittings in 6 and 7.8J0*" It is generally assumed that the electronic coupling Hdadepends exponentially (or nearly so) on the length of the bridge.',26.7,'',13,14 Given the relationships between Hdaand AIP and AEA, this implies that PIP = Ah exp(-Bhn) (3) and PEA = A, exp(-Ben) (4) where n denotes the number of C-C bonds in the bridges, "directly" coupling the D and A groups, and Bh and Be are the exponents for hole and electron transfer, respectively. For the dienes 6-10, n ranges from 4 to 12. In fact, the localized r and u* orbitals are approximately orthogonal to the u orbitals of the adjacent C< bonds, and it is probably more appropriate to view the number of bonds in the main bridges actually *active" in coupling the ethylenic groups at the two ends of the molecule as n - 2. However, conclusions that are reached about the distance (12) Koopmans, T. Physica 1934, 1, 104. (13) Onuchic, J. N.; Beratan, D. N . J. Chem. Phys. 1990, 92, 722. (14) Broo, S.;Larsson, S. Chem. Phys. 1990, 148, 103.
1190 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
dependence of the couplings do not depend on whether or not the two terminal C-C bonds adjacent to the ethylenic groups are counted. Moreover, as will be discussed below, bonds other than those along the main bridges probably make an important contribution to the TB coupling. A fundamental question concerns how the presence of multiple coupling pathways influences the dependence of the splittings on the distance between the D and A groups. Recently, Paddon-Row and Wong" studied the distance dependence of the KT/STO-3G splittings of 6-9 and of other, related, dienes. Least-squares fits of these splittings gave Bh and E, values of 0.47 and 0.60, respectively. These E values indicate that, for both the cation and anion states of the series of compounds 6-9, the TB coupling through norbornyl-type bridges falls off slowly with increasing bridge length. The electron- and holetransfer rates in D-B-A compounds with similar bridges (e.g., 1-5) are expected to vary with the chain length approximately as exp(-2B,n) and exp(-2Bhn), respectively. The available experimental data are consistent with these expectations.2 For example, the E, values deduced from the rate constants for photoinduced ET in 1-5 in different solvents range from 0.46 to 0.63. In the present work we extend the previous theoretical studies of this series of dienes in several important ways. First, the calculations are extended to include 10, in which the two ethylenic groups are separated by 12 C.C u bonds. Second, the split-valence 3-21G, 6-31G, and D95v and the triplezeta 6-31 1G and 6-31+G basis setd5J6are used in addition to the minimal STO-3G basis set employed in previous studies'oJ1of these compounds. The use of basis sets of varying flexibility permits us to determine whether interactions of longer range than can be accounted for with the STO-3G basis set play an important role in the TB coupling. In the inset, we have depicted some of the interactions possible in 6-10. The localized r (or T * ) orbitals are expected to couple
H14
strongly to the bridge through the interaction H I , and, to a lesser extent, through interactions H12and HI& Along the main bridge, the dominant interactions are expected to be of the type H34, H3s, and H36,in order of decreasing importance. (Here it is assumed etc.) In addition, interactions involving that HZs= H34= H45, the methylene bridges, for example, Hl7, H3,, and H7*,might also be expected to be important. The coupling of r orbitals via bridging methylene groups (through a so-called laticyclic hyperconjugation mechanism) has been shown to be important in isomers of 6-12.17 To the extent that interactions between (15) (a) For a discussion of the STO-3G. 3-21G, 6-31G, 6-311G, and 6-31+G basis sets as well as citations to the original references, see: Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; John Wiley and Sons: New York, 1986. (b) The D95v basis sets is described in: Dunning, T. H. J . Chem. Phys. 1970, 53, 2823. The (lOs6p/Ss3p) carbon basis set is described in: Dunning, T. H.; Hay, P. J. In Modern Theoretical Chemistry; Schaefer, H. F., Ed.;Plenum: New York, 1977; Vol. 3. (16) It has been argued that the 6-31 1G basis set is not actually of triplet-{ quality. See: Schaefer, H. F. J . Chem. Phys. 1989, 91, 7305.
Jordan and Paddon-Row nonadjacent bonds, e.g., H35and H7*,are important, the STO-3G or even the 3-21G basis sets could prove inadequate. Third, the calculations are carried out at both the Koopmans' theorem and ASCF levels of theory. Previous studies of 6-9 have been carried out only at the KT level of theory, which neglects the contributions of relaxation and electron correlation to the IP's and EA'S. The ASCF calculationswill permit us to determine whether relaxation effects are important for describing the splittings between the anion and cation states. The present calculations differ also from previous onesloJ1in that the geometries have been optimized at the HF/3-21G rather than the HF/STO-3G level of theory and, thus, should correspond more closely to the experimental geometries. The availability of ab initio splittings, obtained using basis sets of varying flexibility, for the series 6-10 should prove particularly valuable to researchers developing perturbative or model Hamiltonian approaches for describing through-bond interactions.
II.
Computational Details As the first step in this investigation, the geometries of the ground electronic states of the neutral molecules, 6-10, were optimized under the constraint of C, symmetry at the HF/3-21G
level of theory. These geometries were used for all subsequent calculations on the neutral molecules and their ions. Thus, the calculated IPSand EA'S are vertical quantities. The splittings at the ASCF level of theory were obtained from spin-unrestricted Hartree-Fock (UHF) calculations carried out on the two lowest energy cation states and on the two lowest energy anion states of each molecule. H F solutions which localized the charge, and therefore break the spatial symmetry, are possible for the cation and anion states of some of the dienes. Thus, it is important to note that the UHF wave functions for the ionic species maintained C, symmetry for all of the molecules studied. We note also that UHF expectation values of S2 for the ions of 6-10 were found to be between 0.75 and 0.76. Thus, errors due to spin contamination of the wave functions are not expected to be important. To determine whether long-range interactions contribute significantly to the TB coupling, several different basis sets were used. The STO-3G basis set is a minimal basis set and, as such, is only suitable for describing relatively short range interactions: for interactions occurring over distances greater than about 3 A, one would expect there to be sizable errors due to the use of the STO-3G basis set. The other basis sets considered provide either a double-zeta (split-valence) or triple-zeta description of the valence space. In terms of increasing suitability for describing long-range interactions (based on the magnitudes of the exponents of the most diffuse carbon p basis function), the basis sets used here may be ordered: STO-3G, 3-21G, 6-31G, 6-311G, D95v, and 6-31+G. The D95v basis set is Dunning's [9s5p/3s2p]/ [C/2s] contraction of Huzinaga's 9s5p and 4s primitive Gaussian basii sets for carbon and hydrogen, respectivel~.'~~ The other basis sets used are those of Pople and c o - w o r k e r ~ . ~ ~ ~ All six basis sets are used in estimating the splittings at the KT level of theory, although, due to difficulties encountered with SCF convergence, results with the 6-31+G basii set are presented only for 6-8. The STO-3G, 3-21G, and D95v basis sets are used in the calculations of the ASCF splittings. The calculations were carried out with the GAUSSIAN 90 program.ls The anion states of 6-10 are unbound, in the sense that they lie in the continuum of the neutral molecule plus a free electron. In theoretical treatments of such anion states, whether at the KT or a more sophisticated level of theory, it is essential to establish that the wave functions correspond to the anion states rather than to approximations to continuum functions.Ig The latter type of (17) Paddon-Row, M. N . J . Chem. Sac., Perkin Trans. 2 1985, 257. Craig, D. C.; Paddon-Row, M. N.; Patney, H. K. Aust. J . Chem. 1986,39, 1587. (18) GAUSSIAN 90: Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Forseman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M. A,; Binkley, J. S.; Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. Gaussian Inc., Pittsburgh, PA, 1990.
The Journal of Physical Chemistry, Vol. 96, No. 3, I992 1191
Long-Range Interactions in Nonconjugated Dienes
TABLE I: KT and ASCF Estisutea of AIP and AEA for Compounds 61V basisset
Sb KT
7c
6
KT
ASCF
STO-3G 3-21G 6-31G 6-311G D95v 6-31+G
0.02 0.06 0.08 0.11 0.12 0.13
0.99 1.06 1.04 1.02 1.02 1.03
0.99 1.10
STO-3G 3-21G 6-31G 6-311G D95v
0.01 0.08 0.14 0.25 0.34
0.85
0.90
0.86 0.94
0.86 0.89 0.76
0.79
1.01
KT
9
ASCF
0.35 0.38 0.36 0.35 0.34 0.35
0.31 0.35
0.23 0.18 0.15 0.11 0.11
0.22 0.21
0.30
0.15
KT
IP'S 0.141 0.165 0.157 0.154 0.148 0.154 EA'S 0.066 0.091 0.079 0.062 0.069
10
ASCF
KT
ASCF
KT
ASCF
0.105 0.132
0.065 0.085 0.079 0.078 0.075
0.039 0.053
0.0294 0.0427 0.0389 0.0380 0.0373
0.0142 0.0206
0.0057 0.0106 0.0087 0.0065 0.0058
0.0043 0.0084
0.113
0.059 0.085 0.064
0.019 0.032 0.027 0.010 0.021
0.044
0.016 0.027 0.016 0.020
0.0169
0.0056
"Splitting in electronvolts. bThe KT and ASCF splittings are nearly identical for the dimer corresponding to 6, and therefore only the KT splittings have been listed. CCalculationsusing Dunning's (1Os6p/5s3p) carbon basis set (ref 1%) give KT level AIP values of 0.35 and 0.156 eV for 7 and 8, respectively. The corresponding AEA values are 0.09 and 0.053 eV.
solutions, hereafter referred to as discretized continuum (DC) solutions, drop rapidly in energy as increasingly diffuse functions are added to the basis set. With the 6-31+G basis set the lowest energy unfilled orbitals correspond to DC solutions, and for this reason, u+*,u-* splittings are not reported for this basis set. (The + and - subscripts refer to the in-phase and the out-of-phase combinationsof the localized ethylenic u* orbitals.) For the other basis sets, the DC functions lie higher in energy and the lowest lying unfilled bl(a+*) and a2(r-*) orbitals indeed correspond to anion states in a KT sense.19 The MO calculations provide net splittings which include contributions from both TS and TB interactions. In this work the term TS is reserved for the direct interaction between localized ethylenic u or u* orbitals. All interactions proceeding via the intervening bridge, even if they involve relatively long-range interactions such as HZ4,HZs,HI4,and HI,, shown previously in the inset, are referred to as TB in nature. It is instructive to decompose the net splittings into their TS and TB components. Although, there is no unique procedure for doing so,this does not prove to be problematical, since any reasonable scheme predicts that in 6-10 the TB contributions to the splittings are much larger than the TS contributions. In fact, only for 6 do TS interactions make significant contributions to the net splittings. In the present study, the estimates of the TS splittings are obtained from H F calculations carried out on model ethylene dimers, in which the mutual orientation and separation of the two ethylene molecules are the same as between the double bonds in 6-10.208* In the calculations on the dimers, the C - C bond lengths, the exterior C-H bond lengths, and the CCH angles are taken to be the same as in the diene of interest. The interior CH bond lengths are set q u a l to the exterior ones, and the CCH angles involving the interior hydrogen atoms are taken to be the same as the C-C-C angles in the corresponding diene. The TB contributions to the u+,rand u+*,u-+splittings are estimated by subtracting the splittings (19) For sufficiently flexible basis sets, the lowest lying unfilled orbitals correspond to approximations to continuum functions (so-called discretized continuum functions) rather than to negative ions states in a Koopmans' theorem sense. In such cases, there are higher lying virtual orbitals that correspond to the anion states, and the problem is one of identifying the appropriate unfilled orbitals. One way of doing so is to use the stabilization method. [For an application of this method, see: Falcetta, M. F.; Jordan, K.D. J. Am. Chem. Soc. 1991, 113, 2903.1 Stabilization calculations employing the 6-31+G and still more flexible basis sets give r + * , x *splittings for 6 and 7 close to those obtained from the HF/D95v calculations [Jordan, K. D., unpublished results]. (20) (a) Paddon-Row, M. N.;Wong, S.S.;Jordan, K.D. J. Am. Chem. Soc. 1990,112, 1710. (b) Falcetta, M. F.; Jordan, K.D.; McMurry, J. E.; Paddon-Row, M. N. J. Am. Chem. Soc. 1990,112,579. (c) Paddon-Row, M. N.; Jordan, K. D. J. Chem. Soc., Chem. Commun. 1988, 1508. (d) Paddon-Row, M. N.; Wong, S.S.;Jordan, K. D. J. Chem. Soc., Perkin Tram. 2 1990,417. (e) Paddon-Row, M. N.; Wong, S.S.;Jordan, K. D. J . Chem. Soc., Perkin Trans. 2 1990, 425. (f) Heilbronner, E.; Schmelzcr, A. Hela Chim. Acra 1975,58,936. (g) Imamura, A.; Ohsaku, M. Terrahedron 1981, 37, 2191.
of the appropriate dimer from those of the diene of interest.
III. ReaulQ A. Orbital Splitlings. ( i ) Results with the STO-3G and 3-ZIG Basis Sets. Table I summarizes the calculated PIP and PEA values for 6-10 as well as for the ethylene dimer corresponding to 6. We consider first the results obtained at the KT level of theory and using the STO-3G and 3-21G basis sets. It has been firmly established in previous studies that the T+,?T-and u+*,a-* splittings in these dienes are due almost entirely to TB interactions.*JoJ1The present calculations are in full accord with this: splittings obtained with the STO-3G For 6 the ?T+,Rand u+*,?~-* and 3-21G basis sets are over an order of magnitude larger than those for the corresponding ethylene dimer; for 7-10 they are several orders of magnitude larger than for the corresponding dimers. It is for this reason that only for the dimer appropriate for 6 have splittings been included in Table I. The orbital splittings for 6 and 7, obtained with both the STO-3G and 3-21G basis sets, are in fairly good agreement with experiment, with the exception of the PEA value of 7. The calculated u+/u- splittings of 6 and 7 are 10-18% larger than the experimental PIP values. The calculated ?T+*,T-* splittings of 6 are only 2-3% larger than the experimental PEA value, while the calculated u+*,u-* splittings of 7 are smaller (by 10% and 28% with the STO-3G and 3-21G basis sets, respectively) than the experimentally determined AJ3A value of 0.25 eV.'" However, it should be noted that there is considerable uncertainty in the experimental splitting between the two anion states of 7 on account of the fact that the splitting is comparable to the experimental line widths. In fact, only a single peak was seen in the ET spectrum when run in the normal mode (in which the spectrometer is tuned so as to give the total scattering cross section).'OC Although two peaks were resolved when the instrument was tuned so that the elastic backscattering ( 1 8 0 O ) cross section was measured, the reliability of splittings obtained in this manner is unknown. Experimental values for the PIP and AEA values of 8-10 are unavailable, as it would not be possible to resolve the two peaks in the PES and ETS measurements of these compounds. In discussing the trends in the splittings, it is instructive to consider the McConnell mode121 for TB interactions in which the splitting between the a+ and u- levels (or between the a+*and a_*levels) is given by the expression
AE = -2(Tz/D)(t/D)"l
(5)
where T gives the strength of the coupling of the ?T (or u*)orbital of the ethylene chromophore to the bridge (e.g., H I 3in the inset), t gives the strength of the coupling along the bridge (Le., between adjacent localized orbitals of the relevant bridge subunits, such (21) McConnell, H. M. J . Chem. Phys. 1961, 35, 508. As used here, D is defined as positive and t as qegative.
1192 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 TABLE II: Ratios of the KT Snlittim for Consecutive Dienes
2.48 2.30 2.26 2.28 2.28 2.28
2.17 1.94 1.99 1.97 1.98
2.24 1.97 2.02 2.05 2.00
AEA(6)0/ AEA(7)
AEA(7)/ AEA(8)
AEA(8)/ AEA(9)
AEA(9)/ AEA(10)
3.70 (3.81) 5.00 (4.55) 5.72 (4.70) 8.09 (5.96) 6.90 (3.89)
3.33 1.97 1.93 1.75 1.56
3.47 2.84 2.91 6.36 3.25
3.33 3.02 3.13 1.50 3.72
STO-3G 3-21G 6-31G 6-31 1G D95v 6-31+G
2.83 2.79 2.82 2.92 3.00 2.84
basis set STO-3G 3-21G 6-31G 6-311G D95v
(2.77) (2.63) (2.74) (2.60) (2.67) (2.57)
OTwo sets of results are reported for the ratios of the splittings of 6 and 7, the first being obtained by using the total splittings for 6 and the second (in parentheses) by using the estimated TB splittings for 6.
as Ha in the inset), D is the energy difference between the relevant orbitals of the chromophore and the appropriate bridge subunit, which we assume to be that of a C-C bond of the bridge, and n is the number of bridge subunits. Obviously, the values of the parameters entering eq 5 are different for the TB coupling in the A and the u* manifolds. In the derivation of eq 5 it is assumed that all bridge sites are identical, that there is only one coupling pathway, and that T / D
