1182
J. Phys. Chem. 1995, 99, 1182-1193
Superexchange Pathway Calculation of Electronic Coupling through Cyclohexane Spacers Larry A. Curtiss,” Conrad A. Naleway, and John R. Miller
Argonne National Laboratory, Argonne, Illinois 60439 Received: August 24, 1994; In Final Form: November 12, 1994@
The nature of the electronic coupling in donor/acceptor systems with cyclohexane-type spacers, including trans- 1,4-dimethylenecyclohexane,cis- 1,3-dimethylenecyclohexane,2,6-dimethylenedecalin, 2,7-dimethylenedecalin, and 3,16-dimethyleneandrostane,is investigated using a superexchange (SE) pathway method based on a b initio molecular orbital theory with natural bond orbitals (NBOs). The n couplings in anions and cations are examined. The magnitudes of the couplings calculated from the SE pathways method are in reasonable agreement with those from Koopmans’ theorem, the ASCF method, and the AMP2 method. Paths involving hops which skip over bonds make the largest contributions to the total coupling, in agreement with previous studies. The dominant pathway in every case is through C C antibonds. The SE method is used to examine the dependence of the coupling on rotation of the CH2 groups in the dimethylenecyclohexane and dimethylenedecalin donor/acceptor systems. Rotation of the CH2 groups from the (0,O) to the (90,90) conformation decreases couplings by factors of 1.5- 10. The (0,O)couplings are larger because the magnitudes of most paths are larger in (0,O) than in (90,90). This is because interaction of the donor/acceptor groups with C C bonds or CC* antibonds in the bridges is generally larger in (0,O). In 1,3-dimethylenecyclohexane the coupling decrease is the smallest (factor of about 1.5) because axial C H bond paths make a significant contribution in the (90,90) conformation. Finally, the calculated couplings are compared to experimentally derived couplings from molecules containing the same spacers, but with biphenyl and naphthyl donor/acceptor groups. Reasonable agreement is found between theory and experiment on the distance dependence.
I. Introduction Electron transfer (ET) rates in donorlacceptor molecules depend on and d i s t a n ~ e ~ xbetween ~ . ~ - ~ the ~ donor1 acceptor g r o ~ p s . ~ lAlthough -~~ this dependence is known to be largely due to electronic coupling, much remains mysterious about it. For example, the effects of angles on electron transfer rates for isomers of biphenyl-naphthyl decalins? and porphyrin-quinone8s40 compounds are not well understood. Understanding the effects of distance and angles on electronic coupling requires consideration of the material between the electron donor and acceptor and has been the subject of numerous theoretical and experimental studies. Superexchange (SE) pathway calculations have recently been used to investigate the nature of electronic couplings in a number of electron/donor In s u p e r e ~ c h a n g e , ~ orig’-~~ inally formulated for magnetic interaction^,"'"^ electronic mixing between a donor and acceptor occurs indirectly through highenergy intermediates. The SE calculations here use McConnell’s perturbation solution for a single path50 with matrix elements from ab initio molecular orbital theory. The molecular orbital wave functions are transformed into localized orbitals using Weinhold’s natural bond We initially applied this method to the anion of butane 1.4-diyl (-CH*CH2CH2CH2*) and the anion of 1,4-dimethylenecyclohexane(-CH2Ca1&H2*)!3 It was found that the bulk of the interaction comes from pathways which skip over some CC bonds in contrast to the “tight binding” picture in which nearest neighbor interactions are important. In a subsequent more detailed papef15 we applied the method to a series of trans alkyls (CH2)CHn-2(CH2) and obtained similar results; i.e., pathways slupping over bonds make up the bulk of the interaction. The SE couplings were found @Abstractpublished in Advance ACS Absrracrs, January 1. 1995.
0022-3654/95/2099- 1182$09.00/0
to be in reasonable agreement with results from ASCF and Koopmans’ theorem calculations. Liang and Newton44,46and Jordan and P a d d ~ n - R o walso ~~.~~ examined electronic coupling pathways using natural bond orbitals (NBOs) and have also come to the conclusion that nearest neighbor McConnell-type pathways are not very important. Liang and studied electronic coupling in s- and p-type electron and hole transfer in radical ion systems with methylene donor/acceptor groups using pathways analysis based on NBOs. They considered trans alkyl chains, bicyclooctane, and bicyclopentane spacers. Jordan and PaddonRow53used NBOs to examine electronic coupling pathways in dienes with rigid polynorbornyl-type spacers. The partitioning method of L a r ~ s o nprovides ~ ~ . ~ ~a complementary view using canonical orbitals of the bridge instead of the localized orbitals used here. Localized orbitals are used by inference in the Huckel-based pathway model of Beratan and O ~ n i c h i c . ~ ~ . ~ ~ ~ ~ The purpose of the work reported in this paper was to apply the SE pathways method to donorlacceptor systems with cyclohexane spacers in order to obtain insight into how couplings are transmitted through bonds. The SE method is used to probe reasons for the dependence of coupling on orientation of the CH2 groups and the nature of the electron transfer pathways. Included in this paper are results of superexchange calculations on trans- 1,4-dimethylenecyclohexane, cis-1,3-dimethylenecyclohexane,2,6-dimethylenedecalin, 2,7-dimethylenedecalin, and 3,16-dimethyleneandrostane(Figures 1-2). These molecules are models for larger molecules with biphenyl and naphthyl donorlacceptor groups for which experimental data have been reported. The SE couplings are compared with results from calculations based on the ASCF and AMP2 approximations and Koopmans’ theorem. In addition, SE results from different wave functions are compared. Finally, the distance dependence of the coupling for this set of
0 1995 American Chemical Society
Electronic Coupling through Cyclohexane Spacers
J. Plzys. Chem., Vol. 99, No. 4, 1995 1183 and A. In a molecular electron transfer reaction there will be many such chains. The total coupling is the algebraic sum over all chains
26D (8.0)
Figure 1. Structures of 1,4-dimethyIenecyclohexane (1,4C), 1.3dimethylenecyclohexane ( 1,3C), and 2.6-dimethylenedecalin (2.6D).
2,7D(0.0)
3,16A
Figure 2. Structures of 3.16-dimethyleneandrostane (3,16A, donor/ acceptor x orbitals are approximately in the plane of the spacers) and 2.7-dimethylenedecalin (2.7D). four molecules is compared with experiment. In section I1 the methods used are described. Section 111 contains results and discussion. 11. Description of Theoretical Methods
A. Basis Sets. Ab initio molecular orbital t h e ~ r y ~was *,~~ used in all of this work, with the exception of geometry optimizations which were done with molecular mechanics (see below). Calculations were carried out at the HF/STO-3G, HF/ 3-21G, HF/6-3 1G*, and MP2/6-31G* levels. In recent studiesJS.M the 3-21G basis set has been found to give reasonable agreement with couplings calculated using larger basis sets. B. Superexchange Pathways Method. The pathways method is based on the idea of superexchange (SE), the indirect coupling of donor (D) to acceptor (A) wave functions through a chain (path) of high-energy intermediate states. From perturbation theory!’ the coupling of the donor and the acceptor by one such chain (the kth) is given by rr+ I
n
The pu are the couplings between the ith andjth states. The Bi is the energy difference between the ith state and states D
This superexchange picture can be effectively implemented if the ratio po/Bi of the coupling elements to the energy denominators is small. Unfortunately, pu/Bj is large if the representation is made up of atomic orbitals. A more useful representation is one in which the intermediate states of the saturated hydrocarbon are localized o and o* bond orbitals. We have used Fock matrix elements from natural bond orbitals ( N B O ’ S ) , ~ ’which + ~ ~ are localized orbitals. These localized orbitals are obtained by transforming the canonical (i.e., delocalized) self-consistent-field (SCF) molecular orbitals (MO’s) into a set of orthonormal bond orbitals. The NBO’s can be divided into “occupied” and “unoccupied” orbitals. The former include core orbitals, lone pairs, o or n bonds, etc., and the latter include o* or n* antibonds, extra-valence-shell orbitals (Rydbergs), etc. The “unoccupied” orbitals have small occupancies and thus are not the same as virtual MO’s of SCFM O theory, which are strictly unoccupied. These NBO’s are used in the calculation of the electronic coupling interactions in eq 1. The coupling elements, Po, and the energy denominators, Bi, used in eq 1 are obtained from off-diagonal and diagonal elements of the Fock matrix in the NBO representation of the unrestricted Hartree-Fock UHF wave function. The signs obtained for the couplings from the NBO pathways calculations are arbitrary, and the signs given to them in the tables and figures in this paper are on the basis of the Koopmans’ theorem results (see be lo^).^^,^ The method for calculation of the electronic couplings from a superexchange (SE) pathways analysis has been presented in more detail e l s e ~ h e r e . ~ ~ , ~ ~ The SE pathway analysis of coupling in trans alkyls4s was carried out using four different sets of orbitals, two for the cations and two for the anions. These included couplings in the anion from the a orbitals of the anion and the /3 orbitals of the neutral triplet and couplings in the cation from the a orbitals of the neutral triplet and the orbitals of the dication. In ref 60 the SE couplings calculated from the anion and neutral triplet a orbitals tended to have poor convergence with energy threshold for retaining vk in eq 2, while couplings calculated from neutral triplet p and dication orbitals had good convergence. In the present study SE calculations are done using these four sets of orbitals, as well as dianion (for anion coupling) and cation /?(for cation coupling) orbitals in some cases. In the present work we have included contributions to eq 2 from paths having an energy of greater than hartrees unless specified. The hartree threshold was found to give reasonable convergence for the trans alkyls.4s Paths involving hops (forward or backward) between (1) occupied orbitals [referred to as occupied paths 1, (2) unoccupied orbitals [referred to as unoccupied paths], and (3) both occupied and unoccupied orbitals [referred to as mixed paths] are included in the SE results unless specified. Return to previously visited sites (retracing) is not allowed. In ref 45 retracing was found to contribute about 10% to the couplings in most cases for the trans alkyls, although for some of the longer chains it was as high as 50%. Inclusion of retracing did not result in better overall agreement with Koopmans’ theorem couplings. Comparison of the couplings calculated from the SE pathways method is made to couplings calculated from Koopmans’ theorem (KT), the ASCF method, and AMP2 method. Koop-
Curtiss et al.
1184 J. Phys. Clzem., Vol. 99, No. 4, 1995
mans' theorem62provides an approximate way to calculate the splitting parameter A from the differences in the eigenvalues of the molecular orbitals corresponding to the symmetric and antisymmetric combinations of the p orbitals on the CH2 donor and acceptor groups:63
A = E ( # - ) - e(#-)
(3)
The A is associated with the electronic coupling VDA by63
V = A12
(4)
The application of Koopmans' theorem to the six different sets of orbitals, including those of the cation p and anion a orbitals, to obtain couplings, and the meaning of the couplings have been discussed in detail previously.60 Briefly, the KT couplings from the neutral triplet diradical, dication, and dianion orbitals are based on ionization and electron attachment processes which have correct final configurations but incorrect frozen orbitals. The KT couplings from the anion a and cation /3 orbitals are based on ionization and electron attachment processes which have incorrect final configurations but correct frozen orbitals. The validity of the various KT values depends, in part, on the importance of electronic relaxation. The sign convention used for the KT couplings is similar to that used by Newton.63 The interaction is defined as positive when occupancy of the symmetric orbital 4+ from the many-electron wave function is preferable to occupancy of the antisymmetric orbital +-; it is negative if occupancy of 4- is preferable. The ASCF method uses the unrestricted Hartree-Fock (UHF) energies of the ground (E,) and first excited (E,) states, V = A12 = f (E, - E,)/2. This method includes electronic relaxation effects of the excitation which is not included in the KT method or the SE method and should be more realistic. The sign convention used was the same as that used for the Koopmans' theorem splittings. We also use energies from MP2/ 6-31G* calculations and refer to couplings calculated in this way as AMP2 values. The unrestricted Hartree-Fock (UHF) method was used in calculations on open-shell systems (cation, anion, neutral triplet). The resulting wave functions had delocalized donor/acceptor orbitals, i.e., an equal electron distribution on the CH2 groups, for structures with symmetrical donor/acceptor groups. C. Structures. The structures of trans- 1,4-dimethylenecyclohexane (1,4C), cis-1.3-dimethylenecyclohexane(1,3c), and rrans-2,6-dimethylenedecalin(2,6D) used in this study are illustrated in Figure 1. All are equatorial, equatorial (e,e) isomers and have C2h, C, and & symmetries. respectively. The geometries were obtained from molecular mechanics (MM2 ~ p t i m i z a t i o n ) . The ~ ~ . ~bond ~ distances and bond angles were adjusted slightly to obtain structures with symmetry in some cases. Both (0.0) and (90.90) orientations of the CH2 groups were considered. They are illustrated in Figure 1. The structure for 5a-3.16-dimethyleneandrostane is shown in Figure 2. It has C1 symmetry, and the geometry was obtained from an MM2 optimization using the Spartad5molecular orbital program. The lack of symmetry required use of methods to delocalize the donor/acceptor orbitals. This is described in more detail in section 1II.E. The structure of 2,7-dimethylenedecalin (2,7D) is also included in Figure 2. This structure has C, symmetry and was obtained from an MM2 o ~ t i m i z a t i o n .Only ~~ the (0,O) conformation was considered in this study. The calculated couplings for 2,7D are included in the comparison with experiment in section 1V.E. In this study certain assumptions were made concerning the geometries of the systems being considered. All couplings were
TABLE 1: Comparison of Calculated Couplings at the HF/3-21G Level of Calculation (in mhartrees) 1.3C
tvpe
method'.
(0.0)(90.90)
cation SE (triplet a ) KT (tnplet a i 10.42 SE (cation / j ~ KT (cation 13) SE (dication) 11.45 KT idication) 11.60 ASCFh 13.42 ASCFh' 13.34 AMPZhL 10.08 anion (SE (triplet /?I 15.26 KT (triplet /?) 12.59 SE (anion a ) KT ianion a) SE (dianion) KT (dianion) ASCFh 13.21 ASCF'.' 12.22 AMP2h.' 11.07
7 61 6.21 6.58 8.00 7.59 7.04 8.16 7.07
7.27 6.65 6.06
1,4c (0,O)
-2.31 -4.22 10.46 -9.20 -8.20 -7.78 -6.54 -7.26 -6.35 -14.82 -11.91 -7.19 -6.97 -7.74 -8.96 -10.85 -10.75 -9.39
2.7D
2,6D
(90,90) (0.0) (0,O) (90.90) -23.39 -0.23 1.94 - 1.44 -1.11 -0.66 -0.42 -0.60 -0.65
7.88 -1.64 7.43 7.14 6.49 7.81
-2.36
-2.10 -2.62 -0.55 -0.12
-2.70
5.28
-1.81 - 1.74 -2.24 -2.19 -2.49 -2.82 -2.46
-0.16 -0.76 - 1.45 -1.22 -2.06 -0.79 -0.59
-1.11 -1.72
5.18 -2.1s -0.57 -2.04 4.46 - 1.92 -0.44 -0.63 - 1.40 5.07 -2.16
Cutoff of hartrees for SE method. From difference in energies of ground and first excited state of anion or cation. 6-31'3' basis set. calculated at the same geometry for cation, neutral, and anion with the geometries being from MM2 optimizations of the neutral. Only (0.0) and (90,901 conformations were studied (except for 1,4C where several intermediate orientations were examined). Koga et. a1..66 in a study of donor/acceptor molecules containing cyclohexane spacers. found that the maximum in the coupling does not always occur at the same orientation which gives a minimum in the total energy. For some systems such as 1,3C and 2.7D the maximum in the coupling does not occur at the (0,O) conformation, but rather when the CH2 groups are rotated about 30". Thus, the assumptions made in the choice of geometries will introduce some uncertainty in the couplings.
111. Results and Discussion A. Comparison of Superexchange Couplings with KT, ASCF, and AMP2 Values. Table 1 contains a comparison of the couplings for electron and hole transfer calculated using the SE, KT, ASCF. and AMP2 methods for truns-1,4-dimethylenecyclohexane ( 1,4C), cis- 1,3-dimethylenecyclohexane (1,3C), truns-2,6-dimethylenedecalin(2,6D), and 2,7-dimethylenedecalin (2.7D). The agreement between the SE couplings and those calculated from Koopmans' theorem from the same set of orbitals is generally quite good. One dramatic exception is the result from the neutral triplet a orbitals for (90,90) 1,4C where the SE and KT couplings differ by a factor of about 100 (-23.39 vs -0.23 mhartree). This is due to convergence problems in the SE calculation of the neutral triplet a orbita l ~ The . ~ SE ~ couplings are also in reasonable agreement with the ASCF couplings except for this same case. The results in Table 1 indicate that rotation of the CH2 groups has a significant effect on the magnitude of the couplings. The calculated couplings for the (90,90) conformations are considerably smaller than for the (0.0) conformations. The only exception to this is the KT result for 2.6-dimethylenedecalin from the neutral triplet a orbital energies which gives an increase (in the absolute value of the coupling) for the (90,90) orientation over the (0,O) orientation. The reason for the difference with the other results is unclear. Calculation of the couplings at higher levels of theory, Le., HF/6-31G* and MP2/6-3lG*, led to relatively small changes in the couplings (see Table 1) relative to HF/3-21G. The difference between the HF/6-3 lG* and MP2/6-31G* results for
J. Phys. Chem., Vol. 99, No. 4, 1995 1185
Electronic Coupling through Cyclohexane Spacers
OKT
0 XSCF" O
0
20
40
60
80
K
100
120
0
20
40
60
80
100
120
(Y,Y), degrees Figure 3. Couplings (absolute values) for 1,4-dimethylenecyclohexaneas a function of the rotation (with CZsymmetry) of the terminal CHI groups (Y,Y) from Koopmans' theorem (KT), difference in ground and excited state ion energies (ASCF), and superexchange pathway calculations: (a) a orbitals of the anion, (b) /3 orbitals of the neutral triplet, (c) a orbitals of the neutral triplet, and (d) orbitals of the dication. The ASCF values are from the anions, (a) and (b), and the cations, (c) and (d). Results are from 3-21G calculations. Neutral triplet /3 results based on a threshold of hartrees. The pathways results do not include mixed paths. hartrees; all other based on a threshold of
the (0,O) conformation indicates that correlation generally changes the coupling by 10-25%, consistent with Newton's observation^.^^ In the weaker couplings of the (90,90) conformations, such as 2,6-dimethylenedecalin, correlation effects, although still of the same magnitude as for the (0,O) conformation, are a larger fraction of the total. Interestingly, inclusion of correlation effects at the MP2 level decreases the coupling (absolute value) in the case of the cyclohexanes, while increasing the couplings in the case of the decalins. B. 1,4-Dimethylenecyclohexane.I. Effect of CH2 Rotation on Couplings. The effect of rotation of the CH2 groups in 1,4dimethylenecyclohexane on anion coupling has been examined previously by Morokuma et al.66967 They found the coupling to be sensitive to the orientation of the CH2 groups. We also found a similar dependence on orientation in our studym of the basis set dependence of couplings for 1,4-dimethylencyclohexane based on KT and ASCF calculations. In this section we use the SE pathway method to inquire into the reasons for the angular dependence. First, we investigate whether pathway calculations yield an angular dependence in accord with other methods, and then we examine contributions of individual paths and different types of interactions to the coupling. The dependence of the coupling in 1,4 dimethylenecyclohexane from calculations on the orientation of the terminal CH, groups is shown in Figure 3. Included in this figure are couplings calculated using the SE, KT, and ASCF methods (321G basis) from four sets of NBOs (neutral triplet a and ,8 orbitals, anion orbitals, and dication orbitals). The SE couplings from the neutral triplet a orbitals are in poor agreement with the KT and ASCF results. This is due probably to the poor convergence of the SE method with energy threshold for these NBO'S:~ The SE couplings from the anion a wave function are also not in very good agreement, probably for the same reason. In contrast, the SE pathways results for neutral triplet
?!, and dication orbitals are in reasonable agreement with the KT and ASCF couplings and give a good account of the dependence of the coupling with rotation. The three most important pathways (in terms of absolute magnitude for a single pathway) for coupling in anions and cations of (0,O) 1,4-dimethylenecyclohexanefrom the neutral triplet a and ,8 orbitals, anion orbitals, and dication orbitals are shown in Figure 4. The most important path in every case is a two-step one through the central CC* antibonds. The second and third most important pathways are dependent on the set of orbitals used in the SE calculation. For instance, the neutral triplet a orbitals give a McConnell-type pathway (passing through every CC bond on one side of the ring with no bonds "skipped"), while the other sets of orbitals favor paths with hops over bonds to either CC bonds or CC* antibonds. The reason for these differences is discussed in section III.B.2. The contributions of pathways for the (90,90) conformation of 1,4C are also shown in Figure 4. The contributions from the paths decrease in absolute value with change in orientation of the CH2 groups from (0,O) to (90,90), although these are not necessarily the most important (90,90) pathways. Figure 5 shows the dependence of the coupling contribution on the terminal CH2 angles for the first two pathways of the neutral triplet ,8 orbitals. The pathway through the central CC* antibond contributes significantly (about 40%) to the decrease in the coupling as the terminal CHz groups rotate from (0,O) to (90,90). The rest of the decrease in coupling due to CHI rotation is accounted for by contributions from many other pathways. This is illustrated by the results in Table 2 in which the first 20 paths are tabulated for the (0,O) conformation (neutral triplet ,8 orbitals). This table contains a running total of the coupling when the contribution of each successive (0,O) pathway is replaced by that of the identical pathway from the (90,90) conformation. [A similar type of analysis has been done for
1186 J. Phys. Chem., Vol. 99, No. 4, 1995
vo=1.50121 = 0.47 121
Vo = 0.47 121 V,=0.13[2]
v,
"
Curtiss et al.
"
Vo=0.44 141 V,=0.14 [4]
Neutral triplet p (anion) 15 A
v, = 4.07 121 v, = I .26 121
v, = -1.49 121 V, = -0.49 121
Vo = 0.72 141 V,=0.1514]
Neutral triplet a (cation)
%%??P v,= 0.40 [2]
V, = 0.24 121
V, = 0.22 14 1
Dication (cation)
v, = -0.56 121 V,= -0.19 121
v, = 0.M 121 v, = 0.23 [2]
Vo = 2.24 121 = 0.70 121
v,
Figure 4. Three most important SE pathways (3-21G) for coupling in (0.0) 1,4-dimethylenecyclohexane.Contributionsfor the same pathways for the (90,90) conformation are also given. [These are not necessarily the most important (90,90) pathways. See Table 21 Off-diagonal and diagonal Fock matrix elements (underlined) are listed for the neutral triplet /?pathways [(90,90) results in parentheses].
-5 0
10
20
30
40
50
60
70
80
90
100
110
120
(Y,v), degrees
Figure 5. Dependence of first two pathways in Figure 4 (neutral triplet 174-dimethylenecyclohexane (3-21G). Mixed paths are not included in the total coupling.
/?) and total coupling on rotation of CH2 groups of
the first 10 paths of the (90,90) orientation for 1,4C and the (0.0) and (90,90) forms of 1,3C. These tables are included in supplementary material for this article.] Substitution of the first pathway in this manner accounts for 43% of the decrease. Substitution of pathways 2-10 accounts for 37% of the decrease, and pathways 1 1-20 account for 12%. Therefore, substituting the first 20 pathways in this manner accounts for
TABLE 2: Contributions (in mhartree) of SE Pathways for (0,O) 1,4-Dimethylenecyclohexanein Order of Importance from HF/3-21G Calculation Using the /?Orbitals of the Neutral TripleP running total for (0.0) with (90.90) pathwayh (0,O) (90.90) substitut ionC (0,O)coupling with ( 14.22)r no substitutions 1. C2C3* [2] 4.07 1.26(1) 8.60 2. ClC2*,C3C4* [2] - 1.49 -0.49 (2) 10.60 3. C7R. C2R, C2C3* [4] 0.72 20) 7.72 4. C7R, C8R [ 13 -0.65 20) 8.37 5. C2C3*,C4C5*[4] 0.60 -0.19 (6) 5.2 1 (bridge jump) 6. C2C3*,C3R,C3C4*[4] -0.52 -0. I4 (9) 6.73 7. C2C3 [2] 0.39 0.13 (12) 6.21 8. ClC2*,C4C5* [2] 0.33 -0.1 I (18) 5.77 (bridge jump) 9. ClC2 [4] 0.26 20) 4.73 10. C7R. H2R. C2H2*,C3C4* [4] 0.23 CO.10 (>20) 3.81 11. C7R. C2C3* [4] 0.23 X0.20 (>20) 2.89 12. ClC2* [4] 0.22 0.19 (5) 2.77 2.01 13. ClC2, C2C3 [4] 0.19 20) 14. C7R,C2R,C8R [4] -0.19 20) 2.77 2.05 15. ClC2*,C2R,C2C3* [4] 0.18 20) 2.69 16. ClC2*, C3H3*,C4C5* [4] -0.16 20) 17. C7R [2] -0.16 -0.11 (19) 2.79 18. CIC2*, C3H3*, H3R. -0.14 20) 3.35 C8R [4] 19. C7R. C2C3* [4] 0.13 20) 2.83 20. C2R. C2C3* [4] 0.13 20) 2.31 (90.90) coupling (1.16) "The corresponding values for the (90.90) configuration are also listed. The running total is for the (0,O)coupling with the (0.0) pathway contribution replaced by the (90,90) pathway contribution. R = Rydberg, * = antibond; see Figure 1 for numbering; degeneracy in square brackets; rank of (90.90) path is in parentheses. This column contains the coupling with substitutionof each successive (0.0) pathway contribution by the (90,90) pathway contribution. For example, the value of 10.60 mhartrees is obtained by replacing the contributions of the (0.0) paths I [4.07] and 2 [- 1.491by the contributionsof the (90,90) paths 1 [ 1.261 and 2 [-0.491 and taking into account the degeneracies of the paths. This is done in a cumulative manner for all 20 paths. The energy threshold used is hartrees (any path contributing less than hartrees is neglected, Le., its contribution is zero). The total couplings differ with those in Table 1 because the ones in that table were obtained with a threshold of hartrees. 92% of the reduction in rotation of the CH;! groups from the (0,O)to the (90,90) conformation. There are several other features of Table 2 to be noted. Most of the paths involve hops to CC bonds or antibonds. Only two paths (5 and 8) out of the 20 paths listed in the table involve jumps across the ring, and only three (10, 16, and 18) involve CH bonds. Ten of the paths involve Rydberg orbitals. All 20 paths decrease in magnitude in the (90,90) conformation compared to (O,O), although some paths beyond the first 20 are more important in the (90.90) than in the (0,O)conformation. The ordering of the importance of the paths is somewhat dependent on the set of orbitals used in the SE analysis. Figures 4 and 5 and Table 2 therefore suggest an answer to the following question: Why is the coupling smaller for the (90,90) conformation? The magnitudes of the contributions from the pathways listed in Table 2 decrease by factors of up to 5 upon rotation from (0,O)to (90,90). In addition, a few paths are more important in the (90.90) than in the (0.0) conformation. If this is the case, then the coupling should be reduced by at most a factor of 5. However, the total coupling decreases by a factor greater than 10. This additional decrease appears to be due largely to pathways numbered 5 and 8 in
Electronic Coupling through Cyclohexane Spacers Table 2. These pathways are unique because the rotation of the CH;! groups changes the signs of their contributions to the coupling, both from positive to negative leading to losses of 3.16 mhartrees from pathway 5 (four symmetry-equivalent paths) and 0.88 mhartrees from pathway 8. These two pathways are also unusual among those in Table 2 in that they involve "bridge jumps" across the cyclohexane ring. The sign changes are due to the bridge jump as illustrated for path 8 in Table 2. The two matrix elements (in mhartrees) of the (0,O)1,4C donor/ acceptor orbitals (lp) with the CC* bonds (lp,c~C;!*= 67.4; 67.4
V=O.33 mH
-61.4
lp,C4C~*= -67.4) are of opposite sign because the donor/ acceptor orbitals are perpendicular to the plane of symmetry. In contrast, the two matrix elements of the (90,90) 1,4C donor/ acceptor orbitals with the CC* bonds are of the same sign (lp,CIC2* = 41.5; Ip,C4Cs* = 41 S ) because the donor/acceptor orbitals are in the plane of symmetry. The corresponding path with no bridge jump (path 2 in Table 2) gives contributions for (0,O) and (90.90) of the same sign because the two donor/ acceptor matrix elements with the CC* bonds (lp,C1C2*; lp,C3C~*)are of the same sign (see Figure 4) since both CC* bonds are on the same side of the bridge. The reason for the sign change in the contribution from path 5 is similar to that of path 8, although the explanation is more complicated because some of the NBO Fock matrix element signs are arbitrary. In rings of larger diameters and therefore smaller trans-ring interactions, we might expect the angle dependence of total coupling to simply reflect that of the principal pathways. Shepard et al.s4 have recently reported that "cross-talk" interactions are responsible for destructive interference of electronic coupling in polynorbomyl dienes based on NBO analyses. These "cross-talk" interactions are direct, through-space interactions between relays of the dienes and are similar to what we refer to as the bridge jumps in 1,4-dimethylenecycIohexane. The principal reason for larger couplings in the (0,O) conformation is that the magnitudes of contributions from the pathways of the (0,O)conformation in Table 2 are for the most part larger than in the (90.90) conformation. The reason the contributions from the (0.0) pathways are larger is that the interactions of the donor/acceptor orbitals with the bonds of the bridge are larger. For example, Figure 4 shows that the principal pathway (triplet p) contributes (35.4);!/606.5 mhartrees for the (0.0) and (20.2)2/603.3mhartrees for the (90,90). Almost all of the difference is due to the larger couplings of the donor/ acceptor orbitals to the C2C3* antibond. Is it generally true that changes in these off-diagonal interactions donor and acceptor groups are responsible for the change of couplings with angle? This question may be answered by substitution of specific off-diagonal elements in the NBO Fock matrix followed by diagonalization to obtain the Koopmans' theorem estimate of the couplings. The results of this latter procedure are now presented. Selected pi interaction elements from the triplet p Fock matrix for the (90.90) conformation of 1,4C were replaced by elements from the Fock matrix for the (0,O)conformation. The initial coupling for the (90.90) conformation is 2.12 vs 11.91 mhartrees for the (0.0) conformation. Replacement of only the interactions of the lone pair (Ip) D/A orbitals with all other orbitals substantially increased the total coupling to 10.13 mhartrees,
J. Phys. Chem., Vol. 99, No. 4, 1995 1187
TABLE 3: Effect of CHt Group Rotation on Electronic Couplings in 1,3- and 1,4-DimethylenecvclohexaneIons" anion (triplet p)" cation (triplet a)c 1,4c 1,3c 1,4c 1,3c (90.90) Fock matrix 1 1.72 +(O,O) LP'S" -10.13 -4.1 1 10.46 +8 Rydbergs' - 12.15 -4.13 11.71 10.37 -1 1.91 +8 Rydbergf 10.29 -4.05 11.67 (0.0) Fock matrix - 1 1.91 -4.22 12.59 10.42 (0,O)Fock matrixR +(90.90) LP' s -1.91 6.78 -0.09 7.2 1 +8 Rydbergs -1.31 -0.1 1 7.24 6.53 +8 Rydbergs 6.83 -2.09 -0.12 7.27 7.07 -2.12 -0.23 (90.90) Fock matrix 7.61 a Couplings are calculated by diagonalization of Fock matrices for (0,O)and (90.90) conformations and also hybrid Fock matrices in which specific off-diagonal elements are taken from the matrix for the other conformation. The 6-3 lG* basis set was used. Koopman's theorem electronic coupling energies for anion using B orbitals from neutral triplet. Koopman's theorem electronic coupling energies for cation using a orbitals from neutral triplet. Replacement of all couplings (off-diagonal elements of the NBO Fock matrix) associated with the lone pair (donor/acceptor)orbitals with interaction elements from the (0.0) Fock matrix. e Replacement as in footnote d and in addition replacement of all interaction elements associated with the Rydberg orbitals on the terminal (donor/acceptor)carbons. f Replacement as in footnote e and in addition replacement of all interaction elements associated with the Rydberg orbitals on the carbons in the cyclohexane bridge which are attached to the donor/acceptorcarbons. Replacements of (0.0) Fock elements by (90.90) elements as in footnotes d-f.
or about 80% of the increase to 11.91 mhartrees coupling for the (0,O)(see Table 3). Because this was achieved by replacing the interactions of the D/A lone-pair orbitals, we may conclude that larger interactions in the (0,O) conformation are responsible for most of its increased coupling over the (90.90). Additional replacement of all interactions of the eight Rydberg orbitals on the D/A carbon atoms and the eight Rydbergs on the adjacent carbons in the cyclohexane ring restores all of the 11.91 mhartrees found for the pure (0,O) Fock matrix. This exact match is accidental, however, as seen by the too large 12.15 mhartrees coupling obtained when only the Rydbergs on the CH;! donor/acceptor groups are replaced or from other entries in Table 3 for which exact matches are not found. In general, from Table 3 we can see that better pathways due to larger D/A Fock elements are responsible for most of the increases of (0.0) vs (90,90) and that the elements of the Rydberg orbitals on the D/A and adjacent carbon atoms are responsible for much of the remainder. There are exceptions to the generalization that the pathways are larger in the (0,O)than in the (90,90) conformation. The axial CH bonds and antibonds on the carbons adjacent to the D/A groups (carbons 1 and 4) are almost parallel to and interact well with the D/A orbitals in the (90,90) conformation but are prohibited by symmetry from interacting in the (0,O) conformation. Pathways including these bonds are substantial in the (90,90) but do not contribute in the (0.0). The generalizations given above are still correct though because these CH pathways in (90,90) 1,4C nearly cancel each other out. [See Table S1, in supplementary material, which gives the 10 most important pathways for (90,90) 1,4C.] This cancellation does not occur in (90,90) 1,3C where the CH pathways lead to a much weaker angle dependence (see below). 2. NBO Fock Matrix Elements: Accuracy and Convergence of SE Pathways. In Table 4 coupling contributions are listed for four of the more important paths (two occupied and two unoccupied) from the SE calculations based on the six sets of NBOs for the (0,O)and (90,90) 1,4C conformations. In each case the contributions of the paths are decreased in the (90,90)
1188 J . Phys. Chem., Vol. 99, No. 4, 1995
Curtiss et al.
TABLE 4: Contributions (v) of Selected SE Pathways for (0,O) 1,4-Dimethylenecyclohexaneto Coupling at the HF/3-21G Level of Calculation (in mhartreesy pathwayb c2ci*
c2ci
CIC?,C2Ci,CiCI
orbitals used neutral triplet a cation /3 dication neutral triplet fi anion a dianion neutral triplet a cation fi dication neutral triplet ,8 anion a dianion neutral triplet a cation /3 dication neutral triplet /3 anion a dianion neutral triplet a cation /3 dication neutral triplet fi anion a dianion
V 1.30 (0.40) 2.83 (0.90) 2.24 (0.70) 4.07 (1.26) 1.50 (0.47) 2.05 (0.67) -0.22 ( 1 9 8
..A. KT(3-21G) X
KT(6-31G')
+ASCF(3-21G) -A-
ASCFI6-31G*I
Number of Bonds Figure 12. Plot of coupling versus number of bonds for cations with ring spacers as in Figure 1 1. Measured hole transfer rates in cations20 were used to estimate the experimental couplings with the additional assumption that the reorganization energies are the same as those for the anions.
L a r s ~ o n ~calculated ~ , ~ ' the ratio to be 12.6 and 13.6, respectively. Their calculations on this system include naphthyl and biphenyl donor/acceptor groups and are based on the STO-3G basis set. The larger ratios could be due to the use of the minimal STO3G basis set. There are other discrepancies in the comparisons made here. The comparison in Figures 11 and 12 indicates that the calculated couplings are larger than experiment by a factor of about 2 for the anions. This factor is smaller for the 3,16androstane. For the cation this is also true for the ASCF and AMP2 calculations. The simple correction based on MO coefficients for comparison of CH2 with biphenyl and naphthalene may not be adequate. The KT results for the 3,16A cation appear to be too large as were KT results on cations of straight-chain hydrocarbons.60 It appears that KT calculations overestimate couplings for cations at long distances, but the reason for this error is not clear. The calculations of Braga and Larsson do not agree as well with experiment as the calculations presented in Figures 10 and 11, although Braga and Larsson used the actual biphenyl and naphthyl D/A groups. The accuracy of calculations of long-distance couplings and the reasons for the discrepancies noted here will be explored in subsequent work. IV. Conclusions The following conclusions can be drawn from this theoretical study of the electronic coupling in systems spacers based on cyclohexane units. 1. The magnitudes of the couplings calculated from the SE pathways method are in reasonable agreement with those from Koopmans' theorem and the ASCF method, if convergence is attained with energy threshold. This is in agreement with previous studies on other systems.45 The 3-21G basis set gives results in reasonable agreement with larger basis set calculations and those including correlation effects at the second order Moller-Plesset level.
2. Paths involving hops over bonds make the largest contributions, in agreement with previous s t ~ d i e s . j ~ The -~~.~~ dominant pathway in every case is through CC antibonds. There is a significant dependence of the SE analysis on the set of orbitals used which is due largely to the differing energy denominators B,. 3. Rotation of the CH2 ,z donor/acceptor groups from the (0,O) to the (90.90) conformations decreases couplings for all the molecules studied by factors which vary with the method. The decreases are typically by factors of about 10 (1,4C+), 5 (1,4C-), 4 (2,6D-), 2 (2,6D+), and 1.3 (1,3C+ or 1,3C-). The decreases occur because (a) the contributions from most paths decrease by factors of 2-5 in the (90,90) conformation due to poorer interaction of the donor/acceptor orbitals with CC or CC* bonds and (b) the contributions from some pathways change sign upon rotation. The smaller decrease for 1,3C is due to the significant contribution of axial CH bond paths in the (90,90) conformation. Similar paths occur for (90,90) 1,4C but cancel each other out. 4. The interactions (off-diagonal NBO Fock matrix elements) between nearest CC bonds are much larger than the interactions between nearest-neighbor CC* antibonds. This is confirmed herein by the larger spread of energies of the canonical bonding orbitals compared to canonical antibonding orbitals. This observation establishes that the small nearest-neighbor interactions for u* orbitals is not just a peculiar feature of the NBO scheme but is a real property of saturated hydrocarbons. This seems to be a characteristic of the systems containing cyclohexanes as well as other saturated systems such as trans alkyls. 5. The calculated couplings are compared to experimentally derived couplings from molecules containing the same spacers, but with biphenyl and naphthyl donor/acceptor groups. Reasonable agreement is found between theory and experiment on the distance dependence, except for the Koopmans' theorem calculations for steroid cations. For these long spacers the KT estimate is much larger than experiment, although there is reasonable agreement for the anion.
Acknowledgment. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Contract W-3 1-109-ENG38. Supplementary Material Available: Tables of contributions of SE pathways for (90,90) 1,4-dimethylenecyclohexane and (0,O) and (90,90) 1,3-dimethylenecyclohexanes (3 pages). Ordering information is given on any current masthead page. References and Notes (1) Siders, P.; Cave, R. J.: Marcus, R. A. J. Chem. Phys. 1984. 81, 5613-24. (2) Leighton, P.: Sanders. J . K. M. J. Chem. Soc., Chem. Commun. 1985, 24-5. ( 3 ) Cave. R. J.; Siders, P.: Marcus. R. A. J. Phys. Chem. 1986, 90, 1436-44. (4) Closs, G. L.; Calcaterra, L. T.; Green, N. J.: Penfield, K. W.; Miller, J. R. J. Phys. Chem. 1986, 90, 3673-83. ( 5 ) Heiler, D.; McLendon. G.; Rogalsky, P. J. Am. Chem. SOC.1987, 109, 604-606. ( 6 ) Penfield, K. W.; Miller, J. R.; Paddon-Row, M. N.; Cotsaris. E.: Oliver, A. M.; Hush, N. S. J. Am. Chem. Soc. 1987, 109, 5061-5. (7) Miller, J. R. In Radiation Research, Proceedings of the 8th International Congress on Radiation Research; Fielden, J. F. F. E. M.. Hendry, J. H., Scott, D., Eds.: Taylor and Francis: London, 1987: Vol. 2. pp 96-101. (8) Sakata, Y.; Nakashima, S.; Goto, Y.: Tatemitsu. H.: Misumi, S. J. Am. Chem. SOC.1989, 111, 8979. ( 9 ) Wasielewski, M. R.; Niemczyk, M. P.: Svec, W. A,; Pewitt. E. B. J. Am. Chem. SOC. 1985. 107. 1080-2.
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