4050
J. Phys. Chem. 1993,97, 40504058
Theoretical Study of Long-Distance Electronic Coupling in HzC(CH2),2CHz Chains, n = 3-16? Larry A. Curtiss,’ Conrad A. Naleway, and John R. Miller Argonne National Laboratory. Argonne, Illinois 60439-4837 Received: October 6, 1992
The long-range electronic coupling in model chain alkyls H2C(CH2),+2CH2, n = 3-16, and 1,4-dimethylenecyclohexane has been investigated using ab initio molecular orbital theory to assess dependence of the results on basis set and method of calculation. Both anion and cation ?r couplings were examined. Basis sets ranging from minimal STO-3G to triple-zeta plus polarization were used, and diffuse orbitals were added to some of the basis sets. Small basis sets such as the split-valence 3-21G generally gave results in reasonable agreement with the larger basis sets. Couplings were calculated from differences in Hartree-Fock energies or from lower level “Koopmans’ theorem” approximations based on orbital energies of the dianion, anion, neutral triplet diradical, monocation, and dication of the donor-acceptor molecules. The distance dependence is found in some cases to vary significantly with method of calculation, especially at distances greater than 7 A. In most cases the distance dependence was not purely exponential. Small basis sets performed very poorly in calculating long-range direct (“through-space”) interactions but showed dramatic improvement when augmented by “ghost” basis functions located between the interacting groups. Agreement between the 3-21G (+ghost) results and those from larger basis sets provides evidence that direct, through-space interactions are reliably calculated. The direct interactions decrease much more rapidly with distance (0 = 3.0 A-I) than the couplings for the chain alkyls (0 < 1 A-I), demonstrating the importance of superexchange via through-bond interactions. We speculate that the reason that a small basis set such as 3-21G generally does well in the calculation of long-distance couplings in molecules is that basis functions (ghost orbitals) on the intermediate atoms assist in computation of superexchange interactions even when the intermediate atoms themselves are not involved. Finally, electronic couplings in 1,3-dimethylenecyclohexane,1,4dimethylenecyclohexane,and 2,6-dimethylenedecalinare calculated and compared with experiments on molecules with the same spacer groups.
I. Introduction Electronic coupling governs much of the distance and orientation dependence of long-distance electron-transfer rates. The idea that the material between the donor and acceptor has a major role in promoting the electronic coupling interaction is well-established.’ It has been the subject of many experimental and theoretical investigations,but still much remains to be learned about it. It is desirable to develop theoretical methods based on ab initio molecular orbital theory for studying electron transfer, especially in the light of the increasing ability of the ab initio methods to handle large molecules. To develop and apply such methods, it is necessary to determine what level and method of calculation is required to adequately calculate the long-distance coupling. There have been numerous papers reportinginvestigations into electronic coupling based on ab initio molecular orbital theory.] Several of these have dealt with the dependence of the results on the basis sets used.24 Cave et al.z found that Gaussian basis sets with diffuse functions were capable of predicting the expected exponential decay of electronic interactions, even at large internucleardistances. They examined electron transfer in metal dimers XY+ X+Y, where X, Y = Be, Mg, Ca, Zn, or Cd. Jordan and Paddon-Row3 recently reported on the investigation into the ?r+,?r- and a:,?rf couplings of a series of rigid nonconjugated dienes. They used several basis sets including STO-3G, 3-21G, 6-311G, 6-31+G, and D95V and found that the small Gaussian basis sets, though much more modest than those used by Cave et al., did reasonably well at predicting the distance dependence of the couplings. They also found significant deviations from exponential behavior, although at long enough distancesthese deviations became smaller. In a recent paper’ we -+
* Author to whom correspondence should be addressed.
Work supported by the US.Department of Energy, Officeof Basic Energy Sciences, Division of Chemical Sciences, under Contact W-3 1-109-ENG-38.
0022-3654/93/2091-4050$04.00/0
used a superexchange-pathways model based on ab initio molecular orbital theory with the 3-21G basis set to investigate ?r couplings in two model systems, the anion of butane-1,4-diyl (-CH,CH,CH,CH;) and the anion of 1,4-dimethylenecyclohexane (-CH,C,H,,CH;). No attempt was made to assess the reliability of the use of the 3-21G basis set in that work. In this paper we present a systematic investigation of the dependence of long-range ?r electronic coupling on basis set as well as method of calculation in the model chain alkyls H~C(CH~)&HZ,n = 3-16, and 1,4-dimethylenecyclohexane using ab initio molecular orbital theory. The purpose of this work was to determine (1) what size basis set is required to obtain an accurate account of electronic couplings within the HartreeFock approximationand (2) the adequacy of Koopmans’ theorem for evaluation of electronic couplings. Basis sets ranging from minimal STO-3G to triple-zeta plus polarization were used. In addition, diffuse functions were added to some of the basis sets, or basis sets were augmented by placing functions in between donor and acceptor groups where there are no atoms. Couplings were calculated using Koopmans’ theorem applied to various wave functions and the ASCF method. Theoretical methods are presented in section 11, and results on points 1 and 2 are presented and discussed in section 111. Also discussed in section I11 is the distance dependence of anion and cation couplings and a possible reason for why small basis sets such as 3-21G do well in the calculation of long-range electronic coupling of molecules. Finally, in section IV electronic couplings in 1,Cdimethylenecyclohexane, 1,3-dimethylenecycIohexane,and dimethylenedecalin are calculated and compared with experiment. 11. Description of Theoretical Methods
A. Basis Sets. In this work we used ab initio molecular orbital theory5 for the calculation of the couplings. We examined the dependence of the coupling on a series of basis sets of increasing 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 16, 1993 4051
Long-Distance Electronic Coupling size: (1) STO-3G6 minimal basis set; ( 2 ) 3-21G,6 split valence basis set; (3) 6-31G,6 split valence basis set with additional gaussians for the core orbitals; (4) 6-31+G? split valence basis with diffuse (+) sp orbitals on carbon; ( 5 ) 6-3 1G*,6split valence basis with polarization functions on carbon; (6) 6-31++G**,6 split valence basis with polarization functions on carbon and hydrogen, diffuse sp functions on carbon, and diffuse s functions on hydrogen; (7) 6-311G**,6 triple split valence basis with polarization functions on carbon and hydrogen; and (8) TZP,' triple zeta basis with polarization functions on carbon [5s3p] and hydrogen [3s]. B. Methods for Calculating Couplings. The molecules considered here have identical donor and acceptor groups, the simple one-carbon r groups 'CH2. Couplings, V, relevant to electron transfer from -CH2 to 'CHI (anion) and from 'CH2 to +CH2 (cation) are related to the energydifference A between theground and first excited state of the anion or cation by8s9
V = A12 (1) These two states differ by promotion of an electron between the symmetric and antisymmetric orbitals delocalized over the donor and acceptor groups. In the work reported here the couplings are calculated by either the ASCF method or Koopmans' theorem.10 In the ASCF method, the electronic coupling is calculated from the differencein unrestricted Hartree-Fock (UHF) energies (E) of the ground (gr) and first excited (ex) states of the cation 2V= A = i[E(M;,) -E(ML)]
(3) 2 V = A = *[E(M,,) - E ( M i x ) ] We refer to thesecouplingsas ASCFvalues. This method includes electronic relaxation effects. In the method based on Koopmans' theorem (KT), the orbital energies of some chosen wave function are used to derive anion and cation couplings. In this work we have used orbital energies of five different wave functions. The donor and acceptor orbital energy levels of thevarious wave functionsare illustrated in Figure 1 along with the ionization (IP) and electron attachment (EA) processes that are calculated within the KT approximationlo to obtain the couplings. The cation couplings are calculated from these IP's and EA'S as follows. 1. Dication LUMOs. The A is calculated from the difference between the first two EA's of the dication (see Figure 1) within the KT approximation: (4)
The final states of the electron attachment transitions are the ground and first excited states of the cation having the frozen orbitals of the dication. 2. Cation B LUMOs. The A is calculated from the difference between the first two EA energies of the /ILUMOs of the cation (see Figure 1) within the KT approximation: A = EA, -EA, = * [ E ( M l , ) - E(M:x)]
/ EA2
EA,: , ,
I I
, ,
I I
,
I
, ,
I I
: :
, ,
+ e -M'[..(b,)"]
I
!
j
i
EA,
M,+ + e -Mt[..(ag)"]
EA,
..
...:.: ........
2
EA,
Mi + e -Mo[..(b,)a(ag)Pl
EA,
Neutral Triplet p LUMOs
j !EA, EA,: i
I
+e
IP,
M+[..(a,)"] + e
IPI
d-M+[..(b,)"]
Me + e -Mo[..(b,)Yb,)~1
4
;L
Anion
I I
I
.
I
L -
aHOMOs
I
, I
LL-
Mo-
i ;EA2 EA,! j ,
. . .............. ,
Cation PLUMOs
' 'IP, PI!i j
-
* , ,, ,,
..+....... I M-- Mo[..(aJa(b,)~]
+e
IP1
M.-Mo[..(b,)a(bu)b]
+e
IP,
(2)
or anion
A = EA, -EA, = i [ E ( M ; , ) - E ( M L ) ]
Dication
(5)
The final states are frozen orbital (ground-state cation) neutral singlet states. 3. Neutral Triplet a HOMOs. The A is calculated from the differencebetween the first two I P Sof the neutral triplet diradical (see Figure 1) within the KT approximation: A = IP, - IP, = *[E(M;,) - E(Mzx)I (6) The final states are the frozen orbital (neutral triplet diradical) ground and first excited state of the cation. This method has been used by Liang and Newton4for calculationof cation coupling.
Figure 1. Illustration of models used with Koopmans' theorem to obtain the couplings A/2. The four horizontal solid lines in each panel represent the energies of four spin orbitals which are delocalized over the donor and acceptor groups (two a's on the left and two j3's on the right). The short vertical arrows represent the electrons occupying these spin orbitals, the vertical dashed arrows represent electron attachment and ionization energies, and the horizontal dashed lines correspond to zero energy. The symmetry labels on the configurations correspond to those of the n = 4 alkyl chain.
The anion couplings are calculated from the IP's and EA's as follows. 4. Neutral Triplet B LUMO's. The A is calculated from the difference between the first two EA'S of the neutral triplet diradical (see Figure 1) within the KT approximation: A = EA, - EA2 *[E(M,,) - E ( M i x ) ] (7) The final states are the frozen orbital (neutral triplet diradical) ground and first excited state of the anion. This method has been used by Liang and Newton4 for calculation of anion coupling. 5. Anion a HOMOs. The A is calculated from the difference between the first two IP energies of the a HOMOs of the anion (see Figure 1) within the KT approximation:
A = IP, - IP2 = i [ E ( M i , )- E(M:x)]
(8) The final states are frozen orbital (ground-state anion) neutral singlet states. This method was used in our previous study' of pathways in anion coupling based on natural bond orbitals (NBO). 6. Dianion LUMOs. The A is calculated from the difference between the first two IP's of the dianion (see Figure 1) within the KT approximation: A = IPI - IP2 = i [ E ( M , , ) - E ( M i x ) ]
(9) The final states are the frozen orbital (dianion) ground and first excited state of the anion. Braga and Larsson" have used this method to calculate anion couplings. Thus, the KT calculation of cation and anion couplings from neutral triplet diradical, dication, and dianion wave functions correspond to the correct final configurations but with orbitals from a different species, whereas those calculations based on the anion and cation wavefunctionshave incorrect final configurations
4052
The Journal of Physical Chemlstry. Vol. 97, No. 16. 1993
n=4
Curtiss et al.
n=5
A
n=6
n=7
n=8 Figure I Structures for chain alkyls CHz(CHd..FHz. n = 4-8.
withorbitalsfrom thecorrect species. Thevalidityofthevarious KTvalues willdepend on theimportnnceofelectronicrelaxation, in each case. The calculation of couplings via Koopmans’ theorem methods asdscribed abme isquivalent to thedifferencein theeigenvalues of the molecular orbitals corresponding to the appropriate symmetric, V, and antisymmetric, @-. combinations of the p orbitals on the CH2 donor and acceptor groups?
2Y= A = *[e(+-)
-e(&]
(10)
We use a sign convention similar to that used by Newton8 for the electronic coupling V obtained from both the KT and ASCF methods. The interaction is defined as positive when occupncy of thesymmetricorbital from themany-electron wave function is preferable to occupancy of the antisymmetric orbital 4.; it is negative if occupancy of g is preferable. Calculations on the cations, anions, and neuttal triplets were done using the spin unrestricted Hartree-Fock method (UHF). The UHF wave functions for the cations and anions were charge delocalized. The UHF expectation values of S’for the cations ranged from 0.83 to 0.89 in the ground state and were close to 0.75 in the excited state. The expectation values for the anions ranged from 0.75 to 0.82 for basis sets without diffuse functions. In the case of the anions the ground- and excited-state s2 values were nearly the same. The expectation value for the triplets were geirerally close to 2.0. Hence, the ASCF values for the cations could be affected by the spin contamination. C. StNctuns. The structures of the chain alkyls HzC(CHZ),ZCHZ,n = 3-16, wereconstructed such that thecarbons along the backbone are in a trans conformation (CC distance = 1.54A,CHdistances= 1.116A,HCCangles= 109.47°).The structures for n = 4-8 are illustrated in Figure 3. For odd values of n, the structures have C b symmetry, and for even values of n, the structures liuve Clh symmetry. The structures of trans-I.4dimethylenecyclohexane, cis-I ,3-dimethylenecyclohexane.and trans-2,6-dimethylened~linare illustrated in Figure 3. All are equatoria1,quatoriaI (e,e) isomers. They have Czh. C, and Sz symmetries, respectively. The geometries were obtained from molecular mechanics(MM2) optimization.’’ Thebonddistancs and bond angles were adjusted slightly to obtain structures with symmetry. Both (0.0) and (90,90) orientations of 1,Cdimethylenecyclohexane were considered. The (0,O) orientation corresponds to the hydrogens of the terminal CHz groups being in the planeofsymmetry (seeFigure3a), and the (90.90) orientation corresponds to the hydrogens of the terminal CH2groups being
++
L.
(C)
3
~I
Flgnrc 3. Structures of
(a) l,ddimethylcnccyclohexane [(O,O) wnformation]. (b) 1,3dimsthylcn~clohexane,and (E) 1raw2.6-dimsthylenedsalin. Hydrogens on the ring carbons are not shown. perperdicular to the plane of symmetry. The coupling in 1.4dimethylenccyclohexanewas studied previously by Ohta et al.13 In some cnscs we have found that couplings are strongly
dependentonthegeometriessothatcarcfulconsiderntionofboth geometry and vibration may be important toobtaining an a m r a t e assssmentofthecoupling. Wearecurrentlystudyingthisaspect and will report on it separately. In this study all couplings were calculated at the Same geometry.
m. ~anrltr.~ld Disfcpsim A. Dependence on Buis set .adMetbod of c.lnl.tioa. The basis set dependence of the electronic couplingscalculated from Koopman’stheoremand the ASCFmethod wasinvestigatedusing the eight hasis sets described in section 1I.A. The KT results for then = 4 and n = 8 alkyl chains and (0.0) l.4-dimethylenecyclohexane are listed in Table I. Included in this table are results derived from the orbital energies of the dianions. anions, neutral triplet dirndicals, cations, and dications of these three species. The ASCF values for the n = 4 and 8 chains and the (0,O)and (90.90) orientations of 1,4-dimethylenecyclohexaneare listed in Table 11. Also listed in Table I1 are results from second-order Maller-Plesset perturbation theory’ with the 6-31G’ basis set (MP2/6-31G*) toassss theeffectsofcorrelation on thecoupling. Figures 4 and 5 contain the anion and cation coupling from the KT and ASCF methods for the n = 8 chain and (0,O) 1,4-dimethylenccyclohexane,respectively, as a function of basis set size. In general, the cation couplingsfrom the KT and ASCF results are not very dependent on the bnsis set. with the magnitude bccoming slightly smaller with increasing bnsis set size. There are two exceptions. First, the STO-3G basis does poorly in some cascs. For example, in the case of (0.0) l.4-dimethylenccyclohexane it gives a significantly smaller value than the larger basis sets. The other exception is the KT results for the n = 4 chain from the@LUMOs of thecation whichvary over a 12-mH range
Long-Distance Electronic Coupling
The Journal of Physical Chemistry, Vol. 97, No. 16, 1993 4053
TABLE I: Basis Set Dependence of A/2 (in mH) from Koopmans' Theorem for H2C(CH2)2CH2, H2C(CH2)&H2, and (0,O) 1,4-Mmethylenecyclohe~* cation coupling
anion coupling ~
basisset
dication LUMOs n = 4 n = 8 1,4C
STO-3G
-18.03 3-21G -14.96 6-31G -13.54 6-31+G -12.10 6-31G' -13.10 6-31++G** -11.51 6-311G** -12.33 TZP -1 1.74
cation j3 LUMOs n = 4 n - 8 1,4C
neutral tri let a HOM& n = 4 n = 8 1,4C
neutral tri let j3 LUM& n = 4 n = 8 1,4C
-3.57 -13.84 -2.01 -3.78 -24.18 -7.96 -0.50 -18.98 -7.78 -8.25 -3.28 -9.20 -20.06 -7.86 -4.22 -10.65 -7.73 -6.36 -3.02 -9.06 -19.10 -7.51 4 . 2 8 -7.89 -7.84 -2.68 -2.30 -8.13 -18.44 -7.20 -4.56 4 . 8 6 -7.98 -5.26 -2.81 -8.18 -18.23 -7.14 -5.30 -7.44 -8.12 -1.27 -1.90 -8.08 -17.42 -6.81 -5.54 +4.34 -8.20 -3.87 -2.77 -9.46 -17.73 -6.92 -5.44 -2.91 -8.03 -2.68 -2.36 -8.91 -17.61 -6.89 -5.49 -3.41
-2.70 -3.87 -3.60 -3.40 -3.44 -3.25 -3.36 -3.25
-2.68 -4.11 -3.73 -3.19 -3.53 +3.49 -2.25 -2.43
4.17 -11.91 -11.70 +0.26 -11.70 +7.55 -10.62 -10.49
anion a HOMOs n=4 n=8 -17.13 -2.88 -12.41 -4.24 -11.40 4 . 0 5 -15.07 -6.93 -10.33 -3.84 +17.63 +7.47 -9.96 -3.92 -10.07 -3.98
1,4C -3.43 -6.97 -6.79 -5.95 -7.18
-' -7.22 -7.08
~~~~~
dianion HOMOs n = 4 n = 8 1,4C -18.39 -2.15 -3.84 -13.64 -3.42 -8.96 -12.08 -3.16 -8.71 -8.96 -2.68 -' -11.52 -3.00 -8.84 -8.11 -2.48 -8.18 -10.38 -2.96 -9.04 -9.10 -2.70 -8.61
a n = 4 is H2C(CH2)2CH2; n = 8 is H2C(CH2)&H2; and 1,4C is (0,O)1,4-dimethylenecyclohexane;cation fl L U M O s and anion a H O M O s were obtained using cation and anion ground states. See text for sign convention. Unable to obtain converged solution.
TABLE Ik Basis Set Dependence of A/2 (in mH) from Difference in Energies of Ground and First Excited States (ASCF)' 1,4-dimethylenecyclohexane
H2C(CH2)&H2 n-4
(090)
n=8
method basis
anion
cation
HF/STO-3G HF/3-21G HF/6-31G HF/6-31+G HF/6-31G* MP2/6-3 1G* HF/6-3 l++G** HF/6-311GS* HF/TZP ground stateb first excited stateb
-19.95 -13.53 -11.15 -4.48 -10.91 -11.31 +4.12 -8.04 -6.33 2A, 2B"
-23.65 -20.16 -18.93 -17.88 -18.83 -18.84 -17.60 -18.17 -17.84 2Bu 2A,
anion
cation
-2.36 -5.15 -3.82 -6.09 -3.49 -5.66 -2.78 -5.40 -3.32 -5.45 -3.82 -7.43 +3.48 -5.13 -3.25 -5.24 -2.52 -5.15 2A, 2B" 2Bu 2A, See text for sign convention. At 3-21G level. Unable to obtain converged solution. .in
'"
(90,90)
anion
cation
anion
cation
-4.08 -10.85 -10.68 -1.47 -10.75 -9.39
-2.49 -6.54 -6.62 -6.86 -1.26 -6.35 -1.49 -7.47 -7.39 2B, 2Au
-0.02 -1.72 -1 -89 +0.68 -2.04 -1.40
-0.12 -0.42 -0.5 1 -0.57 -0.60 -0.65 -0.66 -0.54 -0.13 2B" 2A,
-C
-1 1.04 -10.20 2AU 2B,
-C
-1.13 -2.16 2A, 2Bu
-15
I
(a) anion -10
-5
0
5
5 5 8 E
r
10
0
1
2
3
4
5
O
(b) cation
KT
6
7
6
1
--
,
10
0
I
Ia HOMOs. t l i ~ l e neutrail t
X ~CF_LvHF,catsnl
t
9
C .--
>
14 12 10
lo
-
8 -
(b) cation
0 K T ( a H O M O S . triplet neutral1 X ~CF_LVHF-cat~nl ,
-
fBK T I B 0 KT IL%?*,
-
LUMOs. cation)
__
- --- -- ----F-p-.-ca. -b.+-4-4 4 -x* - --x- -6 dlcabnl-
6 4 -
0
,
I
I
I
I
I
I
I
I
1
1
2
3
4
5
6
7
8
9
10
Basis set Figure 4. Basis set dependence of electronic couplings, V, in (a) the anion and (b) the cation of C&(CH2)&H2 (n = 8). Basis set numbering is the same as defined in section 1I.A: 1 = STO-3G, 2 = 3-21G, 3 = 6-3 lG, 4 = 6-31+G, 5 6-31G*, 6 6-31++G**, 7 6-311G**, 8 = TZP. (Absolute value of Vis plotted in (b).)
and differ considerably (16 mH) from the ASCF results (see Table I). The most important conclusion to be drawn from the results is that modest basis sets such as the split valence 3-21G basis set generally provide results which agree well with results
2 -
0
1
1
4054 The Journal of Physical Chemistry. Vol. 97. No. 16.1993 dependence of anion couplings on the basis set. This is due to thefact thatattheHFlevel theaddedclectronisnotbound (i.e., the electron affinity is positive). Jordan and Paddon-Row' have found that increasing the size of basis set, especially by adding diffusefunctions,leads todiscretecontinuum solutionswhich are not valid descriptionsof the anion states. This is the reason that the 6-31+G and 6-31++G** basis sets give couplings that are significantly out-of-line with the other basis set results. The couplings from the 3-21G basis are generally in reasonable agreement with the larger basis sets without diffuse functions including the 6-31G*, 6-311G**, and TZP basis sets. Two exceptions are the KT results from the neutral triplet and the ASCF results for then = 4 alkyl, where considerable variation exists over the range of basis sets. Thedependenceoftheanioncoupling on theorientationofthe CHzgroupsof 1,4-dimethylenecyclohexanehas been pointed out by Ohtaet al.')inastudyusing thesplitvalencc4-3lGbasissct. The ASCF results in Table I1 for the cations and anions indicate that thedifferencein thecoupling between the (0.0) and (90.90) orientations given by the 3-21G basis is in g d agreement with that of larger basis sets (ignoring the basis sets with diffuse functions for the anions). For the cation the difference is about 6 mH, while for the anion it is about 9 mH. The MP2/6-31G* results in Table I1 indicate that the correlation effects are usually not large for the splittings of the anions or cations, consistent with a suggestion of Newton.' The largest change is an increase from 5.5 to 7.4 mH for the n = 6 cation. The magnitudes of the cation couplings arc much more dependent on the method of calculation than on the basis set used. The results for the n = 8 chain cation in Figure 4 and for the (0.0) 1.4-dimethylenecyclohexane cation in Figure 5 show that the couplings obtained from the different methods cover a fairlywiderange with the KTvalues from thecation anddication being the lowest and those from the neutral triplet being the highest. The ASCF results, which include electronic relaxation, fall between the KT results. The anion couplings in Figures 4 and 5 are not as dependent on method as those of the cation (neglecting the diffuse basis set results). The KT results are in generalagreementwith the ASCFresultsfrom theneutral triplet radical. The dependencc on method is discussed in more detail in section 1II.D. B. "Ghost" Function Effect.The g d agreement between thecationcouplingscalculated with thesplitvalmce3-21Gbasis set and with the larger basis sets is somewhat surprising. For instancc, the work of Cave et al.2 indicated that Gaussian basis sets with diffuse functions arc required for calculation of longrangeelectroniccouplingon the basisofcalculationson diatomics at different internuclear separations. In this section we present a possible explanation, referred to as the "ghost" basis function effect, for the reasonable performance of a small basis set such as 3-21G in the calculation of the long-distance coupling in
molecules. In Figure 6 the cation couplings obtained from (a) ASCF calculations and (b) Koopmans' thcorem applied to the neutral triplet a orbitals are plotted as a function of distance for the HzC(CH3,zCHz molecules. Both 3-21G and 6-31++G8* results are included in the figure. The 3-21G results arc in good agreement with results from the larger 6-3l++G** basis set which includes diffuse functions, consistent with our conclusion in section II1.A that a modest basis set does well in calculating couplings. Alsoin Figure6barecouplingscalculatedwiththe 3-21Gand 6-31++G** basis sets for the interactions between two CHI molecules positioned at the location of the terminal CHI groups with the third hydrogen added along the CC bond with the same
CHdistanceasinthetenninalCH,groupsasillustratedinFigure 7a. This calculation provides an approximation to the through-
Curtis et al.
0.m1I 2
" " "
4
8
'
8
"
to
'
" '
1
14
18
18
"
12
'
M
CC distance, angstroms FI-6. hpendence of cation coupling V(absoluteva1ue) on distance in CHI(CHZ),ZCHZmolecules using the 3-21G and 6-31++G** basis sets: (a) ASCR (b) Koopmans's thcorem applied to neutral triplet (a HOMO'S). Also in (b) is the coupling between two CHI molcEuIes positionedatthedonorandaoceptorgmupsofthechainallrylsasillustrated in Figure 7a. The numbers in the plot correspond to the value of n.
(b) Figum7. Illustrationof (a) positionsof twoCH~gmupsused tocalculate thedirectinteractioninthen=lalkylchainand(b) thcpositionofCH. between the two CH, group to asscss through-bond interaction. The shaded atoms are ones that are removed from the alkyl chain and on which ghat functions remain in some of the calculations.
space (TS)interaction of the terminal donor and acceptor group of the HzC(CHz),zCHz molecules,Le., the interaction which occurs without the aid of the material h w e e n the donor and acceptor. As expected, the TS interaction falls off much more rapidlythan thesplittingsfor the full molecule(Figure6b) which includes both through-spaccand through-bond (TB) interactions. Asimplermodel for TSinteractionsisdiscussed below (seesection 1II.E). We note that the rapid decrease of 7s interactions is an important factor in couplings, because recent st~diesI,'~~b bad on localized orbitals found that TB pathways involve hops over
Long-Distance Electronic Coupling
The Journal of Physical Chemistry,Vol. 97. No. 16, 1993 4055
I
Fibre 8. Illustration of pathways for anion coupling in 1,Cdimethyl. enecyclohexane (ref I).
one or more bonds. These hops could be considered TS interactions. Two such pathways for (0.0) 1.4-dimethylenecyclohexane are illustrated in Figure 8. The values of Vin Figure 6b for the chain molecules indicate that the 3-21G basis set is inadequate (compared to the larger 6-31++G0* basis set) for TS interactions at distances greater than 4 A. Pathway analysis' of the coupling in a molecule such as 1.4-dimethylenecyclohexanesuggests that paths involving hops of more than 4 A make important contributions to the total elearon coupling. It is surprising then that the 3-21G basis set does so well in the calculationof couplings for 1,4-dimethylenecyclohexaneandother molecules in this study considering its poor performance in calculating the long-distance through-space interactions as illustrated in Figure 6b. The results of calculations that we have carried out suggest that the3-21G hasissetgivesagooda~untofthelong-distance coupling in the alkyl chains and 1,4-dimethylenayclohexane becauseof a ghost functioneffect. TheTS interactiondescribed above for the alkyl chains without the spacers was calculated with basis functions included on the empty carbon and hydrogen sites of the chain between the aacptor and donor groups (see Figure 7a). We refer to the basis functions on the nonexistent atoms as ghost functions. The results in Figure 6b indicate that the 3-21'3 couplings including the ghost basis functions are in surprisingly good agreement with the 6-31++G** couplings out toabout 8 A. Theghost functionsapparentlyprovidetheflexibility required todescribe thetailsofthewave functionsthatarecrucial for an accurate calculation of the coupling. This suggests that the3-21G hasisset makesuseofbasis functions (ghost functions) not directly involved in the pathways (i.e., hops over bonds) to provide a reasonable account of the coupling. This effect is similar to the sc-called "basis set superposition error" (BSSE) in weak intermolecular interactions." The BSSE occurs when the interacting molecules use each other's basis functionstolowertheir energy and thus theenergy ofthecomplex. This results in an overestimateof the binding energy by the small basissets. In contrast, for electroniccoupling,the ghost function effecr enhances the ability of a small basis set to calculate couplings. Theagreement betweenthe3-21G basisset andlarger hasis sets such as 6-31++G** (Tables I and 11; Figure 6) on longdistancecoupling in molecules suggeststhat theghost effects lead to more reliable results. This ghost function effect should
".
.
2
4
6
8
10
12
14
16
18
20
CC distance, angstmm
pigwe 9. Distance depndena of splitting (absolute value) for the CHz(CHz),zCHz chain malsules from different methods of calculation: (a) anion coupling: (b) eation coupling. All results are from the 3-21G basis. The numbers in the plot correspond to the value of n.
alsooccur in thecalculationof the cauphgsof theanions, although this cannot yet be tested in the present systems because of the problems with adding diffuse functions in the anion calculations. The ghost functions enhance substantially the ability of small basis sets to compute electronic couplings in the straight-chain moleculesexamined here. Wecan speculate that theghost effect will be less effective in molecules for which coupling pathways take long hops through empty space, as opposed to space filled withatoms,and thereforebasisfunctions. Examplesmightinclude gauche conformations of the chain molecules studied here or 'C-clamp" shaped molecules. C. E v i d e n e e f o r S u p e ~ c ~ r i . T b n l g h s o a d I a t m f t i o a a Theresults in Figure6b providedramaticcomputational evidence for the importance of superexchange via through-bond interactions. Thecouplingbetween thetwoCH,groupsdecreasesmuch more rapidly than the couplings for the chains which include through-bond interactions. While there is a serious danger that calculations of long-range direct interactions reflect more the distance dependence of the hasis set than reality, the validity of the present calculations is strongly supported by the calculations using the 3-21G and 6-31G++G** basis sets with the ghost functions placed on the missing carbon and hydrogen (ghost) atoms (see Figure 7a). The results from these calculations, referred to as 3-21G(+ghosts) and 6-31++G**(+ghosts). are verysimilartotheresultsfromthe6-3I++G**calculations (see Figure 6h). Further evidence for the superexchange via through-bond interactions comes from acalculation placinga methane molecule between the two CHI groups for then = 7 chain as illustrated in Figure 7b. Acalculation of thecoupling using the631++Go* basis set including ghost functions on the missing atoms indicates that the coupling increases by an order of magnitude compared to the direct interaction between the two CH, groups. This is shown in Figure 6b. D. Distance Depeaaeafc in Chin Molecllka The KT and ASCF 3-2lG couplings for the chain molecules (with even n) are
4056
The Journal of Physical Chemistry, Vol. 97,NO.16, 1993
TABLE 111: Values of B
(,&-I)
Curtiss et al.
for Exwnential Fits to (A/2)2 of Successive Pairs (aEven) of the Alkyl Cham C H Z ( C H ~ , Z C H ~ ~ cation
anion
KT
KT
uairh
neutral
monocation
dication
ASCF
neutral
anion
dianion
ASCF
4,6 6,8 8,lO 10,12 12,14
0.42 0.33 0.40 0.39 0.37
0.11 0.62 0.58
0.44 0.65 0.83 0.76
0.42 0.55 0.67 0.62
0.16 0.60 1.05 0.72
0.31 0.56 0.77 0.94
0.39 0.72 0.96 0.85
0.32 0.70 1.02 0.82
~
3-21G results. Values of n for pairs, e.g., 4,6 means @ is based on n = 4 and n = 6.
plotted in Figure 9 as a function of distance for n up to 16 (18 A) to comparethe dependenceon method of calculation. Included in the figure are results for both cation and anion coupling. Electronic couplings often decrease exponentially with distance:
v = V(0)e-a‘/2 (1 1) In Table I11 the 3-21G /3 values for pairs of molecules are listed for the various methods. The results in Table I11 indicate that none of the distance dependences of the couplings can be fit by a single exponential, with theexception of the KT(neutra1 triplet) results for the cation. For the case of coupling in the anion, the distance dependence is similar from the four methods: /3 is small (0.4 i 0.2) at short distances and increases to 0.86 f 0.2 at longer distances. For the case of coupling in the cations, all methods, except for KT(monocation), give a /3 of around 0.4 at short distances (
