J . Phys. Chem. 1993,97, 3199-321 1
3199
Ab Initio Studies of Electron Transfer. 2. Pathway Analysis for Homologous Organic Spacers' Congxin Liang* and Marshall D. Newton' Department of Chemistry, Brookhaven National Laboratory, Upton, New York 11973 Received: October 9, 1992
Ab initiocalculations of transfer integrals ( TDA)for long-range a- and u-type electron transfer through saturated spacers linking donor (D) and acceptor (A) groups in several radical anion and cation systems have been carried out and analyzed in terms of additive superexchange models. The sensitivity of calculated results to orbital basis and wave function type has been carefully examined. The one-electron Koopmans' theorem approach, based on the neutral triplet diradical parent state and employing the split valence 3-2 1G basis, provides generally reliable results. The minimal STO-3G basis is quantitatively useful in some cases, where it facilitates a compact perturbative analysis of the coupling in terms of competing pathways. For trans alkyl spacers containing one to seven C C bonds, reasonably good global exponential fits are obtained for fall-off of TDAwith D-A separation even though the relative importance of different pathway types (hole, electron, and hybrid) changes markedly over the rangeof spacers. Thecalculated decay coefficients 8, cover a rangeof -0.5-0.9 A-I, being systematically greater for radical anions than for cations (by a factor of 1.1-1.7, depending on wavefunction type), consistent with similar findings by Jordan and Paddon-Row for radical cation and anion systems involving norbornyl spacer groups. The behavior of calculated transfer integrals for spacers composed of bicyclo[ 1.1.l]pentane (bcp) and bicyclo[2.2.2]octane (bco) units is complex, including cases of nonmonotonic variation with number of units (1,2, or 3), and global exponential fits were possible only for a-transfer through the bcp spacers. The calculated lower limit of flr (- 1.6 A-I, based on the first two members of the series) for ?r transfer through the bco spacers is consistent with inferences from experimental fluorescence quenching data. The decay found in the intramolecular systems is similar to that (& 0.9 A-I) exhibited by non-covalently-linked spacers of methane molecules in van der Waals contact. While nearest-neighbor (NN) McConnell-type pathways based on local bonding or antibonding orbitals are found to be unimportant, a McConnell-type model was recovered at a more coarse-grained level based on larger spacer units and representable in a matrix form isomorphic to the standard scalar McConnell expression.
-
I. Introduction Characterizationof the role of electronic structure in facilitating long-rangeelectron transfer has been an area of intensive activity in recent years.IJ-I5 Effective transfer integrals, T D A , whereby indirect coupling of local donor (D) and acceptor (A) sites is established via the electronic manifolds of the intervening medium (the 'bridge" or 'spacer"), have been evaluated quantum chemically for a number of redox systems. In several cases, 1.6.8.9-1l a . ] 3 b ~ 1the 4 computational results have been analyzed using models based on perturbation theory, with the ultimate goal of obtaining compact models which yield the contribution of different electronic "pathways" in an additive fashion. The overall transfer integral may be thought of as the matrix element of the full system Hamiltonian H,which connects the initial and final states of the process of i n t e r e ~ t .To ~ emphasize our focus on electron (or hole) transfer between local D and A sites, we denote these states as I , ~ Dand +A, defining T D Aas
In general, T D A is a many-electron quantity, based on all the electrons of the chosen system. Nevertheless, under suitable approximations, TDAmay be expressed as a one-particle (orbital) matrix element, as discussed below. The theoretical effort to evaluate and analyze T D A has been strongly spurred by the rapidly expanding opportunities for detailed comparisons with experimental data16-23for a number of systems closely related to the model systems for which theoretical studies are feasible. In particular, much interest is focused on obtaining a detailed understanding of the variation of transfer integral magnitudes in systems involving a series of homologous spacer groups, S,, where m is an integer and S a 0022-365419312097-3199304.00/0
spacer unit, and in ascertaining under what circumstances the dependence of TDAvaries exponentially with m: TDA(m) = TDA(O) e x ~ ( ( d / 2 ) m )
(2)
over some range of m. If the effective donor/acceptor separation distance, rDA, is exactly or approximately linear in m, it is often convenient to reexpress eq 2 as T(rDA) = T(~'DA)exp((-8,/2)(rDA - ~ O D A ) ) where rDA is assumed to be of the form
+
rDA= roDA mArDA
(3)
(4)
and P ~ D is A the increment of F D A per unit number of spacer groups. Thus 19 and Br are related by 8, = b/&DA. In eqs 2 and 3, 0 appears as the factor ,912 since in the contextof long-range electron transfer, rate constantsare frequentlyassumed3 to be proportional to (TDA)'.In situations where a "global" exponential behavior is not operative, it may be useful to employ (see below and refs 3 and 7c) a local coefficient, /Imm' or fl,mm', defined in terms of eqs 2 and 3 for a particular pair mm'. The electronic coupling of D and A groups implicit in the transfer integral TDAhas in the past frequently been cast as an effective electron tunnelling phenomenon, and in fact, the decay parameter (eq 3) has been explicitly represented in terms of the Gamow tunneling amplitude, yielding the expression24
p, = 4
T
4
i
(5)
where Vis the height of an effective rectangular barrier separating the D and A groups, m, is the electron mass, and h is Planck's 0 1993 American Chemical Society
Liang and Newton
3200 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 constant. Attempts to model empirical flrvalues using realistic estimates of V,taken as the gap between the D/A level and lowlying states of the spacer, indicate that eq 5 , which rests on the assumptionof a homogenous interveningmedium, often seriously exaggerates the rate of exponential decay (eq 3).16a,20.21 Thus most recent efforts to characterize TDAand 8, have focused on so-called superexchange models.25 which take detailed account of the electronic structure of the spacer group. In the first paper in this series', superexchange coupling was analyzed for radical ion systems involving several acyclic, cyclic, and bicyclic spacer groups comprised of saturated three and fourcarbon chains. Coupling through bicyclic spacer units is of particular interest in view of recent progress in the preparation and analysis of such systems.26 In terms of a one-electron model based on Kmpmans's theorem (KT)2'and implemented according to an additive perturbation scheme using a local orbital basis set defined by thenaturalbondorbital (NBO) procedureof Weinhold et a1.,28 the major contributions to the overall coupling ( T D A ) were found to arise from a large number of primarily non-nearestneighbor pathwayssuperposed in a complex patternofconstructive and destructive interference. These results are similar to those reported by Naleway et al. and by Jordan and P a d d o n - R ~ wfor ~~ other saturated spacer groups. In the present study, we focus on the dependence of superexchange coupling on spacer size by considering homologous spacers comprised of the spacer units dealt with in the earlier study. The size dependence is examined as a function of donor/ acceptor charge (radical anion or cation) and symmetry (ror a) type as well as spacer bonding type, and the extent to which the overall coupling can be captured by compact perturbative models is investigated for different orbital representations. The goal is to identify useful compromises between the extremes of highly localized orbital representations (e.g., atomic orbitals), for which the convergence of perturbation theory is generally poor, and fully delocalized spacer molecular orbitals (MO), for which convergence behavior may be improved, but at the sacrifice of localized (and hence transferrable) chemical building blocks. While most emphasis is placed on intramolecular coupling involving donor and acceptor groups linked by sequences of covalent bonds, comparisonsare also offered with spacers formed by nonbonded units in van der Waals contact. The remainder of the paper commences with a general discussion of superexchange coupling, followed by detailed applications to the several types of electron (or hole) transfer systems outlined above.29 The same superexchange coupling which controls electron transfer is also relevant to a number of other phenomena related to D/A interactions such as energy transfer,16e.16fand charge-transfer,3 ph~toelectron,~ and electron transmission30 spectroscopy. 11. Models for Superexchange
A. General Formulation of SuperexchangeCoupling. Before proceeding to computational details we consider in formal terms the essence of superexchange coupling. We express TDAas a perturbative sum, using a suitable set of zeroth-order or diabatic basis functions consisting of +g) and +f),in which the transferring charge is assumed to be strongly localized, respectively, on the D or A sites, and an auxiliary set, which spans the important intermediatevirtual states generated by charge transfer toor from the spacer,or by intraspacer excitation. Invery general form, the overall coupling, TDA,may be written a ~ 1 . 1 ~ ~
+?),
for n 2 2, where the second sum is over all sets Q,, p = 1-n) and
where the prime above the summation denotes inclusion of intermediate states only (i,, # D, A). The 'hopping integral" f,k is defined as
(7) and the energy gap, A,, between the degenerate states $2' and the state @) is given by
+'Do)
and
(E:') is the expectation value of H with respect to +?)). Thus in cases where perturbation theory is valid (Le., convergent), the net coupling can be displayed as a superposition of a direct (or 'through space") term, tDA, and a set of higher order "pathways", each corresponding to an individual term in the summations (i.e., a given sequence jpr p = 1-n). A pathway involving n intermediate states contributes to the ( n + 1)th order coupling. In the matrix representation, the zeroth order Hamiltonian is given by the E?) and Eg), (as defined following eq 8) and the pertubation contribution is given by the tJk. While quite general in form, eq 6 is limited to pathways originating with +g' and terminating with the first "visit" to $2). Various chemically motivated special cases of eq 6 based on subsetsof the full manifold of pathways will be dealt with below, and in some cases, modified energy denominatorswill be employedl,aa(see also the cases dealt with in refs 8a, 13b, and 14b). Starting from a proper manyelectron framework, we will show how the many-electron quantities in eq 6 may be replaced under certain assumptions by one-electron orbital equivalents. B. Nearest-NeighborCoupling. I . One Localized Srare Per Site. We now make contact with the familiar limiting case provided by the nearest-neighbor (NN) or tight-binding model of the type formulatedby M~Connell.3-8~.~~~ We assume the spacer to be partitioned into a linear sequence of m units S,, j = 1, m, and a corresponding set of states, $?), in each of which the transferring charge is localized (in the virtual sense) on the respectivesitej. The" model includesonlycoupling( 1 ) between adjacent sites. The lowest-order pathway is then given by the following (m + 1)th-order e x p r e ~ s i o n ~ * ~ ~
nt)J+ iA)+ m- I
TDA
= ('DI
I'/
)(
I
IlfmA
(9)
/=I
Equation 9 is somewhat more general than McConnell's original derivation for the limiting case of a homologous spacer where all tJJ+l= t , all AJ = A, and t D l = tmA. In this case, eq 9 implies exponential variation with m, as in eq 2. It is also noteworthy, but not directly o b v i o ~ sthat , ~ eq 9 yields an oscillation of sign with respect to m for many specific cases of interest (i.e., manifestation of the so-called parity rule3'). Equation 9 correspondsto a pathway of the "electron" or "hole" type, when the spacer states $;)' involve, respectively, electron attachment or ionization (as usual, in thevirtual ~ e n ~ e ) . ~ . ~ . ~ ~ ~ . ~ ~ , l 6 2. Case of 2 OrbitalslSite. Equation 9 was presented in rather general terms, without the need for a detailed specificationof the $;I'. To facilitate a more concrete discussion as we proceed to generalize eq 9, we now assume that in the initial and final states, $'Do' and $a"', each unit, S,,has a high-lying occupied orbital, d,!,' and a low-lying empty orbital, 4;. In the "electron"transfer case discussed above, each $;)' then includes a temporary occupation of 4;, while in the "hole" transfer variant of eq 9, each $(') would have an electron removed from 4;. We further assume /or present purposes that H can be expressed as a sum of effective one-electron Hamiltonian operators. Thus the only nonzero tJh are those involving the transfer of a single particle
Ab Initio Studies of Electron Transfer. 2
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3201
Reference ('vacuum') State:
--
-
e
e I I
-
- - - -e - - 1 I
-A;
Relerence ('vacuum') State:
h-transfer (1-particle) -
-A:
4
e I I
- - 4-
-
y(0) D
I
1
1
do) A
(b) h-transler (3-particle)
v($
+- --
i I - -+eh 4I I
--++
I do) I -t ?; +.+-+.z--I I
I I
A;
A
4
(c) hybrid (e, h,) transfer (3-particle) I I
-d;) 1 , +---+ A - - - Le - +---- +eh '
I
I I
-6;
I I
-A;+A;
I I
do) A
I I
- ---& - -
I
I I
(d) hybrid (hl/ep) transler (3-particle)
d:'
I
I
(c') hybrid (e, h,) transfer (3-particle) I I I I +-++ e
d;) +- -- -+z - -
I
- - +-,+
.,:-
1 + -eh
- i-'-
(d') hybrid (h,/e2) transfer (3-particle)
--- -
v$' I I--1 +eh
I
-+L
-
A;. A:
I
--t.eh
--
I
-Ai
e
I I
_ - -4-
--
Figure 1. Schematic depiction of NN superexchange coupling via two spacer sites (SIand SI),each of which has an occupied (6;) and an unoccupied (4') orbital. Orbital occupations relative to the defined reference (or "Cacuum") state are indicated. The excitation energies of the virtual intermediate states relative to the degenerate initial ($:)' and final ($f') states are expressed in terms of the orbital energy differences defined in q s IO and I I . The three primitivesteps (+) in each 'pathway" correspond to electron (e) or hole (h) transfer, or the creation (+eh) or destruction (-eh) of an electron/hole pair. Relative to the vacuum, the various states require the specification of at most, three particles-the added electron and, in cases b-d, the eh pair. Pathway c' is obtained by interchanging the order of the e and +eh steps in pathway c, and pathway d' is obtained by interchanging the e and -ch steps in pathway d. The set of six processes defines all pathways coupling @' and in terms of N N forward-directed (Le., D to A) steps.
Figure 2. Alternative representation of hole-transfer requiring the specification of only a single particle (h). See Figure 1 caption for definition of symbols.
9), and correspondsto a one-particle process, whereas the hybrid pathways (Figure lc,d) involve up to three particles (both e and h) relative to the vacuum level. While the process in Figure 1b is pure hole transfer (Le., involving only holes on the spacer sites), it is seen to be of the three-particle type. Since one expects a close correspondencebetween e and h pathways,j the asymmetry of Figure l a (one-particle) vs Figure l b (three-particle) may be surprising, and in fact, the expected isomorphism may be recovered by simply adopting another vacuum level for hole transfer (Figure 2). Thus Figures l a and 2 bear the expected mirror image oneparticle relationship. While Figure l a 4 displays the four basic types of pathways possible form = 2, there are actually a total of six pathways. The other two are variants of Figure 1c.d obtained, respectively, by interchanging the first and second virtual transfers in I C (Figure IC') and the second and third transfers in Id (Figure Id'). Figures 1 and 2 include the (positive) energy gaps associated with the intermediate states +?). Since H is assumed (see above) to be a sum of one-electron operators, these gaps may be expressed in terms of the following orbital energy gaps:
Note that as defined, Aj < 0 < A;. The various contributions to TDAare assembled in Scheme I, where proper account is taken of the permutation symmetry of the many-electron single-determinant wave functions (this is the only manifestationof many-electron character, since His assumed separable into one-electron terms).)J*
SCHEME I:
T,, Contribution
$2'
between orbitals located on NN sites (Le., units). As a final simplification, we assume no intrasite coupling (Le., between 4; and 4; on site Sj),and restrict attention to the lower order cases in which all transfers are of the "forward" type (i.e., electrons move toward A, or equivalently, holes move toward D).29 A rich diversity of N N pathways is obtained if the occupations of both orbital sets ({@;I and (4;)) are allowed to vary, subject the above assumptions and restrictions. The different types of pathways are illustrated schematically for the case m = 2 in Figure 1. It is of particular interest to identify the minimal number of particles (electrons (e) or holes (h)) necessary to characterize each pathway. Towards this end we define a reference (or "vacuum") configuration and indicate explicitly only those occupation changes relative to it. The resultingvirtual transitions are then of four types: e or h transfer (Le., the passage of an e or h either from or to one of the spacer orbitals 4; or I$;,". respectively), and the creation (+eh) or destruction (-eh) of eh pairs. Figure l a displays pure e transfer (as represented by eq
(d
+ d')
"hlez" -(f~l)(t:S)(rSA)(l/A:
- 1/A5)/(A: -
where r i , = J4,h+:,t:; = J&&, etc., with h denoting the effective one-electron Hamiltonian (i.e., H({riJ)= Zih(ri)). Summation of the contributions from all six pathways yields the following compact expression:
Thus the initial evaluation of TDAin terms of explicit energy gaps
Liang and Newton
3202 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 for the six intermediate many-parricle states +jo) (as displayed in Figure 1 and Scheme I) has been converted to an equivalent one-particleexpression, which is the conventional third-order (for m = 2) perturbativeresult based onfour intermediateone-particle states in the orbital space.33 Generalizing to arbitrary m, we find that the 2 orbital/site NN model yields 2" different pathways, whose relative contributions to TDA(both signs and magnitudes) are controlled by the joint action of the orbital parameters t and A. Clearly, the relative importance of the pure electron and hole pathways is expected to diminish with m. Specific computational results for the case of alkyl spacer units will be presented in Section VA, where h is identified with the Fock one-electron Hamiltonian. While destructiveinterferenceis clearly possible, it is found, for example, that when 4; and 4; correspond, respectively, to local bonding and anti bonding orbitals, the interference is for the most part constructive (section VA and refs 3 and 14c). 3. More General Orbital Models. The optimal subdivision of a spacer into sites or units (often referred to as molecular groups or fragments) involves a tradeoff between the objectives of local (and hence) transferable units and compact perturbational series.'.8a*9cWhen the sites and orbitals are highly localized (e.g., with the qiy) taken as atomic or two-center bonding or antibonding orbitals) one has the possibility of a model cast in terms of quantities which may be transferable among chemically related (e.g., homologous) systems, but the convergence may be problematical. In the other limit, the entire spacer can be taken as a single unit, and the spacer orbitals are the entire set of spacer MO eigenfunctions. This case corresponds to one level of Larsson's partitioning a p p r o a ~ hand ~ ~ ,involves ~~ only secondorder contributions (cf. eq 6). It has the advantage that the 'dilution" effect implicit in the delocalization will tend to reduce the magnitude of the individual hopping integrals, ?D, and t J A , and thus may improve convergenceproperties.',9CHowever, this advantage may be offset since the delocalization will serve to increase the bandwidth of the spacer levels, thus reducing the magnitude of some of the energy gaps.' The general case of NN coupling among a linear sequence of sites, where the number of orbitals assigned to each of m sites is arbitrary, may be representedcompactlyby replacingthe scalar quantities of eq 9 by their matrix analogs, but in a manner which retains the form of the McConnell-type expression:
(n m-
TDA= (tD")(AI)-'
I
(dJ+')(A'+')-') (PA) (1 3)
j= I
where tj.k is the (in general) rectangular matrix of interactions between the orbitals of s i t e j and site k, and where (&)-I is the inverse of the diagonal matrix br, whose elements are the orbital energy gaps for sitej defined analogously to eqs 10 and 1 1.35 The indicated matrix multiplication generates the appropriate superposition of pathway^.)^ Applicationsbased on suitablydefined "group molecular orbitals" (GMOs) will be reported in sections V B and C. C. Beyond Linear NN Pathways. When the model based on NN coupling within a linear sequence of molecular groups is not appropriate (e.g., when multiply connected D/A groups are involved, or when the NN restriction must be relaxed), the matrix formulation of TDA(eq 13) may be straightforwardly generalized by including a sum over the different classes of group sequences deemed necessary. This level, as well as those represented by the orbital expressions presented above (eq 9 and eq 13), is, of course, a special case of the more general eq 6 defined in terms of manyelectron state functions @).
111. Molecular Systems
The radical ions investigated in the present study are based on the homologous species displayed in Figure 3. The radical cations or anions are obtained by placing, respectively, one or three electrons in the indicated terminal orbital pairs. Figure 3a-c depictsintramolecularcoupling mediated, respectively,by spacers of the trans-alkyl (1 ( m ) ) , bicyclo[l.l.l]pentyl (2 ( m ) ) and bicyclo[2.2.2]octyl(3 ( m ) ) type, while a case of intermolecular coupling (see also ref 7d) is provided by a sequence of methane spacer groups in van der Waalscontact (4 (m)). The *-symmetry designation in the labels lr, Zr, and 3x denotes D/A orbitals of the u-type-i.e., perpendicular to the CC bonds linking the terminal D/A methylene groups to the spacer, while the u label for species 2u, 3u, and 4u denotes D/A orbitals directed along the main axis of the spacer. In the remainder of the paper, the r a n d u labels are included in the species labels only when reference is made to specific symmetry types. The local D/A groups employed in the present study (see also refs 1,6, and 12) are intended as prototypes which may serve as models for larger D/A groups; e.g., the CH2 group serves as a local model for the attachment of larger unsaturated D/A groups such as phenyl or naphthyl to the same spacer groups. Of course, in quantitative assessment of coupling one must consider the change in the relevant energy gaps (A?') and in the electron density in the D/A u-orbital adjacent to the spacer when the CH2 model is replaced by other D/A groups of interest. An inspection of the molecular structures and their pointgroup symmetries, which alternate within a given homologous sequence, suggests the possibility (considered in more detail in section V) of distinct patterns for the TDAvalues in the odd and even m sequences. Also, we note that in the translational sense, the homologous 'repeat unit" involves two spacer groups (an adjacent pair). The indicated point group symmetriescorrespondto the spacer conformationsof lowest energy. The barriers for rotation about the internal CC bonds of 1 (m) and those linking the bicyclo units in 2 (m) and 3 (m) are all 1 2 kcal/mol. The conformational dependence of TDAfor systems 2 and 3 are presented in section VB. The relative phases of the D/A orbitals in Figure 3 are drawn so as to correspond to a "bonding" relationship, a designation which of necessity is somewhat arbitrary. For 1~( m )and all the cr cases, the phases correspond to the totally symmetric representation of the indicated point group. For the 2 r ( m ) and 39r ( m ) cases, where the spacer group possesses an intrinsic 3-fold symmetry axis, the 'bonding" designation is assigned on the basis of the analogy with cylindrical symmetry, where and rg correspond to bonding and anti-bonding interactions. The sign of TDAdepends, of course, on the phases assigned to the D/A orbitals, as discussed in the next section.
IV. Computational Details A. Choice of Wave Functions. The types of wave functions one might employ in evaluating TDAmay be characterized accordingto thedegree of state-specific electron relaxation which distinguishes the initial (+o) and final (+A) states in the charge transfer processes associated with the radical ions based on structures 1-4 (Figure 3). Within the frameworkofself-consistent field (SCF) wave functions, one may obtain fully relaxed, charge-localized SCF solutions for +D and +A, as discussed previously'.8d.'0.12,37(all SCF calculations reported in the present study are of the spin-unrestricted (UHF)-'8 type). For the symmetric molecular geometries adopted as appropriate to the transition state for charge transfer (Figure 3), these radical ion solutions are symmetry-broken, forming a degenerate mirror
Ab Initio Studies of Electron Transfer. 2
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3203
D3h
d.
Figure 3. (a) r-type D/A orbitals (the nonbonding orbitals of the terminal CH2 groups) linked by a trans-alkane spacer ((CHz),,,+l)possessing m covalent CC bonds ( m = 1-7). The even and odd m members correspond, respectively, to C2, and C2h point-group symmetry. (b) r-type (2r (m)) and a-type (2u (m))D/A orbitals linked by bicyclo[ l.l.l]pentyl spacer units (m = 1-3), with adjacent units related by a staggered conformation. The even and odd m members correspond, respectively, to C2h and C2,.point group symmetry for the 2 r ( m ) series, and D3d and D3h symmetry for 2a (m) series. (c) Analogous to Figure lb, but for the bicyclo[2.2.2]octyl spacer unit. Departures of the equilibrium spacer framework from the constrained D 3 h symmetry are minor, as discussed in ref 1. (d) Coupling of D/A orbitals (the nonbonding orbitals of the terminal CH2 groups) in a relative u orientation, mediated by a nonbonded sequence of m CH4 spacer units. The even- and odd-m members correspond, respectively, to C2h and C2, point-group symmetry.
image pair (HDO= H A A ) . TDAis obtained directly as
Equation 14 is a slight generalization of eq 1, required when S D A H J$D$A d r # 0; HOD= H A A is the degenerate expectation value of H with respect to $D and $A. As an alternative, a mean-field approach may be adopted by turning to a delocalized-representation and obtaining symmetrydelocalized SCF solutions, $*,l.3 which are approximatelythe i combination of the charge-localized solutions
The transfer integral, TDA,is then obtained from the splitting of the total energies (H++ and H - J of $*:
TDA = (H++- H--)/2 (16) Finally, to make contact with a one-electron model, one may adopt Koopmans' theorem (KT)27and approximate the manyelectron energy difference for the radical ions (eq 16) by the
corresponding orbital energy difference:1J-6*7,8d
where C* are the orbital energies of the close-lying symmetrically delocalized MOs, 4&A, of the neutral triplet diradical parent which are dominated, respectively, by the in-phase and out-ofphase combinations of the local D and A orbitals. In the sense and 46/,, form a defined in section 111, we thus see that bonding and antibonding pair. An analogous pair of MOs distinguishes the delocalized radical ion SCF wave functions $*. Aside from a nearly invariant electronic core, $+has an electron occupying while for @, 4DIA is occupied.' As expressed by eqs 16 and 17, TDAis negative for "normal" orbital ordering (Le., where the occupation of the bonding MO is energetically preferred to occupation of the antibonding counterpart 4DIA) and positive for "inverted" ordering (where occupation of &/4 is preferred). In presenting coupling data below we will continue to employ the sign convention adopted in ref 1, whereby positive quantities always denote normal ordering. Thus the transfer integrals will be presented as -TDA values. The delocalized mean-field representation (either the re-
Liang and Newton
3204 The Journal of Physical Chemistry, Vol. 97, No. 13, 199'3 laxed or the KT variant) generally gives results in reasonable accord with those of the fully-relaxed symmetry-broken framework.l.6.8d.10J7 The delocalized representation also has the advantage of produchg a convenient framework for addressing other manifestations of long-range D/A coupling such as photoelectron (PES) and electron transmission (ETS) spectroscopy.7-17J0Most of the results presented below are obtained at the KT level (eq 17), although comparisons with the other approachesare also provided. The one-electronKT model allows the Fock operator of the neutral diradical species to play the role of the effective one-electron Hamiltonian introduced in section I1 and thus provides a means for implementingthe various orbital pathway analyses discussed there. Theradicalanionsassociatedwith themodelsystemsof interest (Figure 3), are unbound relative to their neutral diradical parents. Nevertheless, judicious use of flexible valence-level atomic basis sets (with avoidance of very diffuse orbitals) leads to reasonably stable SCF anion solutions in the cases we have examined (Le., no evidence of spurious continuum states is apparent). See also refs 1, 7, 17, and 39. The results presented below employ the minimal STO-3G40 basisand thesplit-valence 3-21G4' and 6-3 lG*42bases. In many cases, the STO-3G basis yields TDAvalues in semiquantitative agreement with the results from the more flexible bases. The compact STO-3G basis displays better convergence behavior in the perturbative pathway analyses,' but in some cases, the STO3G TDAvalues are found to differ sharply from the 3-21G and 6-31G* values, whereas the latter two bases almost always yield quite similar TDAvalues. B. Molecular Geometries. As in ref 1, the spacer geometries for the most part are based on optimization at the STO-3G SCF level for molecules closely related to the spacer groups (Le., the parent lralzs-alkanes for 1 ( m ) ;bicyclo[l.l.l]pentane (bcp) and bicyclo[2.2.2]octane (bcp), respectively, for 2 ( m ) and 3 ( m ) ; and methane for 4 ( m ) ) . Optimization of dimers of bcp and bco (obtained by adding H atoms to the termini of 2u (2) and 3u (2) were used to determine the length of the CC bonds linking the bicyclic units and had little effect on their internal geometries (changes of 50.01 A in bond-length and 1 2 " in angles). TheSTO-3G linker bond-length for the bcp dimer (1.52 A) is appreciably greater than the mean experimental value of 1.47 A reported for several related species (the so-called staffanes).26a,bAccordingly, the latter value was employed in the present study. The van der Waals contact of the CH3 and CHI moieties in 4 (m) was defined by setting the C- - -C separations equal to 3.40 A. The adopted molecular geometries yield the followingvariations of rDA with m (see eq 4: 1.27 A (1 ( m ) ) ;3.37 A (2 ( m ) ) ;4.28 A ( 3 ( m ) ) ;and 3.40 A (4 ( m ) ) . The value for 1 ( m ) is an average over the range 1.24-1.30 A for m = 1-7. In all cases, rDA is defined as the separation of the carbon atoms on which the D/A orbitals reside. While alternative indices of size dependence are possible,3 rDA was judged to be most convenient for comparisons among the present set of spacer systems. As in ref 1, the CC bonds joining the terminal CH2 groups to the spacers in 17r ( m ) , 27~( m ) ) ,and 37~( m ) were assigned a bond length of 1.525 A. C. Pathway Analysis. Pathway analysis is carried out at the one-electron KT level using the Fock Hamiltonian (F) based on the neutral triplet diradical parent species. The F operator is represented in the natural bond orbital (NBO) basis,28 which defines the local zeroth-order orbitals introduced in section 11. The set of NBOs includes the D and A orbitals and the twocenter bonding (CC and CH) and antibonding (CC* and CH*) orbitals, as well as the inner-shell carbon 1s orbitals. For the extendedatomicorbital basissets (3-21Gand 6-31G*), additional diffuse extra-valence NBOs are generated. In ref 1 the results of pathway analysis in minimal (STO-3G) and extended (3-21G) bases werecompared (seealsoref 7c). Theextra-valenceorbitals
A
TABLE I: C a p (IA ) for Trans-Alkane Spacer Systems (1 ( m ) ) in a -I( BO) Basis*' bond type
STO-3G
3-21G
CC (internal)d CC (terminal)P CHd CC* (internal)d CC* (terminal)' CH*
0.31-0.33 0.35 0.29-0.30 0.35
0.4M.43 0.45 0.30-0.31 0.32-0.33 0.37 0.42443
0.39 0.39
, The indicated quantities represent the magnitude in hartrees of the lowest-energy gap (A, F D / A- F,,) between the D/A level and the highestlying occupied NBO of the indicated bond type (CC or CH) for the radical cation systems, and between the lowest-lying unoccupied NBO of the given anti-bond type (CC' or CH*) for the radical anion systems. F denotes the Fock matrix in the NBO basis. The ranges indicated in some cases denote small variations among different members of the set (1 (m), m = 1, 7). The NBO Fock matrix elements correspond to the a-spin (for the radical cations) and &spin (for the radical anion) manifolds of the neutral diradical parent (treated at the UHF level).3na Refers to NBOs within the spacer groups. Refers to NBOs linking the spacer group to the terminal CHI D/A groups.
were found to contribute significantly to the important pathways, thus complicating the orbital interpretation of the pathways. Accordingly, in the present paper we consider only results of pathway analysis at the STO-3G level, confining our attention to those cases where the STO-3G basis yields overall T D Avalues which are comparable to those from the more flexible bases. The Fock matrix elements in the NBO representation straightforwardly define the orbital quantities (rJk and A,) which appear in the perturbative expressions given in section I1 (see eqs 6 1 3 ) . The convergence of the perturbative expansions depends strongly on the magnitude of the rjk/AJratios. Typical rJk and A, values are illustrated by presenting the results obtained for 1 ( m ) in Tables I and 11. The different types of gaps (for both radical anions and cations, and for 3-21G as well as STO-3G bases) all lie in the range 0 . 3 4 4 h. While the smallest gaps for occupied orbitals are for the CH NBO's, the net contribution of these NBOs to TDAvalues is nevertheless found to be rather small, with the primary contributors being the CC and CC* orbitals internal to the spacers. The terminal CC and CC* NBOs are of minor importance since they are only weakly coupled (see Table 11) to the D and A orbitals.43 The hopping integral ranges displayed in Table I1 indicate a strong degree of transferability (with respect to different m ) for each type of coupling. In comparing radical anions and cations, it is clear that the differences among the t ] k are considerably more significant than those displayed by the A,. The differences between NN (Le., 1, 2) rJk values for CC and CC* pairs was explained previously' in terms of the relative phases of the hybrid orbital lobes which constitute the NBO's. The data of Tables I and I1 indicate that most of the rJk/AJ ratios will be 10.3 for theSTO-3G basis. The valence-level NBOs for the 3-21G basis yield somewhat larger ratios, while the extravalence orbitals (not shown) yield unacceptable ratios which in some cases exceed unity (see also ref 1). For the bicyclic spacers, the gaps associated with the CC and CC* NBOs are similar to those displayed in Table I (except for the case of a-transfer in the radical cations, where appreciably smaller gaps were found, as in ref l), and the NBO-based perturbative ratios are also 10.3 in most cases. However, the sheer number of pathways which must be included with such a local basis makes it desirable to adopt a more coarse-grained partition of the spacers. Accordingly we maintain the definition of D and A orbitals in termsof the appropriate nonbonded NBOs, but adopt as spacer units the bicyclo moieties for species 2r ( m ) , Za ( m ) ,37r ( m ) ,and 3a ( m ) ,and the methane molecules for the series 4u ( m ) . For Zu ( m ) and 3a ( m ) ,the exocyclic CC groups are also included as spacer units. In the case of 2~ ( m ) and 37r ( m ) ,the exocyclic CC groups do not contribute to TDAsince the
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3205
Ab Initio Studies of Electron Transfer. 2
TABLE II: HoDDinE Intepnls ~
~
(til)
for Trans-Alkane Spacer Systems (1 (m))in a Local (NBO) Basisab
~~~
~
STO-3G NBO pairs C-C
c-c
C-H C-H C-H*
C-C* c-c* C-H C-H* C-H C-H* C-H C-H * C-H*
c-c c-c * c-c c-c C-C* c-c
D/A D/A D/A D/A
c-c
C-C* C-Cwm c-c*,,,,
1,2r 0.105-0.106 0.023-0.026 0.00 1 0.116 0.015 0.018-0.020 0.010-0.01 1 0.123 0.007-0.008 0.014
1,3'
~~~~~
3-21G
I ,2'
1,3'
I ,4'
0.145-0.148 0.025-0.029 0.006 0.1394).I40 0.03 1 0.022-0.024 0.001 0.1334).I40 0.0204.021 0.01 1
0.029-0.030 0.048-0.050 0.029-0.030 0.016-0.017 0.007-0.008 0.008-0).011 0.028-0.029 0.039 0.057 0.036-0.037
0.010-0.01 1
0.017-0.018 0.041
0.004 0.002
1,4c
(A) Internal 0.035 0.007 0.0464048 0.012 0.0404.041 0.01 7-0.01 8 0.019-0.20 0.004 0.016 0.001 0.017-0.019 0.007-0.008 0.024 0.001 0.033 0.008-0.009 0.049 0.01 1-0.012 0.036 0.012-0.013
0.008-0.009 0.0404042 0.003 0.002 0.006-0.007 0.002-0.003 0.005 0.01(M.011 0.012-0).014
(B) D / A and CC or CC* (Radical Cationsy' 0.062 0.092 10.001 50.002
0.014 0.027-0.028
0.002
0.053 0.096 10.002
50.001
(C) D / A and CC or CC* (Radical Anionsy' c-c 0.089 0.020 0.002 0.079 0.023 0.007 D/A 0.001 0.076 0.041 0.002 c-c* 0.071 0.027 D/A 10.001 C-cicrm D/A 10.001 50.006 C-C*,,r, D/A The listed quantities are the magnitudes in Hartrees (h) of the hopping integrals ( t j k E F,a) for the indicated NBO pairs. Blank spaces denote magnitudes
