Chemical Physics 20 (1977) X83-195 0 North-Holland Publishing Company

THE BAND GAP IN LINEAR POLYENES

D.R. YARKONY and R. SILBEY Department of Chemistty. and Cer!rer for Materials Science & Engikeering, Massachusetts Institute of Technology, Cambrkige, Ma. 0.2139, USA

Received 21 June 1976 Revised manuscript received 4 October 19’76

The band gap in the spectrum of long polyenes is considered. It is shown, using a bond ahernation model within a single conf%uration self consistent fieid method, that the gap is determined largely by the difference in the exchange contribu- tion to the ground and excited state energies. This model also predicts a K1 con~~ution to the gap, but of the incorrect sign (when compared to SCF calculations of moderate length polyenes). Secondly, it is shown that the exciton model, when multiple excitations are taken into account, gives a&-r contribution to the band gap, in contrast to thz simplest exciton model.

1. introduction

The nature of the lowest dipole allowed electronic transition in an extended polyene system has been a topic of considerable theoretical and experimental interest. The available experimental evidence [I] indicates that as a function of chain length the excitation energy (bed has the form

where PE, > 0. In this work we consider the analytic solution of simphfied versions of two models, which have been solved numerically elsewhere, in order to understand the origin of the A&-, and Af?, contributions to A,!?.

In section 2 we develop a simple single conjuration approbation to the ground and lowest excited state wave functions. From this model we conclude that electron exchange plays a major role in determining AE,. In section 3, the exciton model of Siipson [2] is considered. There it is shown that if multiple excitations and the non-Bose-Einstein nature of exciton statistics are included in a simplified form of this model, a A,!?, term emerges in A&N). It had been previously supposed that such a term was lacking in exciton models. In section 4, we com- ment on the assumptions used in these model calculations and also consider the relationship of this study to pre- vious work in this area.

2. Bond alternation method (MM)

In this section, we develop a model of the infinite polyene which includes, in an approximate manner, the es- sential features of a single conjuration wavefun~tion description of the system includiig electron-electron re- pulsion and exchange. This approach was motivated by the observation that in previous numerical self-consistent- field (SCF) c~cu~ations [3], the opt~um one~le~tron orbitals showed bond ~tematio~ regardless of the inter- atomic spacing. It will be shown that the inclusion of electron-exchange is essential to the description of the band gap (L\E(oo), producing a gap of u 2.5 eV in a system with minimal bond alternation (I& - &I= 0.12 ev). In a

suiXf?qut?nt section the results of this model calculation will be compared with previous work in f&is area inchnfing the exciton model considered in section 3.

184 D.R. Yarkony. R. Silbey/TTze band gap in linear polyenes

The inftite polyene is modeled by a linear Bravais lattice with 2N + 1 cells each containing two carbon atoms. The intercell (intraceII) spacing is rL (rs) with rL 3 rs. Periodic boundary conditions are assumed and integrals hi- valving the 4N+ 2 locabzed p orbitals are computed within the spirit of the Pariser-Parr approximation [4].

The evaluation of the band gap, the separation of the ground state and fmt excited state, proceeds as follows: (i) A set of appropriate one-electron orbitals is chosen to correspond to the eigenfunctions of a Hiickel-type Fock operator. (Ii) From these orbitals single configuration wave functions for the ground state and all singly-excited states with k = ,F = 0 (the same translational and spin symmetry as the ground state) are constructed and M=E,. - Egs evaluated. (iii) Finally the excited state wave functions are symmetrized with respect to inversion and reflection through the origin of the chain and the fmal expression for M(N) is determined.

The results of the procedure are summarized below with the details of the calculation relegated to the appen- dices. Let pia (i= 1, n = -N, N) denote the localized pz orbital on the ith carbon atom In the nth cell. Further, using the usual Hiickel notation, let

(p~,~lhIpi~)=CG ifi=i’,n=n’, (Ua)

= PS, ifi#i,n =n’, (2.lb)

= PL: ifn#n’andn+i=n’+i’. (2.1 c)

Finally, let

x+0? --$ 6169 + e’%~(i>l , x-6) = 1 [e-i6jj,6) _&(i)] , ti

(2.2a, b)

where 6 = r&L/V + l), and

&j)=@v+ I)-“2 g ei26jn -N<j<N. n=-N

pin 1 (2.3)

Then in the Hiickel approximation the matrix elements of the Fock operator (<x’(j)lFH!~(j))) are given by (here FH = h)

(

a--(Ps + P&OS si ieSi6jcpt - pS)sin Sj

-ieisicpL- I3&in Sj o-t& + fl,)cos Sj ) _ (2.4)

Note that F is diagonal inj. Also observe that in the absence of bond alternation the p(i) diagonabze F. For & + OS the matrix in (4) can be “rotated” to diagonal form. The resulting eigenvectors are

= _!_ 6V) & (e -&63plc) + eis~eiQf.i)~C~)), e-C)=a

l (_e-iGe-icO’)j,(j) + e4ti)j2tj)), (2.5a, b)

and the corresponding eigenvalues

<e*(#F,le*@, = cr F t-J; + $ + 2p,p, cossj)l’*, (2.6a)

with (for Ps, 0~ < 0)

P,-P, . tan2m=- pL+ps, tam,

I 1 --n/2 G 2@(j) d n/2. (2.6b)

From eqs. (2), (S), (6) we see that near the center of the Brillouin zone J?(j) E O*(j) so that bond alternation has little effect on these eigenfunctions of FH’ However near the zone edges (j = %V), 2$(i) + G/2 as N+ OJ (for

‘0% - Ps) I& + 0s) I > 0). Thus even for minimal bond alternation the x’(j) are poor approximations to the true eigenfunctions of FH, the O’($, for ljf huge.

D,R, Ymkony, R. Si~~ey/~e band gup in lhwar poiyenes 185

Now accordhrg to step (ii) of the procedure, the bound electronic state of the polyene is approximated by

J’gs =A _~~<~e~~)~6i~~~* (2.7)

The ZN+ f singlets ~o~espon~g to wave vector k = 0 are formed from the excitation B*(j) j e-@) and are given

by

$tj*j*) =A ml~~~~(~)us~~~)8si~~~-~)‘~. cm

WithEO’-tP)a(tCifi-tj*)iH1~~~I*)}andEs, fS] that

= (~~si~i~~~}, it is a standard rest& from Hartree-Fock theory

AEtj)~Eo’*,j*)- E gs (29a)

= {e-~~l~i~-~~) - {~+~)l~l~~~)} - (~~-~~,~~~~ - 2~~-~~,~i~~), (29b)

where F is now the Fack operator corresponding to I,$,,, F = k + 2l- K. In appendix A the requisite matrix eIe- ments in (9b) are evaluated and found to be

where

(2.1Ja)

From eq. (12) we see that AE(-j) = A@). ThJs degeneracy, which is a consequence of the periodic boundary con- ditions imposed in this model, can be lifted by two interrelated and essentiai equivalent approaches. In the first ag preach we observe that in the analogous linear system the true wave functions carry representations of the point group Cw If PR (PJ) represents the Wigner operator corresponding to reflection (inversion) through the OI%$I,

then it is easily shown (see appendix B) that

(2.13)

186 D.R. Yrrrkony, R Silbeyh%e band gup in linear polyenes

Thus the a~prop~ate~y fretted wavef~ctions are

=--I, 3r0’-Vj*)$e-i2si9C_i3_j*), (2.13a, b)

where ~~6~~~-~~) transforms like ‘B,(‘Ap). The determination of the excitation energy (A&j) = {~~~~t~~~~~~) - E ,J of these states requires the evalua-

tion of the ~tera~tion matrix elements H&+ - = <JtQ+j”)iH! J/(--j* -J*)>. ‘&is calculation is ~rfo~ed in appen- dix C. The excitation energies now follows easily from eqs. (121, (13) and (C.5f:

+ (uv+ If-1 C[(2 5 2)C_(R) - C&Q].? (2N-t 1)-l F ~cos~~~~)~ct,~(~~ + C&z)]/4 R

+ cos firsi(rr - $1 - 4glcj)I [Cl,&) f C&-n)] /41, for 0 G j Q N. (2.14)

EquivaIenGy eqs. [Is), (14) could have been obtained by diagon~~~g the 2 X 2 matrixH@ with matrix elements ~j,io;)=~o3,H1,2~~=~j,_j=~,10)*

The Iowest excited state is obtained by rn~~g eq. (14) with respect toi for fiied EI. However since we are dealing with an approbate model it is sufficient for our purposes to take j = N, VN and evaluate eq. (14) to OP der N-’ giving

=hE&+Q+EJ. (2.15b)

This is the desired result. The first term in eq. (If) (A&$ is the usual Hiickel expression for the band gap. The third term @EJ) which results from both Coulomb and exchange effects (see eq. 10 and appendix A) is easily seen to be of order ~og~~~ -*N-l since for n large C,(n) = l/n. Thus for the N + ~0 limit only the first two terms need be considered. From eq. (10) it is seen that the second term in eq. (15) (AK,), i.e., the correction to the Hiickel vaIue of fS, is due entirely to the exchange ~teraction. It is important to note here that if Z?j) had been used in eq. (9) rather than @), &Q (as well as A.E$ would vanish.

Because of the comp~cated dependence of G(n) on n, A& (the exchange term) is most conveniently treated nume~c~y. We consider two cases one with minimal bond attemation CpS = -1.64 eV, &, = -1 S2 ev) f6] and a second with moderate (ps = -2.39 eV, @L = -2.01 ev) [7] bond alternation (Q = 1.39 A, r~ = I.42 a in both cases) [7]. The results (with Ct,J@ evaluated as in ref. [4]) are displayed in table 1. The salient feature of this tabIe is that it is A& rather than AEK which makes the do~ant co~t~butinn to A,!?. Further while it is clear

Table 1

hEK aH M-1

Minimal bond ~tem~~~n 2.29 0.24 253 Moderate bond ~tema~aN 3.48 0.76 4.24

Ali energies are in electron dolts. The relevant parameters for the two cases are givea in the text AEc extrapolated from data at N= 50,100,2QD.

D.R. Yarkony. R. Silbeyhre band gap itr linear polyenes 187

from the table that Mean be made to agree reasonably well with experiment, it is not our purpose here to pro- duce a quantitatively accurate model. Clearly the approximations involved in this calculation, the approximate evaluation of both l- and 24ectron integrals, the use of approximate SCF orbitah and the neglect of configura- tion interaction argue against such a comparison. Rather what we hope to provide is an improved understanding of the origin of the band gap within a single configuration SCF (or SCF Cf) [8] approach. The important point is that the alternant nature of the orbitals which is “inherent” in an SCF description of $,, leads to a pronounced change in the exchange contribution to the energy of the ground with respect to excited state (the Coulomb con- tributions are identical) which stabilizes $,, and hence the band gap (even for or. z /Is). Thus we conclude that (ii a system where bond alternation can be expected to exist) the Hiickel model cannot Frovide a realistic picture of the band gap since electron exchange plays an essential role in determining AE.

3. Generalized exciton model

The exciton model of polyenes [2,7] assumes that the polyene is made up of N ethylene units (12 = 1, . . . . IV) which in the lowest approximation are non-interacting. The excited state energies of the system are then just those of ethylene; for example, the first singlet state will be at an energy equal to that of the first singlet of ethylene, but will be N-fold degenerate. There will be, in addition, multiple excitation states, where two or more ethylenic units are excited. Let us represent the excited states of this “oriented gas” of ethylenic units by

Ina, mb . ..> = $z@“, . . . R&,r,+# where r$ represents the ath excited state of the rzth unit in the chain and @p re- presents the ground state of the fth unit in the chain- The total hamiltonian will have matrix elements between states which differ by rroo excitations since the perturbation is a two body operator. In the simplest model of this system, we consider only one ethylene excited singlet state; then the hamiltonian may be written in second quan- tized notation (af 10) = in), etc.) as [9]

(3-l)

where the fact that the ground state and double excitation states interact is explicitly taken into account by the term in (fz,*u*, + a,~,). We assume H(0) = 0. The usual approximation is to neglect “counter-rotating” terms in

* * (l), i.e., those in anam and a,,~,,,. However these are just the terms of interest to us. In the above, e. is the exci- tation energy of the ethylene unit andH(n’-m) is the interaction between units n and m. Note that this model is highly simplified and is not expected to give accurate quantitative results. The model is only intended to show how terms in hr ’ can arise in the formula for the energy gap when multiple excitations are included.

The usual method of diagonalizing (1) is to assume that the operators a, and a,T are Boson operators. However, this is not exact and is in error to order N-l . When one correct this and treats the counter-rotating terms correctly, terms in N-l arise m the energy of the first excited state.

First consider the statistics of the operators an and Q;. When the commutator @,,a: - a&) operates on a state in which site n is not excited we find

( Qz,* - a,a,)lO> = IO). (3.2a)

However, when this commutator operates on a state in which n is excited, we find

(a,$ - a,*a,)ln> =-In>, (3.2b)

since a): In) = 0 (which means that two excitations cannot be on the same site). Thus, the commutator is not a pure number but is given by

[a,, a@ = 1 -Z$zn. (3.?c)

188 D.R. Yarkony, R. Silbey/Tlle band gap in linear poiyenes

The other co~utators in the system are Soson4iie:

[a,, a:] = 0, nfm; [n,, a,] = [a$ f&] = 0. (3.3a, b)

Thus, the statistics of these operators are neither Bose-Einstein nor Fermi-Dirac. In cases in which the pverage excitation number is small, the average of the commutator is given by

([a,, Q;] 1= 1-2~4&~ = 1 i-6 (N--l) (3.41

Thus the usual approximation is to assume the excitations are Bosons. [In the case of nearest neighbor interac- tions, there is a t~sformation to Fermion operators. We discuss this below.] We will correct this to lowest order by tmnsforming from excitation operators to Boson operators by a HoIstein-Prhnakoff [lo] transformation or, more generally, by a transformation introduced by Agranovich and Toschich 11 I]

where .

d, = (-Z)“~(u + I)! (3.5c)

The operators brl, bi satisfy Boson commutation rules and the usual methods can be used to treat the h~~to~~. Substituting eq. (5) into eq. (1) results in

H = Hco, + H(‘) + ti2’, (3 x4

where

H@) = fo c bzb, + f c H(n-m)(b, f bz)(b, f bg), H(l)= -Ed cb,*b,*b,b,, (3.6b, c) n Ft. iii n

i

(3.6d)

where V is of sixth and higher order in the Boson operators (b, b*). We will neglect Vi.n what follows. We will

now diagomdize Ho and compute the first perturbation correction to the lowest states of He The zeroth order hamiItonian, Ho, can be diagonalized by a transformation to new Boson operators after first

introducing the wave vector representation (and assuming cyclic boundary conditions and N even)

& = N-112 $ 6, &‘k”, tl=l

k = Zrl/N,

Putting eq. (7) into eq. (6b) and noting that

N-1 2 ei(k-k’)n =. k,k’r n=l

I = -N/2, -N/Z + 1, . . . . 0, I, . . . . N/2 - 1. (3.7)

(3.8a)

(3.8b)

we fmd

We now introduce new Boson operators

D.R. Yarkony. R. SilbyhXc band gap in linear polyenes 189

Ck=ukbk-vkb:k, c; = ukb; - vkb_k, (3.10a, 6)

where

u&$=1. (3.1Oc)

By choosing Uk = cash ok, uk = sinhBk and demanding that Ho be diagonaI in the new representation, we find

exp(-48,) = 1 f ti@)len, \

(3.11)

and the new energies e(k) given by

E(k) = E&l + ti(~)/$-J)l’*. (3.12)

Note that to lowest order in fi(k)/g we have

E(k) = et-j + $rc), (3.13)

which is the result of Simpson, quoted by Szabo et al. [I]. Thus, the inclusion of multiple excitation states changes the excitation energies in the Boson model in a simple manner. We have, in the new representation:

(3.14)

H(l) = --(E&V) x- g ,* b~b~b,b,z6(k + I - m - n), . . I

d2) = -(2N)-’ x Frn n I$k)bF(b_,, + b&Jb,,(b, t brk)li(l t nt - tl - k). -8, ,

(3.15)

(3.16)

At the level of Ho, the band gap is given by the minimum of E(k); there are corrections to this due to H(t) and d2). Since H(l) andH(*) are out of order l/A!, these corrections will, in general, contain terms of this order. Since the exciton model produces a band of closely spaced levels, first order perturbation theory is not appropriate. We will make no attempt to determine the value of the l/rV term within this model.

In the present model, the approximate eigenfunctions of the hamiltonian can be represented by wavefunctions in contrast to methods like the RPA. The transformation from the b operators to the c operators can be accom- plished by a unitary transformation of the type c = exp(S)b exp(-S), where S - (6 -6, - bjtbi). The eigenstates ofH@) are given by exp(-S) operating on 101, b$lO,, . . . . The ground state of H( Of then contains double, qua- druple, etc., excitations of the b type (and hence of the IL type), while the (odd) excited states contain single, triple, etc., excitations. These, then, represent wavefunctions of considerable configuration interaction. These wavefunctions, however, contain an unphysical feature: they allow multiple excitations on a site. This is removed by&‘) Hf2) and Vby introducing a “repulsion” between excitations. It is possible in one case to remove tfris ef- fect completely. We turn to this next.

In the case in which H&-m) is very short range (so that it is zero unIess IZ and /rz are tzecresr neighbors), there is another procedure for treating the hamiltonian of eq. (1). This is to perform a Jordan-Wigner transformation [12] which is discussed in detail for this hamiltonian by L,ieb, Schultz and Mattis [13]. By defining

a, = (-l)““-ld,, a; = (-l)“%,, u,.,-~ = g1 d;di. (3.17)

we fmd that the d, and d,* are Fermion operators. The hamiltonian becomes (ii the case of periodic boundary conditions)

D. R. Yarkony, R. SilbeyjThe band gap in linear poiyenes

N U- -c d?d- = number of excitations,

j=1 ' '

J=H(kl).

(3.1ga)

(3.18b, c)

Transforming to running waves, dk and d& we fmd

H= g (e. + Wcos k)d;dk t Jc sin k(d;dtk - d&k)

+ (J/N) pk, [d;d; eiCk+“‘*) + eikd;dkf + hc] [I f (-I)“]. (3.19)

Neglecting the term proportional to J/N and taking multiple excitations into account by transforming to new Fermion operators, Dk and 0;

Dk = cos &dk - sin QkdEk, D$ = cos &d$ - sin @kd_k, (3.20)

with & chosen so that the Dk operators refer to non-interacting Fermion modes to 0(1/N) we find (aside from a constant term)

H = c [(e. + UCOS k)2 + 4J2 sin2k] 1’2f$Dk f tit), k

(3.21)

with L!(l) al/IV in the space of even number of excitations (which includes the ground state). We thus have a closely spaced band of energy levels with the lowest energy (for J< 0 at k = 0) corresponding to a band gap of me0 - 2151. The corrections to the band gap are again of order l/N (corresponding to corrections to the ground state energy).

We have presented the latter calculation (using a Fermion representation) to show that the unphysical property of the Boson representation (i.e., that more than one excitation can reside on a site) which is removed to low or- der by the Holstein-Primakoff transformation is not responsible for the l/N terms. The Fermion representation does not have this unphysical feature and has a l/Ncorrection also. Unfortunately, the Fermion representation is impossible for longer range interactions, so it cannot be used in general.

The use of periodic boundary conditions is unnecessary in the Fermion representation. In fact, the free end boundary condition is a bit easier to work with.

In conclusion to this section, we reiterate our major fmding. It is simply that treating the exciton model fully (i.e., correcting for statistics and multiple excitations) formally leads to a l/N dependence on the band gap. We have not attempted to compute the numerical value of this correction.

4. Comparison with previous work

As mentioned previously, theoretical determinations of AE have been quite numerous. It is therefore impor- taut to juxtapose the simplified model calculations of this study with previous work in this area and with each other. We begin by establishing the relationship between the widely used Pariser-Parr (PP) integral approximation [4] (which was used in developing the BAM approximation) and the exciton model as described in section 3.

Consider a partitioning of the space of single excitons into vertical (denoted #ohi*)) and charge transfer (denoted $o+i’*)) excitons. We now show that only the space of vertical excitons is non-interacting with the ground state (I,!J~~). Let

{4.la, b)

whew in the spirit of the PP a~~r~~ation

192 D.R. Yarkony, R SilbeylThe bandgap in linear polyenes

In contrast to the vertical exciton model (VEM) the BAM method includes the effects of change transfer exci- tations since it considers the entire space of single excitations consistent with its basis set. However, unlike the situation in the VEM, it is easily verified that BAM excitons interact with the ground state through the exchange part of the Fock operator.

It is possible to circumvent the problem of BAM excitons interacting with JI,,, by substituting the true SCF orbitals for BAM orbitals. However, as was alluded to at the end of section 2, using ground state SCF orbitals to construct both ground and excited state prejudices the calculation in favor of $s,, thus increasing AE. Thus it is preferable, whenever possible, to reduce this prejudice by allowing for configuration interaction (CI) in the space ofsingly excited wavefunctions. These SCF CI calculations which, as we mentioned earlier, motivated the BAM calculation, have been performed numerically (using the PP approximation) by several authors [ 1,3]. These cal- culations exhibit the increase in AE above its Hiickel value, which we have attributed to an exchange interaction. However, these calculations also illustrate a failure of the BAM. From eq. (15) it can be seen that the BAM pre- dicts a U(N) of the form U(N) = aEo - IChV-‘. The numerical work of Szabo et al. [I] on comparatively small, linear polyenes does not show this N-’ dependence within the single configuration approximation. In fact, these authors show that only when a full CI is performed on the space of single excitations does an N-’ depen- dence, with the experimentally observed positive sign, emerge. A possible explanation for this discrepancy is that the orbitals ln the bond alternation model given above are not self consistent; in fact the first excited state has an exchange interaction with the ground state. When the orbitals are-made self consistent, the ground state energy will be lowered, which may explain the inconsistency in the I/N term between our model and the previous numer- ical work. :

Finally we remark that the calculations considered here are basically extensions of the restricted Hartree-Fock

[S] formalism. Thus the possible effects of spin density waves which have been discussed by other authors [14] have been ignored here.

5. Summary

We have considered the first strongly allowed dipole transition in an infmite polyene using simple exciton and single configuration SCF models. In the single configuration SCF model we demonstrated that the altemant na- ture of the SCF orbitals (i.e., the single bond-double bond character of the.Mulliken population) leads to a large exchange contribution to A,!+, even when the degree of bond alternation (as measured by I& - PL I) is minimal. In the exciton model we showed how the non Bose-Einstein statistics of the excitons could be included through the mechanism of the Holstein-Primakoff and Jordan-Wigner transformations. The use of these transformations when coupled with the inclusion of multiple excitation leads to a AE, # 0 contribution to AQV’) which had been previously believed to be lacking in the exciton model. Finally we discussed the assumptions inherent in the models and considered this work in light of previous investigations in this area.

Appendix A

In this appendix the expressions given in eq. (2.10) are developed. The zero differential overlap (zdo) approxi- mation, will be employed throughout, i.e.,

or equivalently @iapjbxl) a 6ii6ab- Thus from definitions (2.3), (2Sa, b):

n

(A.9

(Ada)

(A6b)

(A.&)

(A.%)

Wb)

I94 DA Yarkony. R. SilbeyjThe band gap in Iinear polyenes

where the fact that -29(-/3 = 290 and C&+) =Ci,,{-n]. FinaIIy using the trigonometric identity fur the sum of t.wo cosines reduces (A.7b) to

= (uv” I)-* x0 (2 - 6,,~)COS[26j(rl - ;) - 2#Q)] cos[26m(n - f, - 2&?2)] c&-) (A.7c)

n

= F C, ,2(ti) cos[ZSj(i2 - 5) - 2dG)] 4(n),

where 4(n) is defmed by eq. (2.1 la).

(A.7d)

Appendix B

Here the transformation properties of @) and $0’ -I*) under the operations of reflection and inversion through the origin of the chain are developed.

Let PR(PI) denote the Wigner operator corresponding to reflection (inversion). Then

pRpi n 5 Pi:n I where i = 2,2’ = 1; Plpi,n = pje _n.

Therefore from definition (2,3)

pRPiv) = p&) = -p~Pi(-~~~

so that from deftition (2.5)

PRo*Q) = f [e-‘@@‘pz(-.j] + ei@Q~eis~pl(-j)J

= eisj + [pl(j)ei60 + e-i6Mp2(_j)J

5 ei6i e+(-1) z -p+++(i),

@J)

(B-2)

(BJa)

(B.3b)

(BJG

where in going from (B.3a) to (B.3b) the fact that I$@ = -@I-j) was used. Similarly we find

~~0-0 = _fl-(i)e-iG = -prqj_ (B-4)

From eqs. (13.3) and (B-4) the transformation properties oftbe &u-j*) can be deduced. Eq. (2.8) shows that apart from fully occupied, and therefore totally symmetric, orbitals the spatial part of $q +j*) can be written as

$0’ +i*) * e’(-j) I?‘(-]Js’Q-)s-Q), @=I

so that

(B.6)

(B.7)

Appendix C

Here the matrix eIement (~/~(i+~*)ll$i~J~*+j’*)> =Hi,j’ as used in section 2 is evaluated. Since $Q-+J”> has the form

195