Localized π electron states in polydiacetylene oligomers

Chemical Physics ELSEVIER

Chemical Physics 182 (1994)

131-147

Localized 7~electron states in polydiacetylene oligomers Christian Kollmar Physik Department, Theoretische Physik T38, Boltzmannstrasse,

Technische Universitiit Miinchen, D-85748 Garching, Germany

Received 21 December 1993

Abstract

It is shown that the particular bond alternation pattern of short chain polydiacetylene intermediates gives rise to the occurrence of two nonbonding and nearly degenerate rr molecular orbitals (MOs) which are localized at the chain ends of the longer oligomers. These orbitals evolve with increasing chain length from the highest bonding and the lowest antibonding P MOs leading to a loss of the closed shell structure of the P system, which is characterized by an increasing admixture of a doubly excited configuration to the ground configuration. We present a Hartree-Fock scheme which optimizes the coefficients of this configurational mixing and the MO coefficients simultaneously. The configuration interaction (CI) calculation is reformulated in terms of a simple two-site Hubbard model in a localized basis.

1. Introduction The solid state polymerization reaction of diacetylene monomer crystals [ l] leads to nearly perfect polymer single crystals. The short-chain radical intermediates obtained in the course of the reaction have been extensively studied by ESR [ 21 and ENDOR [3] spectroscopy. A transition from the diradicals (DR) with two unpaired electrons to the dicarbenes (DC) with four unpaired electrons is observed with increasing chain length in agreement with earlier predictions [ 41. These two types of intermediates are shown in Fig. 1. The four unpaired electrons of the dicarbenes give rise to singlet, triplet and quintet states which result from the coupling of two triplet carbenes on opposite chain ends. The electronic structure of these intermediates has been examined in some detail in a preceding paper [5] on the basis of molecular orbital (MO) calculations which confirm the transition from the diradicals to the dicarbenes. According to the ESR results [ 21 we have intermediates with predominant diradical character for chain lengths 2
the number of monomer units, i.e. an oligomer of chain length n consists of 4n carbon atoms. The pcntamer (n = 5) and hexamer (n = 6) correspond to an intermediate situation whereas all longer chains can be considered as dicarbenes. An MO scheme for the diradicals and dicarbenes with two and four nearly degenerateMOs, respectively, is shown in Fig. 2. As shown previously [5] the two singly occupied molecular orbitals (SOMOs) of as and b, symmetry comprise mainly in-plane p orbit& of the three outer carbon atoms on each chain end thus being rather localized. Their shape is basically the same for all chain lengths. Since they are nonbonding and nearly degenerate their character is always open shell. The situation is different for the MOs marked by their symmetry labels b, and a,, in Fig. 2. They belong to the conjugated m system formed by p orbitals perpendicular to the molecular plane. With increasing chain length these orbitals change their character from bonding (bs) and antibonding (aJ to nonbonding and become practically degenerate in the limit of sufficiently long chains (Fig. 2b). This provides for the

C. Kollmar/Chemical

132

l c=c=c=c

/” )cc=c=c=c

R/

R/

/”

sc’ n-2

/”

c=c=c=c, R/

R

~_,,,,~ R/

S=l s=o

I

l c;-czc-c R/

WI

/R

3x n-2 c-c~c-*c, / R

DC, s=2 S=l s=o

Fig. 1, Diradical (DR,) and dicarbene (DC,) intermediates obtained in the solid state polymerization reaction of a diacetylene monomer crystal. The spin multiplicities are shown on the right.

a-

-+I

tt

"Rn a

DC, b

Physics 182 (1994) 131-147

rence of such nonbonding surface states in polydiacetylene oligomers is a consequence of the particular bond alternation pattern of these molecules. The electronic configuration of the 1~system is also affected by the transition from the diradicals to the dicarbenes. For reasons of lucidity we will focus on the IT system alone and disregard the other open shell MOs which complicate the story. For short chains the b, and a, orbitals are doubly occupied and empty, respectively, and well separated in energy so that the b, MO is a closed shell orbital. With increasing chain length the loss of bonding and antibonding character leads to an upward (b,) and downward (a,,) shift, respectively, so that the gap between these orbitals decreases and finally vanishes (Fig. 2b). The situation is analogous to the breaking of a bond e.g. in H,. With increasing distance between the two hydrogens the energy gap between the bonding and antibonding linear combination of the two H 1s atomic orbitals ( AOs) gets smaller and finally approaches zero. It is well known that the Hartree-Fock method cannot describe this bond breaking properly so that a configuration interaction (CI) calculation becomes indispensable. The same problem is encountered in the case of the polydiacetylene oligomers. For short chains the IT system has closed shell

a-

Fig. 2. Schematic MO scheme for the diradicals (a) and dicarbenes (b) with two and four unpaired electrons, respectively. We have also given the symmetry labels according to the point group C,,. The orbitals of s and b, symmetry are mainly formed by in-plane p orbitats of the outer carbon atoms. The additional non-bonding orbitak of a,, and b, symmetry in the case of the dicarbenes belong to the conjugated T system formed by p orbitals perpendicular to the molecular plane.

characteristic transition from the diradicals to the dicarbenes which corresponds to a transition from a closed shell to an open shell IT system with two open shell MOs. The degeneracy of the open shell m MOs is noteworthy because it is not a consequence of the topology of the n network as is usually the case for degenerate 7~ orbitals in alternant hydrocarbons [ 61. Moreover, the b, and a,, MOs are localized near the chain ends in analogy to surface states encountered in solid state physics. We will show in the following that the occur-

+-IIb,b,l

i-i--% x lb,b,l + y ia.&

DR (4

tt

+I-

;: (lb&$

- lGi”l)

DC @I

cc>

Fig. 3. MO scheme for the TI system of diradical (a) and dicarbene (c) oligomers visualizing the energetic shift of the b, and a, MOs with increasing chain length. (b) shows an intermediate situation which corresponds neither to a diradical nor to a dicarbene. We show the electronic configurations giving the main contributions to the ground state of the P system.

C. Kollmar/Chemical

character as indicated in Fig. 3a. Its singlet ground state can then be described fairly well by a single Slater determinant which we denote as 1b,b, I. As the b, and a, MOs shift to the mid gap with increasing chain length there is an increasing admixture of the doubly excited configuration ( a,,ii,, 1 to the ground state (Fig. 3b). Finally, if the two MOs are nearly degenerate for long chains both configurations appear with approximately equal weight so that we end up with the wave function ( 1lfi) ( 1bsb, 1- I a,,&, I ) for the singlet ground state of the 7~system (Fig. 3~). The b, and a,, MOs are then open shell orbitals so that the closed shell Hartree-Fock method is no longer appropriate. In order to describe this transition we introduce a Hartree-Fock method which optimizes the CI coefficients of the configurations I b,b, I and I hii,, I and the MO coefficients simultaneously. This method will be described in the following. In the case of near degenerate MOs shown in Fig. 3c the triplet state obtained by placing two electrons of parallel spin in the degenerate b, and a, MOs is very close in energy to the singlet state shown in Fig. 3c. The triplet state can be represented by the single Slater determinant I baa,, 1 if one chooses the magnetic quantum number M= 1. Then the restricted open shell Hartree-Fock (ROHF) method of Roothaan [7] can be used to calculate the MOs. We will show in the following that these MOs are identical to the ones obtained for thesinglet state (llfi)(]bpbs]-]%&]). One might hope, however, that our more flexible MO approach which allows optimization of the CI coefficients is particularly appropriate for the intermediate state of Fig. 3b. This situation corresponds neither to a diradical with a closed shell n system nor to adicarbene which can be described by the ROHF method for the triplet state of the 7~system. The MO calculations in this contribution will be restricted to the conjugated m system in a Pariser-ParrPople-like fashion. The method and parametrization we use have been developed by Berthier et al. [ 81. A note of caution is appropriate in that context. If we speak of a closed shell m system for the diradicals it should be kept in mind that these molecules have nevertheless open shell character due to the presence of the open shell MOs of ap and b, symmetry not belonging to the 7~system (Fig. 2). This also confuses the terminology referring to the spin multiplicity because we speak of singlet and triplet states of the 7~ system whereas the

Physics 182 (1994) 131-147

133

dicarbenes have four unpaired electrons forming singlet, triplet, and quintet states which in general do not correspond to a spin eigenstate of the two r electrons alone [.5].

2. “Surface states” in polydiacetylene

oligomers

It is a quite peculiar feature of polydiacetylene oligomers to have two nonbonding 7~ MOs for sufficiently long chains (n > 6 according to ESR results [ 21) which do not result from the topology of the molecule as is usually the case in alternant hydrocarbons [ 61. Rather, they have to be considered as surface states as we would like to point out in the following. Let us start by considering the Htickel secular determinant of a linear polyene chain consisting of 2n carbon atoms with alternating bond lengths and resonance integrals ti and tZ, E

t,

0

II

E

t*

0

f,

E

.. .

...

= 0.

. . .

E

t2 0

tz

0

E

4

t,

E

(1)

Here we shifted the zero point of the energy scale to the value of the Coulomb integral (Y by setting e=a--E. If weput E~-t:-t~=2t,tZCOS~

(2)

the roots of ( 1) can be expressed by 0 (0 < 8 < rr ) in the following way [ 91: sin[(n+

1)0] + (t2/fl)

sin(&)

=O.

(3)

It is easily seen by plotting sin [ (n + 1) (31and sin (n 0) that Eq. (3) has at least n - 1 solutions 0 in the interval 0 < 0 < 7~.The corresponding energy eigenvalues E are obtained by inserting these solutions in (2). Note that (2) implies that the energy eigenvalues appear in pairs f E which is a general feature of alternant hydrocarbons [ 61. For large n Eq. (2) gives a quasicontinuum of states with (tl-r2)2<2<(tl+t2)2. In fact, Eq. (3) contains all n roots of Eq. ( 1) if t2/tl < (n + 1) ln. If,

C. Kollmur/Chemical

134

Physics 182 (1994) 131-147

E, n

4, - t* -.

t1-b 0 -.

4, + t*

t,+t*:

*(+E)=

= ++

El

Fig. 4. Energy spectrum of a polyene chain with alternating resonance integrals t, and r,. Apart from the continuum states separated by a band gap 2 1t, - t, 1we obtain two discrete energy levels in the gap for ?Jt, > (n + 1 )n which correspond to surface states.

however, t2/tl > (n+ 1)/n Eq. (3) gives only n - 1 solutions. To find the remaining one Eq. (2) has to be extended by allowing complex arguments of the cosine, i.e. to replace 0 by 8 + i p. The condition that i! be real gives immediately sin esinh p=O.

(4)

If p# 0 this can only be fulfilled if 0 is an integer multiple of T. It is sufficient to consider 0=0, n. It turns out that only 8= 7~ leads to a solution of ( 1). Thus, Eq. (2) has to be replaced by e2-t:-t;=

-2t,t,coshp.

(5)

The corresponding solution of the secular equation ( 1) is now obtained by - (tJti)

sinh(np)

=O.

CZk-,42k-I

dz

c k=l

C2k42k’

(7)

+i is the pL orbital of carbon atom 1. We recognize that the two MOs belonging to a pair + E have the same sign on the odd-numbered and opposite sign on the even-numbered carbons. Again this is a general feature of alternant hydrocarbons [ 61 where the Hiickel MOs always occur in pairs of a bonding and an antibonding orbital with the same sign of the coefficients on one set of carbon atoms (say, the starred carbons [6]) and opposite sign on the remaining set (the unstarred carbons). In the case of a linear chain the starred and unstarred carbons are odd-numbered and even-numbered, respectively. For the pair of surface states we obtain cZk - ( - l)k sinh(kp)

,

cZk-, -(-l)“sinh[(n+l-k)p].

(8)

It is easily recognized that the odd-numbered and evennumbered coefficients have their maximum on the left and right chain end, respectively. For sufficiently long chains the foregoing equations can be simplified by replacing the sinh by a simple exponential. From Eq. (6) we then obtain .

p=ln(t,lt,) Inserting 1 -

C2k-

sinh[ (n+ l)p]

n

c k=l

(9)

(9) in (8) we find for the coefficients l)t(t,lt,)“-k)

(-

(6) C2k

Contrary to Eq. (3)) Eq. (6) has just one solution p # 0 if tJt, > (n + 1) ln and none else. This solution lies in the range 0 < p < ln( Qt,) i.e. in the energy interval 0<&((ti-t2)2accordingtoEq. (S).Thus,thetwo corresponding energy eigenvalues + E are located outside the continuum in the band gap - 1tl - t21 < f E-C 1t, - t2) and correspond to two surface states (remember that each solution gives a pair of states). This is indicated in Fig. 4. Contrary to the continuum states obtained from (3) they are not delocalized over the whole chain but have their dominant contribution near the chain ends. This becomes obvious if we consider the corresponding MO coefficients. Numbering the C atoms of the chain consecutively from the left to the right the MOs corresponding to a pair of eigenvalues f E can be expressed as follows [ 61:

-

( -

l)k(t2/4>k,

(10)

or C7J-i = ( - l)k-‘(t,lr2)k-‘c, C2(n-k)

=

( -

1)k(tlif2)kC2n.

, (11)

This corresponds to an exponential falloff of the coefficients with increasing distance to the corresponding chain end. The rapidity of this falloff and thus the spatial extension of the surface states depends on the ratio t, ltp Inserting (9) in (5) we obtain e2=o.

(12)

Thus, both surface states become nonbonding and degenerate in the limit of long chains with their energy being placed exactly in the mid-gap between the qua-

C. Kollmar/Chemical Physics 182 (1994) 131-147

7

.--- W

(a)

w

____ w

(b)

-

.__-w

(c)

Fig. 5. Bond alternation pattern in a polyene chain consisting of 2n carbon atoms with (a) tzltl > 1 and (c) tJt, < 1. (b) shows the delocalization of the unpaired electrons and their associated bond alternation defects along the chain.

sicontinuum of bonding and antibonding states. It should be noted that these surface states are closely related to those obtained e.g. in mixed crystals with two types of atoms I lo]. In fact, the mathematical treatment within a simple tight-binding model is largely the same although the surface states in our case are due to the particular alternance of resonance integrals ti and t2 contrary to the mixed crystals where they result from the different Coulomb potentials of the two types of atoms. We have seen that the condition for the occurrence of surface states is t2/tl > (n+ 1)/n, i.e. the absolute magnitude of the resonance integral tZ has to be greater than that of ti and the chain length must exceed a certain critical limit (n > ti / ( t2 - I, ) ) . Thus, fZ and t, are resonance integrals for double and single bonds, respectively, in a chain of alternating bond lengths. As can be seen from ( 1) this implies that our chain of 2n carbon atoms contains it single and n - 1 double bonds as illustrated in Fig. 5a by a valence bond structure. This leaves two unpaired electrons on the chain ends corresponding to the two surface states. It is clear that a bond alternation pattern with n double and n - 1 single bonds shown in Fig. 5c and corresponding to t2/tl < 1 is energetically more favorable. Actually, there is no reason for the unpaired electrons left on the chain ends in Fig. 5a to stay there. Rather, they are able to move freely along the chain as mobile bond alternation defects [ 111, which are also known as solitons [ 121. This is indicated in Fig. 5b. If the two unpaired electrons which can be considered as a soliton-antisoliton pair meet on neighboring carbon atoms they “annihilate” each other, i.e. they recombine forming a bond leaving us with the structure of Fig. 5c where all 7~ electrons are perfectly paired. It is well known [ 121 that bond alternation defects exist as local excitations on sufficiently long polyene chains but even then they do not stick to

135

the chain ends so that there are no surface states in polyenes. The situation is obviously different for the polydiacetylenes the backbone of which contains sp hybridized carbons and thus additional localized 7~ bonds formed by inplane p orbitals. Here the unpaired rr electrons stick to the chain ends. The reason is most easily visualized in a valence bond picture as can be seen from Fig. 6. If the unpaired electrons migrate towards the center of the chain they “switch” the structure between themselves and the chain end from acetylene (=CC&-C=) to butatriene (-C=C=C=C-) as indicated in Fig. 6b. The bond alternation defect therefore separates chain segments with non-equivalent structures contrary to the polyenes where we have the same alternation of single and double bonds on both sides of the defect (Fig. 5b). Thus, the bond alternation defect in polydiacetylene oiigomers cannot move freely along the chain because it costs energy to change the structure from acetylene to butatriene [ 131. This forces the unpaired electrons to stay close to the chain ends and prevents them from recombining in the longer oligomers. Only for sufficiently short chains the energy loss due to the creation of the energetically unfavorable butatriene structure can be compensated by the energy gain due to the formation of a m bond in case the two unpaired electrons recombine on neighboring carbons. Due to the more complicated bond alternation pattern an analytical criterium for the occurrence of surface states in polydiacetylene oligomers might not be obtained easily. However, in the limit of sufficiently long chains we can again assume that the surface states are completely nonbonding and degenerate ( E = 0 in a Hiickel approach). This can be verified by inserting E= 0 into the Hiickel eigenvalue equation and calculating the corresponding MO coefficients. Using the bond alternation pattern of Fig. 6a and denoting the resonance integrals for the single, double and triple bonds as ri, t2 and r3, respectively, we obtain for the

Fig. 6. Idealized bond alternation pattern for the dicarbene intermediates of polydiacetylene oligomers (a). The creation of the energetically unfavorable butatriene structure resulting from the delocalization of the unpaired n electrons towards the center of the chain is shown in (b)

C. Kollmar/Chemicul

136

Hiickel MO coefficients of a polydiacetylene consisting of 4n carbon atoms

chain

oc, +t,c* =o, . .. t,C4j-3

+OC,j_,

t3C4j-2+OCy_~

=o,

+t3C4j_1 +t,C4j=O,

... t1c‘+,

+oc,,

=o,

(13)

j is here an integer. It can be seen that the equations for odd- and even-numbered coefficients are decoupled for non-bonding states. From Eqs. ( 13) we find immediately the coefficients on, e.g., the odd-numbered carbons, c4j-l

_

-

c4j-3

5

c4j+l

t3'

c4j-- 1

-

_

fi

(14) t2

or C4j_

1=

c4j+l

=

-

(t,lt3)j(t,lt*)j-‘C,

(t,lt,)j(t,lt,)jc,

,

(15)

Analogous equations are obtained for the even-numbered coefficients. Since 1t2 1> ) t, I, I t3 I > It, I it is easily seen that these coefficients decrease rapidly with increasing distance to the left chain end in case of the odd-numbered coefficients as one would expect for a surface state. Eq. ( 15) is identical to the first of Eqs. ( 11) if we put t3 = t2. Due to the more complicated bond alternation pattern the decay of the coefficients is now determined by two ratios of resonance integrals t, It2 and t, lt3. An analogous behavior is of course observed for the even-numbered coefficients the largest of which is c4,, belonging to the carbon atom on the very right of the chain. The odd- and even-numbered coefficients have therefore their dominant contributions on the left and right chain end, respectively, and fall off rapidly towards the center of the chain. If the chain is sufficiently long the r orbitals formed with these coefficients, i.e. the two terms on the right-hand side of Eq. (7) have very little overlap. This justifies a posteriori our Ansatz E = 0 corresponding to the nonbonding character of the surface states in long chains.

Physics I82 (1994) 131-147

Since only nearest-neighbor interactions are taken into account in Htickel theory a bonding or antibonding MO with E# 0 must have nonvanishing coefficients on two neighboring carbons which comprise an odd-numbered and an even-numbered one in the case of a linear chain. One obtains a bonding or antibonding contribution if these coefficients have equal or opposite sign, respectively, but both must be nonzero. Linear combinations of the two surface states formed with the odd- and evennumbered coefficients according to Eq. (7) therefore result in nonbonding MOs for long chains because we have seen that the odd-numbered coefficients are negligibly small in that half of the chain where the evennumbered coefficients are large and vice versa. However, if the chains get shorter a limit is reached where an overlap of the orbital contributions from the odd-numbered and even-numbered carbons can no longer be avoided so that the formerly nonbonding surface states acquire bonding and antibonding character depending on the sign on the right-hand side of Eq. (7). This explains the shift of the MO energies as indicated in Fig. 3. This simple analytical behavior of the MO coefficients in a Hiickel model is largely retained in more sophisticated MO calculations. The fact that the coefficients obtained in our previous MO calculation [5] do not decrease monotonically contrary to the Htickel coefficients ( 15) is mainly due to the fact that the actual bond alternation pattern as obtained from a geometry optimization [5] is somewhat disturbed at the chain end and does not follow the regular pattern given in Fig. 6a. However, this bond alternation pattern is restored in the interior of the chain so that we always have a falloff of the coefficients in this region in analogy to Eqs. ( 14).

3. Hartree-Fock formalism polydiacetylene oligomers

for the rr system of

In the previous section we used a Hiickel approach to give plausible arguments for the existence of surface states in polydiacetylene oligomers. The occurrence of these states has consequences for the electronic configuration of the r system which we would like to investigate within a Hartree-Fock scheme. As already mentioned there is an increasing admixture of the doubly excited configuration I a&, ( to the ground config-

C. Kollmar/Chemical

137

Physics 182 (1994) 131-147

uration 1b,b, 1 with increasing chain length (Fig. 3) which can only be neglected for very short chains. Thus, within a 7~electron approximation it seems piausible to make the following Ansatz: pzc

(16) The diradicals are characterized -0 whereas we have x=1, - l/ 4 2 for the dicarbenes (Fig. region we expect the magnitude and y to be somewhere in the extremal cases. The total energy (16) can be written as follows: 2n-1 E=2

Zn-12n-I

C

L+

k=l +

by the coefficients x=1/h and y= 3). In the transition of the coefficients x range between these of the wavefunction

C

C

k=l

1=1

(U/c,-Kc,)

2y%,, +X2( 2.&b -&I)

+Y2(2L

+b2h

-L)

Zn-1

2n-1

+2y2

C

(2.L,-&,)

7

(17)

k=l

is the number of monomer units. The indices k and 1 refer to the closed shell ITMOs, i.e. the doubly occupied MOs below the b, level in Fig. 3. The open shell IT MOs of b, and a,, symmetry are referred to by the indices b and a, respectively. The two-electron Coulomb and exchange integrals are defined in the usual

+$

rnE/.L

C

n%”

C

/.L+c“2C + C ncI C kec c”#C

C

C C (2ap”J_-b”“K_) IllE@“E”

(2J,,-K,)

.

(20)

rnsZ/l

The superscripts p, v represent open shells each of which might comprise several MOs which are further distinguished by indices m, n. Following the convention of Roothaan [ 71 we reserve the indices k, Z to the MOs of the closed shell which will be characterized by an index c. n Iris the occupation number of the pth open shell (O
n

1

nb=h2,

na=2y2,

abb=bbb=

-, x2

a =,baa,

A, Y

a’=a=O, bba= - ‘. v

(21)

way, Jk[= [MIllI,

&=

[kllzkl ,

(18)

with

[kllmnl

Note that the occupation numbers are n b = 2, na = 0 and nb= l,na= 1 for the diradicals and dicarbenes, respectively. In the minimization of the energy expression (20) different Fock operators appear for the different shells [ 151,

P=i+

c

(2jk_ik)

Both the MO coefficients and the CI coefficients x and y have to be optimized in such a way that the energy ( 17) reaches a minimum. This corresponds to a simple form of a multiconfiguration self-consistent field (MCSCF) approach [ 141. A general expression for the energy of an open shell system has been given by Edwards and Zerner [ 15 I,

i+““=i+

c

+ c

:

YZC

k

c ?nE

,

(2.&j?;) Y

(2j,--itk) k

+

C

v+c

5

c

(2uFvj;-bcLy~;)

fl2EY

Using (21) we obtain

(22)

C. Kollmar/Chemical

138 zn-I P=i+

(2.& -zQ

c

+x2(2&

-&)

k=l

zn-1 Fb=L+

c

(2jk-I?k)+tib-i?b+

f&,

(2.Tk-Itk)+2&ka+

fkb.

k=l zni”=i-k

c

1

(23)

k=l

Note that these Fock operators depend on the CI coefficients x and y. For the calculation of the MOs we follow the procedure described by Edwards and Zerner [ 151 which can be considered as a generalization of Roothaan’s open shell formalism [ 71. In our case one has to find the eigenvalues and eigenvectors of the following operators:

+P;“(y%a+XVb)

Physics 182 (1994) 131-147

simultaneously. Furthermore, we have to take some precautions that the MOs obtained from the different operators of Eq. (24) do not loose their orthogonality in the course of the iteration procedure. This is achieved in the following way [ 151, within each SCF cycle we start by calculating the eigenvectors of l?’ which we use to form a new density matrix PC. Instead of the operators I?” and ia we take (1 -p‘)i”( 1 -p”) and ( 1 -EC)& 1 -PC), respectively, for the calculation of the open shell MOs of b, and a, symmetry. This insures orthogonality of these MOs to the MOs of the closed shell. Note that the two open shell MOs are always orthogonal because they transform as different irreducible representations of the symmetry group C2,,. We already mentioned that the pure dicarbenes are represented by the CI coefficients x = 1/fi and y= -l/b in Eqs. (17) and (23). In that case the Fock operators of Eq. (23) coincide with the Fock operators of the usual ROHF formalism [ 71 as we will show in the following. It is easily seen that this is the case for the closed shell operator for which we obtain 2n - I

+ (y2P+&b)P],

@“=/is-

c

(tik4k)

k=l

+4 (2fb-kb)+; + (P+y*1”‘a)P) + (y2@+X2P)P])

+P(y*++X*P) (24)

I?‘, I?” and @ serve us to determine the closed shell orbitals and the open shell orbitals of b, and a, symmetry, respectively. Note that the arbitrary constants APU appearing in the matrices of the off-diagonal Lagrangian multipliers according to Eq. (16) of ref. [ 151 have all been set to 8. The projection operators I;‘, e” and pa project onto the subspaces of the closed shell, the first and the second open shell, respectively. In the basis of orthogonalized AOs [ 161 they have the following form:

(2&Q.

(26)

It is less obvious for the open shell operators fib and Fa which now have the following form: znkb=L+

I

c

(2&kk)+uAb-kb-ka,

k=l zn-1 Fa=h^+

(2j,-j?k)+2&J?a-i?b.

c

(27)

k=l

The open shell operator of Roothaan’s ROHF formalism [ 71, on the other hand, writes as follows: 2n?=i+

C

I (2&kk)+

&ka++jb-ftb.

(28)

k=l kec

The crk are the MO coefficients of the kth MO. They are determined self-consistently by the usual iteration procedure. In our particular case, however, not only the density matrices but also the CI coefficients x and y have to be recalculated after each SCF cycle. Thus, the MO coefficients and the CI coefficients are optimized

The operators (27) and (28) are identical if j, = .?, . Now we use again the property (7) for a pair of n MOs. Eq. (7) is strictly valid only in a Hiickel approach but remains a good approximation if one uses more sophisticated MO methods as we have seen previously [ 51. If we summarize the contributions on the odd- and even-numbered C atoms in orbitals c and d, respec-

C. Kollmar/Chemical

tively, Eq. (7) can be rewritten as follows for the pair of b, and a, orbitals: b,=

+c-d), Jz

a,,=l(c+d). fi

(29)

We have seen that the orbitals c and d have another interesting property; they are not only localized on different sets of carbon atoms but also on one particular chain end. They have a limited width of about two monomer units [ 51. Thus, they do not interpenetrate each other for sufficiently long chains (n > 4) so that the MOs b, and 3 become largely non-bonding. The operators Jb and J, can be written as follows:

&= j, =

I du,b:CW,W $9 I dUZa,*(2>d2) $.

Using (29) and the zero differential mation we obtain .&,=.?, =

From (28) MOs y= -

I

du2 ,c*(2)c(2)

(30) overlap approxi-

+d*(2)d(2)]

$.

(31) Eq. (3 1) we can see that the operators (27) and are identical. Thus our approach gives the same as the usual ROHF method if x= 1 /fi and 1Ifi, i.e. for dicarbenes.

4. Numerical

results

In a previous paper [ 51, we performed a geometry optimization for the short-chain oligomers based on a restricted open shell Hartree-Fock (ROHF) calculation using the AM1 Hamiltonian implemented in the MOPAC6 program package [ 171. In that contribution the pentamer and hexamer were found to have acetylene structure assuming four open shell orbitals in the ROHF formalism. At this time we were not aware of the fact that there exists a second butatriene-like equilibrium geometry for both oligomers if one uses only two open shell orbitals in the ROHF calculation cor-

Physics 182 (1994) 131-147

139

responding to the two unpaired electrons of a diradical. This second equilibrium geometry has not been found previously because the convergence of the SCFprocess is very sensitive to the choice of the starting geometry in the case of only two open shell orbitals. Note that both equilibrium geometries have been calculated within the same configuration space, i.e. the 36 microstates obtained by distributing four electrons among the four MOs marked by their symmetry labels in Fig. 2. They differjust by their MO basis which has been found using either two (diradicals) or four (dicarbenes) open shell orbitals corresponding to, respectively, a closed shell 7~system or an open shell 7~system with two open shell n MOs. The bond lengths for the two equilibrium geometries obtained by the different MO methods are shown in Fig. 7 for the tetramer, pentamer and hexamer. Disregarding the chain ends where the bond alternation pattern is always disturbed we obtain butatriene structure with bond lengths of, respectively, 1.44, 1.32 and 1.26 A for the single bond, the outer double bonds and the central double bond of a monomer unit for all three oligomers in the case of the calculation with two open shell MOs. The short bond length of the central double bond arises from the fact that there is an additional isolated 7~bond formed by in-plane p orbitals of the sp hybridized carbons in the center of a single monomer unit, whereas the bond alternation of the conjugated system leads to single and double bond lengths which are shorter and longer, respectively, than those of isolated C-C single and double bonds (see ref. [ 131 for a more detailed discussion of the bond alternation in polydiacetylene oligomers). In the case of four open shell orbitals comparison of Figs. 1 and 7 shows that we have acetylene structure with bond lengths of 1.40, 1.35 and 1.20 A for the single, double and triple bonds, respectively, in the interior of the pentamer and hexamer chains. The bond lengths of the tetramer, however, deviate slightly from these values. The transition of the diradicals to the dicarbenes corresponds to a transition from a closed to an open shell IT system. Now we consider a gradual transition between the equilibrium geometries obtained by the two different calculational methods. If the position vectors of the carbon atoms in the equilibrium geometries of the diradical and dicarbene states are given by $” and F, respectively, we can characterize an intermediate geometry as follows:

C. Kollmar/Chemical

140

1.212

1.303

c-%c-c

1.212 c-c-&-c

1.212 1’1448 1.258 F-&C-c-c

9.448

1.215 c-c-,c-c p.444

-

V.396

1.216

\1.@4 yy6c-c-c

1.400

\.1948

l,200

c&c-c

c

\1’394

1.203

t&c-c-c

n=6

1.258

---

1.330

C-C-J1.316

1.204

~,~5c-C-c

---

c,l.448

1.258 $,&c-c-c

C___

1.328

n=5

1.258

1.303 y,;;8c-

1.216

c,&-2c-%ioc,l.364

1.258 1.317

141x \’

n=4

1.316 \1.444 c ___

ck6C_C

1.212

1.314

c-y,;6c-c

1.303

%15c-c-c

’

Physics 182 (1994) 131-147

1.402

\1.347

1.201

c-,c-c-c

1.316 \‘+l

1.400 \I.348 C___

c --(a>

(W

Fig. 7. Carbon-carbon bond lengths of the tetramer (n = 4), pentamer (n = 5) and hexamer (n = 6) as obtained by a geometry within a ROHF calculation using the MOPAC6 program with (a) two open shell MOs and (b) four open shell MOs.

ri

=rDR +f(ry

-lyR),

O
1.

(32)

It is easily seen that f = 0 and f = 1 correspond to the butatriene and acetylene structure, respectively. NOW a IT MO calculation is performed for 0
optimization

obtained in this way for the tetramer, pentamer and hexamer are shown in Fig. 8 by dotted (RHF basis) and dashed (ROHF basis) lines. The addition of the nuclear repulsion makes these curves look a little bit _~____-__________

2 2 ti 5

-140.0 .@_.+L~~

./

/

-i -140’4 w -140.6L 0.0

0.1

,.>.>','I""" 0.2 0.3 0.4

0.5

0.6

0.7

0.8

0.9

I 1 1.0

I 1.0

-174.5-c--?T-T 0.0 0.1

0.2

0.3

0.4

I 0.5

0.6

0.7

0.6

0.9

-207.5ivT--T 0.0 0.1

0.2

0.3

0.4

0.5

0.6

I 0.7

I 0.8

0.9

y

0

Fig. 8. n electron energies of (a) the tetramer, (b) pentamer and (c) hexamer as a function of the geometry parameterf. The results of the RHF, the ROHF and the improved MO method are given by dotted, dashed and full lines, respectively.

C. Kollmar/Chemical

nicer although they still do not reflect the behavior of the total energy of the molecule as a function off because the energy of the remaining valence electrons not taken into account in the 7~electron approximation varies also withf. Remember that f=O and f= 1 correspond to energy minima in the MOPAC calculation. Thus, the curves in Fig. 8 only serve us to compare rr electron energies obtained by different methods. It can be seen from Fig. 8 that the closed shell method, i.e. the RHF basis leads to the lowest energy atf= 0 in all cases. For the tetramer the energy of the RHF basis remains below the energy for the ROHF basis in the whole energy range 0 6. The D values of the pentamer and hexamer are thus too small to correspond to a dicarbene. But they are also too large

Physics 182 (1994) 131-147

141

to correspond to a diradical (f= 0). In this case the 7~ system has largely closed shell character so that there are no unpaired n electrons which could give a contribution to the zero field splitting. The magnetic dipolar interaction of the remaining two unpaired electrons is negligibly small because they are located on different chain ends and thus too far apart from each other to exhibit significant fine structure splitting. Note that the relatively large D value of the dicarbenes arises from the fact that each of the unpaired 7~electrons can interact with another unpaired electron located on the same chain end (Fig. 1). One might therefore doubt if either of the two equilibrium geometries shown in Fig. 7 for the pentamer and hexamer is realistic. The two equilibrium geometriesf= 0 andf= 1 have been obtained by using different methods for the calculation of the MO basis. In the intermediate region, i.e. where the dotted and dashed curves in Figs. 8b and 8c intersect neither of the two methods might be appropriate. With regard to Eq. ( 16) of the previous section the RHF method corresponds to fixed CI coefficients x = 1, y = 0 which is reasonable forf= 0. It was shown in the previous section that the ROHF method is analo ous to choosing CI coefficients of x = 1 Ifi, y = - 1I $ 2 in the case of adicarbene cf= 1) . In the intermediate region, however, a more flexible MO approach might be needed which does not keep the CI coefficients fixed, but optimizes the MOs and the CI coefficients simultaneously, thus providing for a better MO basis in the intermediate region. The technical details of such an improved MO method have been presented in the preceding section. In the following we examine the effect of this approach on the electronic energy of the IT system especially in the intermediate region. Does it lead to a substantial lowering or even removal of the energy barrier betweenf= 0 andf= 1 so that a new equilibrium geometry can be obtained which might account better for the experimental results? The rr electron energies obtained by the improved MO method are given in Fig. 8 by full lines. For the tetramer this curve largely parallels the one obtained with the RHF basis over the whole range 0
142

C. Kdlmar/Chemical

Physics 182 (1994) 131-147

Table I Cl coefficents x and Y obtained by the MO method presented in this paper for the tetramer and pentamer. (DR) and dicarbene (DC) states exist depending on the value of the geometry parameterf n=4 f

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

In the case of the pentamer diradical

n=5 X

0.991 0.990 0.988 0.986 0.984 0.981 0.977 0.972 0.964 0.954 0.938

Y

-0.134 -0.144 -0.155 -0.167 -0.180 -0.196 -0.214 - 0.237 - 0.265 -0.301 - 0.348

DR

DC

x

Y

0.987 0.985 0.983 0.981 0.977 0.970 0.962

-0.161 -0.171 -0.182 -0.195 -0.211 - 0.243 - 0.274

shell ground state configuration 1b,b, 1 contributes between 98% cf=O) and 88% cf= 1) to the wavefunction ( 16). No dicarbene solution is obtained within the geometry range considered here. Again the situation is different for the pentamer and hexamer. First of all, the improved method does not lead to a substantial energy lowering in the intermediate region as one might have assumed. In the “diradical region” (e.g. 0 0.6 if one allows the (Y(spinup) and p (spin-down) HOMO to differ [ 181. For the hexamer such a twofold RHF solution is obtained for f=0.4 and 0.5 as can be seen by looking at the solid curves in Fig. 8c. It depends on the choice of the starting orbitals in the SCF procedure which of the two solu-

X

Y

0.808 0.776 0.758 0.747 0.739

-

0.589 0.630 0.652 0.665 0.674

tions is found. Starting with the RHF MOs we obtain a diradical solution with x being much larger in absolute magnitude than y. Using the ROHF MOs as starting orbitals results in a dicarbene solution with x being comparable in absolute magnitude toy. Note that apart from f= 0.6 for the pentamer and f= 0.4, 0.5 for the hexamer there is a unique Hartree-Fock solution no matter which MO set is used in starting the iterative SCF procedure of the improved MO method. We conclude that the improved MO method does not result in a state intermediate between a diradical and a dicarbene for a geometry intermediate between the butatriene and acetylene structure. Rather, it largely reproduces the results of either the RHF or ROHF method depending on the particular value of the geometry parameterf. Since the energy contribution of the remaining valence electrons is not much affected by the method used for the calculation of the IT electron states for a fixed geometry it can be seen from Fig. 8 that the improved MO method cannot be expected to lead to a new intermediate equilibrium geometry for the pentamer and hexamer as one might have inferred from the intermediate D values observed in the ESR experiments [ 21. For a pure dicarbene, i.e. near f= 1 for the pentamer and hexamer the results of the improved MO method are almost exactly identical to the ROHF results as already discussed in the preceding section.

C. Kollmar/Chemical

Physics 182 (1994) 131-147

143

Table 2 MO coefficients of the b, and a,, MOs on the odd-numbered carbons (orbital c)

1

=,

0.21

3 -0.36

5

I

9

11

13

0.49

- 0.25

0.13

-0.06

0.03

It can be seen from Fig. 8 that the geometry range where a dicarbene is obtained increases with increasing chain length. For the tetramer there is no dicarbene in the whole range 0 4. We give only the coefficients on the oddnumbered carbons because those on the even-numbered carbons are close to zero (vice versa for orbital d on the right chain end). Contrary to the Htickel coefficients given by Eq. (15) which decrease monotonically the coefficients in Table 2 show an initial increase before they fall off towards the center of the chain. As we already mentioned this is due to the disturbed bond alternation pattern at the chain ends.

the CI calculation can be reformulated in terms of a simple two-site Hubbard model if we change the basis from the MOs b, and a, to the orbitals c and d according to Eq. (29). The term “site” comprises a whole set of carbon atoms, either the odd-numbered or the evennumbered ones corresponding to the location of orbitals c and d, respectively. Then we can form states with the two unpaired electrons on the same site (charge transfer state, index CT) or on different sites (covalent states, index CV) . The triplet state is of course always covalent because the Pauli principle prevents two electrons of the same spin from being on the same site. Denoting the states by their symmetry labels we have

5. A model Hamiltonian for tbe transition diradical-dicarbene

t=~(‘A~“IEi(‘A~)=~((~~~~~~a,a,)

So far the CI coefficients n and y served us to indicate the character of the intermediates. Now we show that

U=(‘A~lAI’A~)-(‘A~VIAllA~V)

3B:V)=

lcdl = lb,a,,l

‘By)=

L fi

,

(IcEI - ldd

I)

=~(1d-,gI+l’+ul~.

(33)

Here we have chosen the magnetic quantum number M= 1 for the triplet state. The Hubbard parameters t and U are now defined as follows:

-(b&g

=2(a&

Ifilh$,>)

9

Ifilb,~,)=2taub,Ia,b,l

.

(34)

C. Kollmar/Chemical

144

U is the energy difference between the charge transfer state and the covalent state. It is easily related to the parameters of the MO calculation as is obvious from Eq. (34) which shows that it is just twice the exchange integral between the a,, and b, MOs. Using the SlaterCondon rules for the evaluation of matrix elements between Slater determinants and Eq. (29) the transfer integral t can be reduced to a one-electron integral, t=f((au]F]aa,)-(b,]fi]b,))=(c]p]d).

(35)

Here we have used the zero differential overlap approximation which results in J,, = &, = Jab (compare to Eq. (3 1) ) and the following Fock operator: I’=h^+

c

.

(2&&)

(36)

k

The sum in Eq. (36) extends over the closed shell orbitals, i.e. those located below the b, orbital in Fig. 3. It can be seen from Eq. (35) that the transfer integral t provides for an electron transfer between orbitals c and d. For the dicarbenes the MOs can be obtained by the ROHF formalism [7] with the Fock Hamiltonians (26) and (28). In that case the transfer integral t is directly related to the energy gap between the a, and bZ MOs. This is due to the fact that the Fock operator F in Eq. (35) can be replaced by the Fock operator (28) because the additional terms contained in the latter cancel if we form the difference in Eq. (35). In this case we therefore end up with r= f[E(a,,)

--Vb,)l

U=2[a,b,

]a,b,l

, (37)

Thus, the Hubbard parameters r and U can be obtained easily from the MO results. E(a,,) and E(b,) are MO energies. The Hubbard Hamiltonian giving the CI matrix in the basis of the orbitals c and d has the following form:

+ U(n,rn,l

+n,r%l)

.

(38)

and a are creation and annihilation operators, respectively. Spin-up and spin-down electrons are symbolized by arrows. The number operators are defined in the usual way, e.g.

U+

Physics 182 (1994) 131-147

+ net =ucfuct *

(39)

Using (38) and (33) we obtain the following tonian matrix: ]‘A:“) Ci

]‘A;V) ]‘A?) 13BzV)

I

]‘Ay) 2;

Hamil-

13BzV) 0

2t

lJ

0

0

0

0

(40) I

Here the energy of the ‘AZ” state marks the zero point of the energy scale. We have omitted the ‘B, state because only charge transfer states mixing with one of the covalent states are of interest to us. The 3B, state corresponds to the configuration used in the ROHF formalism. Now the character of the intermediates is indicated by the ratio 4tllJ. If this ratio is sufficiently small the singlet ground state is dominated by the covalent state ‘Ay with little admixture from the charge transfer state. This is characteristic of a dicarbene as becomes obvious from the first of Eqs. (33) which corresponds toEq. (16) withx= -l/fiandy=l/fi.Thesmall admixture of the charge transfer state leads to an energy lowering of the singlet state so that we obtain the following singlet-triplet gap in second-order perturbation theory: E(3B,) +(‘A,)

=4t2/U.

(41)

Since the magnitude of the transfer integral t depends on the overlap between orbitals c and d as can be seen from Eq. (35) it decreases drastically with increasing chain length so that the gap converges to zero. At this point it should be noted that the discussion in terms of singlet and triplet states of the n system alone is somewhat misleading because we have two additional unpaired electrons not belonging to the 7~system in orbitals denoted as a and b which are localized on the left and right chain end, respectively. These orbitals are strongly reminiscent of the non-bonding orbitals in allyl. It has been shown previously [.5] that the three in-plane p,, orbitals of the three outer carbon atoms on each chain end represent a kind of “7~” system by themselves in analogy to that of the ally1 radical. Thus, the non-bonding orbitals a and b are mainly formed by p_”orbitals on carbon atoms 1 and 3 for orbital a (see Fig. 9) and carbon atoms 4n and 4n - 2 for orbital b [ 51. We therefore end up with altogether four unpaired

C. Kollmar/Chemical

145

Physics 182 (1994) 131-147

electron system. Introducing the allyl-like orbitals a and b we obtain the following covalent states [ 51: I’Ai”)=i

cabd -&

(2aa/3P+2

/~/?CXCX

A

-c-c

fP UC
Fig. 9. Carbenic configurations of two unpaired electrons on the left chain end of a dicarbene. The unpaired T electron in orbital c has contributions on the pr orbitals of carbon atoms I and 3 whereas the second unpaired electron in orbital a is almost completely localized in the in-plane pv orbitals of the same carbons (ref. [ 51). A corresponding picture applies to the right chain end (orbitals b and d) electrons for the dicarbenes. The unpaired IT electrons on different chain ends couple via the transfer term in (38). This coupling is weak. Much more important is the exchange interaction which couples a IT radical electron on one chain end to the remaining unpaired electron on the same chain end leading to a triplet carbene [ 51. This interaction is characterized by a twoelectron exchange integral which we denote as K. It arises from two unpaired electrons being in orthogonal p orbitals on the same carbon and favors parallel spin alignment as indicated in Fig. 9. Thus, in our previous paper [5] we extended the Hamiltonian (38) by an additional Heisenberg term taking into account the exchange interaction between two electrons on the same chain end. It has the following form [ 51: &

= -2K($,.s^,

+s;.&,)

.

(42)

Here the index pairs 1,2 and 3,4 refer to the two electrons on the left and right chain end, respectively. It is favorable to choose the spin basis of the four-electron system in such a way that it consists of eigenstates to the exchange operator (42). The operator (42) commutes with $ = (s1, +i2)* and 3; = (slj +&)’ so that the spin states are also eigenfunctions to the latter operators with spin quantum numbers S1 =0, S,=O (singlet) and S, = 1, S, = 1 (triplet). In the following we couple only spin functions with S, = S2 = 1 corresponding to a triplet state of both electron pairs, i.e. a triplet carbene on each chain end. Since K > 0 this leads to the energetically lowest eigenstate of (42). The vector coupling of the two triplet functions on the two chain ends yields singlet, triplet and quintet states of the four

+ papCY+ pcurXp+cr~~a)

.

(43)

These wavefunctions replace the covalent states of Eq. (33). a is the antisymmetrization operator. Note that the Hamiltonian (42) acts only on the spin part of the wavefunction (43). Since the transfer integral I is small for the dicarbenes ( 1t1 -SC K,lJ) the transfer term in the Hamiltonian (38) can be considered as a perturbation. It leads to a small admixture of the following charge transfer states to the covalent states [ 51: - I’AF)=$( IcCa6I + IcCbCil +ddabI + Iddbal), - 13BzT)=i( - I&a61 + lccb?il - Iddab + IdabZl) . (44) Note that there is now an additional triplet state with charge transfer character which mixes with one of the covalent states. It results from the triplet state of the two electrons being in orbitals a and b (compare Eq. (44) tothestate I’AF) ofEq. (33)). So far this is just a brief repetition from our previous paper [ 51. What we would like to emphasize in the present context is the fact that the covalent states in (43) except for the quintet do not correspond to a pure spin state of the unpaired 7~electrons. Rather they contain a mixture of singlet and triplet functions of the IT electrons. This is easily seen if we form the expectation value of the spin operator s^’ for the n electrons (electrons 1 and 4 according to the notation of Eq. (43) ) , (‘A?

@;I ‘A;“)

=(‘A;“)(&

+&)*1’A;“)=;,

(3B;V Ij;13B:“) =(3B:vI(s;

+s^4)*13B;“)=1,

C. Kollmar/Chemical

146

=(5A;V](S1,

+?J2]5A;V)=2.

(45)

Thus, only the quintet state contains a pure IT triplet whereas the singlet state of the n system contributes 75% and 50%, respectively, to the overall singlet and triplet states. This singlet character is crucial for the energy lowering due to the admixture of charge transfer states. The larger the singlet character of the rr system the larger the admixture of charge transfer states. Taking the sum of the Hamiltonians (42) and (38) and considering the transfer term in (38) as a perturbation we obtain the following energies in second-order perturbation theory [ 51: E(‘A,)=

3t2

-K-

U+K’ SK,

E(3B,) = -KE(5A,) = -K

.

(46)

Since K -=x U we notice that the energy lowering of the singlet, triplet and quintet states arising from the transfer term in Eq. (38) is 75%, 50% and O%, respectively, of the gap (4 1) which corresponds exactly to the contribution of the rr singlet to these states.

6. Conclusions It has been shown by means of a simple Hiickel model that the transition from diradical to dicarbene intermediates in polydiacetylene oligomers is due to the occurrence of nonbonding 7~MOs which are localized on the chain ends (“surface” states). This is a consequence of the particular bond alternation pattern in polydiacetylene where the presence of sp instead of sp2 hybridized carbon atoms in the main chain leads to the existence of two energetically non-equivalent structures (butatriene and acetylene structure). The qualitative shape of the nonbonding Htickel MOs is retained in more sophisticated MO calculations as performed in this contribution (see Table 2) and a preceding paper [ 51. Comparing the MO coefficients of the nonbonding MOs shown in Table 2 with those given in our previous paper one may note that there is some deviation although the qualitative shape of the MOs is of course

Physics 182 (1994) 131-147

the same. To the main part this is not a consequence of the fact that we use a n electron method whereas the AM1 Hamiltonian [ 171 has been used in our previous paper. Rather it is due to the different ROHF approach used in the earlier calculation. That calculation was based on the half electron method [ 191 instead of the more accurate ROHF approach of Roothaan [7] applied in the present contribution. Using the half electron method within the rr electron approximation we get results which are very close to the earlier ones. The transition from the diradicals to the dicarbenes leads to a decrease of the gap between the highest bonding and the lowest antibonding rr MO which converges to zero with increasing chain length so that these MOs become approximately degenerate and nonbonding. This is accompanied by an increasing admixture of a doubly excited configuration to the ground configuration of the 7~ system. We tried to account for the gradual transition from a closed shell to an open shell rr system by a particular MO approach which allows simultaneous optimization of the CI and MO coefficients thus corresponding to a simple version of a multiconfiguration self-consistent field (MCSCF) method [ 141. Such an approach might also be appropriate to describe the breaking of a bond because the bonding and antibonding MOs in that case behave like the b, and a,, MOs in Fig. 3. In the present case, however, it turned out that it did not result in a substantial change of the rr electron energies as compared to the conventional Hartree-Fock methods, i.e. the RHF method for the closed shell IT system of the diradicals and the ROHF method for the open shell n system of the dicarbenes. Thus, it does not account for the smooth transition from the diradicals to the dicarbenes observed in ESR experiments where the pentamer and hexamer correspond to an electronic state intermediate between a diradical and a dicarbene. However, it is found both experimentally and theoretically that all oligomers of chain lengths n < 5 have predominant diradical character. We have used a very limited CI calculation for the description of the 1~system of the dicarbenes. Thus, we just consider the correlation of the unpaired electrons as described by the simple model Hamiltonian of section 5. The remaining electrons are assumed to form a fixed “core” which is not affected by the particular spin state of the unpaired electrons. This is of course only an approximation. In fact, the energy gaps

C. Kollmar/Chemical

between the lowest singlet, triplet and quintet states of the dicarbenes calculated by the limited CI method are too small as compared to the experimental values [ 5 ] even if the observed rapid decrease of these gaps with increasing chain length is reproduced qualitatively by the theoretical calculations. A more extensive CI calculation including excitations of the closed shell electrons is needed if one wants to get better results for the energy gaps [ 201. It might be expected that such a CI calculation will widen the energy gaps because correlation effects not taken into account in our simple model are supposed to lower the energy of a low-spin state more strongly than that of a high-spin state corresponding to the same orbital configuration. Note that correlation of electrons with the same spin is already included in a simple Hartree-Fock calculation. In our simple model, the size of the closed shell core plays no role because the closed shell electrons enter only indirectly into the calculation of the model parameter f of section 5 by providing a Coulomb potential in the operator (36) used in the formation of the transfer integral t=(c]$]d) (Eq. (35)).ThemodelparameterU,on the other hand, depends exclusively on the form of the open shell MOs b, and a,, (c and d in the localized basis) as can be seen from Eq. (37). However, in extending the CI calculation to closed shell MOs one is faced with the problem of size inconsistency of a truncated CI expansion because the number of closed shell electrons increases with the chain length. It is not clear how this will affect the gaps between the lowlying multiplets. Finally, it should be noted that the geometry optimization in our previous paper [ 51 on which the present calculation is also based applies to molecules in the gas phase. It can be expected that the crystalline environment in which the oligomers are embedded has some influence on their geometrical arrangement. Actually, there is a misfit of the oligomers in the crystal matrix of a partially polymerized crystal [ 211, which leads to some crystal strain acting on the oligomers. This effect may be hard to quantify.

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