Journal of MolecularStructure (Theochem), 136 (1986) 155-164 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

MOLECULAR TOPOLOGY IN EXCITED STATES

PREDRAG ILIC

Chemistry Department, Montana State University, Bozeman, MT 59717 (U.S.A.)

BORIS SINKOVIC

Chemistry Department, University of Hawaii, Honolulu, HI 96822 (U.S.A.)

NENAD TRINAJSTIC

The Rugier Bo&ovie Institute, 41001 Zagreb, Croatia (Yugoslavia)

(Received 7 June 1985)

ABSTRACT

Perturbation theoretical formalism is applied to topological (graph theoretical) reso- nance energy model in order to study the conjugacy of annulenes in excited states.

INTRODUCTION

Strict conjugacy is a property which confers a remarkable uniformity on the compounds which possess it. Within a descriptive formalism of chemical bonding, conjugated compounds are described as rigid spatial structures built of alternating single and double bonds. In more technical terms, electrons occupy ll MO’s and are delocalized over the whole carbon-atom lattice. With these two views are associated two theoretical methods of considerable historical and actual value. One is founded on a total delocalization of electrons and is known as the Free Electron Molecular Orbital Method, FEMO [ 11. The other approach is based on the condition that a conjugated compound is merely a lattice of uniformly connected carbon atoms; and this approach was used in this work. Existence of this uniformity has been shown to justify a certain abstract mathematical representation of conjugated compounds.

In this representation carbon-atom framework is abstracted and given nonmetric linear structure possessing a weak topology [2]. Carbon atoms (or nitrogen, oxygen, etc. atoms) are represented by vertices or 8-simplices [2] forming a set usually denoted as VG [3]. The relation EG represents all chemically defined interatomic connections and is known as a set of edges or l-simplices [2]. The class VG and the relation EG make an intersection denoted as a graph G [4]. The relation EC reflects a metric of the MO method physically underlying the applied mathematical structure. At the most simple, yet most effective, level this relation is non-reflexive, symmetric and

0166-1280/86/$03.50 o 1986 Elsevier Science Publishers B.V.

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non-transitive [5]. It is essential to understand that graphs of all existing conjugated compounds are planar structures. Thus, for instance, it can be shown [ 6 ] that the graphs of the so-called three-dimensional aromatic com- pounds [7] are perfectly planar structures. It is exactly that planarity, or the dimensionality, of the surface the particular graph is embedded in which determines fundamental properties of all MO models suitable for a represen- tation of conjugated cyclic compounds [8]. The most important property, of course, is that the fundamental group of the maximal torus covering a two dimensional surface is a cyclic group generated by the imaginary unit ‘5” [ 91. An immediate consequence is that essential properties of those structures are suitably expressed in terms of ring structure modulo 4. The somewhat more familiar form of this statement is the well known “4n + 2” rule in the Hiickel MO method.

Subsequent representation of a conjugated system entails an even lower dimensionality. Using a specific combinatorial procedure [lo] the graph is mapped into a scalar index. That index is a certain invariant (or an approxi- mate invariant) of a graph representing a specific property of the graph and ultimately of the conjugated compound. In the approach presented here that index represents the cyclic structural characteristics of a compound; it may be called a planar cyclicity index, PCI, of a graph G. At this point the follow- ing assumptions are usually made (though not usually stated): (i) the planar cyclicity index is proportional to the sum of the one-electron eigenvalues; (ii) the sum of energy eigenvalues is proportional to the molecular energy. The second assumption has been shown to be generally valid [ 111 while the first assumption is true only when a particular linear part of the total molec- ular electronic energy is considered. It is exactly that part of the electronic energy we are concerned with in this communication. Due to the essential step of using a graph representation of a conjugated compound that quantity was named topological resonance energy [12]. The method has been thoroughly and multiply described in the literature [ 133 . It is obvious that the existence of an uniformity in bonding and of a cyclic structure are essen- tial for the validity of this approach. Fortunately, this requirement has been closely fulfilled in a majority of conjugated compounds in the ground state. Hence the relative success of that method. In this communication some limitations and bounds of this approach are analysed, with particular con- sideration of systems where fundamental requirements for an application of the method are only approximately fulfilled. These are, for example, some excited states of conjugated compounds.

It may be illustrative to mention the simplest system of this type, i.e., ethene. It is well established [ 14 3 that this compound undergoes significant geometrical changes with an electronic excitation. Expressed in an alternate

way - the bonding pattern is mutated. The rigidity of the structure, inherent to the double bond, is remarkably relaxed resulting in rotation around the carbon-carbon bond in the excited state being rather an easy process. Conse- quently the molecule of ethene may be represented as in Fig. 1. Ethene by

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Fig. 1. Representation of rotation about the carbon-carbon bond in ethene.

itself is of little use in this work since we rely more on similarities in the general pattern of geometrical changes which seem to pervade, to some extent, to cases of considerably more complex conjugated compounds. It has been postulated [15] and experimentally assessed [ 161 that a certain relaxation of a rigidity in the carbon lattice occurs in excited states of cyclic conjugated compounds. It should be clear that the purpose of this com- munication is not to analyse closely the possible processes in a particular electronic excitation, for example, from an aspect of adiabaticity, non- adiabaticity, vibronic coupling, mixing of vibrational modes etc. A graph representing an excited state may be considered as a structure which is static in space and time, that is, static in the space of the averaged distortions of a carbon lattice accompanying a particular electron transition, and static in the time of the duration of that excited state. More than one electron excitation may also be considered, with a restriction to the Il, II * subset of MO%. In broad terms, therefore, a certain change in bonding of a conjugated com- pound, which may be caused by electron excitation, is considered. A conse- quence of this process, which is essential to the model used here, is that in such ’ systems uniformity of a relation EG is no longer a good assumption. The problem may be treated by using at least two approaches: (i) para- metrizing the overlap integral associated with the centers connected by the “weakened” bond, or (ii) treating that change as a perturbation in the system.

METHOD

The use of parameters in graph representation is well known [17]. It allows an extension of the representation to a substantially larger number of conjugated compounds. On the other hand it also means a certain departure from the inherently non-parametric topological approach to conjugacy. Therefore, the perturbation method is used here and applied to representative conjugated annulenes.

A graph combinatorial method for calculating resonance energy depends essentially on a reference structure of a cyclic conjugated compound [13]. In general, this structure is hypothetical, it exists only in the specific combinatorial map. It has been shown, however, [lS] that in the case of

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annulenes, corresponding reference structures are real, existing and easily constructed graphs [19]. Therefore a perturbation method must be applied to both a cyclic graph and its noncyclic reference graph. In the following text the indices “C” and “R” are used for “cyclic” and “reference”, respec- tively. Other notation used is as follows: $, molecular orbitals for a cyclic system; A, molecular orbit& for a reference system; $, atomic orbitals for a cyclic system; x, atomic orbitals for a reference system, and p, q, atomic centers incident to the perturbed bond.

The total energy (ll electrons) is calculated as a difference between the total energies of the cyclic and reference systems, i.e.,

AEck) = (J/k\& + H’l$k) - (AkIH,J + H’(Ak) (1)

with the first order term explicitly given by,

-1 ClaClb(xalHOkb) a. b

-c wxpI~0lx&-” ai c ~l,~lb(x,~~O~xb)~~~ {(XplHOkq)) (2) P.9 a,b

In this formalism [Nlannulenes possess either (N - 2)/2 or (N - 1)/2 doubly degenerate eigenvalues for even-number and odd-number systems, respec- tively. The regular perturbation treatment for nondegenerate levels is there- fore applied, with the formalism given by (2), and what is called the singular perturbation treatment for degenerate levels [ 201. The formalism in the latter case is somewhat more involved and, for example, for the degenerate pair of levels U, u it is given by

A,Z$U, = ,z$c; + 6E(,C) - &.a) - &$P) (3) AE’” = ,?$cL + 6E;c) _ E(UR) _ G@’ (3’)

where E$ and Et:4 are, by the fundamental condition of the perturbation treatment, identical. The terms SE:‘) and 6E$‘) are obtained as solutions to the following quadratic equation

6E;y; = 1/2{UU+ VV? d{(UU+ VV)2 - 4[(UU)(VV) -(UV)(VU)]}} (4)

which is the characteristic polynomial of the secular determinant,

UU-6E UV det =

I (5)

VU VV-6E

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The element UU is given as

x a {cc c,,hiH0l C cdvl . 6 U~pl~oldbq)l a b

and analogously for VV, while UV = VU and are given by

(6)

uv = (1 Cua$JolH,,l 1 C,b$b) + 1 a{(@PIHoI~d-’ a b P,Q

x a {( x Cua$alHOi 1 Cub&8 ’ 6 {(t+dHOi@q)) (7) B b

It should be noted that, while using the singular perturbation treatment for calculation of the GEtC? (3, 3’), only the regular perturbation treatment is used for the corresponding values &EL:;; i.e., there is no degeneracy in the eigenvalues of the reference structures of annulenes.

APPLICATION

The following problem now arises. When using the perturbation formalism the centers p, q, connected by the perturbated bond, have not been defined. The position of that bond on an annulene ring is irrelevant due to the fact that an [N]annulene is a N-fold permutationally degenerate object. How- ever, the reference structure of an annulene is represented by a terminally weighted noncyclic graph. Typical graphs of this type are given in Fig. 2. These graphs show lower and substantially different symmetry from that of the corresponding cyclic graph. Furthermore, it is generally known that clos- ing a chain into ring(s) or partitioning a ring into chain(s) is site dependent [ 211. In the case of the perturbation treatment used here [22] the dis- similarity of cyclic and reference graphs can generate serious problems. The benzene molecule may be used to illustrate this point.

Previous theoretical and experimental work [23] on similar systems was examined and from this it can be assumed that, in the excited state, the benzene skeleton may undergo symmetric changes on opposite bonds. This is

~,8’q*s\,~N (b )

? 3 N-l

Fig. 2. Terminally weighted noncyclic graph of [Nlannulene.

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6

5

4 0

1 6+! 1

WI I

2 4e 2

3 3

Fig. 3. Schematic representation of the bond changes occurring in benzene.

shown schematically in Fig. 3. Transforming this picture into technical terms it is necessary to induce numerical variations in the particular pair of resonance integrals. This must be done simultaneously on both cyclic and reference models. To make the process more clear it may be said that the corresponding bonds in the reference structure are undergoing a conrotatory torsion through an angle 8. The value of AE in the first row of Table 1 is practically unchanged for the topological resonance energy of benzene [13]. At 0 = 89.99”, how- ever, AE should approach the topological resonance energy of a hexagonal conjugated system which is substantially perturbed so that it more closely resembles two weakly connected three-center fragments. The calculated AE, however, indicates a system which is even more stabilized by conjugacy than benzene. If different pairs of perturbed bonds are used the results given in Tables 2 and 3 are obtained. The AE values in Table 2 do not seem to reflect the significant geometrical evolution of the hexagonal system but their trend is at least more reasonable than those in Table 1. The same values in Table 3 are even more promising. However, the same question pervades - which are the true perturbation calculation topological resonance energy values? In order to be sure that this erratic behavior is not symptomatic to the benzene system exclusively, these calculations were repeated for a system which may be considered as antipodal to the benzene molecule [24], i.e., cyclobuta- diene. The representing graph and the changes induced on it are shown in Fig. 4. The application of the perturbation treatment to cyclobutadiene yielded the results given in Tables 4 and 5. The trend shown by the AE values in Table 4 is of proper sense. and magnitude for a small-ring system such as cyclobutadiene [25], but with more pronounced numerical changes than expected. Table 5 shows an even more acceptable trend.

TABLE 1

Resonance integrals in the cyclic structure, angle of torsion in the reference noncyclic structure and calculated values of AE for benzene

0 (deg.1 AE

-0.00001 0.26 0.28028 -0.33333 48.19 0.31262 -0.66667 70.58 0.34498 -0.99999 89.99 0.37730

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TABLE 2

P 656 28 = 0 (deg.) AE

-0.00001 0.26 0.28028 -0.33333 48.19 0.25038 -0.66667 70.58 0.22046 -0.99999 89.99 0.19056

TABLE 3

fi PI1 34 = TV (deg.1 AE

-0.00001 0.26 0.28028 -0.33333 48.19 0.09856 -0.66667 70.58 -0.08316 -0.99999 89.99 -0.13243

TABLE 4

PI2 = P 34 0 (deg.) AE

-0.00001 0.26 -1.21846 -0.33333 48.19 0.01306 -0.66667 70.58 1.24458 -0.99999 89.99 2.47606

TABLE 5

P*, = &I I.J (deg.) AE

-0.00001 0.26 -1.21846 -0.33333 48.19 -1.21846 -0.66667 70.58 -1.21846 -0.99999 89.99 -1.21846

4 1 4+

-/;

3 2 3”2

Fig. 4. Schematic representation of the bond changes occurring in cyclobutadiene.

As indicated earlier (vide supra) the reason for inconsistencies among different series of calculations is the dissimilar symmetry of the molecular and reference graphs. Perturbed topological resonance energies for an annu- lene are constant on the chosen place of perturbation whilst the analogous

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energies for a reference system (which is a structure with one symmetry element) are quite dependent on that choice. The total energy, given in Tables l-5 as AE, is a difference between two series of energies. This is the reason for dramatic variations among sequences of calculated AE values. In some cases, however, even more crude inconsistencies were overlooked. For example, in Tables 3 and 5 the AE values were arrived at by perturbing the bonds between centers 1 and 6 in the benzene system and between centers 1 and 4 in the cyclobutadiene system. However, neither bond exists in corre- sponding reference structures. In terms of the simple physical model used here this means a rotation around a nonexisting bond. It is thus obvious that the perturbation treatment applied simultaneously to pairs of graph repre- sentatives cannot be used by treating the molecular and associated reference graphs as purely algebraic objects. A more sensible application of the method should be based on physical facts related to the changes in both cyclic and corresponding linear systems. In the case of benzene this means that a specific perturbation results in a degradation of a cyclic hexagonal system into two ally1 fragments connected at both ends. Simultaneous perturbation of the reference system produces two ally1 fragments connected at one end. When based on this model, perturbation calculations produce reasonable numerical values for AE forming a meaningful sequence, as given in Table 6. Application of the same reasoning to the cyclobutadiene system gave the results shown in Table 7. The high positive trend in AE values for the perturbed cyclobuta- diene system clearly corroborates the existence of the low-lying, photo- chemically induced, transition state in the dimerization reaction of olefins [26] _ It is interesting to note that, within the presented formalism, the topology of that state is more properly represented by the scheme shown in Fig. 5.

TABLE6

P x1.23.34 = P 45.56.61 6 (deg.) AE

-0.00001 0.26 0.28028 -0.33333 48.19 -0.06444 -0.66667 70.58 -0.40920 -0.99999 89.99 -0.75392

TABLE7

P 12.34 = 34.41 P 0 (deg.) AE

-0.00001 0.26 -1.21846 -0.33333 48.19 -0.60273 -0.66667 70.58 0.01306 -0.99999 89.99 0.62880

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hlJ hV -

Fig. 5. Schematic representation of the photochemicaily induced transition state in the dimerization of cyclobutadiene.

CONCLUSION

The existence of noncyclic graphs and the possibility for their construction [18], as reference combinatorial structures for annulene molecular graphs, represent an interesting and valuable extension of the topological representa- tion of chemical phenomena. The subsequent application of perturbation formalism however, should be performed with due concern for the physical model of the system under consideration.

ACKNOWLEDGEMENT

One of the authors (P. I.) gratefully acknowledges NIH financial help through the grant GM31824.

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19 Beside annulenes there are other monocyclic graphs that possess simple, easily con- structed reference graphs. One of the authors (P. I.) is indebted for this communication to Prof. W. C. Hemdon (El Paso, TX, U.S.A.).

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