Factorization of Interaction Graphs with n-fold Symmetry: Some Applications in PPP Calculation and Molecular Vibrational Analysis

JAYANTA SARKAR Okersa High School, Okersa, Burdwan-713514, West Bengal, India

ASOK K. MUKHERJEE" Department of Chemistry, University of Burdwan, Burdwan-713104, West Bengal, India

Received August 25, 1995; revised manuscript received December 14, 1995; accepted December 28, 1995

ABSTRACT m The (6 and iF matrices in the molecular vibration problem, the secular matrix in Hiickel calculation including some nonneighbor interactions, and the Fock matrices at any stage of iteration in the Parker-Parr-Pople (PPP) calculations on cis- and trans-butadiene, benzene, and s-triaminobenzene systems have been shown to be factorizable by representing them graphically and then applying the generalized technique of splitting of graphs with n-fold symmetry. 0 1996 John Wiley & Sons, Inc.

Introduction

t the Huckel molecular orbital (HMO) level A graph theory (GT) has found extensive appli- cations [ 1-31, an interesting and useful application being in the analysis of carbon clusters [4-81. Graph factorization techniques were developed by Hall [9], McClelland [lo, 111, and D'Amato [12, 131 primarily for studying the eigenspectral features of conjugated molecules in the HMO level. A de- tailed review of these methods and their applica-

*To whom correspondence should be addressed.

tion in the analysis of HMO eigenspectra was done later by DAmato, Gimarc, and Trinajstic [141. Davidson [151 and more recently Shen [161 gener- alized the above methods to the case of n-fold rotationally symmetric graphs. While the former authors [9-131 concentrated on Huckel graphs, Shen [16] used some graphs of general type where an edge stands for an "interaction" between the vertices connected by it; the nature of the interac- tion was not, however, specifically mentioned in this work. It is, therefore, felt that although the existing graph factorization techniques are very elegant, their utility in quantum chemical prob- lems beyond the HMO level remains to be shown.

International Journal of Quantum Chemistry, Vol. 60, 825-832 (1 996) 0 1996 John Wiley & Sons, Inc. CCC 0020-7608 I96 1040825-08

SARKAR AND MUKHERJEE

The Davidson-Shen [15, 161 algorithm for fac- torization of graphs is in fact a compact graphical form of carrying out a unitary transformation on circulant matrices. Such matrices, owing to their special structure, are amenable to interesting mathematical manipulation, as was shown many years ago by Lowdin, Pauncz, and De Heer [17] in the calculation of inverses of overlap matrices of cyclic systems.

The applicability of GT in HMO theory rests upon the fact that in this theory there is one orbital per atom (site). In the Pariser-Parr-Pople (PPP) theory [ 18-20] also, the molecular orbitals (MOs) involve one orbital per site and hence extension of GT applications to PPP level is natural. The first application along this line was made by McLach- lan [21, 221 who demonstrated that a Coulson- Rushbrooke-like [23] pairing theorem holds for the self-consistent field (SCF) T-MOS of alternant hy- drocarbons. A GT analysis of the difference be- tween HMO and PPP bond orders has been done fairly recently by Randie, PlavsiC, and Trinajstie [24]. Many discussions of the use of graphical vertex alternancy in the PPP model can be found in the literature [25-321. Novel color-pairing sym- metries which involve hybrids of point group and alternancy symmetries have been developed and used in a number of model hamiltonians (which require one orbital per site) by Klein and iivkovie 1331.

The object of the present study is to show the applicability of the generalized Davidson-Shen [ 15, 161 factorization technique on some interaction graphs representing secular matrices in real chemi- cal calculations. Examples will include factoriza- tion of (a) Wilson’s [34] [F and G matrices in a molecular vibration analysis, (b) the HMO secular matrix of bicyclooctatetraene where some relevant nonneighbor interactions have been included, and (c) the PPP Fock matrices of cis- and trans- butadiene, benzene, and s-triaminobenze system at any stage of SCF iteration.

~ -~ ~-

Brief Description of the Factorization Technique

The cases used in this study are representable by graphs with an n-fold axis of symmetry with no vertex lying on the axis. For such a graph with nN vertices the entire factorization technique is

based on the following labeling scheme:

1. Choose any vertex for assigning label = 1. 2. The vertices obtained by the operations

C,, Ci, . . . , C:-’ on vertex 1 are labeled 1 + N, 1 + 2N,. . . , etc., respectively.

3. Choose any vertex from the remaining unla- beled ones and call it the second vertex (label = 2). Its images under C,,C;,. . ., are la- beled 2 + N, 2 + 2N,. . . , respectively.

This procedure is repeated until all the vertices are exhausted. Under such a labeling scheme the adja- cency matrix (A) of the graph becomes a parti- tioned circulant matrix:

A = . (1)

It has been shown that A is isospectral with the block-diagonal matrix,

1

Factorization of the Interaction Graphs Representing the 1F and G Matrices in the Vibrational Analysis of Pyramidal AB,-”ype Molecules

The internal coordinates of the system are shown in Figure 1. The [F matrix, constructed by Wilson’s method [341, is

1 f r f i r , f i r , 0 fro f r o I f r r , f r f i r , f r o 0 f r o

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INTERACTION GRAPHS WITH n-FOLD SYMMETRY

G1

G2

FIGURE 1. Interaction graphs for vibrational analysis of pyramidal A6,-type molecules: G, represents the F matrix and G, the G matrix.

This [F may be regarded as the adjacency matrix of the graph G, (Fig. 1) whose vertices have been so labeled that their adjacencies are in accordance with [F. The vertices indicated by hollow circles carry weight f0 and the dark ones carry weight fr. The edges indicated by solid lines are of weight frrr, and those indicated by broken lines are of weight frO. This G, is the "interaction graph" for the problem under consideration. Obviously, it has a C, axis.

For factorization using C, according to the scheme outlined above, we note that 3 N = 6 and

relabel the vertices as follows:

Old label New label

With the new labels the adjacency matrix of G, becomes

5, = [frO f r fro f#]. B2 = [$;; :],

5 3 = [ f ; , fro ( ) I . and

where the relation 1 + w + w 2 = 0 has been used. Thus f is block-diagonalized to the form,

The interaction graph representing the G matrix for AB, can be similarly constructed and factor- ized. Following the convention used by Cotton

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SARKAR AND MUKHERJEE

[35], the G matrix is

G =

Only the upper semimatrix is shown in (6) and the corresponding interaction graph (G,) is shown in Figure 1. In this diagram each vertex indicated by a hollow circle carries weight g&,, and those indi- cated by dark circles are of weight g:r each. The edges indicated by solid, broken, and wavy lines are of weight g,'r., g,'e, and gf8(2), respectively; each are in the periphery of weight g&,(l). Graph G, also has a C, axis and the old and new vertex labels are the same as those in GI. The adjacency matrix of G, is

A, =

which is again a circulant-partitioned matrix. Hence according to the present scheme, it is isospectral with the block-factored matrix consist- ing of the blocks,

where we have used the facts that 1 + w + w 2 = 0 and that the characteristic polynomial of a 2 X 2 matrix remains unchanged if the off-diagonal ele- ments k , and k , are replaced by d m .

Factorization of the Interaction Graph in a Huckel Calculation of Bicyclooctatetraene Including Relevant Distant Interactions

The structure of the molecule is shown in Fig- ure 2. A detailed HMO treatment was given by Wilcox [36]. If we neglect the interaction between nonbonded atoms completely (as in the usual HMO theory), the m-energy level pattern of this molecule becomes like that of three isolated ethylene molecules. Cotton [ 371 has recalculated the rr-en- ergy levels including all interactions between dis- tant sp2 c atoms; full D,, symmetry was em- ployed in this work. We show here that the same results can be obtained very easily using only the C, axis of symmetry. The resonance integral be- tween double-bonded C atoms is, as usual, p; that between A-type and p-type C atoms is kp; and that between A- and vtype C atoms is k ' p where 0 < ( k , k') < 1. Taking a as the zero and p as the

I I

I 1

G3 (b)

( C 1

FIGURE 2. (a) Molecular structure and (b) interaction graph for Huckel calculation including distant interactions. (c) McClelland factorization of Gi and G; .

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INTERACTION GRAPHS WITH n-FOLD SYMMETRY

unit of energy, the interaction graph representing this situation is G, (Fig. 21, where thick edges stand for C=C with edge-weight = 1; light edges stand for the resonance integral between h- and p-type sp2 C atoms, the corresponding edge- weight being k p = k; the edges in G, shown by dotted lines and carrying a weight of k ' p = k' indicate the resonance integral between A- and ptype C atoms.

Graph G, has a C, axis of symmetry, and on its basis the vertices of G, have been labeled follow- ing the present scheme. The adjacency matrix of G, is then found to be

The diagonal factor blocks will then be

. (9)

and 2 2

i = O c B i + l w i = i = O c [EBii lW2i = [ l - k '

(lob)

where the fact that 1 + o + w 2 = 0 has been uti- lized. These reduced matrices may be regarded as the adjacency matrices of the graphs G; and G3. They can be easily factorized using McClelland's scheme [ 10, 111, as shown in Figure 2, into isolated vertices whose weights are the eigenvalues of G l and G;. The eigenspectrum of G, comprises that of G; and of G; taken twice.

Thus, the T-MO energies of bicyclooctatetraene are 1 + 2 k + Zk', -1 - 2k' + 2k, 1 - k' - k, 1 - k' - k , -1 + k' - k, and - 1 + k' - k. These are identical with the result obtained by Cotton [371 (to compare, put a = 0, p = 1, p' = k p = k, and p" = k ' P = k ' ) .

Factorization of PPP Fock Matrices

BUTADIENE

For both cis- and trans-butadiene there are two types of C atoms, terminal and intermediate. At

any stage of SCF iteration, the Fock matrix must have two types of diagonal elements:

where a, b, c, and d are the consecutive labels of the C atoms from any end. The intrinsic topology of the molecule also suggests that there will be four types of off-diagonal elements in the Fock matrix corresponding to four types of pairwise interaction:

Fnb = Fed = k,, Fa, = F b d = k,, (12) Fb, = k,, and F,d = k,.

The interaction graph (G,), shown in Figure 3 will, therefore, have twofold symmetry and the present labeling scheme requires the labels a, b, c, and d to be changed to 1, 2, 4, and 3, respectively. The adjacency matrix of this graph, i.e., the Fock matrix at any of stage of iteration is, therefore, partitioned as:

k l 'm: ki I 2

G4

(a)

Squaring I 4 Q k2+(w ( b )

FIGURE 3. (a) Interaction graph (GJ for PPP calculation of cis- and trans-butadiene; (b) factorization of a two-vertex graph by a vertex-alternation scheme.

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Hence we get two factor blocks:

Bl + B, = [ k + k4 kl + k 3 ] . k , + k , k' + k , '

(14) k - k4 k , - k3

k , - k 3 k' - k2 D1 - B, =

Thus, the Fock matrices of both cis- and trans- butadiene are block-factored in the same fashion. This is expected since their point groups, C,, and C,,, respectively, are isomorphic.

Each of the factor blocks corresponds to a graph with two vertices of weights k, and h, joined by an edge of weight k. Its eigenvalues,

k [ k 2 + {(h, - h,)/2}2]1'2 hl + h, 2

are easy to determine by a vertex alternation pro- cedure [38] shown in Figure 3.

The eigenvector matrix U for G4 is the matrix product PQ + Q - where

Here 0 is the 2 X 2 unit matrix, P is the Davidson-Shen transformation matrix for twofold symmetry, and J, and 9- are, respectively, the Jacobi transformation matrices for the blocks [EB, + D, and B, - B,. As the Jacobi method diagonal- izes a 2 x 2 matrix in a single step, the eigenvector matrix for G4 at any stage of SCF iteration is also obtained in a single step.

BENZENE

At any stage of SCF iteration the PPP Fock matrix for benzene can be expressed in terms of the interaction graph G, in Figure 4. It has a sixfold symmetry axis, 6 N = 6, and the present scheme makes the vertex labels consecutive, viz. 1,2,3,. . . ,6. The partitioned Fock matrix is

; kl?. k13 k14 k13 k12

t k , Z k13 I k14 k13

, (16)

where the Fock matrix elements have been repre- sented as

and for symmetry,

Each block of A, consists of a 1 x 1 matrix and so the present scheme converts A, into a completely diagonal matrix. Thus,

5

C B j + l = [h + 2k1, + 2k1, + kl41 (19) i = O

gives the eigenvalue h + 2k,, + 2k,, + k14. At the HMO level, k,, = k,, = 0 and k = a (the Coulomb integral of sp2 C atom) and k,, = p (the resonance integral between adjacent sp2 C atoms) so that one HMO eigenvalue is a + 2p. The PPP orbital en- ergy corresponding to this eigenvalue is obtained by directly using the Fock matrix elements. The block Cg=o Bi+ wi , where w = exp(2n-i/6), gives the 1 X 1 matrix [ h + k,, - k, , - k,,] on simplifi- cation, the matrix element itself being the eigen- value. At the HMO level the eigenvalue is h + k,, = a + p. The results of such calculations are as follows:

HMO Block PPP eigenvalue eigenvalue

5 C k + 2k12 + 2kI3 + k14 + 2 P

i = O 5 c ' i + l W i + k l Z - k13 - k14 c-w+P

C D j + l w Z i h - k12 - k13 - k14 a - P

i = O 5

i = O 5

C B i + l ~ 3 i i = O

5

k - 2k,, + 2k13 - k14 (Y - 2p

C B i + l ~ ~ ~ k - k, , - k1, + k14 ( Y - P

C B i + l ~ ~ ~ k + k1, - k,, - k14 a + P

i = O 5

i = O

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INTERACTION GRAPHS WITH n-FOLD SYMMETRY

FIGURE 4. Interaction graph for s-triaminobenzene-like systems; all the interaction lines have not been shown for the sake of clarity.

S-TRIAMINOBENZENE-LIKE SYSTEMS

Such systems have a threefold axis of symmetry with 3 N = 9. The interaction graph (G,) with its vertices labeled according to the present scheme is shown in Figure 4, where, for the sake of clarity, all the edges representing the various Fock matrix elements ( F j j ) have not been shown. The diagonal elements are of three types:

h, = F,, = F44 = F77; h, = FZ2 = F5, = Faa; h, = F,, = F66 =

(20)

The topology of the molecule also suggests that there will be nine types of off-diagonal elements as indicated in the following:

Starting Vertex Ending Vertex

1 2 3 4 5 6 8 9 7

2 4 7 3 5 6 8 9

3 4 7 5 6 8 9

4 5 6 7 8 9 5 7 6 8 9 6 7 8 9 7 8 9 8 9

kl k 2 k 3 k4 k5 k6 k7 k 8 k9

The edges starting at a particular vertex and end- ing at any vertex in a particular column of these off-diagonal elements are of equal weight. Thus for the column corresponding to k,, we have

k , = F,, = F,, = F24 = F4, = F57 = F7,. (21)

And k2, k, etc. are similarly defined. Now the Fock matrix for the system at any stage of SCF iteration is found to be partitioned as

1 2 3 4

F = 5 6 7 8 9

1 2 3 4 5 6 7 8 9 hl k l k2 k, k4 k5 k, k, k2 k, h2 k, k, k7 k, k4 k7 ka k2 k, h, k2 ka k9 k, k, k9 k, k l k2 h l k l k2 k, k4 k, k4 k7 k8 kl h2 k6 kl k7 k 8

k5 k8 k9 k 2 k6 h3 k2 kt3 k9

k , k , k5 k , kl k2 h, kl k2

kl k7 k 8 k4 k7 kt3 kl h2 k6

k 2 k8 k9 k5 k 8 k9 k2 k6 h3

This gives the block factors

1 1

h, + 2k, 2k1 + k, 2 k 2 + k,

2 k 2 + k, 2k, + k, h, + 2 k 9 2

2 2k1 + k4 h2 + 2 k 7 2k8 + k6 ,

i = O

2 c B i + , W i = c B i + , W 2 i i = O i = O

h, - k, k4 - k, k5 - k2 k, - k, h2 - k7 k, - k, , k, - k, k6 - ka h, - k9

where we have utilized the facts that 1 + w + w 2 = 0 and w 3 = 1. Owing to the identity of the last two blocks, the eigenspectrum of IF will always show twofold degeneracy.

Concluding Remarks

In the usual quantum chemical methods matri- ces are block-diagonalized by constructing symme- try-adapted linear combinations of the basis func- tions and then constructing new matrices in the new basis set. This procedure requires considera- tion of all the elements of symmetry of the point group of the system. Alternatively, suitable uni- tary transformations are carried out on the given matrix, but this requires a good amount of numeri- cal computation. In the present approach only one element of symmetry, namely, a C, axis is utilized. The given graph is at first represented graphically, and as soon as a "diagram" for the matrix is obtained and a C, axis is found in it, the pres-

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SARKAR AND MUKHERJEE

ent labeling scheme converts the matrix into a circulant-partitioned matrix. This, in fact, is a com- pact and convenient way of doing a large number of elementary row-column operations on the given matrix. Obviously, all the elements of symmetry of the point group are not needed. Subsequent con- struction of the component blocks is then an easy job and has its mathematical basis in certain prop- erties of circulant matrices as can be found in the original paper (Shen [16]).

ACKNOWLEDGMENT

The authors thank the learned referee for his valuable suggestions. One of us (A. K. M.) also thanks the UGC, New Delhi, for financial assis- tance extended through DSA.

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