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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. 46. 635-649 (1993)

Self-Returning Walks and Fractional

Electronic Charges of Atoms in Molecules

D. BONCHEV

Higher Institute of Chemical Technology, Burgas 8010, Bulgaria

L.B. KIER

Virginia Commonwealth University, Richmond, Virginia 23298

0. MEKENYAN

Higher Institute of Chemical Technology, Burgas 8010, Bulgaria

Abstract

Three hierarchically ordered topological factors, i.e., atom connectivity, centrality, and cyclicity, were

found to control the number of self-returning walks (SRWS) associated with every atom in the molecule.

The reversal of their order was observed in a few cases where the central location of atoms had a stronger

influence than did their connectivity on the number of SRWS. Three atomic topological indices, i.e., the

Morgan extended connectivity, the Balaban, Mekenyan, and Bonchev hierarchical extended connectivity,

the RandiC atomic path code, were found to closely match the ordering of atoms in molecules determined

by their number of SRWS. New atomic graph invariants fi = lim SRW)/SRW” and fi = fi . SRW2 were

specified and may find application in QSAR and QSPR. The ti indices are nonintegers close to atomic valence.

The fi indices represent the limit of the number of SRWS of length n for the atom i, SRW), normalized by

dividing it by the total number of SRWS for the molecule. In the case of Huckel MO considerations, these

invariants were shown to be numerically equal to the partial electronic charges of the lowest occupied

molecular orbital (LOMO). A new class of isocodal atoms (atoms having the same number of SRW”s) was

observed, i.e., atoms that become isocodal only at n * 1. A number of open questions following from

these findings were formulated, including the possibility for a topological modeling of electron correlation.

0 1993 John Wiley & Sons, Inc.

1. Introduction

One of the most challenging problems in chemistry is the derivation of properties of

chemical species from their structure. During the last 25 years, graph theory has proved

to be a powerful tool in solving this problem. Various molecular descriptors based on

the chemical graph theory (termed topological indices) provide interesting correlations

with physical properties and biological activities of chemical compounds [ 1 - 151.

Some indices describing attributes of molecular topology, such as molecular skeleton

branching [ 16-20], cyclicity [21], and complexity [22-241, as well as the quantitative

0 1993 John Wiley & Sons, Inc.

CCC 0020-7608/93/050635- 15

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BONCHEV, KIER, AND MEKENYAN

structure-property (QSPR) and structure-activity relationships (QSAR) obtained using

these indices, are regarded as semiempirical, rather than rigorous theoretical, results.

Hence, why topological indices work so well is still an open question. The lack of a

direct link with quantum mechanical ideas and molecular orbital theory has led to the

late and reluctant recognition of this new branch of theoretical chemistry.

Recently, self-returning walks in molecular graphs have attracted some attention.

These walks describe molecular structure by means of closed circuits that start from

and end on the same atom after traversing other atoms and bonds in different patterns.

The self-returning walks were found, by the method of moments, to be the first

topological indices directly linked to quantum mechanics. This method of moments, a

powerful technique developed in solid-state physics [25-281, was shown by Burdett

to be of prime importance for the topological control of the structures of molecules

[29-351. Some structure-property correlations with the number of self-returning

walks (SRWS) of different lengths have been reported [36-381, and the relationship of

SRWS to the Wiener index has also been studied [39]. Furthermore, such walks have

been used in specifying isomer comparabilities, which is a problem of importance in

optimizing correlation samples of compounds in OSPR/QSAR techniques [40,41].

Most of the above-cited studies used SRWS as global molecular descriptors. Our

attention in the present work, however, was centered on the role of these walks in

characterizing atoms in molecules. Such atom-centered graph invariants have potential

applications in predicting characteristics such as atomic reactivities, charges, and

NMR shifts. Here, we sought to explore the basic topological molecular features that

determine the number of SRWS. In addition, we searched for other known topological

invariants or. procedures that order atoms in molecules as the atomic SRWs do.

2. The Method of Moments and Self-Returning Walks

A molecule can be characterized by a discrete spectrum of energy levels

El, E2,. . . , Ek. Its nth moment of energy is specified by the equation

p,, = xi El = Tr(H”).

The second in quality Eq. (1) follows from the invariancy of the trace of the

corresponding Hamiltonian matrix upon the diagonalization used in solving the

secular determinant. The trace of the nth power Hamiltonian has a simple topological

interpretation: It equals the weighted sum of all SRWs of length n in the molecule,

beginning and ending with the same orbital:

The term is the product of ‘‘a” elements of the Hamiltonian matrix. The weights

associated with the path are the interaction integrals Hiaib involving the overlapping

orbitals ia and ib. The simplest weighting results from a one-electron Hiickel-type

model in which all Hinib = p, if ia and ib are pW orbitals located on atoms of a

m-bonded network. For such systems, the nth moment of the j orbital is determined

simply by the number of walks that start at this orbital and return to it in n steps,

S R W ~ , traversing one “bond” in each step:

p u j ” = SRWjnpn.

(3)

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

637

a) p2= 6p2

0.829 2.171

P

2.171 0.829

Figure 1.

SRWS of lengths 2 and 4, the respective second and fourth moments, and

topological valencies of the atoms in the butadiene molecule.

Figure 1 illustrates the second and fourth moments of the butadiene molecule by

presenting all SRWS of lengths two and four of the two nonequivalent atoms. Clearly,

the second moment of each atom equals its connectivity (the number of nearest

neighbors).

The knowledge of the moments provides information on the density of states

[42,43]. The details on the molecular application of this technique, known as the

continued fraction method, can be found in [29]. We did not use this method, however,

but instead developed a different procedure, which originally aimed to produce the

ordering of atoms in a molecule according to the number of SRWS of length n associated

with each atom. (The latter has been calculated by L. H. Hall's MOLCONN computer

program.) These atomic topological indices, S R W ~ , were normalized by dividing them

by the total number of such walks in the molecule:

f : = S R W ? / x S R W y = SRW;/SRW".

i

With the increase in n, the fl values converge to a certain limit:

(4)

The f; fractions thus specified represents the relative occurrence of the SRWS of the

atom i among all SRWS in the molecular graph. These are graph invariants that order

the atoms in molecules in a way that depends on molecular topology only. On the

other hand, they may be supposed to represent a kind of fractional atomic charges

mirroring the distribution of one electron charge over all the atoms in the molecule.

This interpretation of the fi index was hypothesized by assuming that each SRW is in

a certain way associated with the movements of an electron; consequently, the larger

the number of SRWS for a given atom, the larger the electronic charge ascribable to

this atom. Then, after a convenient normalizing procedure, one could hope to arrive

at the partial electronic charges of different molecular orbitals (MOs).

Unpublished observations revealed that a specific procedure for coding chemical

compounds [44,45] based on hierarchically ordered extended connectivities (see

Section 5 below) orders the atoms in v-electronic molecules similarly to their ordering

according to the value of the coefficients of the lowest occupied Hiickel molecular

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BONCHEV, KIER, AND MEKENYAN

orbital (LOMO). The examination of a number of .zr-electronic molecules confirmed a

similar guess for the f i atomic indices, derived in this study from a different type of

atom connectivities. It was shown that these indices produce an atom ordering in the

molecule that coincides with the ordering of atoms according to their LOMO-coefficient

values. Moreover, fi itself was found to be numerically equal (with the accuracy up

to the last fifth digit after the decimal point as calculated by the computer program)

to the partial LOMO charges:

f i = c?,LoMo.

This numerical equality observed for 71 n--electron molecules is supposed to have

a general validity; its rigorous derivation, however, remains an open question. It

should be mentioned that the continued fraction method mentioned above proceeds in

a different way by using matrices, the entries of which are different atomic moments

[29]. The result is the simultaneous generation of all eigenvalues and the coefficients

of all MOS. The solution cannot be partitioned in a way to produce solely the coefficient

of one MO. On the other hand, our fractional charges are graph invariants and do not

depend on whether T- or a-orbitals are taken into consideration, i.e., they are not

necessarily related to the Hiickel treatment of T-electronic molecules.

Another invariant, closely related to Eq. (6), may be introduced by multiplying the

fractional atomic charge fi by the second moment of the molecule:

(6)

ti = f i SRW2.

(7)

We termed this index the “corrected second moment,” or the “topological valence,”

of atoms (this is a noninteger close in value to the integer second moment or to

the chemical valence of the atoms). The topological valence for the atoms in the

butadiene molecule is given in Figure l(c). In a parallel study, this index was

shown to correlate with the CND0/2 atomic charges in alkanes [46]. Thus, the new

atomic graph-theoretical invariants f i and ti proved to have practical importance.

They may also prove useful in modeling atomic reactivity, NMR-Shifts, and charges

in structure-activity and structure-property studies.

A question may arise as to why Eq. (6) should be of any importance when it

is well known that the LOMOS are of no importance to chemistry. We suppose that

the unexpected finding in Eq. (6) indicates a possible closer interplay of molecular

topology and electronic structure. Keeping in mind that the partial LOMO charges were

derived from the number of SRWS of atoms after a convenient normalizing procedure,

one may anticipate other procedures to produce the frontier orbital partial charges or

even those of all MOs. Moreover, if one hypothesizes once a certain association of each

SRW with a specific electronic movement, one may also be prompted to model electron

correlation purely topologically. Such preliminary studies are in progress [47]. Any

advancement along this avenue could be greatly facilitated if the topological factors

influencing the number of atomic SRWS were better known. Such factors are discussed

in Sections 3 and 5 below.

3. Factors Influencing the Numbers of Self-Returning Walks in Different Atoms

Seventy-one n--electronic compounds [a choice influenced by Eq. (6)] were selected

for a study of the S R W ~ , including cyclic, monocyclic, catafused, and perifused

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

639

polycyclic hydrocarbons, as well as branched and bridged polycyclic hydrocarbons.

They are shown in Figure 2 with their atoms ordered according to the decrease in

their fi values. The compound numbering corresponds to that given in [48]. This large

(a)

Figure 2. Molecular graphs of 71 r-electronic hydrocarbons.

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BONCHEV, KIER, AND MEKENYAN

c ' i ,

0

6

I

4

( b)

Figure 2. (Continued)

sample of data provided a basis for a detailed analysis of the factors that influence

the number of SRws. The number of these walks of length 2 coincides with the vertex

degree, or to a-skeleton valency, which classifies the atoms as primary, secondary,

tertiary, or quaternary. This led to the following conjecture about the atoms in a

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

64 1

(c)

Figure 2. (Continued)

molecule:

fprimary < fsecondary < ftertiary < f quaternary.

Otherwise, it was supposed that the fourth moments of atoms, e.g., could only

distinguish some of the atoms with the same second moments but not enable a

reordering of atoms with different second moments. Or, more generally, it was

conjectured that if fl < fj", then fl" < fj""

This, however, proved to be only a very general trend and not the rule. In fact, it

holds for the ultimate ordering of 490 of 503 nonequivalent atoms in the examined

compounds. In the remaining 13 cases (compounds 9, 10, 13, 23, 35-38, 41, and

68; Fig. 3), the second topological factor, i.e., the central location or atom centrality

had a stronger influence than did the major factor, i.e., the atom connectivity on the

number of SRWS. (The atom centrality was rigorously evaluated according to a set of

three hierarchical criteria based on the distances in a molecular graph, namely, the

vertex maximal distance, the vertex total sum of distances to the other vertices, and

the occurrence of the longest distances [49-521). In one of these cases, a primary

atom is classified before two secondary atoms, whereas in all the remaining 11 cases,

a secondary atom has more SRWS than does a tertiary one. Interestingly enough, the

fourth to tenth moments do not produce any of the above-mentioned reorderings, and

for some cases, moments higher than the 30th one are needed (compounds 37 and 38).

In a few cases, reordering occurred for atoms that have the same second moments,

differ in their fourth moments, and then at some rather high moment reverse their

ordering. Such is the case with compound 20, in which the ultimate ordering of atoms

2 and 3 occurs as late as the 70th moment. This is generally related to the slower

conversion of f 1 values in larger molecules, but some specific structural patterns, like

those of two benzene rings separated by a long bridge (compounds 20, 22, and 24),

cause an additional convergence slowdown.

The opposite case could also occur, although rarely: A n atom having its limit fi

value as early as the second moment:

(8)

holds for any n.

f .

11

= f ' .

(9)

Besides the trivial case with the C3 radical, Eq. (9) was observed to hold for position

2 in Cs (pentadienyl radical), where the increase in the f i of the central atom is

exactly compensated for by the corresponding decrease in the terminal atom. Such a

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BONCHEV, KIER, AND MEKENYAN

0

n

8 -0

2<3

Figure 3. Ten molecular graphs in which atom centricity influences the number of SRWS

more strongly than does the atom valency.

compensation, however, does not occur at position 3 in Cg. Thus, the problem arises

of finding the conditions that control the occurrence of equality (9).

Careful inspection of Figure 1 revealed a third factor that becomes important when

two atoms have the same connectivity and centricity. This factor, cyclicity, gives

priority to atoms belonging to a cycle rather than to those incorporated in side chains

or bridges. Thus, in Figure 4(a), atoms 4 are equally distant and atoms 5 are more

remote from the central atom 1 than are atoms 6, which, however, do not belong to a

cycle. This priority is kept for cycles having three to six atoms, but it does not hold

for larger cycles, which do not differ much from open chains [Fig. 4(b)].

4. “Accidental” Degeneracies in the Moments of Atoms

MO theory shows that some .rr-electronic molecules have identical Huckel orbital

coefficients of geometrically nonequivalent atoms. Graph theory represents such

molecules by endospectral graphs and terms the nonequivalent vertices having the

same number of SRWS isocodal vertices 153-571. The problem was analyzed by Burdett

et al. [32], who pointed out that this “accidental” degeneracy does not result from some

geometrically hidden symmetry factors. Rather, it arises from the way the molecule

is assembled via the connectivity of atoms and implies that for such a pair of atoms

a and b the moments of energy pu“, = pz are equal for any n.

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

643

24

5

Figure 4. Examples of molecules in which the presence of cycles and their size influences

the number of SRWS and the atomic ordering that they produce.

We have found five such pairs of atoms in four of the examined molecules. In

Figure 5(a), they are denoted by the same numbers and are underlined (compounds 7,

12, 17, and 19). All have fractional charges obeying the similar condition f,“ = fi

for any n. However, a second class of “accidental” degeneracies was observed for

which the two moments become equal only at some rather higher n value:

limf,“ = fa = fb = limf,”

flS1

f,“ # fi

for

flBl

2 5 n < lim .

(10)

These cases are shown in Figure 5(b) (compounds 10, 22, 54, and 94). It has to be

emphasized that in all cases the convergence is extremely slow. The limit is reached

for compound 10 at n = 90 and for compound 54 at n = 100, whereas for compounds

22 and 94, the limit is not reached within this range of n values (Table I).

An interesting extreme case of this class deals with the cr-electronic system

illustrated in the molecular graph shown below:

Quite surprisingly, all nonterminal atoms that include a triplet of nonequivalent

atoms have the same fractional charge, with the limit being reached at n > 70

(Table 11). Evidently, the possibility of predicting the atomic positions at which this

type of “accidental” degeneracy occurs is another open problem related to the atomic

SRWs.

5. Other Graph Invariants That Produce Atom Ordering

Similar to That Resulting from Self-Returning Walks

Since this question has important theoretical and practical aspects, it seems

worthwhile to examine which of the atomic topological indices used so far produces

a result similar to that of S R W ~ , the only index having a direct link with quantum

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BONCHEV, KIER, AND MEKENYAN

Figure 5. Examples of the two classes of “accidental” degeneracies of atoms: (a) the known

cases for which pi = pg holds for any n; (b) the new class for which the moments of energy

of a pair of atoms differ at low n values and converge at high n values.

mechanics. If such an index could be found, it would be of use in case of larger

molecules requiring high order moments, i.e., more complex calculations.

For most indices, poor correlations with SRW; have been obtained: The atom

orderings thus produced differ drastically from that taken as a reference. This was

observed for indices based on the distances in molecular graphs: the distance number

[58], the information index on the distribution of atomic distances [59], and the RandiC

distance code [60,61]. Surprisingly enough, a poor correlation with SRW; was found

for molecular connectivity indices 1F

and *X of Hall and Kier [62], which correlate

TABLE I. Convergence, with the increase in atomic moments, of the fractional atomic charges for a pair

of atoms having a new type of accidental degeneracy.

Atomic moment n

Compound

no. Atoms

2 6 20 30 40 60 80 100

~ ~~

1 0

a

b

a

b

a

b

a

b

.045

,045

. 0 7 5

.050

. 0 5 3

. 0 5 3

. 0 5 9

. 0 5 9

5 4

22

94

.044

.034

.098

.062

.044

.042

.047

.044

,029

. 0 2 3

. l o 6

. 0 9 1

.040

. 0 2 9

.042

.032

.024

.022

.lo5

.098

.040

,028 ,028 .0293

,041

,040

,037 ,038 .0385

,022 ,0214

,021 ,0213

,104 ,1036

.lo2 .lo33

.039 ,0389

.0399

,02134

,02133

.lo36

. l o 3 5

.0384

.0303

.0399

.0386

,02133

,02133

.lo355

.lo354

.03800

,03120

.03987

,03862

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

645

TABLE

11. Fractional charges of the three nonterminal carbons in 2,6-dimethylheptane for different atomic

moments n.

Atomic moment n

Atom 2 10

20 30 40 50 70 90

2

3

4

,188

.125

,125

,183

,151

,135

,171

.162

,158

,168

.166

.164

,1669

,1664

,1661

,1667

,1666

,1665

.16667

.16666

,16666

,16667

,16667

,16667

highly with the DND0/2 electronic charges in alkanes [47,62]. An example illustrating

this failure is given below for compound 8 from Figure 2:

(a) fi ordering; (b) ‘X ordering; (c) 2X ordering.

Still, there are three procedures that produce relatively satisfactory atom orderings.

One of them makes use of the RandiC atomic path code [63,64]; like the atomic

distance code, it is a four-digit sequence representing the count of paths (not distances

that are the shortest paths) of length 1, 2,3, and 4. For the 503 nonequivalent atoms in

the sample of 71 compounds, a correct ordering was found for 42 compounds. In the

remaining compounds, 31 pairs of nonequivalent atoms were classified as equivalent.

Reverse ordering also resulted in 26 pairs and three triplets of atoms. The large number

of atom degeneracies, however, clearly indicated that four digits in the code do not

suffice. In many cases, the counts of paths of lengths 5 and 6 corrected the atom

ordering, although in some cases, paths of lengths up to 9 were required. The total

number of compounds with correctly ordered atoms thus increased to 45. Nevertheless,

the ordering of 22 pairs and six triplets of atoms still did not correspond to the SRWY

ordering, and the “accidental” degeneracy of eight pairs of nonequivalent atoms could

not be reproduced.

A similar result was obtained for the atom ordering produced by the HOC

(Hierarchically Ordered extended Connectivities) procedure for canonical numbering

and coding of chemical compounds [44,45]. It is a modification of the Morgan [65]

extended connectivity (EC) procedure used by the Chemical Abstract Service in coding

chemical compounds, which quite surprisingly reproduced the best atom ordering

according to the number of SRWS. These two procedures are illustrated in Figure 6.

The so-called EC is obtained by summing the connectivities (i.e., SRW:) of the nearest-

neighboring atoms, then repeating this step iteratively until a constant atom ordering is

obtained in two consecutives steps. Alternatively, the HOC procedure does the same in

a hierarchical manner, i.e., the ECS are used only to distinguish between atoms within

the groups of equivalent atoms found in the previous iteration. This also means that the

hierarchical inequalities (8) will be followed, thus making it impossible for the HOC

procedure to reproduce the 13 cases of pairs of atoms in which centrality was shown to

be a stronger factor than was atom connectivity. Contrary to this, the Morgan EC values

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BONCHEV, KIER, AND MEKENYAN

Figure 6. Illustration of the atomic ordering as produced by the Morgan extended connectiv-

ity (EC) and Balaban, Mekenyan, and Bonchev hierarchically ordered extended connectivity

(HOC) methods.

produced correct atom ordering in 10 of 13 cases. Furthermore, a comparison of the

total number of compounds having all their atoms correctly ordered by the Morgan

EC vs. HOC procedures (57 vs. 46) also indicated that the Morgan EC procedure is

more satisfactory. In addition, only 10 pairs of atoms were wrongly ordered by the

Morgan EC procedure, and in six more pairs, this procedure produced oscillations in the

consecutive iteration steps between the correct and reversed pairs ordering. It should

be noted, however, that to obtain these results we had to modify the requirement for

stopping the iterative procedure because the same ordering in two consecutives steps

does not prevent a reordering in the other steps to follow. No oscillations resulted from

the HOC procedure, but 18 pairs and 11 triplets were still incorrectly ordered. The three

procedures just discussed failed, however, in reproducing the S R W ~ ordering for some

large asymmetric molecules like nos. 12 and 13 or symmetric ones like nos. 10, 14,

22, 23, and 68.

6. Conclusion

The present study aimed to answer several questions, the most important being how

molecular topology controls the number of SRWS at different atoms and, in this way,

the electronic charges of atoms within Huckel-type T- or a-approximations. Some

preliminary results have already been reported by us and others [29,33,39], e.g., that

a branched graph vertex has more such walks than does a nonbranched one or, more

generally, that the branching and cyclicity of a molecular skeleton tend to increase

their number. Yet, in this paper, three topological factors were for the first time clearly

specified to control the atomic moments: atom connectivity, centrality, and cyclicity

with a strong hierarchical relationship between them. Atom connectivity is the major

factor because the number of nearest-neighboring atoms equals the number of SRWs of

length 2. In case several atoms have the same second moments, they may then differ by

the number of their second neighbors (EC), which is important for the fourth moments,

or they can have different third neighbors, which reflects in the sixth moments, etc.

Still, the different second moments of two atoms do not automatically cause all higher

moments to keep the same hierarchical order of these atoms. The centric location of

atoms (or atom centrality) was shown to be a dominant factor in 2-3% of the cases,

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SELF-RETURNING WALKS OF ATOMS IN MOLECULES

647

a tendency that gains in strength with the increased size of the molecule. The third

factor, which is inferior to the first two, gives priority to atoms incorporated into cycles

rather than into side chains or bridges for equal or comparable atom connectivity and

centrality. The cyclic priority pertains mainly to smaller cycles having up to six atoms

and is of less value for larger cycles, which behave more like open chains.

The main role played by atom connectivity also explains why the Morgan EC index

[65], the HOC index [44,45], and the molecular path code [63,64] order the atoms in

molecules in a way rather close to the ordering according to atomic moments. All these

procedures start with atom connectivities and only then differ in the way they account

for the topology of the rest of the molecule. At this point, it would be worthwhile

to emphasize the importance of a certain topological procedure that reproduces the

correct atom ordering. According to Einstein’s general relativity principle, it is the

atom ordering in space that is conserved as the only molecular invariant upon any

continuous space transformation.

In answering the questions discussed in the foregoing sections, some new problems

arose related to the convergence of the fi atomic invariants specified by Eqs. (4) and

(5). A new class of accidental degeneracies was found for which the pairs of degenerate

atoms have equal fi values at very high order moments only. Another type of atom

was specified for which the fi limit is reached as early as in second moment. Thus,

the question is raised whether the existence (or nonexistence) of such types of atoms

in molecules may be predicted by some general theoretical rules or concepts.

The interesting numerical finding that the fi atomic invariants are equal to the

partial LOMO atomic charges [Eq. (6)] also raises a number of questions. It is not the

LOMO itself that is of interest (it is of almost no importance to chemistry); rather, the

mere fact that a purely topological procedure making use of the atomic SRWS is in

a state to reproduce the coefficients of one molecular orbital raises hopes that, after

certain modification, such a procedure could produce the frontier orbitals or all MOS.

In an attempt to explain why the fi indices cfi is, in fact, relative occurrence of the

SRWs of atom i to all SRWs in the molecule) could really represent partial electronic

charges, we hypothesized that each such walk is associated with a certain electron

movement. This idea does not simply posit that a larger number of SRWS associated

with an atom means a larger electronic charge at this atom. It also gives a starting

point for developing procedures for a topological modeling of electron correlation that

uses the atomic SRWs as a basic tool. Work along these two lines is in progress [47].

Acknowledgments

D. B. wishes to express his appreciation for the most stimulating hospitality of

Lemont B. Kier during his stay at Virginia Comonwealth University, as well as for

valuable discussions with Jack W. Fraser and for support of this work from Sterling

Drug, Inc. (Malvern, PA) and Allied Signal Corporation, Engineering Materials

Research (Desplaines, IL). The authors are also indebted to Dr. J. Burdett (University

of Chicago) and Mr. Nikhil Joshi (Virginia Commonwealth University) for reading

the manuscript and for their valuable comments.

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BONCHEV, KIER, AND MEKENYAN

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Received August 11, 1992

Revised manuscript received November 9, 1992

Accepted for publication December 23, 1992

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