'
Symposium on Groph Theory in Chemistry
6
2
4
An Example Molecular Orbital Calculation Using the Sachs Graph Method Jerry Ray Dias University of Missouri, Kansas City, MO 64110-2499 Introduction The LCAO method only requires one to know how to set up and solve secular determinants . While this is not very difficult in itself. noncomnuter solution of large determinants become exceedingly time consuming and error Drone. Thus. our eoal in the followine sections is to orovide bne with the methodologies to circumvent solviig large secular determinants. We will do this by translating molecules into molecular graphs and formulating the LCAO method into graph theoretical terminology and mathematics. By this approach, one will be able to streamline their computations and learn to extract the desired information about molecules with a facility that allows them to eonceputualize chemical information with ease.
Table 1. Glossary of Terms
-
Graph Theoretical Terminology Molecular Graph Notation
Throughout this work the carbon and hydrogen atoms and the prr bonds in all molecular graphs (graphs representing molecules) will be omitted and only the C-C obond skeleton will be shown. The various types of (carbon) vertices referred to herein are methylene, shown as a primary (degreeone) graph vertex,
-/ -C\
methine shown as a secondary (degreetwo) graph vertex, and
=c carbon, shown as a tertiary (degree-three) graph vertex.
=c
H H
/
H
\
/ \
As examples, the methyl radical will be represented by a point and ethene by a line. Alternant hydrocarbon polyenes (bipartite molecular graphs) have no odd size rings and can be maximally labelled with nonadjacent stars. Odd carbon hydrocarbons will always be radical species when electronically neutral. The radical nature of odd carbon polyenes will be implied. Table 1presents a glossary of terms used in this paper Presented in part at the Pedagogical Symposium on Graph Theory in Chemistry, 4th Chemical Congress of North America and 202nd ACS National Meeting, New York City, August 2S30, 1991.
X
fourth coefficientin the characteristic polynomial sixth coefficient in the characteristicpolynomial HMO Coulomb integral (of carbon) number of branches on a trigonal ring number of branches on a tetraganal ring HMO exchange integral cycle or circuit of size n eigenvedor meffkient for atom rand thejth eigenvalue number of vertices of degree i number of edges with a vertex of degree i at one end and a vertex of degreej at the other end an edge of weight k energy level or HMO eigenvalue total pn energy a molecular or isaeonjugate graph a molecular graph with a single weighted vertex a molecular graph with a single weighted edge the weight of a heteroatom vertex [h = (araeYP1 the weight of an edge [k = linear polycoujugated system of size n number of (carban atom) vertices characteristicpolynomial of a molecular graph number of C-C cr-bond edges number of rings (cycles)of n vertices number of comhinatorial pairs of trigonal rings vertex (E - a)@ = graph eigenvalue
Determining Characteristic Polynomials Characteristic Polynomials
Expansion of the Huckel molecular orbital (HMO) secular determinant for a molecular graph gives the characteristic polynomial
of the corresponding conjugated system where I is the identity matrix,X = (e- a)@, and A is the adjacency matrix ( I ) . The matrix A is an n x n matrix whose (id)entry is 1if vertices ui and uj are adjacent in G and 0 otherwise. The Volume 69 Number 9 September 1992
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characteristic polynomial of an N carbon atom system has the following form
Table 2. Equations for the Calculation of Characteristic Polynomials of Molecular Graphs
N
P(G;X)=
C
=0
( 1 3 -
n=o
where a, are coefficientsthat can be obtained without having to solve the HMO secular determinant. The mots of the characteristic polynomial give the eigenvalues (energy levels) of the corresponding molecular graph. A major thrust of this paper is the development of facile procedures for solving the above characteristic polynomial for obtaining energy, bond order, and electron density parameters for most molecular graphs of chemical interest. Most of these procedures will owe their origin to the elegant method, which is totally general, involving enumeration of all Sachs graphs (2). Sachs Graphs
for edgelvertex-weightedgraphs
The determination of the coefficients a, of the characteristic polynomial P(G; X) by the Sachs graph theoretical method is a formalized variant of the procedure first presented by Coulson (3).This method is summarized by the following formula
.
(1"
P(G,; X) = P(G- ek;XI - ~ ' P ( G-~(ek);X) - 2kD(Gk -&;XI
for edge-weightedgraphs
= -&l)@v=l
for right mirror-plane fragment
d"
where 0 S n S N ,s is a Sachs graph, S, is a set of all Sachs graphs with exactly n vertices, c(s) is the total number of Kz and C, components, and rls) is the total number of rings (cycles)ins. All combinations of lKz and k c , such that 21 + km = n are enumerated. By definitionan= 1. Since only Kz and C, components are allowed, S, = $(the empty set) giving al = 0.The set S2of all Sachs graphs on two vertices leads to the value of a2= 9, which is equal to the negative number of graph edges (abonds). Only C, components or C, (n = odd) components together with Kz components are allowed for m = odd. Consider the cvclooentadienvl , (C.. .~ . .) system. The set of Sachs graphs ofS3;s zkm since there are no trieonal rincs in the cvclooentadienvl molecule. For S4 the fofiowing e&xmerates"alls a c k graihs on four vertices
where P(L,; X) = 1and P(L1;X) = X P(G - v,; Xj) for atom rand eigenvaluej and C. components on a given number of vertices ranging from 1to N,. This method obviously considers only 1-2 interactions by Kz and cyclic interactions by C,. This arises quite naturally from the symmetric property of the adjacency matrix where only nearest neighbors are input. Alternant hydrocarbons (AH'S) have no odd size rings and, therefore, all the odd coefficients ofthe characteristic polynomials are zero. A single Cq. component makes a negative contribution to the even weficientsofAHh Ingeneral, the number of combinations of Sachs eraohs for the odd coeficients of nonAH's are smaller than for even ones. Also, the number of Sachs graphs associated with the tail coefficient
- .
which give a 4 = 5 (-1)220 = 5 . For S6,there is only a single Cg mmponent giving a5 = (-112 = -2. Thus, the characteristic polynomial for the cyclopentadienyl molecule is
For larger molecular graphs the manual enumeration of all Sachs graphs, represents a labor-intensive,error-prone process. We will circumvent this through the use of a combination of algebraic, combinatoric, and discrete mathematics. General equations for the a4 and as (Table 2)have been derived by the author (4). Using these equations along with embedding (5),mirror plane fragmentation (61, and other graph theoretical decomposition schemes (7, 8), we can show how to amve at the characteristic polynomials for most molecular graphs of chemical interest. When it is obvious from the context C, will be used for PC,; X ) and L, will be used for P(L,; XI. Table 3 above summarizes all the Sachs graphs needed for computing the coefficients up to as. For a given molecular graph of size N o , one determines all the various combinations of subgraphs containing mutually disjoint K2 696
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Table 3. The Set of Sachs Graphs Used for Calculating the Coefficients to the Characteristic Polynomials of Molecular Graphs
Coefficient
Set of Sachs Graphs
of nonAH's are frequently sufficiently small that one will usually have no difficulty in their determination. The Fourth and Sixth Coefficients
Using combinatoric and discrete mathematics the author previously derived general expressions for a4 and a6 (9).From the Sachs graph formulation, a4will consist of all combinations of two disjoint Kz subgraphs and one Cqsuhgraph. For conjugated hydrocarbons a 4 = ( u 2 ) ( q 2 - 9 q + 6 ~ , ) - 2 r , - d , -...
without enumerating Sachs graphs since eq 3 is totally general for molecular graphs up to N, = 6. The set of Sachs graphs on seven vertices is
(1)
where q is the number of abonds (edges 1, N,is the number of carbon vertices, r4 is the number of tetragonal rings, and dl is the number ofvertices of degree-1(methylene groups) in the molecular graph. The sixth coefficientfor conjugated hydrocarbons is given by all disjoint combinations of three Kz, one Kzand one C4, one Ce, and two C3 subgraphs. Thus,
where e ( i j )is the number of edges between vertices of degree i on one end and degree j at the other end, a4is the number of branches on a tetragonal ring, and r i is the number of combinatorial pairs of trigonal rings. With these relationships for a4 and as, the characteristic polynomial for any molecular graph having six carbon vertices or less is specifically given by eq 3 in Table 2. Consider benzene with the Cs molecular graph. From eq 3, we obtain
since benzene has no odd-size rings. Solving eq 1and eq 2 with q = 6, N, = 6, r6 = 1, and all other parameters zero gives
Benzene is an even alternant hydrocarbon (EAH).Alternant hydrocarbons (AH'S)have no odd-size rings and consequently all odd coefficients aa as, a 7 , ... are zero. The odd coefficients in the characteristic polynomials will be only required for nonalternant hydrocarbons (nonAH's) and will usually have smaller values than the even ones. It is useful to distinguish between even and odd carbon alternant and nonalternant hydrocarbons. Both even and odd carbon alternaut hydrocarbons (EAH's and OAH's) have characteristic polynomials with only even coefficientsof alternating signs. OAH's will always have at least one zero eigenvalue (i.e., a NBMO). ~ccordingtothe Sachs graph formalism, odd coefficients only occur for molecular graphs having at least one oddsize ring, i.e., nonAH's. Since odd coefficients are smaller than even ones, it is usually quite easy to count the Sachs graphs for smaller molecular graphs. For larger molecular graphs, one can use eq 1for a4 and eq 2 for a6 on appropriate subgraphs produced by deletingthe odd-size rings. For example, if the sole odd-size ring in a molecular graph is C,, then the a,+z, a,+4, and a,& coefficients to the relevant characteristic polynomial can he obtained by determining the az, a4, and a6 coefficients for the subgraph obtained upon deleting C, from the molecular graph. Consider cyclopropabenzene (10). From eq 1 to eq 3 in Table 2, we obtain
which gives a ? = (-112 + (-I)~ 2 = 4.We now obtain as for cyclopropabenzene by two different methods. The set of Sachs graphs on five vertices is
which gives as = 3(-1)22 = 6. Alternatively, by deleting C3 from cyclopropabenzenewe obtain
For L4, a2 = -q = -3 and a4 = -1. Thus, a5(C?Hs)= az(L4). a3(Cs)= -3(-2) = 6. Similarly, the second term to a? above is given by a4(L4) as(C3)= 162) = -2. A more complicated example is provided by planar bicyclo[7.1.0ldecapentaene (13)as follows.
.
Deletion of C3 from CloHs gives L7 for which a z = -6, a4 = 10 and a6 = -4. Thus, a&,&,)
= a2(L,) .a3(C3)= -6(-2) = 12
The very last coeftieient alo can be easily computed by determining all the Sachs graphs on 10 vertices giving ale = (-112 + 2(-115 = -4. Determination of a8(CloH8)can be accomplished by deleting the six distinct Kz subgraphs and determining a 6 for the six different remaining subgraphs. This gives a, = 29. The Tail Coefficient
The absolute value of the last coefficient amof the characteristic polynomial of an even AH devoid of 4n rings (n = integer) is equal to the square of the number of Kekule structures of the corresponding molecular graph, that is Volume 69 Number 9 Se~tember1992
697
For example, benzene has K = 2 and l as l = 4. Similarly, the sum of the squares of the unnormalized integral coefficients to the NBMO (eigenvector coefficients for E = 0) of an OAH gives the tail coefficient (anr.~)to the corresponding characteristic polynomial. For nonAH, one will have to enumerate all the Sachs graphs on N vertices as it was done above for bicyclo[7.1.0ldecapentaene. In summary, all molecular graphs of size N, C 8 and no symmetry can have their HMO parameters easily determined by the methods outlined in this section. It can be shown that some molecular graphs can have additional eigenvalues determined by a method called embedding. Obtaining additional eigenvalues by embedding will permit us to cross-check our computed characteristic polynomials for small molecular graphs and frequently to obtain characteristic polynomials for even larger ones.
unnormalized wave functions (g= Clql + C2q2+ C3q3)for the allyl eigenvalues of E = 0, and are
a
-a
To obtain normalized eigenveetors, we can use either of the following two equations for each eigenvalue:
Eigenvectors Introduction
In this section we wlll show how to use the same inspection methods and eouations (Table 2) to determine the eigenvectors (wave Gnction coefficients) belonging to each eigenvalue (jth energy level) of the characteristic polynomial. Two principal methods will be presented (11-12). The first method (11)involves deleting all mutually exclusive paths from vertex 1 to vertex r in a molecular graph to give a set of remaining subgraphs. The sum of the characteristic polynomials of these subgraphs gives the eigenvector polynomial for the C, coefficient corresponding to the vertex r of the oarent molecular eraoh. The second method (12, involve;deleting each verrex; of a molecular graph. The characteristic polynomial of the subgraph obtained by deletingvertexr divided by the derivative of the characteristic ~olvnomialofthe molecular eraoh when evaluated for eachkigenvalue will give the corks&mling normalized C+ coefficient. Path Deleting Procedure for DeterminingEigenvectors
Consider aUyl and its secular determinant as follows:
Deleting vertex 1gives ethene and correspondsto the scoring out the fust row and first column of the above determinant, namely
Similarly, deleting the path containing vertex 1and vertex 2 leaves a methyl radical and corresponds to scoring out the first row and second column to the allyl secular determinant and multiplying the resulting cofactor by -1, which gives
Finally, deleting the path 1-2-3leaves the null graph which corresponds to scoring out the first row and third column of the allyl secular determinant to give
These are just the unnormalized eigenvector polynomials of allyl, which give the specific eigenvectors by substituting in the corresponding eigenvalues. Thus, the respective 698
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The second equation is the product of the characteristic polynomial of the subgraph obtained by deleting vertex 1 from a precursor molecular graph and the derivative of the characteristic polynomial of the molecular graph. Multiplying each corresponding coefficient by the normalizing factor N gives the desired normalized ei envectors. In the case of allyl, the first equation for E = 2 glves
s".
Alternatively, using the second equation for
As another example, consider the following styrene molecular graph
Using equations previously given (Table 2), the characteristic polynomial of styrene is easily deduced by inspection
Note that styrene can be embedded by ethene and allyl giv& L the other four eigenvaling eigenvalues of E = f1 and ; ues can be obtained easily by iteration with a hand calculator or by any programmable calculator/computer. The eigenvector polynomials for styrene as numbered above are
When it is obvious from the context,. L. will be used for PGJ . Solution of these eigenvector olynomials for E = 4 gives c1= s*, c2= -2, ~3 = &, cCI = ~8 = 0, ~5 = ~7 = &, and C6 = 2. Dividing each of these coefficients by 4 gives the corresponding normalized vectors. In this example, all C , = 0 for E = f1, although styrene is not degenerate in these eigenvalues. Nevertheless, the corresponding eigenvectors in this case are easily deduced from its ethene embedding. Ideally, one should select an initial vertex for eigenvalue evaluation that does not have a zero coefficient for any of the eigenvalues. Sometimes this is unavoidable and more than one vertex would have to be used to generate the complete eigenvector set. If styrene is renumbered as follows. ~
0, -fiwe obtain
For Xevaluated at E =
c~(G) =
or
= 114
C , = C, = 112
and C i ( j Z i = 1 / 2 or C2=*l%
As another example, consider the following molecule: P(G,X)=P-~X~-&+~& +8X-19P-4x+5
then the eigenvector polynomials become
c, = x 7 - s X 6 + 9 2 - 3 x c2=P-&+&-2
c3=P-&+ x c4=W'-&+2
P'@X)=8~~-5&-1& +9&+2&-38~-4
Using the equations in Table 2, one obtains all the coefficients to the above characteristic polynomials, except a7 and as. From the set of Sachs graphs on seven vertices
c, = P - w a + x c,=x'-P c,=P-x c,=P-&+7P-2
w h i c h f o r ~ = f l g i v e C ~ = f l , C ~ = C ~ = C ~ = C ~71, = O , C ~where = an and a4 are given by equations in Table 2. From C4 = -1, and C8 = 1. the set of Saehs graphs on eight vertices The graph theoretical procedure for determining an unnormalized eigenvector set for a molecular graph having degenerate eigenvalues requires one to take the derivative of the eigenvector polynomials and then solve them as one obtains above. 4 0 a , = (-1) 2
Vertex Deleting Procedure for Determining Eigenvectors This procedure leads directly to normalized eigenvectors and is based on Ulam's subgraphs. An Ulam subgraph of graph G is obtained by deleting a vertex u with all incident edges from G. The derivative P'(G; X ) is equal to the sum of the characteristic polynomials of all the Ulam subgraphs (13). It has been shown ( 1 2 ) that the normalized eigenvector coefficient of the rth vertex corresponding to the jth eigenvalue is given by
c2.- P(G - u,; Xj) P(G,q)
"-
Thus, the collection of Ulam subgraphs provides a graph theoretical route for obtaining eigenvectors. One shortcoming of this procedure is that the sign of C , must be determined by other means. For AH, one knows that these eigenvector coefficients are all positive for the most positive eigenvalue and one can use the Pairing Theorem to obtain the signs for these coefficients for the AMO's from the BMO's. Consider ally1 as numbered previously Using this method gives
+ ( - 1 ) ~ =2 5~
The eigenvector polynomials per the vertex deletion procedure are
c$= x 7 - 7 P + 1 & - & - 6 x + 2 P(G; X)
cij=(x2-1)(P-52+~-2) P(G; x) -2)(9-d+1) cgj= ( 2 - 3 ~P(G; 4
c; = x 7 - 7 P - W 4 + 1 3 P + 6 x ' - L v - 2 P(G; X)
lwa+&-4x-2 c,2 --x 7 - 7 P - W ' +P(G; X)
It is not necessary to know the signs of the above coefficients if we wish to use them for computing electron density ( Q ) or charge density (1- Q ) . The electron density on atom r is given by w.2
a, = C ujc; j=1
and
where uj is the number of electrons in the jth energy level. The occupied energy levels are 2.3590, 1.8158,0.6765, and Volume 69 Number 9 September 1992
699
0.6180 with two electrons in each. The first eigenvector polynomial above gives a charge density of +O.28 (Q = 0.72) for atom 1, and the last eigenvector polynomial gives a charge density of -0.17 (Q = 1.17) for atom 5. Thus, the molecule above is principally a zwitterion with +0.84 positive charge on the cyclopmpenyl ring and -0.84 negative charge on the cyclopentadienyl ring, that is, the neutral species below makes a 16%contribution to the overall hybrid structure.
vectors bewmes the preferred method when one needs to obtain bond orders. For the allyl radical, one obtains PIZ=Pz3= 2 (112). (JZi2) + 1 .(U2).0 = f i t 2 which is the same for the allyl carbocation and and carbanion.
.
Consider the zwitterionic molecule again, renumbered (at right) to avoid starting at the zero coefficients. The unnormalized eigenvedor polynomials per the path deleting procedure are
4
c,=x7-~X6-2yr+i3Xa+&-5x-z
For this zwitterionic molecule all the eigenvector coefficients have positive values for the 2.3590 energy level: for the 0.6180,-1.0, and-1.6180 energy levels the corresponding sign@of the eigenvector coefficients can be obtained by inipe&on. This iibecause these eigenvalues result from a type of semi-embeddingof ethene in the cyclopmpenyl ring . . . and butadiene in the &clo~entadienvlring. It is worthwhile to note that all h e examples that we have dealt with thus far were implicitly assumed to be totally coplanar. If the last zwitterionic molecule assumed a ueruendicular orientation of the cvclooro~envland cyclopentadienyl rings, then its zwitterionic character should aooroach 100%. However. in eoine from a coolanar to perp&cular arrangement of it: rings its tot& electronic pn energy (En) would decrease from 10.94 to 10.47P. A key concept that forms the basis of Dewar's perturbation molecular orbital (PMO) method is that the nonstarred positions of O m s have zero eigenvector coefficients (zero nodes) for the unpaired zero eigenvalue and that the eigenvedor coefficients to each nonstarred position must sum to zero. For OAHs this unpaired zero eigenvalue (energy level) is the frontier molecular orbital. Embedded fragment submaphs (5)are separated by zero nodes (zero eigenvector &&cients) and require opposite sims in order to have the eirrenvector coefficientsattached toeach node sum to zero.
. .
. .
.
.
A
.
Determinationof Bond Order
To determine the bond orderP, between adjacent carbon atoms r a n d s , one needs the signs of the eigenvedor coefficients. This is because the antibonding contributions for the relevant eigenvalues must be subtradedfromthe bonding contributions. This is done automatically in the following bond order equation om
z .JGJGj
pm= j=o
where uj is the number of electrons in the jth energy level. Thus, the path deleting procedure for determining eigen-
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Journal of Chemical Education
The occupied energy levels for this zwitterionic molecule are 2.3590, 1.8158, 0.6765, and 0.6180 with two electrons in each. For the 5-6 bond. one obtains a bond order of0.44. which can be compared &the bond order of 0.45 for the bond of butadiene. Summary As a finale to this paper, the author invites the readers to substantiate the charge densities given below (14,15).
Literature Cited 1. 2. 3. 4. 5. 6. 7.
8. 9. 10.
~nnajs&N.Ckmical Gmph Them;CRC h e : Bas Faton. FL. 1992.
seeha,H.Publ. Molh I s m , 11,119
Coullon, C.A. P a . CombndgoPhilm. Sm. 1860,46,2M. Dias J.R. J Math. C k m 1867,4, 127 Dias, J. R J. C k m . Edvc 1885.64, 21W. H d . G . C. k . F d y S o e 1SK7.53, 513. MeCklland, B. J. J. C k m S m . F m d w h m . 11 1W4,70, 145%Dim, J. R. J. C k m . Edue lW.66.1012. Dim,J. R.%r Chlm.Acto l W , 7 6 , 1 5 % 5Mdm. Struct. (Theachem) 1988.165, 125. LVhato, S. S. Theor. Chim Ado ISB9,53.319. Diss,J . R. That Chim Aclo 1986 68,107. Halton B.;RandaU, C. J.; Gained, G. J.; Straw P.J. J. A m C h . Sm. ISM. 108.
