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Bounds For Total π-Electron Energy
Volume 24, number 2
15 January 19’74
CHEMICAL PHYSICS LETTERS
BOUNDS @OR TOTAL n-ELECTRON ENERGY Ivan GUTMAN Institute “Ruder Bofkovif *,P.O. B. IO1 6, Zagreb. 0oatia. Yugoslavia Received 15 October 1973
Upper and lower bounds for the tbtal x-electron energy of conjugated hydrocarbons in tbe Hrtckel appro:iimation are presented.
The dependence of the Hiickel total n-electron energy Q of conjugated hydrocarbons on the molecular structure has been investigated in a recent paper [I] where further references can be found. The notation of ref. [l] will be adopted here also. Thus, N and v are the numbers of carbon atoms and carbon-carbon bonds, respectively_ A is the adjacency matrix of the molecular graph [2] and Xi (i = 1,2, ._.,N) are its eigkvalues (= the spectrum of the graph). By convention x1 > x2 > ...a xN. Hence, E=2
(1)
1) (det A)*lN GE* < 3Nv _
g +=O,
(2)
=&I.
(3) from (1) and (2) it follows
(4).
(6)
(7)
Using eq. (3) and the well known formula det A = lTy=,xi, it can be checked without difficulties that D/N is just the difference between the arithmetic and geometric mean of the numbers xf_ Therefore D >, 0. bi us denote 2& (l&i -l-$1)* by Q and con-.’ :
2Nv-Ex=Q,
E=$ IXJ. i=l
l)D,
where
sider the identity
cQ@4-~/2*l,
(5)
In this work we would like to present some additional upper and lower bounds. For reasons which will be clear later, the inequalities (5) will be transformed to _-
D = 2~ - N(det A)2/N.
PI
i=l ,$-x;
2v + N(N-
0 G 2Nu -E2 <(N-
N/2 C Xi i=2
and
The following estimates for E have been obtained by McClelland [3]:
Volume 24,number2
~MICAL
Table 1
and Y the value of E decreases with the increase of the width of the occupied (or cnoccupjed) rr-energy band. Let there be No zeros in she spectrum of the molecular graph. Then N
Q=N,,
(9)
cx,'+Q', i=-1
where Q’ indicates the summation over pairs of nonzero eigenvahres. Since 9’2 0, from eqs. (3) and (8) it follows E2 G 2(N- N&J.
15 January 1974
PHYSICS LETTERS
(10)
which is a generahzation of McClelland’s upper bound. Note that simple graph-theoretical aJgorithms are avdable [4] for enumeration of No. It is evident from eq. (8) that in order to fmd bounds for E it is sufficient to determine inequalities for Q. Fortunately, there is a result of Kober [S] which can be used for this purpose. Let Ma and Mg
Molecule
Lower bound
butadiene hexatriene cyclobutadiene benzene fulvene naphthalene azulene fulvalene pentalene heptalene
a
upper
4.47 6.63 4.00 7.88
6.93 12.97 12.23 11.99 6.00 7.21
bound
Exact E
4.47 7.21 4.00 8.19 7,75 14.39 14.23 14.17 10.39 16.12
4.47 6.99 4.00 8.00 7.47 13.68 13.36 12.80 10.46 15.62
completely analogous way of reasoning as in the
derivation
ofsq.
(8) it can be shown that
N/2 2NV-E2=4
C
(Xi-Xj)2
(15)
i
and ifin eq. (1 I) Ii and PZare replaced by respectively,
be the arithmetic and geometric mean, respectively, for a set of non-negative numbers tj (i = 1.2, ....II).
$
and N/Z,
Then [S] 20<2Nu-Ez
-fjn)*4(n-l)n(M
w
a -M
(12)
end from (8) DC2Nv--E*Q;(N-1)D.
technique given in ref. (21. Because of (14) the rekftiod (16) is valid for even N only, but an analogous formula for odd N can be derived in a similar way. However, UrJsis not necessary because if the hydrocarbon poserses an odd number of carbon atoms the graph G’ can be considered instead of the molecular graph G.
(‘3) @
Thus, ordy the upper bound for E is better than in (6). A further hnprovemant of (6) can be gained if it is assumed that (14)
Eq. (14) holds exactfy for aJtemant hydrocarbons, but is a&o a sufHckntJy good approximation for most of the nonattemants (but see pentalene Jn table 1). UsJng
284
06)
Numerical examples are given in table 1. The value of c”ludet A can be calculated by a simple graph-theoretical
RepJacJng fi and n by ~12andN, respectively, and using the notation of eq. (7) one obtains D
<(N-2)0.
).
a0 G
G’
Then N(Cr) = N(C) + J and, of course, E(G’) = E(G). Often is also det A(G) = det A(G’) = 0. For example
0
r\ U G
0
-0 G’
Volume24, number2
CHEMICALPHYSICS LETI’ERS
Formula (16) gives 81/* < E(G’) < 8v* and thus E(G) = 81j2. The total Irelectron energy is an important property of conjugated molecules and it is of general interest to recognize the structure factors determiniig its value. The McClelland-type formulae (6), (13) and (16) have one common feature - they emphasize the fact that a great part of E is determined solely by the number of atoms and bonds (i.e., by the empirical formula of the compound). For illustration compare columns 3 and 4 in table I since the upper bound of eq. (16) seems to be rather near the exact E value. The differences rarely exceed 5%. The role of det A is also interesting because there is a close relationship [2] between the number of Kekule structures (K) and det A. Thus, for benzenoid hydrocarbons det A = (-f12 K2. Since for N being large enough (det A)2/N
=
1
=O
ifN,
= 0;
iflv,>O,
(17)
it comes out that the number of RekuIb structures has
15 January 1974
only a small contribution to E. A more pronounced inon E has the presence (N,, > 0) or absence (ZVO= 0) of zeros in the spectrum of the molecular
fluence
graph (41. Finally, the effect of the ring size is completely neglected in the McClelland-type formulae. Although this may be a rdativeIy small contribution to E (say * 5%), it has a dominant influence on the chemical behaviout of hydrocarbons. Some results in this direction were given in a previous work [6].
References [l] I. Gutman and N. TrinajstiC, Chem. whys. Letters 17 (3972) 535. [7-l A. Graovac, I. Gutman, N. Trinajstid and T. %vkoviE, Theoret. C&n. Acta 26 (1972) 67. [3] B.J. McClelland. I. Chem. Pbys. 54 (1971) 640. 14) T. %vkovic’,Croat. Chem. Acta 44 (1972) 35 1; D. CvetkoviE. I. Gutman and N. Trinajstid, Croat. Cbem. Acta (1972) 36.5. [S] H. Kober, Proc. Am. Math. Sot. 9 (1958) 452. [61 I. Gutman and N. Trinajstif, Chem. Fhys. Letters 20 (I 973) 257.
15 January 19’74
CHEMICAL PHYSICS LETTERS
BOUNDS @OR TOTAL n-ELECTRON ENERGY Ivan GUTMAN Institute “Ruder Bofkovif *,P.O. B. IO1 6, Zagreb. 0oatia. Yugoslavia Received 15 October 1973
Upper and lower bounds for the tbtal x-electron energy of conjugated hydrocarbons in tbe Hrtckel appro:iimation are presented.
The dependence of the Hiickel total n-electron energy Q of conjugated hydrocarbons on the molecular structure has been investigated in a recent paper [I] where further references can be found. The notation of ref. [l] will be adopted here also. Thus, N and v are the numbers of carbon atoms and carbon-carbon bonds, respectively_ A is the adjacency matrix of the molecular graph [2] and Xi (i = 1,2, ._.,N) are its eigkvalues (= the spectrum of the graph). By convention x1 > x2 > ...a xN. Hence, E=2
(1)
1) (det A)*lN GE* < 3Nv _
g +=O,
(2)
=&I.
(3) from (1) and (2) it follows
(4).
(6)
(7)
Using eq. (3) and the well known formula det A = lTy=,xi, it can be checked without difficulties that D/N is just the difference between the arithmetic and geometric mean of the numbers xf_ Therefore D >, 0. bi us denote 2& (l&i -l-$1)* by Q and con-.’ :
2Nv-Ex=Q,
E=$ IXJ. i=l
l)D,
where
sider the identity
cQ@4-~/2*l,
(5)
In this work we would like to present some additional upper and lower bounds. For reasons which will be clear later, the inequalities (5) will be transformed to _-
D = 2~ - N(det A)2/N.
PI
i=l ,$-x;
2v + N(N-
0 G 2Nu -E2 <(N-
N/2 C Xi i=2
and
The following estimates for E have been obtained by McClelland [3]:
Volume 24,number2
~MICAL
Table 1
and Y the value of E decreases with the increase of the width of the occupied (or cnoccupjed) rr-energy band. Let there be No zeros in she spectrum of the molecular graph. Then N
Q=N,,
(9)
cx,'+Q', i=-1
where Q’ indicates the summation over pairs of nonzero eigenvahres. Since 9’2 0, from eqs. (3) and (8) it follows E2 G 2(N- N&J.
15 January 1974
PHYSICS LETTERS
(10)
which is a generahzation of McClelland’s upper bound. Note that simple graph-theoretical aJgorithms are avdable [4] for enumeration of No. It is evident from eq. (8) that in order to fmd bounds for E it is sufficient to determine inequalities for Q. Fortunately, there is a result of Kober [S] which can be used for this purpose. Let Ma and Mg
Molecule
Lower bound
butadiene hexatriene cyclobutadiene benzene fulvene naphthalene azulene fulvalene pentalene heptalene
a
upper
4.47 6.63 4.00 7.88
6.93 12.97 12.23 11.99 6.00 7.21
bound
Exact E
4.47 7.21 4.00 8.19 7,75 14.39 14.23 14.17 10.39 16.12
4.47 6.99 4.00 8.00 7.47 13.68 13.36 12.80 10.46 15.62
completely analogous way of reasoning as in the
derivation
ofsq.
(8) it can be shown that
N/2 2NV-E2=4
C
(Xi-Xj)2
(15)
i
and ifin eq. (1 I) Ii and PZare replaced by respectively,
be the arithmetic and geometric mean, respectively, for a set of non-negative numbers tj (i = 1.2, ....II).
$
and N/Z,
Then [S] 20<2Nu-Ez
-fjn)*4(n-l)n(M
w
a -M
(12)
end from (8) DC2Nv--E*Q;(N-1)D.
technique given in ref. (21. Because of (14) the rekftiod (16) is valid for even N only, but an analogous formula for odd N can be derived in a similar way. However, UrJsis not necessary because if the hydrocarbon poserses an odd number of carbon atoms the graph G’ can be considered instead of the molecular graph G.
(‘3) @
Thus, ordy the upper bound for E is better than in (6). A further hnprovemant of (6) can be gained if it is assumed that (14)
Eq. (14) holds exactfy for aJtemant hydrocarbons, but is a&o a sufHckntJy good approximation for most of the nonattemants (but see pentalene Jn table 1). UsJng
284
06)
Numerical examples are given in table 1. The value of c”ludet A can be calculated by a simple graph-theoretical
RepJacJng fi and n by ~12andN, respectively, and using the notation of eq. (7) one obtains D
<(N-2)0.
).
a0 G
G’
Then N(Cr) = N(C) + J and, of course, E(G’) = E(G). Often is also det A(G) = det A(G’) = 0. For example
0
r\ U G
0
-0 G’
Volume24, number2
CHEMICALPHYSICS LETI’ERS
Formula (16) gives 81/* < E(G’) < 8v* and thus E(G) = 81j2. The total Irelectron energy is an important property of conjugated molecules and it is of general interest to recognize the structure factors determiniig its value. The McClelland-type formulae (6), (13) and (16) have one common feature - they emphasize the fact that a great part of E is determined solely by the number of atoms and bonds (i.e., by the empirical formula of the compound). For illustration compare columns 3 and 4 in table I since the upper bound of eq. (16) seems to be rather near the exact E value. The differences rarely exceed 5%. The role of det A is also interesting because there is a close relationship [2] between the number of Kekule structures (K) and det A. Thus, for benzenoid hydrocarbons det A = (-f12 K2. Since for N being large enough (det A)2/N
=
1
=O
ifN,
= 0;
iflv,>O,
(17)
it comes out that the number of RekuIb structures has
15 January 1974
only a small contribution to E. A more pronounced inon E has the presence (N,, > 0) or absence (ZVO= 0) of zeros in the spectrum of the molecular
fluence
graph (41. Finally, the effect of the ring size is completely neglected in the McClelland-type formulae. Although this may be a rdativeIy small contribution to E (say * 5%), it has a dominant influence on the chemical behaviout of hydrocarbons. Some results in this direction were given in a previous work [6].
References [l] I. Gutman and N. TrinajstiC, Chem. whys. Letters 17 (3972) 535. [7-l A. Graovac, I. Gutman, N. Trinajstid and T. %vkoviE, Theoret. C&n. Acta 26 (1972) 67. [3] B.J. McClelland. I. Chem. Pbys. 54 (1971) 640. 14) T. %vkovic’,Croat. Chem. Acta 44 (1972) 35 1; D. CvetkoviE. I. Gutman and N. Trinajstid, Croat. Cbem. Acta (1972) 36.5. [S] H. Kober, Proc. Am. Math. Sot. 9 (1958) 452. [61 I. Gutman and N. Trinajstif, Chem. Fhys. Letters 20 (I 973) 257.