NEW BOUNDS AND APPROXIMATIONS FOR TOTAL π-ELECTRON ENERGY – A CRITICAL TEST

The quality and correlating ability of some recently deduced bounds and approximate formulas for total π-electron energy are tested on a sample of 106 Kekuléan benzenoid hydrocarbons. It was found that not a single new approximate formula and not a single new bound is better than those from the 1970s.


Introduction
The total π-electron energy, calculated by mens of the the Hückel molecular orbital (HMO) approximation, is a quantum-theoretical characteristics of conjugated molecules that has been extensively studied in the last 50 or more years [4,7]. It is attracting the attention of theoretical chemists and mathematicians thanks to its explicit, but mathematically non-trivial, dependence on molecular topology [13]. In the same time, total the π-electron energy is one of the most reliable results that can be deduced from HMO theory, and is in reasonably good agreement with experimentally determined thermodynamic properties of conjugated molecules, especially their enthalpies [6,10].
When the HMO total π-electron energy is expressed in the units of the HMO carbon-carbon resonance integral β, then it is directly related to the eigenvalues of the molecular graph [4,10]. For the majority of conjugated hydrocarbons (including the benzenoid systems studied in the present work), the total π-electron energy satisfies the following expression, which in the same time serves as the definition of graph energy [13]: In Eq. (1), G is the molecular graph (for details see [10]), whose eigenvalues are In what follows, the number of vertices and edges of the graph G will be denoted by n and m, respectively. It should never be forgotten that n is the number of carbon atoms of the underlying conjugated hydrocarbon, whereas m is the number of its carbon-carbon bonds.
By A we denote the adjacency matrix of the graphs G (for details see [10]).
Within the theory of total π-electron energy, numerous bounds and approximate expressions for E have been discovered, many of which in terms of n, m, and det A.
The first upper and lower bounds for E were obtained in 1971 by McClelland [14]: and and are valid for all graphs. Eventually, these bounds were improved in 1974 as [5] and where the upper bound (4) holds for all graphs, whereas the bounds (5) and (6)  their comparative studies can be found in our earlier works [6,9,11]. In particular, in [9], 10 (n, m)-type approximate formulas for E were tested. In [6], 24 (n, m)type and additional 15 (n, m, K)-type formulas were examined. In [11], a total of 44 (n, m)-type approximations for E were dealt with.
Quite recently, a number of new bounds and approximate expressions for E were established [1][2][3]. In [3], the following lower bound was obtained which requires that det A ̸ = 0.
The bound (8) is valid for all graphs. The validity of the bound (9) is restricted to bipartite graphs. The bounds (10) and (11) are applicable only for graphs with nonzero determinant of the adjacency matrix. In addition, (11) holds only for bipartite graphs.
What has not been done in the papers [1][2][3] was the checking of the quality of the bounds (8)- (11), relative to the earlier known bounds, and testing the accuracy of the approximations a E * + b, where E * is an expression on the right-hand side of Eqs. (8)- (12) and (15), (16). We do this in the present work.

Testing the new approximate formulas and bounds for total π-electron energy
All earlier tests of approximate formulas for total π-electron energy [6,9,11] were done on a representative set of 106 Kekuléan benzenoid hydrocarbons from the book [15].
We did the same also in the present work.
The molecular graphs of benzenoid hydrocarbons contain no odd-membered cycles and are therefore bipartite [10]. If G is the molecular graph of a benzenoid hydrocarbon, then according to the Dewar-Longuet-Higgins' theorem [4,8], det A = (−1) n/2 K 2 and therefore where K is the number of Kekulé structures of the corresponding benzenoid molecule [8]. The sample considered by us consisted of Kekuléan benzenoids, i.e., of benzenoids for which K > 0. Therefore, all estimates and approximate expressions for E, mentioned in the preceding section (except, of course, Eqs. (13) and (14)), are applicable for our sample. These estimates and approximate expressions are thus formulas of (n, m)-or (n, m, K)-type, and say something about the dependence of total π-electron energy on the number of carbon atoms (n), carbon-carbon bonds (m), and Kekulé structures (K). Recall that n, m, and K are believed to be the three most important structural parameters, determining the value of total π-electron energy and thus the thermodynamic properties of (strain free) benzenoid hydrocarbons [7,8,12] The results obtained for approximating the total π-electron energy as where E * is one of the above specified formulas of (n, m)-or (n, m, K)-type, and where a and b are determined by least-squares fitting, are presented in Table 1.  Table 1. Statistical parameters obtained by approximating the total π-electron energy as a E * + b for the set of 106 benzenoid hydrocarbons from the book [15]. E * is the expression on the right-hand side of one of the above given relations; R is the correlation coefficient; ARE and GRE are the average and greatest relative errors (in %); asterisk indicates that the (greatest) error of benzene is skipped.
We also tested the quality of the above specified upper and lower bonds for the total π-electron energy. The results obtained are illustrated by the data presented in Tables 2 and 3. Recall that these all pertain to Kekuléan benzenoid hydrocarbons, which are typical and more often encountered polycyclic conjugated molecules.
Therefore, from the examples found in Tables 2 and 3 we may conclude about the quality and applicability of the bounds for E in the case of all chemically relevant graphs.  Table 3. Examples illustrating the quality of the lower bounds (3), (6)-(9); the compounds same as in Table 2.

Discussion and concluding remarks
From Table 1 we see that the best approximations of the form E ≈ a E * + b for the total π-electron energy of benzenoid hydrocarbons are the (n, m)-type McClelland's formula, with E * given by (2), and its two (n, m, K)-type improvements, with E * given by (4) and (5). Recall that all these were discovered in the 1970s.
The newly proposed approximations are weaker than these old ones, some slightly, some significantly. Outstandingly bad is the formula with E * from Eq. (15).
From the data given in Table 2 we see that McClelland's upper bound (2) and its improvements (4) and (5), all from the 1970s, are far better than the upper bounds (10) and (11) put forward in 2013. As expected, (11) is slightly better than (10).
Anyway, we may safely conclude that the new upper bounds (10) and (11) are useless (at least for applications in chemistry) and should be ignored in future considerations of both total π-electron energy and graph energy.
The data in Table 3 reveal that the improved McClelland's lower bound (6), from the 1970s, has not been superseded by any of the bounds discovered in 2013.
In particular, the lower bound (7) is slightly weaker than the original McClelland's estimate (3) and is thus weaker also than (6). From a purely mathematical point of view, the bound (8) is better than (3), and (9) is better than (8). Numerical tests show that (at least in the case of typical molecular graphs) the improvements of (8) over (3) and of (9) over (8) are insignificant and chemically negligible.
The final conclusion of the present work is that the recently communicated bounds and approximate formulas for total π-electron energy, in spite of their mathematical appeal, are of no practical value for chemical applications, and shed hardly any new light on the dependence of total π-electron energy on molecular structure.