Total Π-electron Energy Research Articles

Polynomial matrix method for the estimation of π-electron energies of some linear conjugated molecules

The polynomial matrix method (PMM) is applied to the study of the characteristic polynomials of two classes of molecular graphs: Rn and Sn. Various identities and inequalities for these polynomials are deduced. Lower and upper bounds for the total π-electron energy of Sn are obtained. An alternative formulation of the PMM is proposed.

ChemInform Abstract: EFFECT OF CYCLES ON TOTAL Π‐ELECTRON ENERGY OF ALTERNANT CONJUGATED HYDROCARBONS

Chemischer InformationsdienstVolume 10, Issue 31 Physical Organic Chemistry ChemInform Abstract: EFFECT OF CYCLES ON TOTAL Π-ELECTRON ENERGY OF ALTERNANT CONJUGATED HYDROCARBONS I. GUTMAN, I. GUTMANSearch for more papers by this author I. GUTMAN, I. GUTMANSearch for more papers by this author First published: July 31, 1979 https://doi.org/10.1002/chin.197931079AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume10, Issue31July 31, 1979 RelatedInformation

Effect of cycles on total π-electron energy of alternant conjugated hydrocarbons

Mathematical properties of the function eƒ(G, C), the effect of a cycle on total π-electron energy of a conjugated molecule, are analysed. The main results of the paper are summarized in propositions (1)–(7). The conditions under which the Hückel rule holds are discussed and an approximate formula for total π-electron energy is derived, which is an additive function of the eƒ(G, C). Several rules are obtained for the sign of ef(G, C).

Topological studies on heteroconjugated molecules

A graph-theoretical analysis of certain π-electron properties of alternant molecules with one heteroatom is given. Topological formulas for total π-electron energy, π-electron charge density, bond order and various polarizabilities are derived. The main results of the paper are summarized in Rules 1-7.

A Topological Formula for Total π-Electron Energy

Abstract Total π-electron energy E is presented as a sum of first-neighbour interaction terms λ and contributions of cycles ζ [Equation (2)]. The main conclusion of the work is that the total effect of λ's is a constant, independent of the molecular structure. Therefore, E is an additive function of ζ′s only.

A class of approximate topological formulas for total π-electron energy

A class of approximate topological formulas, Eqs. (12), (15), (25), and (27) for total π-electron energy E is derived. These formulas reflect the dependence of E on a number of structural details of the molecule: the number of atoms, bonds, rings, and nonbonding MO’s, the ring size, the algebraic structure count, and the branching. The formulas are obtained using analytical considerations and contain no fitting parameters.

Acyclic systems with extremal H�ckel ?-electron energy

The main result of the present work is the proof that among acyclic polyenes CnHn+2, the linear isomer H2C=(CH)n−2=CH2 has maximal HMO π-electron energy. The 1,1-divinyl isomer (H2C=CH)2C(CH)n−6=CH2 has maximal π-energy among branched acyclic systems. Among trees withn vertices, the star has minimal energy. A number of additional inequalities for HMO total π-electron energy of acyclic conjugated systems are proved.

Quantitative approach to hückel rule the relations between the cycles of a molecular graph and the thermodynamic stability of a conjugated molecule

Contribution of the rings to the total π-electron energy (CE) is calculated for a number of polycyclic conjugated hydrocarbons. The Hückel 4m+2 rule is tested quantitatively in this way. It is shown that (4m)-membered rings always destabilize, while (4m + 2)-membered rings almost always stabilize the molecule. The magnitude of this effect is, however, subject to considerable variations. The main factors which determine the magnitude of CE are discussed. Examples of the alternant hydrocarbons are found for which the (4m + 2)-membered cycles have a destabilizing effect, and therefore violate the Hückel rule. The odd membered cycles are shown to have a negligibly small effect on the stability of the conjugated molecules.

Effect of a Ring on the Stability of Polycyclic Conjugated Molecules

Abstract A graph theoretical method is developed which enables to calculate the contribution of a partic­ular ring to the total π-electron energy of a conjugated molecule.

Topological studies on heteroconjugated molecules. The stability of alternant systems with one heteroatom

The stability of alternant conjugated molecules, containing one heteroatom, is studied by means of graph theory. The Coulson integral representation of total π-electron energy yields a formula for the difference between the π-energy of a heteroconjugated molecule and its parent hydrocarbon. The integral is calculated approximatively and a simple topological formula is derived, enabling the prediction of the stability of a given heteroconjugated molecule relative to its parent hydrocarbon.

A linear relationship between the total π-electron energy and the characteristic polynomial

A linear relationship between the total π-electron energy ( E π) and the coefficients of the characteristic polynomial of the molecular graph is analysed. It is deduced that (a) the main part of E π depends on the number of bonds, (b) the Hückel 4 m+2 rule holds generally, but corrections regarding the role of the size of the rings are to be added, (c) the perimeter rule can be justified, and (d) E π is increased proportionally to the number of Kekulé and Dewar resonance forms. A quantitative measure of all these topological factors is estimated.

ChemInform Abstract: PREDICTIONS OF MOLECULAR GEOMETRIES AND ELECTRONIC SPECTRA OF COMPLEX UNSATURATED MOLECULES FROM MC‐LCAO‐MO METHOD WITH PARTICULAR REFERENCE TO TRIPLET‐TRIPLET TRANSITIONS OF NAPHTHALENE

An MC-LCAO-MO approach which has been proposed for open-shell systems of unsaturated hydrocarbons having degenerate MO's is applied to naphthalene, calculating its molecular geometry and electronic spectrum. The results are compared with those obtained by the usual semi-empirical SCF-CI method and with experiment. As for benzene, anthracene, phenanthrene and triphenylene, the bond lengths and the π-electron energies in their ground states are calculated in the same manner. Most of the calculated bond lengths are in fairly good agreement with experiment. The total π-electron energies of the ground states obtained by the MC-LCAO-MO and SCF-CI methods agree within about 0.01 eV when CI is included and within about 0.1 eV when CI is not invoked. It is found that the electronic spectrum of naphthalene obtained by the present method is in good agreement with that derived from the SCF-CI method and also explains most part of experiments. A detailed discussion is given on the calculated triplet-triplet absorption spectrum and its intensity distribution of naphthalene.

On the Coulson integral formula for total π-electron energy

The Coulson integral formula for total π-electron energy ( E π) is analysed, and an elementary derivation of it is presented. Three novel Coulson-type formulae are obtained and one is used for deriving the McClelland (approximate) expression for E π. An approximation of a particular formula is proposed which results in a linear relationship between E π and the coefficients of the characteristic polynomial of the molecular graph.

Aromatic substitution

It is demonstrated that a previously developed topological expression for the total π-electron energy of aromatic molecules provides a good qualitative account of localization energies. The logarithm of the ratio of the algebraic structure counts for the ground state and the localized reaction intermediate is the principle energy determining factor. This expression fails for hydrocarbons with unusually small HOMO-LUMO separations. An alternative topological expression that includes a correction for this situation provides an excellent non-empirical explanation for the successful empirical use of Dewar reactivity numbers in correlating exact localization energies. The present analysis provides an explanation for the success of the traditional resonance structure counting technique in predicting relative rates of aromatic substitution. The analysis develop applied only to alternant hydrocarbons.

Predictions of molecular geometries and electronic spectra of complex unsaturated molecules from MC-LCAO-MO method with particular reference to triplet-triplet transitions of naphthalene

An MC-LCAO-MO approach which has been proposed for open-shell systems of unsaturated hydrocarbons having degenerate MO's is applied to naphthalene, calculating its molecular geometry and electronic spectrum. The results are compared with those obtained by the usual semi-empirical SCF-CI method and with experiment. As for benzene, anthracene, phenanthrene and triphenylene, the bond lengths and the π-electron energies in their ground states are calculated in the same manner. Most of the calculated bond lengths are in fairly good agreement with experiment. The total π-electron energies of the ground states obtained by the MC-LCAO-MO and SCF-CI methods agree within about 0.01 eV when CI is included and within about 0.1 eV when CI is not invoked. It is found that the electronic spectrum of naphthalene obtained by the present method is in good agreement with that derived from the SCF-CI method and also explains most part of experiments. A detailed discussion is given on the calculated triplet-triplet absorption spectrum and its intensity distribution of naphthalene.

Bounds For Total π-Electron Energy

Upper and lower bounds for the total π-electron energy of conjugated hydrocarbons in the Hockel approximation are presented.

The nonexistence of topological formula for total ?-electron energy

A definition of the topological formula for total π-electron energy (E) is given and its necessary analytical from is shown to be a linear combination of graph invariants. It is demonstrated that no exact topological formula for E can exist, i.e. that E cannot be expressed as a function of graph invariants. The proof of the nonexistence of the topological formula is based on Ulam's conjecture.

Graph theory and molecular orbitals. The loop rule

Graph theory is applied to the study of the dependence of total π-electron energy, π-electron charge distribution and free valency indices of conjugated hydrocarbons on molecular topology. It is shown that the number of loops in the molecular graph determines these quantities.

Additivity of Heats of Combustion, LCAO Resonance Energies, and Bond Orders of Conformal Sets of Conjugated Compounds

The notion of ``conformal sets'' of aromatic and other conjugated compounds is introduced. To a good approximation, it is found that the sum of the heats of combustion of the compounds in one set equals the sum for the compounds in the other set. Similar additivity occurs for their LCAO-calculated total π-electron energies, LCAO bond orders and free valences. Bond lengths and dissociation energies are also considered. Conformal sets are defined in terms of their number and quality of self-returning random walks on the atoms of the conjugated compounds. Even-electron nonalternant and alternant compounds are included, as are, in certain cases, radicals. Some insight into the additivity for LCAO properties is obtained by relating the coefficients in LCAO secular equations to random walks and using contour integral formulas connecting LCAO properties with secular determinants.