Klaus Ruedenberg

Free‐Electron Network Model for Conjugated Systems. I. Theory

The free‐electron model for conjugated systems is consistently developed as the limiting case of a three‐dimensional quantum‐mechanical treatment of the π electrons in such systems. Joint conditions (for branching points) and boundary conditions (for free end points) are derived and the hermiticity of the Hamiltonian is shown. A matrix formulation of the theory is established which makes the application to large systems feasible, and at the same time leads to a close analogy with the LCAO model (LCAO MO treatment considering only nearest neighbor interactions). Quantities analogous to the quantities q (the charge in an atomic orbital) and p (the bond order) are defined, and special attention is given to alternant conjugated systems for which a population theorem, analogous to the one in LCAO theory, is valid.

Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. III. A Unified Treatment of the Hybrid, Coulomb, and One‐Electron Integrals

The problem of the two‐center hybrid integrals is solved in a general way for Slater‐type atomic orbitals with integral effective quantum numbers. No other restrictions are placed upon effective quantum numbers and effective nuclear charges. The integrals are expressed linearly in terms of one type of auxiliary functions. A method for finding the coefficients is given, and the latter are computed for all cases involving orbitals with quantum numbers 1 and 2. The auxiliary functions are discussed in detail, and methods given for their computation. The treatment developed for the hybrid integrals is shown to be applicable also to the two‐center Coulomb integrals and to the one‐electron integrals.

Electronic Interaction in the Free‐Electron Network Model for Conjugated Systems. II. Spectra of Aromatic Hydrocarbons

A theoretical calculation is presented for the main uv transitions of the polyacenes, azulene, phenanthrene, the isomers of naphthacene and the peri‐condensed systems pyrene and coronene. A configuration interaction treatment, based on antisymmetrized products of free‐electron molecular orbitals, is carried out, using the methods developed for this purpose in the previous paper. Transition energies, intensities and polarizations compare well with experiment. Particular emphasis is placed upon illuminating the various factors influencing the spectra in the series of molecules considered.

Mobile Bond Orders in Conjugated Systems

A comparative study is made of the following bond orders: MO bond orders of Coulson, MO bond orders of Mulliken, MO bond populations of Platt‐Ruedenberg‐Scherr, VB bond orders of Pauling. The comparison is carried through on the basis of theoretical derivations and on the basis of numerical calculations for a large number of molecules. The main results are: an accurate functional connection between the Coulson and Mulliken bond orders; the characterization of a new type of bond order based on bond populations, and related linearly to bond lengths; a MO expression for VB bond orders.

Electronic Interaction in the Free‐Electron Network Model for Conjugated Systems. I. Theory

The free‐electron network model for conjugated systems is extended to include electronic interaction. The free‐electron wave functions are considered as molecular orbitals arising from an average one‐electron potential, and a configuration interaction treatment with these MOs is developed. The integrals occurring in this procedure, in particular the electron interaction integrals, are analyzed and a convenient method for their computation in the FE model is suggested. The integral evaluation requires certain semiempirical parameters which are determined from the benzene spectrum.

Energy Levels, Atom Populations, Bond Populations in the LCAO MO Model and in the FE MO Model. A Quantitative Analysis

Quantitative results for fourteen representative organic molecules are derived and analyzed on the basis of the LCAO MO model and on the basis of the FE MO model. Calculated are, besides the energy levels, the atom populations and the bond populations, quantities defined in a recent paper as an accurate measure of the distribution of electrons.The results of the LCAO MO model and of the FE MO model are then compared, and for the more important quantities considerable agreement is found. The origin of the agreements and disagreements is shown to be different from the one generally supposed.Because of their general usefulness, complete tables are given for the eigenvalues and eige nvectors of the fundamental topological matrices associated with the molecules investigated.

Theorem on the Mobile Bond Orders of Alternant Conjugated Systems

A generalization of Halls theorem for Coulson bond orders in alternant molecules is derived. The new relation expresses general bond order‐like quantities in terms of the topological molecular matrix. Specialization to the Mulliken bond orders furnishes an exact relation between the Coulson and the Mulliken bond order matrices in the ground state of alternants.