Molecular Orbital Theory for Large Molecules. Approximation of the SCF LCAO Hamiltonian Matrix¹

Abstract

Model SCF MO calculations for small molecules have been employed to establish methods and parameters for obtaining Hamiltonian matrix elements of larger related molecules, with all electrons included. Overlap and kinetic energy elements are calculated exactly, while the SCF models yield values of diagonal Hamiltonian elements (as), and parameters which are used in a modified Mulliken approximation to generate the off-diagonal potential-energy parts of the Hamiltonian matrix. A general discussion of the Mulliken approximation is given. SCF a's are found not to vary linearly with net atomic charge, nor are they adequately approximated by free-atom parameters. It is shown that the assumption that Hamiltonian elements are proportional to overlap integrals neglects elements several electron volts in magnitude. A new formula is presented for calculating these zero-overlap elements.Molecular orbitals (MO), expressed as linear combinations of atomic orbitals (LCAO), are the basis of a large body of theoretical molecular structural discussion, including predictions, correlations, and interpretations. The LCAO MO self-consistent-field (SCF) method3 is becoming increasingly more useful as multi-centered integrals become manageable upon large computers, but the present limits of exact applications of this method are molecules of about the size of ethane and diborane. It is therefore no wonder that chemists have turned to more approximate methods4 for complex molecules. Many of these latter methods have failed to consider the explicit form of the oneelectron Hamiltonian, and nearly all of them have required some evaluation of parameters from experiment. The extended Hiickel theory,5 which goes (1) Paper I in a series of four papers.(2) National Science Foundation Predoctoral Fellow, 1964-1966 (3) Current applications of Hartree-Fock theory use the LCAO formulation given by C. C. J. Roothaan, Rev. Mod. Phys., 23, 69 (1951).(4) (a) For a recent review of these methods, see "The Quantum

Decoupled Hartree‐Fock methods. II. Calculation of the potential curves of diatomic molecules

“…The Mulliken approximation serves in its original form to the simplification (see Newton [8]). The above equation decouples the diatomic overlap density matrix into one-electron contributions.…”

High Temperature Chemistry: Stabilities and Structures of High Temperature Species

“…However, the continued testing and refinement (157) of semiempirical MO calculations, together with their critical comparison with experimental and ab initio data, present some hope that MO theory may eventually be of assistance to high temperature chemistry on a broad basis. The following work may be cited as examples of the application, or otherwise, of semiempirical MO theory to the calculation of properties for species of interest to high temperature chemis try.…”

On invariance requirements in approximate SCF MO Theory

“…Discussion of the rotational invariance question with reference to one-electron core-Hamiltonian matrix elements can be found in Refs. [2] and [5].…”