The Matrix Form of the Noncanonical Theory of Molecular Orbitals

摘要

The eigenblock equation for a Fockian- or Huckel-type Hamiltonian matrix of a molecule being an alternative form of the Brillouin theorem and determining the noncanonical molecular orbitals (NCMOs) was generalized to the case of nonorthogonal initial basis sets. As in the Lowdin's partitioning technique, the initial set of basis orbitals was divided into two subsets. This allowed the problem to be formulated in terms of these subsets and the respective submatrices. A perturbative approach was developed for the solution of the new matrix equation in terms of entire submatrices (blocks) of both Fockian and overlap matrices without specifying either the structures or dimensions of these submatrices. Two multidimensional molecular orbitals (MMOs) containing the subsets of occupied and vacant noncanonical MOs (NCMOs), respectively were defined for any molecule, and these MMOs were expressed in the form of linear combinations of two subsets of basis orbitals with matrix coefficients. A linear combination of this type may be considered as a generalization of the usual LCAO approximation for an MO of a two-level system. The relevant common form of the bond-order matrix was shown to be obtainable on the basis of the projector to a single occupied MMO. These results allowed us to conclude that the generalized matrix problem obtained in this article forms the basis of the noncanonical MO theory of electronic structures of molecules in its matrix representation. The new matrix problem along with its solution was compared to related problems and methods.