  The article contains an overview of authors achievements in development of
alternative quantum-chemical approaches oriented towards revival of the
classical tradition of qualitative chemical thinking instead of obtaining
numerical results. The above-mentioned tradition is concluded to be based
mainly on principles (rules) of additivity, transferability and locality of
molecular properties. Accordingly, model Hamiltonian matrices are used in the
approaches under development (called quasi-classical alternatives), wherein
algebraic parameters play the role of matrix elements and these are assumed to
be transferable for similar atoms and/or atomic orbitals in addition. Further,
passing to delocalized descriptions of electronic structures (as usual) is
expected to be the main origin of difficulties seeking to formulate
quasi-classical alternatives. In the framework of the canonical method of
molecular orbitals (MOs), delocalization is shown to be partially avoidable by
invoking a recently-suggested approach to secular (eigenvalue) equations for
model Hamiltonian matrices, wherein the usual initial imposing of the
zero-determinant condition is replaced by a certain reformulation of the
problem itself. The most efficient way of achieving the same end, however, is
shown to consist in passing to non-canonical one-electron problems. The latter
may be exemplified by the block-diagonalization problem for the relevant
Hamiltonian matrix following from the Brillouin theorem and yielding
non-canonical (localized) MOs and by the commutation equation for the
respective one-electron density matrix (charge- bond order matrix). In this
connection, most of attention is paid in the article to perturbative solutions
of the above-mentioned non-canonical problems and to their implications,
including common quantum-chemical descriptions of entire classes of chemical
compounds.
