Theoretical foundations of purely semiempirical quantum chemistry

All the purely semiempirical quantum chemical theories of molecular electronic structure, such as the Pariser-Parr-Pople theory and its all valence electron generalizations like MINDO, assume the existence of an abstract true effective Hamiltonian which acts only within the space spanned by a minimum basis set of valence shell orbitals. Instead of following the customary procedure of attempting to determine the properties of this true effective Hamiltonian by fitting its matrix elements to experiment, this effective Hamiltonian is derived from the full N-electron molecular Schrödinger equation under the condition that a complete configuration interaction calculation within the valence shell using this true effective Hamiltonian reproduces the exact ground and valence state energies. It is shown that this true effective Hamiltonian also exactly reproduces the projection of the exact ground and valence state wavefunctions on the space spanned by the minimum valence shell basis in the fixed core approximation. Explicit expressions are presented enabling the ab initio evaluation of the matrix elements of this true effective Hamiltonian in the valence shell basis, the true parameters. A discussion is presented of how these true parameters may be evaluated by slight modifications of standard methods such as degenerate Rayleigh-Schrödinger perturbation theory (linked cluster form), Nesbet's hierarchy method, and related cluster function approaches. For the first time, this enables the ab initio evaluation of the true parameters which are empiricized in the semiempirical theories, thereby providing systematic methods for the testing of all of the assumptions of these semiempirical theories and for the development of new and improved parametrization schemes. A detailed discussion is also given concerning the nonclassical aspects of the true parameters.