Semiempirical effective Hamiltonians from a first principles’ viewpoint

We extend the formal apparatus for generating effective electronic Hamiltonians by a unitary (canonical) transformation on the Coulomb Hamiltonian to model spaces having various numbers of valence electrons. This formal structure is then used to analyze the problem of deducing effective Hamiltonians from empirical data. It is found that certain ambiguities generally arise since there is no way of empirically establishing the unitary transformation within the model space. At best, one can find a family of unitarily equivalent effective Hamiltonians. Examples are given for the π-electron system of ethylene, and the implications of this point of view are discussed for understanding semiempirical theories of electronic molecular structure.