When the (nearly) exact solutions of a Hamiltonian are known, as occurs, for instance, for the electronic Hamiltonian of ${\mathrm{Li}}_2$, the definition of an effective (valence) Hamiltonian may proceed through the direct use of the Bloch equations. Even when intruder-state problems (due, for instance, to the crossing between a valence and a Rydberg state) are no longer divergent behaviors in the quasidegenerate perturbation expansion, they still produce an ambiguity in the choice of the wave operator. Two definitions are proposed, a ‘‘diabatic’’ one, which selects the eigenvectors having the largest components on the model space, and which leads to a discontinuous effective Hamiltonian in the regions of avoided crossings, and an ‘‘adiabatic’’ definition, which follows the same adiabatic root and tends to lose any meaning on one side of the avoided crossing. Both definitions are unsatisfactory if transferability of the effective operators is expected. The recently proposed intermediate Hamiltonians solve this dilemma by realizing some kind of diabatization of the energies of the states crossed by the intruder state. The analysis is illustrated on two molecular problems: the definition of a valence-only effective Hamiltonian for ${\mathrm{Li}}_2$ and the obtaining of a Heisenberg effective Hamiltonian for acetylene.

• Received 17 February 1987

DOI:https://doi.org/10.1103/PhysRevA.35.4930