Studies in perturbation theory XIII. Treatment of constants of motion in resolvent method, partitioning technique, and perturbation theory
Per‐Olov Löwdin · International Journal of Quantum Chemistry · 1968
Abstract After a brief survey of some basic concepts in the theory of linear spaces, the eigenvalue problem is formulated in the resolvent technique based on the introduction of a reference function φ and a complex variable ℰ. This leads to a series of fundamental concepts including the trial wave function, the inhomogeneous equation, and finally the transition and expectation values of the Hamiltonian, of which the former renders a “bracketing function” for the energy. In order to avoid the explicit limiting procedures in this approach, the eigenvalue problem is then reformulated in terms of the partitioning technique which, in turn, leads to a closed form of infinite‐order perturbation theory. The eigenvalue problem is greatly simplified if the Hamiltonian H has a constant of motion Λ or has symmetry properties characterized by the group G = {g}, and the question is now how these simplifications can be incorporated into the partitioning technique and into perturbation theory. In both cases, there exists a set of projection operators {Qk} which lead to a splitting of the Hilbert space into subspaces which have virtually nothing to do with each other. It is shown that, in the partitioning technique, it is sufficient to consider one of these subspaces at a time, and the results are then generalized to perturbation theory. It turns out that the finite‐order expansions are no longer unique, and the commutation rules connecting the various forms are derived. The infinite‐order results are finally presented in such a form that they are later suitable for the evaluation of upper and lower bounds to the energy eigenvalues.