STATIONARY PERTURBATION-THEORY .1. SURVEY OF BASIC CONCEPTS

Stationarity of the energy expectation value as required in the variational approach can in most cases of practical interest be formulated with reference to a unitary 'variational group' and its associated Lie algebra. In terms of a basis of this Lie algebra, Brillouin (hypervirial) conditions and a Hessean are defined. The formulation of stationary perturbation theory (as well as of multiple perturbation theory) in the Lie-algebraic framework is straightforward. If the operator Y, which describes the perturbation of the wave function, is expandable in the Lie algebra of the variational group, a Hellmann-Feynman theorem holds (as a special case of Wigner's [2n + 1] rule) and the first order operator Y1 can be obtained from minimization of a Lie-algebraic Hylleraas functional. Under the same condition for two perturbations, Dalgarno's exchange theorem holds. An analysis of the spectrum the Hessean leads to a generalization of the RPA method for any chosen variational group. Any variational group automatically generates a model excitation spectrum. Sometimes one wants to formulate the variational approach in terms of two or more (independent and noncommuting) variational groups. An example is coupled-MC-SCF theory. One must specify the order in which operators of the two groups act, but otherwise there is not much change with respect to the case of a single group. Time-dependent stationary perturbation theory, based on Frenkel's stationarity principle, is possible on similar lines. Singularities related to an indefinite phase, which plague traditional time dependent perturbation theory are automatically avoided. In the framework of stationary perturbation theory the dipole length and dipole velocity formulas for a transition element are equivalent. For a time-dependent Hamiltonian there is no unique definition of the corresponding energy. There are various possibilities to define a 'pseudo-energy'. One of these definitions is consistent with a special form of a time-dependent Hellmann-Feynman theorem. For a perturbation periodic in time so-called Floquet states exist and stationary time-dependent perturbation theory starting from a stationary state of the unperturbed problem automatically leads to these. For Floquet states a genuine stationarity condition can be derived, that is based on the concept of a generalized Hilbert space and that does not suffer from the shortcomings of Frenkel's principle. The perturbation formalism for these is surprisingly close to that of the time-independent theory. For degenerate and near-degenerate states a quasidegenerate generalization of stationary perturbation theory is possible.