Perturbation expansion of variational principles at arbitrary order

Abstract

When perturbation theory is applied to a quantity for which a variational principle holds (eigenenergies of Hamiltonians, Hartree-Fock or density-functional-theory energy, etc. ), diferent variationalperturbation theorems can be derived. A general demonstration of the existence of variational principles for an even order of perturbation, when constraints are present, is provided here. Explicit formulas for these variational principles for even orders of perturbation, as well as for the "2n+1 theorem, " to any order of perturbation, with or without constraints, are also exhibited. This approach is applied to the case of eigenenergies of quantum-mechanical Hamiltonians, studied previously by other methods. PACS number(s): 31.15. -p, 02.30.Mv, 02.70. -c, 71.10.+x I. INTKODUCTIC)N In aa. early study of the two-electron atomic systems, published in 1930, Hylleraas [1] observed that the knowledge of an eigenfunction and its erst-order derivative with respect to some perturbation allows one to build easily not only the erst derivative of its eigenenergy with respect to the same perturbation, but also its second and even third derivative.This result is one instance of a "2n+1 theorem, " stated in this seminal paper: the (2n+1)-order derivative of the eigenenergies of some Hamiltonian can be calculated from the knowledge of the eigenfunction and its derivatives up to order n (see also Wagner [2]). In the same paper, Hylleraas also noticed that an expression for the second-order derivative of the eigenergy was variational (minimal) with respect to deviation of the erst-order derivative of the wave function from its exact value. Since that time, theoretical studies of physical systems submitted. to small perturbations have been numerous. A large spectrum of perturbations such as small atomic displacements, electric fields, magnetic fields, small length scale changes, "transmutation" of elements, etc. , eventually time dependent, have been considered, in the framework of widely used ab initio approaches such as the density-functional theory [3 -7] (DFT), Hartree-Fock formalism [8 -14], X approximation [15], multiconfiguration self-consistent field formalism (MCSCF) [9,14,16,17], configuration interaction technique [9,14,17], coupledcluster [9, 14 expansion, and also Moeller-Plesset expansion [9,13, 14 . It is noticeable that a variational principle lies at the heart of most of these approaches. In that case, a nice interplay between a variational principle (or an extremal principle) and perturbation theory, found by Sinanoglu [18,19], allows one to find generalization of the Hylleraas results.In order to situate the present work in the proper context, the second. section will give a brief overview of arbitrary order perturbation expansion of variational principles. The interesting results obtained previously will be recalled. While the 2n+1 has been demonstrated in the case of a variational principle under constraints, by Epstein [20], the existence of even-order variational principles in such a case has not been demonst...

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