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The Journal of Chemical Physics

Resolvent Operator Formulation of Stationary State Perturbation Theory

Robert Yaris

Published Apr 1, 1964• 9 citations
Publisher

Abstract

By starting with an exact operator equation and using different methods of expanding the resolvent operator, the Schrödinger, Wigner—Brilloin, similarity transformation, gauge transformation, and first-order perturbation iteration method, perturbation expansions are generated in a rigorous and straightforward manner. It is also shown how additional perturbation and perturbation iteration methods can be generated.

DOI
10.1063/1.1725419

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The Journal of Chemical Physics•Feb 15, 1982•9 citations
By starting with an exact operator equation and using different methods of expanding the resolvent operator, the Schrödinger, Wigner—Brilloin, similarity transformation, gauge transformation, and first-order perturbation iteration method, perturbation expansions are generated in a rigorous and straightforward manner. It is also shown how additional perturbation and perturbation iteration methods can be generated.
The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.
A formulation of many-body perturbation theory starting from the operator equation (H,Q + )= omega Q + is presented. The method of solution is based on an operator scalar product, (X/Y)=Tr(X + Y), which allows the use of resolvent and partitioning techniques to establish Rayleigh-Schrodinger or Brillouin-Wigner perturbation theory for the excitation energy and excitation operator, Q + . The excitation operator contains all information about the two states involved in the transition. Specific results are given for removal or addition of an electron and for excitations of particle-hole type and comparison with the Green's function methods is made.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.
In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.
In this paper, we develop a time-dependent approach to many-body perturbation theory for closed shells based on the resolvent of the Schrödinger equation. We introduce a quantity S(t) = i<φ‖exp [−i(H−E0)t]‖ψ≳/<φ‖ψ≳, where φ and ψ are, respectively, the unperturbed and exact wave functions for the system and E0 is the unperturbed energy. The Fourier transform of S(t), S(ω) = <φ‖(ω+E0− H)−1‖ψ≳/ <φ‖ψ≳, is a matrix element of the resolvent containing the exact function ψ and, thus, has a pole at ω = (E−E0), the correlation energy. Starting from a time-dependent perturbation expansion of S(t) via the Gellman–Low adiabatic theorem, we have obtained a Dyson-like equation: S−1(ω) = S−10(ω)+Σ<φ‖ψ≳ for S(ω). Such a derivation requires judicious grouping of terms of the perturbation series for S(t). It has been shown that specific regroupings of the terms of S(t) into appropriate ’’top’’ and ’’bottom’’ parts and corresponding time-integration procedures yield a Σ which generates the Brillouin–Wigner (BW) or Rayleigh–Schrödinger (RS) energy series. Σ for both the cases are identified as the ’’top’’ parts. The characteristic features of the BW series, namely, the appearance of disconnected and ω-dependent diagrams is to be contrasted with that of the RS series, which contains connected and ω-independent diagrams only, and both emerge naturally as a consequence of the dissection procedure into appropriate top and bottom parts.