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The Journal of Chemical Physics

Resolvent operator approach to many-body perturbation theory. I. Closed shells

Ajit Banerjee, Debashis Mukherjee, Jack Simons

Published Feb 15, 1982• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.443170

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Resolvent operator approach to many-body perturbation theory. II. Open shells

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In this paper, we develop a time-dependent approach to many-body perturbation theory for open shells based on the resolvent of the Schrödinger equation. We introduce, analogous to the closed-shell case, quantities Sij(t) = i<φi‖e−i(H−Ec)t ‖ψj>/<ψj0‖ψj>, where ψj0 and ψj are the jth unperturbed and exact functions, respectively. The ψj0’s can be expressed in terms of ’’model space’’ functions φi = Ωi+‖φ>, where the Ωi+ are appropriate creation/annihilation operator products acting on a conveniently chosen closed-shell vacuum φ. These φi’s are not necessarily degenerate with respect to the unperturbed Hamiltonian H0. Ec is the exact (correlated) energy of the vacuum φ. The Fourier transforms Sij(ω) of Sij(t) have the form Sij (ω)=<φi‖(ω +Ec−H)−1‖ψj>/<ψj0‖ψ j> and thus have poles at energy differences (Ej−Ec), i.e., relative to the exact vacuum energy. Using the time-dependent perturbation expansion of S(t), we obtain a Dyson-like equation N̄−1(ω)=N̄0–1(ω)+Σ, where N̄ is defined as N̄ij (ω) = <φi‖(ω+Ec−H)−1‖φj> and N̄0 is the corresponding unperturbed component. Knowledge of the combining coefficients Cji in ψj0=ΣJCjiφi is thus not required for finding the poles. We arrive at the Dyson-like equation by first eliminating closed diagrams and then regrouping the remaining terms in the perturbation series for S into ’’top’’ and ’’bottom’’ parts. Regrouping appropriate to the Brillouin–Wigner (BW) case together with an associated time-integration procedure yields ΣBW which consists of disconnected and ω-dependent diagrams. This is shown to yield the open-shell BW series in the Bloch–Horowitz form. An alternative regrouping procedure and use of the ’’folding technique’’ of Johnson and Baranger leads to a ΣRS which is ω-independent, Hermitian, contains connected diagrams only, and is, thus, size-consistent.

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Journal of Physics B Atomic and Molecular Physics•Oct 1, 1977•14 citations
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.
In this paper, we develop a time-dependent approach to many-body perturbation theory for open shells based on the resolvent of the Schrödinger equation. We introduce, analogous to the closed-shell case, quantities Sij(t) = i<φi‖e−i(H−Ec)t ‖ψj>/<ψj0‖ψj>, where ψj0 and ψj are the jth unperturbed and exact functions, respectively. The ψj0’s can be expressed in terms of ’’model space’’ functions φi = Ωi+‖φ>, where the Ωi+ are appropriate creation/annihilation operator products acting on a conveniently chosen closed-shell vacuum φ. These φi’s are not necessarily degenerate with respect to the unperturbed Hamiltonian H0. Ec is the exact (correlated) energy of the vacuum φ. The Fourier transforms Sij(ω) of Sij(t) have the form Sij (ω)=<φi‖(ω +Ec−H)−1‖ψj>/<ψj0‖ψ j> and thus have poles at energy differences (Ej−Ec), i.e., relative to the exact vacuum energy. Using the time-dependent perturbation expansion of S(t), we obtain a Dyson-like equation N̄−1(ω)=N̄0–1(ω)+Σ, where N̄ is defined as N̄ij (ω) = <φi‖(ω+Ec−H)−1‖φj> and N̄0 is the corresponding unperturbed component. Knowledge of the combining coefficients Cji in ψj0=ΣJCjiφi is thus not required for finding the poles. We arrive at the Dyson-like equation by first eliminating closed diagrams and then regrouping the remaining terms in the perturbation series for S into ’’top’’ and ’’bottom’’ parts. Regrouping appropriate to the Brillouin–Wigner (BW) case together with an associated time-integration procedure yields ΣBW which consists of disconnected and ω-dependent diagrams. This is shown to yield the open-shell BW series in the Bloch–Horowitz form. An alternative regrouping procedure and use of the ’’folding technique’’ of Johnson and Baranger leads to a ΣRS which is ω-independent, Hermitian, contains connected diagrams only, and is, thus, size-consistent.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
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.