Abstract A unitary group formulation of the many‐electron problem is employed to give explicit representations of state vectors which are convenient for the discussion of progerties derived from propagatorcalculations. New results are obtained concerning the nature of various random‐phase‐like approximations and ground state representatives are generated from consistency requirements for the spectral resolution of the polarization propagator. The explicit solution admits the calculation of ground state average values for arbitrary operators and a variational upper bound to the ground state energy.
Abstract A unitary group formulation of the many‐electron problem is employed to give explicit representations of state vectors which are convenient for the discussion of progerties derived from propagatorcalculations. New results are obtained concerning the nature of various random‐phase‐like approximations and ground state representatives are generated from consistency requirements for the spectral resolution of the polarization propagator. The explicit solution admits the calculation of ground state average values for arbitrary operators and a variational upper bound to the ground state energy.
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.
Some of the difficulties encountered in the approximation of Green's functions are reviewed with particular emphasis on the self-consistent polarization propagator. Some new basic theorems are presented, involving a general procedure, which lead to both a derivation and alternate prescriptions for the computations. As demonstrated by Linderberg and Öhrn the ground state is an AGP (antisymmetrized geminal power). An energy optimized AGP ( E AGP ) in a Brueckner orbital representation can be used to generate a manifold of excited states| K ⟩ = O K + |AGP⟩, O K |AGP⟩ = 0 with the excitation operators, O + K , and killers, O K , being combinations of particle-hole and hole-particle operators. The SCPP can be expressed then as ⟨⟨ X, Y ⟩⟩ E = - ⟨AGP| XT 0 (- E + E AGP ) Y + YT 0 ( E + E AGP ) X |AGP⟩ in terms of the resolvent T 0 ( ) = P ( - PHP ) -1 with P spanned by the states| K ⟩⟨ K |, in the sense of Löwdin. Our results establish a natural relationship between hermiticity, completeness, non-redundance of operator manifolds with stationarity and killer conditions.
A general method for constructing bases for operator manifolds for any propagator, which satisfy ``vacuum annihilation conditions'' (VAC's) is developed. This approach is based on the observation that if the transformation of the unperturbed ground state to the correlated ground state is represented as a rotation in the Fock space, the corresponding rotation induced in the basis of the concerned operator space would generate a basis which satisfies VAC's on the correlated ground state. The associated requirements for the Hermiticity of superoperator Hamiltonian would also be met in this new basis. The proposed method is noniterative in that, once the form of the ground-state function is specified, the expansion of the operator manifold satisfying VAC's on the ground state does not require any iterative readjustment. The resultant propagators in this approach are fully linked. It is shown that this theory is equivalent to a propagator formalism in terms of hole- and particle-creation and/or annihilation operators with a modified effective Hamiltonian. The self-consistent electron and polarization propagators are considered as examples, and their underlying perturbative structures are analyzed. The role of density shift operators and higher-rank operators are discussed.
We derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalize Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the N-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.