It is shown that the coupled-cluster theory can be interpreted as an application of the similarity-transformation theory for the quantum-mechanical eigenvalue problem. The transformed Hamiltonian H = e-THeT is introduced with the cluster operator T. The Hamiltonian H is expanded into different clusters according to the number of interacting particles. The n-body part of the cluster operator T is determined from the condition that the n-body part of H should be decoupled between two n-body states with configurations of n occupied and n unoccupied orbits. It is proved that the problem of solving the decoupling equation can be reduced to solving an eigenvalue equation for a subsystem Hamiltonian. it is also proved that there exist one- and two-body subsystem equations with certain one- and two-body average fields such that they determine the one- and two-body parts of the cluster operator and consequently lead to the exact ground-state energy.
It is shown that the coupled-cluster theory can be interpreted as an application of the similarity-transformation theory for the quantum-mechanical eigenvalue problem. The transformed Hamiltonian H = e-THeT is introduced with the cluster operator T. The Hamiltonian H is expanded into different clusters according to the number of interacting particles. The n-body part of the cluster operator T is determined from the condition that the n-body part of H should be decoupled between two n-body states with configurations of n occupied and n unoccupied orbits. It is proved that the problem of solving the decoupling equation can be reduced to solving an eigenvalue equation for a subsystem Hamiltonian. it is also proved that there exist one- and two-body subsystem equations with certain one- and two-body average fields such that they determine the one- and two-body parts of the cluster operator and consequently lead to the exact ground-state energy.
The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.
Normal ordered exponential operators have been used extensively in open-shell formulations of coupled cluster theory. The inverse of such an operator is known to exist, but a closed form explicit expression for the inverse is not available. We will address the evaluation of many-body similarity transformations generated by normal ordered exponential transformation operators without explicit use of the inverse. The similarity transform can be obtained as the solution of a linear system of equations that can be solved trivially using backward substitution. In addition a closed form diagrammatic expression for the similarity transformed operator is presented. Using the many-body similarity transformation strategy a simple and more general formulation of Fock space coupled cluster theory is presented which is akin in spirit to the formulation by Stolarczyk and Monkhorst [Phys. Rev. A 32, 725, 743 (1985); 37, 1908, 1926 (1988)], but which on the other hand is completely equivalent to the conventional wave operator formulation of Fock space coupled cluster theory (under suitable conditions). Other possible applications of the many-body similarity transformation will be briefly discussed.
We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from many-body coupled-cluster theory, to obtain form factors and other observables.
A theory is formulated for solving a many-Fermion system in the framework of unitary-transformation method. A unitary transformation of a product form Πnexp S(n) with n-body operator S(n) is introduced. The operator exp S(n) is determined on the principle that the transformed Hamiltonian should not have interactions which induce n-particle-n-hole excitation. It is proved that the operator S(n) can be given in terms of eigenstates of n-body subsystem Hamiltonian and the true gound state can be given by Πn exp S(n)|φ0>, where |φ0> is the unperturbed free ground state.
This article gives a discussion of the application of the extended coupled-cluster method (ECCM) to the excited states of a general quantum many-body system. The direct eigenvalue equations for the excitation amplitudes of both the ket and the bra eigenstates are derived in the biorthogonal basis obtained by a double similarity transformation. The equations correspond to the diagonalization of a matrix involving second-order functional derivatives of the average-value functional for the Hamiltonian with respect to the basic ECCM amplitudes. The same excitation spectrum is obtained by considering small oscillations around the equilibrium. The problem with its associated effective Hamiltonian has the structure of a generalized random-phase approximation. By diagonalizing the effective Hamiltonian we perform a canonical or symplectomorphic coordinate transformation into normal coordinates in the symplectic ECCM phase space. In this coordinate system the exact average-value functional for the Hamiltonian has a structure analogous to that of classical lattice dynamics or the phenomenological Ginzburg-Landau theory. At all stages the method satisfies the property of quantum locality, which in real space shows up as a definite quasilocality. Due to this property the method allows, for example, the treatment of mesonlike excitations in the presence of topological objects or in other symmetry-broken equilibrium states.