By use of basic results of diagrammatic perturbation theory and combinatorial arguments, a formula is derived for the number of topologically distinct, connected, time ordered Goldstone diagrams contributing to the nth order perturbation energy. A general perturbation consisting of one- and two-particle operators is considered. The result is extended to an effective one-particle operator that implicitly includes all equal-time contractions and to multiple one-particle perturbations that enter only a restricted number of times in any order; the former is important for a Hartree–Fock zero order function and the latter for a system with external or internal field perturbations. It is pointed out that the method for computing the number of disconnected diagrams is quite general and independent of the nature of the perturbation and that the connected topologies which possess unusual symmetry (i.e., those for which all sets of vertex space–time labels are not distinct) are of a very specific and easily identified type. Numerical results for the various cases considered are given through 5th order.
By use of basic results of diagrammatic perturbation theory and combinatorial arguments, a formula is derived for the number of topologically distinct, connected, time ordered Goldstone diagrams contributing to the nth order perturbation energy. A general perturbation consisting of one- and two-particle operators is considered. The result is extended to an effective one-particle operator that implicitly includes all equal-time contractions and to multiple one-particle perturbations that enter only a restricted number of times in any order; the former is important for a Hartree–Fock zero order function and the latter for a system with external or internal field perturbations. It is pointed out that the method for computing the number of disconnected diagrams is quite general and independent of the nature of the perturbation and that the connected topologies which possess unusual symmetry (i.e., those for which all sets of vertex space–time labels are not distinct) are of a very specific and easily identified type. Numerical results for the various cases considered are given through 5th order.
A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.
Connections between the Brueckner and Goldstone forms of the linked-cluster theorem are clarified in an elementary fashion employing a finite many-body system of electrons interacting via two-body forces as an illustrative example. It is shown that the order-by-order cancellations of Brueckner unlinked terms in the many-body wave function and energy occur in the Goldstone development upon the additions of the different time orderings of Feynman-Goldstone diagrams having disconnected vacuum parts. The factoring of the vacuum amplitude from the wave function, upon which customary presentations of the theorem focus attention, is seen to be irrelevant to the cancellations of Brueckner unlinked terms, and has to do, rather, with the presence of secular and normalization terms in the time-dependent perturbation functions for an adiabatically switched static perturbation. Similarly, the equivalence of the vacuum amplitude with the exponential of all connected vacuum diagrams, originally demonstrated by Feynman in the case of hole theory, is seen to be irrelevant to the cancellations of Brueckner unlinked terms in the energy, which occur in the Goldstone development upon the summations over all time orderings of disconnected vacuum diagrams. The distinction between Brueckner unlinked terms on the one hand, and secular and normalization terms on the other, is confused in customary presentations of the linked-cluster theorem by the use of exponential switching, following Gell-Mann and Low, and is clarified in the present development by considering the Friedrichs limit of ideal adiabatic switching.
A generalization of many-fermion Feynman-Goldstone diagrams is derived based on a time-dependent unperturbed Hamiltonian along with a time-dependent reference state. In lowest order, the time-dependent Hartree-Fock equations follow immediately from the same arguments which lead to static Hartree-Fock when the basis is time independent. Systematic corrections to the time-dependent Hartree-Fock mean-field approximation are obtained through higher order diagrams.[NUCLEAR STRUCTURE Generalization of Feynman-Goldstone diagrams to a time-dependent reference state. Diagrammatic derivation of the time-dependent Hartree-Fock equations and their systematic corrections.]
The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.
A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams-renormalization diagrams-whose resolvent lines are displaced upward, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.