We explicitly calculate the four leading-order terms of the formal asymptotic series for large |ε| (where ε denotes the external energy parameter) of the single-particle Green function G σ(ε), σ ϵ {−s, −s + 1, …, s}, and the three leading-order terms of that of the self-energy operator ∑σ(ε), pertaining to systems of spin-s fermions (s is a half-integer) in d-dimensional spatial space, interacting through an arbitrary two-body potential v(r-r'). These contributions, which are expressed in terms at most of a three-body static ground-state correlation function, are amenable to accurate numerical calculation through employing correlated many-particle ground-state wavefunctions such as determined within, for example, the quantum Monte Carlo framework. Such calculations will provide indisputably reliable information with regard to the spectral function of the single-particle excitations at high energies of correlated systems, as well as energy moments of this spectral function, and thus help to assess the reliability of theoretical approaches that are applied in studying such systems. We give especial attention to d = 3 and v ≡ v C, the long-range Coulomb potential, for which case we explicitly calculate the five leading-order terms of the regularized large-|ε| asymptotic series of ∑σ(ε). Our considerations reveal some interesting aspects which are very specific to the behaviour of v C(r-r') at both small and large values of ||r-r'||. In particular, we show that in these systems an inhomogeneity, even on the atomic scale, in the particle spin polarization density gives rise to a pronounced effect, directly discernible in the inverse photoemission spectra; we show this effect to be absent in models with v bounded at origin and those in which v ≡ v C but Umklapp processes are neglected, unless the ground state possesses long-range magnetic order. Our analyses shed light on the importance of the non-local part of the self-energy operator and disclose that some of the strictly non-local contributions to this operator transform into local contributions upon replacing v ≢ v C by v C, implying that a local approximation to ∑σ(ε), which for v ≢ v C proves accurate, is necessarily less accurate for v ≡ v C. These findings establish a fundamental limitation of the so-called ‘dynamical mean-field’ approximation to ∑σ(ε), which is strictly local, specifically in applications where v ≡ v C. We further explicitly establish some of the shortcomings of ∑σ(ε) as calculated within the framework of the many-body perturbation theory. In this context we demonstrate the empirically well-known inadequacy of the dynamically screened exchange self-energy operator in particular for describing the photoemission and inverse photoemission spectra of interacting systems at intermediate and large transfer energies and put forward a workable scheme that rids this self-energy of its fundamental defects. We present ample explicit analyses of our results in terms of uncorrelated many-body wavefunctions.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
On the basis of an exact perturbational expression for the interacting one-particle Green function
G corresponding to bosons / fermions in terms of the bare interaction potential
v and permanents / determinants of the non-interacting one-particle Green function
G0, we deduce four recursive perturbation expansions for the self-energy
Σ. With
W denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of
Σ in terms of (i) self-energy diagrams and
(v,G0), (ii)
G-skeleton self-energy diagrams and
(v,G), (iii)
W-skeleton self-energy diagrams and
(W,G0), and (iv)
G- and
W-skeleton self-energy diagrams and
(W,G). For the calculation of
W, we rely on a similar exact perturbational expression for the interacting two-particle Green function
G2 as for
G. From this expression, we deduce four recursive perturbation expansions for the polarization function
P, necessary for the calculation of
W, that are similar to those for
Σ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-
21 fermions on a lattice in arbitrary
d spatial dimensions. In two further appendices, we present methods and short programs for determining the
νth-order diagrams corresponding to the perturbation expansions of
G in terms of
(v,G0), and
Σ in terms of
(v,G0) and
(v,G) on the basis of the cycle decompositions of the elements of the symmetric group
S2ν.
The one-particle Green function of a many-electron system is traditionally formulated within the self-energy picture. A different formalism was recently proposed, in which the self-energy is replaced by a dynamical exchange-correlation field, which acts on the Green function locally in both space and time. It was found that there exists a fundamental quantity, referred to as the dynamical exchange-correlation hole, which can be interpreted as effective density fluctuations induced in a many-electron system when a hole or an electron is introduced into the system, as in photoemission and inverse photoemission experiments. The dynamical exchange-correlation potential is simply the Coulomb potential of this exchange-correlation hole, which fulfils a sum rule and an exact constraint, identical to those satisfied by the static exchange-correlation hole in density-functional theory. The proposed formalism has been applied to a number of model systems such as the half-filled one-dimensional Hubbard model, the one-dimensional antiferromagnetic Heisenberg model, and the single-impurity Anderson model. The dynamical exchange-correlation hole and field of the homogeneous electron gas have also been studied with the view of constructing a density-functional approximation such as the local-density approximation. The availability of simple but accurate approximations for the exchange-correlation potential would circumvent costly computations of the traditional self-energy. The formalism may also provide new perspectives and insights into the many-body problem.
In this second work in a series of papers the coupled-cluster or
exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator
S2 which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.
We investigate several approximations for self-bound many-fermion systems, each of which employs the Green's-function formalism together with a suitable factorization to close the otherwise infinite hierarchy of coupled integral equations. The particular approximations, which attempt to include the important two-body correlations for short-range forces, differ in the choice of propagator $\ensuremath{\Lambda}$ for two-particle intermediate states in the
T matrix used to calculate the two-body correlation function. The advantages and shortcomings of the choice ${\ensuremath{\Lambda}}_{00}\ensuremath{\sim}{{G}_{1}}^{0}{{G}_{1}}^{0}$ have been noted previously and are not treated here. We show that the choice ${\ensuremath{\Lambda}}_{10}\ensuremath{\sim}{{G}_{1}}^{0}{G}_{1}$, which appears in a "natural" factorization, is formulated in a manner which preserves much of the simplicity of the ${\ensuremath{\Lambda}}_{00}$ approximation. (Numerical calculations using the ${\ensuremath{\Lambda}}_{10}$ approximation are presented by Foster in the following paper.) The ground-state energies for three $\ensuremath{\Lambda}$ approximations (${\ensuremath{\Lambda}}_{00}$, ${\ensuremath{\Lambda}}_{10}$, and ${\ensuremath{\Lambda}}_{11}\ensuremath{\sim}{G}_{1}{G}_{1}$) are expanded through third order in the interaction strength to facilitate comparison with each other and with the Brueckner-Goldstone series. Numerical comparison of these third-order contributions for a square-well potential is presented. ${\ensuremath{\Lambda}}_{10}$ is in dramatically closer agreement with the full perturbation theory (for ${{k}_{F}}^{0}c\ensuremath{\lesssim}1$) than are the other $\ensuremath{\Lambda}$ approximations or Brueckner theory.