The necessity to determine the static self-energy part Σ(∞) in many-body Green’s function studies of atomic and molecular ionization introduces difficulties affecting both the accuracy and the efficiency of the method. We show how this bottleneck can be overcome under an approximation obtained by truncating the Dyson expansion for the one-particle Green’s function G(ω). Here the essential computational step consists of an inversion or a Lanczos diagonalization of constant Hermitian secular matrices associated with the dynamical self-energy part M(ω). Both methods are very practical and efficient as is demonstrated in an exemplary application to the CO molecule. The same approximation and numerical techniques apply also to the problem of extracting ground state information from G(ω), i.e., correlation energy and expectation values of single-particle operators.
The necessity to determine the static self-energy part Σ(∞) in many-body Green’s function studies of atomic and molecular ionization introduces difficulties affecting both the accuracy and the efficiency of the method. We show how this bottleneck can be overcome under an approximation obtained by truncating the Dyson expansion for the one-particle Green’s function G(ω). Here the essential computational step consists of an inversion or a Lanczos diagonalization of constant Hermitian secular matrices associated with the dynamical self-energy part M(ω). Both methods are very practical and efficient as is demonstrated in an exemplary application to the CO molecule. The same approximation and numerical techniques apply also to the problem of extracting ground state information from G(ω), i.e., correlation energy and expectation values of single-particle operators.
In order to properly understand the utility of many-body perturbation theory as applied to finite systems, we use the configuration interaction approach to compute the electronic self-energy. The validity of the commonly used GW approximation from many-body perturbation theory is tested by comparing the self-energy explicitly computed according to the diagrammatic expansions to that from the inversion of the Dyson equation. It is constructed as a functional of the interacting Green function G and screened Coulomb interaction W. W is explicitly computed through the diagonalization of the many-body Hamiltonian and, thus, takes polarization effects into account beyond the random phase approximation. The Na9+ cluster and the dissociation of the C2 molecule are discussed as examples. We find that the GW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as Na9+ clusters. The C2 molecule is a pathological system with multideterminantal ground state where the GW approach ceases to be valid.
In this paper, we propose a generic and systematic approach for study of the electronic structure for atoms or molecules. In particular, we address the issue of single particle states, or orbitals, which should be one of the most important aspects of a quantum many-body theory. We argue that the single-particle Green function provides a most general scheme for generating these single particle states or orbitals. We call them the correlated atomic or molecular orbitals to make a distinction from those determined from Hartree−Fock equation. We present the calculation of the single particle properties (i.e., the electron affinities (EA′s) and ionization potentials (IP′s)) for the H2O molecule using the correlated molecular orbitals in the context of quantum chemistry with a second-order self energy. We also calculate the total ground state energy with a single Slater wavefunction determined only from the hole states. Comparisons are made with available experimental data as well as with those from the Hartree−Fock or density functional theory (DFT) calculations. We conclude that the correlated atomic or molecular orbital approach provides a strictest and most powerful method for studying the single-particle properties of atoms or molecules. It also gives a better total energy than do the Hartree−Fock and DFT even at the single Slater determinant level. It promises that a correlation theory based on the correlated atomic or molecular orbitals will become an approach which possesses the advantages and also overcomes their shortcomings of current quantum chemistry methods based on either the conventional quantum many-body theory or the DFT.
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.
We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O( n 4 ) and O( n 5 ) of molecular size ( n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O( n 5 ) and O( n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.
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 report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases, GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.