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arXiv:1208.6103

Correlation Energy Divergences in Metallic Systems

James J. Shepherd, Andreas Grüneis

Published Aug 30, 2012• 1 citation
arXivHTMLPDF

Abstract

We numerically examine divergences of the total energy in metallic systems of approximate many-body theories using Hartree--Fock as a reference, including perturbative (M\oller-Plesset, MP), coupled cluster (CC) and configuration interaction (CI) approaches. Controlling for finite size effects and basis set incompleteness error by comparison with energies from the random phase approximation (RPA), we suggest convincingly that non-perturbative coupled cluster theories are acceptable for modelling electronic interactions in metals whilst perturbative coupled cluster theories are not. Data are provided from the RPA with which it is possible to test other approximate finite-basis methods for divergences with only modest computational cost.

DOI
10.48550/arxiv.1208.6103

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We numerically examine divergences of the total energy in metallic systems of approximate many-body theories using Hartree--Fock as a reference, including perturbative (M\oller-Plesset, MP), coupled cluster (CC) and configuration interaction (CI) approaches. Controlling for finite size effects and basis set incompleteness error by comparison with energies from the random phase approximation (RPA), we suggest convincingly that non-perturbative coupled cluster theories are acceptable for modelling electronic interactions in metals whilst perturbative coupled cluster theories are not. Data are provided from the RPA with which it is possible to test other approximate finite-basis methods for divergences with only modest computational cost.
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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 GWGWGW 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 GGG and screened Coulomb interaction WWW. WWW 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+\mathrm{Na}_{9}{}^{+}Na9​+ cluster and the dissociation of the C2{\mathrm{C}}_{2}C2​ molecule are discussed as examples. We find that the GWGWGW approximation yields accurate results for weakly correlated systems with a predominantly single-determinant ground state such as Na9+\mathrm{Na}_{9}{}^{+}Na9​+ clusters. The C2{\mathrm{C}}_{2}C2​ molecule is a pathological system with multideterminantal ground state where the GWGWGW approach ceases to be valid.
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
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