Recent calculations using coupled cluster on solids have raised the discussion of using a N –1/3 power law to fit the correlation energy when extrapolating to the thermodynamic limit, an approach which differs from the more commonly used N –1 power law, which is, for example, often used by quantum Monte Carlo methods. In this paper, we present one way to reconcile these viewpoints. Coupled cluster doubles calculations were performed on uniform electron gases reaching system sizes of 922 electrons for an extremely wide range of densities (0.1 < r s < 100.0) to study how the correlation energy approaches the thermodynamic limit. The data were corrected for the basis set incompleteness error and use a selected twist angle approach to mitigate the finite size error from shell filling effects. Analyzing these data, we initially find that a power law of N –1/3 appears to fit the data better than a N –1 power law in the large system size limit. However, we provide an analysis of the transition structure factor showing that N –1 still applies to large system sizes and that the apparent N –1/3 power law occurs only at low N .
High Resolution Image Download MS PowerPoint Slide Finite size error is commonly removed from coupled cluster theory calculations by N –1 extrapolations over correlation energy calculations of different system sizes ( N ), where the N –1 scaling comes from the total energy rather than the correlation energy. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of N –2/3 should also be present in the correlation energy when using the conventional Coulomb interaction. The rationale for this is that the total energy goes as N –1 and the exchange energy goes as N –2/3; thus, the correlation energy should be a combination of these two power laws. Further, in coupled cluster theory, these power laws are related to the low G scaling of the transition structure factor, S ( G ), which is a property of the coupled cluster wave function calculated from the amplitudes. We show here that data from coupled cluster doubles calculations on the uniform electron gas fit a function with a low G behavior of S ( G ) ∼ G . The prefactor for this linear term is derived from the exchange energy to be consistent with an N –2/3 power law at large N . Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of S ( G ) ∼ G 2, consistent with an N –1 scaling of the exchange-correlation energy. We then look for the presence of an N –2/3 power law in the energy. To do so, we first develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the uniform electron gas. Then, we collect data from a wide range of electron numbers and densities to systematically test five methods using N –1 scaling, N –2/3 scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both N –1 and N –2/3 scaling perform better than either alone, especially when the prefactor for N –2/3 scaling can be found from exchange energy calculations.
A perturbation theory is developed for the correlation energy
Ec[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+
tsumi${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(
ri). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]=
ec,2[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$
ec,3[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$
ec,4[n]+..., where each
ec,j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy
ec,2[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}
Ec[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as
Ec[n] in finite insulators and semiconductors.Approximations to
Ec[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
The variational and diffusion quantum Monte Carlo methods are used to calculate the correlation energy of the paramagnetic three-dimensional homogeneous electron gas at intermediate to high density. Ground-state energies in finite cells are determined using Slater-Jastrow-backflow trial wave functions, and finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Our correlation energies in the thermodynamic limit are more accurate than previous results. The present diffusion quantum Monte Carlo energies together with our recently reported [Phys. Rev. B 105, 245135 (2022)] results at low density, are used to parametrize the correlation energy of the electron gas using a functional form that satisfies the exact asymptotic behavior at high density.