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arXiv:2107.14434 · Physical review. B./Physical review. B

Accurate total energies from the adiabatic-connection fluctuation-dissipation theorem

Nick Woods, Mike Entwistle, Rex W. Godby

Published Sep 17, 2021• 4 citations
arXivHTMLPDF

Abstract

In the context of inhomogeneous one-dimensional finite systems, recent numerical advances [Phys. Rev. B 103, 125155 (2021)] allow us to compute the exact coupling-constant dependent exchange-correlation kernel ${f}_{\text{xc}}^{\ensuremath{\lambda}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ within linear response time-dependent density-functional theory. This permits an improved understanding of ground-state total energies derived from the adiabatic-connection fluctuation-dissipation theorem (ACFDT). We consider both one-shot and self-consistent ACFDT calculations, and demonstrate that chemical accuracy is reliably preserved when the frequency dependence in the exact functional ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is neglected. This performance is understood on the grounds that the exact ${f}_{\text{xc}}[n]$ varies slowly over the most relevant $\ensuremath{\omega}$ range (but not in general), and hence the spatial structure in ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is able to largely remedy the principal issue in the present context: self-interaction (examined from the perspective of the exchange-correlation hole). Moreover, we find that the implicit orbitals contained within a self-consistent ACFDT calculation utilizing the adiabatic exact kernel ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ are remarkably similar to the exact Kohn-Sham orbitals, thus further establishing that the majority of the physics required to capture the ground-state total energy resides in the spatial dependence of ${f}_{\text{xc}}[n]$ at $\ensuremath{\omega}=0$.

DOI
10.1103/physrevb.104.125126

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In the context of inhomogeneous one-dimensional finite systems, recent numerical advances [Phys. Rev. B 103, 125155 (2021)] allow us to compute the exact coupling-constant dependent exchange-correlation kernel fxc\ensuremathλ(x,x\ensuremath′,\ensuremathω){f}_{\text{xc}}^{\ensuremath{\lambda}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})fxc\ensuremathλ​(x,x\ensuremath′,\ensuremathω) within linear response time-dependent density-functional theory. This permits an improved understanding of ground-state total energies derived from the adiabatic-connection fluctuation-dissipation theorem (ACFDT). We consider both one-shot and self-consistent ACFDT calculations, and demonstrate that chemical accuracy is reliably preserved when the frequency dependence in the exact functional fxc[n](\ensuremathω=0){f}_{\text{xc}}[n](\ensuremath{\omega}=0)fxc​[n](\ensuremathω=0) is neglected. This performance is understood on the grounds that the exact fxc[n]{f}_{\text{xc}}[n]fxc​[n] varies slowly over the most relevant \ensuremathω\ensuremath{\omega}\ensuremathω range (but not in general), and hence the spatial structure in fxc[n](\ensuremathω=0){f}_{\text{xc}}[n](\ensuremath{\omega}=0)fxc​[n](\ensuremathω=0) is able to largely remedy the principal issue in the present context: self-interaction (examined from the perspective of the exchange-correlation hole). Moreover, we find that the implicit orbitals contained within a self-consistent ACFDT calculation utilizing the adiabatic exact kernel fxc[n](\ensuremathω=0){f}_{\text{xc}}[n](\ensuremath{\omega}=0)fxc​[n](\ensuremathω=0) are remarkably similar to the exact Kohn-Sham orbitals, thus further establishing that the majority of the physics required to capture the ground-state total energy resides in the spatial dependence of fxc[n]{f}_{\text{xc}}[n]fxc​[n] at \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0.
The exact exchange-correlation (xc) kernel ${f}_{\text{xc}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ of linear response time-dependent density functional theory is computed over a wide range of frequencies for three canonical one-dimensional finite systems. Methods used to ensure the numerical robustness of fxc{f}_{\text{xc}}fxc​ are set out. The frequency dependence of fxc{f}_{\text{xc}}fxc​ is found to be largely due to its analytic structure, i.e., its singularities at certain frequencies, which are required in order to capture particular transitions, including those of double excitation character. However, within the frequency range of the first few interacting excitations, fxc{f}_{\text{xc}}fxc​ is approximately $\ensuremath{\omega}$ independent, meaning the exact adiabatic approximation ${f}_{\text{xc}}(\ensuremath{\omega}=0)$ remedies the failings of the local density approximation and random phase approximation for these lowest transitions. The key differences between the exact fxc{f}_{\text{xc}}fxc​ and its common approximations are analyzed, and cannot be eliminated by exploiting the limited gauge freedom in fxc{f}_{\text{xc}}fxc​. The optical spectrum benefits from using as accurate as possible an fxc{f}_{\text{xc}}fxc​ and ground-state xc potential, while maintaining exact compatibility between the two is of less importance.
The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation-dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange-correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.
We have calculated the correlation energy of the homogeneous electron gas (HEG) and the dissociation energy curves of molecules with covalent bonds from an efficient implementation of the adiabatic connection fluctuation dissipation expression including the exact-exchange (EXX) kernel. The EXX kernel is defined from first-order perturbation theory and used in the Dyson equation of time-dependent density-functional theory. Within this approximation (RPAx), the correlation energies of the HEG are significantly improved with respect to the random phase approximation (RPA) up to densities of the order of ${r}_{s}\ensuremath{\approx}10$. However, beyond this value, the RPAx response function exhibits an unphysical divergence and the approximation breaks down. Total energies of molecules at equilibrium are also highly accurate, but we find a similar instability at stretched geometries. Staying within an exact first-order approximation to the response function, we use an alternative resummation of the higher-order terms. This slight redefinition of RPAx fixes the instability in total energy calculations without compromising the overall accuracy of the approach.
The interacting and the Kohn-Sham static density-density response functions for different one-dimensional two-electron singlet systems are reconstructed numerically. From their inverse we obtain the exact static exchange-correlation kernel. This quantity represents the adiabatically exact approximation of the frequency-dependent exchange-correlation kernel that is crucial for time-dependent linear density-response theory. We investigate its performance for nonlocal perturbations and analyze its sum rule properties. We also compute the adiabatically exact transition energies that follow from the static kernel within linear-response theory.
A hierarchy of universally applicable density functional methods based on the adiabatic-connection fluctuation-dissipation theorem is introduced that can be systematically improved to treat electronic structures with increasing accuracy. Methods at the first rank of the hierarchy already can compete with established high-level wave-function methods and exhibit an attractive ratio of accuracy to computational effort. Strong correlation can be treated in a black-box manner based on a single Slater determinant: the Kohn-Sham determinant.
Thermal density functional theory calculations often use the Mermin-Kohn-Sham scheme, but employ ground-state approximations to the exchange-correlation (XC) free energy. In the simplest solvable nontrivial model, an asymmetric Hubbard dimer, we calculate the exact many-body energies and the exact Mermin-Kohn-Sham functionals for this system and extract the exact XC free energy. For moderate temperatures and weak correlation, we find this approximation to be excellent. We extract various exact free-energy correlation components and the exact adiabatic connection formula.