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The Journal of Chemical Physics

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

Lucas M. Everhart, Julio A. Derteano, Jefferson E. Bates

Published Feb 25, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0080382

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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.
We provide the expression of the exchange-correlation potential in the adiabatic connection fluctuation-dissipation (ACFD) framework, for arbitrary time-dependent (TD) kernels. We investigate the asymptotic behavior of the ACFD potential in three relevant approximations: the random-phase approximation, the exact-exchange kernel in two-electron systems, and the adiabatic local-density approximation. We show that these potentials have the expected $\ensuremath{-}{1/r+Q/r}^{3}\ensuremath{-}\ensuremath{\alpha}{/(2r}^{4})$ tail (in closed-shell systems with spherical symmetry), where Q and $\ensuremath{\alpha}$ depend on the TD kernel and reflect the physics included in each approximation. We also discuss approximate ACFD potentials that are much simpler to compute than the exact ones while being likely of reasonable accuracy.
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 fxc[n]{f}_{\text{xc}}[n]fxc​[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 fxc[n]{f}_{\text{xc}}[n]fxc​[n] at $\ensuremath{\omega}=0$.
Ensemble density functional theory (eDFT) is an exact time-independent alternative to time-dependent DFT (TD-DFT) for the calculation of excitation energies. Despite its formal simplicity and advantages in contrast to TD-DFT (multiple excitations, for example, can be easily taken into account in an ensemble), eDFT is not standard, which is essentially due to the lack of reliable approximate exchange-correlation (xcxcxc) functionals for ensembles. Following Smith et al. [Phys. Rev. B 93, 245131 (2016)], we propose in this work to construct an exact eDFT for the nontrivial asymmetric Hubbard dimer, thus providing more insight into the weight dependence of the ensemble xcxcxc energy in various correlation regimes. For that purpose, an exact analytical expression for the weight-dependent ensemble exchange energy has been derived. The complementary exact ensemble correlation energy has been computed by means of Legendre-Fenchel transforms. Interesting features like discontinuities in the ensemble xcxcxc potential in the strongly correlated limit have been rationalized by means of a generalized adiabatic connection formalism. Finally, functional-driven errors induced by ground-state density-functional approximations have been studied. In the strictly symmetric case or in the weakly correlated regime, combining ensemble exact exchange with ground-state correlation functionals gives better ensemble energies than when calculated with the ground-state exchange-correlation functional. However, when approaching the asymmetric equiensemble in the strongly correlated regime, the former approximation leads to highly curved ensemble energies with negative slope which is unphysical. Using both ground-state exchange and correlation functionals gives much better results in that case. In fact, exact ensemble energies are almost recovered in some density domains. The analysis of density-driven errors is left for future work.
Recently it has been shown how to employ the adiabatic connection (AC) formalism to obtain correlation energy for multireference wavefunctions [K. Pernal, Phys. Rev. Lett. 120, 013001 (2018)]. Approximations to the exact AC formulation have been based on assuming that a one-electron reduced density matrix is constant along the AC path and by employing the extended random phase approximation. In this paper, the importance of these approximations is examined by comparing approximate AC integrands with their exact counterparts obtained for the hydrogen molecule in its ground and excited states. Encouraging results obtained for H2 indicate that AC is a viable and promising approach to a correlation energy problem not only for ground but also for excited states of electronic systems.
When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange-correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn-Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange-correlation potential from a coupling-constant path integration.