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Register a new userPerformance of the Constrained Minimization of the Total Energy in Density Functional Approximations: the Electron Repulsion Density and Potential
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Added by
Nektarios Lathiotakis
Publication date
2018
fields
Physics
and research's language is
English
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In the constrained minimization method of Gidopoulos and Lathiotakis (J. Chem. Phys. 136, 224109), the Hartree exchange and correlation Kohn-Sham potential of a finite $N$-electron system is replaced by the electrostatic potential of an effective charge density that is everywhere positive and integrates to a charge of $N-1$ electrons. The optimal effective charge density (electron repulsion density, $rho_{rm rep}$) and the corresponding optimal effective potential (electron repulsion potential $v_{rm rep}$) are obtained by minimizing the electronic total energy in any density functional approximation. The two constraints are sufficient to remove the self-interaction errors from $v_{rm rep}$, correcting its asymptotic behavior at large distances from the system. In the present work, we describe, in complete detail, the constrained minimization method, including recent refinements. We also assess its performance in removing the self-interaction errors for three popular density functional approximations, namely LDA, PBE, and B3LYP, by comparing the obtained ionization energies to their experimental values for an extended set of molecules. We show that the results of the constrained minimizations are almost independent of the specific approximation with average percentage errors 15%, 14%, 13% for the above DFAs respectively. These errors are substantially smaller than the corresponding errors of the plain (unconstrained) Kohn-Sham calculations at 38%, 39%, and 27% respectively. Finally, we showed that this method correctly predicts negative values for the HOMO energies of several anions.
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We review and expand on our work to impose constraints on the effective Kohn Sham (KS) potential of local and semi-local density functional approximations. In this work, we relax a previously imposed positivity constraint, which increased the computational cost and we find that it is safe to do so, except in systems with very few electrons. The constrained minimisation leads invariably to the solution of an optimised effective potential (OEP) equation in order to determine the KS potential. We review briefly our previous work on this problem and demonstrate with numerous examples that despite well-known mathematical issues of the OEP with finite basis sets, our OEP equations are well behaved. We demonstrate that constraining the screening charge of the Hartree, exchange and correlation potential not only corrects its asymptotic behaviour but also allows the exchange and correlation potential to exhibit nonzero derivative discontinuity, a feature of the exact KS potential that is necessary for the accurate prediction of band-gaps in solids but very hard to capture with semi-local approximations.
In this chapter, we provide a review of ground-state Kohn-Sham density-functional theory of electronic systems and some of its extensions, we present exact expressions and constraints for the exchange and correlation density functionals, and we discuss the main families of approximations for the exchange-correlation energy: semilocal approximations, single-determinant hybrid approximations, multideterminant hybrid approximations, dispersion-corrected approximations, as well as orbital-dependent exchange-correlation density functionals. The chapter aims at providing both a consistent birds-eye view of the field and a detailed description of some of the most used approximations. It is intended to be readable by chemists/physicists and applied mathematicians.
Recently a novel approach to find approximate exchange-correlation functionals in density-functional theory (DFT) was presented (U. Mordovina et. al., JCTC 15, 5209 (2019)), which relies on approximations to the interacting wave function using density-matrix embedding theory (DMET). This approximate interacting wave function is constructed by using a projection determined by an iterative procedure that makes parts of the reduced density matrix of an auxiliary system the same as the approximate interacting density matrix. If only the diagonal of both systems are connected this leads to an approximation of the interacting-to-non-interacting mapping of the Kohn-Sham approach to DFT. Yet other choices are possible and allow to connect DMET with other DFTs such as kinetic-energy DFT or reduced density-matrix functional theory. In this work we give a detailed review of the basics of the DMET procedure from a DFT perspective and show how both approaches can be used to supplement each other. We do so explicitly for the case of a one-dimensional lattice system, as this is the simplest setting where we can apply DMET and the one that was originally presented. Among others we highlight how the mappings of DFTs can be used to identify uniquely defined auxiliary systems and auxiliary projections in DMET and how to construct approximations for different DFTs using DMET inspired projections. Such alternative approximation strategies become especially important for DFTs that are based on non-linearly coupled observables such as kinetic-energy DFT, where the Kohn-Sham fields are no longer simply obtainable by functional differentiation of an energy expression, or for reduced density-matrix functional theories, where a straightforward Kohn-Sham construction is not feasible.
When a molecule dissociates, the exact Kohn-Sham (KS) and Pauli potentials may form step structures. Reproducing these steps correctly is central for the description of dissociation and charge-transfer processes in density functional theory (DFT): The steps align the KS eigenvalues of the dissociating subsystems relative to each other and determine where electrons localize. While the step height can be calculated from the asymptotic behavior of the KS orbitals, this provides limited insight into what causes the steps. We give an explanation of the steps with an exact mapping of the many-electron problem to a one-electron problem, the exact electron factorizaton (EEF). The potentials appearing in the EEF have a clear physical meaning that translates to the DFT potentials by replacing the interacting many-electron system with the KS system. With a simple model of a diatomic, we illustrate that the steps are a consequence of spatial electron entanglement and are the result of a charge transfer. From this mechanism, the step height can immediately be deduced. Moreover, two methods to approximately reproduce the potentials during dissociation suggest themselves. One is based on the states of the dissociated system, while the other one is based on an analogy to the Born-Oppenheimer treatment of a molecule. The latter method also shows that the steps connect adiabatic potential energy surfaces. The view of DFT from the EEF thus provides a better understanding of how many-electron effects are encoded in a one-electron theory and how they can be modeled.
We provide a rationale for a new class of double-hybrid approximations introduced by Bremond and Adamo [J. Chem. Phys. 135, 024106 (2011)] which combine an exchange-correlation density functional with Hartree-Fock exchange weighted by $l$ and second-order M{o}ller-Plesset (MP2) correlation weighted by $l^3$. We show that this double-hybrid model can be understood in the context of the density-scaled double-hybrid model proposed by Sharkas et al. [J. Chem. Phys. 134, 064113 (2011)], as approximating the density-scaled correlation functional $E_c[n_{1/l}]$ by a linear function of $l$, interpolating between MP2 at $l=0$ and a density-functional approximation at $l=1$. Numerical results obtained with the Perdew-Burke-Ernzerhof density functional confirms the relevance of this double-hybrid model.
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