Kohn-Sham Exchange-correlation Potential Research Articles

Accurate real-time evolution of electron densities and ground-state properties from generalized Kohn-Sham theory

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation (xc) potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalised by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.

Semianalytical wavefunctions and Kohn–Sham exchange-correlation potentials for two-electron atomic systems in two-dimensions

We propose accurate semianalytical wavefunctions for two-electron atomic systems in two-dimensions. These wavefunctions take into account the cusp conditions arising due to the Coulombic nature of external potential and electron–electron interaction, and the screening effects. The first wavefunction is motivated by the Le Sech wavefunction for two-electron atoms and ions in three-dimensions while the second one is constructed to give more variational freedom to the Le Sech wavefunction and is obtained by solving a Schrödinger like equation self-consistently. The results for energies, densities and other quantities are highly accurate and therefore may play an essential role in developing and testing density functional methods for two-dimensional systems.

Universal nature of different methods of obtaining the exact Kohn–Sham exchange-correlation potential for a given density

Finding the external potential or the Kohn–Sham (KS) potential for a given ground state density is a fundamental inverse problem in density functional theory. Furthermore, it is important in generating the exact exchange-correlation potential for a density which can then serve as a benchmark for testing the accuracy of an exchange-correlation energy functional. Over the years different methods have been proposed to do the density-to-potential inversion and all of these appear to be disjoint. In this paper we show that these methods are connected to and can be derived from one general algorithm that utilizes two forms of the equation for the density; these are the KS equation in terms of orbitals and the Euler equation in terms of the density. We obtain the condition for the convergence of the general method and show that all the methods considered by us satisfy this condition. We demonstrate the method and its flexibility by obtaining the KS potential for some spherical systems in a variety of ways.

Uniform electron gas limit of an exact expression for the Kohn–Sham exchange-correlation potential

Previously, we derived an exact formula for the Kohn–Sham exchange-correlation potential corresponding, in the basis-set limit, to the Hartree–Fock electron density of a given system. This formula expresses the potential in terms of the occupied Hartree–Fock and Kohn–Sham orbitals and orbital energies. Here, we show that, when applied to the Hartree–Fock description of a uniform electron gas, the formula correctly reduces to the exchange-only local density approximation.

Neural-network Kohn-Sham exchange-correlation potential and its out-of-training transferability.

We incorporate in the Kohn-Sham self-consistent equation a trained neural-network projection from the charge density distribution to the Hartree-exchange-correlation potential n → VHxc for a possible numerical approach to the exact Kohn-Sham scheme. The potential trained through a newly developed scheme enables us to evaluate the total energy without explicitly treating the formula of the exchange-correlation energy. With a case study of a simple model, we show that the well-trained neural-network VHxc achieves accuracy for the charge density and total energy out of the model parameter range used for the training, indicating that the property of the elusive ideal functional form of VHxc can approximately be encapsulated by the machine-learning construction. We also exemplify a factor that crucially limits the transferability-the boundary in the model parameter space where the number of the one-particle bound states changes-and see that this is cured by setting the training parameter range across that boundary. The training scheme and insights from the model study apply to more general systems, opening a novel path to numerically efficient Kohn-Sham potential.

Contracted Schrödinger equation and Kohn–Sham effective potentials

ABSTRACTThe precise relationship between the exact expressions for the Kohn–Sham exchange-correlation potential, , deduced by Buijse, Baerends and Snijders and by Ryabinkin, Kohut and Staroverov is clarified. These two expressions differ nontrivially by a single term which in the former method generally involves the three-electron reduced density matrix, whereas in the latter approach the same term is expressed using the two-electron reduced density matrix at most. The link between the two expressions turns out to be the 1,3-contracted Schrödinger equation. An essential feature of these and other similar expressions for is that they produce different results in finite-basis-set calculations and become equivalent only in the basis-set limit.

Improved method for generating exchange-correlation potentials from electronic wave functions.

Ryabinkin, Kohut, and Staroverov (RKS) [Phys. Rev. Lett. 115, 083001 (2015)] devised an iterative method for reducing many-electron wave functions to Kohn-Sham exchange-correlation potentials, vXC(𝐫). For a given type of wave function, the RKS method is exact (Kohn-Sham-compliant) in the basis-set limit; in a finite basis set, it produces an approximation to the corresponding basis-set-limit vXC(𝐫). The original RKS procedure works very well for large basis sets but sometimes fails for commonly used (small and medium) sets. We derive a modification of the method's working equation that makes the RKS procedure robust for all Gaussian basis sets and increases the accuracy of the resulting exchange-correlation potentials with respect to the basis-set limit.

Response to "Comment on 'Kohn-Sham exchange-correlation potentials from second-order reduced density matrices'" [J. Chem. Phys. 145, 037101 (2016)

Response to "Comment on 'Kohn-Sham exchange-correlation potentials from second-order reduced density matrices'" [J. Chem. Phys. 145, 037101 (2016)

Comment on “Kohn-Sham exchange-correlation potentials from second-order reduced density matrices” [J. Chem. Phys. 143, 244116 (2015)

Comment on “Kohn-Sham exchange-correlation potentials from second-order reduced density matrices” [J. Chem. Phys. 143, 244116 (2015)

Exact expressions for the Kohn–Sham exchange-correlation potential in terms of wave-function-based quantities

ABSTRACTSeveral workers have deduced various exact expressions for the Kohn–Sham exchange-correlation potential in terms of quantities computable from the interacting and noninteracting wave functions of the system. We show that all these expressions can be obtained by one general method in which the interacting N-electron wave function is expanded in products of one- and (N − 1)-electron functions. Different expressions correspond to different choices of the latter functions. Our analysis unifies and clarifies the previously proposed exact treatments of the exchange-correlation potential, and suggests new ways of expressing this quantity.

Kohn-Sham exchange-correlation potentials from second-order reduced density matrices.

We describe a practical algorithm for constructing the Kohn-Sham exchange-correlation potential corresponding to a given second-order reduced density matrix. Unlike conventional Kohn-Sham inversion methods in which such potentials are extracted from ground-state electron densities, the proposed technique delivers unambiguous results in finite basis sets. The approach can also be used to separate approximately the exchange and correlation potentials for a many-electron system for which the reduced density matrix is known. The algorithm is implemented for configuration-interaction wave functions and its performance is illustrated with numerical examples.

Accurate and Efficient Approximation to the Optimized Effective Potential for Exchange

We devise an efficient practical method for computing the Kohn-Sham exchange-correlation potential corresponding to a Hartree-Fock electron density. This potential is almost indistinguishable from the exact-exchange optimized effective potential (OEP) and, when used as an approximation to the OEP, is vastly better than all existing models. Using our method one can obtain unambiguous, nearly exact OEPs for any reasonable finite one-electron basis set at the same low cost as the Krieger-Li-Iafrate and Becke-Johnson potentials. For all practical purposes, this solves the long-standing problem of black-box construction of OEPs in exact-exchange calculations.

Near-Origin Structure of the Hooke's Atoms

The Hooke's atoms with two or more than two electrons give rise to an interesting quantum mechanical model with valuable practical applications. In this work, we study the electronic properties near the origin of the harmonic potential. It is seen that the spherically averaged density, ρ̄, exhibits an interesting character — it has only even order terms in its small r expansion. The spherical average of the Hartree potential, v̄H, and the spherical average of the Kohn—Sham exchange-correlation potential, v̄xc, are also shown to have the same property — all odd order terms in their expansions vanish. Furthermore, the analysis and results extend also to the case of two-dimensional models. While only models interacting via. the Coulomb potential are primarily considered in the article, the results also extend to models interacting via. other potentials (viz. Van der Waals potential).

Localized versus extended systems in density functional theory: Some lessons from the Kohn-Sham exact exchange potential

A long-standing puzzle in density functional theory is the issue of the long-range behavior of the Kohn-Sham exchange-correlation potential at metal surfaces. As an important step toward its solution, it is proven here, through a rigorous asymptotic analysis and an accurate numerical solution of the optimized-effective-potential integral equation, that the Kohn-Sham exact exchange potential decays as $\text{ln}(z)/z$ far into the vacuum side of an extended semi-infinite jellium. In contrast with the situation in localized systems, such as atoms, molecules, and slabs, this dominant contribution does not arise from the so-called Slater potential. This exact exchange result provides a strong constraint on the suitability of approximate correlation-energy functionals.

Density functional theory and the band gap problem

How can the fundamental band gap of an insulator be predicted? As a difference of ground-state energies, the fundamental gap seems to fall within the reach of density functional theory, yet the predicted gaps from band structure calculations within the local density approximation (LDA) are about 40% too small. It is argued here that even the exact Kohn-Sham potential veff(r), which generates the exact density in a self-consistent-field calculation, generates a band structure which underestimates the gap. Within the context of the band gap problem, several recent developments in the density-functional theory of many-electron systems are reviewed: (1) The Langreth-Mehl approximation to the Kohn-Sham exchange-correlation energy and potential, based upon the Langreth-Perdew wavevector analysis of the density gradient expansion. This functional leads to more accurate ground-state energies and densities than those of the LDA with little change in the calculated band structures of solids. (2) The derivative discontinuity of the exchange-correlation energy, which is responsible for substantial underestimation of the fundamental gap by even the exact Kohn-Sham potential. (3) The self-interaction correction, which yields accurate gaps in insulators only by virtue of its orbital-dependent potential. (4) The density response function of the uniform electron gas, which suggests that the LDA gives a good estimate of the exact Kohn-Sham potential for a semiconductor with a weak periodic potential. In short, several very different (but admittedly approximate) numerical calculations suggest that most of the error in the LDA fundamental gap would persist in the gap of the exact Kohn-Sham band structure. This error would persist in any attempt to calculate the gap from LDA total energy differences for clusters of increasing size.

Reconstruction of Density Functionals from Kohn-Sham Potentials by Integration along Density Scaling Paths.

We demonstrate by specific examples that if a Kohn-Sham exchange-correlation potential is given explicitly in terms of the electron density and its derivatives, then one can easily reconstruct the parent density functional by evaluating analytically (or numerically with one-dimensional quadratures) the van Leeuwen-Baerends line integral (Phys. Rev. A 1995, 51, 170-178) along a path of (coordinate)-scaled densities. The choice of a density scaling path amounts to defining the gauge of the resultant exchange-correlation energy density. The well-known Levy-Perdew virial relation for exchange potentials can be viewed as an analytical line integral along the electron-number-conserving uniform density scaling path. Energies obtained from model exchange-correlation potentials should be interpreted with caution because the reconstructed density functional is unique (up to a gauge transformation) only if the model Kohn-Sham potential is a functional derivative.

Chemical hardness and the discontinuity of the Kohn-Sham exchange-correlation potential

Chemical hardness, identified as the difference between the vertical first ionization potential I and the vertical electron affinity A, is analyzed in the context of the ionization theorems to derive expressions for its evaluation at different levels of approximation that arise as a direct consequence of the derivative discontinuity of the exchange-correlation potential. The quantities involved in these expressions incorporate indirectly the effects of the discontinuity, but their values may be calculated with any functional of the local density approximation, generalized gradient approximation, or optimized effective potential type, with or without derivative discontinuity, and with or without the correct asymptotic behavior. By comparison with the vertical energy difference values of I and A, which requires the calculation of the N-, (N-1)-, and (N+1)-electron systems, it is found, for a set of 14 closed shell molecules, that the difference between the eigenvalues of the highest occupied molecular orbitals of the N- and (N+1)-electron systems leads to rather accurate values, when the correct asymptotic behavior is incorporated, and that a second-order one-body perturbation approach that only requires information from the N-electron system leads to reasonable values.

Localized exchange-correlation potential from second-order self-energy for accurate Kohn-Sham energy gap

A local Kohn-Sham (KS) exchange-correlation potential is derived by localizing the second-order self-energy operator, using approximations to the linear response Sham-Schlüter equation. Thanks to the use of the resolution-of-identity technique for the calculation of the self-energy matrix elements, the method is very efficient and can be applied to large systems. The authors investigate the KS energy gaps and lowest excitation energies of atoms and small- and medium-size molecules. Reference KS energy gaps (from accurate densities) of atoms and small molecules can be reproduced with great accuracy. For larger systems they found that the KS energy gap is smaller than the one obtained from the local-density approximation, showing the importance of an ab initio correlation in the Kohn-Sham potential.

Electrostatic exchange-correlation charge density in Be and Ne: quantal density functional theoretic analysis

Using classical electrostatics, the total effective integrated charge-density function is calculated for Be and Ne using the multiplicative potentials derived from (1) Hartree and (2) Hartree–Fock approximation to quantal density functional theory (3)exchange-only optimized effective potential and (4) Kohn–Sham exchange-correlation potential using the quantum Monte Carlo density. The evolution of effective integrated charge-density function for these atoms is examined as the electron correlation is built up stepwise from its absence to the stage of its near complete presence. These results provide a deeper understanding of the Kohn–Sham exchange-correlation potential in terms the correspondingly defined integrated charge-density functions based on the Poisson equation.

The self-consistent implementation of exchange-correlation functionals depending on the local kinetic energy density

The self-consistent derivation and implementation of the true Kohn–Sham exchange-correlation potential (local, multiplicative and common for all orbitals) corresponding to a meta-GGA functional depending on the local kinetic energy density is reported. Our approach is based on the optimized effective potential (OEP) method. The method is compared to the less rigorous method of taking functional derivatives with respect to the molecular orbitals. The notable differences between these two approaches are illustrated by calculating nuclear shielding constants for 17 small main-group molecules.