Effective Kohn-Sham Potential Research Articles

Machine learning of kinetic energy densities with target and feature smoothing: Better results with fewer training data.

Machine learning (ML) of kinetic energy functionals (KEFs), in particular kinetic energy density (KED) functionals, is a promising way to construct KEFs for orbital-free density functional theory (DFT). Neural networks and kernel methods including Gaussian process regression (GPR) have been used to learn Kohn-Sham (KS) KED from density-based descriptors derived from KS DFT calculations. The descriptors are typically expressed as functions of different powers and derivatives of the electron density. This can generate large and extremely unevenly distributed datasets, which complicates effective application of ML techniques. Very uneven data distributions require many training datapoints, can cause overfitting, and can ultimately lower the quality of an ML KED model. We show that one can produce more accurate ML models from fewer data by working with smoothed density-dependent variables and KED. Smoothing palliates the issue of very uneven data distributions and associated difficulties of sampling while retaining enough spatial structure necessary for working within the paradigm of KEDF. We use GPR as a function of smoothed terms of the fourth order gradient expansion and KS effective potential and obtain accurate and stable (with respect to different random choices of training points) kinetic energy models for Al, Mg, and Si simultaneously from as few as 2000 samples (about 0.3% of the total KS DFT data). In particular, accuracies on the order of 1% in a measure of the quality of energy-volume dependence B'=EV0-ΔV-2EV0+E(V0+ΔV)ΔV/V02 (where V0 is the equilibrium volume and ΔV is a deviation from it) are obtained simultaneously for all three materials.

Complete solution to the inverse Kohn-Sham problem: From the density to the energy

A complete solution to the inverse problem of Kohn-Sham (KS) density functional theory is proposed. Our method consists of two steps. First, the effective KS potential is determined from the ground-state density of a given system. Then, the knowledge of the potentials along a path in the space of densities is exploited in a line integration formula to determine numerically the KS energy of that system. A possible choice for the density path is proposed. A benchmark in the case of a simplified yet realistic nuclear system is shown to be successful, so the method seems promising for future applications.

Self-interaction-corrected Kohn-Sham effective potentials using the density-consistent effective potential method.

Density functional theory (DFT) and beyond-DFT methods are often used in combination with photoelectron spectroscopy to obtain physical insights into the electronic structure of molecules and solids. The Kohn-Sham eigenvalues are not electron removal energies except for the highest occupied orbital. The eigenvalues of the highest occupied molecular orbitals often underestimate the electron removal or ionization energies due to the self-interaction (SI) errors in approximate density functionals. In this work, we adapt and implement the density-consistent effective potential method of Kohut, Ryabinkin, and Staroverov [J. Chem. Phys. 140, 18A535 (2014)] to obtain SI-corrected local effective potentials from the SI-corrected Fermi-Löwdin orbitals and density in the Fermi-Löwdin orbital self-interaction correction scheme. The implementation is used to obtain the density of states (photoelectron spectra) and HOMO-LUMO gaps for a set of molecules and polyacenes. Good agreement with experimental values is obtained compared to a range of SI uncorrected density functional approximations.

Reduction of Hartree-Fock Wavefunctions to Kohn-Sham Effective Potentials Using Multiresolution Analysis.

We present a highly accurate numerical implementation for computing the Kohn-Sham effective potentials for molecules based on a Hartree-Fock wavefunction and density, following the RKS approach of Staroverov and co-workers [ J. Chem. Phys. 2014, 140, 18A535]. Potentials and orbitals are represented in a multiresolution wavelet basis, avoiding basis set incompleteness-related issues. Together with the RKS method, the often occurring problems of oscillating potentials are removed. The MRA implementation of the RKS method allows the generation of molecular Kohn-Sham potentials of benchmark quality. Numerical data for atoms up to Kr and a number of molecules are given, with a special emphasis on the role of nodal planes in the calculations, as showcased in HCN and benzene.

Potential Functional Embedding Theory with an Improved Kohn-Sham Inversion Algorithm.

Potential functional embedding theory (PFET) is a rigorous theory that can yield a unique, self-consistent embedding potential shared by different subsystems treated at different levels of theory. Application of PFET has been limited by the time-consuming and sometimes unstable optimized effective potential (OEP) procedure. Here, we improve the performance of PFET by replacing the OEP algorithm with a new method to reconstruct the effective Kohn-Sham (KS) potential. We propose a direct, efficient KS inversion algorithm to solve for the effective KS potential and then employ the resulting algorithm in PFET. We benchmark our KS inversion algorithm against the recently reported modified Ryabinkin-Kohut-Staroverov (mRKS) procedure. Numerical examples show that, with sufficiently large basis sets, our KS inversion algorithm generates almost as accurate results as the mRKS procedure does, except in the vicinity of atomic nuclei, and that it requires less computational time. Three types of chemical interactions then were tested using the new KS inversion algorithm in PFET; the energetics computed from the updated formalism compare well to benchmarks.

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.

Connections between variation principles at the interface of wave-function and density-functional theories.

A recently proposed variation principle [N. I. Gidopoulos, Phys. Rev. A 83, 040502(R) (2011)] for the determination of Kohn-Sham effective potentials is examined and extended to arbitrary electron-interaction strengths and to mixed states. Comparisons are drawn with Lieb's convex-conjugate functional, which allows for the determination of a potential associated with a given electron density by maximization, yielding the Kohn-Sham potential for a non-interacting system. The mathematical structure of the two functionals is shown to be intrinsically related; the variation principle put forward by Gidopoulos may be expressed in terms of the Lieb functional. The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework.

Do fractionally incremented nuclear charges improve time-dependent density functional theory excitation energies as reliably as fractional orbital populations?

Gaiduk et al. (Phys Rev Lett 108:253005, 2012) showed that one can improve local, semilocal, and hybrid approximations to the Kohn–Sham effective potentials of atoms and molecules by removing a system-independent fraction of electron charge from the highest occupied molecular orbital (HOMO); if the corrected Kohn–Sham potential is used for adiabatic linear-response time-dependent density functional theory (TDDFT) calculations, accurate Rydberg excitation energies are obtained. One may ask whether the same effect could also be achieved by fractionally increasing the positive charges of the nuclei. We investigate this question and find that a small increase in nuclear charges can indeed substantially reduce errors in TDDFT Rydberg excitation energies. However, the optimal magnitude of the charge increase is system-dependent. In addition, the procedure is ambiguous for molecules, where one has to decide how to distribute the additional charge among individual nuclei. These two drawbacks of the fractional nuclear charge method make it disadvantageous compared to the HOMO depopulation technique.

Nonadditive kinetic potentials from inverted Kohn–Sham problem

AbstractThe nonadditive kinetic potential is a key element in density‐dependent embedding methods. The correspondence between the ground‐state density and the total effective Kohn–Sham potential provides the basis for various methods to construct the nonadditive kinetic potential for any pair of electron densities. Several research groups used numerical or analytical inversion procedures to explore this strategy which overcomes the failures of known explicit density functional approximations. The numerical inversions, however, apply additional approximations/simplifications. The relations known for the exact quantities cannot be assumed to hold for quantities obtained in numerical inversions. The exact relations are discussed with special emphasis on such issues as: the admissibility of the densities for which the potential is constructed, the choice of densities to be used as independent variables, self‐consistency between the potentials and observables calculated using the embedded wavefunction, and so forth. The review focuses on how these issues are treated in practice. The review is supplemented with the analysis of the inverted potentials for weakly overlapping pairs of electron densities—the case not studied previously.

On augmented Kohn–Sham potential for energy as a simple sum of orbital energies

ABSTRACTWe have recently observed that the ground-state energy may be obtained directly as a simple sum of augmented Kohn–Sham orbital energies, where it was ascertained that the corresponding one-body shifted Kohn–Sham effective potential has appealing features. With this in mind, eigenvalue and virial constraints are deduced for approximating this potential.

Fragment-based treatment of delocalization and static correlation errors in density-functional theory.

One of the most important open challenges in modern Kohn-Sham (KS) density-functional theory (DFT) is the correct treatment of systems involving fractional electron charges and spins. Approximate exchange-correlation functionals struggle with such systems, leading to pervasive delocalization and static correlation errors. We demonstrate how these errors, which plague density-functional calculations of bond-stretching processes, can be avoided by employing the alternative framework of partition density-functional theory (PDFT) even using the local density approximation for the fragments. Our method is illustrated with explicit calculations on simple systems exhibiting delocalization and static-correlation errors, stretched H2 (+), H2, He2 (+), Li2 (+), and Li2. In all these cases, our method leads to greatly improved dissociation-energy curves. The effective KS potential corresponding to our self-consistent solutions displays key features around the bond midpoint; these are known to be present in the exact KS potential, but are absent from most approximate KS potentials and are essential for the correct description of electron dynamics.

Reduction of Electronic Wave Functions to Kohn-Sham Effective Potentials.

A method for calculating the Kohn-Sham exchange-correlation potential v(XC)(r) from a given electronic wave function is devised and implemented. It requires on input one- and two-electron reduced density matrices and involves construction of the generalized Fock matrix. The method is free from numerical limitations and basis-set artifacts of conventional schemes for constructing v(XC)(r) in which the potential is recovered from a given electron density, and is simpler than various many-body techniques. The chief significance of this development is that it allows one to directly probe the functional derivative of the true exchange-correlation energy functional and to rigorously test and improve various density-functional approximations.

Stability conditions for exact-exchange Kohn-Sham methods and their relation to correlation energies from the adiabatic-connection fluctuation-dissipation theorem.

The occurrence of instabilities, in particular singlet-triplet and singlet-singlet instabilities, in the exact-exchange (EXX) Kohn-Sham method is investigated. Hessian matrices of the EXX electronic energy with respect to the expansion coefficients of the EXX effective Kohn-Sham potential in an auxiliary basis set are derived. The eigenvalues of these Hessian matrices determine whether or not instabilities are present. Similar as in the corresponding Hartree-Fock case instabilities in the EXX method are related to symmetry breaking of the Hamiltonian operator for the EXX orbitals. In the EXX methods symmetry breaking can easily be visualized by displaying the local multiplicative exchange potential. Examples (N2, O2, and the polyyne C10H2) for instabilities and symmetry breaking are discussed. The relation of the stability conditions for EXX methods to approaches calculating the Kohn-Sham correlation energy via the adiabatic-connection fluctuation-dissipation (ACFD) theorem is discussed. The existence or nonexistence of singlet-singlet instabilities in an EXX calculation is shown to indicate whether or not the frequency-integration in the evaluation of the correlation energy is singular in the EXX-ACFD method. This method calculates the Kohn-Sham correlation energy through the ACFD theorem theorem employing besides the Coulomb kernel also the full frequency-dependent exchange kernel and yields highly accurate electronic energies. For the case of singular frequency-integrands in the EXX-ACFD method a regularization is suggested. Finally, we present examples of molecular systems for which the self-consistent field procedure of the EXX as well as the Hartree-Fock method can converge to more than one local minimum depending on the initial conditions.

Ground-state energy as a simple sum of orbital energies in Kohn-Sham theory: a shift in perspective through a shift in potential.

It is observed that the exact interacting ground-state electronic energy of interest may be obtained directly, in principle, as a simple sum of orbital energies when a universal density-dependent term is added to w([ρ];r), the familiar Hartree plus exchange-correlation component in the Kohn-Sham effective potential. The resultant shifted potential, w[over ¯]([ρ];r), actually changes less on average than w([ρ];r) when the density changes, including the fact that w[over ¯]([ρ];r) does not undergo a discontinuity when the number of electrons increases through an integer. Thus, the approximation of w[over ¯]([ρ];r) represents an alternative direct approach for the approximation of the ground-state energy and density.

Removal of Basis-Set Artifacts in Kohn-Sham Potentials Recovered from Electron Densities.

Kohn-Sham effective potentials recovered from Gaussian-basis-set electron densities exhibit large oscillations and asymptotic divergences not found in exact potentials and in functional derivatives of approximate density functionals. We show that the detailed structure of these oscillations and divergences is almost exclusively determined by the basis set in terms of which the reference density is expressed, and is almost independent of the density-functional or wave function method used for computing the density. Based on this observation, we propose a smoothening scheme in which most basis-set artifacts in a Kohn-Sham potential recovered from a given density are removed by subtracting the oscillation profile of the exchange-only local-density approximation potential computed in the same basis set as the reference density. The correction allows one to obtain smooth Kohn-Sham potentials from electron densities even for small Gaussian basis sets and greatly reduces discrepancies between the original (input) density and the density obtained from the reconstructed potential.

Determination of Kohn–Sham effective potentials from electron densities using the differential virial theorem

We present an accurate method for constructing the Kohn-Sham effective potential corresponding to a given electron density in one-dimensional and spherically symmetric systems. The method is based on the differential virial theorem--an exact relation between the effective potential, the electron density, and the kinetic energy density. A distinctive feature of the proposed technique is that it employs a size-consistent bosonic reference potential to ensure the correct asymptotic behavior of the resulting Kohn-Sham potential. We describe a practical implementation of our method and use it to obtain high-quality exchange-correlation and correlation potentials of the neon and argon atoms from ab initio densities generated in large Slater- and Gaussian-type basis sets.

Structural and electronic properties of Na/Cu(111) at different coverages by first principles

Nanostructures are presently enjoying an increasing interest in the field of materials science. In particular, importance is given to ordered monolayers prepared by deposition of atoms on a crystalline surface. The growth of these superlattices can be controlled so as to obtain an ordered structure by means of the lateral interaction of adatoms lying on the metal surface. The objective of our study is to investigate the structural and electronic properties using DFT total-energy calculations; we employ a jellium-like model to describe the substrate but we also take into account the presence of discrete surface states that are known to affect the lateral interaction. Our treatment of the substrate is based on the model proposed by E.V. Chulkov et al. [Surf. Sci. 437, 330 (1999)]; in this model one constructs a mono-dimensional potential so as to reproduce some important electronic properties of the metal surface, such as i) the energy gap in the projected bulk band-structure and ii) the energy position of surface states. We put into practice Chulkov potential implementing into an existing plane-waves code (ABINIT, URL http://www.abinit.org) an ionic potential, so as to obtain a self-consistent Kohn-Sham effective potential which corresponds to the Chulkov one. Using this effective potential in a fully three-dimensional code we are able to study the adsorption process and the interaction between adsorbates. We illustrate some details of our implementation of the Chulkov model and we present our results about the simple system of Na adatoms on a Cu(111) surface for different coverages. In particular, we compare electronic properties and adsorption energies with those obtained within a standard jellium model substrate and with those obtained for Na adsorption on a realistic Cu(111) surface.

Self-healing diffusion quantum Monte Carlo algorithms: Direct reduction of the fermion sign error in electronic structure calculations

We develop a formalism and present an algorithm for optimization of the trial wave-function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project-out a multi-determinant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the non-interacting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave-function with better nodal structure and (b) we argue that the noise in the fixed-node wave-function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust towards the ones of the exact many-body ground state in a simulated annealing-like process. We propose a method to improve both single determinant and multi-determinant expansions of the trial wave-function. We test the method in a model system where benchmark configuration interaction calculations can be performed. Comparing the DMC calculations with the exact solutions, we find that the trial wave-function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave-functions orthogonal to the exact ground state. In the optimization process we find an optimal non-interacting nodal potential of density-functional-like form whose existence was predicted earlier[Phys.Rev. B {\bf 77}, 245110 (2008)]. We obtain the exact Kohn-Sham effective potential from the DMC data.

A Kohn–Sham system at zero temperature

A one-dimensional Kohn–Sham system for spin particles is considered which effectively describes semiconductor nanostructures, and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn–Sham potential and obtain W1,2-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow us to apply Schauder's fixed point theorem. In the case of vanishing exchange–correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.

A Determination Method of the Work function using the Slab Model with a First-Principles Electronic Structure Calculation

We have reinvestigated the method to determine the work function of metals by the first-principles electronic structure calculation using the plane-wave expansion method with pseudo-potentials. Changing width of a vacuum layer of a slab model, we obtained the Fermi level measured from the effective Kohn-Sham potential at the center of the vacuum layer. By extrapolating the values of the Fermi level to the infinite vacuum-layer limit, the Fermi energy is determined. We propose a fitting function for the extrapolation. This function works well for determination of the highest occupied level and the lowest unoccupied level of materials with a surface relative to the vacuum level in the Kohn-Sham scheme. The obtained work functions for [100], [110] and [111] surfaces of three noble metals, Rh, Pd and Pt are comparable to the known theoretical estimation as well as the experimental values. [DOI: 10.1380/ejssnt.2008.103]