Exchange and correlation energy in density functional theory

Several different versions of density functional theory (DFT) that satisfy Hohenberg–Kohn theorems are characterized by different definitions of a reference or model state determined by an N‐electron ground state. A common formalism is developed in which exact Kohn–Sham equations are derived for standard Kohn–Sham theory, for reference‐state density functional theory, and for unrestricted Hartree–Fock (UHF) theory considered as an exactly soluble model Hohenberg–Kohn theory. A natural definition of exchange and correlation energy functionals is shown to be valid for all such theories. An easily computed necessary condition for the locality of exchange and correlation potentials is derived. While it is shown that in the UHF model of DFT the optimized effective potential (OEP) exchange satisfies this condition by construction, the derivation shows that this condition is not, in general, sufficient to define an exact local exchange potential. It serves as a test to eliminate proposed local potentials that are not exact for ground states. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 521–525, 2000     

Self-consistent field calculations using two-body density functionals for correlation energy component: I. Atomic systems

Self-consistent field calculations are done using two-body density functionals for the correlation energy. The corresponding functional derivatives are obtained and used in pseudo-eigenvalue equations analogous to the Kohn–Sham ones. The examples studied include atomic systems from He to Ar. The values obtained for ionization potentials, electron affinities, dipole polarizabilities, and virial ratios from these calculations are given, and the effect of exchange is addressed. The results obtained are in good agreement with experimental values, and are of the same quality as those given by accurate exchange-correlation functionals. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1887–1898, 1998     

Correlation factor approach to the correlation energy functional

A global survey of the correlation factor energy functionals and its application to atomic and molecular properties is made. Its performances are compared with those of the density functional theory (DFT) correlation energy functionals, and some interesting conclusions from previous publications are reinforced here; namely, after removing the one-Slater-determinant hypothesis from the Kohn–Sham method, all DFT correlation functionals are able to provide reasonable results in any circumstance, with an additional restriction, for systems having a quasi-degenerate wave function, the DFT correlation functionals must depend explicitly on the on-top density. Acknowledgement.This work has been done under the support of the Spain DGICYT, project n⁰ BQU2001-0883.     

An analysis of nonlocal density functionals in chemical bonding

In this work, we carry out an analysis of the gradient-corrected density functionals in molecules that are used in the Kohn–Sham density functional approach. We concentrate on the special features of the exchange and correlation energy densities and exchange and correlation potentials in the bond region. By comparing to the exact Kohn–Sham potential, it is shown that the gradient-corrected potentials build in the required peak in the bond midplane, but not completely correctly. The gradient-corrected potentials also exhibit wrong asymptotic behavior. Contributions from different regions of space (notably bond and outer regions) to nonlocal bonding energy contributions are investigated by integrating the exchange and correlation energy densities in various spatial regions. This provides an explanation of why the gradient corrections reduce the local density approximation (LDA ) overbinding of molecules. It explains the success of the presently used nonlocal corrections, although it is possible that there is a cancellation of errors, too much repulsion being derived from the bond region and too little from the outer region. © John Wiley & Sons, Inc.     

Density functional theory for strongly-interacting electrons: perspectives for physics and chemistry

Improving the accuracy and thus broadening the applicability of electronic density functional theory (DFT) is crucial to many research areas, from material science, to theoretical chemistry, biophysics and biochemistry. In the last three years, the mathematical structure of the strong-interaction limit of density functional theory has been uncovered, and exact information on this limit has started to become available. The aim of this paper is to give a perspective on how this new piece of exact information can be used to treat situations that are problematic for standard Kohn-Sham DFT. One way to use the strong-interaction limit, more relevant for solid-state physical devices, is to define a new framework to do practical, non-conventional, DFT calculations in which a strong-interacting reference system is used instead of the traditional non-interacting one of Kohn and Sham. Another way to proceed, more related to chemical applications, is to include the exact treatment of the strong-interaction limit into approximate exchange-correlation energy density functionals in order to describe difficult situations such as the breaking of the chemical bond.     

Comparing ab initio density-functional and wave function theories: the impact of correlation on the electronic density and the role of the correlation potential

The framework of ab initio density-functional theory (DFT) has been introduced as a way to provide a seamless connection between the Kohn-Sham (KS) formulation of DFT and wave-function based ab initio approaches [R. J. Bartlett, I. Grabowski, S. Hirata, and S. Ivanov, J. Chem. Phys. 122, 034104 (2005)]. Recently, an analysis of the impact of dynamical correlation effects on the density of the neon atom was presented [K. Jankowski, K. Nowakowski, I. Grabowski, and J. Wasilewski, J. Chem. Phys. 130, 164102 (2009)], contrasting the behaviour for a variety of standard density functionals with that of ab initio approaches based on second-order M?ller-Plesset (MP2) and coupled cluster theories at the singles-doubles (CCSD) and singles-doubles perturbative triples [CCSD(T)] levels. In the present work, we consider ab initio density functionals based on second-order many-body perturbation theory and coupled cluster perturbation theory in a similar manner, for a range of small atomic and molecular systems. For comparison, we also consider results obtained from MP2, CCSD, and CCSD(T) calculations. In addition to this density based analysis, we determine the KS correlation potentials corresponding to these densities and compare them with those obtained for a range of ab initio density functionals via the optimized effective potential method. The correlation energies, densities, and potentials calculated using ab initio DFT display a similar systematic behaviour to those derived from electronic densities calculated using ab initio wave function theories. In contrast, typical explicit density functionals for the correlation energy, such as VWN5 and LYP, do not show behaviour consistent with this picture of dynamical correlation, although they may provide some degree of correction for already erroneous explicitly density-dependent exchange-only functionals. The results presented here using orbital dependent ab initio density functionals show that they provide a treatment of exchange and correlation contributions within the KS framework that is more consistent with traditional ab initio wave function based methods.     

Left–right and dynamic correlation

Within density functional theory, it is natural to separate the correlation energy into two parts: left–right correlation and dynamic correlation. Left–right correlation arises from the exchange part of functionals, and dynamic correlation arises from the correlation part of functionals. We examine the nature of these correlation energies as molecules are distorted. We observe that such a natural separation is not possible using the methods of quantum chemistry. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Coupled‐cluster reaction barriers of : An application of the coupled‐cluster//Kohn–Sham density functional theory model chemistry

In this work, we report a theoretical investigation concerning the use of the popular coupled‐cluster//Kohn‐Sham density functional theory (CC//KS‐DFT) model chemistry, here applied to study the entrance channel of the reaction, namely by comparing CC//KS‐DFT calculations with KS‐DFT, MRPT2//CASSCF, and CC//CASSCF results from our previous investigations. This was done by performing single point energy calculations employing several coupled cluster methods and using KS‐DFT geometries optimized with six different functionals, while conducting a detailed analysis of the barrier heights and topological features of the curves and surfaces here obtained. The quality of this model chemistry is critically discussed in the context of the title reaction and also in a wider context. © 2013 Wiley Periodicals, Inc.     

Quantum hydrodynamic model of density functional theory

In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sham equation is a non-linear Schrödinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following Cai et al. (J Math Phys 53:103503, 2012). The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn–Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This work provides a new possible way to solve DFT problems.     

The importance of Colle–Salvetti for computational density functional theory

The purpose of this presentation is to show the importance of the Colle–Salvetti (Theor Chim Acta 37:329, 1975) paper in the development of modern computational density functional theory. To do this we cover the following topics (1) the Bright Wilson understanding (2) the Kohn–Sham equations (3) local density exchange (4) the exchange-hole (5) generalised gradient approximation for exchange (Becke and Cohen) (6) left–right correlation and dynamic correlation (7) the development of the Lee–Yang–Parr dynamic correlation functional from the Colle–Salvetti paper (8) the early success of GGA DFT. Finally we observe that the the BLYP and OLYP exchange-correlation functionals are not semi-empirical; this may explain their great success.     

Comparative performance of exchange and correlation density functionals in determining intermolecular interaction potentials of the methane dimer

We have calculated the interaction potentials of the methane dimer for the minimum-energy D(3d) conformation using the density functional theory (DFT) with 90 density functionals chosen from the combinations of nine exchange and 10 correlation functionals. Several hybrid functionals are also considered. While the performance of an exchange functional is related to the large reduced density gradient of the exchange enhancement factor, the correlation energy is determined by the low-density behavior of a correlation enhancement factor. Our calculations demonstrate that the correlation counterpart plays an equally important role as the exchange functional in determining the van der Waals interactions of the methane dimer. These observations can be utilized to better understand the seemingly unsystematic DFT interaction potentials for weakly bound systems.     

DFT performance for the hole transfer parameters in DNA π stacks

Recently, we showed that unoccupied Kohn‐Sham (KS) orbitals stemming from DFT calculations of a neutral system can be used to derive accurate estimates of the free energy and electronic couplings for excess electron transfer in DNA (Félix and Voityuk, J Phys Chem A 2008, 112, 9043). In this article, we consider the propagation of radical cation states (hole transfer) through DNA π‐stacks and compare the performance of different exchange‐correlation functionals to estimate the hole transfer (HT) parameters. Two different approaches are used: (1) calculations that use occupied KS orbitals of neutral π stacks of nucleobases, and (2) the time‐dependent DFT method which is applied to the radical cation states of these stacks. Comparison of the calculated parameters with the reference data suggests that the best results are provided by the KS scheme with hybrid functionals (B3LYP, PBE0, and BH&HLYP). The TD DFT approach gives significantly less accurate values of the HT parameters. In agreement with high‐level ab initio results, the KS scheme predicts that the hole in π stacks is confined to a single nucleobase; in contrast, the spin‐unrestricted DFT method considerably overestimates the hole delocalization in the radical cations. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Performance of range‐separated hybrid exchange–correlation functionals for the calculation of magnetic exchange coupling constants of organic diradicals

The prediction of magnetic behavior is important for the design of new magnetic materials. Kohn–Sham density functional theory is popular for this purpose, although one should be careful about choosing the right exchange–correlation functional. Here, we perform a statistical analysis to test different range‐separated hybrid density functionals for the calculation of magnetic exchange coupling constants J of fourteen organic diradicals. Our analysis suggests that in absolute terms the MN12SX functional performs best among the series of twelve functionals studied here (including the popular B3LYP), followed by N12SX functionals along with Scuseria's HSE series of functionals. LC‐ PBE was found to be the least accurate, which is in contrast with its good performance for calculating J for transition metal complexes. The HSE family of functionals and B3LYP are the only functionals to reproduce the qualitative trends of the coupling constants correctly for the ferromagnetically coupled diradicals under study. © 2017 Wiley Periodicals, Inc.     

Quadrupole and octupole Cauchy moments of the atoms through argon

Quadrupole and octupole Cauchy moments of the atoms through argon are calculated using the hydrodynamic formulation of time-dependent Kohn–Sham theory. The exchange-correlation energy density functional is approximated by a gradient expansion for atoms that has an explicit dependence upon the number of electrons. The first-order corrections to the Kohn–Sham amplitudes and phases are found by seeking variational solutions of the derived sequential set of functionals. The trial functions employed contain both linear and nonlinear variational parameters and are thus flexible enough to provide rapid convergence to the multipole polarizabilities. The resulting Cauchy moments provide information that allows the calculation of various properties that result from the linear interaction of atoms with a time-varying electric field. © 1994 John Wiley & Sons, Inc.     

Acceleration of catalyst discovery with easy,fast, and reproducible computational alchemy

The expense of quantum chemistry calculations significantly hinders the search for novel catalysts. Here, we provide a tutorial for using an easy and highly cost‐efficient calculation scheme, called alchemical perturbation density functional theory (APDFT), for rapid predictions of binding energies of reaction intermediates and reaction barrier heights based on the Kohn‐Sham density functional theory (DFT) reference data. We outline standard procedures used in computational catalysis applications, explain how computational alchemy calculations can be carried out for those applications, and then present benchmarking studies of binding energy and barrier height predictions. Using a single OH binding energy on the Pt(111) surface as a reference case, we use computational alchemy to predict binding energies of 32 variations of this system with a mean unsigned error of less than 0.05 eV relative to single‐point DFT calculations. Using a single nudged elastic band calculation for CH₄ dehydrogenation on Pt(111) as a reference case, we generate 32 new pathways with barrier heights having mean unsigned errors of less than 0.3 eV relative to single‐point DFT calculations. Notably, this easy APDFT scheme brings no appreciable computational cost once reference calculations are performed, and this shows that simple applications of computational alchemy can significantly impact DFT‐driven explorations for catalysts. To accelerate computational catalysis discovery and ensure computational reproducibility, we also include Python modules that allow users to perform their own computational alchemy calculations.     

Examination of several density functionals in numerical Kohn–Sham calculations for atoms

We studied several exchange‐only and exchange–correlation energy density functionals in numerical, i.e., basis‐set‐free, nonrelativistic Kohn–Sham calculations for closed‐shell ¹S states of atoms and atomic ions with N electrons, where 2≤N≤120. Accurate total energies are presented to serve as reference data for algebraic approaches, as do the numerical Hartree–Fock results, which are also provided. Gradient‐corrected exchange‐only functionals considerably improve the total energies obtained from the usual local density approximation, when compared to the Hartree–Fock results. Such an improvement due to gradient corrections is not seen in general for highest orbital energies, neither for exchange‐only results (to be compared with Hartree–Fock results), nor for exchange–correlation results (to be compared with experimental ionization energies). © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 227–241, 2001     

About the compatibility between ansatzes and constraints for a local formulation of orbital‐free density functional theory

Functional properties that are exact for the Hohenberg–Kohn functional may turn into mutually exclusive constraints at a given level of ansatz. This is exemplarily shown for the local density approximation. Nevertheless, it is possible to reach exactly the Kohn–Sham data from an orbital‐free density functional framework based on simple one‐point functionals by starting from the Levy–Perdew–Sahni formulation. The energy value is obtained from the density‐potential pair, and therefore does not refer to the functional dependence of the potential expression. Consequently, the potential expression can be obtained from any suitable model and is not required to follow proper scaling behavior.     

Some questions on the exchange contribution to the effective potential of the Kohn–Sham theory

The current success of Density Functional Theory applications hinges upon the availability of explicitly density-dependent functionals to self-consistently solve a set of one-electron equations, the Kohn–Sham (KS) equations, which determine the occupied orbitals and its associated electronic density. In KS theory, a local exchange potential is proposed as part of an effective potential. This potential is compared to the exchange operator of the Hartree–Fock theory, which is of a non-local nature. The present paper discusses the variational framework of the KS equations, and the equivalence between both exchange potentials within a correlation-free theory. The common difficulties of current local exchange functionals to correctly simulate the non-locality of the exchange energy density in chemical systems are also analyzed and explained through an exactly solvable model. We give then numerical arguments and conclude by analyzing the performance of various commonly used approximations to exchange functionals.     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.