Kohn-Sham Exchange Research Articles

Jump discontinuities of finite-basis-set exchange–correlation potentials at atomic nuclei

The kinetic energy density of electrons and the gradient of the electron density have pronounced jump discontinuities at the positions of the atomic nuclei in molecules. Certain exact relations then imply that molecular Kohn–Sham exchange–correlation potentials may also be discontinuous at atomic nuclei. Here, we confirm that exchange–correlation potentials derived from Hartree–Fock and correlated wavefunctions within Slater-type basis sets do exhibit such discontinuities. Despite their persistence even in large basis sets, these discontinuities are almost certainly artifacts of basis set finiteness and are expected to disappear in the basis-set limit. The findings imply that imposing electron–nucleus cusp conditions in spherically averaged form may not always be appropriate.

THEORETICAL STUDYING SPECTRAL CHARACTERISTICS OF Tm ATOM WITHIN OPTIMIZED RELATIVISTIC MANY-BODY THEORY

Theoretical studying Rydberg spectrum of complex lanthanide atom of Tm has been performed within the relativistic many-body perturbation theory and generalized relativistic energy approach. The zeroth approximation of the relativistic perturbation theory is provided by the optimized Dirac-Kohn-Sham-Breit ones. Optimization has been fulfilled by means of introduction of the special gauge parameter to the Fock and Kohn-Sham exchange potentials and further minimization of the gauge-non-invariant contributions into radiation width of atomic levels with using relativistic orbital sets, generated by the corresponding zeroth approximation Hamiltonian. The calculated energies and widths of autoionization resonant states 4f--1j 6s(J12)nsnp[J] of the Tm atom with n=25-35 are presented and compared with known theoretical results, received within other approaches. Two main types of the Rydberg autoionization resonances decay, namely, the classic Beutler-Fano decay channel and a new Ivanov et al reorientation-type decay channel ar4e studied. It is noted that, for example, for autoionization resonant states with the considered values of n, the decay of resonances occurs along both channels, and it is extremely difficult to understand a priori which of them is dominant.

Numerically stable inversion approach to construct Kohn-Sham potentials for given electron densities within a Gaussian basis set framework.

We present a Kohn-Sham (KS) inversion approach to construct KS exchange-correlation potentials corresponding to given electron densities. This method is based on an iterative procedure using linear response to update potentials. All involved quantities, i.e., orbitals, potentials, and response functions, are represented by Gaussian basis functions. In contrast to previous KS inversion methods relying on Gaussian basis sets, the method presented here is numerically stable even for standard basis sets from basis set libraries due to a preprocessing of the auxiliary basis used to represent an exchange-correlation charge density that generates the exchange-correlation potential. The new KS inversion method is applied to reference densities of various atoms and molecules obtained by full configuration interaction or CCSD(T) (coupled cluster singles doubles perturbative triples). The considered examples encompass cases known to be difficult, such as stretched hydrogen or lithium hydride molecules or the beryllium isoelectronic series. For the stretched hydrogen molecule, potentials of benchmark quality are obtained by employing large basis sets. For the carbon monoxide molecule, we show that the correlation potential from the random phase approximation (RPA) is in excellent qualitative and quantitative agreement with the correlation potential from the KS inversion of a CCSD(T) reference density. This indicates that RPA correlation potentials, in contrast to those from semi-local density-functionals, resemble the exact correlation potential. Besides providing exchange-correlation potentials for benchmark purposes, the proposed KS inversion method may be used in density-partition-based quantum embedding and in subsystem density-functional methods because it combines numerical stability with computational efficiency.

Semilocal approximations to the Kohn-Sham exchange potential as applied to a metal surface

Several semilocal exchange potentials usually employed in the framework of density-functional theory (DFT) are tested and compared with their exact counterpart, the exchange Optimized Effective Potential (OEP), as applied to the jellium-slab model of a metal-vacuum interface. Driven by their explicit dependence on the ground-state density, its gradient, and its kinetic-energy density, the three analyzed semilocal exchange potentials approach their respective asymptotic limits faster than in the case of the OEP, all of them having an asymptotic scaling of the form $-\alpha\,e^2/z + V_{\infty}$, with $\alpha < 1$. Here we provide the leading analytic asymptotics of the three model potentials under study, and we find that none of them exhibits the exact OEP slab asymptotics $-\;e^2/z$. While the so-called Becke-Roussel potential's leading asymptote is close to its exact OEP counterpart, the other two model potentials under study approach a material-dependent positive constant value far into the vacuum, resulting in considerably overestimated ionization potentials.

THEORETICAL STUDYING SPECTRAL CHARACTERISTICS OF Zn-LIKE IONS ON THE BASIS OF RELATIVISTIC MANY-BODY PERTURBATION THEORY

A theoretical study of the spectroscopic characteristics of Zn-like multiply charged ions is carried out within the framework of the relativistic many-body perturbation theory. The optimized Dirac-Kohn-Shem approximation was chosen as the zero approximation of the relativistic perturbation theory. Optimization has been fulfilled by means of introduction of the parameters to the Kohn-Sham exchange and correlation potentials and further minimization of the gauge-non-invariant contributions into radiation width of atomic levels with using relativistic orbital set, generated by the corresponding zeroth approximation Hamiltonian.

THEORETICAL STUDYING EXCITED STATES SPECTRUM OF THE YTTERBIUM WITHIN THE OPTIMIZED RELATIVISTIC MANY-BODY PERTURBATION THEORY

Theoretical studying spectrum of the excited states for the ytterbium atom is carried out within the relativistic many-body perturbation theory with ab initio zeroth approximation and generalized relativistic energy approach. The zeroth approximation of the relativistic perturbation theory is provided by the optimized Dirac-Kohn-Sham ones. Optimization has been fulfilled by means of introduction of the parameter to the Kohn-Sham exchange potentials and further minimization of the gauge-non-invariant contributions into radiation width of atomic levels with using relativistic orbital set, generated by the corresponding zeroth approximation Hamiltonian. The obtained theoretical data on energies E and widths W of the ytterbium excited states are compared with alternative theoretical results (the Dirac-Fock, relativistic Hartree-Fock, perturbation theories) and available experimental data. Analysis shows that the theoretical and experimental values ​​of energies are in good agreement with each other, however, the values ​​of widths differ significantly. In our opinion, this fact is explained by insufficiently accurate estimates of the radial integrals, the use of unoptimized bases, and some other approximations of the calculation.

Numerically stable optimized effective potential method with standard Gaussian basis sets.

We present a numerically stable optimized effective potential (OEP) method based on Gaussian basis sets. The key point of the approach is a sequence of preprocessing steps of the auxiliary basis set used to represent exchange or correlation potentials, the Kohn-Sham (KS) response function, and the right-hand side of the OEP equation in conjunction with a representation of exchange or correlation potentials via exchange or correlation charge densities whose electrostatic potentials generate the potentials. Due to the preprocessing, standard Gaussian basis sets from basis set libraries can be used in OEP calculations. As examples, we present numerical stable computational setups based on aux-cc-pwCVXZ basis sets with X = T, Q, 5 for the orbitals and aux-cc-pVDZ/mp2fit and aux-cc-pVTZ/mp2fit auxiliary basis sets and use them to calculate KS exchange potentials with the exact exchange-only KS method for various atoms and molecules. The resulting exchange potentials not only are numerically stable and physically reasonable but also show convergence with increasing quality of the orbital basis sets. The effect of incorporating exact conditions that the KS exchange potential has to obey is discussed. Moreover, it is briefly demonstrated that the presented approach not only works for KS exchange potentials but equally well for correlation potentials within the direct random phase approximation. Besides for OEP methods, the introduced preprocessing of auxiliary basis sets should also be beneficial in procedures to calculate back effective KS potentials from given electron densities.

Accurate effective potential for density amplitude and the corresponding Kohn–Sham exchange–correlation potential calculated from approximate wavefunctions

Over the past few years it has been pointed out that direct inversion of accurate but approximate ground state densities leads to Kohn–Sham exchange–correlation (xc) potentials that can differ significantly from the exact xc potential of a given system. On the other hand, the corresponding wavefunction based construction of exchange-correlation potential as done by Baerends et al and Staroverov et al obviates such problems and leads to potentials that are very close to the true xc potential. In this paper, we provide an understanding of why the wavefunction based approach gives the exchange–correlation potential accurately. Our understanding is based on the work of Levy, Perdew and Sahni (LPS) who gave an equation for the square root of density (density amplitude) and the expression for the associated effective potential in the terms of the corresponding wavefunction. We show that even with the use of approximate wavefunctions the LPS expression gives accurate effective and exchange–correlation potentials. Based on this we also identify the source of difference between the potentials obtained from a wavefunction and those given by the inversion of the associated density. Finally, we suggest exploring the possibility of obtaining accurate ground-state density from an approximate wavefunction for a system by making use of the LPS effective potential.

Fragment Exchange Potential for Realizing Pauli Deformation of Interfragment Interactions.

In fragment-based methods, the lack of explicit short-range exchange interactions between monomers can result in unphysical deformation in charge density. In this study, we describe a fragment exchange potential (XFP) to explicitly account for interfragmental Pauli deformation. In our implementation, a Kohn-Sham exchange potential is adopted along with the Yukawa potential. The method has been validated by comparison of the computed exchange energies using the XFP potential with results obtained from antisymmetrized fragmental orbitals on the S66×8 data set containing 528 bimolecular interactions of equilibrium and arbitrary geometries. It was also found that it is only necessary to deploy numerical grids on atoms within their van der Waals contacts, significantly reducing the small, albeit extra, computational cost. We anticipate that the XFP presented here may be applied to molecular dynamics simulations of macromolecules using a fragment-based quantum mechanical potential with improved SCF convergence and computational accuracy.

Accurate Prediction of the S1 Excitation Energy in Solvated Azobenzene Derivatives via Embedded Orbital-Tuned Bethe-Salpeter Calculations.

By employing the Bethe-Salpeter formalism coupled with a nonequilibrium embedding scheme, we demonstrate that the paradigmatic case of S1 band separation between cis and trans in azobenzene derivatives can be computed with excellent accuracy compared to experimental optical spectra. Besides embedding, we show that the choice of the Kohn-Sham exchange correlation functional for DFT is critical, despite the iterative convergence of GW quasiparticle energies. We address this by adopting an orbital-tuning approach via the global hybrid functional, PBEh, yielding an environment-consistent ionization potential. The vertical excitation energy of 20 azo molecules is predicted with a mean absolute error as low as 0.06 eV, up to three times smaller compared to standard functionals such as M06-2X and PBE0, and five times smaller compared to recent TDDFT results.

Robust and accurate hybrid random-phase-approximation methods.

A fully self-consistent hybrid dRPA (direct random phase approximation) method, named sc-H[γ]dRPA, is presented with γ = 1/3. The exchange potential of the new method contains a fraction γ of nonlocal Hartree-Fock-like exchange besides the exact local Kohn-Sham (KS) exchange potential. The sc-H[γ]dRPA method, in contrast to a straightforward self-consistent dRPA method within the KS formalism, does not suffer from convergence problems for systems with small eigenvalue gaps. Moreover, the sc-H[γ]dRPA method yields distinctively more accurate reaction, isomerization, and transition state energies than other dRPA approaches, e.g., the frequently used non-self-consistent dRPA method using orbitals and eigenvalues from a KS calculation with the exchange-correlation potential of Perdew, Burke, and Ernzerhof (PBE). The sc-H[γ]dRPA method outperforms second-order Møller-Plesset perturbation theory and coupled cluster singles doubles methods while exhibiting a more favorable scaling of computational costs with system size. A value of γ = 1/3 is shown to be a good choice also for a dRPA@PBE[γ] method, which is a non-self-consistent dRPA method using orbitals and eigenvalues from the hybrid PBE0 method with an admixture of γ = 1/3 of exact exchange instead of the 25% of the PBE0 functional. The dRPA@PBE[γ] method yields reaction, isomerization, and transition state energies that are as good as the sc-H[γ]dRPA ones but is computationally simpler and more efficient because it does not require the self-consistent construction of the dRPA correlation potential. The direct sc-H[γ]dRPA, on the other hand, in contrast to all standard density-functional methods, yields qualitatively correct correlation potentials.

Relativistic quantum average-atom model with relativistic exchange potential

We present a relativistic quantum average-atom model where bound and free electrons are treated self-consistently. We propose a relativistic exchange potential that generalizes the nonrelativistic Kohn-Sham exchange potential. The relativistic exchange potential is considered in detail. We interpolate smoothly between the nonrelativistic and ultrarelativistic limits. We neglect the correlation effect. Numerical examples are given.

ТЕОРЕТИЧНЕ ВИВЧЕННЯ СПЕКТРУ ІТЕРБІЮ НА ОСНОВІ РЕЛЯТИВІСТСЬКОЇ БАГАТОЧАСТКОВІ ТЕОРІЇ ЗБУРЕНЬ: ДВІЧІ ЗБУДЖЕНІ ВАЛЕНТНІ СТАНИ

Theoretical studying spectrum of doubly excited valence states of the ytterbium is carried out within the relativistic many-body perturbation theory and generalized relativistic energy approach. The zeroth approximation of the relativistic perturbation theory is provided by the optimized Dirac-Kohn-Sham ones. Optimization has been fulfilled by means of introduction of the parameter to the Kohn-Sham exchange potentials and further minimization of the gauge-non-invariant contributions into radiation width of atomic levels with using relativistic orbital set, generated by the corresponding zeroth approximation Hamiltonian

Revealing the field-counteracting term in the exact Kohn-Sham correlation potential

We construct the (within numerical limits) exact correlation potential of Kohn-Sham density functional theory for a one-dimensional model chain of four hydrogen atoms from a highly accurate, correlated wave function. Specifically we investigate how the correlation potential changes when an external linear electrical field is applied. This model situation is of great interest to Kohn-Sham theory because it is well known that the exact Kohn-Sham exchange potential shows ultranonlocality in this situation, with the exchange potential responding opposite to the applied external potential. Here we show that the correlation potential shows a field-counteracting term as well. However, we also find that self-consistent exact exchange captures a major part of this term, contrary to the exact exchange potential that results from the exact orbitals. The results obtained here are relevant for providing guidance in the construction of improved approximate exchange correlation functionals.

Challenges for semilocal density functionals with asymptotically nonvanishing potentials

The Becke-Johnson model potential [A. D. Becke and E. R. Johnson, J. Chem. Phys. 124, 221101 (2006)] and the potential of the AK13 functional [R. Armiento and S. K\"ummel, Phys. Rev. Lett. 111, 036402 (2013)] have been shown to mimic features of the exact Kohn-Sham exchange potential, such as step structures that are associated with shell closings and particle-number changes. A key element in the construction of these functionals is that the potential has a limiting value far outside a finite system that is a system-dependent constant rather than zero. We discuss a set of anomalous features in these functionals that are closely connected to the nonvanishing asymptotic potential. The findings constitute a formidable challenge for the future development of semilocal functionals based on the concept of a nonvanishing asymptotic constant.

Orbital nodal surfaces: Topological challenges for density functionals

Nodal surfaces of orbitals, in particular of the highest occupied one, play a special role in Kohn-Sham density-functional theory. The exact Kohn-Sham exchange potential, for example, shows a protruding ridge along such nodal surfaces, leading to the counterintuitive feature of a potential that goes to different asymptotic limits in different directions. We show here that nodal surfaces can heavily affect the potential of semilocal density-functional approximations. For the functional derivatives of the Armiento-K\"ummel (AK13) [Phys. Rev. Lett. 111, 036402 (2013)] and Becke88 [Phys. Rev. A 38, 3098 (1988)] energy functionals, i.e., the corresponding semilocal exchange potentials, as well as the Becke-Johnson [J. Chem. Phys. 124, 221101 (2006)] and van Leeuwen--Baerends (LB94) [Phys. Rev. A 49, 2421 (1994)] model potentials, we explicitly demonstrate exponential divergences in the vicinity of nodal surfaces. We further point out that many other semilocal potentials have similar features. Such divergences pose a challenge for the convergence of numerical solutions of the Kohn-Sham equations. We prove that for exchange functionals of the generalized gradient approximation (GGA) form, enforcing correct asymptotic behavior of the potential or energy density necessarily leads to irregular behavior on or near orbital nodal surfaces. We formulate constraints on the GGA exchange enhancement factor for avoiding such divergences.

Energetics of the AK13 semilocal Kohn-Sham exchange energy functional

The recent nonempirical semilocal exchange functional of Armiento and K\"ummel [Phys. Rev. Lett. 111, 036402 (2013)], AK13, incorporates a number of features reproduced by higher-order theory. The AK13 potential behaves analogously with the discontinuous jump associated with the derivative discontinuity at integer particle numbers. Recent works have established that AK13 gives a qualitatively improved orbital description compared to other semilocal methods, and reproduces a band structure closer to higher-order theory. However, its energies and energetics are inaccurate. The present work further investigates the deficiency in energetics. In addition to AK13 results, we find that applying the local-density approximation (LDA) non-self-consistently on the converged AK13 density gives very reasonable energetics with equilibrium lattice constants and bulk moduli well described across 13 systems. We also confirm that the attractive orbital features of AK13 are retained even after full structural relaxation. Hence, the deficient energetics cannot be a result of the AK13 orbitals having adversely affected the quality of the electron density compared to that of usual semilocal functionals; an improved orbital description and good energetics are not in opposition. This is also confirmed by direct calculation of the principal component of the electric field gradient. In addition, we prove that the non-self-consistent scheme is equivalent to using a single external-potential-dependent functional in an otherwise consistent, nonvariational Kohn-Sham density-functional theory (KS DFT) scheme. Furthermore, our results also demonstrate that, while an internally consistent KS functional is presently missing, non-self-consistent LDA on AK13 orbitals works as a practical nonempirical computational scheme to predict geometries, bulk moduli, while retaining the band structure features of AK13 at the computational cost of semi-local DFT.

Acoustic magnons in the long-wavelength limit: Investigating the Goldstone violation in many-body perturbation theory

Collective spin excitations in magnetic materials arise from the correlated motion of electron-hole pairs with opposite spins. The pair propagation is described by the transverse magnetic susceptibility, which we calculate within many-body perturbation theory from first principles employing the full-potential linearized augmented-plane-wave formalism. Ferromagnetic materials exhibit a spontaneously broken global rotation symmetry in spin space leading to the appearance of acoustic magnons (zero gap) in the long-wavelength limit. However, due to approximations used in the numerical scheme, the acoustic magnon dispersion exhibits a small but finite gap at $\mathrm{\ensuremath{\Gamma}}$. We analyze this violation of the Goldstone mode and present an approach that implements the magnetic susceptibility using a renormalized Green function instead of the Kohn-Sham one. This much more expensive approach shows substantial improvement of the Goldstone-mode condition. In addition, we discuss a possible correction scheme, which involves an adjustment of the Kohn-Sham exchange splitting, which is motivated by the spin-wave solution of the one-band Hubbard model. The new exchange splittings turn out to be closer to experiment. We present corrected magnon spectra for the elementary ferromagnets Fe, Co, and Ni.

One-electron self-interaction and the asymptotics of the Kohn–Sham potential: an impaired relation

One-electron self-interaction and an incorrect asymptotic behavior of the Kohn–Sham exchange–correlation potential are among the most prominent limitations of many present-day density functionals. However, a one-electron self-interaction-free energy does not necessarily lead to the correct long-range potential. This is shown here explicitly for local hybrid functionals. Furthermore, carefully studying the ratio of the von Weizsäcker kinetic energy density to the (positive) Kohn–Sham kinetic energy density, τW/τ, reveals that this ratio, which frequently serves as an iso-orbital indicator and is used to eliminate one-electron self-interaction effects in meta-generalized-gradient approximations and local hybrid functionals, can fail to approach its expected value in the vicinity of orbital nodal planes. This perspective article suggests that the nature and consequences of one-electron self-interaction and some of the strategies for its correction need to be reconsidered.

Kohn-Sham orbitals and potentials from quantum Monte Carlo molecular densities

In this work we show the possibility to extract Kohn-Sham orbitals, orbital energies, and exchange correlation potentials from accurate Quantum Monte Carlo (QMC) densities for atoms (He, Be, Ne) and molecules (H2, Be2, H2O, and C2H4). The Variational Monte Carlo (VMC) densities based on accurate Jastrow Antisymmetrised Geminal Power wave functions are calculated through different estimators. Using these reference densities, we extract the Kohn-Sham quantities with the method developed by Zhao, Morrison, and Parr (ZMP) [Phys. Rev. A 50, 2138 (1994)]. We compare these extracted quantities with those obtained form CISD densities and with other data reported in the literature, finding a good agreement between VMC and other high-level quantum chemistry methods. Our results demonstrate the applicability of the ZMP procedure to QMC molecular densities, that can be used for the testing and development of improved functionals and for the implementation of embedding schemes based on QMC and Density Functional Theory.