Publications

2022
First-principles derivation and properties of density-functional average-atom models
T. J. Callow, Hansen, S. B. , Kraisler, Eli , and Cangi, Attila . 2022. First-Principles Derivation And Properties Of Density-Functional Average-Atom Models. Phys. Rev. Research, 4, Pp. 023055. Abstract

WDM regime chart

Finite-temperature Kohn-Sham density functional theory (KS-DFT) is a widely-used method in warm dense matter (WDM) simulations and diagnostics. Unfortunately, full KS-DFT-molecular dynamics models scale unfavourably with temperature and there remains uncertainty regarding the performance of existing approximate exchange-correlation (XC) functionals under WDM conditions. Of particular concern is the expected explicit dependence of the XC functional on temperature, which is absent from most approximations. Average-atom (AA) models, which significantly reduce the computational cost of KS-DFT calculations, have therefore become an integral part of WDM modeling. In this paper, we present a derivation of a first-principles AA model from the fully-interacting many-body Hamiltonian, carefully analyzing the assumptions made and terms neglected in this reduction. We explore the impact of different choices within this model—such as boundary conditions and XC functionals—on common properties in WDM, for example equation-of-state data, ionization degree and the behavior of the frontier energy levels. Furthermore, drawing upon insights from ground-state KS-DFT, we discuss the likely sources of error in KS-AA models and possible strategies for mitigating such errors.

Publisher's version                                         arXiv version

2021
Charge-Transfer Steps in Density Functional Theory from the Perspective of the Exact Electron Factorization
Jakub Kocák, Kraisler, Eli , and Schild, Axel . 2021. Charge-Transfer Steps In Density Functional Theory From The Perspective Of The Exact Electron Factorization. J. Phys. Chem. Lett., 12, Pp. 3204. Abstract
Steps in the potential created due to charge transfer, along with the conditional wavefunction

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 factorization (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 are proposed. 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.

Publisher's version                    arXiv version

 

From Kohn-Sham to many-electron energies via step structures in the exchange-correlation potential
Eli Kraisler, Hodgson, Matthew J. P. , and Gross, E. K. U. . 2021. From Kohn-Sham To Many-Electron Energies Via Step Structures In The Exchange-Correlation Potential. J. Chem. Theory Comput., 17, Pp. 1390. Abstract
Accurately describing excited states within Kohn–Sham (KS) density functional theory (DFT), particularly those which induce ionization and charge transfer, remains a great challenge. Common exchange-correlation (xc) approximations are unreliable for excited statesTeaser from article JCTC 17. 1390 (2021)

owing, in part, to the absence of a derivative discontinuity in the xc energy (Δ), which relates a many-electron energy difference to the corresponding KS energy difference. We demonstrate, analytically and numerically, how the relationship between KS and many-electron energies leads to the step structures observed in the exact xc potential in four scenarios: electron addition, molecular dissociation, excitation of a finite system, and charge transfer. We further show that steps in the potential can be obtained also with common xc approximations, as simple as the LDA, when addressed from the ensemble perspective. The article therefore highlights how capturing the relationship between KS and many-electron energies with advanced xc approximations is crucial for accurately calculating excitations, as well as the ground-state density and energy of systems which consist of distinct subsystems.

Publisher's version       arXiv version

 

2020
Asymptotic behavior of the exchange-correlation energy density and the Kohn-Sham potential in density functional theory: exact results and strategy for approximations

This paper is an invited review article for the Isreal Journal of Chemistry, which is part of the special issue: Computational Materials Science in Israel.

Abstract: Density functional theory (DFT) is nowadays the leading theoretical framework for quantum description of materials from first principles. Being an exact theory on one hand and computationally efficient on the other hand, DFT allows to address large and complex many‐electron systems and accurately predict their properties. The predictive power of DFT critically depends though on an accurate approximation to the generally unknown exchange‐correlation (xc) energy functional. Approximations can be constructed from first principles by satisfying known properties of the exact functional.

The exchange-correlaion tree

In this work I review two such exact properties: the asymptotic behavior of the xc energy density per particle and the asymptotic behavior of the Kohn‐Sham potential, in finite many‐electron systems. The derivation of the asymptotic forms for both quantities is reviewed, employing the concepts of the abatic connection and of the xc hole with relation to the first quantity and the exact electron factorization approach for the second one.  Furthermore, it is shown that the correct asymptotic behavior of one of the aforementioned quantities does not guarantee a correct behavior of the other. These quantities are related via the xc hole response function, which is defined, examined and its exact exchange part is analytically derived. The extent to which existing xc approximations satisfy the named exact properties is reviewed and the relationship between correct asymptotics and freedom from one‐electron self‐interaction in DFT is discussed. Finally, a strategy for development of advanced approximations for exchange and correlation with a correct asymptotic behavior is suggested.

 

Publisher's version                              ChemRxiv version

Discontinuous behavior of the Pauli potential in density functional theory as a function of the electron number

The Pauli potential is an essential quantity in orbital-free density functional theory (DFT) and in the exact electron factorization method for many-electron systems. Knowledge of the Pauli potential allows the description of a system relying on the density alone, without the need to calculate the orbitals.

 

In this work, we explore the behavior of the exact Pauli potential in finite systems as a function of the number of electrons, employing the ensemble approach in DFT. Assuming the system is in contact with an electron reservoir, we allow the number of electrons to vary continuously and to obtain fractional as well as integer values. We derive an expression for the Pauli potential for a spin-polarized system with a fractional number of electrons, and we find that when the electron number surpasses an integer, the Pauli potential jumps by a spatially uniform constant, similarly to the Kohn-Sham potential. The magnitude of the jump equals the Kohn-Sham gap. We illustrate our analytical findings by calculating the exact and approximate Pauli potentials for Li and Na atoms with fractional numbers of electrons.

Publisher's version                              ChemRxiv version

2019
Asymptotic behavior of the Hartree-exchange and correlation potentials in ensemble density functional theory
Tim Gould, Pittalis, Stefano , Toulouse, Julien , Kraisler, Eli , and Kronik, Leeor . 2019. Asymptotic Behavior Of The Hartree-Exchange And Correlation Potentials In Ensemble Density Functional Theory. Phys. Chem. Chem. Phys., 21, Pp. 19805-19815. Abstract
We report on previously unnoticed features of the exact Hartree-exchange and correlation potentials for atoms and ions treated via ensemble density functional theory, demonstrated on fractional ions of Li, C, and F. We show that these potentials, when treated separately, can reach non-vanishing asymptotic constant values in the outer region of spherical, spin unpolarized atoms. In the next leading order, the potentials Effective screening charge as a function of no. of electrons

resemble Coulomb potentials created by effective charges which have the peculiarity of not behaving as piecewise constants as a function of the electron number. We provide analytical derivations and complement them with numerical results using the inversion of the Kohn–Sham equations for interacting densities obtained by accurate quantum Monte Carlo calculations. The present results expand on the knowledge of crucial exact properties of Kohn–Sham systems, which can guide development of advanced exchange–correlation approximations.

Publisher's version                              ChemRxiv version

 

2017
How interatomic steps in the exact Kohn-Sham potential relate to derivative discontinuities of the energy
Matthew J. P. Hodgson*, Kraisler*, Eli , Schild, Axel , and Gross, E. K. U. . 2017. How Interatomic Steps In The Exact Kohn-Sham Potential Relate To Derivative Discontinuities Of The Energy. J. Phys. Chem. Lett., 8, Pp. 5974. Abstract

Accurate density functional calculations hinge on reliable approximations to the unknown exchange-correlation (xc) potential. The most popular approximations usually lack features of the exact xc potential that are important

for an accurate prediction of the fundamental gap and the distribution of charge in complex systems. Two principal features in this regard are the spatially uniform shift in the potential, as the number of electrons infinitesimally surpasses an integer, and the spatial steps that form, for example, between the atoms of stretched molecules. Although both aforementioned concepts are well known, the exact relationship between them remained unclear.Here we establish this relationship via an analytical derivation. We support our result by numerically solving the many-electron Schrödinger equation to extract the exact Kohn–Sham potential and directly observe its features. Spatial steps in the exact xc potential of a full configuration-interaction (FCI) calculation of a molecule are presented in three dimensions.

Publisher's version          arXiv version

 

2015
Effect of ensemble generalization on the highest-occupied Kohn-Sham energy level
Eli Kraisler*, Schmidt*, Tobias , Kümmel, Stephan , and Kronik, Leeor . 2015. Effect Of Ensemble Generalization On The Highest-Occupied Kohn-Sham Energy Level. J. Chem. Phys., 143, Pp. 104105. https://aip.scitation.org/doi/full/10.1063/1.4930119. Abstract

Shifts in ho KS nergy levels, due to the ensemble generalization, in the up and down channelsThere are several approximations to the exchange-correlation functional in density-functional theory, which accurately predict total energy-related properties of many-electron systems, such as binding energies, bond lengths, and crystal structures. Other approximations are designed to describe potential-related processes, such as charge transfer and photoemission. However, the development of a functional which can serve the two purposes simultaneously is a long-standing challenge. Trying to address it, we employ in the current work the ensemble generalization procedure proposed by Kraisler and Kronik [Phys. Rev. Lett. 110, 126403 (2013)]. Focusing on the prediction of the ionization potential via the highest occupied Kohn-Sham eigenvalue, we examine a variety of exchange-correlation approximations: the local spin-density approximation, semi-local generalized gradient approximations, and global and local hybrid functionals. Results for a test set of 26 diatomic molecules and single atoms are presented. We find that the aforementioned ensemble generalization systematically improves the prediction of the ionization potential, for various systems and exchange-correlation functionals, without compromising the accuracy of total energy-related properties. We specifically examine hybrid functionals. These depend on a parameter controlling the ratio of semi-local to non-local functional components. The ionization potential obtained with ensemble-generalized functionals is found to depend only weakly on the parameter value, contrary to common experience with non-generalized hybrids, thus eliminating one aspect of the so-called “parameter dilemma” of hybrid functionals.

Elimination of the asymptotic fractional dissociation problem in Kohn-Sham density functional theory using the ensemble-generalization approach
Many approximations within density-functional theory spuriously predict that a many-electron system can dissociate into fractionally charged fragments. Here, we revisit the case of dissociated diatomic molecules, known to exhibit this problem when studied within standard approaches, including the local spin-densityEnergy of the dissociated Li...F molecule vs q -- a fraction of transferred charge

approximation (LSDA). By employing our recently proposed [E. Kraisler and L. Kronik, Phys. Rev. Lett. 110, 126403 (2013)] ensemble generalization we find that asymptotic fractional dissociation is eliminated in all systems examined, even if the underlying exchange correlation (xc) is still the LSDA. Furthermore, as a result of the ensemble-generalization procedure, the Kohn-Sham potential develops a spatial step between the dissociated atoms, reflecting the emergence of the derivative discontinuity in the xc energy functional. This step, predicted in the past for the exact Kohn-Sham potential and observed in some of its more advanced approximate forms, is a desired feature that prevents any fractional charge transfer between the system's fragments. It is usually believed that simple xc approximations such as the LSDA cannot develop this step. Our findings show, however, that ensemble generalization to fractional electron densities automatically introduces the desired step even to the most simple approximate xc functionals and correctly predicts asymptotic integer dissociation.

Publisher's version                         arXiv version

 

2014

The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional--the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.

We present and test a new approximation for the exchange-correlation (xc) energy of Kohn-Sham density functional theory. It combines exact exchange with a compatible non-local correlation functional. The functional is by construction free of one-electron self-interaction, respects constraints derived from uniform coordinate scaling, and has the correct asymptotic behavior of the xc energy density. It contains one parameter that is not determined ab initio. We investigate whether it is possible to construct a functional that yields accurate binding energies and affords other advantages, specifically Kohn-Sham eigenvalues that reliably reflect ionization potentials. Tests for a set of atoms and small molecules show that within our local-hybrid form accurate binding energies can be achieved by proper optimization of the free parameter in our functional, along with an improvement in dissociation energy curves and in Kohn-Sham eigenvalues. However, the correspondence of the latter to experimental ionization potentials is not yet satisfactory, and if we choose to optimize their prediction, a rather different value of the functional's parameter is obtained. We put this finding in a larger context by discussing similar observations for other functionals and possible directions for further functional development that our findings suggest.

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.

2013
Density-functional theory (DFT) is an exact alternative formulation of quantum mechanics, in which it is possible to calculate the total energy, the spin, and the charge density of many-electron systems in the ground state. In practice, it is necessary to use uncontrolled approximations that can mainly be verified against experimental data. Atoms and ions are simple systems, where the approximations of DFT can be easily tested. We have calculated within DFT the total energies, spin, and higher ionization energies of all the ions of elements with 1⩽Z⩽29. We find the calculations in close agreement with experiment, with an error of typically less than ca. 1% for 1⩽Z⩽29. Surprisingly, the error depends on the electronic configuration of the ion in both local spin density approximation and Perdew-Burke-Ernzerhof general gradient approximation and independent of both self-interaction correction and relativistic corrections. Larger errors are found for systems in which the spin-spin correlation is significant, which indicates the possible benefit from an orbital-dependent formulation of the correlation energy functional.
In the exact Kohn-Sham density-functional theory, the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of density-functional theory. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density dependent. This opens the door to calculations of the ionization potential and electron affinity, even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.
Transition voltage spectroscopy (TVS) has become an accepted quantification tool for molecular transport characteristics, due to its simplicity and reproducibility. Alternatively, the Taylor expansion view, TyEx, of transport by tunneling suggests that conductance–voltage curves have approximately a generic parabolic shape, regardless of whether the tunneling model is derived from an average medium view (e.g., WKB) or from a scattering view (e.g., Landauer). Comparing TVS and TyEx approaches reveals that TVS is closely related to a bias-scaling factor, V0, which is directly derived from the third coefficient of TyEx, namely, the second derivative of the conductance with respect to bias at 0 V. This interpretation of TVS leads to simple expressions that can be compared easily across primarily different tunneling models. Because the basic curve shape is mostly generic, the quality of model fitting is not informative on the actual tunneling model. However internal correlation between the conductance near 0 V and V0 (TVS) provides genuine indication on fundamental tunneling features. Furthermore, we show that the prevailing concept that V0 is proportional to the barrier height holds only in the case of resonant tunneling, while for off-resonant or deep tunneling, V0 is proportional to the ratio of barrier height to barrier width. Finally, considering TVS as a measure of conductance nonlinearity, rather than as an indicator for energy level spectroscopy, explains the very low TVS values observed with a semiconducting (instead of metal) electrode, where transport is highly nonlinear due to the relatively small, bias-dependent density of states of the semiconducting electrode.
2010
The total energies and the spin states for atoms and their first ions with Z=1–86 are calculated within the the local spin-density approximation (LSDA) and the generalized-gradient approximation (GGA) to the exchange-correlation (xc) energy in density-functional theory. Atoms and ions for which the ground-state density is not pure-state v-representable are treated as ensemble v-representable with fractional occupations of the Kohn-Sham system. A recently developed algorithm which searches over ensemble v-representable densities [E. Kraisler et al.Phys. Rev. A 80, 032115 (2009)] is employed in calculations. It is found that for many atoms, the ionization energies obtained with the GGA are only modestly improved with respect to experimental data, as compared to the LSDA. However, even in those groups of atoms where the improvement is systematic, there remains a non-negligible difference with respect to the experiment. The ab initio electronic configuration in the Kohn-Sham reference system does not always equal the configuration obtained from the spectroscopic term within the independent-electron approximation. It was shown that use of the latter configuration can prevent the energy-minimization process from converging to the global minimum, e.g., in lanthanides. The spin values calculated ab initio fit the experiment for most atoms and are almost unaffected by the choice of the xc functional. Among the systems with incorrectly obtained spin, there exist some cases (e.g., V, Pt) for which the result is found to be stable with respect to small variations in the xc approximation. These findings suggest a necessity for a significant modification of the exchange-correlation functional, probably of a nonlocal nature, to accurately describe such systems.
2009
In the framework of Kohn-Sham density-functional theory, systems with ground-state densities that are not pure-state v-representable (PSVR) in the noninteracting reference system occur frequently. In the present contribution, an algorithm, which allows the solution of such systems, is proposed. It is shown that the use of densities which do not correspond to a ground state of their noninteracting reference system is forbidden. As a consequence, the proposed algorithm considers only noninteracting ensemble v-representable densities. The Fe atom, a well-known non-PSVR system, is used as an illustration. Finally, the problem is analyzed within finite-temperature density-functional theory, where the physical significance of fractional occupations is exposed and the question of why degenerate states can be unequally occupied is resolved.
2000
כא"ם מושרה בסליל כתוצאה מתנועת מגנט-מוט דרכו Induced voltage in a coil as a result of a rod magnet motion through it

מובא ניסוי חדש שמטרתו להבהיר את התלות של הכא"מ המושרה, הנוצר בסליל כתוצאה מתנועת מגנט-מוט דרכו, במספר פרמטרים: מספר הליפופים בסליל, כיוון הליפופים, רדיוס הסליל, וגודל מהירות תנועת המגנט וכיוונה.

A new experiment is presented aimed at clarifying the dependence of the induced voltage formed in a coil as a result of a motion of a rod magnet through it, on a number of parameters: he number of turns in the coil, the direction of the turns, the radius of the coil, the speed of the magnet and direction of its movement.

 

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