Kohn-Sham Potential Research Articles

Are Exact Exchange-Correlation Potentials Continuous at Atomic Nuclei in Molecules?

The Kohn-Sham exchange-correlation potential of a given system can be written exactly in terms of several quantities including the interacting and noninteracting kinetic energy densities of the electrons. Electronic kinetic energy densities are sharply discontinuous at the atomic nuclei in molecules, which implies that exact exchange-correlation potentials of molecules might also be discontinuous at the nuclear positions. This surprising possibility is made even more likely by the fact that molecular exchange-correlation potentials derived from wave functions within Slater-type basis sets do have jump discontinuities. We deduce analytically the exact behavior of the relevant quantities near atomic nuclei and conclude that, in the basis-set limit, all jump discontinuities cancel out nontrivially so that exact exchange-correlation potentials are continuous after all.

Partially deorbitalized meta-GGA

Mejia-Rodriguez and Trickey recently proposed a procedure for removing the explicit dependence of meta-GGA exchange-correlation energy functionals Exc on the kinetic energy density τ. We present a simple modification to this approach in which the exact Kohn-Sham τ is used as input for Exc but the functional derivative of τ with respect to the density ρ, required to calculate the potential term ∫d3r′δExc∕δτ(r′)∣ρ⋅δτ(r′)∕δρ(r), is evaluated using an approximate kinetic energy density functional. This ‘half-way’ strategy ensures that the Kohn-Sham potential is a local multiplicative function (as opposed to the non-local potential of a generalized Kohn-Sham approach) while preserving the inherent non-locality of the functional itself. Electronic structure codes can be easily modified to use the new method. We validate it by quantifying the accuracy of the predicted lattice parameters, bulk moduli, magnetic moments and cohesive energies of a large set of periodic solids. An unanticipated benefit of this method is to gauge the quality of approximate kinetic energy functionals by checking if the self-consistent solution is indeed at the variational minimum.

Ensemble Ground State of a Many-Electron System with Fractional Electron Number and Spin: Piecewise-Linearity and Flat-Plane Condition Generalized.

Description of many-electron systems with a fractional electron number (Ntot) and fractional spin (Mtot) is of great importance in physical chemistry, solid-state physics, and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system and characterize the dependence of the energy and the spin-densities on both Ntot and Mtot, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at a constant Ntot, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.

Reaction rate constant: a theoretical description from local temperature.

Application of various descriptors based on electron density and its associated quantities to quantify chemical reactivity within the conceptual density functional theory has recently come into spotlight. Among others and particularly relevant to our study, local temperature based on electron density as well as kinetic energy density, as a measure of the kinetic energy of an electron moving in the Kohn-Sham potential of systems, should be mentioned. In this work, we propose to use the local temperature for describing the reaction rate constant, where our main idea originates from the point that the smaller the local temperature at the reaction center, the easier the electron removal, leading to a larger rate constant. On the basis of theoretical considerations, it is proved that the rate constant variations caused by the substituent effects can well be proportional to the local temperature at the reaction center. In order to numerically validate our proposed approach, we have taken the phenol derivatives with the available experimental rate constants of their O-methylation reaction as working models. The reason for this choice is that one of the most versatile approaches for labeling biologically active compounds with the 11C nuclide for positron emission tomography (PET) is methylation by methyl iodide including 11C nuclide, [11C]MeI, where methylation of phenolic oxygen with [11C]MeI is utilized to label some important tracers for PET studies. Our results unveil that the local temperature changes at the reaction center of the aforementioned reaction are reasonably correlated with the rate constant variations. Hopefully, incorporating the proposed correlations between the local temperature and the kinetics data into a computer control algorithm not only provides a simple tool for predicting the rate constant of the O-methylation reaction for other substituted phenols, but also, as a part of the chemical artificial intelligence, the optimum [11C]MeI labeling conditions for a wide variety of phenol derivatives can be controlled.

Orbital-free density-functional theory for metal slabs.

Orbital-Free Density-Functional Theory (OF-DFT) is known to represent a promising alternative to the standard Kohn-Sham (KS) DFT, as it relies on the electron density alone, without the need to calculate all KS single-particle orbitals and energies. Here, we investigate the behavior of the main ingredients of this theory, which are the non-interacting kinetic-energy density (KED) and the Pauli potential, for metal slabs. We derive explicit density functionals for these quantities in the quantum limit where all electrons are in the same slab discrete level of energy, and we present numerical calculations beyond this quantum limit for slabs of various widths. We have found the first explicit KED functional for a realistic many-particle fermionic system, which we prove to be generally valid with no assumption about the KS potential. We also discuss the total non-interacting kinetic energy and the corresponding enhancement factor, which represent basic quantities for the practical implementation of OF-DFT.

A multi-GPU implementation of Real-Time Time-Dependent Auxiliary Density Functional Theory for the investigation of nanosystems irradiations

We report a new Multi-GPU (Graphical Processor Unit) implementation of real-time time-dependent Auxiliary Density Functional Theory (DFT) for simulations of attosecond electronic dynamics in molecular systems subjected to strong perturbations. Our code relies on the Kohn-Sham formalism of DFT and has been implemented in the deMon2k Fortran code. We expand single-particle wave functions (i.e molecular orbitals) as linear combinations of Gaussian-type-orbitals centered on atoms. The density matrix propagation is carried out on GPU while the Kohn-Sham potential is operated on CPUs (Central Processor Unit) with the help of variationally fitted densities. We propose a parallelization strategy using the MAGMA/CUDA libraries to calculate the exponential of dense Hermitian matrices entering the mathematical definition of the propagator, here using Taylor expansions. We report performance benchmarks on water droplets and on fullerenes (C50 to C540). They show a clear advantage of GPU over CPU (using the Scalapack library). The benchmarks also show the benefit of using more than one GPU for systems comprised of up to more than 10,000 basis functions. There, a speed-up of almost 40 between pure 40 CPU and four 4 GPU is obtained. Attosecond electron dynamics simulation in molecular systems comprised of several thousands of electrons becomes amenable to routine simulations in our code. We assess the accuracy of the GPU implementation considering various applications, namely, the calculation of extreme UV absorption spectra with non-Hermitian dynamics, the response of C180 to an electric perturbation, and finally the irradiation of a DNA/protein complex by a 0.4 MeV proton. The results demonstrate the robustness of the implementation. This work also paves the way for future even more efficient implementations. Program SummaryProgram title: deMon2k-RT-TDDFTCPC Library link to program files: https://doi.org/10.17632/whwc5zkst5.1Developer's repository link: DOI: 10.5281/zenodo.8301468Licensing provisions: CeCILL Free Software License Agreement v2.1Programming language: FortranExternal routines/libraries: MPI, BLAS/LAPACK, CUDA, MAGMASupplementary material: User manual, installation instructions, test cases. Nature of problem: Electron dynamics within molecular systems is amenable to simulation with Real-Time Time-Dependent Density Functional Theory (RT-TD-DFT). The methodology consists in propagating on the attosecond time scale the single particle electronic wave functions on the underlying Kohn-Sham potential [1,2]. In deMon2k-RT-TDDFT, the propagation is coupled to the Auxiliary DFT framework for fast evaluation of the potential [3]. A remaining computational bottleneck is the matrix exponential calculation of the Hermitian Hamiltonian matrix. This task is achieved in deMon2k via diagolanization or expansions formulas (Taylor, Chebychev or Baker-Campbell-Hausdorff). The operation is cumbersome on central-processor-unit machines for large matrices (e.g. > 5000 × 5000), which corresponds to molecular systems comprising around 2,000 electrons. Alternatives are needed to overcome this bottleneck.Solution method: We propose to carry out the matrix exponentiation on graphical-processor-units thanks to the MAGMA[4]/CUDA libraries. The code permits a drastic reduction of the computational cost, opening the attosecond dynamic simulations of large molecular systems comprised thousands of electrons with controlled accuracy. The method is tested here for Taylor expansions. Additional comments including restrictions and unusual features: The code described in this article opens the realization of RT-TD-ADFT simulations. The code is linked to the Master version 6.1.6 of deMon2k (http://www.demon-software.com), which must be pre-installed by the user.

Machine-learning Kohn–Sham potential from dynamics in time-dependent Kohn–Sham systems

The construction of a better exchange-correlation potential in time-dependent density functional theory (TDDFT) can improve the accuracy of TDDFT calculations and provide more accurate predictions of the properties of many-electron systems. Here, we propose a machine learning method to develop the energy functional and the Kohn–Sham potential of a time-dependent Kohn–Sham (TDKS) system is proposed. The method is based on the dynamics of the Kohn–Sham system and does not require any data on the exact Kohn–Sham potential for training the model. We demonstrate the results of our method with a 1D harmonic oscillator example and a 1D two-electron example. We show that the machine-learned Kohn–Sham potential matches the exact Kohn–Sham potential in the absence of memory effect. Our method can still capture the dynamics of the Kohn–Sham system in the presence of memory effects. The machine learning method developed in this article provides insight into making better approximations of the energy functional and the Kohn–Sham potential in the TDKS system.

A non-self-consistent tight-binding electronic structure potential in a polarized double-ζ basis set for all spd-block elements up to Z = 86.

Existing semiempirical molecular orbital methods suffer from the usually minimal atomic-orbital (AO) basis set used to simplify the calculations. Here, a completely new and consistently parameterized tight-binding electronic structure Hamiltonian evaluated in a deeply contracted, properly polarized valence double-zeta basis set (vDZP) is described. The inner-shell electrons are accounted for by standard, large-core effective potentials and approximations to them. The primary target of this so-called density matrix tight-binding method is to reproduce the one-particle density matrix P of a molecular ωB97X-V range-separated hybrid density functional theory (DFT) calculation in exactly the same basis set. Additional properties considered are orbital energies, dipole polarizabilities and dipole moments, and dipole polarizability derivatives. The key features of the method are as follows: (a) it is non-self-consistent with an overall fixed number of only three required matrix diagonalizations; (b) only AO overlap integrals are needed to construct the effective Hamiltonian matrix; (c) new P-dependent terms emulating non-local exchange are included; and (d) only element-specific empirical parameters (about 50 per element) need to be determined. The method globally achieves a high accuracy for the target properties at a speedup compared to the ωB97X-V/vDZP reference of about 3-4 orders of magnitude. It performs robustly for difficult transition metal complexes, for highly charged or zwitterionic systems, and for chemically unusual bonding situations, indicating a generally robust approximation of the (self-consistent) Kohn-Sham potential. As an example application, the vibrational Raman spectrum of an entire protein with 327 atoms with respect to the DFT reference calculation is shown. This method may be used out-of-the-box to generate molecular/atomic features for machine learning applications or as the basis for accurate high-speed DFT methods.

Analytical Forces for the Optimized Effective Potential Calculations.

The optimized effective potential (OEP) equation is an ill-conditioned linear system when using finite basis sets. Without any special treatment, the obtained exchange-correlation (XC) potential may have unphysical oscillations. One way to alleviate this problem is to regularize the solutions; however, a regularized XC potential is not the exact solution to the OEP equation. As a result, the system's energy is no longer variational against the Kohn-Sham (KS) potential, and the analytical forces cannot be derived from the Hellmann-Feynman theorem. In this work, we develop a robust and nearly black-box OEP method to ensure that the system's energy is variational against the KS potential. The basic idea is to add a penalty function that regularizes the XC potential to the energy functional. Analytical forces can then be derived based on the Hellmann-Feynman theorem. Another key result is that the impact of the regularization can be much reduced by regularizing the difference between the XC potential and an approximate XC potential rather than regularizing the XC potential. Numerical tests show that forces and the energy differences between systems are not sensitive to the regularization coefficient, which indicates that in practice accurate structural and electronic properties can be obtained without extrapolating the regularization coefficient to zero. We expect this new method to be found useful for calculations that employ advanced, orbital-based functionals, especially for applications that require efficient force calculations.

Strong field driven extreme nonlinear photoemission from individual single-walled carbon nanotubes

Photoemission from nanostructures under strong fields has attracted significant interests in materials science and optoelectronic applications. Here, we report the extreme nonlinear photoemission in single-walled carbon nanotubes (SWCNTs) and its underlying mechanisms related to the microscopic carrier excitation and the sequential emission of occupied-state electrons. We show that the characteristic photoemission responses, e.g., nonlinear slope, of individual SWCNTs strongly depend on the unique electronic structures near the Fermi level, i.e., localization of the emitting states and their tunneling probabilities. The transition of photoemission mechanisms giving rise to the extreme nonlinear behavior was elucidated using Kohn-Sham potentials and the extended Fowler-Nordheim theory. This paper provides insights into photoemission dynamics of SWCNTs on the ultimate atomic length scale and attosecond time scale, will hopefully help design materials for tunable optoelectronic responses.

Nuclear cusps and singularities in the nonadditive kinetic potential bifunctional from analytical inversion

The non-additive kinetic potential $v^{\text{NAD}}$ is a key quantity in density-functional theory (DFT) embedding methods, such as frozen density embedding theory and partition DFT. $v^{\text{NAD}}$ is a bi-functional of electron densities $\rho_{\rm B}$ and $\rho_{\rm tot} = \rho_{\rm A} + \rho_{\rm B}$. It can be evaluated using approximate kinetic-energy functionals, but accurate approximations are challenging. The behavior of $v^{\text{NAD}}$ in the vicinity of the nuclei has long been questioned, and singularities were seen in some approximate calculations. In this article, the existence of singularities in $v^{\text{NAD}}$ is analyzed analytically for various choices of $\rho_{\rm B}$ and $\rho_{\rm tot}$, using the nuclear cusp conditions for the density and Kohn-Sham potential. It is shown that no singularities arise from smoothly partitioned ground-state Kohn-Sham densities. We confirm this result by numerical calculations on diatomic test systems HeHe, HeLi$^+$, and H$_2$, using analytical inversion to obtain a numerically exact $v^{\rm NAD}$ for the local density approximation. We examine features of $v^{\rm NAD}$ which can be used for development and testing of approximations to $v^{\rm NAD}[\rho_{\rm B},\rho_{\rm tot}]$ and kinetic-energy functionals.

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

AbstractIn 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.

Secondary Kinetic Peak in the Kohn-Sham Potential and Its Connection to the Response Step.

We consider a prototypical 1D model Hamiltonian for a stretched heteronuclear molecule and construct individual components of the corresponding KS potential, namely, the kinetic, the N - 1, and the conditional potentials. These components show very special features, such as peaks and steps, in regions where the density is drastically low. Some of these features are quite well-known, whereas others, such as a secondary peak in the kinetic potential or a second bump in the conditional potential, are less or not known at all. We discuss these features building on the analytical model treated in Giarrusso et al. J. Chem. Theory Comput. 2018, 14, 4151. In particular, we provide an explanation for the underlying mechanism which determines the appearance of both peaks in the kinetic potential and elucidate why these peaks delineate the region over which the plateau structure, due to the N - 1 potential, stretches. We assess the validity of the Heitler-London Ansatz at large but finite internuclear distance, showing that, if optimal orbitals are used, this model is an excellent approximation to the exact wave function. Notably, we find that the second natural orbital presents an extra node very far out on the side of the more electronegative atom.

Building Kohn–Sham Potentials for Ground and Excited States

We analyze the inverse problem of Density Functional Theory using a regularized variational method. First, we show that given k and a target density $$\rho $$ , there exist potentials having kth bound mixed states which densities are arbitrarily close to $$\rho $$ . The state can be chosen pure in dimension $$d=1$$ and without interactions, and we provide numerical and theoretical evidence consistently leading us to conjecture that the same pure representability result holds for $$d=2$$ , but that the set of pure-state v-representable densities is not dense for $$d=3$$ . Finally, we present an inversion algorithm taking into account degeneracies, removing the generic blocking behavior of standard ones.

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.

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

BREMS: Partial-wave calculation of spectra and angular distributions of electron-atom bremsstrahlung at electron energies less than 30 MeV (New Version Announcement)

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.

Chemical potential, derivative discontinuity, fractional electrons, jump of the Kohn-Sham potential, atoms as thermodynamic open systems, and other (mis)conceptions of the density functional theory of electrons in molecules.

Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. The origin of this notion has been the identification of the Lagrange multiplier μ = ∂E/∂N in the Euler-Lagrange variational equation for the ground state density as the chemical potential of the electrons. We first discuss why the Lagrange multiplier in this case is an arbitrary constant and therefore cannot be a physical characteristic of an atom or molecule. The switching of the energy derivative ("chemical potential") from -I to -A when the electron number crosses the integer, called integer discontinuity or derivative discontinuity, is not physical but only occurs when the nonphysical noninteger electron systems and the corresponding energy and derivative ∂E/∂N are chosen in a specific discontinuous way. The question is discussed whether in fact the thermodynamical concept of a chemical potential can be defined for the electrons in such few-electron systems as atoms and molecules. The conclusion is that such systems lack important characteristics of thermodynamic systems and do not afford the definition of a chemical potential. They also cannot be considered as analogues of the open systems of thermodynamics that can exchange particles with an environment (a particle bath or other members of a Gibbsian ensemble). Thermodynamical (statistical mechanical) concepts like chemical potential, open systems, grand canonical ensemble etc. are not applicable to a few electron system like an atom or molecule. A number of topics in DFT are critically reviewed in light of these findings: jumps in the Kohn-Sham potential when crossing an integer number of electrons, the band gap problem, the deviation-from-straight-lines error, and the role of ensembles in DFT.

Spherical vs non-spherical and symmetry-preserving vs symmetry-breaking densities of open-shell atoms in density functional theory.

The atomization energies of molecules from first-principles density functional approximations improve from the local spin-density approximation to the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) to the strongly constrained and appropriately normed (SCAN) meta-GGA, and their sensitivities to non-spherical components of the density increase in the same order. Thus, these functional advances increase density sensitivity and imitate the exact constrained search over correlated wavefunctions better than that over ensembles. The diatomic molecules studied here, singlet C2 and F2 plus triplet B2 and O2, have cylindrically symmetric densities. Because the densities of the corresponding atoms are non-spherical, the approximate Kohn-Sham potentials for the atoms have a lower symmetry than that of the external (nuclear) potential so that the non-interacting wavefunctions are not eigenstates of the square of total orbital angular momentum, breaking a symmetry that yields a feature of the exact ground-state density. That spatial symmetry can be preserved by a non-self-consistent approach in which a self-consistent equilibrium-ensemble calculation is followed by integer re-occupation of the Kohn-Sham orbitals as the first of several steps. The symmetry-preserving approach is different from symmetry restoration based on projection. First-step space- (and space-spin-) symmetry preservation in atoms is shown to have a small effect on the atomization energies of molecules, quantifying earlier observations by Fertig and Kohn. Thus, the standard Kohn-Sham way of calculating atomization energies, with self-consistent symmetry breaking to minimize the energy, is justified at least for the common cases where the molecules cannot break symmetry. Unless symmetry breaking is allowed in the molecule, SCAN strongly underestimates the atomization energy of strongly correlated singlet C2.

Readdressing molecular dissociation within the Kohn–Sham formalism of density-functional theory: simple models and a different point of view

Within the Kohn–Sham (KS) formalism of density-functional theory (DFT), most available exchange-correlation (XC) density-functionals are known to yield incorrect molecular dissociation even when dealing with simple molecules as H. It is known that the KS potential must develop a peak and a step in the region between the isolated atoms, in order to prevent electrons to hop from one atom to another, and hence avoiding fractionally charged dissociated atoms. Considering one-dimensional models of two-electron homo and heteronuclear molecules, we here (i) readdress the limit of dissociated molecules, with a different point of view on the formation of the peaks and steps in the KS potential and (ii) investigate a simple approach to treat not necessarily dissociated molecules, by employing single-particle orbitals (not KS orbitals). Our results indicate that an accurate description of dissociation can be achieved.