Kohn-Sham Research Articles

A Critical Evaluation of the Hybrid KS DFT Functionals Based on the KS Exchange-Correlation Potentials.

We have developed a critical methodology for the evaluation of the quality of hybrid exchange-correlation (XC) density functional approximations (DFAs) based on very fundamental quantities, i.e., Kohn-Sham (KS) XC potentials, self-consistent electron densities, first ionization potentials (IPs), and total energies. Since the XC potentials, the primary objects in the current study, are not directly accessible for the hybrids, we calculate them by inverting the KS electron densities. Utilizing this methodology, we tested 155 hybrid DFAs available in the LIBXC library using FCI and CCSD(T) methods as a reference. We have found that a group of functionals produces very decent XC potentials, mainly those with a large mixture of Hartree-Fock exchange. Moreover, the value of IP strongly depends on the XC potential quality. On the other hand, we show that the XC energy is dominated by functional-driven error, which in some cases leads to substantial errors in electronic densities. The study shows new directions for constructing more accurate XC functionals within the KS-DFT framework.

On the Simulation of Photoreactions Using Restricted Open-Shell Kohn-Sham Theory.

It is a well-established standard to describe ground-state chemical reactions at an ab initio level of multi-electron theory. Fast reactions can be directly simulated. The most widely used approach is density functional theory for the electronic structure in combination with molecular dynamics for the nuclear motion. This approach is known as ab initio molecular dynamics. In contrast, the simulation of excited-state reactions at this level of theory is significantly more difficult. It turns out that the self-consistent solution of the Kohn-Sham equations is not easily reached in excited-state simulations. The first program that solved this problem was the Car-Parrinello molecular dynamics code, using restricted open-shell Kohn-Sham theory. Meanwhile, there are alternatives, most prominently the Q-Chem code, which widens the range of applications. The present study investigates the suitability of both codes for the molecular dynamics simulation of excited-state motion and presents applications to photoreactions.

Application of the Adiabatic Connection Random Phase Approximation to Electron-Nucleus Hyperfine Coupling Constants.

The electron-nucleus hyperfine coupling constant is a challenging property for density functional methods. For accurate results, hybrid functionals with a large amount of exact exchange are often needed and there is no clear "one-for-all" functional which describes the hyperfine coupling interaction for a large set of nuclei. To alleviate this unfavorable situation, we apply the adiabatic connection random phase approximation (RPA) in its post-Kohn-Sham fashion to this property as a first test. For simplicity, only the Fermi-contact and spin-dipole terms are calculated within the nonrelativistic and the scalar-relativistic exact two-component framework. This requires to solve a single coupled-perturbed Kohn-Sham equation to evaluate the relaxed density matrix, which comes with a modest increase in computational demands. RPA performs remarkably well and substantially improves upon its Kohn-Sham (KS) starting point while also reducing the dependence on the KS reference. For main-group systems, RPA outperforms global, range-separated, and local hybrid functionals─at similar computational costs. For transition-metal compounds and lanthanide complexes, a similar performance as for hybrid functionals is observed. In contrast, related post-Hartree-Fock methods such as Møller-Plesset perturbation theory or CC2 perform worse than semilocal density functionals.

Towards chemical accuracy using a multi-mesh adaptive finite element method in all-electron density functional theory

Chemical accuracy serves as an important metric for assessing the effectiveness of the numerical method in Kohn–Sham density functional theory. It is found that to achieve chemical accuracy, not only the Kohn–Sham wavefunctions but also the Hartree potential, should be approximated accurately. Under the adaptive finite element framework, this can be implemented by constructing the a posteriori error indicator based on approximations of the aforementioned two quantities. However, this way results in a large amount of computational cost. To reduce the computational cost, we propose a novel solver for the Kohn–Sham equation by using the multi-mesh adaptive technique, in which the Kohn–Sham equation and the Poisson equation are solved in two different meshes on the same computational domain, respectively. With the proposed method, chemical accuracy can be achieved with less computational consumption compared with the adaptive method on a single mesh, as demonstrated in a number of numerical experiments.

Revisiting Artifacts of Kohn-Sham Density Functionals for Biosimulation.

We revisit the problem of unphysical charge density delocalization/fractionalization induced by the self-interaction error of common approximate Kohn-Sham (KS) density functional theory functionals on simulation of small to medium-sized proteins in a vacuum. Aside from producing unphysical electron densities and total energies, the vanishing of the HOMO-LUMO gap associated with the unphysical charge delocalization leads to an unphysical low-energy spectrum and catastrophic failure of most popular solvers for the KS self-consistent field (SCF) problem. We apply a robust quasi-Newton SCF solver [ Phys. Chem. Chem. Phys. 2024, 26, 6557] to obtain solutions for some of these difficult cases. The anatomy of the charge delocalization is revealed by the natural deformation orbitals obtained from the density matrix difference between the Hartree-Fock and KS solutions; the charge delocalization not only can occur between charged fragments (such as in zwitterionic polypeptides) but also involves neutral fragments. The vanishing-gap phenomenon and troublesome SCF convergence are both attributed to the unphysical KS Fock operator eigenspectra of molecular fragments (e.g., amino acids or their side chains). Analysis of amino acid pairs suggests that the unphysical charge delocalization can be partially ameliorated by the use of some range-separated hybrid functionals but not by semilocal or standard hybrid functionals. Last, we demonstrate that solutions without the unphysical charge delocalization can be located even for semilocal KS functionals highly prone to such defects, but such solutions have non-Aufbau character and are unstable with respect to mixing of the non-overlapping "frontier" orbitals. Caution should be exercised when unexpectedly small (or vanishing) HOMO-LUMO gaps and atypical SCF convergence patterns (e.g., oscillatory) are observed in KS DFT simulations in any context (bio or otherwise).

Density functional theory from spherically symmetric densities: Ground and excited states of Coulomb systems.

Recently, Theophilou [J. Chem. Phys. 149, 074104 (2018)] proposed a peculiar version of the density functional theory by showing that the set of spherical averages of the density around the nuclei determines uniquely the external potential in atoms, molecules, and solids. Here, this novel theory is extended to individual excited states. The generalization is based on the method developed in the series of papers by Ayers, Levy, and Nagy [Phys. Rev. A 85, 042518 (2012)]. Generalized Hohenberg-Kohn theorems are proved to the set of spherically symmetric densities using constrained search. A universal variational functional for the sum of the kinetic and electron-electron repulsion energies is constructed. The functional is appropriate for the ground state and all bound excited states. Euler equations and Kohn-Sham equations for the set are derived. The Euler equations can be rewritten as Schrödinger-like equations for the square root of the radial densities, and the effective potentials in them can be expressed in terms of wave function expectation values. The Hartree plus exchange-correlation potentials can be given by the difference of the interacting and the non-interacting effective potentials.

Local existence and uniqueness of solutions to the time-dependent Kohn–Sham equations coupled with classical nuclear dynamics

We prove the short-time existence and uniqueness of solutions to the initial-value problem associated with a class of time-dependent Kohn–Sham equations coupled with Newtonian nuclear dynamics, combining Yajima's theory for time-dependent Hamiltonians with Duhamel's principle, based on suitable Lipschitz estimates. We consider a pure power exchange term within a generalisation of the so-called local-density approximation, identifying a range of exponents for the existence and uniqueness of H2 solutions to the Kohn–Sham equations.

Impact of SOFC anode carbon deposition and hydrogen adsorption on the cell performance by molecular simulation.

This study primarily investigates the changes in carbon adsorption capacity and hydrogen adsorption capacity on the anode catalyst surface when using methane fuel and mixed gas fuel as the anode fuel for SOFC systems. To reduce the carbon adsorption capacity of the commonly used anode catalyst-nickel-based catalysts-towards hydrocarbon fuels, copper and gold are doped into the nickel-based catalysts to compare the effects on carbon and hydrogen adsorption capacities. Moreover, aside from calculating the carbon and hydrogen adsorption capacities, this project also evaluates the impact of mixed gas effects and doping effects on SOFC performance through the analysis of hydrogen diffusion coefficients and performance polarization curves. The findings reveal a noteworthy enhancement in the diffusion coefficient of syngas within the Au-doped Ni catalyst, showing an improvement of up to 45.46% at 973K. Furthermore, the electrical power generated by syngas in the Au-doped Ni catalyst at 973K demonstrates an increase of up to 12.06%. This study primarily employs DFT to calculate the carbon and hydrogen adsorption energies on methane, utilizing CASTEP for the calculations. During these calculations, the adsorption energy is determined through a three-layer surface approach, in conjunction with the Kohn-Sham equations, combining the Generalized Gradient Approximation and ultrasoft pseudopotentials for TS-search calculations. On the other hand, this project will analyze the diffusion coefficient of hydrogen on the anode catalyst using MD methods combined with the ReaxFF potential field, with GULP being utilized to complete all dynamics calculation theories. Finally, the project will analyze the performance of SOFC cells, incorporating relevant numerical equations with Matlab for numerical analysis.

Structural and optoelectronic properties of three different phases of Zn2V2O7 compound under pressure: FP-LAPW approach

The structural, electronic, and optical properties of the zinc pyrovanadate (Zn2V2O7) under hydrostatic pressure are investigated using the first-principle calculations. The Kohn–Sham equations are solved by the full potential linearized augmented plane wave approach. We have chosen the Perdew-Burke-Ernzerhof generalized gradient approximation for calculation of the exchange-correlation potentials. By the structural optimization, it is found that α-Zn2V2O7 has higher stability than β - and γ-Zn2V2O7. Calculated density of states spectra show that the hydrostatic pressure has a considerable effect on the O-p, Zn-d and V-d states on the top of valence band. By increasing pressure, the band gap value increases from α to β phase, then decrease at γ phase. By pressure, the absorption spectra are shifted towards high energies as the blue shift. The metallic character of α and β− Zn2V2O7 is observed at 5–10 eV. The plasmon energies increase from α to β phase, with pressure. There is a close agreement between the calculated absorption spectra and the experiment. The high plasmon energy and the wide absorption energy range makes this compound suitable for the optoelectronic devices.

MISTER-T: An open-source software package for quantum optimal control of multi-electron systems on arbitrary geometries

We present an open-source software package, MISTER-T (Manipulating an Interacting System of Total Electrons in Real-Time), for the quantum optimal control of interacting electrons within a time-dependent Kohn-Sham formalism. In contrast to other implementations restricted to simple models on rectangular domains, our method enables quantum optimal control calculations for multi-electron systems (in the effective mass formulation) on nonuniform meshes with arbitrary two-dimensional cross-sectional geometries. Our approach is enabled by forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency. We provide several examples of the versatility and efficiency of the MISTER-T code in handling complex geometries and quantum control mechanisms. The capabilities of the MISTER-T code provide insight into the implications of varying propagation times and local control mechanisms to understand a variety of strategies for manipulating electron dynamics in these complex systems. Program summaryProgram Title: MISTER-TCPC Library link to program files:https://doi.org/10.17632/psymy4ddnw.1Licensing provisions: GNU General Public License 3Programming language: MATLABSupplementary material: animated movies of total electron densities under the influence of optimal control fields for (1) an asymmetric double-well potential for long propagation times, (2) an asymmetric double-well potential for short propagation times, and (3) a triple-well potential with a position-dependent effective mass.Nature of problem: The MISTER-T code solves quantum optimal control problems for interacting electrons within a time-dependent Kohn-Sham formalism. It can handle two-dimensional systems with arbitrary cross-sectional geometries within the effective mass formulation. The user-friendly code uses forward and backward propagator integration methods to evolve the Kohn-Sham equations with a pseudoskeleton decomposition algorithm for enhanced computational efficiency.Solution method: iterative solution of the quantum optimal control equations using finite element methods, effective mass formulation, pseudoskeleton decomposition, sparse matrix linear algebra, and nonuniform fast Fourier transforms.

Extended N-centered ensemble density functional theory of double electronic excitations.

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

Meta-GGA study of 2D AlN/BN planer heterostructure and performance enhancement via strain engineering.

The 2D AlN/BN planer heterostructure is a promising wide band gap semiconductor, but systematic studies of its bandgap and optical characteristics under applied strain are scarce. Here, the engineering property of 2D AlN/BN comprising bandgap nature transition and optical absorption capability (from unstrained to strained) have been investigated using density functional theory calculations. The formation energy calculations confirm the stability of the simulated nanoheterostructure. The electronic band structure calculations demonstrate that nanoheterostructure is an indirect bandgap material with a large bandgap of 5.26 eV, which can be modified effectively by applying strain. According to the calculations, the transition from indirect to direct band gap behavior has been observed at +15% biaxial strain with 2.71 eV band gap energy. Meanwhile, calculations for optical absorption and dielectric function reveal that the system has significant absorption peaks in the ultraviolet region which are very sensitive to applied strain. As strain increases, the first absorption peaks are shifted towards a lower energy range from 5.73 eV (Ꜫ= 0 %) to 3.76 eV (Ꜫ = +15%), which features an enhancement of optical absorption for solar and solar-blind regions. Furthermore, we determined that the band edge positions in 2D AlN/BN straddled the water redox potential under strain, indicating its effectiveness as a proficient photocatalyst. These characteristics make 2D AlN/BN planer nanoheterostructure a promising candidate for applications in optoelectronics and photocatalytic water splitting performance. First principles computations based on density functional theory were employed to carry out all the calculations with a self-consistent approach. For solving the Kohn-Sham equations, the first principles dependent full-potential linearized augmented plane wave scheme were adopted. For addressing the exchange-correlation effects, the generalized gradient approximation of PBEsol functional was used. To prevent interaction between the periodic images, we have inserted a vacuum region of 10 Å in the z-direction. Non-negligible weak dispersion corrections in nanoheterostructure were considered by using the DFT-D3 method of Grimme's. The locally modified Becke-Johnson (lmBJ) exchange potential has also been applied to compute electronic and optical properties in this research to obtain more accurate information.

Calculating Potential Energy Surfaces with Quantum Computers by Measuring Only the Density Along Adiabatic Transitions.

We show that chemically accurate potential energy surfaces (PESs) can be generated from quantum computers by measuring only the density along an adiabatic transition between different molecular geometries. In lieu of using phase estimation, the energy is evaluated by performing line-integration using the inverted real-space time-dependent density functional theory Kohn-Sham (KS) potential obtained from the geometry-varying densities of the full wave function. The accuracy of this method depends on the validity of the adiabatic evolution itself and the potential inversion process (which is theoretically exact but can be numerically unstable), whereas the total evolution time is the defining factor for the precision of phase estimation. We examine the method with a one-dimensional system of two electrons for both the ground and first triplet states in first quantization, as well as the ground state of three- and four-electron systems in second quantization. It is shown that few accurate measurements can be utilized to obtain chemical accuracy across the full potential energy curve, with a shorter propagation time than may be required using phase estimation for a similar accuracy. We also show that an accurate potential energy curve can be calculated by making many imprecise density measurements (using a few shots) along the time evolution and smoothing the resulting density evolution. Finally, it is important to note that the method is able to classically provide a check of its own accuracy by comparing the density resulting from a time-independent KS calculation using the inverted potential with the measured density. This can be used to determine whether longer adiabatic evolution times are required to satisfy the adiabatic theorem.

Basis Set Requirements of σ-Functionals for Gaussian- and Slater-Type Basis Functions and Comparison with Range-Separated Hybrid and Double Hybrid Functionals.

σ-Functionals belong to the class of Kohn-Sham (KS) correlation functionals based on the adiabatic-connection fluctuation-dissipation theorem and are technically closely related to the random phase approximation (RPA). They have the same computational demand as the latter, with the computational effort of an energy evaluation for both methods being lower than that of a preceding hybrid DFT calculation for typical systems but yield much higher accuracy, reaching chemical accuracy of 1 kcal/mol for quantities such as reactions and transition energies in main group chemistry. In previous work on σ-functionals, rather large Gaussian basis sets have been used. Here, we investigate the actual basis set requirements of σ-functionals and present three setups that employ smaller Gaussian basis sets ranging from quadruple-ζ (QZ) to triple-ζ (TZ) quality and represent a good compromise between accuracy and computational efficiency. Furthermore, we introduce an implementation of σ-functionals based on Slater-type basis sets and present two setups of QZ and TZ quality for this implementation. We test the accuracy of these setups on a large database of various physical properties and types of reactions, as well as equilibrium geometries and vibrational frequencies. As expected, the accuracy of σ-functional calculations becomes somewhat lower with a decreasing basis set size. However, for all setups considered here, calculations with σ-functionals are clearly more accurate than those within the RPA and even more so than those of the conventional KS methods. For the smallest setup using Gaussian-type basis functions and Slater-type basis functions, we introduce a reparametrization that reduces the loss in accuracy due to the basis set error to some extent. A comparison with the range-separated hybrid ωB97X-V and the double hybrid DSD-BLYP-D3 shows that σ functionals outperform in accuracy both of these accurate and, for their class, representative functionals.

GPAW: An open Python package for electronic structure calculations.

We review the GPAW open-source Python package for electronic structure calculations. GPAW is based on the projector-augmented wave method and can solve the self-consistent density functional theory (DFT) equations using three different wave-function representations, namely real-space grids, plane waves, and numerical atomic orbitals. The three representations are complementary and mutually independent and can be connected by transformations via the real-space grid. This multi-basis feature renders GPAW highly versatile and unique among similar codes. By virtue of its modular structure, the GPAW code constitutes an ideal platform for the implementation of new features and methodologies. Moreover, it is well integrated with the Atomic Simulation Environment (ASE), providing a flexible and dynamic user interface. In addition to ground-state DFT calculations, GPAW supports many-body GW band structures, optical excitations from the Bethe-Salpeter Equation, variational calculations of excited states in molecules and solids via direct optimization, and real-time propagation of the Kohn-Sham equations within time-dependent DFT. A range of more advanced methods to describe magnetic excitations and non-collinear magnetism in solids are also now available. In addition, GPAW can calculate non-linear optical tensors of solids, charged crystal point defects, and much more. Recently, support for graphics processing unit (GPU) acceleration has been achieved with minor modifications to the GPAW code thanks to the CuPy library. We end the review with an outlook, describing some future plans for GPAW.

Fragment-Based Deep Learning for Simultaneous Prediction of Polarizabilities and NMR Shieldings of Macromolecules and Their Aggregates.

Simultaneous prediction of the molecular response properties, such as polarizability and the NMR shielding constant, at a low computational cost is an unresolved issue. We propose to combine a linear-scaling generalized energy-based fragmentation (GEBF) method and deep learning (DL) with both molecular and atomic information-theoretic approach (ITA) quantities as effective descriptors. In GEBF, the total molecular polarizability can be assembled as a linear combination of the corresponding quantities calculated from a set of small embedded subsystems in GEBF. In the new GEBF-DL(ITA) protocol, one can predict subsystem polarizabilities based on the corresponding molecular wave function (thus electron density and ITA quantities) and DL model rather than calculate them from the computationally intensive coupled-perturbed Hartree-Fock or Kohn-Sham equations and finally obtain the total molecular polarizability via a linear combination equation. As a proof-of-concept application, we predict the molecular polarizabilities of large proteins and protein aggregates. GEBF-DL(ITA) is shown to be as accurate enough as GEBF, with mean absolute percentage error <1%. For the largest protein aggregate (>4000 atoms), GEBF-DL(ITA) gains a speedup ratio of 3 compared with GEBF. It is anticipated that when more advanced electronic structure methods are used, this advantage will be more appealing. Moreover, one can also predict the NMR chemical shieldings of proteins with reasonably good accuracy. Overall, the cost-efficient GEBF-DL(ITA) protocol should be a robust theoretical tool for simultaneously predicting polarizabilities and NMR shieldings of large systems.

Inverting the Kohn–Sham equations with physics-informed machine learning

Electronic structure theory calculations offer an understanding of matter at the quantum level, complementing experimental studies in materials science and chemistry. One of the most widely used methods, density functional theory, maps a set of real interacting electrons to a set of fictitious non-interacting electrons that share the same probability density. Ensuring that the density remains the same depends on the exchange-correlation (XC) energy and, by a derivative, the XC potential. Inversions provide a method to obtain exact XC potentials from target electronic densities, in hopes of gaining insights into accuracy-boosting approximations. Neural networks provide a new avenue to perform inversions by learning the mapping from density to potential. In this work, we learn this mapping using physics-informed machine learning methods, namely physics informed neural networks and Fourier neural operators. We demonstrate the capabilities of these two methods on a dataset of one-dimensional atomic and molecular models. The capabilities of each approach are discussed in conjunction with this proof-of-concept presentation. The primary finding of our investigation is that the combination of both approaches has the greatest potential for inverting the Kohn–Sham equations at scale.

Two-Electron Scenario of High-Order Harmonics Generation by Atom in an Intense Infrared Field and Attosecond Pulse

The two-electron scenario is proposed for the high-order harmonic generation (HHG) by an atom interacting with an intense infrared field and attosecond pulse. The two-electron dynamics involved in the proposed scenario is realized for the specific attosecond pulse, which can excite the resonance between the valence and deeper shells. Within the numerical solution of the time-dependent Kohn–Sham equations, we analyze the contribution of the single [Phys. Rev. A 98, 063433 (2018)] and two-electron scenarios of HHG by decreasing the duration of attosecond pulse having carrier frequency detuned from the atomic resonance. Favorable conditions are formulated for the realization of a two-electron scenario, which causes the enhancement of the harmonic yield in the spectral range exceeding the cutoff energy of the HHG spectrum in the infrared field.

Natural determinant reference functional theory.

The natural determinant reference (NDR) or principal natural determinant is the Slater determinant comprised of the N most strongly occupied natural orbitals of an N-electron state of interest. Unlike the Kohn-Sham (KS) determinant, which yields the exact ground-state density, the NDR only yields the best idempotent approximation to the interacting one-particle reduced density matrix, but it is well-defined in common atom-centered basis sets and is representation-invariant. We show that the under-determination problem of prior attempts to define a ground-state energy functional of the NDR is overcome in a grand-canonical ensemble framework at the zero-temperature limit. The resulting grand potential functional of the NDR ensemble affords the variational determination of the ground state energy, its NDR (ensemble), and select ionization potentials and electron affinities. The NDR functional theory can be viewed as an "exactification" of orbital optimization and empirical generalized KS methods. NDR functionals depending on the noninteracting Hamiltonian do not require troublesome KS-inversion or optimized effective potentials.

A first-principles theoretical study of structural, electronic, and magnetic properties of lead-doped alloys of praseodymium bismuth compounds PrPbxBi1-x

The magnetic, electronic, and structural properties of the cubic phase of lead-doped alloys of praseodymium bismuth compounds with the generic formula PrPbxBi1-x (x = 0, 0.25, 0.50, 0.75, and 1.0) have been reported in this paper by employing the formalism of density functional theory (DFT). For the analysis of physical properties, we have executed the full-potential linearized augmented plane wave plus local orbit (FPLAPW+lo) technique, while the exchange-correlation potentials in the Kohn-Sham equation (KSE) are implemented within the generalized gradient approximation (GGA) extended by the Perdew-Burke-Ernzerhof (PBE) correction. The structural parameters, lattice constants, volume, bulk modulus, pressure derivatives, and energy have been computed with the Wein2k code by fitting total energy through Murnaghan's equation of state. The structural stability of the compounds has been reported from the spin-polarized calculations. The electronic energy bands and total and partial densities of states of the compounds have been calculated in both majority and minority spins, depicting them as metallic. The similar spectrum intensities of the Pr(5d+4f) and (Pb +Bi)2p states account for the majority of the contribution to the density of states near the Fermi energy level. The spin magnetic moments computed for the supercell of the doped compounds have indicated that they are magnetic materials. From the comparison of spin magnetic moments in the PrBi compound, we noticed an improvement in the magnetic moments after doping lead into the PrBi compound.