Kohn-Sham Scheme Research Articles

Strong Correlation in Kohn-Sham Density Functional Theory

We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.

Intrinsic uncertainty on ab initio phase diagram and compound formation energy calculations: BCC Mo–Fe as a test case

AbstractAb initio electronic structure calculations, within the Kohn–Sham scheme of the density functional theory, are often considered reliable and a powerful tool to provide ground state information on intermetallic compounds. More recently, it has been used in multi‐scale intermetallic phase diagram calculations with particular importance when experimental data is lacking. In spite of this recent success, they rely on basic choices of the computational solution of the complicated quantum mechanical problem. Therefore, the calculated phase diagrams may depend on these choices. Here, we concentrate on the influence of some methodological aspects of the ab initio calculations on the resulting phase diagram, for a given statistical mechanics approach. We use the Full Potential‐Linear Augmented Plane Wave and the Projector Augmented Wave Methods to perform electronic structure calculations combined with the cluster variation method in the irregular tetrahedron approximation to calculate the BCC Mo–Fe phase diagram. It is shown that all calculated phase diagrams present similar qualitative features (an asymmetric miscibility gap), but quantitative variations are found depending on some of the basic assumptions adopted for the electronic structure calculations. Based on these results, a “natural” accuracy of the order of ± 1 kJ mol−1 can be estimated for ab initio compound formation energies.

Self-interaction correction in a real-time Kohn-Sham scheme: Access to difficult excitations in time-dependent density functional theory

We present a real-time Kohn-Sham propagation scheme for the self-interaction correction (SIC). The multiplicative Kohn-Sham potential is constructed in real-time and real-space based on the generalized optimized effective potential equation. We demonstrate that this approach yields promising results for a wide range of test systems, including hydrogen terminated silicon clusters, conjugated molecular chains, and molecular charge-transfer systems. We analyze the nature of excitations by calculating transition densities from the time evolution and by evaluating the time-dependent exchange-correlation potential. A properly constructed Kohn-Sham SIC potential shows a time-dependent field-counteracting behavior. These favorable characteristics of the exchange-correlation potential may be lost in approximations such as the SIC-Slater potential.

Communication: Monte Carlo calculation of the exchange energy

In recent generalized Kohn-Sham (GKS) schemes for density functional theory (DFT) Hartree-Fock type exchange is important. In plane waves and grid approaches the high cost of exchange energy calculations makes these GKS considerably more expensive than Kohn-Sham DFT calculations. We develop a stochastic approach for speeding up the calculation of exchange for large systems. We show that stochastic error per particle does not grow and can even decrease with system size (at a given number of iterations). We discuss several alternative approaches and explain how these ideas can be included in the GKS framework.

Generalization of internal density-functional theory and Kohn-Sham scheme to multicomponent self-bound systems, and link with traditional density-functional theory

We generalize the recently developped "internal" Density Functional Theory (DFT) and Kohn-Sham scheme to multicomponent systems. We obtain a general formalism, applicable for the description of multicomponent self-bound systems (as molecules where the nuclei are treated explicitely, atomic nuclei and mix of 3He and 4He droplets), where the fundamental translational symmetry has been treated correctly. The main difference with traditional DFT is the explicit inclusion of center-of-mass correlations in the functional. A large part of the paper is dedicated to the application to molecules, which permits among other to clarify the approximations that underly traditional DFT.

The First and Second Static Electronic Hyperpolarizabilities of Zigzag Boron Nitride Nanotubes. An ab Initio Approach through the Coupled Perturbed Kohn–Sham Scheme

The coupled perturbed Kohn-Sham (CPKS) computational scheme for the evaluation of electric susceptibility tensors in periodic systems, recently implemented in the CRYSTAL code, has been extended to third-order. It is, then, used to obtain static electronic hyperpolarizabilities of zigzag BN nanotubes for the first time. This procedure, which is fully analytic in all key steps, requires a double self-consistent treatment for taking into account the first- and second-order response of the system to the applied field. The performance of different functionals is compared and the B3LYP hybrid is ultimately chosen for calculations on nanotubes having radii as large as R = 20 Å (6-200 atoms in the unit cell). Such large radii are sufficient to give the pure longitudinal component of the (hyper)polarizability tensors to within 1% of the "exact" hexagonal BN monolayer limit. Other tensor components involving the transverse direction converge more slowly. They can, however, be extrapolated to the monolayer limit to within 4% accuracy except for the pure transverse second hyperpolarizability, which has an error of 13% in that limit.

Polarizability and hyperpolarizability of BN zigzag nanotubes calculated by the coupled perturbed Kohn-Sham scheme

Linear and nonlinear electric dipole susceptibilities are evaluated for infinite periodic zigzag BN nanotubes utilizing primarily the coupled perturbed Kohn-Sham scheme recently implemented in the crystal code. The effect of different functionals, basis set, and computational parameters is examined. Most of the calculations are done at the B3LYP/6-31G* level. For electronic linear polarizabilities, substantial differences compared to the uncoupled sum-over-states scheme are found. Much larger radii were considered than in earlier studies, thereby permitting accurate comparison with corresponding properties of the hexagonal monolayer. In addition, we confirmed the dielectric shell model for the linear polarizability, but with a significantly different shell thickness than previously thought. Vibrational (ionic) contributions to the nonlinear susceptibilities are calculated. In doing so, the finite-field--nuclear-relaxation (FF--NR) method was employed for the transverse components of the (6,0), (9,0), and (12,0) nanotubes. Aside from being computationally more efficient than other procedures, this method includes anharmonicity effects through first order and, as shown, is readily applied to key dynamic as well as static properties (and yields the static linear polarizability as well). Our calculated nonlinear vibrational susceptibilities sometimes exceed, or even greatly exceed, the corresponding static electronic susceptibility. In such cases, the relative magnitude of the vibrational contribution grows substantially with tube radius over the range considered. Future plans include extending these FF--NR calculations to large nanotubes and to the longitudinal (periodic) direction as well.

Exchange functional by a range-separated exchange hole

An approximation to the exchange-hole density is proposed for the evaluation of the exact exchange energy in electronic structure calculations within the density-functional theory and the Kohn-Sham scheme. Based on the localized nature of density matrix, the exchange hole is divided into the short-range (SR) and long-range (LR) parts by using an adequate filter function, where the LR part is deduced by matching of moments with the exactly calculated SR counterpart, ensuring the correct asymptotic $\ensuremath{-}1/r$ behavior of the exchange potential. With this division, the time-consuming integration is truncated at a certain interaction range, largely reducing the computation cost. The total energies, exchange energies, exchange potentials, and eigenvalues of the highest-occupied orbitals are calculated for the noble-gas atoms. The close agreement of the results with the exact values suggests the validity of the approximation.

A DFT Study of the Reactivity Indexes of Ionic [4 + 2+] Diels-Alder Cycloaddition to Nitrilium and Immonium Ions

The global electrophilicity index, defined within the conceptual density functional theory (DFT), was used to classify the dienes and dienophiles currently used in Diels-Alder reactions on a unique scale of electrophilicity. The index, obtained within the Kohn-Sham scheme, is based on the HOMO and LUMO energies. A systematic study of the global reactivity indexes of the reagents involved in formal [4 + 2+] Diels-Alder cycloaddition reactions of nitrilium and immonium ions with isoprene is presented. Keywords: Diels-Alder, reactivity indexes, nitrilium, immonium, electrophilicity, nucleophilicity, Huckel molecular orbital (HMO) theory, electrofugality and nucleofugality, Berny analytical gradient, isoprene

Electronic Transport Calculations Using Maximally-Localized Wannier Functions

I present a method to calculate the ballistic transport properties of atomic-scale structures under bias. The electronic structure of the system is calculated using the Kohn-Sham scheme of density functional theory (DFT). The DFT eigenvectors are then transformed into a set of maximally localized Wannier functions (MLWFs) [N. Marzari and D. Vanderbilt, Phys. Rev. B 56 (1997) 12847]. The MLWFs are used as a minimal basis set to obtain the Hamitonian matrices of the scattering region and the adjacent leads, which are needed for transport calculation using the nonequilibrium Green's function formalism. The coupling of the scattering region to the semi-infinite leads is described by the self-energies of the leads. Using the nonequilibrium Green's function method, one calculates self-consistently the charge distribution of the system under bias and evaluates the transmission and current through the system. To solve the Poisson equation within the scheme of MLWFs I introduce a computationally efficient method. The method is applied to a molecular hydrogen contact in two transition metal monatomic wires (Cu and Pt). It is found that for Pt the I–V characteristics is approximately linear dependence, however, for Cu the I–V characteristics manifests a linear dependence at low bias voltages and exhibits apparent nonlinearity at higher bias voltages. I have also calculated the transmission in the zero bias voltage limit for a single CO molecule adsorbed on Cu and Pt monatomic wires. While a chemical scissor effect occurs for the Cu monatomic wire with an adsorbed CO molecule, it is absent for the Pt monatomic wire due to the contribution of d-orbitals at the Fermi energy.

Frozen density embedding with hybrid functionals

The Kohn-Sham equations with constrained electron density are extended to hybrid exchange-correlation (XC) functionals. We derive the frozen density embedding generalized Kohn-Sham (FDE-GKS) scheme which allows to treat the nonlocal exact-exchange in the subsystems. For practical calculations we propose an approximated version of the FDE-GKS in which the nonadditive exchange potential is computed at a semilocal level. The proposed method is applied to compute the ground-state electronic properties of small test systems and selected DNA base pairs. The results of calculations employing the hierarchy of XC functionals BLYP/B3LYP/BHLYP and PBE/PBE0 are presented, in order to analyze the effect of nonlocal exchange contributions, and compared with reference coupled-cluster singles and doubles results. We find that the use of hybrid functionals leads to a significant improvement in the description of ground-state electronic properties of the investigated systems. The semilocal version of the FDE-GKS correctly reproduces the dipole and the electron density distribution of the exact GKS supramolecular system, with errors smaller than the ones obtained using conventional semilocal XC functionals.

DFT performance for the hole transfer parameters in DNA π stacks

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

Publisher’s Note: Time-dependent internal density functional theory formalism and Kohn-Sham scheme for self-bound systems [Phys. Rev. C80, 054614 (2009)

Publisher’s Note: Time-dependent internal density functional theory formalism and Kohn-Sham scheme for self-bound systems [Phys. Rev. C<b>80</b>, 054614 (2009)

Energy transfer and Förster’s dipole coupling approximation investigated in a real-time Kohn-Sham scheme

We present a scheme to investigate energy transfer by real-time propagation of the Kohn-Sham equations. The scheme's purpose is to check and go beyond the dipole coupling approximation underlying a F\orster-type energy transfer, and to obtain information about the coupling on the grounds of the density-functional theory. We observe deviations from the dipole coupling approximation for small molecules.

Elongation cutoff technique at Kohn–Sham level of theory

AbstractThe elongation cutoff technique ELG/C belongs to fragmentation methods. It uses the concept of locality and takes into account the sparsity of Kohn–Sham (KS) matrix in regionally localized molecular orbital basis set. In this article, a linear‐scaling implementation of ELG/C method at Kohn–Sham (KS) level of theory is presented. Its efficiency and accuracy is discussed for model system. The ELG/C KS scheme is based on local exchange‐correlation space approximation. Such approximation improves efficiency of the method and does not introduce a significant error. The current analysis includes the overall CPU (central processing unit) time and its most time consuming steps. The obtained results indicate that the ELG/C is a very efficient and accurate computational scheme. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

A massively multicore parallelization of the Kohn‐Sham energy gradients

In a previous article [Brown et al., J Chem Theory Comput 2009, 4, 1620], we described a quadrature-based formulation of the Kohn-Sham Coulomb problem that allows for efficient parallelization over thousands of small processor cores. Here, we present the analytic gradients of this modified Kohn-Sham scheme, and describe the parallel implementation of the gradients on a numerical accelerator architecture. We demonstrate an order-of-magnitude acceleration for the combined energy and gradient calculation over a conventional single-core implementation.

Direct mapping between exchange potentials of Hartree–Fock and Kohn–Sham schemes as origin of orbital proximity

It is found that for closed-l-shell atoms, the exact local exchange potential v(x)(r) calculated in the exchange-only Kohn-Sham (KS) scheme of the density functional theory (DFT) is very well represented within the region of every atomic shell by each of the suitably shifted potentials obtained with the nonlocal Fock exchange operator for the individual Hartree-Fock (HF) orbitals belonging to this shell. This newly revealed property is not related to the well-known steplike shell structure in the response part of v(x)(r), but it results from specific relations satisfied by the HF orbital exchange potentials. These relations explain the outstanding proximity of the occupied HF and exchange-only KS orbitals as well as the high quality of the Krieger-Li-Iafrate and localized HF (or, equivalently, common-energy-denominator) approximations to the DFT exchange potential v(x)(r). Another highly accurate representation of v(x)(r) is given by the continuous piecewise function built of shell-specific exchange potentials, each defined as the weighted average of the shifted orbital exchange potentials corresponding to a given shell. The constant shifts added to the HF orbital exchange potentials, to map them onto v(x)(r), are nearly equal to the differences between the energies of the corresponding KS and HF orbitals. It is discussed why these differences are positive and grow when the respective orbital energies become lower for inner orbitals.

Time-dependent internal density functional theory formalism and Kohn-Sham scheme for self-bound systems

We generalize to the time-dependent case the stationary Internal DFT / Kohn-Sham formalism presented in Ref. [14]. We prove that, in the time-dependent case, the internal properties of a self-bound system (as an atomic nuclei) are all defined by the internal one-body density and the initial state. We set-up a time-dependent Internal Kohn-Sham scheme as a practical way to compute the internal density. The main difference with the traditional DFT / Kohn-Sham formalism is the inclusion of the center-of-mass correlations in the functional.

Calculation of the dielectric constant ϵ and first nonlinear susceptibility χ(2) of crystalline potassium dihydrogen phosphate by the coupled perturbed Hartree–Fock and coupled perturbed Kohn–Sham schemes as implemented in the CRYSTAL code

The high-frequency dielectric varepsilon and the first nonlinear electric susceptibility chi((2)) tensors of crystalline potassium dihydrogen phosphate (KH(2)PO(4)) are calculated by using the coupled perturbed Hartree-Fock and Kohn-Sham methods as implemented in the CRYSTAL code. The effect of basis sets of increasing size on varepsilon and chi((2)) is explored. Five different levels of theory, namely, local-density approximation, generalized gradient approximation (PBE), hybrids (B3LYP and PBE0), and HF are compared using the experimental and theoretical structures corresponding not only to the tetragonal geometry I4d2 at room temperature but also to the orthorhombic phase Fdd2 at low temperature. Comparison between the two phases and their optical behavior is made. The calculated results for the tetragonal phase are in good agreement with the experimental data.

Density functional theory and Kohn-Sham scheme for self-bound systems

We demonstrate how the separation of the total energy of a self-bound system into a functional of the internal one-body Fermionic density and a function of an arbitrary wave vector describing the center-of-mass kinetic energy can be used to set up an ``internal'' Kohn-Sham scheme.