Kohn-Sham Procedure Research Articles

Solution of the v-representability problem on a one-dimensional torus

Abstract We provide a solution to the v-representability problem for a non-relativistic quantum many-particle system on a one-dimensional torus domain in terms of Sobolev spaces and their duals. Any one-particle density that is square-integrable, has a square-integrable weak derivative, and is gapped away from zero can be realized from the solution of a many-particle Schrödinger equation, with or without interactions, by choosing a corresponding external potential. This potential can contain a distributional contribution but still gives rise to a self-adjoint Hamiltonian. Importantly, this allows for a well-defined Kohn-Sham procedure but, on the other hand, invalidates the usual proof of the Hohenberg-Kohn theorem.

Calculation of the Infrared Intensity of Crystalline Systems. A Comparison of Three Strategies Based on Berry Phase, Wannier Function, and Coupled-Perturbed Kohn–Sham Methods

Three alternative strategies for the calculation of the IR intensity of crystalline systems, as determined by Born charges, have been implemented in the Crystal code, using a Gaussian type basis set. One uses the Berry phase (BP) algorithm to compute the dipole moment; another does so, instead, through well localized crystalline orbitals (Wannier functions, WF); and the third is based on a coupled perturbed Hartree–Fock or Kohn–Sham procedure (CP). In WF and BP, the derivative of the dipole moment with respect to the atomic coordinates is evaluated numerically, whereas in CP it is analytical. In the three cases, very different numerical schemes are utilized, so that the equivalence of the obtained IR intensities is not ensured a priori but instead is the result of the high numerical accuracy of the many computational steps involved. The main aspects of the three schemes are briefly recalled, and the dependence of the results on the computational parameters (number of k points in reciprocal space, toleranc...

Derivative Discontinuity in the Strong-Interaction Limit of Density-Functional Theory

We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on strongly interacting systems, this functional yields exact features crucial for important applications such as quantum transport. In particular, the steplike structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accurate quantitative agreement with exact many-body chemical potentials. While it can be shown that a sharp derivative discontinuity is present only in the infinitely strongly correlated limit, at finite correlation regimes we observe a slightly smoothened discontinuity, with qualitative and quantitative features that improve with increasing correlation. From the fundamental point of view, our results obtain the derivative discontinuity without making the assumptions used in its standard derivation, offering independent support for its existence.

Analytic density functionals with initial-state dependence and memory

We analytically construct the wave function that, for a given initial state, produces a prescribed density for a quantum ring with two noninteracting particles in a singlet state. In this case the initial state is completely determined by the initial density, the initial time derivative of the density and a single integer that characterizes the (angular) momentum of the system. We then give an exact analytic expression for the exchange-correlation potential that relates two noninteracting systems with different initial states. This is used to demonstrate how the Kohn-Sham procedure predicts the density of a reference system without the need of solving the reference system's Schr\"odinger equation. We further numerically construct the exchange-correlation potential for an analytically solvable system of two electrons on a quantum ring with a squared cosine two-body interaction. For the same case we derive an explicit analytic expression for the exchange-correlation kernel and analyze its frequency dependence (memory) in detail. We compare the result to simple adiabatic approximations and investigate the single-pole approximation. These approximations fail to describe the doubly excited states, but perform well in describing the singly excited states.

Analytic derivatives for the XYG3 type of doubly hybrid density functionals: Theory, implementation, and assessment

We present a theoretical development of the equations required to perform an analytic geometry optimization of a molecular system using the XYG3 type of doubly hybrid (xDH) functionals. In contrast to the well-established B2PLYP type of DH functionals, the energy expressions in the xDH functionals are constructed by using density and orbital information from another standard Kohn-Sham (KS) functional (e.g., B3LYP) for doing the self-consistent field calculations. Thus, the xDH functionals are nonvariational in both the hybrid density functional part and the second-order perturbation part, each of which requires formally to solve a coupled-perturbed KS equation. An implementation is reported here which combines the two parts by defining a total Lagrangian such that only a single set of the Z-vector equations need to be solved. The computational cost with our implementation is of the same order as those for the conventional Møller-Plesset theory to the second order (MP2) and B2PLYP. Systematic test calculations are provided for covalently bonded molecules as well as compounds involving the intramolecular nonbonded interactions for the main group elements. Satisfactory performance of the xDH functionals demonstrates that the extra computer time on top of the conventional KS procedure is well-invested, in particular, when the standard KS functionals and MP2 as well, are problematic.

Energy-gap opening and quenching in graphene under periodic external potentials

We investigated the effects of periodic external potentials on properties of charge carriers in graphene using both the first-principles method based on density functional theory (DFT) and a theoretical approach based on a generalized effective spinor Hamiltonian. DFT calculations were done in a modified Kohn-Sham procedure that includes the effects of the periodic external potential. Unexpected energy band gap opening and quenching were predicted for the graphene superlattice with two symmetrical sublattices and those with two unsymmetrical sublattices, respectively. Theoretical analysis based on the spinor Hamiltonian showed that the correlations between pseudospins of Dirac fermions in graphene and the applied external potential, and the potential-induced intervalley scattering, play important roles in energy-gap opening and quenching.

Intrinsic-density functionals

The Hohenberg-Kohn theorem and Kohn-Sham procedure are extended to functionals of the localized intrinsic density of a self-bound system such as a nucleus. After defining the intrinsic-density functional, we modify the usual Kohn-Sham procedure slightly to evaluate the mean-field approximation to the functional, and carefully describe the construction of the leading corrections for a system of fermions in one dimension with a spin-degeneracy equal to the number of particles $N$. Despite the fact that the corrections are complicated and nonlocal, we are able to construct a local Skyrme-like intrinsic-density functional that, while different from the exact functional, shares with it a minimum value equal to the exact ground-state energy at the exact ground-state intrinsic density, to next-to-leading order in $1/N$. We briefly discuss implications for real Skyrme functionals.

Kohn-Sham inversion and iterative energy minimization

A scheme is presented for inverting the Kohn-Sham procedure so that the local potential is obtained that has a given electron density as its ground state. The central idea is to constrain the electron density using a position-dependent Lagrange multiplier and perform iterative minimization with respect to the orbitals. The required potential is obtained from the Lagrange multiplier. The method is then applied to the iterative minimization of the Kohn-Sham energy functional. In contrast to conventional mixing schemes presently employed, at each stage of the minimization orbitals and an electron density are consistent and a variational upper bound of the energy is obtained. The convergence of the iterations onto the ground state can be very rapid. Real-space implementation with applications to Li clusters, and $\mathrm{C}{\mathrm{H}}_{4}$ and $\mathrm{C}{\mathrm{O}}_{2}$ molecules is presented.

Model density approach to the Kohn–Sham problem: Efficient extension of the density fitting technique

AbstractWe present a novel procedure for treating the exchange‐correlation contributions in the Kohn–Sham procedure. The approach proposed is fully variational and closely related to the so‐called “fitting functions” method for the Coulomb Hartree problem; in fact, the method consistently uses this auxiliary representation of the electron density to determine the exchange‐correlation contributions. The exchange‐correlation potential and its matrix elements in a basis set of localized (atomic) orbitals can be evaluated by reusing the three‐center Coulomb integrals involving fitting functions, while the computational cost of the remaining numerical integration is significantly reduced and scales only linearly with the size of the auxiliary basis. We tested the approach extensively for a large set of atoms and small molecules as well as for transition‐metal carbonyls and clusters, by comparing total energies, atomization energies, structure parameters, and vibrational frequencies at the local density approximation and generalized gradient approximation levels of theory. The method requires a sufficiently flexible auxiliary basis set. We propose a minimal extension of the conventional auxiliary basis set, which yields essentially the same accuracy for the quantities just mentioned as the standard approach. The new method allows one to achieve substantial savings compared with a fully numerical integration of the exchange‐correlation contributions. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Gold-Thiolate Clusters: A Relativistic Density Functional Study of the Model Species Au13(SR)n, R = H, CH3, n = 4, 6, 8

Abstract The binding of sulfanyl and alkylsulfanyl model ligands to gold clusters was studied for the case of Au13(SR)n with R = H, CH3 and n = 4, 6, 8. Accurate all-electron electronic structure calculations and geometry optimizations of these gold-thiolate clusters have been performed with a scalar relativistic Kohn-Sham procedure as implemented in the density functional program PARAGAUSS. In all structures obtained, bridge coordination was preferred for both types of ligands; no higher coordinated sites where occupied. While in many cases ligand decoration did not change the overall structure of the Au13 core, also more open structures with Au-Au distances elongated beyond the bulk value have been obtained. The effects due to increasing ligand decoration were small: a small decrease of the binding energy per ligand does not exclude higher ligand coverages. The differences between the model ligands SH and SCH3 were consistent in all cases considered: SCH3 exhibits weaker binding and a slightly smaller charge separation between cluster core and ligand shell, which amounts up to about 1.5 e for 8 ligands. Overall, the Au13 core of the clusters was found to be quite flexible. This can be rationalized by the fact that the calculated binding energy per ligand is comparable or even exceeds the binding energy per atom in Au13.

Construction of local hybrid exchange-correlation potentials and their evaluation for nuclear shielding constants

The optimized effective potential (OEP) method has been applied to convert ‘traditional’ non-local and non-multiplicative hybrid exchange-correlation potentials into local and multiplicative Kohn–Sham potentials. The latter have been evaluated in calculations of nuclear shielding constants for 22 small main-group molecules. The results obtained with local potentials are in much better agreement with experiment than those from their non-local counterparts. Exact-exchange admixture between 0.4 and 0.7 provides the best agreement with the experimental shielding constants. This should be contrasted to much lower exact-exchange admixture in a recent multiplicative Kohn–Sham procedure. With local hybrid potentials no coupling terms due to exact-exchange admixture have to be evaluated in the perturbational treatment.

Variational method for inverting the Kohn-Sham procedure

A procedure based on a variational principle is developed for determining the local Kohn-Sham (KS) potential corresponding to a given ground-state electron density. This procedure is applied to calculate the exchange-correlation part of the effective Kohn-Sham (KS) potential for the neon atom and the methane molecule.

Energy and angular momentum transfer in the excitation of electron-hole pairs by slow dimers

We calculate the transfer of energy and angular momentum through electron-hole pair excitations for a slow dimer in an electron gas. We show that the Kohn-Sham procedure can be used under the adiabatic conditions that prevail in the shifted Fermi sphere approximation. We obtain the low-energy limit of the friction coefficient and average angular momentum transfer from a self-consistent calculation for the embedded dimer. We apply our theory to ${\mathrm{H}}_{2}$ and LiH molecules. We use our results to evaluate the role of electron-hole pair excitations when a molecule approaches a metal surface.

The Hartree product and the description of local and global quantities in atomic systems: A study within Kohn–Sham theory

The Hartree product is analyzed in the context of Kohn–Sham theory. The differential equations that emerge from this theory are solved with the optimized effective potential using the Krieger, Li, and Iafrate approximation, in order to get a local potential as required by the ordinary Kohn–Sham procedure. Because the diagonal terms of the exact exchange energy are included in Hartree theory, it is self-interaction free and the exchange potential has the proper asymptotic behavior. We have examined the impact of this correct asymptotic behavior on local and global properties using this simple model to approximate the exchange energy. Local quantities, such as the exchange potential and the average local electrostatic potential are used to examine whether the shell structure in an atom is revealed by this theory. Global quantities, such as the highest occupied orbital energy (related to the ionization potential) and the exchange energy are also calculated. These quantities are contrasted with those obtained from calculations with the local density approximation, the generalized gradient approximation, and the self-interaction correction approach proposed by Perdew and Zunger. We conclude that the main characteristics in an atomic system are preserved with the Hartree theory. In particular, the behavior of the exchange potential obtained in this theory is similar to those obtained within other Kohn–Sham approximations.

Time-dependent Kohn-Sham formalism

Within density-functional theory a time-dependent perturbation theory is derived that yields exact orbital-dependent expressions for the time-dependent exchange-correlation potential in the form of a perturbation series. This leads to an exact time-dependent Kohn-Sham scheme based on perturbation theory. In first order an ``exchange-only'' Kohn-Sham procedure is obtained, that is a generalization of a recently presented time-dependent optimized potential method. The exact, in higher order, but quite complicated expressions for the exchange-correlation potential may serve as a starting point for the construction of approximate exchange-correlation potentials. An adiabatic connection or coupling constant path between a time-dependent physical system and its corresponding Kohn-Sham system is introduced.

Exchange-correlation potentials

We describe our implementation of the Zhao, Morrison, and Parr method [Phys. Rev. A 50, 2138 (1994)] for the calculation of molecular exchange-correlation potentials from high-level ab initio densities. The use of conventional Gaussian basis sets demands careful consideration of the value of the Lagrange multiplier associated with the constraint that reproduces the input density. Although formally infinite, we demonstrate that a finite value should be used in finite basis set calculations. The potential has been determined for Ne, HF, N2, H2O, and N2(1.5re), and compared with popular analytic potentials. We have then examined how well the Zhao, Morrison, Parr potential can be represented using a computational neural network. Assuming vxc=vxc(ρ), we incorporate the neural network into a regular Kohn–Sham procedure [Phys. Rev. A 140, 1133 (1965)] with encouraging results. The extension of this method to include density derivatives is briefly outlined.

Exact treatment of exchange in Kohn-Sham band-structure schemes.

A Kohn-Sham (KS) scheme for solids treating exchange exactly is introduced. The scheme emerges from a recently introduced exact formal KS procedure [A. G\"orling and M. Levy, Phys. Rev. A 50, 196 (1994)]. The exact density-functional exchange potential is obtained in terms of its Fourier components. The method can easily be implemented on the basis of existing Hartree-Fock schemes for solids employing plane waves as basis sets. \textcopyright{} 1996 The American Physical Society.

Generalized Kohn-Sham schemes and the band-gap problem.

As an alternative to the standard Kohn-Sham procedure, other exact realizations of density-functional theory (generalized Kohn-Sham methods) are presented. The corresponding generalized Kohn-Sham eigenvalue gaps are shown to incorporate part of the discontinuity ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{xc}}$ of the exchange-correlation potential of standard Kohn-Sham theory. As an example, a generalized Kohn-Sham procedure splitting the exchange contribution to the total energy into a screened, nonlocal and a local density component is considered. This method leads to band gaps far better than those of local-density approximation and to good structural properties for the materials Si, Ge, GaAs, InP, and InSb. \textcopyright{} 1996 The American Physical Society.

Variational charge relaxation in ionic crystals: An efficient treatment of statics and dynamics.

An ab initio variationally induced breathing (VIB) description of the energetics and dynamics of ionic crystals is developed within the spherical-ion-pair approximation and the Gordon-Kim ansatz. In the VIB method, the total crystal charge density is given by the overlap of Kohn-Sham ionic charge densities. Spherical charge deformation of the ions is accomplished by adding an effective many-body crystal site potential to the atomic potential used in the Kohn-Sham procedure. Parameters defining this effective site potential are treated variationally such that the total crystal electronic energy is a minimum for a given structural configuration. We have explored a number of forms of the effective potential, including Watson-sphere types where both the charge and radius of the sphere are variationally determined. In addition, we have investigated anion-cation charge transfer by incorporating an additional variational parameter. We also present a formulation of the lattice dynamics of ionic crystals based on the VIB prescription in which the electronic variational parameters are treated as dynamical variables within the adiabatic approximation. With the computed phonon density of states, the structural parameters of a crystal can then be determined at any temperature and pressure (or stress condition) by mini- mizing the quasiharmonic Gibbs free energy with respect to the structural parameters along the electronic adiabatic surface. The complete pressure- and temperature-dependent thermal and elastic properties of a crystal can also be determined. We have applied the VIB procedure to calculate the equations of state, elastic properties, phonon dispersion relations, and phase stability of the alkaline earth oxides (MgO, CaO, and SrO). The calculated properties are generally in quite good agreement with experiment.Spherical charge deformation within the ionic description is responsible for a considerable improvement over rigid-ion elastic constants and bulk moduli, but some of the remaining discrepancies cannot in principle be eliminated within a spherical-ion model. In particular, the effect of spherical breathing on the optical vibrational frequencies is not dramatic and further improvement (e.g., reduction of LO-TO splitting) will require the inclusion of nonspherical charge relaxation (polarizability).

An implementation of a Kohn—Sham density functional program using a Gaussian-type basis set. Application to the equilibrium geometry of C 60 and C 70

An implementation of the Kohn—Sham procedure with orbitals expanded in a Gaussian-type basis is presented. It has been built into the direct-SCF program of a TURBOMOLE package, from which it inherits the ability to exploit all finite point groups. While the one-electron and Coulomb part of the problem are handled exactly as in the Hartree—Fock case, the exchange-correlation part is treated by a three-dimensional numerical integration. The code is designed to exploit the sparseness inherent in the problem and nearly everything is done in terms of matrix operations. An analytic geometry gradient has been implemented on the same lines. Calculations on the equilibrium geometry of the fullerenes C 60 and C 70 are reported.