Hartree-Fock Results Research Articles

On the shape of spherically averaged Fermi-hole correlation functions in density functional theory. 1. Atomic systems

We have compared, for atomic systems, the spherically averaged Fermi-hole correlation function [Formula: see text] in the Hartree–Fock theory with the corresponding function [Formula: see text] employed in local density functional theory. It is shown that, in contrast to [Formula: see text], the function [Formula: see text] behaves qualitatively incorrectly at positions r1 of the reference electron far from the nucleus. Furthermore, we have shown that the qualitatively incorrect behaviour of [Formula: see text] can be remedied by an approximate expansion of [Formula: see text] in powers of s, where s is the inter-electronic distance. However, such an expansion must be conducted in two regions due to the discontinuity of [Formula: see text] as a function of s at the atomic nucleus. Based on the two-region expansion of [Formula: see text] we have developed an alternative approximate density functional expansion [Formula: see text] for the spherically averaged Fermi-hole correlation function. The corresponding exchange energy density functional yields values for the exchange energies of atoms in good agreement with Hartree–Fock results. Keywords: atomic exchange energy, density functional theory, Fermi hole.

Exchange-only density-functional theory in an exactly soluble model.

We show that the exact Kohn-Sham wave functions and exact Kohn-Sham exchange potential ${v}_{x}^{\mathrm{KS}}$ may be calculated for any spherically symmetric system in which the electron states are described by orbitals having only two different radial wave functions. In this exchange-only theory the total energy, electron density, and maximum single-particle energy eigenvalue are all identical to the corresponding Hartree-Fock ground-state results. We present results for an external harmonic-oscillator potential with two filled subshells corresponding to a total of eight (unpolarized) electrons or a total of four (spin-polarized) electrons for a wide range of spring constants. Comparisons are made between the results obtained from the exact KS theory and those obtained from employing the local-spin-density (LSD) and Slater (SLA) exchange approximation and from the exact Hartree-Fock results with respect to wave functions, densities, single-particle eigenvalues, exchange energy, and the calculation of Hartree-Fock expectation values for total energy and single-particle energies employing single-particle density-functional wave functions. Of particular interest is the result that ${v}_{x}^{\mathrm{KS}}$ and ${v}_{x}^{\mathrm{LSD}}$ are similar in shape, but not in magnitude, which results in LSD orbitals which closely approximate the exact KS orbitals even though the single-particle eigenvalues are significantly in error and are in fact less accurate than those given by the SLA.

Correlation theory of antiferromagnetism in 2D Cu-O planes

We present a theory of antiferromagnetic ordering (AF) in a 2D extended Hubbard model of the Cu-O system based on an equation of motion method which projects out Mott-Hubbard bands. In particular, we discuss the AF order as a function of doping (δ) away from half-filling. We find in “mean field” that the system is AF ordered for small δ and the disappearance of AF is much different than the Hartree-Fock result and more in line with experiment. Limitations of the theory are also discussed.

Nuclear moments and charge radius changes in platinum isotopes

Isotope shift (IS) and hyperfine structure (HFS) measurements have been performed for185, 187, 189, 191, 195Pt using the PILIS (Post ISOCELE Laser Isobar Separator) apparatus installed at the ISOCELE facility. Magnetic and quadrupole moments have been deduced from the HFS results. The charge radius changes determined for these odd nuclei from the IS results, added to the σ values of the even-A nuclei are compared to the results of lattice Hartree-Fock+BCS calculations for asymmetric solutions:185Pt is confirmed to be prolate shaped whereas187, 189, 191Pt are likely triaxial in their ground states.

Intracule densities and electron correlation in the hydrogen molecule

The spherically averaged intracule density I(r) is calculated with minimal and split-valence basis sets in the ground state of the hydrogen molecule. Restricted Hartree-Fock calculations indicate a high probability of interelectronic distances (r) of the order of 1 au as the internuclear separation (R) goes to infinity due to the incorrect dissociation of H2. The inadequacy of the restricted Hartree-Fock I(r) at large R is further illustrated by calculating (1/r) as a function of R. Configuration interaction calculations are shown to yield the correct qualitative form for I(r) at large R and when combined with the Hartree-Fock results yield the approximate Coulomb hole in H2 as a function of internuclear separation. Explicit formulae for the evaluation of I(r) and its moments with s-type Gaussian basis functions are included.

NMR chemical shifts of CX and XCY molecules (X, Y = O, S, Se, Te): A comparison of coupled Hartree-Fock, semi-empirical rex and experimental results

Carbon-13 and selenium-77 NMR results are reported for CSe 2 dissolved in nematic liquid crystals. The obtained chemical shift anisotropies, Δσ, are 506 ± 30 ppm and 2210 ± 120 ppm for 13C and 77Se, respectively. The error limits include the estimated uncertainty from the unknown Δ J(CSe). Coupled Hartree-Fock (CHF) calculations are reported for the shielding tensors for the entire series CX and CX 2, X=OSe, and for OCS. The degree of incompleteness of the basis is tested using sum rules. All shielding tensors are calculated using at least two gauge origins. Dipole and quadrupole moments and electric field gradients are also given. Relativistic extended Hückel (REX) calculations are also reported, both for the paramagnetic chemical shift terms and for the spin-spin coupling tensors.

Density functional theory at finite temperatures

A rigorous foundation of the density functional theory, providing a description of many-body systems at finite temperatures, is given. The general results of Hohenberg-Kohn-Mermin are confirmed by using local-scale point transformations. We show how to obtain a constructive and fully practical procedure for building up approximate functionals of the local density ϱ( r). The explicit form of these functionals and the corresponding Euler-Lagrange equations are derived without assuming any particular form of the two-body forces. It is shown that the kinetic energy density contains only two terms: the original Weizsäcker term and nonlocality term. Special attention is paid to a formulation of the local density approach, whereby the temperature-dependent Hartree-Fock results are exactly reproduced.

Use of local-density pseudopotentials in Hartree-Fock calculations.

Norm-conserving pseudopotentials obtained from a local-density approximation (LDA) are implemented in an unrestricted Hartree-Fock (UHF) formalism. A series of tests is devised to ascertain the utility and validity of this methodology. Calculations using LDA and coreless Hartree-Fock (CHF) pseudopotentials are contrasted and compared to full-core Hartree-Fock results. We find that for light atoms (Zl21) the LDA-derived potentials yield accurate results when used in UHF calculations. The methodology reproduces experimental results as accurately as full-core calculations. LDA pseudopotentials are found to yield accurate wave functions for use in many-body-theory corrections to the correlation energy. By including correlation corrections in this manner the methodology reproduces experimental results to within 0.2 eV.

Oscillator strengths for the alkaline-earth atoms using rotor-vibrator and configuration-interaction wave functions.

We have calculated oscillator strengths for transitions between low-lying states of the alkaline-earth atoms using both configuration-interaction (CI) and simple optimized molecular rotor-vibrator (RV) wave functions for the $e$-core-$e$ system. Our CI results compare favorably with experimental and other accurate values, indicating that these functions used in previous studies of electron correlation are reliably accurate. Most RV results agree better with accurate values than do Hartree-Fock results, thus showing the success of the RV model in its first test of calculating an experimental observable.

Local-density description of nuclear systems at finite temperature

A density functional approach for describing properties of nuclear systems at finite temperatures is developed by using local-scale point transformations. Within a mean-field approximation it leads to a fully practical procedure for solving the temperature-dependent Hartree-Fock problem in terms of the nuclear local-density distribution. The numerical results obtained in a large scale of temperatures for the nucleus,208Pb with Skyrme-type forces SkM are seen to reproduce quite accurately the average as well as the microscopic temperature-dependent Hartree-Fock results. Finite-temperature random-phase approximation sum rule expressions are applied for investigating nuclear scaling incompressibility and giant-monopole isoscalar and isovector energies as functions of temperature.

Calculation of the hyperfine splitting constants for diatomic molecules using numerical wavefunctions

Isotropic and anisotropic hyperfine splitting constants have been calculated for the 2Σ states of BeH, MgH, CN, and CP. All wavefunctions were calculated numerically using our partial-wave procedure. Spin polarization has been introduced with selected single excitations from the restricted Hartree-Fock (RHF) wavefunctions, followed by complete or partial orbital optimization. For comparison the results of spin-unrestricted Hartree-Fock (SUHF) calculations are presented. Comparison is also made with RHF results as well as with available configuration interaction (CI) and experimental results.

The open-shell coupled-cluster method: Ionization potentials and electron affinities of the alkali atoms, Li to Cs

The open-shell coupled-cluster method with single and double excitations (CCSD) is applied to the alkali atoms, Li to Cs, with moderately large basis sets. The calculated electron affinities are all within 0.01 eV of experiment, with valence-shell correlation contributing almost all the correction to the (poor) Hartree–Fock results. Errors in the 2S ionization potentials range from 0.02 (Li) to 0.22 eV (Cs); with relativistic corrections estimated from numerical Hartree–Fock, IPs are within 0.1 eV of experiment. 2P IPs are even more accurate (0.02–0.04 eV).

Photoionization cross section and resonance structure of atomic yttrium.

Many-body perturbation theory has been used to calculate the photoionization cross section of neutral yttrium from threshold at 6.52--100.0 eV. The resonance structure due to 4p excitations to 4d states has also been calculated. Hartree-Fock results are also presented for comparison.

Approximation methods for many-electron atoms from a field theory point of view

A summary of a field-theoretic approach to the study of the energies and radiative transition matrix elements of many-electron atoms is presented. The approach can deal with both non-relativistic and relativistic atoms. The method provides a unified description of a number of well-known approximations for calculating energy eigenvalues, and can easily be generalized to include new types of approximations and higher order correlations. Insistence on gauge invariance of transition amplitudes at every level of approximation leads to a generalization of the previously used Hartree-Fock results for transition amplitudes. It involves non-local effective currents.

Electron transfer and spin-flip processes in atom-atom collisions from variationally improved time-dependent Hartree-Fock results.

A variational functional for transition amplitudes is applied to atom-atom collisions using a time-dependent formalism. The transition amplitude is a functional of two independent trial wave functions with specified initial and final conditions, respectively. Transition probabilities are obtained by evaluating the functional in terms of standard time-dependent Hartree-Fock (TDHF) trial functions. The TDHF method is implemented by expanding the molecular orbitals as linear combinations of traveling atomic orbitals. The procedure requires no other molecular integrals than those needed to construct the Fock operator. The present method is applied to He+${\mathrm{He}}^{2+}$ and ${\mathrm{H}}^{+}$+${\mathrm{He}}^{+}$ collisions with specified nuclear trajectories using a minimal atomic-orbital basis set. Within this model an exact, close-coupling-type solution is also feasible, and has been obtained for comparison purposes. Numerical values of probabilities and of cross sections obtained from the variational functionals for elastic, spin-flip, and one- and two-electron transfer processes show substantial improvements over the standard TDHF results.

Comparative study of the electronic structure of an aluminum atom with different local-spin-density exchange-correlation functionals.

Ab initio self-consistent spin-polarized calculations have been performed numerically for the total energy of an aluminum atom, to assess the various approximate forms of the exchange-correlation functionals used frequently in local-spin-density approximation. Transition-state calculations of all the ${\mathrm{Al}}^{\mathrm{n}+}$ ions are reported and compared with the restricted Hartree-Fock results and the experimental values. Calculations of the total energy of the ionic ground state confirms the accuracy of the transition-state method. The total energy is calculated employing an indirect path via information obtained from the ionization potentials and is found to be better than that given by the conventional method.

Investigation of three exchange energy formulas, Hartree–Fock, LDA, and 10/9 LDA

Three different formulas for the exchange energy are compared through comprehensive self-consistent calculations on atoms and diatomic molecules. The formulas considered are the Hartree–Fock exchange, the local density approximation (LDA) exchange, and the first order approximation to the new Ghosh–Parr exchange formula which is equivalent to 10/9 LDA. The 10/9 LDA results are vastly superior to the LDA results and comparable or better than the Hartree–Fock results. Possible reasons for this are discussed.

Properties of the BCl molecule studied by fourth-order many-body perturbation theory

Properties of the ground state (X 1Σ) of the BCl molecule are calculated in complete fourth-order many-body perturbation theory. The equilibrium bond distance is determined to be Re = 1.740 A (Re(exp.) = 1.716 A). Spectroscopic constants are evaluated from the potential curve and agree well with available experimental data. Static electric properties are computed by the finite-external-electric-field perturbation method. Apparently, no experimental information on these properties is available. Fully numerical Hartree–Fock results are used to estimate basis-set effects on the calculated quantities.

Comparison of relativistic CI and MCRHF calculations of oscillator strengths and excitation energies for the ns21S0-nsnp3P1,1P1transitions in neutral cadmium and mercury

The oscillator strengths and excitation energies for the ns21S0-nsnp 3P1, 1P1 transitions in neutral cadmium and mercury calculated in the configuration interaction (CI) approach based on numerical relativistic SCF wavefunctions are compared with the results of a multiconfiguration relativistic Hartree-Fock (MCRHF) scheme, which employs the same set of configurations. It is found that with careful choice of the method used to generate the valence orbitals, the relativistic configuration interaction treatment yields results comparable with those of the MCRHF approach. The influence on oscillator strengths and excitation energies of the additional nd2 and npnd configurations, which could not be included in MCRHF calculations because of difficulties with the convergence of the SCF process, is also studied and is found to be very small.

Various functionals for the kinetic energy density of an atom or molecule.

Various approximate density functionals for the kinetic energy density of atoms and molecules are analyzed. These include the results of a gradient expansion to first and second orders and a form recently derived from a new Green's function approximation [W. Yang, preceding paper, Phys. Rev. A 34, 4575 (1986)]. All the approximate functionals studied diverge to minus infinity at a nucleus, due to the ${\ensuremath{\nabla}}^{2}$\ensuremath{\rho} term that is in them, while the exact functional is positive and finite everywhere. Away from nuclei, however, the Hartree-Fock results are well reproduced, including the atomic shell structure. New functionals are proposed to correct the divergent behavior, and accurate total kinetic energy values are obtained from a new formula for kinetic energy density ${t}_{\mathrm{MP}(\mathrm{r})={C}_{k}\mathrm{\ensuremath{\rho}}{(\mathrm{r})}^{5/3}}$ +(1/72)\ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\rho}(r)${\ensuremath{\Vert}}^{2}$/\ensuremath{\rho}(r)+( 1/12)${\ensuremath{\nabla}}^{2}$\ensuremath{\rho}(r), with a divergence correction.