Mean-field Operator Research Articles

Quasilocal density functional theory in nuclei and its extension to include pairing correlations

We propose first a generalization of the Density Functional Theory. This theory leads to single-particle equations of motion with a quasilocal mean-field operator, which contains a quasiparticle position-dependent effective mass and a spin—orbit potential. The energy density functional is constructed using the extended Thomas—Fermi approximation. Some ground-state properties of doubly magic nuclei are considered within the framework of this approach. Calculations are performed using the finite-range Gogny D1S forces, and the results are compared with the exact Hartree—Fock calculations. Next, we present an extension of the density functional theory to include pairing correlations without formal violation of the particle-number condition. This problem, which is nonlocal, can be simplified by a suitable quasilocal reduction, which is also briefly discussed in this paper.

Density-functional calculations of relativistic spin-orbit effects on nuclear magnetic shielding in paramagnetic molecules

Terms arising from the relativistic spin-orbit effect on both hyperfine and Zeeman interactions are introduced to density-functional theory calculation of nuclear magnetic shielding in paramagnetic molecules. The theory is a generalization of the former nonrelativistic formulation for doublet systems and is consistent to O(alpha4), the fourth power of the fine structure constant, for the spin-orbit terms. The new temperature-dependent terms arise from the deviation of the electronic g tensor from the free-electron g value as well as spin-orbit corrections to hyperfine coupling tensor A, the latter introduced in the present work. In particular, the new contributions include a redefined isotropic pseudocontact contribution that consists of effects due to both the g tensor and spin-orbit corrections to hyperfine coupling. The implementation of the spin-orbit terms makes use of all-electron atomic mean-field operators and/or spin-orbit pseudopotentials. Sample results are given for group-9 metallocenes and a nitroxide radical. The new O(alpha4) corrections are found significant for the metallocene systems while they obtain small values for the nitroxide radical. For the isotropic shifts, none of the three beyond-leading-order hyperfine contributions are negligible.

Quasilocal density functional theory and its application within the extended Thomas-Fermi approximation

In this paper we propose a generalization of the density functional theory. The theory leads to single-particle equations of motion with a quasilocal mean-field operator, which contains a quasiparticle position-dependent effective mass and a spin-orbit potential. The energy density functional is constructed using the extended Thomas-Fermi approximation and the ground-state properties of doubly magic nuclei are considered within the framework of this approach. Calculations were performed using the finite-range Gogny D1S forces and the results are compared with the exact Hartree-Fock calculations.

Spin-free relativistic no-pair ab initio core model potentials and valence basis sets for the transition metal elements Sc to Hg. II

This is the second part of a report on spin-free relativistic no-pair ab initio core model potentials for the transition elements Sc to Hg. In the first part [J. Chem. Phys. 110, 3678 (1999)], we introduced the no-pair ab initio model potential method and supplied model potentials for [Mg], [Zn], and [Cd,4f] cores of first-, second-, and third-row transition metals, respectively. At the Hartree–Fock level excellent agreement between all-electron and model potential results was observed for late transition metal oxides, whereas the performance of the model potentials was slightly less satisfactory for early transition metal oxides. In this paper we will present small-core model potentials corresponding to [Ne], [Ar,3d], and [Kr,4d,4f] cores, respectively. The performance of the model potentials is tested extensively in calculations on the diatomic oxides VO, NbO, TaO, NiO, PdO, and PtO, both at the Hartree–Fock level and when electron correlation is included by means of coupled-pair functional methods. Further we investigate the requirements on valence and intermediate basis sets used to represent the exchange and no-pair operators.

Spin−Orbit Coupling Patterns Induced by Twist and Pyramidalization Modes in C2H4: A Quantitative Study and a Qualitative Analysis

A study of the spin−orbit coupling (SOC) mechanisms which couple the triplet ππ* state (T1) to the singlet ground state (S0) in ethylene is carried out at a variety of computational levels and basis sets, using the full Breit−Pauli (BP) SOC Hamiltonian, the one-electron mean-field (MF) operator, and the approximate one-electron operator based on an effective nuclear charge, Z*. The basis set and wave functions requirements needed for good quality SOC calculations are elucidated by studying the SOC interaction using single- and multireference CI as well as MCSCF wavefunctions, with basis sets ranging from the minimal STO-3G all the way to an extended one with quadruple ζ and polarization quality. Two archetype distortion modes of ethylene were considered: a twist mode which changes the symmetry from D2h to D2 and then to D2d and pyramidalization modes which change the ethylene symmetry to C2v (syn-pyramidalization) or C2h (anti-pyramidalization), as well as Cs (i.e., a mono-pyramidalization distortion). I...

Ab initio calculations of the ${} ^2{\bi P}_{{\b 1}\over {\b 2}} \hbox{-}{} ^2{\bi P}_{{\b 3} \over {\b 2}} $ splitting in the thallium atom

The splitting between the \({} ^2P_{3 \over 2} \) and the \({} ^2P_{1 \over 2} \) terms in the thallium atom has been calculated at the perturbation theory level and by spin-orbit CI calculations, using both the Breit-Pauli and the no-pair form of the microscopic spin-orbit Hamiltonian. The importance of the spin-other-orbit contribution to the spin-orbit splitting is investigated, and it is also shown that an averaging procedure of the kinematic factors in the expression for the spin-other-orbit integrals in the no-pair spin-orbit Hamiltonian yields highly accurate results. A slightly modified version of a previously proposed mean-field spin-orbit method is shown to have an accuracy of a few wave numbers. Perturbation theory is found to give a term-splitting which is too low by more than 1000 cm−1, while spin-orbit CI with the no-pair form of the spin-orbit operator with the averaged spin-other-orbit term, and the no-pair mean-field operator, gives results in good agreement with experimental data.

A mean-field spin-orbit method applicable to correlated wavefunctions

Starting from the full microscopic Breit-Pauli or no-pair spin-orbit Hamiltonians, we have devised an effective one-electron spin-orbit Hamiltonian in a well defined series of approximations by averaging the two-electron contributions to the spin-orbit matrix element over the valence shell. In addition the two-electron integrals were restricted to comprise only one-centre terms. The validity of these approximations has been tested on several palladium containing compounds. Excellent agreement of the matrix elements of the mean-field operator with corresponding full results is observed; deviations amount to a few cm −1 in absolute value or at most 0.2% on a relative scale. The newly defined mean-field operator can thus safely be employed to evaluate spin-orbit effects in transition metal containing compounds.

Exact asymptotics in a mean field model with random potential

For a mean field operator with a random potential, asymptotic properties of the eigenvalues and eigenfunctions are studied and applied to investigate the longerm behavior of the solutions of a corresponding large system of differential equations. The total mass of the system is approximately concentrated in the record point of the random potential (complete localization). A more detailed inspection of the peaks shows that there is a phase transition: Only in the case of a moderate increase of time relatively to the growth of the space size the model behaves similarly to the system without “diffusion”. But also in the non-moderate case the asymptotic height of peaks can exactly be described.

On the Role of the Nonlocal Hartree-Fock (HF) Exchange in Narrow-Band Materials

The influence of the nonlocal Hartree-Fock (HF) exchange in narrow-band materials with finite band gaps is analyzed. The mean-field Hamiltonian in the crystal orbital (CO) basis contains two k-dependent matrix elements responsible for the dispersion of the one-electron levels: the classical tight-binding integrals (kinetic energy of the electrons) that decay exponentially as well as the (non)local exchange. The asymptotic behavior of the two-electron potential is determined by the fall-off of the intercell bond-order matrices; their range exceeds significantly the spatial extension of the k-dependent one-electron integrals. The analytic structure of the HF dispersions of narrow-band systems (weak intercell interactions) is largely influenced by the magnitude of the Fermi-correlation beyond the direct neighbors. The associated ε(k) curves differ strongly from idealized tight-binding relations. The bands are broadened and show enhanced energy gradients in certain domains of k-space. Nonlinearities in the ε(k) relations are a direct consequence of finite neighbor's approximations adopted for the evaluation of the lattice sums. The analytic structures of such HF bands are intermediate between idealized tight-binding relations of covalent solids, on one side, and HF dispersions of metals in (nearly-)free electron-gas approximations, on the other, that show divergent ε(k) gradients at the Fermi level. The exchange influence in insulating narrow-band materials is restricted to the filled one-particle space; this is demonstrated by a perturbational analysis. The crucial importance of reliable numerical integration procedures for the determination of the intercell bond-order matrices is pointed out. Standard techniques may lead to artificial periodicities pretending unphysical decay properties of the electronic exchange. Dispersion patterns of a simple one-orbital model are analyzed as a function of the mutual strength of the kinetic hopping integrals and the HF exchange as well as the spatial extension of the k-dependent two-electron potential. The validity of the theoretical expectations deduced from simple model calculations is studied for two complex polymers. Important one-electron properties of one-dimensional (1 D) porphyrinato nickel(II) derivatives (2 and 3) are investigated by means of semiempirical SCF (self-consistent-field) HF INDO (intermediate neglect of differential overlap) CO calculations. The lattice spacings of 2 and 3 differ by 0.31 Å. This geometrical distinction allows for an inversion of the relative importance of the k-dependent one- and two-electron contributions to the mean-field operator. The exchange influence on the width of the HF dispersion in 3 exceeds the one-electron part by nearly one order of magnitude. It is shown that band structure properties of narrow-band systems are neither properly described by one-electron models of the Wolfsberg-Helmholtz-type nor by bare (unscreened) HF dispersions. The width of a mean-field band calculated within a nonlocal exchange approximation has to be corrected for quasi-particle (QP) interactions beyond the HF scheme (i.e., long-range and short-range correlations and relaxations) as well as for electron (optical) phonon interactions. The phononic coupling leads to a narrowing of the band width via Franck-Condon-like vibrational overlaps; this part is independent of the theoretical details of the electronic structure investigation. Important physical consequences of the exchange-control in narrow-band solids are shortly discussed.