Generalized Hartree-Fock Equation Research Articles

Stability of the complex generalized Hartree-Fock equations

For molecules with complex and competing magnetic interactions, it is often the case that the lowest energy Hartree-Fock solution may only be obtained by removing the spin and time-reversal symmetry constraints of the exact non-relativistic Hamiltonian. To do so results in the complex generalized Hartree-Fock (GHF) method. However, with the loss of variational constraints comes the greater possibility of converging to higher energy minima. Here, we report the implementation of stability test of the complex GHF equations, along with an orbital update scheme should an instability be found. We apply the methodology to finding the local minima of several spin-frustrated hydrogen rings, as well as the non-collinear molecular magnet Cr3, illustrating the utility of the broken symmetry GHF method and some of its lesser-known nuances.

A dense Bose fluid at zero temperature: condensation and clusters in liquid 4He

We present a full set of wave equations describing a dense Bose fluid, applicable both to non- ideal gases and to liquid 4He. The phonon spectrum in liquid 4He is found and the fraction of condensed particles is calculated at zero temperature for a wide range of densities. The theory also yields the ground-state energy for the quantum liquid 4He in agreement to high accuracy with Monte Carlo simulations and experimental data at low pressure. We also present the derivation of a generalized Hartree-Fock equation describing roton clusters in low temperature liquid 4He, allowing us to confirm that, at low enough temperatures and for a wide range of pressures, the stable clusters consist of 13 bound atoms.

Koopmans’ theorem in the ROHF method: Canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.

Towards an optimal zeroth-approximation for excited state energy

A method is suggested to take into account the orthogonality constraints and to calculate the energy of excited states (ESs). It allows us to reduce a constrained minimization to an unconstrained one. Both general theory and single-determinant approximation have been considered in detail. Generalized Hartree-Fock equations for ES molecular orbitals were obtained. The method enabled us to carry out a basis set correction to a given ES and to construct the optimal zeroth-approximation for the Møller-Plesset perturbation theory of ES energy.

Functional analysis concepts and Fartree-Fock instability conditions

Concepts of functional analysis, namely, regular points, tangent subspaces, constraint surfaces, Lagrangian matrix restricted to the tangent subspace of a constraint surface, are presented in connection with the Hartree-Fock (HF) problem. The energy functional in LCAO approximation is considered to be a polynomial function of several variables subject to subsidiary conditions. General HF equations and instability conditions for the unrestricted Hartree-Fock (UHF) solutions are derived from this standpoint.

Theory of inhomogeneous quantum systems. III. Variational wave functions for Fermi fluids.

We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid $^{3}\mathrm{He}$, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.

Local electronic structures in disordered systems

The local electronic structure at a region ${C}^{M}$ within a disordered system, characterized by a definite composition ${t}^{M}$, is described by a partially averaged Green matrix ${\stackrel{-}{\stackrel{-}{G}}}^{{t}^{M}}(\ensuremath{\epsilon})$, which is obtained in principle by performing a weighted average over all compositions in the rest of the system. It is shown that essentially correct information about ${\stackrel{-}{\stackrel{-}{G}}}^{{t}^{M}}(\ensuremath{\epsilon})$ can be extracted from intermediate quantities that become available when adopting the traveling cluster approximation, or a simplified form thereof, for calculating the average properties of the system. When the partially averaged Green matrices are known, a number of problems are easily solved: The effect of short-range order on the electronic properties of the system can be evaluated, the local-bond-order matrices become available and generalized Hartree-Fock equations for disordered systems can be formulated, and different kinds of defects at specified locations can be treated. Examples of applications are provided, referring to a linear infinite chain in the tight-binding approximation, for which exact results are known from numerical calculations. As an example of defects the semi-infinite chain is considered with different compositions at its terminal end.

Charge current waves in a one-dimensional Coulomb gas

As a first step toward a solution of the general Hartree–Fock equations for a one-dimensional Coulomb system we construct charge current waves by pairing restricted Hartree–Fock orbitals with functions from their orthogonal complement. The degree of pairing is determined by the solution of the corresponding gap equation. Special attention is paid to the screening problem.

Density fluctuations in infinite Fermi systems

Within the framework of many-body theory, a generalized Hartree-Fock equation is derived assuming a periodic density fluctuation. In a one-dimensional model, self-consistent solutions are calculated. A band structure of the single-particle spectrum and several minima of the total energy appear as novel features.NUCLEAR MATTER Static density fluctuations, generalized Hartree-Fock method.

Pairing and general spin orbitals for the electron gas

The solutions of the general Hartree-Fock equations for the electron gas are discussed with reference to Fukutome's classification. Solutions of different character are constructed by pairing occupied and virtual restricted Hartree-Fock spin orbitals. The results of illustrative calculations for k-independent pairing are described and discussed.

Numerical solution of general Hartree-Fock equations for atoms

Two methods, M1 and ∼M2 , for improving numerical Hartree-Fock (HF) radial functions during an SCF iteration are considered. It is shown that when rotations are introduced into the SCF process, functions can be improved one at a time, without direct concern over orthonormality conditions. Convergence characteristics for various classes of HF problems are considered. Generally M1 has the faster rate of convergence but when no exchange is present ∼M2 is preferable becoming, in effect, inverse iteration with a difference correction. In all cases, except the hydrogenic one, the asymptotic rate of convergence of the SCF process is linear.

A new method for evaluating the density matrix and its application to the ground state form factors of 4He and 16O

The form factors of 4He and 16O are calculated using wave functions given by solutions of a generalized Hartree-Fock equation. No free parameters exist in the theory except the parameters of various phenomenological nucleon-nucleon potentials. The c.m. motion is treated exactly in the framework of a many body theory. The results strongly depend on the nucleon-nucleon potentials.

The Change of the Vacuum in Superheavy Quasi-Molecules

In collisions of very heavy ions (Z1 + Z2 > 172) molecular electronic orbitals are formed. Generalized Hartree-Fock equations for this case are derived from field theoretic principles. It is shown that the lowest ground state (i.e. the vacuum) becomes charged for Z > Zcr and R < Rcr. If the K-shell is vacant, positron production sets in. The energy distribution of the positrons and the cross-section for this process are calculated.

The calculation of nuclear rotational states based on the generalized Hartree-Fock approximation

The generalized Hartree-Fock method is applied to a description of rotating nuclei, and the generalized Hartree-Fock equation is solved with the aid of self-consistent perturbation calculations. Numerical calculations are carried out for the rare-earth nuclei, and the results are compared with experiment. It is shown that good agreement with experiment is obtained.

Spin-Dependent Potential Functions in a Generalized Hartree-Fock Equation

The exchange term in the generalized Hartree-Fock equation found by the Green's-function method is investigated with an electron gas in a metal in mind. This term contains a generalized exchange charge density and a generalized exchange potential function. The latter not only is shielded, but contains effects depending on the spin of the particle being described by the Hartree-Fock equation. This spin dependence is calculated for spin deviations periodic in the lattice, and also for the case of static spin density waves. The original Overhauser spin-density-wave integral equation, arising from a perturbation in the exchange charge density, now must incorporate the corresponding perturbation in the exchange potential function. It is shown that the two perturbations are of the same order of magnitude, and that the constant solution of the integral equation is no longer valid under the same type of approximation.

Exchange Narrowing ofdBands in Ferromagnets

A semiphenomenological Hamiltonian based on the generalized Hartree-Fock equations for a two-component spinor is developed which describes the interaction of an electron in the $d$ band with the spin waves of a ferromagnet. Solutions to the effective Hamiltonian are examined in the tight-binding approximation using a self-consistent separation of the electron and spin-wave variables. The dynamic interaction of the electron with the spin waves gives rise to a band-narrowing factor $\mathrm{exp}(\ensuremath{-}\ensuremath{\zeta})$ analogous to that obtained by Yamashita and Kurosawa for the polaron. $\ensuremath{\zeta}$ increases with temperature and estimates show that it may be on the order of unity for a typical ferromagnet at low temperatures.