Modified Hartree Fock Research Articles

Hermitian “chemical” Hamiltonian: an alternative ab initio method

Some previous results of the present author are combined in order to develop a Hermitian version of the “Chemical Hamiltonian Approach.” In this framework the second quantized Born–Oppenheimer Hamiltonian is decomposed into one- and two-center components, if some finite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones, which become exact only in the limit of complete basis sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and diatomic energy terms which were introduced in our previous “chemical energy component analysis” scheme. The corresponding modified Hartree–Fock–Roothaan equations are also derived; they do not contain any three- and four-center integrals, while the non-empirical character of the theory is conserved. This scheme may be useful also as a “layer” in approaches like ONIOM.

Double excitations from modified Hartree Fock subsequent minimization scheme

Doubly excited states have nowadays become important in technological applications, e.g., in increasing the efficiency of solar cells and therefore, their description using ab initio methods is a great theoretical challenge as double excitations cannot be described by linear response theories based on a single Slater determinant. In the present work we extend our recently developed Hartree-Fock (HF) approximation for calculating singly excited states [M. Tassi, I. Theophilou, and S. Thanos, Int. J. Quantum Chem. 113, 690 (2013)] in order to allow for the calculation of doubly excited states. We describe the double excitation as two holes in the subspace spanned from the occupied HF orbitals and two particles in the subspace of virtual HF orbitals. A subsequent minimization of the energy results to the determination of the spin orbitals of both the holes and the particles in the occupied and virtual subspaces, respectively. We test our method, for various atoms, H2 and polyene molecules which are known to have excitations presenting a significant double excitation character. Importantly, our approach is computationally inexpensive.

Finite temperature correction to the Thomas–Fermi approximation for a Bose–Einstein condensate: comparison between theory and experiment

We observe experimentally a deviation of the radius of a Bose–Einstein condensate from the standard Thomas–Fermi prediction, after free expansion, as a function of temperature. A modified Hartree–Fock model is used to explain the observations, mainly based on the influence of the thermal cloud on the condensate cloud.

Density functional scheme for calculating the ground-state pair density

We present an approximate scheme for calculating the pair density (PD) by means of the extended constrained-search theory. The single-particle equation of the auxiliary noninteracting system is derived on the basis of the variational principle with respect to the pair density. It is the modified Hartree–Fock (HF) equation that additionally contains the kinetic contribution of the exchange-correlation energy functional as the correlation term. The present scheme can provide a better total energy than that of the HF approximation.

A proposal of the approximate scheme for calculating the pair density

The density functional scheme for calculating the pair density is presented by means of the constrained-search technique. The resultant single-particle equation takes the form of a modified Hartree–Fock equation, which contains the kinetic contribution of the exchange-correlation energy functional as the correlation potential. A practical form of the kinetic contribution is also proposed with the aid of the scaling relations of the kinetic energy functionals.

Antiferromagnetic ordering of itinerant systems in modified mean-field theory

This is an analysis of the itinerant model for antiferromagnetism, in which is included both on-site and inter-site electron correlations. We also consider the band degeneration, which brings into the Hamiltonian the on-site exchange interactions. The Green function technique is used and the coherent potential approximation (CPA) decoupling for the on-site Coulomb repulsion. This decoupling combined with the modified Hartree–Fock approximation for the inter-site interactions generates the change in shape of the spin bands with growing interaction constants, which is described by the correlation factors and which decreases the kinetic energy of the system. The effective Hartree field and the gain in kinetic energy due to the on-site and inter-site correlation factors drive the antiferromagnetism. The on-site and inter-site interactions act together towards the antiferromagnetism. Their cooperation decreases the interaction constants required for the antiferromagnetic ordering. This new approach allows for the antiferromagnetic instability in the purely itinerant model at the half-filling in the split band limit. This situation describes the high temperature superconducting cuprates.

Magnetic ordering of itinerant systems in modified mean field theory: antiferromagnetism

We analyze the itinerant model for antiferromagnetism, which was developed previously by Plischke, Mattis, Brouers and Mizia. In this model we include both; single-site and two-site electron correlations. Additionally, including band degeneration into the model allows for considering intra-atomic exchange interactions in the Hamiltonian. The modified Hartree–Fock approximation for the two-site interactions is used. This approximation gives the spin band narrowing, which is the same for both spin directions and affects the possibility of antiferromagnetic ordering. We use the Green function technique and CPA decoupling. This allows for the change in shape of the spin bands, described by the correlation factors, which decreases the kinetic energy of the system. The effective Hartree field and the gain in kinetic energy due to the on-site and inter-site correlation factors drive the antiferromagnetism.

Magnetic ordering of itinerant systems in modified CPA approximation

We will analyze the itinerant model for ferromagnetism with both single-site and two-site electron correlations. We will include band degeneration into the model. This will allow us to consider the on-site exchange interactions in the Hamiltonian. The modified Hartree–Fock approximation for the two-site interactions will be used. This approximation will give the relative one spin band broadening with respect to the other, in addition to the shift in position of majority and minority spin bands. Next, we will use the coherent potential approximation technique (CPA) with two self-energies. One will describe the on-site correlation and the second one the inter-site correlation. We will separate from these self-energies, the linear terms arising from Hartree–Fock approximation which will be applied to both; the on-site and inter-site interactions. These terms will contribute to the effective molecular field and the remaining ones (non-linear) will contribute to the correlation factors. The Green function technique and CPA decoupling will allow for the change in the shape of spin bands, which has been described by the correlation factors and which will decrease the kinetic energy of the system. The gain in kinetic energy due to the on-site and inter-site correlation factors will drive the ferromagnetism and significantly reduce the effective Hartree field necessary to create this ordering.

Magnetopolarons in quasi-one-dimensional quantum-well wires

The interaction of quasi-one-dimensional electrons and longitudinal-optical (LO) phonons in the presense of a quantizing magnetic field is calculated. Results are detailed presented for the polaron correction to the subband energies, the polaron cyclotron mass and the polaron effective mass. The calculations are done by using different approaches, Rayleigh–Schrödinger perturbation theory, Wigner–Brillouin perturbation theory, improved Wigner–Brillouin perturbation theory and Green's function technique. In the framework of a diagrammatic approach a modified Hartree–Fock approximation is developed, which allows the calculation of the quasi-particle properties of lower-dimensional semiconductor nanostructures in the subband space. It is shown that the Rayleigh–Schrödinger perturbation theory, improved Wigner–Brillouin perturbation theory and the modified Hartree–Fock approximation well describe the ground-state renormalization, but only the improved Wigner–Brillouin perturbation theory and the modified Hartree–Fock approximation well describe the bend-over and pinning behaviour of the subbands at the boundary of the phonon continuum. Rayleigh–Schrödinger and Wigner–Brillouin perturbation theories fail for the calculation of the renormalization of the excited subband energies in quantum-well wires with strong confining potential even for vanishing magnetic field and polaron momentum. In addition to the symmetric scattering processes, which are the only processes involved in the self-energy of the old-fashioned perturbation theories, Green's function technique allows also to calculate the contribution of all asymmetric scattering processes of two electrons by exchanging an optical phonon.

Exact modified‐Hartree–Fock scheme through perturbation expansion of density matrices

In the modified-Hartree–Fock (MHF) approach, as defined by Baroni and Tuncel (1983) (BT), the external potential is supplemented by a specific correlation potential, such that the density (determined with the HF method) of an N-electron system in this modified potential is the same as the ground-state density of the exact many-body solution for this system in the original potential. The MHF equations may be viewed as describing a model noninteracting N-electron system [analog of the Kohn–Sham (KS) system], moving in a one-body effective potential—the sum of three local terms: external, electrostatic, and BT correlation, and of the nonlocal HF exchange. The present study introduces an adiabatic link between the fully interacting system and this MHF noninteracting system by scaling the two-body electron–electron interaction with a factor α in the range [0, 1], similarly as it was done by Levy and Perdew (1985) to link with the KS noninteracting system. An appropriate α-dependent functional F of the density n is defined using Levy constrained-search formulation of the density-functional theory. Density matrices serve as an indispensable tool. A term in F, representing the (1−α) fraction of the electron repulsion energy, leads to an effective one-body nonlocal potential in an α-dependent Hamiltonian, the construction of which guarantees the density to be independent of α. Its ground-state solution serves for calculating F. This solution can be determined by means of the perturbation theory, with the unperturbed (α=0) Hamiltonian generating the MHF determinantal wave function, in analogy to the Görling and Levy (1993) approach with the KS function as the unperturbed one. Terms of the perturbation expansion for the BT correlation energy and potential can be calculated self-consistently in this way. Expressions for the BT correlation energy are obtained also in a form of integrals over the coupling parameter, involving α-dependent density matrices. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 469–483, 1998

Double resonance spectroscopy of NaI and KI: the van der Waals states Ω=1 and Ω=2 of the first atomic asymptote

The energy region of the lowest atomic asymptote of NaI and KI was investigated in a high-resolution Λ-type optical double resonance experiment with the highly excited covalent C0 + state as resonant intermediate. For the first time two van der Waals states with electronic angular momenta Ω=1 and Ω=2 were observed correlating to the ground states of sodium and potassium with iodine. The electronic system C0 +−Ω=2 shows a significant transition moment because of the heterogeneous coupling with Ω=1 at large rotational quantum numbers. For the new states the Dunham parameters and the RKR potential curves were determined. In addition, for the Ω=1 states modified Hartree—Fock dispersion potentials, were calculated by a least-squares fit of the observed energy levels and the dissociation energy of both molecules could be derived with an accuracy of a few wavenumbers.

The modified Hartree–Fock self‐consistent field equation

AbstractA one‐electron correlation operator is introduced into the Hartree–Fock self‐consistent field equation. The correlation operator is derived from the second‐order perturbation theory. Energies of atomic and molecular systems calculated from this modified Hartree–Fock equation are equal to that from second‐order perturbation of Hartree–Fock equation. The modified equation can also be solved self‐consistently by the LCAO approximation. We also presented the modified expressions for other operators.

A self-definining a b i n i t i o procedure modeling the (100) face of diamond

The (100) face of diamond is investigated within the cluster approach. The edge effects introduced by the use of a cluster of small size are minimized with the help of an ab initio model calculation. The model is implemented by two techniques: (1) a modified Hartree–Fock SCF wave function for the cluster, which approximately includes a representation of the extended solid, and (2) a projection procedure. For comparison, full SCF calculations are also performed. A Gaussian lobe basis set is used in all of the cases. The electronic structure for a carbon on the (100) face is presented. Results of Mulliken population analysis are reported and interpreted in terms of electronic density contour maps of the difference between the model and SCF calculations. The results show that the model gives a more uniform distribution of the charge across the surface than the SCF.

Modified virtual orbitals of Hartree–Fock method in calculation of second‐order perturbation energy

AbstractAn advantage of modified virtual orbitals of Hartree–Fock method is discussed in the calculation of the second‐order perturbation energy. All the modified virtual orbitals can be fitted for the intermediate virtual states in the perturbation expansion, only if the molecular orbitals are expanded in terms of infinite basis functions and the set of molecular orbitals is infinite and complete. If the molecular orbitals are expanded in terms of finite basis functions, only the modified virtual orbitals with lower energies are appropriate to describe the intermediate virtual states, but the modified virtual orbitals with higher energies become inadequate. To explain the usefulness of the modified virtual orbitals, the calculation by the modified Hartree–Fock method without CI are compared with the calculation by the traditional Hartree–Fock method with complete CI.

Effect of correlations on nuclear muon capture

The process μ−+12C→12B+vμ is studied using the modified Hartree Fock wavefunction obtained with the unitary-model-operator-approach starting from the realistic hardcore nucleon-nucleon interaction, with the aim of testing the wavefunctions and obtaining a numerical value for the induced pseudoscalar coupling constant (gP). These observables, namely, the partial capture rate to the12B(1+; g.s), its recoil nuclear polarisation and the total capture rate, which exhaust the available experimental data in the above process have been calculated and compared with the other theoretical and experimental results.

Structural and energetic predictions for anions from a modified Hartree—Fock procedure

The performance of a modifed Hartree—Fock (MHF) procedure which differs from conventional HF theory through the inclusion of continuum functions in t

Effects of electron correlation on physical quantities expressed as second-order perturbation energies

A physical quantity of a molecule which can be expressed as second-order perturbation energy is investigated, by using the modified Hartree–Fock operator proposed by Huzinaga and Arnau. The calculation shows that the electron correlation in the ground electronic state is much more important than that in the intermediate excited states. The electron correlation in the ground state increases the absolute value of the calculated quantity, if the intermediate excited states are the triplets. The reverse effect can be expected, if the intermediate excited states are the singlets.

Independent-particle-model potentials for atoms and ions with36<Z≤54and a modified Thomas-Fermi atomic energy formula

Using the ab initio energy-minimization procedure of Bass, Green, and Wood, we determine two potential parameters, $\ensuremath{\xi}$ and $\ensuremath{\eta}$, characterizing the independent-particle-model potential of Green, Sellin, and Zachor (GSZ) for atoms and positive ions with $36<Z\ensuremath{\le}54$. This extends earlier modified-Hartree-Fock (MHF) calculations of Szydlik and Green and of Green, Garvey, and Jackman. We find that both of the parameters in question display, to a good approximation, a linear dependence on the degree of ionization $Z\ensuremath{-}N$ for fixed numbers of electrons $N$. The slopes and $y$ intercepts associated with the linear dependence of $\ensuremath{\xi}$ display marked shell-like behavior, while those associated with $\ensuremath{\eta}$ vary rather smoothly with $N$. Our determinations of total energies are usually within 50 ppm of earlier Hartree-Fock calculations for those cases in which such calculations exist. Using the entire collection of energies and GSZ minimization parameters now available, we reexamine a modified version of the Thomas-Fermi statistical model (MTF) due to Green, Sellin, and Darewych. We show that this model is capable of yielding the linear $Z\ensuremath{-}N$ dependence of the GSZ parameters which we found empirically in the MHF work. By numerical adjustment of the coefficients of our MTF model, we obtain energies of stable atoms and ions, as well as GSZ potential parameters which are in good agreement with the MHF calculations.

Multiple Solutions to the Hartree–Fock Problem. II. Molecular Wavefunctions in the Limit of Infinite Internuclear Separation

The restricted Hartree–Fock equation for a closed-shell molecule is investigated in the limit of infinite internuclear separation. The conventional molecular equation reduces, in the limit, to a set of modified Hartree–Fock equations for the individual fragments. The modification consists of admitting fractional occupation numbers for occupied orbitals. The individual fragment equations are coupled by conditions on the orbital energies and on the fractional “charges.” The self-consistent field equation, being nonlinear, admits a variety of mathematically acceptable solutions. For lithium hydride a wavefunction corresponding to ionic fragments Li+ and H− is one such limiting solution. Contrary to popular notion, it is not the solution of lowest energy even within the restricted SCF formalism. Limiting solutions are obtained for H2, LiH, and CH4. The separated hydrogen molecule is shown to have an SCF energy which is exactly one-quarter that of the helium atom. The long-range interaction of two hydrogen atoms is described by an inverse first-power potential according to Hartree–Fock theory. The Z dependence of correlation energy at infinite separation is analyzed in terms of a charge-transfer process. It is argued that in passing from the equilibrium separation to infinity the ground-state SCF wavefunction for a molecule such as lithium hydride becomes at first more and then less ionic.

Pressure Calculations and the Virial Theorem for Modified Hartree-Fock Solids and Atoms

Pressure Calculations and the Virial Theorem for Modified Hartree-Fock Solids and Atoms