Unrestricted Hartree-Fock Theory Research Articles

Electronic structure of dangling bonds in amorphous silicon studied via a density-matrix functional method

A structural model of hydrogenated amorphous silicon containing an isolated dangling bond is used to investigate the effects of electron interactions on the electronic level splittings, localization of charge and spin, and fluctuations in charge and spin. These properties are calculated with a recently developed density-matrix correlation-energy functional applied to a generalized Anderson Hamiltonian, consisting of tight-binding one-electron terms parametrizing hydrogenated amorphous silicon plus a local interaction term. The energy level splittings approach an asymptotic value for large values of the electron-interaction parameter U, and for physically relevant values of U are in the range 0.3-0.5 eV. The electron spin is highly localized on the central orbital of the dangling bond while the charge is spread over a larger region surrounding the dangling bond site. These results are consistent with known experimental data and previous density-functional calculations. The spin fluctuations are quite different from those obtained with unrestricted Hartree-Fock theory.

Spin-Unrestricted Formalism for a Partially Restricted Hartree–Fock Approach

An alternative method of building a partially restricted single Slater determinant with the number of α electrons equal to the number of β electrons has been proposed. Using a simple to implement method of taking the orthogonality constraints into account the amended unrestricted Hartree–Fock (UHF) equations for molecular orbitals have been obtained. The formalism deals as does the UHF theory with the cubic equations with respect to the LCAO coefficients whereas a partially restricted HF approach developed in [10] leads to the fifthdegree equations. The minimal extra calculations are required to implement this scheme in existing UHF codes. Unlike the known Roothaan coupling formalism the proposed approach is suitable for use with the well-established unrestricted Moller–Plesset perturbation theories in incorporating the electron correlation effects. Some peculiarities of using the method are illustrated by considering a model molecular system.

A theoretical study of the polarized neutron scattering from Cs3CoCl5

Spin-orbit interactions play an essential part in elucidating the magnetic structures which are measured by the polarized neutron diffraction technique. This work extends our previous Hartree–Fock theory, with a one-electron spin-orbit term limitation, to incorporate both one- and two-electron terms exactly. This new theory, which is based on the current density, has been applied to calculate the low-temperature magnetic structure factors of the Cs3CoCl5 crystal. The crystal is assumed to be assembled from noninteracting molecular fragments. The calculated structure factors were compared directly with those observed from experiment. The agreement between these theoretical results and the experimental data shows great improvement compared with those from the usual unrestricted Hartree–Fock theory and with those from the previous generalized Hartree–Fock theory including one-electron terms. To examine the electron correlation effects, we use ab initio wave-function-based correlation methods, unrestricted second-order Møller–Plesset perturbation (UMP2) and quadratic configuration interaction including single and double excitations (QCISD), and also density functional theoretical methods, Xα, local-spin density approximation, generalized gradient approximation, Becke’s three-parameter hybrid methods, and modified half-and-half hybrid methods, to calculate magnetic structure factors. Results indicate that the effects of spin-orbit interactions are comparable to those of electron correlation, and both must be included in reproducing the experimental data. To consider both these effects simultaneously we employed a simple model. With only three parameters, this model reproduced the observed data almost exactly as reflected by a χ2 goodness of fit close to unity.

Determination of easy axis in a chemical ferromagnet, 4-(p-chlorobenzylideneamino)-TEMPO (TEMPO = 2,2,6,6-tetramethylpiperidin-1-oxyl)

The trapping sites for muon and muonium in ferromagnetic p-Cl–Ph–CH=N–TEMPO [(4-(p-chlorobenzylideneamino)–TEMPO (TEMPO = 2,6,6-tetramethyl-piperidin-1-yloxyl)] and the hyperfine interaction tensors for these sites are obtained using first-principles Unrestricted Hartree–Fock theory. The calculated hyperfine interactions are used to compare the calculated zero field muon spin rotation (μSR) frequencies for different choices for the easy axis and the observed frequency. It has been concluded that the two trapping centers that can best explain the observed μSR frequency are a trapped singlet muonium near the radical oxygen and a trapped muon site near the chlorine. The direction of the easy axis also is determined to be the b-axis of the monoclinic lattice. This direction for the easy axis is confirmed by determining the direction of the distributed magnetization in the molecular solid which leads to a minimum dipole–dipole interaction energy. The consequences of this agreement for the easy axis direction by two independent procedures are discussed.

Homonuclear 3d transition-metal diatomics: A systematic density functional theory study

The equilibrium bond lengths, harmonic vibrational frequencies, and dissociation energies of the ground state homonuclear 3d transition-metal diatomics (scandium through copper) were determined using six density functional or hybrid Hartree–Fock/density functional theory (HF/DFT) methods and unrestricted Hartree–Fock theory. Results are compared to other theoretical studies and to experimental values when available. The accuracy of the DFT results is found to be highly dependent upon the functional employed, with the pure DFT methods, BLYP and BP86, often performing significantly better than the hybrid HF/DFT methods. For the van der Waals complex Mn2, all six functionals predict the ground state to be high-spin, disagreeing with experiment; the true (antiferromagnetic) ground state was not found for any functional. Average errors for theoretical geometries and vibrational frequencies are for B3LYP, 0.053 Å (2.4%) and 122 cm−1 (31.1%); for B3P86, 0.051 Å (2.4%) and 122 cm−1 (31.3%); for BHLYP, 0.077 Å (4.1%) and 208 cm−1 (49.3%); for BLYP, 0.024 Å (1.3%) and 98 cm−1 (24.5%); for BP86, 0.020 Å (1.1%) and 104 cm−1 (25.6%); and for LSDA, 0.056 Å (3.0%) and 158 cm−1 (37.9%). No functional gives results directly comparable for all nine species. Dissociation energy results are severely overestimated in many instances and negative in others. Anecdotal reports of success for density functional theory for these systems may have been overblown.

Hartree–Fock investigation of muon trapping in the chemical ferromagnet 4-( p-chlorobenzylideneamino)-TEMPO

First-principles unrestricted Hartree–Fock theory is used to obtain the trapping sites for muon and muonium in ferromagnetic p-Cl–Ph–CHN-TEMPO (4-( p-chlorobenzylideneamino)- 2,2,6,6-tetramethylpiperidin-1-yloxyl) and the hyperfine interaction tensors for these sites. Using the calculated hyperfine interactions to fit the two experimentally observed muon spin rotation frequencies, it has been concluded that the two most likely candidates for explaining the experimental data are a muon trapped at the chlorine site and a singlet muonium state at the radical oxygen. The direction of the easy axis is also determined.

On R�denberg's integral approximations and their unrestricted and combined use in molecular orbital theories of Hartree-Fock type

Ruedenberg's well-known letter of 1951 contains two implications which still have been employed so far in numerical computations: (1) Whenever all types of attraction and repulsion integrals are subject to the Ruedenberg approximations in its simplest form partially known already from Mulliken, the attractive, the Coulomb, as well as the exchange part of the restricted Hartree-Fock-Roothaan equation can be led back to the calculation of corresponding diagonal elements, only. Using Ruedenberg's more elaborate concepts, which are invariant with respect to the rotation of local coordinate axes, the complete Fock-matrix representation can be constructed equivalently from only its own diagonal blocks, each belonging to one atom. Similar formulas are valid for the unrestricted Hartree-Fock theory of Pople and Nesbet. (2) If, however, one prefers to calculate all types of one- and two-center integrals exactly as suggested in Ruedenberg's headline, the original simplicity of both representations is lost. Instead, one is led to more complicated expressions, which arise from the fact that Ruedenberg's integral formulas, when applied to certain kinds of three-center repulsion integrals, imply considerable oversimplifications. In spite of this critical result, Ruedenberg's ideas offer an extension together with an interpretation of the semiempirical Wolfsberg and Helmholz recipe (better known frommore » Hoffmann's extended Hueckel theory), on the one hand, and of the neglect of differential overlap schemes ZDO and NDDO, on the other, from a common point of view.« less

Nonlinear optical properties of the rhombic B4-cluster

The equilibrium geometry, energy, dipole polarizability, and second hyperpolarizability of the rhombic B4 -cluster with D2h symmetry are calculated by restricted (RHF) and unrestricted (UHF) Hartree-Fock theory; Møller-Plesset perturbation theory (MPn); coupled-cluster theory with singles, doubles, and a perturbational treatment for connected triples [CCSD(T)]; Brueckner doubles with a perturbational treatment for connected triples [BD(T)]; and complete active space SCF (CASSCF) and restricted active space SCF (RASSCF) methods, using three different basis sets. The multireference methods show that excited state configurations contribute appreciably to the ground state wave function. Accordingly, the RHF method yields results that differ greatly from those of the correlated methods, even for the geometry. UHF gives more reliable geometries, but suffers from high spin contamination. The electric properties calculated at reasonably highly correlated levels are qualitatively comparable for both the single reference and multireference descriptions, although differences between CCSD(T) and BD(T) are larger than usually reported in the literature and properties calculated at the MPn (n=2, 3, 4) series show much damped oscillatory behavior, especially for the components along the long in-plane axis. It is found that inclusion of f-functions in the basis set do not have a large effect on the electric properties. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 679-687, 1999.

Optical transitions and EPR properties of two-coordinated Si, Ge, Sn and relatedH(I),H(II),andH(III)centers in pure and doped silica fromab initiocalculations

The photoabsorption and photoluminescence properties of two-coordinated Si, Ge, and Sn defect centers and the magnetic interactions of the corresponding $[=\mathrm{S}\mathrm{i}\ensuremath{-}\mathrm{H}{]}^{\ifmmode\bullet\else\textbullet\fi{}},$ $[=\mathrm{G}\mathrm{e}\ensuremath{-}\mathrm{H}{]}^{\ifmmode\bullet\else\textbullet\fi{}},$ and $[=\mathrm{S}\mathrm{n}\ensuremath{-}\mathrm{H}{]}^{\ifmmode\bullet\else\textbullet\fi{}}$ structures, the $H(\mathrm{I}),$ $H(\mathrm{II}),$ and $H(\mathrm{III})$ centers, respectively, have been studied by means of ab initio quantum-chemical methods. Using cluster models and configuration interaction wave functions, we have determined the transition energies to the lowest singlet and triplet states and the corresponding emissions and lifetimes. The computed optical properties are in close agreement with those proposed for the two-coordinated centers in pure and doped silica. The hyperfine interactions of the paramagnetic $H(\mathrm{I}),$ $H(\mathrm{II}),$ and $H(\mathrm{III})$ centers have been studied using unrestricted Hartree-Fock and density-functional theory approaches. Also, in this case, the proposed assignments are consistent with the calculations. The results provide strong theoretical support to the proposal of the two-coordinated Si as the origin of one component of the 5.0 eV ${B}_{2}$ absorption band in amorphous silica.

Simple spin correction of unrestricted density-functional calculation.

The unrestricted Hartree-Fock and unrestricted Kohn-Sham calculations generally result in spin-contaminated solutions. Moreover, the energies from these calculations cannot be directly compared with the results of corresponding restricted calculations since the latter yield higher energies due to restrictions imposed on the form of the wave function. We present here a simple method of correcting the mixed spin energies resulting from unrestricted Hartree-Fock or density-functional theory calculations and removing the foreign spin components. The method allows for elimination of higher-multiplet components from the given mixed spin state solution by performing unrestricted calculations at the same fixed geometry for the higher multiplets and the state under consideration. The performance of the method is illustrated with several examples of density-functional calculations of radical species. The current method is also variational in nature and can be further extended in a self-consistent field fashion. \textcopyright{} 1996 The American Physical Society.

An ab initio calculation of magnetic structure factors for Cs3CoCl5 including spin–orbit and finite magnetic field effects

Spin–orbit interaction plays a significant role in determining the magnetic density in some transition metal complexes. We present a new ab initio technique, based on an extension of unrestricted Hartree–Fock theory, which includes nonperturbatively these spin–orbit effects, and simultaneously also the effects of a finite magnetic field. We also present a new and efficient method for calculating magnetic structure factors, based on the current density rather than magnetic dipole moment density, for a crystal composed of noninteracting molecular fragments. These structure factors are directly comparable to polarized neutron diffraction experiments. Results for the Cs3CoCl5 crystal are compared with experiment and previous studies. Without one-electron spin–orbit coupling terms, the magnitudes of the predicted structure factors are on average 10–15 % too low, whereas, with the spin–orbit terms, the magnitudes are 25–30% too high. Using an effective nuclear charge for Co in the spin–orbit term brings the results into much better agreement, and suggests that the two-electron spin–orbit shielding terms omitted in the present work are important. For over one quarter of the reflections studied, the magnetic contribution to the structure factors is more than 20% of the nuclear contribution.

Time-dependent unrestricted Hartree-Fock theory for the multiphoton ionization of atoms.

Time-dependent unrestricted Hartree-Fock (TDUHF) theory is formulated for the multiphoton ionization of a model atom. The single-particle states are nonorthogonal and the quantum variational principle requires additional normalization terms. For a model He atom, the three-dimensional (3D) electrostatic interaction is replaced by a 1D soft-core interaction that allows the construction of the two-electron atom in only two dimensions. The resulting 1D formulation of the TDUHF equations contains numerous exchange and overlap terms. The TDUHF multiphoton ionization rates are found to be in better agreement with the results of a direct solution of the time-dependent Schr\odinger equation on a 2D lattice than the rates predicted by a 1D formulation of the standard TDHF equations.

Interaction of H2 and He with metal atoms, clusters, and ions.

The binding energy, equilibrium geometry, and electronic structure of molecular hydrogen and helium atoms interacting with Ni, Cu, Co, and Al atoms and cations have been calculated using self-consistent-field linear combination of the atomic-orbitals--molecular-orbital method. The exchange interaction between the electrons is treated using the unrestricted Hartree-Fock theory. Correlation is included using M\oller-Plesset perturbation theory up to the fourth order. The accuracy of the calculation is tested by comparing the results of the ground-state spin and binding energy with available experiments on dimers. While a neutral metal atom is found to dissociate the ${\mathrm{H}}_{2}$ molecule, the metal cations bind the ${\mathrm{H}}_{2}$ molecule associatively and with rather large binding energy. The binding of the He atom to a metal cation is found to be analogous to that of a ${\mathrm{H}}_{2}$ molecule since both possess closed electronic shells. The number of hydrogen molecules and He atoms that can be bound to a metal cation at a given temperature and gas density is studied using the effective-medium approximation and density-functional theory. The results are compared with recent experiments.

There are no unfilled shells in unrestricted Hartree-Fock theory.

We prove that in an exact, unrestricted Hartree-Fock calculation each energy level of the Hartree-Fock equation is either completely filled or completely empty. The only assumption needed is that the two-body interaction is---like the Coulomb interaction---repulsive; it could, however, be more complicated than a simple potential; e.g., it could have tensor forces and velocity dependence. In particular, the Hartree-Fock energy levels of atoms and molecules, often called shells, are never partially filled.

An ab-initio electronic structure study of the chemisorption of small molecules in aluminum cluster ions

A systematic ab-initio study of the chemisorption of small diatomic molecules (H2) in aluminum cluster ions, Aln+ (with 2 ≤ n ≤ 5) is reported. Results from both unrestricted Hartree-Fock (with two different basis sets) and density functional theory calculations (at the local spin density approximation and nonlocal corrections to it) are compared, discussing peculiarities of each technique. We report the equilibrium geometries of the adducts formed, the enthalpy of adsorption and predict the chemisorption nature (molecular versus dissociative). General agreement with available experimental data is obtained.

Kinetic and Exchange Energy Densities of an Inhomogenous Electron Liquid Related in Hartree—Fock Theory and Other Models

Abstract March and Santamaria have recently proposed non-local generalizations of kinetic and exchange energy densities of an inhomogeneous electron liquid and have related these through restricted Hartree–Fock theory. Prompted further by the work of Lee, Lee and Parr, the above results are generalized to include the unrestricted Hartree–Fock theory. Finally, approximations of the type required to relate to the work of Lee et. al. are referred to.

The physics of Mott electron localization

Although rare earth metals and compounds are well known to have localized 4f electrons, the concept of “localized electrons” in a periodic system remains a notorious source of confusion. Similar issues arise for 3d electrons and have been studied most intensively for 3d transition metal oxides — the traditional Mott insulator materials. We review a conceptual picture of the Mott insulator state which clarifies the concept of localized electrons, and which has recently been implemented with much success. The key feature is strong orbital as well as spin polarization, within the context of unrestricted Hartree—Fock theory using screened Coulomb and exchange parameters to represent the intra-atomic d—d interactions. The failure or success of local-density approximations is directly related to the presence or absence of orbital polarization. Some practical considerations and recent calculations are discussed.

Electronic and hyperfine structures of hydrocarbon radical cations

Ab initio calculations on alkane and alkene radical cations with 2 to 6 carbon atoms are reported. Geometry optimizations were performed on both the unrestricted Hartree-Fock (UHF) and second-order perturbation theory (MP2) levels, with subsequent hyperfine structure (HFS) calculations on the optimized geometries, by means of relatively large (30000–60000) configuration-interaction calculations including single and double excitations (SDCI). By comparison with experimental HFS data, it was deduced that the MP2 optimizations are able to provide accurate geometries in most cases, but that the basis-set requirements are very stringent, at least for smaller radicals. For the alkene cations, it is found that the HFS is dominated by the region immediately surrounding the double bond, with virtually no influence from protons which are not bonded to the α or β carbon atoms.

Molecular view of the interfacial adhesion in aluminum-silicon carbide metal-matrix composites

The binding energies, electron charge transfer, bond lengths, and core level shifts of Al-Al, Al-Si, Al-C, and Si-C dimers have been calculated self-consistently using the linear combination of atomic orbitals-molecular orbital theory. The exchange interactions are treated using the unrestricted Hartree–Fock theory and correlation corrections are included through the Möller–Plesset perturbation scheme up to fourth order. The results are used to understand the nature and strength of bonding at the interface of Al and SiC crystals. The strong bonding of Al-C dimers compared to Al-Al and Al-Si is shown to be responsible for the aluminum carbide formation at the interface. The charge transfer between the constituent atoms in the dimer and the accompanying core level shifts are also shown to be characteristic of what has been observed at the Al/SiC interface.

Static Dipole Polarizabilities of C, N, O and F: The Importance of Spin Projection

ABSTRACTThe static dipole polarizability for the ground state of C(3Π ), N(4∑+), O(3Π ) and F(2∑+ ) have been calculated using ab initio molecular orbital techniques. The polarizabilities are obtained from unrestricted Hartree-Fock and Many-Body Perturbation Theory wave functions with and without spin projection, which are computed by finite-field differentiation. The effect of spin contamination is shown to be important on the calculation of static dipole polarizabilities for the ground state of C, N, O and F.