Bond length alternation in cyclic polyenes. I. Restricted Hartree–Fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian and its relationship to the singlet stability problem for symmetry adapted Hartree–Fock solutions for these systems is investigated using the restricted Hartree–Fock method. The σ-energy contribution is approximated by a quasiharmonic empirical potential. It is shown that the restricted Hartree–Fock energies favor the cyclic polyene distortion and an estimate of the distortion and of the stabilization energy for infinite linear polyenes is obtained.     

Bond length alternation in cyclic polyenes. II. Unrestricted hartree–fock method

The problem of bond length alternation in cyclic polyene models as described by the Pariser–Parr–Pople π-electron Hamiltonian, together with an empirical quasi harmonic σ-core potential is investigated using the unrestricted Hartree–Fock wave function employing different spatial orbitals for different spins. It is shown that in contrast to the restricted Hartree–Fock method, which favors bond alternation in large cyclic polyenes, the unrestricted Hartree–Fock method stabilizes the symmetric structures with equidistant internuclear separation. An assessment of the amount of correlation error recovered by the unrestricted Hartree–Fock procedure is examined and the qualitatively different behavior of the cyclic polyene models when described by restricted and unrestricted Hartree–Fock wave functions is discussed from this viewpoint.     

The modified Hartree–Fock self-consistent field equation

A 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.     

Simple derivation of conditions for instability in the Hartree–Fock and projected Hartree–Fock schemes

The conditions for instability of solutions of Hartree–Fock and projected Hartree–Fock equations are derived in a form involving finite real symmetric matrices. These conditions are also expressed in terms of the Fock–Dirac density matrix, both at the spin–orbital and at the orbital level. The particular variations which give rise to the so-called singlet and triplet instabilities are described.     

Extended Hartree–Fock self-consistent field with some results for berylliumlike atomic systems

The general methods of deriving the extended Hartree–Fock equations are described. The rules for going over from the energy expression in the ordinary method of calculation to that in an extended one are reformulated and illustrated. The extended Hartree—Fock equations for berylliumlike atomic systems based on the use of nonorthogonal radial orbitals are given and solved. The numerical values of overlap integrals and total energies are given and discussed.     

Solution of periodic Poisson's equation and the Hartree–Fock approach for solids with extended electron states: Application to linear augmented plane wave method

We formulate a Hartree–Fock‐LAPW method for electronic band structure calculations. The method is based on the Hartree–Fock–Roothaan approach for solids with extended electron states and closed core shells where the basis functions of itinerant electrons are linear augmented plane waves. All interactions within the restricted Hartree–Fock approach are analyzed and in principle can be taken into account. In particular, we obtained the matrix elements for the exchange interactions of extended states and the crystal electric field effects. To calculate the matrix elements of exchange for extended states, we first introduce an auxiliary potential and then integrate it with an effective charge density corresponding to the electron exchange transition under consideration. The problem of finding the auxiliary potential is solved by using the strategy of the full potential LAPW approach, which is based on the general solution of periodic Poisson's equation. Here, we use an original technique for the general solution of periodic Poisson's equation and multipole expansions of electron densities. We apply the technique to obtain periodic potentials of the face‐centered cubic lattice and discuss its accuracy and convergence in comparison with other methods. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.     

Hartree–Fock instabilities and electronic properties

Hartree–Fock instabilities are investigated for about 80 compounds, from acetylene to mivazerol (27 atoms) and a cluster of 18 water molecules, within a double ζ basis set. For most conjugated systems, the restricted Hartree–Fock wave function of the singlet fundamental state presents an external or so‐called triplet instability. This behavior is studied in relation with the electronic correlation, the vicinity of the triplet and singlet excited states, the electronic delocalization linked with resonance, the nature of eventual heteroatoms, and the size of the systems. The case of antiaromatic systems is different, because they may present a very large internal Hartree–Fock instability. Furthermore, the violation of Hund's rule, observed for these compounds, is put in relation with the fact that the high symmetry structure in its singlet state has no feature of a diradical‐like species. It appears that the triplet Hartree–Fock instability is directly related with the spin properties of nonnull orbital angular momentum electronic systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 483–504, 2000     

Spin-Projected extended Hartree–Fock equations. II. Odd-electron systems

The spin-projected extended Hartree–Fock equations discussed in Part I for an even number of electrons are given here for the odd-electron case.     

Coulomb-Hole–Hartree–Fock functional

We have extended to molecules a density functional previously parametrized for atomic computations. The Coulomb-hole–Hartree–Fock functional, introduced by Clementi in 1963, estimates the dynamical correlation energy by the computations of a Hartree–Fock-type single-determinant wave function, where the Hartree–Fock potential was augmented with an effective potential term, related to a hard Coulomb hole enclosing each electron. The method was later revisited by S. Chakravorty and E. Clementi [Phys. Rev. A 39 , 2290 (1989)], where a Yukawa-type soft Coulomb hole replaced the previous hard hole; atomic correlation energies, computed for atoms with Z = 2 to Z = 54 as well as for a number of excited states, validated the method. In this article, we parametrized a function, which controls the width of the soft Coulomb hole, by fitting the first and second atomic ionization potentials of the atoms with 1 ? Z ? 18. The parametrization has been preliminarily validated by computing the dissociation energy for a number of molecules. A few-determinant version of the Coulomb-hole–Hartree–Fock method, necessary to account for the nondynamic correlation corrections, is briefly discussed. © 1994 John Wiley & Sons, Inc.     

Spin-Free Quantum Chemistry. XXI. Hartree–Fock theory

The theory of the open-and closed-shell restricted Hartree–Fock method is given a unitary group formulation. Both extremum and stability conditions are employed, the former leading to a generalized Brillouin theorem.     

The relationship between the unrestricted and projected Hartree–Fock methods in a simple three-electron model system

Various forms of the unrestricted Hartree–Fock (UHF ) scheme are studied for a simple three-electron model system, represented by the PPP (Pariser–Parr–Pople) π-electron model of the allyl radical. Both spin and space symmetry are violated in the UHF trial wave function, either individually or simultaneously. A comparison with the projected Hartree–Fock (PHF ) schemes studied earlier is made and the effect of the order in which various symmetries are broken in both UHF and PHF schemes is studied. The effectiveness of various schemes follows from a comparison of the correlation energy and the wave function is obtained by various UHF (or projected UHF ) and PHF schemes, in the whole range of the coupling constant, with the corresponding quantities given by the exact solution of the model. Finally, the implications of the stability of the restricted HF solutions for the behavior of various single- and multiparameter UHF and PHF schemes are briefly outlined and exemplified on the studied model.     

Parallelization of four‐component calculations. I. Integral generation,SCF, and four‐index transformation in the Dirac–Fock package MOLFDIR

The treatment of relativity and electron correlation on an equal footing is essential for the computation of systems containing heavy elements. Correlation treatments that are based on four‐component Dirac–Hartree–Fock calculations presently provide the most accurate, albeit costly, way of taking relativity into account. The requirement of having two expansion basis sets for the molecular wave function puts a high demand on computer resources. The treatment of larger systems is thereby often prohibited by the very large run times and files that arise in a conventional Dirac–Hartree–Fock approach. A possible solution for this bottleneck is a parallel approach that not only reduces the turnaround time but also spreads out the large files over a number of local disks. Here, we present a distributed‐memory parallelization of the program package MOLFDIR for the integral generation, Dirac–Hartree–Fock and four‐index MS transformation steps. This implementation scales best for large AO spaces and moderately sized active spaces. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1176–1186, 2000     

Corrections to coupled Hartree–Fock theory: the rearrangement effect

In this communication we discuss a method of incorporating corrections to the coupled Hartree–Fock (CHF ) formalism by introducing the so-called “rearrangement effect.” In this we take account of the relaxation of the core orbitals when excitations from a starting Hartree–Fock wave function occur. The magnitude of this correction numerically is found to be quite significant for the polarizabilities of two-electron atomic systems, results for which are reported.     

Some remarks on multiconfiguration time-dependent Hartree–Fock theory

Recent progress in multiconfiguration time-dependent Hartree–Fock (MC –TDHF ) theory is reviewed. The generality of a formalism first put forward by Moccia [Int. J. Quantum Chemistry 8 , 293 (1974)] is emphasized, and illustrative applications are discussed. A calculation of the frequency-dependent electric-dipole polarizability of the helium atom shows that even with a modest amount of CI, it is possible to achieve an accuracy comparable to that obtained by use of Hylleraas-type wave functions.     

Formulation of unrestricted and restricted Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) method, dealing with Bogoliubov orbitals, which consist of particle and hole part, can provide states with pair correlations associated with Cooper pairs. The dimension of HFB Fock matrices can be reduced by restrictions of spin states of Bogoliubov orbitals similarly to ordinary Hartree–Fock (HF) equations such as restricted HF (RHF), unrestricted HF (UHF), and generalized HF (GHF). However, there are few studies of moderate restricted HFB equations such as UHF‐based HFB equations. In this article, formulation and calculations of restricted HFB equations are described. The solutions of general and restricted HFB equations are compared. Pair correlations taking account of restricted and general HFB equations are discussed. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Near-Dirac–Hartree–Fock results for first-row atoms calculated with GTO basis sets

Relativistic basis sets for first-row atoms have been constructed by using the near-Hartree–Fock (nonrelativistic) eigenvectors calculated by Partridge. These bases generate results of near-Dirac–Hartree–Fock quality. Relativistic total and orbital energies, relativistic corrections to the total energy, and magnetic interaction energies for the first-row atoms have been presented. The smallest Gaussian expansions (13s8 p expansions) yield Dirac–Hartree–Fock total energies accurate through six significant digits, while the largest expansions (18s13p expansions) give these energies accurate through seven significant digits. These results are more accurate than some of the results reported earlier, particularly for the open-shell atoms, indicating that the basis employed is reasonably economical for relativistic calculations. © 1995 John Wiley & Sons, Inc.     

Some properties of the bivariational Hartree–Fock scheme for complex symmetric many-particle operators

The bivariational Hartree–Fock scheme for a general many-body operator T is discussed with particular reference to the complex symmetric case: T^? = T*. It shown that, even in the case when the complex symmetric operator T is real and hence also self-adjoint, the complex symmetric Hartree–Fock scheme does not reduce to the conventional real form, unless one introduces the constraint that the N-dimensional space spanned by the Hartree–Fock functions ? should be stable under complex conjugation, so that ?* = ?α. If one omits this constraint, one gets a complex symmetric formulation of the Hartree–Fock scheme for a real N-electron Hamiltonian having the properties H = H* = H^?, in which the effective Hamiltonian H_eff (1) may have complex eigenvalues ε_k. By using the method of complex scaling, it is indicated that these complex eigenvalues—at least for certain systems—may be related to the existence of so-called physical resonance states, and a simple example is given. Full details will be given elsewhere.     

New derivation of (in)stability conditions for projected Hartree–Fock solutions

A simple and straightforward derivation of (in)stability conditions for projected Hartree–Fock solutions is carried out and compared to the existing approaches to the problem.     

Adapted Gaussian basis sets for atoms Cs to Lr based on the generator coordinate Hartree–Fock method

We have applied a discretized version of the generator coordinate Hartree–Fock method to generate adapted Gaussian basis sets for atoms Cs (Z=55) to Lr (Z=103). Our Hartree–Fock total energy results, for all atoms studied, are better than the corresponding Hartree–Fock energy results attained with previous Gaussian basis sets. For the atoms Cs to Lr we have obtained an energy value within the accuracy of 10⁻⁴ to 10⁻³ hartree when compared with the corresponding numerical Hartree–Fock total energy results. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 858–865, 1998