Improved uncoupled Hartree–Fock (IUCHF) perturbation methods and bounds for the second-order energy in coupled Hartree–Fock perturbation theory

Different kinds of improved uncoupled Hartree–Fock methods are proposed for the calculation of second-order perturbation energies. Using these methods inequalities are derived for the error of the uncoupled procedure with geometric approximation.     

Numerical Hartree–Fock results for atoms Cs through Lr

Accurate nonrelativistic numerical Hartree–Fock results are reported for the heavy atoms Cs (Z = 55) through Lr (Z = 103) in their ground states. © 1995 John Wiley & Sons, Inc.     

Gaussian overlap approximation in the projected Hartree–Fock method

A method for the approximate calculation of matrix elements with respect to projected Hartree–Fock wave functions is proposed. The method is tested on some calculations in the many-parameter AMO method. It is found that the approximation reduces the amount of work, involved in the evaluation of the energy, by a factor of five and that it reproduces the exact values to within a few per cent.     

Exact and approximate Hartree–Fock calculations for extended metallic systems

Starting out with the electron gas, we make a survey of the reasons for the singularity in the derivative of the orbital energy with respect to the wave number at the Fermi level for a realistic extended metallic system. Some properties of the occupation function are reviewed and it is pointed out that the direct reason for the singularity resides in a divergent lattice sum originating in the exchange part of the orbital energy. Numerical aspects are discussed, in particular with reference to the difficulty in detecting this singularity in actual computations.     

Hartree–Fock wave functions with a modified GTO basis for atoms

We have solved the atomic Hartree–Fock equations by using the algebraic approach, expanding the single-particle radial wave function in terms of a modified Gaussian type orbitals (GTOs) basis. Several atomic properties such as Kato's cusp condition for the electron density or the correct asymptotic behavior of the electron momentum density distribution are accurately verified. Additionally the energy of the atomic ground state can be obtained by using a smaller number of basis functions than in standard GTO expansions. This study has been performed for several atoms of the first three rows. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 59–64, 1997     

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.     

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 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.     

The Hartree–Fock ground state of atomic two-electron systems and the Wilson–Silverstone 1s orbital

For the Hartree–Fock ground state of atomic two-electron systems, the variational function of Wilson and Silverstone, ?(r) = (a + kr)^?1 exp(-kr) / (4π)^1/2, can be optimized in two complementary ways. For small values of the atomic number Z, all intergrals have been calculated numerically and optimization can be performed accurately. However, as Z increases, loss of significant figures is increasingly detrimental to the optimization process. For sufficiently large values of Z, the integrals may be replaced by asymptotic expansions in terms of (2a)^?. As a result of optimization, the parameters and expectation values can be given as expansions in terms of (32Z)^?1/2. Both methods yield good results for Z ≈ 25, so that the whole range of Z can be treated accurately. The results have been compared with those derived from other analytical two-parameter functions. It is found that ?(r) is indeed the outstanding two-parameter function, at least for small and intermediate values of Z.     

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.     

Extension of the Coulomb–Hole–Hartree–Fock theory to molecules

The Coulomb–Hole–Hartree–Fock method introduced by E. Clementi in the early 1960s and reparametrized more recently by S. Chakraworty and E. Clementi to compute the correlated electronic energy in atomic systems, is here extended to compute molecules. The new parametrization is obtained empirically by fitting first and second atomic ionization potentials from He to Ca and a few diatomic molecules. The present formulation makes use of either one or more determinants in order to ensure proper dissociation products, following the early proposal of G.C. Lie and E. Clementi in the context of density functional computations for molecular systems. The new formulation is tested against the dissociation energies of a large number of molecules and it is found satisfactory. © 1995 John Wiley & Sons, Inc.     

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.     

On uncoupled approximation with approximate form of the Hartree–Fock operator

It is shown that the uncoupled approximation of Dalgarno with the approximate form of the Hartree–Fock operator gives no bound for the second order energy.     

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.     

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.     

Decoupled Hartree–Fock methods. I. Calculations for atoms with orthogonalized orbitals

By a proper approximation of the interaction term in a many-electron Hamiltonian the Hartree-Fock equations are decoupled. Making use of this simplification one obtains a good initial guess for the wave function with minimal computational work. Refining the procedure, the exact HF limit can be achieved. Total energies, ionization potentials and excitation energies for light atoms are calculated.     

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.