The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

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.     

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.     

Spline methods for multiconfiguration Hartree–Fock calculations

The earlier numerical multiconfiguration Hartree–Fock atomic structure package was not designed with high-performance computers in mind. In this paper, some new algorithms based on spline–Galerkin methods are described that are appropriate for concurrent/vector architectures. The goal is to improve the level of numerical accuracy by several orders of magnitude using fewer basis functions than points in a numerical grid. Of critical importance is the robustness of the code: The most serious problems in the numerical implementation were associated with orthogonality constraints. In a spline basis approach, the orthogonality requirements can be integrated into quadratically convergent update procedures. These procedures are evaluated for a number of cases.     

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.     

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.     

Coupled Hartree–Fock calculation of the electric dipole hyper-polarizabilities: He sequence

The variation perturbation method in the coupled Hartree-Fock (HF) scheme has been used to calculate the fourth-order energies of He, Li⁺ and Be²⁺ ions subjected to a uniform external electric field. The hyper-polarizabilities γ_h which are related to the fourthorder energies have been calculated neglecting intra-atomic correlation. For He the observed value is γ_h = 53.6 ± 4.0 a.u. The present estimate in the coupled and uncoupled HF schemes yield 36.2 and 45.3 a.u. respectively. The other calculations using correlated ground state wave function gave values in the range 42-44 a.u. The discrepancies between our result and those of others have been discussed.     

Unrestricted Hartree Fock perturbation theory

The coupled and uncoupled Hartree Fock perturbation theories are generalised using unrestricted Hartree Fock theory. The variational approach to coupled theory is discussed and various corrections to the uncoupled theory are considered. Trial calculations are carried out to obtain atom-atom polarizabilities for Allyl and the first excited triplet state of cis-Butadiene, and to obtain estimates of spin and charge densities for various heterocyclics.

---

Zusammenfassung Die gekoppelte and die ungekoppelte Hartree-Fock Störungstheorie werden mit Hilfe der uneingeschränkten Hartree-Fock-Theorie verallgemeinert. Das Variationsverfahren für die gekoppelte Theorie wird diskutiert und verschiedene Korrekturen für die ungekoppelte Theorie werden untersucht. Es werden Testrechnungen durchgeführt, um die Atom-Atom-Polarisierbarkeiten des Allylradikals und des ersten angeregten Tripletzustandes von cis-Butadien, und um die Spin- und Ladungsdichten verschiedener Heterozyklen abzuschätzen.

     

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     

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.     

Equations de Hartree–Fock dependant du temps

The time-dependent Hartree–Fock (TDHF ) equations are derived up to the second order when the system is perturbed by a monochromatic plane wave. The solutions of the equations are subjected to the orthonormalization conditions satisfied by the orbitals. In the equations, these conditions are expressed by the appearing of coefficients λ playing the part of Lagrangian multipliers. Relations between the coefficients λ are established. These relations are equivalent to the above-mentioned orthonormalization conditions. This equivalence enables us to substitute for the solution of an integrodifferential equation system subject to constraint conditions, that of a free system. The TDHF equations obtained determine the first- and second-order orbital perturbations, which no doubt verify the orthonormalization conditions. These orbitals can be used in the calculation, up to second order, of different nonlinear optical effects.     

Half-projected Hartree–Fock natural orbitals for defining CAS–SCF active spaces

We have extended the range of systems to which the half-projected Hartree–Fock (HPHF ) method has been applied, including the triplet state of the wave function. In our implementation, DIIS overcomes the convergence difficulties reported in earlier studies. HPHF allows generation of a symmetry-broken wave function in regions of the potential energy surface where the RHF wave function is triplet-stable. The fractionally occupied natural orbitals (FONOS ) of the HPHF wave function are good starting vectors for CAS –SCF calculations. A CAS –SCF in the space defined by the HPHF FONOS should be used instead of the unrestricted natural orbital CAS –SCF method in regions of triplet stability and for small active space problems. We draw extensive comparisons between the results of both the UNO –CAS and HPNO –CAS methods and those of full CAS –SCF calculations. © 1993 John Wiley & Sons, Inc.     

Ground-state multiplicity by spin-extended Hartree–Fock method

New possibilities of the spin-extended Hartree–Fock method in determining the sequence of energy levels are analyzed and demonstrated by the example of some oxygen-contained compounds.     

Hartree–Fock MO–LCAO equations with charge-dependent atomic integrals

The Hartree–Fock equations are derived in the MO -LCAO approximation for the case when the integrals (except overlap integrals) over the atomic orbitals are charge-dependent. It is shown that inclusion of the overlap matrix in the iterative procedure gives equations which are too complicated for the simple model under consideration. The approach is applied to the VESCF method in the PPP scheme.     

Multiple time scale Hartree–Fock molecular dynamics

This is the first application of a rigorous, established multiple time-step method to ab initio molecular dynamics. The resulting algorithm is conceptually simple and easy to implement, but very effective. It translates the large mass differences present in ab initio molecular dynamics into substantial savings in computer time while retaining high accuracy. This transforms ab initio molecular dynamics from a desirable but prohibitively expensive possibility into a viable method, at least for short-time phenomena in small systems or for otherwise inaccessibly complicated potential energy surfaces. © 1993 John Wiley & Sons, Inc.     

Triazene proton affinities: A comparison between density functional,Hartree–Fock,and post-Hartree–Fock methods

The consistency of three density functional computational implementations (DMol, DGauss, and deMon) are compared with high-level Hartree–Fock and Møller–Plesset (MP) calculations for triazene (HN?NNH₂) and formyl triazene (HN?NNHCOH). Proton affinities on all electronegative sites are investigated as well as the geometries of the neutral and protonated species. Density functional calculations employing the nonlocal gradient corrections show agreement with MP calculations for both proton affinities and geometries of neutral and protonated triazenes. Local spin density approximation DMol calculations using numerical basis sets must employ an extended basis to agree with other density functional codes using analytic Gaussian basis sets. The lowest energy conformation of triazene was found to be nonplanar; however, the degree of nonplanarity, as well as some bond lengths, is dependent on the basis set, electron correlation treatment, and methods used for the calculation. © 1994 by John Wiley & Sons, Inc.

1 This article is a U.S. Government work and, as such, is in the public domain in the United States of America.

     

Convergence properties of Hartree–Fock SCF molecular calculations

Hartree-Fock equations are viewed as nonlinear algebraic equations that can be solved iteratively. Provided we assume the existence of a solution, valuable properties of convergence may be assessed. The close connection between convergence of the SCF procedure and stability properties of the solution is shown from a nonapproximate standpoint. The convergence features of level-shifting convergence-forcing techniques are analyzed. The connection between this nonlinear algebraic approach and the related gap equation is displayed and the example of the restricted Hartree-Fock hydrogen molecule is discussed.     

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.