Perturbative calculation of the Hartree–Fock interaction energy using orthogonalized orbitals

A many‐body perturbation theory based on the partitioning of the dimer Hamiltonian, formulated in an orthogonalized basis set, is used for the calculation of interaction energies at the Hartree–Fock (HF) level. Numerical results for the (HF)₂ and (H₂O)₂ systems in selected geometries are presented. The interaction‐energy components are compared with the results obtained from the standard supermolecular approach and the intermolecular perturbation theory based on the biorthogonal basis set. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 81–88, 1999     

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.     

Local Hartree–Fock orbitals using a three‐level optimization strategy for the energy

Using the three‐level energy optimization procedure combined with a refined version of the least‐change strategy for the orbitals—where an explicit localization is performed at the valence basis level—it is shown how to more efficiently determine a set of local Hartree–Fock orbitals. Further, a core–valence separation of the least‐change occupied orbital space is introduced. Numerical results comparing valence basis localized orbitals and canonical molecular orbitals as starting guesses for the full basis localization are presented. The results show that the localization of the occupied orbitals may be performed at a small computational cost if valence basis localized orbitals are used as a starting guess. For the unoccupied space, about half the number of iterations are required if valence localized orbitals are used as a starting guess compared to a canonical set of unoccupied Hartree–Fock orbitals. Different local minima may be obtained when different starting guesses are used. However, the different minima all correspond to orbitals with approximately the same locality. © 2013 Wiley Periodicals, 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.     

Relativistic Dirac–Fock calculations for closed-shell molecules

A new code has been written to perform relativistic Dirac–Fock self-consistent field (SCF) calculations on closed-shell molecules of any symmetry. The choice of the basis set allows us to work at different levels of approximation depending on the precision required. Calculations on the H₂Po molecule show that accurate results on specific problems like geometry optimization can be obtained by evaluating the two-electron integrals on half the basis spinors. © 1994 by John Wiley & Sons, Inc.     

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.     

Natural energy orbitals and the one-particle Green's function

It is shown how the properties of the one-particle Green's function lead naturally to the definition of the so-called natural energy orbitals. These orbitals allow the fully correlated total energy of a system to be written in Hartree–Fock-like fashion and might therefore provide a bridge between sophisticated correlated wave functions and approximate theories of chemical structure and reactivity based on a Hartree–Fock-like energy expression. Moreover these orbitals form the basis for a self-consistent scheme to calculate the one-particle Green's function. The relation between these natural energy orbitals and the extended Koopmans' theorem is considered. Finally it is shown that the exactness of the lowest extended Koopmans' ionization potential implies the linear independence of the corresponding Dyson orbital from all other Dyson orbitals.     

Variation–iteration method in momentum space: Determination of Hartree–Fock atomic orbitals

A method using the Svartholm iterative procedure to solve atomic Hartree–Fock equations in momentum space is defined and applied to the ground states of Be and B⁺. The calculated atomic orbital properties follow a monotonic and stable convergence, but with rates of convergence depending on each property. The evolution of the orbitals during the iterations is explained by the combined actions of the variational principle, the Svartholm iterative procedure, and the momentum space representation. © 1993 John Wiley & Sons, Inc.     

Extended Hartree–Fock calculations for the helium ground state

The extended Hartree–Fock (EHF) wave function of an n-electron system is defined (Löwdin, Phys. Rev. 97 , 1509 (1955)) as the best Slater determinant built on one-electron spin orbitals having a complete flexibility and projected onto an appropriate symmetry subspace. The configuration interaction equivalent to such a wavefunction for the ¹S state of a two-electron atom is discussed. It is shown that there is in this case an infinite number of solutions to the variational problem with energies lower than that of the usual Hartree–Fock function, and with spin orbitals satisfying all the extremum conditions. Two procedures for obtaining EHF spin orbitals are presented. An application to the ground state of Helium within a basic set made up of 4(s), 3(p₀), 2(d₀) and 1 (f₀) Slater orbitals has produced 90% of the correlation energy.     

Electronic structure calculations of 1,3-dipolar cycloadditions using density functional and Hartree–Fock methods

Local spin density (LSD ) methods were used to study the concerted 1,3-dipolar cycloadditions for fulminic acid plus acetylene, fulminic acid plus ethylene, and nitrone plus ethylene. Cartesian Gaussian double-zeta split-valence basis sets augmented with one set of polarization functions (DZVP ) were used for the LSD calculations. The LSD calculations were performed with the LSD exchange functional (Dirac) and with the Vosko, Wilk, and Nusair correlation energy functional (VWN ). Nonlocal spin-density corrections (NLSD ) were estimated with the exchange functional of Becke and the correlation energy functional of Perdew (VWN + BP ) and Becke, and the correlation energy functional of Lee, Yang, and Parr (B-LYP ). Vibrational frequencies were computed at the VWN and B-LYP levels by numerical differentiation of the analytical first derivatives of the energy. Each of these reactions was examined using Hartree–Fock and Møller–Plesset perturbation theory for comparison. Geometry optimizations were carried out at the Hartree–Fock level with the 6-311G (d,p) basis set, and correlation energies were computed up to the MP 4SDTQ /6-311G (d,p) level of theory. For the reactions of fulminic acid plus acetylene, fulminic acid plus ethylene, and nitrone plus ethylene, our best estimated density functional barrier heights are 7.8 ± 1.5, 8.9 ± 0.3, and 11.05 ± 1.9 kcal/mol, respectively. These results are in reasonable agreement with the correlated wave-function calculations and experimental estimates. © 1994 John Wiley & Sons, Inc.     

Analysis of the half-projected Hartree–Fock function: Density matrix,natural orbitals,and configuration interaction equivalence

The half-projected Hartree–Fock function for singlet states (HPHF ) is analysed in terms of natural electronic configurations. For this purpose the HPHF spinless density matrix and its natural orbitals are first deduced. It is found that the HPHF function does not contain any contribution from odd-times excited configurations. It is seen in addition, in the case of the singlet ground states, this function is approximately equivalent to two closed-shell configurations, although the nature of the excited one depends on the nuclear geometry. An example is given in the case of the LiH ground state. Finally, the application of this model for studying systems of more that two atoms is criticized.     

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.     

Parallel pseudospectral electronic structure: I. Hartree–Fock calculations

We present an outline of the parallel implementation of our pseudospectral electronic structure program, Jaguar, including the algorithm and timings for the Hartree–Fock and analytic gradient portions of the program. We also present the parallel algorithm and timings for our Lanczos eigenvector refinement code and demonstrate that its performance is superior to the ScaLAPACK diagonalization routines. The overall efficiency of our code increases as the size of the calculation is increased, demonstrating actual as well as theoretical scalability. For our largest test system, alanine pentapeptide [818 basis functions in the cc-pVTZ(-f) basis set], our Fock matrix assembly procedure has an efficiency of nearly 90% on a 16-processor SP2 partition. The SCF portion for this case (including eigenvector refinement) has an overall efficiency of 87% on a partition of 8 processors and 74% on a partition of 16 processors. Finally, our parallel gradient calculations have a parallel efficiency of 84% on 8 processors for porphine (430 basis functions). © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1017–1029, 1998     

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.     

Use of VN−1 potentials in Green's function calculations

An analysis of the leading effects of a propagator approach to the calculation of resonance energies is presented. By comparison with ΔSCF results, a simple approximation formula has been developed which contains the main features of the V^N?1 potential option and the diagonal Tamm–Dancoff approximation. This result is expected to be a useful guide for further refinements of the many-body aspects of the theory.     

New derivation of the Waller–Hartree–Fock spatial wave function

A new derivation is given for the Waller–Hartree–Fock double-determinantal spatial wave function. One starts from the single-determinant wave function in which a orbitals are doubly occupied, and decomposes it into a sum of products of spatial and spin functions. The spatial product of the first genealogical spin eigenfunction is a double-determinantal function. The derivation is based on the simple form of U_1?(P) when the representation matrix is obtained from the genealogical spin eigenfunction.     

New method for approximate Hartree–Fock calculations using density approximations and coulomb field corrections. I

A method is proposed that reduces the computational effort of HF calculations considerably by reducing the number of two-electron integrals that have to be calculated. The following concepts are used: (i) approximation of the electron density by only few functions for the Coulomb part of the HF matrix; (ii) modification of this approximate density, to improve its Coulomb field; (iii) in the exchange part, a basis function χ is replaced by a function \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \chi $\end{document} consisting of fewer Gaussian lobes; (iv) the error caused by this replacement is reduced by a modification of the densities \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \chi _i \tilde \chi _j $\end{document} in the exchange integrals. The computation time of the integral part is reduced by a factor 6 for molecules containing five first-row atoms as, e.g., CF₄, if one uses a 7S/3P basis set contracted to (5, 1, 1/3). The integral time increases roughly with n³, if n is the number of Gaussian lobes.     

Uncoupled Hartree–Fock calculations of the polarizability and hyperpolarizabilities of nitrophenols

The polarizability and hyperpolarizabilities of nitrophenols as model compounds for studying nonlinear optics have been investigated at the Hartree-Fock level of approximation by means of the Dalgarno Uncoupled Hartree-Fock (DUHF) or Sum Over Orbitals (SOO) method. The additive character and the charge transfer effects in α, β, and γ have been analyzed in terms of the σ and π molecular orbital contributions, the contribution of the individual π molecular orbitals, and the contribution of the highest occupied and the lowest unoccupied molecular orbitals. Within the SOO approach, the reliability of the Two-Level Model has been tested and the influence of the rotation of the nitro group and of the presence of the intramolecular hydrogen bonding in ortho-nitrophenol have been studied. The results show that the present method is a reliable and efficient tool for the prediction of trends in the molecular polarizability and hyperpolarizabilities of large molecules. © 1995 by John Wiley & Sons, Inc.