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.     

Conformational analysis of melatonin at Hartree–Fock ab initio level

Melatonin is a hormone, which is synthesized and secreted by the pineal gland during darkness and has been implicated in the regulation of several neural and endocrine processes. Conformational analysis of melatonin was performed at Hartree–Fock ab initio level. With the help of STO-3G and 6-31G* basis sets, 192 and 128 conformers were obtained, respectively. In both cases, there were eight conformers with nearly planar 3-amido-side chain and the most stable conformation found had folded structure in accord with previously published results. The present paper compares the Hartree–Fock conformational space of melatonin with the results of active conformation models and conformational analyses published in the literature.     

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     

Splicing I: Using mixed basis sets in ab initio calculations

The effect of mixing (or “splicing”) extended and minimal basis sets on molecular properties such as geometries, Mulliken charges, dipoles, and internal rotation barriers was studied for several test molecules. The effect is gauged by comparison with full extended basis set calculations. It is found that splicing improves most properties relative to full minimal basis set calculations, and little accuracy is lost if the splicing is done in a judicious manner.     

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.     

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.     

Exact-exchange Hartree–Fock calculations for periodic systems. I. Illustration of the method

A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO ) self-consistent-field (SCF ) ab initio Hartree–Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron–electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k , as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.     

On the possibility of improving the integral evaluation section in ab initio Hartree–Fock calculations on large molecules

A modified integral package for evaluation of two-electron integrals over Gaussian basis functions is described. Modifications are implemented in the MOLECULE program system and are especially suited for the study of large molecules and molecular complexes.     

Molecular structure and vibrational spectra of indole and 5-aminoindole by density functional theory and ab initio Hartree–Fock calculations

The molecular geometry and vibrational frequencies of indole and 5-aminoindole in the ground state have been calculated by using the Hartree–Fock and density functional method with 6-311++G(d,p) basis set. The optimized geometrical parameters obtained by DFT calculations are in good agreement with the experimental values. Comparison of the observed fundamental vibrational frequencies of indole and 5-aminoindole with the calculated results by density functional and Hartree–Fock methods indicates that B3LYP is superior to the scaled Hartree–Fock approach for molecular vibrational problems. The theoretical spectrograms for FT-IR spectrum of 5-aminoindole have been constructed.     

Use of the mixed basis method for ab initio SCF MO calculations

A previously described mixed basis method for performing SCF MO calculations has been applied to the benzene molecule and to the allyl cation. A basis set of Slater type orbitals is used to calculate the one-electron and one-centre two-electron integrals whilst the many-centre two-electron integrals are approximated by small gaussian type orbital expansions of the STO basis. Comparison of the results with all-gaussian basis sets and literature values indicates that the mixed basis 2-GTO approximation is inadequate for molecules of this size because of the consistent underestimation of the electron repulsion integrals. The use of gaussian exponents chosen by a least-squares procedure rather than variationally gives better mixed basis results, but the indications are that a 4-GTO expansion is necessary for reliable mixed basis calculations. A method for more accurate integral evaluation by gaussian expansions of orbital products is suggested.

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Zusammenfassung Eine früher beschriebene Methode mit gemischter Basis zur Durchführung von SCF-MO-Berechnungen wurde auf das Benzolmolekül sowie das Allylkation angewendet. Ein Basissatz aus Slaterorbitalen wird verwendet, um die Einelektronen- und die Einzentrenzweielektronenintegrale zu berechnen, während die Mehrzentrenzweielektronenintegrale mit Hilfe von Entwicklungen der Slaterorbitale nach wenigen Gaußfunktionen angenähert werden. Ein Vergleich mit Ergebnissen von Rechnungen mit Gesamtbasissätzen von Gaußfunktionen und mit Literaturwerten zeigt, daß der gemischte Basissatz 2-GTO ungeeignet für Moleküle von dieser Größe ist, da die Elektronenabstoßungsintegrale durchgehend zu klein berechnet werden. Die Verwendung von Exponenten der Gauß-funktion, die mit Hilfe einer Methode der kleinsten Quadrate gewonnen wurde, ergibt bei der gemischten Basis bessere Ergebnisse als diejenigen, die nach der Variationsmethode gewonnen wurden; es zeigt sich jedoch, daß 4-GTO-Entwicklungen für angemessene Berechnungen mit gemischter Basis notwendig werden. Eine Methode für eine genauere Integralberechnung mit Hilfe der Entwicklung von Orbitalprodukten nach Gaußfunktionen wird vorgeschlagen.

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Résumé Application à la molécule de benzène d'une méthode de base mixte précédemment décrite pour effectuer les calculs SCF MO. Une base d'orbitales de Slater est utilisée pour calculer les intégrales monoélectroniques et les intégrales biélectroniques monocentriques; les intégrales biélectroniques polycentriques sont approchées au moyen d'une expression des orbitales de Slater en orbitales gaussiennes. La comparaison des résultats avec ceux obtenus en bases gaussiennes et avec ceux donnés dans la littérature indique que l'approximation 2-GTO n'est pas adaptée aux molécules de cette taille car elle provoque une sous-estimation constante des intégrales de répulsion électronique. L'emploi d'exposants gaussiens choisis par un procédé de moindres carrés plutôt que par une méthode variationnelle donne de meilleurs résultats en bases mixtes mais il s'avère nécessaire d'utiliser un développement du type 4-GTO. On propose une méthode pour l'évaluation plus précise des intégrales par développement gaussien des produits d'orbitales.

     

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.     

Molecular structure, IR spectra of 2-mercaptobenzothiazole and 2-mercaptobenzoxazole by density functional theory and ab initio Hartree–Fock calculations

Quantum chemistry calculations have been performed using Gaussian03 program to compute optimized geometry, harmonic vibrational frequency along with intensities in IR and Raman spectra and atomic charges at RHF/6-31+G*, B3LYP/6-31+G* and B3LYP/6-31++G* levels for 2-mercaptobenzothiazole (MBT, C₇H₅NS₂) and 2-mercaptobenzoxazole (MBO, C₇H₅NOS) in the ground state. The scaled harmonic vibrational frequencies have been compared with experimental FT-IR and FT-Raman spectra. The results show that the scaled theoretical vibrational frequencies is very good agreement with the experimental values. A detailed interpretation of the infrared and Raman spectra of 2-mercaptobenzothiazole and 2-mercaptobenzoxazole was reported. Comparison of calculated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibrational modes.     

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.     

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.     

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.     

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.