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.     

Convergence problems in the solution of SCF equations

Some problems connected with the convergence of iterative solutions of the Hartree-Fock equations for the open shell systems are discussed. The nonuniqueness of the Hartree-Fock operator form is used. The method of iterating by the operator's power is developed for obtaining solutions of the Hartree-Fock equations. Some particular results for the molecules Li, Li are presented.     

Accelerated convergence in SCF calculations and the level shifting technique

The possibility of improving the convergence rate in SCF calculations by exploiting the essential arbitrariness of the diagonal matrix elements of the Fock operator formed in the MO basis is studied. It is shown that it is possible to accelerate convergence in many cases by adopting a different form of level shifting technique (called the reverse level shifting technique by us).     

Convergence of LCAO density matrices in Hartree—Fock calculations on extended model chains

Some mathematical implications of the extended nature of model chains are reviewed to call attention to their importance in electronic structure calculations at the restricted Hartree-Fock Roothaan level.     

A hybrid technique of orthonormality constrained orbital optimization in SCF calculations

A recently proposed orthonormality constrained orbital optimization technique is operationally modified further by coupling it to a gradient biased method, namely the steepest descent procedure of McWeeny. The hybrid technique developed in this way is shown to have better convergence properties in closed and unrestricted open-shell calculations. The technique can be adapted to MCSCF procedures as well. The important role played by "orbital symmetries" in the operation of the method is analysed. Similarities and differences of the present method with the orthogonal gradient method are pointed out. Possible avenues of circumventing convergence difficulty that one may encounter in pathological cases, particularly in ab initio calculations involving extended basis set, are suggested.     

A new SCF procedure and its applications to ab initio calculations of the states of the fluorosulphate radical

Ab initio calculations using a small Gaussian basis set, including 3d orbitals on the sulphur atom, have been performed on the fluorosulphate radical and the related ions SO₃F⁺ and SO₃F^?. A new SCF procedure is described and applied to the open shell cases discussed here. The results are compared with recent CNDO calculations and with the experimental transition energies of the radical.     

Test applications of a new SCF method for excited states

In order to test a recently proposed technique for deriving orthogonality-constrained HF wave functions for excited states, several applications to molecular systems, have been made and the results compared with those provided by other SCF techniques.     

Large scale Hartree-Fock calculations with conventional SCF algorithm. Influence of integral and index compression on Fock matrix construction

It is shown that a compression of two-electron integrals and their indices significantly improves efficiency of the conventional self-consistent field (SCF) algorithm for a solution of the Hartree-Fock equation by decrease the Fock matrix calculation time. The improvement is reached not only due to a reduction of the integral file size, but mainly because data compression reduces or even can eliminate a cache conflict in data transfer from the hard drive to the main computer memory. Thus, the conventional SCF algorithm with the data compression becomes very efficient and permits to carry out large-scale Hartree-Fock calculations. The largest Hartree-Fock calculations have been performed for RNA 433D structure from the PDB data bank with 6080 basis functions formed from 6-31G basis on a workstation with 1 GHz Alpha processor.     

Vector coupling coefficients for calculations of transition-metal atoms and ions by the SCF coupling operator method

We derived the necessary conditions to which the vector coupling coefficients (VCC ) a and b describing atomic L,S-multiplets of the configurations d^N (1 ≤ N ≤ 9), should satisfy. Special attention is paid to the states of non-Roothaan type for which VCC depend on the choice of degenerate d-orbitals basis set determined within the accuracy up to an orthogonal transformation u. It is shown that for such states the direct sum of matrices ‖a‖ and ‖b‖ must be the non-symmetric matrix. Obtained VCC were used for the ab initio calculations (basis set (14s9p5d)/[8s4p2d] from [15]) on first-row transition atoms (from Sc to Cu) to compare to similar calculations [16], in which the Peterson's VCC have been used, and with calculations [15] carried out by the atomic SCF program [4] as well.     

Unconditional convergence in SCF theory: a general level shift technique

A level shift procedure is described in the framework of one-operator SCF formalisms. This procedure permits one to obtain in any case unconditional convergence to a stationary energy.     

Further applications of a new biosensor method for dating cellulosic finds

Using a cellulosic material dating method recently proposed by three of the authors of the present article further applications to real samples are discussed. In the first instance, to wood samples, that is, to a type of sample for which the method was specifically developed but with the samples differing widely in age, and then to textile or paper samples. Of course in the latter two cases the results obtained are still quite preliminary, above all because of the difficulty of procuring certainly dated samples of this type.     

Near-minimum basis set SCF calculations on HCl as a source of transferable parameters

Ab initio LCAO-MO-SCF calculations on HCl yield an optimum exponent of 1.93 for 3d Slater-type orbitals with a single radial function when best atom exponents are used for the minimum basis set orbitals of chlorine and 1.2 is used for the 1s orbital of hydrogen. Hamiltonian matrix elements and other theoretical parameters which are used in approximate MO calculations on large molecules are tabulated. One-electron properties and electron density contour maps are compared to those computed from other wave functions. The accumulation of electrons in the A-H bonds, as well as the occupation of the 3d orbitals, decreases through the isoelectronic series SiH₄, PH₃, H₂S, HCl.

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Zusammenfassung Ab initio LCAO-MO-SCF-Rechnungen für HCl ergeben einen optimalen Exponenten von 1,93 für 3d-Slaterorbitale mit einer einzigen Radialfunktion, wenn gleichzeitig beste Atomexponenten für den minimalen Basissatz am Chloratom und 1,2 als Exponent für die 1s-Orbitale am Wasserstoffatom angenommen werden. Die Hamilton-Matrixelemente und andere theoretische Parameter, die bei Näherungs-MO-Rechnungen an großen Molekülen gebraucht werden, sind tabelliert. Ein-Elektron-Eigenschaften und Elektronendichte-Diagramme werden mit solchen aus anderen Rechnungen verglichen. Die Konzentration von Elektronen in den A-H-Bindungen sowie die Besetzung der 3d-Orbitale nimmt in der isoelektrischen Reihe SiH₄, PH₃, H₂S, HCl ab.

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Résumé Calculs ab-initio LCAO-MO-SCF sur HCl donnant un exposant optimal de 1,93 pour les orbitales 3d de Slater a fonction radiale unique lorsque l'on utilise les meilleurs exposants pour la base minimale du chlore et 1,2 pour l'orbitale 1s de l'hydrogène. Les éléments de matrice hamiltoniens et les autres paramètres théoriques utilisés dans les calculs approchés sur les grandes molécules ont été tabulés. Les propriétés monoélectroniques et les cartes de densité électronique sont comparés à ceux obtenus à l'aide d'autres fonctions d'onde. L'accumulation des électrons dans les liaisons A-H, et l'occupation des orbitales 3d, décroît le long de la série isoélectronique SiH₄, PH₃, H₂S et HCl.

     

Optimization of the basis functions in SCF MO calculations optimized one-center SCF MO basis set for HCL

Equations have been derived which give the first derivatives of the energy, calculated by the expansion SCF MO method, with respect to the parameters contained in the basis functions. The amount of computations needed has a reasonable limit and is particularly reduced if the parameters are the orbital exponents of STO basis functions. Knowledge of such derivatives has been exployted to optimize a one center basis set, comprised of 71 STO functions, for a SCF MO wave function of HCl. A molecular energy of –460.05810 au has been obtained at a bond length of 2.424 au. The resulting wave function, has been used to calculate several observables of interest.

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Zusammenfassung Ausdrücke für die ersten Ableitungen der SCF-Energie nach den in den Basis-Funktionen enthaltenen Parametern werden abgeleitet. Die zusätzlichen Rechnungen sind gering, wenn die Parameter die Orbitalexponenten von Slater-Funktionen sind. Die Kenntnis solcher Ableitungen wird dazu ausgenutzt, eine Einzentren-Basis von Slater-Funktionen für eine SCF-Wellenfunktion des HCl-Moleküls zu optimieren. Die elektronische Molekül-Energie ergab sich zu –460.05810 AE bei einem Kernabstand von 2.424 AE. Mehrere bedeutsame Erwartungswerte wurden berechnet.

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Résumé Des équations ont été établies donnant les dérivées premières de l'énergie par rapport aux paramètres des fonctions de la base utilisée dans la méthode SCF MO. La quantité de calculs nécessaire est raisonnablement bornée; elle est particulièrement réduite s'il s'agit des exposants orbitaux des fonctions de base de Slater. La connaissance de ces dérivées a été utilisée pour optimiser une base à un centre, comprenant 71 orbitales de Slater, pour la fonction d'onde SCF MO de HCl. L'énergie moléculaire obtenue est –460.05810 u.a. pour une longueur de liaison de 2.424 a.u. La fonction d'onde résultante a été utilisée pour le calcul de plusieurs observables intéressantes.

     

Symmetry breaking in the Hartree–Fock approximation for binuclear transition metal compounds—a theoretical investigation based on a variable model operator

The validity of the Hartree–Fock (HF ) approximation in bis(π-pentadienyl)dinickel ( 1 ) and in cyclopentadienyl-allyl-cyclobutadiene-dinickel ( 2 ) has been investigated by means of the Thouless instability conditions in the computational framework of a variable model Hamiltonian. Singlet, nonsinglet (triplet), and nonreal instabilities in 1 and 2 have been studied as a function of the one-electron resonance integral β and as a function of the one- and two-center elements of the electron–electron interaction. The one-center integrals of Coulomb (γ) and exchange-type (K) have been modified by a multiplicative factor; the two-center integrals (γ) have been calculated by means of an exponential interpolation formula with a variable decay amplitude. Additionally the Thouless conditions have been studied for nuclear deformations. The stability domain of the HF solution in the model space spanned by the variable INDO Hamiltonian has been analyzed. The nature of the many-body interactions in the unstable region depends strongly on the parametrization of the model operator. HF instabilities in the high-density region (long-range forces) of 1 have their origin in individual particle–hole fluctuations while negative roots for short-range forces (low-density region) are similar to collective excitations in many-body systems (strong off-diagonal coupling). The opposite behavior is encountered in the Ni complex 2 . The physical origin of these different types of correlation processes are analyzed in a simple two-electron two-orbital model. The nature of the HF fluctuations in 1 and 2 , the importance of spatial and spin correlation, and the coupling of symmetry breaking of the electronic wave function with nuclear deformations are compared with the nature of phase transitions in solid-state systems.     

The problems of reaction kinetics calculations based on thermoanalytical curves

The work, results and estimations are summarized which are related to reaction kinetic calculations using thermoanalytical curves. Mathematical operations applied to the calculations are studied. Calculation methods are evaluated on the basis of our test results. The effect of test conditions on reaction kinetic parameters is analysed. Calculation of half-period is also investigated in the study.     

Test of an integral approximation scheme based on semiorthogonalized orbitals in LCAO SCF MO CI calculations of naphthalene

Ab initio LCAO SCF MO CI calculations of naphthalene are carried out with a minimal basis set to test an integral approximation scheme proposed in a previous paper. When 71.3 and 53.0% of the two-electron integrals are neglected, the errors in the SCF total energy are only ?0.0534 and ?0.0006 a.u., respectively. In the latter case, the maximum absolute errors of the orbital energy and the gross AO population are 0.007 a.u. and 0.001, respectively. Even in the former case the errors of the excitation energies are less than 0.0004 a.u. Errors of oscillator strengths are also examined and are found to be tolerably small.     

SCF MO CI calculations on the electronic spectra of fluorenone and related compounds

The electronic spectra of fluorenone, 9-iminofluorene and 9-ethylidenefluorene are studied by means of the SCF MO CI calculation. It is shown that the lowest singlet-singlet transition of fluorenone located at 380 m. can be assigned to a - ^* transition (¹ B ₂¹ A ₁). The nature and location of the lowest triplet state are also discussed.

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Zusammenfassung Die Elektronenspektren von Fluorenon, 9-Iminofluoren und 9-Äthylidenfluoren werden mit Hilfe der SCF MO CI-Methode studiert. Es wird gezeigt, daß der niedrigste Sigulett-Singulett-Übergang in Fluorenon bei 380 m einem ^*-Übergang (¹ B ₂¹ A ₁) zugeordnet werden kann. Die Natur und Lage des niedrigsten Triplett-Übergangs werden ebenfalls studiert.

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Résumé Les spectres électroniques de la fluorenone, du 9 iminofluorène et du 9-éthylidène fluorène sont étudiés à l'aide de calculs SCF-MO-CI. On montre que la plus basse transition singuletsingulet de la fluorénone, située à 380 m, peut être attribuée à une transition ^* (¹ B ₂¹ A ₁ La nature et la position du plus bas état triplet sont aussi étudiées.

     

SCF CI MO calculations on the base–base interactions in dinucleoside monophosphates

In order to elucidate further details of RNA conformations we have studied base stacking in dinucleoside monophosphates (DNP's) using UV difference spectra and the hypochromic effect. SCF CI MO calculations according to PPP and MIM approximations were carried out for six pyrimidine-containing DNP's in each of several stacking geometries. The calculated and plotted difference spectra were fitted to the experimental spectra. Different DNP's showed distinct geometries in the range of the helix angle of 35°. From our results we conclude that there is a microstructure in the helix. This would imply an additional content of information in this macromolecule, a higher precision in nucleic acid interactions, and make possible the prediction of the conformation of polynucleotides.     

Further calculations on the internal rotation spectrum of phenol

The influence of a small deformation of C?O angle in phenol (tilt), into the rotational far-infrared (FIR ) spectrum is analyzed using several approaches. In all of them, the CNDO /2 method is used to determine the potential energy functions. In a first step, the C? O bond and the rotation axis are both supposed to coincide with the C₂ symmetry axis of the phenyl group. With this assumption the torsional frequencies are calculated in both the symmetric and asymmetric rotor approximations. In a second step, the tilt of the C? OH bond is determined theoretically and found to be ?3°, measured from the C₂ symmetry axis, the C? OH bond crossing this axis, Using this second geometry, and taking as the rotation axis the C₂ axis, the torsional frequencies are again determined in both approximations. An improvement of the calculated transition energies is encountered at each stage of the calculation, when compared with experimental data. Finally the importance of the introduction of a tilt into the FIR torsional frequency calculations is discussed.