The MC SCF theory: Method of one-electron Hamiltonian

The one-electron Hamiltonian method is developed to solve the variational equations of the MC SCF theory. The many-parameter family of the one-electron Hamiltonian is derived and conditions for parameters to provide convergence of the SCF procedure to the energy minimum are obtained. A computation scheme based on the use of the one-electron Hamiltonian is described.     

Improved convergence of self-consistence procedures in the MC SCF theory

To obtain optimized orbitals within the MC SCF theory, the energy surface near a chosen point is approximated by a quadratic function of independent matrix elements of a small orthogonal orbital transformation. The method of a second-order one-electron Hamiltonian (OEH) is developed on the basis of this approximation. A procedure is proposed to define step coordinates, insuring a rapid descent along an average-energy surface also in the cases when the matrix of second energy derivatives has eigenvalues negative or close to zero. The results obtained in applying the OEH method for the calculation of ground and triplet states of uracile in the π-electron approximation are discussed. When a complete matrix of the second energy derivatives is used, the self-consistence procedure is quadratically convergent. An exponential, yet rapid enough convergence is provided by a simplified computation scheme neglecting cross derivatives.     

Scaled one-electron hamiltonian model for open-shell LCAO–MO–SCF calculations: Comparison with the restricted open-shell method of roothaan

The performance of a recently proposed scaled one-electron Hamiltonian (SOEH ) model is tested against parallel sets of restricted open-shell calculations by the method of Roothaan. It is found that the energy calculated by SOEH model, in general, lies slightly higher than the energy computed by the restricted open-shell method of Roothaan lending credibility to the application of variational argument to the scaled pseudoenergy functional (E^av) for deriving the SOEH model. The numerical stability of the converged SOEH energy with respect to changes in trial vectors indicates the reliability of the method. The SOEH model is shown to perform well in the calculation of geometries of radicals and ions. The convergence behavior of the SOEH model is compared with that of the restricted open-shell method of Roothaan.     

Coupled multiconfiguration self-consistent field (MC SCF) perturbation theory

The analytical form of the perturbation theory for the MC SCF method of Veillard and Clementi is presented. The appropriate second-order energy functional which takes into account the self-consistency requirements, leads to a set of coupled first-order perturbed equations determining the perturbed configuration coefficients and orbitals. The second-order energy formula derived from this functional can be given a clear physical interpretation. The present analytical approach is compared with the finite perturbation MC SCF scheme. The possibility of the approximate solution of the coupled MC SCF perturbation equations is also discussed and the so-called uncoupled procedures are devised. In the limit of the single determinant wave function the present formulae are shown to be equivalent to the appropriate Hartree-Fock perturbation results. The differences between the one-configuration SCF and the MC SCF approach are illustrated by the calculation of the electric dipole polarizability of. HZ in the CNDO/2 approximation. It is shown that the one-configuration SCF approaches cannot account for the correct asymptotic properties of the second-order energy for large internuclear distances. This feature of the SCF perturbation theories does not depend on the specific approximations of the CNDO/2 scheme and is corrected by using the MC SCF perturbation theory.     

Molecular structural formulas as one-electron density and hamiltonian operators: the VIF method extended

The valency interaction formula (VIF) method is given a broader and more general interpretation in which these simple molecular structural formulas implicitly include all overlaps between valence atomic orbitals even for interactions not drawn in the VIF picture. This applies for VIF pictures as one-electron Hamiltonian operators as well as VIF pictures as one-electron density operators that constitute a new implementation of the VIF method simpler in its application and more accurate in its results than previous approaches. A procedure for estimating elements of the effective charge density-bond order matrix, Pmunu, from electron configurations in atoms is presented, and it is shown how these lead to loop and line constants in the VIF picture. From these structural formulas, one finds the number of singly, doubly, and unoccupied molecular orbitals, as well as the number of molecular orbitals with energy lower, equal, and higher than -1/2Eh, the negative of the hydrogen atom's ionization energy. The VIF results for water are in qualitative agreement with MP2/6311++G3df3pd, MO energy levels where the simple VIF for water presented in the earlier literature does not agree with computed energy levels. The method presented here gives the simplest accurate VIF pictures for hydrocarbons. It is shown how VIF can be used to predict thermal barriers to chemical reactions. Insertion of singlet carbene into H2 is given as an example. VIF pictures as one-electron density operators describe the ground-state multiplicities of B2, N2, and O2 molecules and as one-electron Hamiltonian operators give the correct electronegativity trend across period two. Previous implementations of VIF do not indicate singly occupied molecular orbitals directly from the pictorial VIF rules for these examples. The direct comparison between structural formulas that represent electron density and those that represent energy is supported by comparison of a simple electronegativity scale, chiD=N/n2, with well-known electronegativity scales of Pauling, Mulliken, and Allen. This scale comes from the method used to calculate Pmumu for sp3 hybridized period-two elements and is comparable to electronegativity because it has the same form as <1/r> for hydrogenic orbitals. It therefore provides a physical basis for the representation of one electron density and Hamiltonian operators by the same VIF picture.     

An efficient second-order MC SCF method for long configuration expansions

A new second-order optimisation procedure for general MC SCF wavefunctions is described. The method shows greatly improved convergence as compared to previous methods. Using a determinant-based direct CI procedure which avoids the construction of a formula tape, very long complete active space (CAS SCF) wavefunctions can be handled. Energy averages of several states can also be optimised. Sample calculations for CH₂, FeO, and the vinoxy radical CH₂CHO with up to 178916 configurations are presented.     

Force in SCF theories. MC SCF and open-shell RHF theories

A theorem reported for Hartree-Fock SCF theory is shown to be valid for general MC SCF and open-shell RHF theories - a sufficient condition for these wavefunctions to satisfy the Hellmann-Feynman theorem is that the basis set includes the derivative AO ^?X_r/^?X_rfor any basis X_r. The new force approach is applicable to wider fields including electronic processes in chemical reactions. Test calculations are given for some simple systems.     

MC SCF optimization using the direct,restricted step,second-order norm-extended optimization method

The potential of a direct, restricted step, second order MC SCF computer program is illustrated with a large configuration space calculation containing 24156 configuration state functions on the ³B₁ state of methylene.     

Multiple stationary point representations in MC SCF calculations

By using a complete second-order Newton-Raphson multiconfigurational self-consistent field (MC SCF) procedure combined with the Fletcher restricted step constraint algorithm and a modification of the surface walking procedure of Simons et al., an MC SCF energy hypersurface at fixed geometry has been examined in considerably more detail than had been done previously. By calculational example, it is shown that there may exist several MC SCF stationary points which fulfill all four structural criteria we require of a state for being a “good” representation of an exact state. The problem with the existence of several stationary point solutions may be reduced if care is taken in the selection of the MC SCF configuration space. Calculational examples also demonstrate that near-lying stationary points exist which fulfill some, but not all, of these four structural criteria. Hence, stationary points should be obtained with a global MC SCF method which automatically eliminates convergence to as many as possible of these unwanted stationary points. Upon convergence, structural criteria which are not automatically fulfilled should be examined in detail.     

Molecular MC SCF calculations by direct minimization

Calculations on the H₂CO ground state and excited states at the equilibrium planar geometry, using the single excitation MC SCF method, are described for two basis sets. The results obtained by a full calculation including all the electrons are compared with those obtained by a pseudopotential version of the method including only the valence electrons. The results agree quite well both with each other and with the observed values.     

Eigenspace manipulation in SCF theory

The procedure proposed by Silverstone and others to modify virtual levels in SCF theory is extended as a technique to be used in any type of generalized secular equations. Some useful results are obtained.     

Interaction energy calculation scheme employing one-electron hamiltonian approximation for the evaluation of short-range interactions

A semiempirical scheme for the calculation of intermolecular energy is presented. A distinctive feature of the scheme is the employment of the one-electron Hamiltonian approximation in EHT parametrization for the calculation of exchange repulsion and charge transfer energies. Electrostatic, induction and dispersion components are calculated according to known approximate formulas containing point multipole moments and bond polarizabilities. The proposed scheme is applied to the calculation of binding energies and equilibrium geometries of various molecular dimers.     

Self-consistent-field theory for one-electron properties

The SCF-formalism is modified to include as constraints in the variational procedure physical quantities defined by one-electron operators. Introducing empirical values for the expectation values of these operators, pseudo-eigenvalue equations are obtained whose solution will lead to self-consistent orbitals which are constrained to achieve the chosen empirical values.

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Zusammenfassung Die SCF Methode ist abgewandelt, um als Nebenbedingung in der Variationsmethode physikalische Eigenschaften einzuschließen, die durch Ein-Elektron-Operatoren definiert sind. Durch Einführung von empirischen Werten für die Erwartungswerte dieser Operatoren erhält man Pseudo-Eigenwert-Gleichungen, deren Lösung zu SCF-Orbitalen führt, die die gewählten empirischen Werte geben.

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Résumé On a modifié le formalisme de champ auto-cohérent à fin de pouvoir introduire les valeurs de propriétés physiques comme conditions sécondaires dans la méthode de variation. La solution des équations à valeurs propres, qu'on y obtient, permet d'obtenir des orbitales qui donnent les valeurs empiriques choisies.

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This work has been supported in part by the National Research Council of Canada.     

Equivalence-restricted open-shell SCF theory

A method is presented for generating open-shell equivalence-restricted SCF orbitals in high-symmetry situations using Roothaan–Hartree–Fock programs which are adapted for lower symmetry.     

A comparison of one-electron properties calculated from Gaussian SCF and CI wavefunctions

Ground state LCAO-MO-SCF single- and multiconfiguration wavefunctions of NH₃ and CH ₃ ^– , constructed from a large contracted Gaussian basis set, have been analyzed in terms of one-electron expectation values.Both sets of values agree fairly well with the experimental data in the case of NH₃. It is expected therefore that the results obtained for CH ₃ ^– are also reliable.

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Zusammenfassung Einfach- und Mehrfachdeterminanten-LCAO-MO-SCF-Funktionen für NH₃ und CH ₃ ^– wurden zur Berechnung von Einelektroneneigenschaften herangezogen. Die Übereinstimmung ist in beiden Fällen bei NH₃ gut, man darf daher für CH ₃ ^– dasselbe erwarten.