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.     

A comparison of different methods in determining energies and geometries of open-shell systems

Three methods of obtaining eigenvectors for open-shell systems, namely the Roothaan restricted open-shell method, the “half-electron” method, and the use of ground state orbitals, are compared with each other both on a formal basis, and by contrasting theab initio energies predicted by these schemes for a series of radicals and triplets. The r.m.s. improvement in the energy of eight radicals by use of the Roothaan method rather than the half-electron approximation amounts to 4.8 kcal mole^?1, whereas the r.m.s. improvement for nineteen triplet states is 8.2 kcal mole^?1. Optimum geometries predicted by these two methods do not differ appreciably. The use of closed-shell eigenvectors rather than those of the half-electron method leads to very large errors for triplet states whose electron density distributions differ appreciably from those of the ground state.     

The determination of SCF LCAO solutions for open shell configurations

A series of calculations has been carried out using the McWeeny and Roothaan Open Shell methods for determining LCAO-SCF solutions. The work reported here suggests that in order to obtain convergence to the absolute minimum with the McWeeny method it is often necessary to provide a very good initial approximation and that the rate of convergence could be improved by replacing the Steepest Descent by a more powerful minimization technique. Further the Roothaan method does not converge in all cases considered and it appears that Extrapolation techniques accelerate convergence but do not induce convergence in cases displaying oscillatory behaviour.

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Zusammenfassung LCAO-SCF-Rechnungen an offenen Schalen wurden nach den Methoden von Roothaan und McWeeny durchgeführt. Um bei dem McWeenyschen Verfahren Konvergenz zum absoluten Minimum zu erhalten, ist oft eine schon sehr gute Anfangsnäherung nötig. Die Methode des steilsten Abstiegs ist keineswegs optimal. Extrapolationsmethoden können wohl eine Konvergenz beschleunigen, nicht aber das beim Roothaan-Verfahren manchmal auftretende Oszillieren unterdrücken.

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Résumé On a exécuté des calculs LCAO SCF en couches ouvertes avec les méthodes de Roothaan et de McWeeny. Dans la méthode de McWeeny il est souvent nécessaire d'avoir une bonne approximation initiale, pour obtenir une convergence vers le minimum absolu. La méthode de descente le plus escarpé n'est pas optimale. Les méthodes d'extrapolation peuvent accélérer la convergence en effet, mais ils ne suppriment pas l'oscillation, qui arrive quelquefois dans la méthode de Roothaan.

     

Validity of <Emphasis Type="Italic">B</Emphasis>-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations

The validity of B-splines as a universal basis set for atomic Hartree–Fock–Roothaan calculations is studied. In order to accomplish our aim, the ground-state energies of neutral atoms He–Xe, cations Li ⁺–Xe ⁺, and anions H ^-–I ^- with the nuclear charge Z=54 are calculated by the Hartree–Fock–Roothaan method with the B-spline sets. All radial functions of the atoms and singly charged ions are expanded by common B-spline sets regardless of atomic systems and symmetries of atomic orbitals. The energies obtained by the best B-spline set are in excellent agreement with ten-digit numerical Hartree–Fock results.     

Roothaan's open shell theory from the viewpoint of an orthogonal group

Explicit equations for the constants a and b of the energy as the average expectation value for the minimal domain of states giving the Roothaan energy functional are derived in terms of the total spin value and the “seniority” quantum number. The state with off-diagonal long-range order can appear in the framework of the Roothaan scheme for the quasidegenerate system as an alternative to the Hund rule. Admissible many-electron states are established for all configurations of the icosahedral symmetry group systems, and corresponding a and b coefficients are calculated.     

Coupled hartree–fock calculation of static and shielding factors for open shell atomic systems

In this paper a calculation of the static quadrupole polarizabilities and the shielding factors of the atoms and ions of the 2p open shell systems is presented. A fully coupled Hartree–Fock method using the Roothaan formalism is adopted. Calculations have been done for the ground as well as for some valence excited states. An antishielding effect is observed for all the atoms and ions under study.     

Combined Open Shell Hartree–Fock Theory of Atomic–Molecular and Nuclear Systems

In this study, the combined Hartree–Fock (HF) and Hartree–Fock–Roothaan equations are derived for multideterminantal single configuration states with any number of open shells of atoms, molecules and nuclei. It is shown that the postulated orbital-dependent energy and Fock operators are invariant to the unitary transformation of orbitals. This new methodology is based entirely on the spin-restricted HF theory. As an application of combined open shell theory of atomic–molecular and nuclear systems presented in this paper, we have solved Hartree–Fock–Roothaan equations for the ground state of electronic configuration C(1s ²2s ²2p ²) using Slater type orbitals as a basis.     

Roothaan INDO CI spin density calculation

Spin densities have been calculated by the Roothaan INDO CI method in the parametrization of Pople and Kaufman. The dependence of the spin densities on both the type and the number of configurations in CI has been followed. Results have been compared with the UHF and spin-projected UHF methods.     

Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

An alternative computational strategy for direct space hartree–fock calculations on extended model chains

Based on previous analyses of slowly convergent exchange lattice sums entering the configuration space restricted Hartree–Fock–Roothaan scheme for chain systems, an alternative computational strategy is developed. Within the present formalism, the traditionally used finite Fourier transform of k-dependent LCAO density matrices are by-passed and an advantageous computational organization is obtained.     

Evaluation of Two- and Three-center Overlap Integrals Over Complete Orthonormal Sets of Ψ<Superscript><Emphasis Type="Italic">α</Emphasis></Superscript>-ETOs using their Addition Theorems

With the help of addition theorems for complete orthonormal sets of Ψ^α-ETOs in momentum space (α = 1,0,−1,−2,...) introduced by the author, the general expansion formulas are established for the two- and three-center overlap integrals occurring when Hartree-Fock–Roothaan and explicitly correlated methods, respectively, are employed. The relationships obtained are valid for the arbitrary quantum numbers, screening constants and location of Ψ^α-ETOs.     

Computation of ionization and various excited state energies ofhelium and helium‐like quantum dots

In the effective mass approximation, we calculated the wave functions and some energy states of helium and helium‐like quantum dots (QDs) with impurity charges Z = 0, 1, 2, 3, and 4. In addition, we carried out the ionization energies of these QDs as a function of dot radius, and we investigated the influence of impurity on the ionization energy. We utilized the method that is a combination of quantum genetic algorithm (QGA) and Hartree‐Fock Roothaan (HFR). The results are in a good agreement with literature results. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Quadratically convergent iteration procedure for geminal determinations

A general formalism is presented for the determination of the optimum natural orbitals within the various strongly orthogonal geminals of a wavefunction describing a 2N-electron closed-shell molecule or atom. The relationship to Hartree–Fock–Roothaan theory is established; the algorithm that is developed is quadratically convergent to the desired result, and does not ignore off diagonal Lagrangian multipliers, or require an infinite series of 2 × 2 orthogonal transformations of the original basis.     

Quasi-relativistic approximation in the SCF-MO-LCAO method

A quasi-relativistic approach to the MO-LCAO method is formulated taking into account the relativistic effects with an accuracy up to (v/c)² terms, the relativistic part of the electronic interaction in the Hamiltonian being neglected. In the framework of this approximation a set of SCF equations of the Roothaan form is derived; here only the relativistic analogue to the closed shell systems with one-determinant wave functions is considered. In so doing three types of relativistic corrections arise which are quite similar to those of the Pauli equation for one-electron atoms. The new matrix elements appearing due to these corrections can be reduced to some common integrals, which have to be calculated with relativistic radial atomic functions. The method allows a semi-empirical approach to the problem and does not require the Dirac four-component atomic functions (unknown in the most cases), thus making possible approximate quasi-relativistic electronic structure calculations of heavy-atom compounds.     

Rotational invariance of INDO theories including d-orbitals into the basis set

It is proved that in general the INDO approximation to the full Roothaan theory does not lead to expressions which are invariant under a rotation of local atomic axes. However, when only s- and p-functions are used in the atomic basis set, the equations obtained are invariant due to the special properties of the p-functions. When d-orbitals are included into the basis set, rotational invariance is lost but can be restored if a supplementary approximation is introduced.     

Coupled Hartree Fock calculation of static dipole polarizabilities and shielding factors of open shell systems

Static dipole polarizabilities and shielding factors for the 2p open shell atomic systems are presented using coupled Hartree Fock theory in the framework of the Roothaan formalism. Calculations have been performed for the ground as well as for some valence excited states. A variational approach has been adopted for the determination of the first-order perturbed functions. The results are compared with those obtained from the correlated calculations and other techniques. The shielding factor values are in excellent agreement with the theoretical N/Z ratio.     

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.     

The dynamic “level shift” method for improving the convergence of the SCF procedure

This article is concerned with the improvement of convergence of the iterative procedure for solving the Hartree–Fock–Roothaan equations by the dynamic “level shift” method.     

Frequency-dependent polarizability of open-shell atomic systems

The time-dependent variational principle is used to calculate the frequency-dependent dipole polarizabilities of 2p open-shell atomic systems. A general theory is built up to include frequency-dependent perturbations using the Roothaan–Hartree–Fock (HFR) formalism. The excitation energies corresponding to the low-lying excited states of the system are obtained from the poles of the dynamic polarizability values. A fairly accurate knowledge of transition oscillator strengths is also obtained from a knowledge of the polarizability values near the poles. The excitation energies and oscillator strengths are compared with available data.     

Application of combined Hartree–Fock–Roothaan theory to molecules with arbitrary number of closed and open shells

In this study, the applicability of the combined Hartree–Fock–Roothaan (CHFR) theory of atomic-molecular and nuclear systems (Guseinov, J Math Chem 42:177, 2007) to the molecules is demonstrated using minimal basis set of Slater type orbitals (STO). As an example of application of CHFR theory, the calculations have been performed for the ground state of electronic configuration of methylene molecule CH ₂ which has two open shells. The results of computer calculations for the orbital, kinetic and total energies, linear combination coefficients of symmetrized molecular orbitals and virial ratios are presented.