The spin-orbit interaction in the configuration nlN n'l using nonorthogonal radial orbitals

Formulas for the calculation of matrix elements of the simplified operator of the spin-orbit interaction for the configuration n₁l^Nn₂l are deduced for the case when radial orbitals are nonorthogonal. The coefficients of the radial integrals are tabulated for the configuration p^Np. The fine structure of the excited configuration of the carbon atom is calculated.     

The general scheme of using non-orthogonal radial orbitals in a complex electronic configuration of the atom

A theory for using non-orthogonal radial orbitals between shells with identical orbital quantum numbers, in the case of complex configurations, is presented. Construction of the antisymmetric wave function of the whole configuration, with the help of antisymmetrical wave functions of individual shells, is described. General methods of calculating matrix elements of one- and two-electron operators are given.     

On the limiting radial distribution function for hydrogenic orbitals

An exact reduced limiting expression for the generalized radial distribution functionD _n (r) is derived and compared with quantum distributions for various degrees of excitation. It represents the quantum result at large quantum numbers significantly better than a prior empirical representation of the universal reduced distribution and gives a somewhat larger electronic partition function for the hydrogen atom than that based on the previous distribution.     

Valence type vacant orbitals for configuration interaction calculations

A method is proposed to determine the valence type vacant orbitals, which are suitable for CI calculations and for the initial guess orbitals in MC SCF calculations. The method was applied to calculate the ionization energies of series of molecules and to draw the potential energy curves of various states of N₂ and N⁺₂.     

Iterative natural orbitals for configuration interaction using perturbation theory

Approximate natural orbitals are determined iteratively from CI expansions constructed using first-order perturbation theory in order to investigate the possibility of eliminating the complete transformation of MO integrals on each iteration. Results on LiH and H₂O are compared with fully variationally determined NO's to assess questions of convergence.     

Virtual orbitals for obtaining rapid convergence in configuration interaction calculations

A perturbational method is used to generate virtual orbitals intended to yield rapid convergence in configuration interaction calculations. It is shown that the problem of finding the orbitals and pair functions which maximize the interaction with the Hartree-Fock wavefunction can be solved exactly. The resulting first-order semi-internal virtual orbitals are determined by a simple closed expression. All further semi-internal excitations vanish to first order. As an example, the method of this work is applied to the semi-internal orbitals of the ground state of the carbon +1 ion. The resulting first order orbitals compare very well to corresponding variational virtual orbitals.     

A radial probability density function for analysis of canonical molecular orbitals

A one‐dimensional probability density function, analogous to the atomic radial density for the hydrogen atom, r²R_nl(r), is defined for an arbitrary three‐dimensional density. It is obtained numerically by taking the derivative of a cumulative probability distribution with respect to the cubic root of the volume enclosed by each in a series of isosurfaces. Each point in the function is associated with a unique isosurface, and the isosurface associated with the maximum of the defined function represents the most probable isosurface with respect to the putative radius. This function therefore provides an objective selection criterion for a single isosurface to represent a three‐dimensional density. This technique is applied to set of canonical molecular orbitals. The selected threshold value varies from orbital to orbital, but the enclosed probability falls in the range of 20% to 55% for the reported orbitals. In all cases, the enclosed probability is much smaller than the common choices found in the literature. The concomitant smaller volume often makes possible a more localized interpretation and helps to clarify the conventional delocalized interpretation of molecular orbitals. For example, the isosurface plots selected by this method distinguish the formally bonding orbital in He₂ from the true bonding orbital in H₂. Examples from N₂, F₂, HF, H₂O, C₂H₆, and Ni(CO)₄ are also presented. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 310–321, 2000     

Hartree-Fock orbitals for complex-scaled configuration interaction calculation of highly excited Feshbach resonances

We examine a complex-scaled configuration interaction [(CS)CI] for highly excited Feshbach resonances, where we study the 2s(2) resonance of helium as a test case. Sizable full-CI calculations are reduced by using a correctly defined minimum active space. We compare the convergence of the minimum active space for conventional Hartree-Fock (HF) orbitals obtained as solutions to Hermitian HF equations, to the convergence of minimum active space for complex orbitals obtained as solutions to complex-scaled HF equations. Ground-state optimized orbitals are compared to a simple modification of the HF method using the excited-state mean-field potential.     

Direct selected multireference configuration interaction calculations for large systems using localized orbitals

A selected multireference configuration interaction (CI) method and the corresponding code are presented. It is based on a procedure of localization that permits to obtain well localized occupied and virtual orbitals. Due to the local character of the electron correlation, using local orbitals allows one to neglect long range interactions. In a first step, three topological matrices are constructed, which determine whether two orbitals must be considered as interacting or not. Two of them concern the truncation of the determinant basis, one for occupied/virtual, the second one for dispersive interactions. The third one concerns the truncation of the list of two electron integrals. This approach permits a fine analysis of each kind of approximation and induces a huge reduction of the CI size and of the computational time. The procedure is tested on linear polyene aldehyde chains, dissociation potential energy curve, and reaction energy of a pesticide-Ca(2+) complex and finally on transition energies of a large iron system presenting a light-induced excited spin-state trapping effect.     

The use of average natural orbitals for configuration interaction calculations on the Boron Hydride molecule

The use of average natural orbitals is compared to some better known methods of performing limited and restricted CI calculations. It is found that a moderately extensive restricted valence shell CI computation using a subset of these orbitals is an efficient and accurate method for the calculation of state wavefunctions. Total and electronic excitation energies have been calculated for the BH molecule.Taken in part from a Ph. D. thesis submitted to the University of Toronto in 1971.     

Virtual orbital transformation prior to configuration interaction with localized orbitals

A virtual orbital transformation is proposed involving pairing of localized occupied orbitals with virtual orbitals. The virtual orbitals are transformed so that each virtual orbital is as “close” as possible to its occupied counterpart, where closeness is the inverse of the particular definition of localization. The appropriate transformation is derived for the special case of Foster–Boys localization, and an illustrative CNDO /2 calculation on HNO is presented. INDO CI results on the series N₂, CO, BF indicate that use of this transformation may reduce the number of energetically significant configurations.     

The role of radial nodes of atomic orbitals for chemical bonding and the periodic table

The role of radial nodes, or of their absence, in valence orbitals for chemical bonding and periodic trends is discussed from a unified viewpoint. In particular, we emphasize the special role of the absence of a radial node whenever a shell with angular quantum number l is occupied for the first time (lack of "primogenic repulsion"), as with the 1s, 2p, 3d, and 4f shells. Although the consequences of the very compact 2p shell (e.g. good isovalent hybridization, multiple bonding, high electronegativity, lone-pair repulsion, octet rule) are relatively well known, it seems that some of the aspects of the very compact 3d shell in transition-metal chemistry are less well appreciated, e.g., the often weakened and stretched bonds at equilibrium structure, the frequently colored complexes, and the importance of nondynamical electron-correlation effects in bonding.     

Theory of two shells of atomic electrons using non-orthogonal radial orbitals

A theory for handling non-orthogonal radial orbitals of two shells of atomic electrons based on the mathematical apparatus of irreducible tensor operators is presented. The general expressions for one- and two-electron operator matrix elements are given.     

Complete active-space configuration interaction with optimized orbitals: Application to Li2

We performed CAS –CI calculations on Li₂ using a set of molecular orbitals (MO ) optimized with a procedure that, in the case of highly symmetric molecules, permits extraction of a small set of MO out of a large set of atomic orbitals (AO ). The dimension of the CAS –CI space was of about 12 million symmetry-adapted determinants. We determined some spectroscopic constants of Li₂ with three different atomic basis sets of increasing quality. The values obtained with the largest atomic basis set are very close to the experimental results. © 1995 John Wiley & Sons, Inc.     

Nonorthogonal density-matrix perturbation theory

Recursive density-matrix perturbation theory [A.M.N. Niklasson and M. Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] provides an efficient framework for the linear scaling computation of materials response properties [V. Weber, A.M.N. Niklasson, and M. Challacombe, Phys. Rev. Lett. 92, 193002 (2004)]. In this article, we generalize the density-matrix perturbation theory to include properties computed with a perturbation-dependent nonorthogonal basis. Such properties include analytic derivatives of the energy with respect to nuclear displacement, as well as magnetic response computed with a field-dependent basis. The theory is developed in the context of linear scaling purification methods, which are briefly reviewed.     

On radial weighting effects in gaussian expansions of self-consistent field atomic orbitals

Gaussian expansions of the SCF functions for the first row atoms, boron through fluorine, in ground and low-lying electronic states have been generated under a wide range of radial weighting conditions by a full least-squares procedure. Typical results are presented and the quality of the wavefunctions obtained are analyzed in terms of regional electron densities and a variety of expectation values including energies. A novel method for recursive evaluation of repeated integrals of the error function, F _l (,), is adopted and analyzed. These integrals are central quantities in the least-squares procedure employed.

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Zusammenfassung Entwicklungen nach Gaußfunktionen für SCF-Funktionen von Elementen der ersten Reihe (Bor bis Fluor) wurden für den Grundzustand und niedrige angeregte Zustände einer Methode der kleinsten Quadrate berechnet, wobei eine Reihe von radialen Gewichtsfaktoren benutzt wurden. Einige typische Ergebnisse werden mitgeteilt und die Qualität der erhaltenen Wellenfunktion wird mit Hilfe von regionalen Elektronendichten und einer Reihe von Erwartungswerten sowie der Energie geprüft. Eine neue Methode für die rekursive Auswertung der mehrfachen Integrale der Fehlerfunktion F _l (, ) wird angewendet und analysiert. Diese Integrale sind von zentraler Bedeutung in der benutzten Methode der kleinsten Quadrate.

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Résumé Engendrement de représentations gaussiennes des fonctions SCF pour les atomes de la première ligne, du bore au fluor, dans les états électroniques fondamentaux et faiblement excités, avec un large éventail de conditions de pondération radiale obtenues par une procédure de moindres carrés. Des résultats typiques sont présentés et les qualités des fonctions d'onde obtenues sont analysées en fonction des densités électroniques par régions et de différentes valeurs moyennes dont les énergies. Adoption et analyse d'une nouvelle méthode pour l'évaluation récursive d'intégrales répétées de la fonction d'erreur F _l (,). Ces intégrales sont des quantités centrales dans les procédures de moindres carrés utilisées.

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Research performed under the auspices of the U.S. Atomic Energy Commission.

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Summer student program participant, 1971, from Swarthmore College, Swarthmore, Pa.     

PCILON. perturbative configuration interaction using localized orbitals and numerical integration. I. Numerical integration techniques for the calculation of Hamiltonian matrix elements between localized orbitals

Modern techniques for multidimensional numerical integration, Korobov's and Sobol's formulas namely, are used for the direct computation of matrix elements between the localized molecular orbitals needed for a configuration interaction calculation by a perturbation method. A minimal orbital basis of Slater functions is used for formaldehyde and ethylene taken as example. The resulting precision is satisfactory.     

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

 The numerical properties of the radial part of overlap integrals with the same screening parameters in the form of polynomials in p = ξR over Slater-type orbitals have been studied and obtained by using three different methods. For that purpose, the characteristics of auxiliary functions were used first, then Fourier transform convolution theorem, and recurrence relations for the basic coefficients of A ^s _{n l λ, n ′ l ′λ} were used. The calculations of the radial part of overlap integrals with the same screening parameters were made in the range 1 ≤ n ≤ 75, 1 ≤ n′ ≤ 75, and 10⁻⁶ ≤ p. Received: 18 January 2001 / Accepted: 5 April 2001 / Published online: 27 June 2001