Linear Approximation of Basis Set Orbital Exponents for Isoelectronic Series of Atoms

The orbital exponents of Slater type atomic orbitals (AOs) in isoelectronic series of atoms may be approximated by the linear dependence on the nuclear charge using a technique developed for optimization of AO basis sets in Hartree–Fock–Roothaan calculations. This approach yields the analytical Hartree–Fock wave functions for any ion in the isoelectronic atomic series without optimization of orbital exponents. The approximated linear equations for atomic orbital basis sets of B, C, O, and F in the ground state are presented as an example.     

Variable ζ calculation for self-consistent screening parameters in ab initio molecular theory

A resolution of Roothaan's HF–SCF–LCAO–MO equations is proposed in which atomic orbital exponents (ζ) are made dependent on the molecular charge distribution and included in the self-consistent scheme. Screening parameters so obtained are self-consistent with the molecular orbital coefficients and compare closely to optimum orbital exponents found by other methods. The technique is applied to the ground, lowest positive, and lowest negative ion states of the hydride series LiH, BH, and HF. Calculated potential curves are used to define purely theoretical values for the vertical and adiabatic ionization energies and electron affinities. Predictions are compared to experimental values where available.     

Optimization of basis sets for isoelectronic series of closed-shell atoms in Hartree-Fock-Roothaan calculations

Optimization of the nonlinear parameters (orbital exponents) of basis functions in Hartree-Fock-Roothaan calculations may be canied out with a high degree of accuracy. A scheme using second-order methods is suggested for optimization of the orbital exponents of Slater type basis functions defining the ground state of closed-shell atoms. An exact fomula is derived for calculating the partial second derivatives of energy with respect to nonlinear parameters in temis of the density matrix. In bases of isoelectronic series, the orbital exponents are shown to be the linear functions of the charge of the ion nucleus. Optimization calculations are reported for He, Be, Ne, and Mg atoms and their isoelectronic series. Translated fromZhumal Struktumoi Khimii, Vol. 41, No. 2, pp. 217–228, March–April, 2000     

Technique d'optimisation de l'energie SCF en fonction des exposants de slater

We describe a method to optimize simultaneously Slater orbital exponents. The procedure is based on Newton's method and requires the derivatives of the energy as a function of the exponents. The calculation of these derivatives is described explicitly. The method has been applied to the hydrogen molecule.     

Self‐consistent shielding in atoms and molecules

The orbital exponents of trial wave functions for simple systems can be found from the potential energy terms alone. Shielding of the nuclear charge by one electron on another is determined by the relative values of the nuclear–electron attraction and the electron–electron repulsion. For two electrons in the same orbital, the shielding is divided equally. For different orbitals, only the inner electron shields the outer. The systems tested are first‐row atoms, using Slater orbitals. It appears that if this approach can be generalized, it may not be necessary to calculate kinetic energies in chemical systems, since they will be determined by the orbital exponents. This would be useful if trial wave functions were not available, but trial electron density functions were. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Reduced hamiltonian orbitals. II. Optimal orbital basis sets for the many-electron problem

Optimal orbital exponents are approximated by minimization of the reduced Hamiltonian orbital ground state energy. They appear to be as good as and are obtained at much less expense than the values derived by the usual SCF exponent optimization scheme. Partitioning of energy into 0-energy, 1-energy, and 2-energy (Absar and Coleman, Int. J. Quant. Chem. 10 , 319 (1976); Chem. Phys. Lett. 39 , 60 (1976)) is used to study the variation in the electronic energy surface upon variation of orbital exponents. The 1-energy operator, the natural orbitals of which are the reduced Hamiltonian orbitals, is compared with the SCF operator.     

Compact natural orbital expansions for the helium ground state

Using the natural orbital representation and optimizing the exponents in the Slater-type orbital basis, configuration interaction type wave functions for the helium atom are given which combine compactness and high accuracy.     

Polynomial expansion of basis sets: Alternatives to fully optimized exponents

Alternatives based on polynomial expansions of gaussian basis set exponents are introduced and evaluated. The formulas presented here outperform methods based upon the even-tempered formula or combinations of it. They closely match the performance of other methods based upon larger polynomial expansions of the logarithm of the exponents using the same or one less parameter per orbital angular symmetry.     

Configuration interaction study of H2 and H3 systems

The configuration interaction method has been applied to the H₂ and H₃ systems. The effect of increasing the size of the atomic Slater-type orbital basis has been studied. A minimization procedure with respect to orbital exponents has been carried out.     

A technique for orbital exponent optimization in ab initio HF?SCF?LCAO?MO calculations

A technique for Slater orbital exponent optimization in an HF? SCF? LCAO? MO calculation is proposed in which orbital exponent variation is incorporated into the SCF scheme. This is accomplished by rewriting Slater's rules so that the shielding terms depend on the molecular charge distribution through the elements of the population matrix. The SCF scheme then includes a calculation of a new set of orbital exponents from the coefficients of self-consistent molecular orbitals obtained from the previous set of exponents. The process is iterated until the energy attains its lowest value. The technique is illustrated by minimal basis calculations on LiH, BH, and HF. Near optimization is obtained with considerably less effort than is necessary for other reported techniques. Aside from interesting properties, the technique can be important for extended basis calculations where exponent optimization is a difficult task.     

Efficient diffuse function-augmented basis sets for anion calculations. III. The 3-21+G basis set for first-row elements,Li–F

The relatively small diffuse function-augmented basis set, 3-21+G, is shown to describe anion geometries and proton affinities adequately. The diffuse sp orbital exponents are recommended for general use to augment larger basis sets.     

MO-Berechnungen an Heterocyclen, 19. Mitt.: Zum Einfluß unterschiedlicher Orbitalexponenten für 2 s-und 2 p-STO auf die Sequenz der Orbitalenergien in CNDO-Typ-Rechnungen

CNDO-typ calculations based on a different choice of orbital exponents for 2s- and 2p-STO are able to reproduce almost exactly the order ofHartree-Fock orbital energies from high-accuracyab initio calculations. A uniform symmetry consideration for monocyclic molecules represents a useful method to correlate and compare the results. Some comments are given concerning the interpretation of UPS by means ofKoopmans theorem.

18. Mitt.:G. Kluge, undM. Scholz, Z. Chem., im Druck.     

Double orbital exponent SCF functions for H2O,NH3, CH4

The results of an investigation on best double orbital exponents for Hydrogen in H₂O, NH₃, CH₄ are reported. An error analysis for calculations with extended basis sets is presented. This analysis is based on the hypothesis that the errors on the integrals are small so that it is possible to use statistical methods.     

Photoionization cross sections of H2(+) and H2 with complex Gaussian-type basis functions optimized for the frequency-dependent polarizabilities

Within the framework of the complex basis function method, the photoionization cross sections of H(2)(+) and H(2) were calculated based on the variational principle for the frequency-dependent polarizabilities. In these calculations, complex orbital exponents of Gaussian-type basis functions for the final state continuum wavefunctions were fully optimized for each photon energy with the numerical Newton-Raphson method. In most cases, the use of only one or two complex Gaussian-type basis functions was enough to obtain excellent agreement with previous high precision calculations and available experimental results. However, there were a few cases, in which the use of complex basis functions having various angular momentum quantum numbers was crucial to obtain the accurate results. The behavior of the complex orbital exponents as a function of photon energy was discussed in relation to the scaling relation and the effective charge for photoelectron. The success of this method implies the effectiveness of the optimization of orbital exponents to reduce the number of basis functions and shows the possibility to calculate photoionization cross sections of general molecules using only Gaussian-type basis functions.     

Configuration interaction calculations on the 2P ground state of boron atom and C+ using Slater orbitals

Configuration Interaction (CI) calculations on the ground ²P state of boron atom are presented using a wave function expansion constructed with L‐S eigenfunction configurations of s‐, p‐, and d‐Slater orbitals. Two procedures of optimization of the orbital exponents have been investigated. First, CI(SD) calculations including few types of configurations and full optimization of the orbital exponents led to the energy ?24.63704575 a.u. Second, full‐CI (FCI) calculations including a large number of configuration types using a fixed set of orbital exponents for all configurations gave ?24.63405222 a.u. using the basis [4s3p2d] and 2157 configurations, and to an improved result of ?24.64013999 a.u. for 3957 configurations and a [5s4p3d] basis. This last result is better than earlier calculations of Schaefer and Harris (Phys Rev 1968, 167, 67), and compares well with the recent ones from Froese Fischer and Bunge (personal communication). In addition, using the same wave functions, CI calculations of the boron isoelectronic ion C⁺ have been performed obtaining an energy of ?37.41027598 a.u. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Angular dependence of Gaussian-Lobe orbitals. I. Analysis of standard p- and d-orbitals

Multipole expansions of Gaussian-lobe atomic orbitals around their centers are theoretically investigated in order to study the exact angular dependence of such functions. Analytical expressions of the multipole coefficients are derived for standard lobe orbitals. It is shown that the average-square values of multipole components are related to a unique orbital parameter λ. The numerical values of p- and d-components are given for selected λ and the choice of this parameter is discussed on the basis of symmetry and computational arguments. The transferability of optimized atomic exponents from harmonic (or Cartesian) functions to lobe functions is established so that the possibility of applying the Gaussian-lobe orbital approach in chemical studies is greatly extended.     

Photoionization cross sections with optimized orbital exponents within the complex basis function method

We show a new direction to expand the applicability of the complex basis function method for calculating photoionization cross sections through the imaginary part of the frequency-dependent polarizability. Based on the variational stability of the frequency-dependent polarizability, we made nonlinear optimizations of complex orbital exponents in basis functions representing continuum wave functions, and obtained fairly accurate results for H atom with only one or two complex basis functions particularly with dipole velocity gauge. Results were almost independent of whether Slater-type or Gaussian-type orbitals are used, implying the applicability to general many electron problems. The method was also applied to the (1)S (1s)(2) --> (1)P (1s)(1)(kp)(1) cross section of He atom and the optimized complex orbital exponents were related to those of H atom through the scaling property. The nonlinear optimizations have converged smoothly and the cross sections were in excellent agreement with experiment throughout wide photon energies, which suggest the effectiveness of the approach for many-electron systems.     

Molecular states of atoms. II

Orbital energy parameters, previously obtained from atomic valence state energies, are used in calculating approximate wave functions for their orbitals. The radial factors of these wave functions are expressed as linear combinations of three Gaussian type orbitals with selected exponents, the coefficients being determined by normalisation and reproduction of the kinetic energy and interelectron repulsion parameters. Wave functions of universal form are obtained for the non-transition elements up to xenon. Each calculated s orbital wave function (except 1s) has a radial node, as is appropriate if there is a p orbital in the same shell with none.     

Atomic orbital basis sets for use with effective core potentials

Basis sets developed for use with effective core potentials describe pseudo‐orbitals rather than orbitals. The primitive Gaussian functions and the contraction coefficients in the basis set must therefore both describe the valence region effectively and allow the pseudo‐orbital to be small in the core region. The latter is particularly difficult using 1s primitive functions, which have their maxima at the nucleus. Several methods of choosing contraction coefficients are tried, and it is found that natural orbitals give the best results. The number and optimization of primitive functions are done following Dunning's correlation‐consistent procedure. Optimization of orbital exponents for larger atoms frequently results in coalescence of adjacent exponents; use of orbitals with higher principal quantum number is one alternative. Actinide atoms or ions provide the most difficult cases in that basis sets must be optimized for valence shells of different radial size simultaneously considering correlation energy and spin‐orbit energy. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 516–520, 2000     

Basis Set Orbital Relaxation in Atomic and Molecular Hydrogen Systems

Orbital relaxation (OR) amounts to variation of the orbital exponents in hydrogen molecules and ions relative to the exponents of the isolated atom; it is represented as the sum of the one- and two-center contributions depending on the effective atomic charge and on the presence of other atoms in the molecule. The procedure for isolating the contributions of the exponent includes treatment of the OR of hydrogen in a special set of neutral and charged atoms and molecules with certain multiplicities of their electronic states. Within the framework of the spin-unrestricted Hartree-Fock method, we found and discussed the optimal values of the exponents of the basis orbitals of hydrogen atoms and molecules using the minimal split valence-shell basis set, the basis set that includes the polarization function, and the expanded set of grouped natural orbitals. A simple energy model is suggested for OR. Expressions are derived for evaluating the exponents of the relaxed orbitals in hydrogen-containing systems.Original Russian Text Copyright © 2004 by A. I. Ermakov, A. E. Merkulov, A. A. Svechnikova, and V. V. Belousov__________Translated from Zhurnal Strukturnoi Khimii, Vol. 45, No. 6, pp. 973–978, November–December, 2004.