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.     

Hartree-Fock-Roothaan calculations of dipolar polarizabilities for closed-shell atoms

Methods are suggested for optimization of Slater type atomic orbitals in polarizability calculations of closed-shell atoms using “bound” perturbation theory (algebraic version of the Hartree-Fock method). The size and composition of the basis set of atomic orbitals providing the Hartree-Fock limit for perturbation parameters are considered. Dipolar polarizabilities are calculated for He, Be, Ne, and Mg atoms and their isoelectronic series. Translated fromZhurnal Strukturnoi Khimii, Vol. 41, No. 3, pp. 439-448, May–June, 2000.     

Calculations of Isoelectronic Series of He Using Noninteger n‐Slater Type Orbitals in Single and Double Zeta Approximations

Using integer and noninteger n-Slater type orbitals in single- and double-zeta approximations, the Hartree-Fock-Roothaan calculations were performed for the ground states of first ten cationic members of the isoelectronic series of He atom. All the noninteger parameters and orbital exponents were fully optimized. In the case of noninteger n-Slater type orbitals in double zeta basis sets, the results of calculations obtained are more close to the numerical Hatree-Fock values and the average deviations of our ground state energies do not exceed 2×10^－6 hartrees of their numerical results.     

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     

The energetic effects of p,d, and f Gaussian polarization functions on closed-shell AHn oxygen and sulfur hydrides

A series of LCAO -MO -SCF calculations, using various basis sets of Gaussian-type functions, has been made in order to study the effects of p, d, and f polarization functions for a 10-electron isoelectronic series of oxygen hydrides and for an 18-electron isoelectronic series of sulfur hydrides. Conclusions from these results suggest that meaningful proton affinities cannot be calculated without the inclusion of a d function on the heavy atom and a p function on the hydrogen atoms.     

Isoelectronic changes in energy of quark atoms and molecules via the Levy equation

Summary Z-transition state calculations based on the Levy equation suggest that the isoelectronic changes in energy of quark atoms,Q, (ordinary atoms with extra nuclear charge in units of ±1/3 and/or ±2/3) can be expressed quantitatively in terms of the electrostatic potential at the nucleus of an isoelectronic ordinary atom. Numerical tests within the local density functional theory are presented for the quark atoms of Li-F. Theab initio MO (molecular orbital) calculations using STO-5G basis on the C₂ molecule and its quark derivatives lead to similar conclusions.     

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     

Hartree–Fock High-Precision Analytical Functions of Open-Shell Atoms

The algorithm of high-precision optimization of basis functions suggested previously for calculating the analytical Hartree–Fock orbitals of closed-shell atoms is generalized to open-shell systems described by the Roothaan method (1960). Expressions for the first (free gradient) and second (Hesse matrix) derivatives of the system's energy with respect to the nonlinear parameters (orbital exponents) of the basis functions are derived in terms of density matrices for the filled and open shells. An algorithm is proposed for high-precision optimization of the nonlinear parameters using these equations based on Murtagh–Sargent and Newton minimization procedures. To illustrate the application of this algorithm, we give optimization of the basis sets of Slater type functions for atoms from the second row, as well as for Al, Si, P, K, Sc, and Fe atoms. The analytical Hartree–Fock orbitals giving nearly Hartree–Fock energies are calculated with a high degree of accuracy.     

Use of noninteger n‐Slater type orbitals in combined Hartree–Fock–Roothaan theory for calculation of isoelectronic series of atoms Be to Ne

The ground state calculations in the combined Hartree–Fock–Roothaan approach are performed for the neutral and the first 20 cationic members of the isoelectronic series of atoms from Be to Ne using noninteger n‐Slater type orbitals. For the total energies obtained, only a small deviation has been found. At the same time, the size of the present noninteger n‐Slater type orbitals is smaller than that of the usual extended integer n‐Slater functions in literature. All of the nonlinear parameters are fully optimized. The relationship between optimized parameters and atomic number Z is also investigated. For each atom, the total energies are given in tables. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

MIDI! basis set for silicon, bromine, and iodine

The MIDI! basis set is extended to three new atoms: silicon, bromine, and iodine. The basis functions for these heteroatoms are developed from the standard 3-21G basis set by adding one Gaussian-type d subshell to each Si, Br, or I atom. The exponents of the d functions are optimized to minimize errors in the geometries and charge distributions that these basis functions yield when they are used in Hartree-Fock calculations with all atoms represented by the MIDI! basis. The MIDI! basis is defined to use five spherical d functions in a d subshell. We present a detailed comparison of such calculations to calculations employing six Cartesian d functions in each d subshell; these studies show that 5D and 6D options give very similar results for molecular geometries and dipole moments, not only for compounds containing Si, Br, and I but also for compounds containing N, O, F, P, S, and Cl. The MIDI! basis set is also tested successfully for hypervalent Si compounds. Received: 7 January 1998 / Accepted: 7 January 1998     

Ab initio calculations of chloride complexes of Au, Hg, Tl, Pb, and Bi in anomalous oxidation states (2S1/2 electron state)

Ab initio calculations of chloride complexes of Au, Hg, Tl, Pb, and Bi in anomalous oxidation states (²S_1/2 electron state) were carried out by the Becke-Lee-Yang-Parr density functional method using the Dunning-Hay LanL2DZ basis set. Optimum geometric parameters and electronic characteristics of MCl _n (H₂O) _m ⁿ (n=1–4 andm=0,4,5) complexes were determined. In each of the considered series the spin, population on the central metal atom decreases as its atomic number increases. The energy of transition of the unpaired electron to the lowest unoccupied MO decreases in the same order. The unpaired electron occupies an orbital that is mostly a linear combination of the s-orbital of the metal atom and the p-orbital of the Cl atom (the antibonding σ-orbital of the M−Cl bond). Distinctions in the changes in spectral properties of aquacomplexes and chloride complexes in isoelectronic series, observed as the degree of oxidation of the metal atom increases, were explained. The results of calculations are in agreement with the experimental data obtained by ESR and optical spectroscopy. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1049–1055, June, 1999.     

Contracted polarization functions for the atoms Ca, Ga–Kr, Sr, and In–Xe

 Contracted Gaussian-type function sets are proposed for polarization functions of the atoms Ga–Kr and In–Xe. We also report polarization functions for Ca and Sr. A segmented contraction scheme is used for its compactness and computational efficiency. The contraction coefficients and orbital exponents are fully optimized to minimize the deviation from accurate atomic natural orbitals. The present polarization functions yield more than 99% of atomic correlation energies predicted by accurate natural orbitals of the same size. Received: 23 February 2001 / Accepted: 19 April 2001 / Published online: 13 June 2001     

Calculations using a set of evenly spaced S-type gaussian functions

A basis set of evenly spaced S-type Gaussian functions with common exponents is examined. Formulas for common one- and two-electron integrals are derived. Because of thesymmetry of this basis set, a very compact two-electron integral list is produced. The number of two-electron integrals that must be stored is approximately eight times the number of basis functions. Use of this basis set in an SCF calculation is examined. Numerical results show that this approach works well for molecules containing only small atoms such as hydrogen, helium, or lithium, but that the method has problems with the core orbitals of heavier atoms. Procedures for augementing this basis set in calculations involving heavier atoms are examined.     

Valence p functions for alkali and alkaline-earth atoms

Contracted Gaussian-type function (CGTF) basis sets are reported for valence p orbitals of the six alkali and alkaline-earth atoms Li, Be, Na, Mg, K, and Ca for molecular applications. These sets are constructed by Roothaan–Hartree–Fock calculations for the ns → np excited states of atoms, in which both linear and nonlinear parameters of CGTFs are variationally optimized. The present CGTF sets reproduce well the numerical Hartree–Fock ns → np excitation energies: the largest error is 0.0009 hartrees for Li. New CGTFs are tested with diatomic Li₂, Na₂, K₂, and MH molecules, where M = Li, Be, Na, Mg, K, and Ca, by self-consistent-field (SCF) and multiconfiguration SCF calculations. The resultant spectroscopic constants compare well with those of more elaborate calculations and are sufficiently close to experimental values, supporting the efficiency of the present set for the valence p orbitals. Received: 9 July 1998 / Accepted: 17 September 1998 / Published online: 1 February 1999     

Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules

We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.     

An intermediate neglect of differential overlap model for second-row transition metal species

An intermediate neglect of differential overlap (INDO ) model for elements of the second transition series is described. Values of exponents for the radial portion of the Slater basis atomic orbitals, the orbital ionization energies, and the beta (or bonding) parameters for these elements are reported. Use of the model to investigate equilibrium geometries, relative energy ordering of geometric isomers, spectroscopic transition energies, energy ordering of state wave functions of different symmetry, trends in dipole moments, and equilibrium stretching frequencies is reported. The results indicate that the INDO method is capable of yielding quality results for species containing atoms of the second transition series.     

Double even tempering of orbital exponents: Application to Roothaan-Hartree-Fock calculations for He through Xe in Slater-type basis sets

Summary Double even tempering (DET) of orbital exponents is proposed as a useful generalization of even tempering (ET). The DET scheme uses two sets of basis functions for each angular momentum. The two sets have different principal quantum numbers and their exponents are generated by two different geometric sequences. Roothaan-Hartree-Fock (RHF) calculations on the atoms from He through Xe using both ET and DET Slater-type basis sets of the same size are carried out to demonstrate the substantial improvement offered by the DET scheme. The DET scheme reduces the maximum deviation of the RHF energies relative to the Hartree-Fock limit from 1.4 to 0.3 millihartrees.     

Correlating basis sets for the H atom and the alkali-metal atoms from Li to Rb

 Contracted Gaussian-type function sets are developed for correlating p, d, and f functions for a valence electron of the hydrogen atom and alkali-metal atoms from Li to Rb. A segmented contraction scheme is used for its compactness and efficiency. Contraction coefficients and exponents are determined by minimizing the deviation from the K orbitals of the atoms. The present basis sets yield an accuracy comparable to the correlation-consistent basis set for the hydrogen atom and also give a similar high accuracy for the alkali-metal atoms. In the calculations of spectroscopic constants of alkali hydrides, the decontraction of the p function plays an important role, especially for LiH. The contributions of d and f functions are nontrivial for KH and RbH. Received: 6 September 2002 / Accepted: 13 November 2002 / Published online: 19 March 2003 Acknowledgements. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education of Japan. Correspondence to: T. Noro e-mail: tashi@sci.hokudai.ac.jp