Physical nature of the chemical bond II. Valence atomic orbital and energy partitioning studies of linear nitriles

The valence atomic orbitals (VAO 's) of several linear nitriles are determined using non-empirical SCF –LCAO –MO wave functions expanded in a minimal (CN^?, HCN, FCN, C₂N₂), double-zeta (CN^?, HCN), or double-zeta + polarization (HCN) basis of Slater atomic orbitals (AO 's). The molecular energy of each system (except the double-zeta + polarization HCN system) is partitioned according to the procedure of Ruedenberg to obtain numerical values of nitrile C and N atomic and C?N bond components of the energy. In addition, the nitrile results are compared with minimal AO basis results obtained previously by other authors for homonuclear diatomics, diatomic hydrides and H₂O. The numerical data are used to test the internal self-consistency of the various definitions entering the partitioning method, i.e. whether or not analogous quantities assume similar values in chemically similar situations. The analysis of nitrile SCF –MO wave functions in terms of the set of VAO 's characteristic of the system under consideration is shown to be a promising approach to the problem of extracting useful information from the wave functions. In general, numerical results for the nitrile systems studied are fairly consistent with the concepts on which the partitioning method is based: promotion, quasi-classical interaction, sharing penetration, sharing interference and charge transfer. However, the VAO expansions for several energy components need to be investigated further and possibly revised.     

Relativistic Gaussian basis sets for radon through plutonium

Summary Relativistic Gaussian basis sets of neutral atoms Rn-Pu and ions Th⁺⁴, U⁺³ and Pu⁺³ in the configurations of average energies are presented. The exponent parameters of the basis sets are determined by least-squares fitting to the numerical Dirac-Fock wave functions. The total energies obtained are within 0.155 a.u. of the Dirac-Fock limits and the qualities of the basis sets are between double-zeta and triple-zeta in the valence parts. Using the exponent parameters the Breit interaction energies have been calculated by perturbation theory and the self-consistent field treatment.     

Ab initio study of low-lying electronic states of the FNO2 molecule

Ab initio electronic structure calculations are reported for low-lying electronic states, ¹A₁, ¹A₂, ³A₂, ¹B₁, ³B₁, ¹B₂, and ³B₂ of the FNO₂ molecule. Geometric parameters for the ground state ¹A₁ are predicted by MRSDCI calculations with a double-zeta plus polarization basis set. The vertical excitation energies for these electronic states are determined using MRSDCI/DZ+P calculations at the ground-state equilibrium conformation. The oscillator strengths and radiative lifetimes for some electronic states are calculated based on the MRSDCI wave functions. © 1993 John Wiley & Sons, Inc.     

General variation method for the relativistic calculations of atoms and molecules

Use of the general variation method of Weinstein and MacDonald for the relativistic calculation of atoms and molecules is proposed. It is shown from the numerical calculations for hydrogenlike atomic systems that this method is useful in judging an accuracy of energies and wave functions obtained with a relativistic Hamiltonian whose spectra are not bounded. It is also shown that this method can be used to find spurious solutions such as 1p_½ or 2d_3/2 appearing in atomic systems. Problems in extending the method to many-electron atoms and molecules are discussed.     

Bound D‐states of helium atom under Debye screening

We have investigated the 1snd^1,3D (3 ≤ n ≤ 7) state energies of helium atom embedded in weakly coupled plasma environments using the Rayleigh–Ritz variational method. The effect of the plasma environment is taken care of using a Debye screening model. A correlated wave function involving exponential expansion has been used to represent correlation between the charge particles. The bound 1snd^1,3D (3 ≤ n ≤ 7) state energies of helium for various Debye lengths along with the excitation energies of few singlet and triplet states are reported. Our results are useful references to atomic physics, plasma physics, and astrophysics research communities. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Analytical density-dependent representation of Hartree—Fock atomic potentials

A procedure to represent atomic electron charge densities [L. Fernandez Pacios, J. Phys. Chem., 95 , 10653 (1991); J. Phys. Chem., 96 , 7294 (1992)] is here generalized to obtain simple analytical functions for potential energy contributions. Based upon suitable functions to describe atomic electron densities in a physically meaningful form, the procedure is developed to define density-dependent analytical expressions for the electrostatic (classical) and exchange (quantum) potentials by means of proper approximate functionals. Calculations of correlation energies by using various density-functional approaches are also performed. The whole scheme is used to represent Hartree–Fock limit atomic wave functions by Clementi–Roetti. This way, a set of analytically simple, nonbasis set-dependent functions are defined with the aim to be further implemented in energy decomposition schemes for molecular interactions studies using atomic instead of electronic building blocks. © 1993 John Wiley & Sons, Inc.     

Use of noninteger n‐generalized exponential type orbitals with hyperbolic cosine in atomic calculations

The efficiency of noninteger n‐generalized exponential type orbitals (NGETO) r^n*?1 e with hyperbolic cosine (HC) cosh (βr^μ) as radial basis functions in atomic ground state total energy calculations is studied. By the use of these functions, the combined Hartree‐Fock‐Roothaan calculations have been performed for some closed and open shell neutral atoms and their anions and cations with Z ≤ 21. The performance of new basis functions within the minimal basis framework has been compared with numerical Hartree‐Fock (NHF) results. Our total energy values are significantly close to NHF results. The presented minimal basis total energies obtained from the noninteger NGETO with HC are notably better than minimal basis functions total energies previously reported in the literature. It is found that the accuracy of new noninteger NGETO with HC almost correspond to the accuracy of the conventional double‐zeta functions. All the nonlinear parameters are fully optimized. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Analysis of fourth-order Møller–Plesset limit energies: the importance of three-electron correlation effects

Fourth-order M?ller–Plesset (MP4) correlation energies are computed for 28 atoms and simple molecules employing Dunning's correlation-consistent polarized-valence m-zeta basis sets for m=2, 3, 4, and 5. Extrapolation formulas are used to predict MP4 energies for infinitely large basis sets. It is shown that both total and partial MP4 correlation energies can be extrapolated to limit values and that the sum of extrapolated partial MP4 energies equals the extrapolated total MP4 correlation energy within calculational accuracy. Therefore, partial MP4 correlation energies can be presented in the form of an MP4 spectrum reflecting the relative importance of different correlation effects. Typical trends in calculated correlation effects for a given class of electron systems are independent of the basis set used. As first found by Cremer and He [(1996) J Phys Chem 100:6173], one can use MP4 spectra to distinguish between electron systems with well-separated electron pairs and systems for which electrons cluster in a confined region of atomic or molecular space. MP4 spectra for increasing size of the basis set reveal that smaller basis set calculations underestimate the importance of three-electron correlation effects for both classes by overestimating the importance of pair correlation effects. The minimum size of a basis set required for reliable MP4 calculations is given by a valence triple-zeta polarized basis, which even in the case of anions performs better than a valence double-zeta basis augmented by diffuse functions. Received: 14 June 2000 / Accepted: 16 June 2000 / Published online: 24 October 2000     

Energy expressions for atomic configurations in the L–S coupling scheme

The matrix elements of the spin-free Hamiltonian between two atomic configuration state functions (CSF 'S ) in the L–S coupling scheme are expressed in terms of the atomic integrals F^k's and G^k's. Using these general expressions, the matrix elements have been obtained for all the atomic configurations with three valence electrons that have not been solved so far by earlier methods. The scope for applying this new approach to obtain the Auger line energies and the promotion energies of metals that involve more than two partially filled shells is indicated. The energy expressions for some of the relevant configurations are tabulated.     

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.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.     

A CI non-empirical pseudopotential calculation of the vertical valence excited states of the iodine molecule

The non-empirical atomic pseudopotential proposed by Durand and Barthelat has been used, together with the CIPSI algorithm for large scale CI, to calculate the vertical transition energies of the iodine molecule, in a valence extended (double-zeta + d) basis set. All the valence excited states were considered. The mixing of configurations is very important especially for the Σ⁺_g, Π_g and Π_u symmetries. The experimentally known transition energies are calculated within a 1 eV error, despite the lack of diffuse orbitals and spin-orbit interaction. Some qualitative Mulliken's estimates are discussed. A new ³Σ⁺_g state from the 10 σ_u → 11 σ_u single excitation is predicted in the 9 eV region.     

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.     

Localized impurity states in the Hartree-Fock,LCAO approximation II. The F Center in KCI

We apply the techniques of a previous paper (I) to the F center in KCl. Our purpose is to place the application of Hartree-Fock methods to the F center on a firm theoretical basis by calculating in a consistent manner the magnitude and effect of approximations commonly made in less complete treatments. It is shown that the familiar point-ion approximations and crystal-field approximations with partial consideration of exchange effects are special cases of our results. We compute wave functions and energies step by step for each of the various levels of approximation possible with our model. It is found that the functions resulting from the point-ion model are not good approximations to the final wave functions. Our results show that exchange effects with at least the first two shells of nearest neighbors should be considered since they are of the same order of magnitude as terms in the point-ion model. Overlaps of the F-center function with ion functions out to sixth neighbors are considered. The absorption energy for the F center is calculated to be 0.1619 Ry as compared with the experimentally observed energy of 0.170 Ry. The magnetic hyperfine structure contact terms are calculated for the first two shells of nearest neighbor ions, using approximate orthogonalized functions, and found to be 29.7 Mc/h for the nearest neighbor K⁺ ions and 10.9 Mc/h for the next nearest neighbor Cl^? ions. The experimentally observed values are 21.6 and 7.0, respectively. Given these differences and the excessively low values of the one-electron energies, it is concluded that electronic and ionic polarization effects in the ionized crystal states must be considered to calculate accurate F-center wave functions and absolute energy levels.     

Analytical Hartree–Fock wave functions subject to cusp and asymptotic constraints: He to Xe,Li+ to Cs+, H− to I−

Analytical, variational approximations to Hartree–Fock wave functions are constructed for the ground states of all the neutral atoms from He to Xe, the cations from Li⁺ to Cs⁺, and the stable anions from H⁻ to I⁻. The wave functions are constrained so that each atomic orbital agrees well with the electron–nuclear cusp condition and has good long‐range behavior. Painstaking optimization of the exponents and principal quantum numbers of the Slater‐type basis functions allows us to reach this goal while obtaining total energies that, at worst, are a few microHartrees above the numerical Hartree–Fock limit values. The wave functions are freely available by anonymous ftp from okapi.chem.unb.ca or upon request to the authors. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 491–497, 1999     

Improved Hylleraas calculations for ground state energies of lithium ISO–electronic sequence

The ²S ground state of lithium iso–electronic sequence is calculated by the use of Hylleraas-type wave functions. A 92 term one-spin wave function was used for lithium atom calculations. The energy obtained was ?7.478031 a.u. as compared with the previous best value of ?7.478025 a.u. calculated by Larsson. In addition, improved energies for Z = 4 to 8 were calculated by the use of 60 term wave functions. This work thus provides the lowest ab initio ground state energies for lithium sequence to date.     

High-quality Gaussian basis sets for fourth-row atoms

Summary Energy-optimized Gaussian basis sets of triple-zeta quality for the atoms Rb-Xe have been derived. Two series of basis sets are developed; (24s 16p 10d) and (26s 16p 10d) sets which we expand to 13d and 19p functions as the 4d and 5p shells become occupied. For the atoms lighter than Cd, the (24s 16p 10d) sets with triple-zeta valence distributions are higher in energy than the corresponding double-zeta distribution. To ensure a triple-zeta distribution and a global energy minimum the (26s 16p 10d) sets were derived. Total atomic energies from the largest basis sets are between 198 and 284E _H above the numerical Hartree-Fock energies.     

A simplex optimized INDO calculation of 13C chemical shifts in hydrocarbons

The variable-size simplex optimization method is used to reparametrize the I + A and β parameters of an INDO approximation to the perturbed Hartree–Fock calculation of ¹³C chemical shifts in hydrocarbons. The absolute shifts for 39 nuclei in a set of molecules containing up to four carbons are reproduced within a standard error of 9.9 ppm for an unconstrained optimization and to a standard error of 10.0 ppm for an optimization constrained to yield gross atomic charges in agreement with double-zeta ab initio calculations.