Slater-type Functions Research Articles

Scaling the Coulomb interaction in calculations of electron spectra of transition metal complexes

UDC 541.6 + 539.194 A simple technique of scaling two-electron integrals in ab initio calculations of the electronically excited states of transition metal complexes is proposed. This technique uses the fact that one-center two-electron integrals depend linearly on the scaling factor when Slater type functions are subjected to scaling transformation. This leads to a linear dependence of the d-d transition energy on the "scale" of Coulomb interaction, which allows one to affect the calculation result by varying the Slater exponential. To test the technique, ab initio configuration interaction and full active space calculations of the low excited states of the CrF63-, MnF 2-, and VF 3- complexes are performed. For transition elements, a basis of Slater type effective functions chosen from the optical spectra of the atoms and ions of transition elements is used. It is shown that in the STO-6G basis with effective exponentials, experimental transitions are reproduced with an accuracy of about 2000 cm -! even with the use of small active space determined by the orbitals localized on the central atom of the complex.

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{sup +} to Cs{sup +}, and the stable anions from H{sup {minus}} to I{sup {minus}}. 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 one 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.

The use of ab initio quantum molecular self-similarity measures to analyze electronic charge density distributions

Exact quantum molecular overlaplike and Coulomb-like self-similarity measures are studied in a selected series of molecules with the same number of electrons. It is found that quantum molecular overlap self-similarity measures can be used to estimate the concentration of electronic charge in molecules. A good linear relationship between the overlap self-similarity measure and the volume is found for molecules with the same number of electrons when the atoms of the systems being compared belong to the same row of the periodic table. Finally, an upper bound for the quantum molecular overlap self-similarity measure of molecules with a number of electrons up to 54 is given from the atomic quantum self-similarity measures obtained using Slater-type functions. © 1996 John Wiley & Sons, Inc.

Hyperbolic cosine functions applied to atomic Roothaan–Hartree–Fock wave functions

A hyperbolic cosine function cosh ( βr) is incorporated into a Slater-type radial function r n−1 exp (− αr) with noninteger principal quantum number n. The new radial basis functions r n−1 exp (− αr)cosh ( βr) are applied to Roothaan–Hartree–Fock calculations of atoms within the minimal-basis framework. The results of a systematic study on the neutral atoms from He ( Z=2) to Lr ( Z=103) in their ground state show that the total energy errors of the parent Slater-type functions, relative to the numerical Hartree–Fock values, are reduced to half or less by the addition of cosh ( βr). Orbital energies are also improved. The present accuracy of the minimal-basis description of atoms supersedes the previous one achieved in the literature.

Nonlinear Least-Squares Fitting of Numerical Relativistic Atomic Wave Functions by a Linear Combination of Slater-Type Functions for Atoms withZ= 1–36

New relativistic atomic ground-state wave functions calculated by using the Dirac–Fock program package GRASP92 [Parpia et al. (1996). Comput. Phys. Commun. 94, 249–271; Su & Coppens (1997). Acta Cryst. A53, 749–762, (1998). Acta Cryst. A54, 357] for atoms H through Kr (Z = 1–36) have been fitted by a linear combination of Slater-type functions using a nonlinear least-squares procedure. These analytical expressions allow derivation of closed-form expressions for the X-ray scattering factors of the core and valence electrons and for the atomic densities and electrostatic properties of the pseudoatoms, the latter being required for the evaluation of physical properties from accurate X-ray diffraction data. All results are accessible via http://wings.buffalo.edu/~chem9982.

Accurate universal basis set for H through Xe for Hartree–Fock calculations

We have applied the generator coordinate Hartree–Fock method to generate Slater-type functions for the atoms from H through Xe. The Griffin–Hill–Wheeler–Hartree–Fock equations are integrated numerically generating a universal basis set for these atoms. When compared with the values obtained by Koga et al. [Phys. Rev. A 47 (1993) 4510] using a smaller but fully-optimized basis sets, it is generally observed that our Hartree–Fock ground state energies are equal for the atoms from He ( Z=2) to Mg ( Z=12), larger for the atoms from Al ( Z=13) to Kr ( Z=36), and lower for the atoms from Rb ( Z=37) to Xe ( Z=54). Moreover, our energy results compare satisfactorily with the corresponding atom-optimized and numerical Hartree–Fock calculations of Koga et al. [J. Chem. Phys. 103 (1995) 3000].

Hyperbolic cosine functions applied to atomic Roothaan-Hartree-Fock wavefunctions: further improvements

A hyperbolic cosine function is incorporated into a Slater-type radial function with a noninteger principal quantum number n. The new radial basis functions are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal-basis framework. Our systematic study on the neutral atoms from He (Z = 2) to Lr (Z = 103) in their ground states shows that the incorporation of greatly improves the minimal-basis approximation, and the present minimal-basis total energies are lower than the conventional double-zeta energies obtained from Slater-type functions with integer n. Orbital energies are also improved. The present minimal-basis wavefunctions surpass the conventional double-zeta wavefunctions in their accuracy, despite fewer numbers of variational parameters.

Calculations on diatomic molecules with Slater-type orbital basis sets

In this work, a general method of calculating multicenter integrals over Slater-type functions is presented and numerical calculations on diatomic molecules, H2 and N2, are carried out. Our results are then compared with those obtained with Gaussian-type functions. The method, which is essentially an addition theorem for Slater-type functions, is outlined and some of its mathematical and numerical properties are presented. As regards our numerical results, although such very small diatomics are of limited interest in chemistry, they are to be considered as the necessary step that will allow us to tune up the computational machinery. Furthermore, the conclusions drawn throughout this work will be used for more general algorithms which will allow us to treat efficiently more complicated systems, namely, three, four, and many more atomic systems. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 89–93, 1998

Double-zeta Slater-type basis sets with noninteger principal quantum numbers and common exponents

A double-zeta basis set of extended Slater-type functions, whose principal quantum numbers are allowed to have noninteger values, is reported for the atoms He to Ar in their ground state. The total energies are lower than those from conventional and improved double-zeta basis sets, in which the principal quantum numbers are restricted to integer values. For the same atoms, we develop a new type of double-zeta basis set that combines the use of noninteger principal quantum numbers and the use of common exponents in Slater-type functions. The new double-zeta basis sets result in an improvement of the energy and a reduction of the computational time.

Nonadiabatic electron transfer rate of hydrated [formula omitted] redox couples in solution: a novel scheme for the hydration potential function

Two new theoretical models are presented for accurately determining various kinetic parameters of electron transfer processes. By relating the electron transfer process to the hydration process of an ion, two accurate potential functions are determined in terms of experimental spectroscopic and the hydration thermodynamic data. They are used to depict the energy dependence of the reacting system on the separation between the central ion and the inner-sphere water molecules and the solvent reorganization. The activation energy is obtained in terms of the proposed activation model and hydration potential functions. It is calibrated using the corresponding experimental value extracted from the experimental rate constant, and is also compared with literature values. The electronic transmission coefficient is calculated using the Landau-Zener formalism. The slopes of the potential energy surfaces are obtained from the proposed accurate hydration potential functions. The coupling matrix element is calculated using the two-state model and the numerical integral method with perturbed Slater-type double-zeta wave functions. The theoretical values of the electron transfer rate constant are calculated and compared with the experimental value. The applicability of these models is discussed.

A hydration function scheme of electron transfer kinetics for hydrated redox couples in solution and its application in the [formula omitted] system

This paper presents a hydration potential function scheme for the accurate determination of various electron transfer kinetics parameters. By comparing the common features of the electron transfer process and the hydration process of an ion, two hydration potential functions are determined in terms of the experimental spectroscopic and hydration thermodynamic data, and are then used to depict the energy dependence of the reacting system on the separation between the central ion and the innersphere water molecules and the solvent reorganization. The activation energy is obtained in terms of the proposed new activation model and the hydration potential functions, and is calibrated using the corresponding experimental value extracted from the experimental rate constant, and is also compared with those from other methods. The electronic transmission coefficient is calculated using the Landau-Zener formalism. The involved slopes of the potential energy surfaces are obtained from the proposed accurate hydration potential functions. The coupling matrix element is calculated using the two-state model and the numerical integral method over Slater-type double-zeta wave functions obtained using the perturbation method. The theoretical value of the electron transfer rate is calculated and compared with the experimental value. The applicability of the relevant models is also discussed.

Generalized exponential functions applied to atomic calculations

The use of a generalized exponential function rv−1 exp(−ζrμ) as a radial basis function in atomic calculations is studied with our special interest in the variationally optimum value of the parameter μ, since special cases of μ = 1 and μ = 2 correspond respectively to the radial parts of commonly-used Slater-type and Gaussian-type functions. Roothaan-Hartree-Fock calculations are performed for ground-state neutral atoms with atomic number Z = 2–54, singly-charged cations with Z = 3–55, and anions with Z = 1–53 within the single-zeta (or minimal basis) framework. For all the species examined, the optimtum μ values are found to be smaller than unity and increase towards unity as the atomic number increases. The present results support the use of Slater-type functions when μ is restricted to be an integer, but suggest from the variational point of view that even the exponential decay of Slater-type functions is too “strong” within the single-zeta approximation.

Noninteger principal quantum numbers increase the efficiency of Slater-type basis sets: singly charged cations and anions

For the singly charged cations to and anions to in their ground state, Roothaan - Hartree - Fock calculations are carried out using a single-zeta (or minimal) basis set of extended Slater-type functions whose principal quantum numbers are allowed to be variationally optimum noninteger values. The resultant total energies are substantially lower than those obtained from the conventional single-zeta method which implicitly restricts the quantum numbers to be integer values. In the case of and ions, for example, the improvements amount to 11.4 and 11.0 Hartrees, respectively. The noninteger principal quantum numbers also improve the orbital energies. In particular, unphysical positive orbital energies predicted by the conventional single-zeta method for 56 atomic orbitals change to realistic negative values (with only three exceptions) by the use of noninteger principal quantum numbers without increasing the number of basis functions. Ionization potentials and electron affinities are improved as well within the single-zeta approximation.

Hydration Model for the Energy Barrier in Self-Exchange Electron Transfer Reactions in Solution

This paper presents two new theoretical models for accurately determining activation energy and reorganization energy for an electron exchange reaction in solution. The hydration process of an ion is considered and two accurate potential functions (Morse function and anharmonic oscillator potential function) are defined in terms of experimental spectroscopic and hydration thermodynamic data. These functions are then used to depict the energy dependence of the reacting system on the separation between the central ion and the innersphere water molecules and the solvent reorganization. The activation energy and reorganization energy are obtained in terms of the proposed activation and reorganization models and the hydration potential functions. The experimental activation energy is corrected by taking into account the actual electronic transmission coefficient. The slopes of the potential energy surfaces are obtained from the proposed accurate hydration potential functions, and the coupling matrix element is determined by the two-state model and numerical integral method over the perturbed d-electron double-zeta Slater-type wave functions. The theoretical values of the activation energy are compared with the experimental values, and the relationship between the activation energy and the reorganization energy is tested. The applicability of these models are also discussed.

Noninteger principal quantum numbers increase the efficiency of Slater-type basis sets: heavy atoms

For the atoms Cs (Z = 55) through Lr (Z = 103) in their ground state, Roothaan-Hartree-Fock calculations are performed using a single-zeta (or minimal) basis set of extended Slater-type functions whose principal quantum numbers are allowed to be variationally optimum noninteger values. The resultant total energies are superior to those obtained from the conventional single-zeta method which implicitly restricts the quantum numbers to be integer values. In the case of the Lr atom, the improvement amounts to 56.8 E h. The noninteger principal quantum numbers improve the orbital energies; unphysical positive d and f orbital energies predicted by the conventional single-zeta method change to realistic negative values by the use of noninteger principal quantum numbers.

Gaussian- and Slater-type bases for ground and certain low-lying excited states of positive and negative ions of the atoms H through Xe based on the generator coordinate Hartree–Fock method

We applied a discretized version of the generator coordinate Hartree–Fock (GCHF) method to generate Gaussian- and Slater-type functions for mono positive and mono negative ions of the atoms H through Xe. The basis sizes for Slater-type functions are (12s, 10p, 10d) for positive ions, and (13s, 11p, 10d) for negative ions. In the case of Gaussian-type functions the bases are (18s, 12p, 11d) for both positive and negative species. Ground and excited state Hartree–Fock energies are calculated with these bases and the results compared with the best atom-optimized calculations and numerical HF results available. A discussion on the role of weight functions in the evaluation of electronic energies emphasizes the integral character of the GCHF method. Key words: Slater-type bases, Gaussian-type bases, generator coordinate Hartree–Fock, atomic ions.

Theoretical determination of energies and widths of autoionizing states of the Be atom

The energy and width of the autoionizing state of the Be atom were calculated. A series of multireference configuration-interaction (MRCI) calculations with an extensive Slater-type function (STF) basis set was carried out and the full CI energy was estimated for each of the neutral and ionized states. The estimated energy of is with respect to the ionized state. The photoionization cross section from the state was calculated by using the R-matrix method. The resultant energy and width are and , respectively, in reasonable agreement with the observed values of and . The energies and widths of the series up to n = 11 were also calculated.

Retardation long-range potentials between two helium atoms.

The retardation long-range potentials between two 1${\mathrm{}}^{1}$S (or 2${\mathrm{}}^{3}$S) helium atoms are calculated. The complex dynamic multipole polarizabilities of 1${\mathrm{}}^{1}$S and 2${\mathrm{}}^{3}$S states of helium are obtained through a variation-perturbation approach with B-spline and Slater-type basis functions in a configuration-interaction scheme. They are used to calculate the retardation long-range potentials. The retardation coefficients are examined. Their relative magnitudes are different for different systems. The possibility of an excited state of helium dimer from the 1${\mathrm{}}^{1}$S--2${\mathrm{}}^{3}$S ${\mathrm{He}}_{2}$ state is estimated. \textcopyright{} 1996 The American Physical Society.

Convergence accelerators in the computation of molecular integrals over Slater-type basis functions in the two-range one-center expansion method.

The so-called nonlinear transformations may be of substantial help in many practical problems including the so-called multicenter integrals. In this work Wynn's epsilon algorithm and the Levin's u transformation have been used to improve the convergence of the series representation of three-center two-electron integrals obtained via the so-called two-range one-center expansion method.

Improved Double-Zeta Description for the Atoms Li through Xe

Abstract The conventional double-zeta (DZ) approximation uses two Slater-type functions χnlm’s with different exponents for the description of each occupied atomic orbital φnlm, where the two principal quantum numbers n’s are assumed to be the same. We show that the removal of this implicit restriction improves the DZ functions nontrivially for the ground-state atoms Li through Xe. The largest improvement 0.0267 a.u. in the atomic energy is found for Pd. The valence orbital energies are generally improved, particularly for the 4d-orbital of the fourth-row atoms. It is also found that the conventional DZ functions given in the literature can be further improved by reoptimization of the exponents.