Slater-type Orbitals Research Articles

Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers

Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers

Calculation of electric multipole transition radial matrix elements, oscillator strengths and Einstein coefficients over nonrelativistic radial wave function using Slater type orbitals

In this study, a new method is proposed for evaluating electric multipole transition (radial) matrix elements of the generalized type H nl , n ′ l ′ k in hydrogenic atom and ions using the Slater type orbitals (STOs). The formula obtained allows the determination of all multipole transition matrix elements between two different nonrelativistic radial wave functions R nl and R n ′ l ′ . A comparative study carried out between the results of analytical computations and other numerical simulations shows that the methods agree well and emphasizing thus the effectiveness and accuracy of the proposed analytical expressions. The simple equation thus obtained has been found to be remarkable accurate and has shown a wide range of applicability.

Hartree–Fock self-consistent field method for atoms with two open shells of identical symmetry

A generalization of the Roothaan–Bagus method (Roothaan–Hartree–Fock atomic theory) on atoms with open shells of identical symmetry is given. Using orbital exponents of Slater-type atomic orbitals optimized with high accuracy by methods for the minimization of the first and second orders, energy values for atoms with two open s-type shells are calculated within the limits of the Roothaan–Hartree–Fock atomic theory.

The use of unsymmetrical one-range addition theorems of Slater type orbitals for the calculation of intermolecular Coulomb interaction energy

The formulae at Hartree–Fock level are derived for the energy of Coulomb interaction between molecules containing any number of closed and open shells. These formulae are expressed in terms of linear combination coefficients of molecular orbitals and multicenter integrals of Slater type orbitals and interaction potentials produced by the charges of the molecules. As an example of application, by the use of unsymmetrical one-range addition theorems of Slater type orbitals the Hartree–Fock–Roothaan calculations have been performed for the energy of interaction produced by the ground states of two molecules H2O.

Efficient algorithm for asymptotics-based configuration-interaction methods and electronic structure of transition metal atoms

Asymptotics-based configuration-interaction (CI) methods [G. Friesecke and B. D. Goddard, Multiscale Model. Simul. 7, 1876 (2009)] are a class of CI methods for atoms which reproduce, at fixed finite subspace dimension, the exact Schrödinger eigenstates in the limit of fixed electron number and large nuclear charge. Here we develop, implement, and apply to 3d transition metal atoms an efficient and accurate algorithm for asymptotics-based CI. Efficiency gains come from exact (symbolic) decomposition of the CI space into irreducible symmetry subspaces at essentially linear computational cost in the number of radial subshells with fixed angular momentum, use of reduced density matrices in order to avoid having to store wave functions, and use of Slater-type orbitals (STOs). The required Coulomb integrals for STOs are evaluated in closed form, with the help of Hankel matrices, Fourier analysis, and residue calculus. Applications to 3d transition metal atoms are in good agreement with experimental data. In particular, we reproduce the anomalous magnetic moment and orbital filling of chromium in the otherwise regular series Ca, Sc, Ti, V, Cr.

Experimental and theoretical Compton profiles of Be, C and Al

The results of Compton profile measurements, Fermi momentum determinations, and theoretical values obtained from a linear combination of Slater-type orbital (STO) for core electrons in beryllium; carbon and aluminium are presented. In addition, a Thomas–Fermi model is used to estimate the contribution of valence electrons to the Compton profile. Measurements were performed using monoenergetic photons of 59.54 keV provided by a low-intensity Am-241 γ-ray source. Scattered photons were detected at 90° from the beam direction using a p-type coaxial high-purity germanium detector (HPGe). The experimental results are in good agreement with theoretical calculations.

Extremely compact formulas for the Fourier transform of a product of two-centre Slater-type orbitals

A compact formula for the Fourier transform of a product of Slater-type orbitals on different centres is derived. The integral is reduced to a finite one-dimensional integration over non-oscillatory hypergeometric functions of type . The formula is valid for all quantum numbers and does not involve the reduced Bessel functions that are usually used to evaluate these integrals. Reduced formulas are calculated for some special directions in the reciprocal space. Also, some useful identities for the Fourier transforms of a product of Slater-type orbitals with correlated sets of parameters are obtained. In order to illustrate simple and efficient use of the presented results, we have applied them to graphene.

Including Charge Penetration Effects in Molecular Modeling.

Electrostatic effects are often the dominant component of intermolecular interactions, but they are often modeled without accounting for charge penetration effects due to the finite extent of electronic orbitals. Here, we propose a new scheme to include charge penetration effects in electrostatic modeling, and we parametrize it and illustrate it by employing the electronically embedded combined quantum mechanical and molecular mechanical (QM/MM) method. It can also be extended to other molecular modeling approximations that include electrostatic effects. The method, which is based on introduction of a single parameter for each element, is simple in concept and implementation, modest in cost, and easily incorporated into existing codes. In the new scheme, the MM atomic charge density of an atom in a molecule is represented by a screened charge rather than by a point charge. The screened charge includes a point charge for the nucleus, core electrons, and inner valence electrons, and a smeared charge for the outer valence electron density, which is distributed in a Slater-type orbital representing the outer part of the atomic charge distribution such that the resulting pairwise interactions are still analytic central potentials. We optimize the exponential parameters of the Slater-type orbitals for 10 elements, in particular H, C, N, O, F, Si, P, S, Cl, and Br, to minimize the mean unsigned error (MUE) of the QM/MM electrostatic and induction energies with respect to the Hartree-Fock electrostatic energies and the sum of induction and induction-exchange energies calculated by symmetry-adapted perturbation theory (SAPT). The resulting optimized exponential parameters are very physical, which allows one to assign parameters to all nonmetal elements (except rare gases) with atomic number less than or equal to 35. For a test set of complexes, the improved description of MM charge densities reduces the error of electrostatic interactions between QM and MM regions in the QM/MM method from 8.1 to 2.8 kcal/mol and reduces the error of induction interactions from 1.9 to 1.4 kcal/mol.

Evaluation of potential of electric field produced by molecule using symmetrical one-range addition theorems for Coulomb-Yukawa like correlated interaction potentials of integer and noninteger indices

Using symmetrical one-range addition theorems the series expansion formulae in terms of multicenter charge density expansion coefficients for noninteger n Slater type orbitals (STO), parameters of Coulomb-Yukawa like correlated interaction potentials (CIP) of noninteger indices and linear combination coefficients of molecular orbitals are established for the potential of electrostatic field produced by the charges of molecule. The final results are useful for the study of interaction between atomic-molecular systems containing any number of closed and open shells when the Hartree–Fock–Roothaan (HFR) approximation and the explicitly correlated methods based upon the use of STO as basis functions and Coulomb–Yukawa like CIP are employed. As an example of application, the calculations have been performed for the Coulomb interaction potential produced by the ground state of CH 2 molecule \({(1a_1^2 2a_1^2 1b_1^2 3a_1^1 1b_1^1,^3B_1 )}\).

Use of binomial coefficients in fast and accurate calculation of Löwdin-α radial functions

In the present paper, we introduce a new algorithm for the development of efficient numerical methods for the analytical solution of the Lowdin-α radial function. The final results are expressed through the binomial coefficients, which enable fast and accurate calculation of the Lowdin-α radial function. The comparisons of the obtained results with the literature have shown that the presented approach can be used for the fast calculation of multicenter multielectron integrals over Slater-type orbitals (STOs). The efficiency of the algorithm is discussed and its performance with several examples is demonstrated.

Nodal surfaces of the wave functions of the hydrogen molecule in the triplet state 3Σu +

For molecular hydrogen in the triplet state 3Σu+, the nodal surfaces of the wave function corresponding to the minimum basis set of Slater orbitals in the Hartree—Fock approximation and those of the wave function used in calculations by the diffusion quantum Monte Carlo method were plotted and analyzed. Taking account of the condition for antisymmetrical wave function of the triplet state 3S of He atom, the Hartree-Fock approximation in the minimum basis set of one-electron orbitals is inappropriate for a priori determination of the nodal surfaces of many-electron wave functions (MWF). An MWF quantum chemical method developed by the authors is outlined. The alternative nodal surfaces for H2 (3Σu+) a priori specified in this method are presented.

Calculation of multicenter electric field gradient integrals over Slater-type orbitals using unsymmetrical one-range addition theorems

The electric field induced within a molecule by its electrons determines a whole series of important physical properties of the molecule. In particular, the values of the gradient of this field at the nuclei determine the interaction of their quadrupole moments with the electrons. Using unsymmetrical one-range addition theorems introduced by one of the authors, the sets of series expansion relations for multicenter electric field gradient integrals over Slater-type orbitals in terms of multicenter charge density expansion coefficients and two-center basic integrals are presented. The convergence of the series is tested by calculating concrete cases for different values of quantum numbers, parameters and locations of orbitals.

Time-dependent density functional theory gradients in the Amsterdam density functional package: geometry optimizations of spin-flip excitations

An implementation of time-dependent density functional theory (TDDFT) energy gradients into the Amsterdam density functional theory program package (ADF) is described. The special challenges presented by Slater-type orbitals in quantum chemical calculation are outlined with particular emphasis on details that are important for TDDFT gradients. Equations for the gradients of spin-flip TDDFT excitation energies are derived. Example calculations utilizing the new implementation are presented. The results of standard calculations agree well with previous results. It is shown that starting from a triplet reference, spin-flip TDDFT can successfully optimize the geometry of the four lowest singlet states of CH2 and three other isovalent species. Spin-flip TDDFT is used to calculate the potential energy curve of the breaking of the C–C bond of ethane. The curve obtained is superior to that from a restricted density functional theory calculation, while at the same time the problems with spin contamination exhibited by unrestricted density functional theory calculations are avoided.

Legendre expansion of the Debye interaction

AbstractThe Legendre expansion of the Debye interaction has been derived in a form that is suited to evaluate two‐electron matrix elements between Slater type orbitals using a Gauss‐Laguerre quadrature algorithm. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

First-principles calculation of parameters of electron paramagnetic resonance spectroscopy in solids

The hyperfine A-tensor and Zeeman g-tensor parameterize the interaction of an 'effective' electron spin with the magnetic field due to the nuclear spin and the homogeneous external magnetic field, respectively. The A- and g-tensors are the quantities of primary interest in electron paramagnetic resonance (EPR) spectroscopy. In this paper, we review our work [E.S. Kadantsev, T. Ziegler, J. Phys. Chem. A 2008, 112, 4521; E. S. Kadantsev, T. Ziegler, J. Phys. Chem. A 2009, 113, 1327] on the calculation of these EPR parameters under periodic boundary conditions (PBC) from first-principles. Our methodology is based on the Kohn-Sham DFT (KS DFT), explicit usage of Bloch basis set made up of numerical and Slater-type atomic orbitals (NAOs/STOs), and is implemented in the 'full potential' program BAND. Our implementation does not rely on the frozen core approximation. The NAOs/STOs basis is well suited for the accurate representation of the electron density near the nuclei, a prerequisite for the calculation of highly accurate hyperfine parameters. In the case of g-tensor, our implementation is based on the method of Van Lenthe et al. [E. van Lenthe, P. E. S. Wormer, A. van der Avoird, J. Chem. Phys. 1997, 107, 2488] in which the spin-orbital coupling is taken into account variationally. We demonstrate the viability of our scheme by calculating EPR parameters of paramagnetic defects in solids. We consider the A-tensor of 'normal' and 'anomalous' muonium defect in IIIA-VA semiconductors as well as the S2 anion radical in KCl host crystal lattice.

Density Functional Calculations of E2 and SN2 Reactions: Effects of the Choice of Method, Algorithm, and Numerical Accuracy.

Herein we provide a detailed account on how the potential energy surfaces of the E2 and SN2 reactions of X(-) + CH3CH2X (X = F, Cl) depend on various methodological and technical choices in density functional calculations. We cover a choice of density functionals (OLYP, various M06-types, and the new SSB-D), basis sets (up to quintuple- and quadruple-ζ for Gaussian- and Slater-type orbitals, respectively, plus polarization and diffuse functions), and other aspects of the computations (among others: nonrelativistic versus zeroth-order regular approximation relativistic; numerical integration accuracy; all-electron versus frozen core; self-consistent field (SCF) versus post-SCF). The program codes ADF and NWChem are used for calculations with Slater- and Gaussian-type basis sets, respectively. The fluoride systems (X = F) appear to not only depend extremely sensitively on the basis set size (especially the presence of diffuse functions) but also on other technical settings, especially in the case of hybrid meta-generalized gradient approximation functionals. This work complements a recent contribution (Y. Zhao, D. G. Truhlar, J. Chem. Theory Comput. 2010, 6, 1104) and provides recommendations for density functionals, basis sets, and technical settings.

Calculation of One-Electron Multicenter Integrals of Slater Type Orbitals and Coulomb–Yukawa Like Correlated Interaction Potentials with Integer and Noninteger Indices Using Unsymmetrical One-Range Addition Theorems

Abstract Using expansion formulae for multicenter charge densities of integer and noninteger n Slater type orbitals (ISTO and NISTO) established by one of the authors with the help of unsymmetrical one-range addition theorems of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETO), where α is the frictional quantum numbers (α = 1,0, - 1, - 2, … ), we perform the calculations for one-electron multicenter integrals of Slater type orbitals (STO) and Coulomb–Yukawa like correlated interaction potentials (CIP) with integer and noninteger indices (ICIP and NICIP). The convergence of the series is tested by calculating concrete cases for arbitrary values of indices and screening constants of orbitals and potentials. It should be noted that the results for one-electron multicenter integrals presented in this work were not shown in our previous publications. As an example of application, the calculations are performed for the ground state of electronic configuration of molecule BH2 that has the open shell.

Localized donor states in axially symmetrical heterostructures

We present a general method for calculating the energy spectrum of donors confined in heterostructures with axial symmetry in the presence of magnetic and electric fields applied along the symmetry axis. The donor’s wave functions are chosen as a product of the Slater orbitals and an envelope function that is a solution of a one-dimensional differential equation, which we derive starting from Schrödinger’s variational principle. We calculate the energies of the ground and some excited states of a donor confined in multiple quantum wells and a nanowire superlattice as functions of the donor position, electric and magnetic fields for structures with different numbers and widths of the wells and the barriers. Our method could be applicable to a variety of complex quantum-confined semiconductor structures for which more rigorous approaches require extensive numerical calculations.

Spin-State-Corrected Gaussian-Type Orbital Basis Sets

Recently, we reported that the basis set has a profound influence on the computed values for spin-state splittings [J. Phys. Chem. A 2008, 112, 6384]. In particular, small Gaussian-type orbital (GTO) basis sets were shown to be unreliable for the prediction of them. Here, we report simple modifications of the small Pople-type Gaussian-type orbital basis sets (3-21G, 3-21G*, 6-31G, 6-31G*), which correct their faulty behavior for the spin-state energies. We have investigated the basis sets for a set of 13 first-row transition-metal complexes for which reliable reference data have been obtained at the OPBE/TZ2P(STO) level. For several systems, we have used single and double spin-contamination corrections to avoid ambiguity of the results because of spin contamination, that is, the energies and geometries were obtained for the pure spin states. The spin ground states as predicted by the spin-state-corrected GTO basis sets (s6-31G, s6-31G*) are in complete agreement with the reference Slater-type orbital (STO) data, while those of the original basis sets and a recent modification by Baker and Pulay (m6-31G*) are not for all cases. The spin-state-corrected GTO basis sets also improve upon the original and modified basis sets for the accuracy of geometry optimization, while the accuracy of the vibrational frequencies is as good or better. At a limited additional cost, one therefore obtains very reliable results for these important spin-state energies.

Excitation and ionization energies of substituted anilines calculated with density functional theory

Valence electron singlet excitation energies (VEExE), valence electron ionization energies (VEIE), core electron binding energies (CEBE), and non-resonant X-ray emission energies of substituted anilines and related molecules were calculated using density functional theory (DFT). The energy calculations were done with TZP basis set of Slater Type Orbitals. PW86x-PW91c, turned out to be the best XC functional among eight functionals tested for time dependent DFT (TDDFT) calculation of the singlet excitation energies of the substituted anilines. Using the XC functional, average absolute deviation (AAD) from experiment was 0.223 eV for eighteen cases with maximum absolute deviation of 0.932 eV. The valence electron ionization energies of the substituted benzenes were calculated by ΔSCF method with PW86x-PW91c. AAD from experiment was 0.21 eV. The CEBEs were calculated with the previously established method, named as scheme 2003. ΔCEBE(SMS),, sum of mono substituted (SMS) CEBE shift , and mutual interference effect (MIE) were defined and their values were calculated. Magnitude of MIE provides the degree of mutual interference between two substituents in a phenyl ring. Average absolute value of MIE was ca. 0.1 eV for the three isomers of phenetidine. Using the calculated valence electron ionization energies and the core electron binding energies of one of the phenetidines, some X-ray emission energies were calculated.