Slater Functions Research Articles

Construction of Оne-Range Addition Theorems for Noninteger Slater Functions Using Self-Friction Exponential Type Orbitals and Polynomials

The addition theorems of Slater type orbitals (STOs) presented in literature are generally complicated to theoretically examine the electronic structure of atoms and molecules. The computational deficiencies in use of these theorems arise from the separation of integral variables. In this work, to eliminate these calculation efforts, a large number of independent one-range addition theorems for $$\chi $$ -noninteger Slater type orbitals ( $$\chi $$ -NISTOs) in terms of $$\chi $$ -integer STOs ( $$\chi $$ -ISTOs) is presented by using complete orthogonal basis sets of $${{L}^{{(p_{l}^{*})}}}$$ -self-friction (SF) polynomials ( $${{L}^{{(p_{l}^{*})}}}$$ -SFPs), $${{\psi }^{{(p_{l}^{*})}}}$$ -SF exponential type orbitals ( $${{\psi }^{{(p_{l}^{*})}}}$$ -SFETOs), $${{L}^{{(\alpha \text{*})}}}$$ -modified SFPs ( $${{L}^{{(\alpha \text{*})}}}$$ -MSFPs), and $${{\psi }^{{(\alpha \text{*})}}}$$ -modified SFETOs ( $${{\psi }^{{(\alpha \text{*})}}}$$ -MSFETOs) introduced by one of the authors. Here, $$p_{l}^{*} = 2l + 2 - \alpha \text{*}$$ and $$\alpha \text{*}$$ are the integer $$(\alpha \text{*} = \alpha ,\; - {\kern 1pt} \infty < \alpha \leqslant 2)$$ or noninteger ( $$\alpha \text{*} \ne \alpha ,$$ $$ - \infty < \alpha < 3$$ ) SF quantum numbers based on the Lorentz damping theory. The expansion coefficients of series for the one-range addition theorems are expressed through the analytical relations for the overlap integrals of $$\chi $$ -NISTOs with the same screening parameters. As an application, the calculations of overlap integrals with the different screening constants of $$\chi $$ -NISTOs are performed.

Kinetic Energy of Electrons and Potential Energy of Atoms with Open Electron Shell in the Basis of the Slater Functions

The kinetic energy of electrons and the potential energy of a nitrogen atom with open electron shell are calculated in this work in the basis of the Slater functions using the Hartree–Fock–Roothaan (HFR) method. Analytical expressions are derived for the kinetic and potential energies of the nitrogen atom with open electron shell. Twenty independent determinant wave functions corresponding to the 1s22s22p3 electronic configuration of the nitrogen atom have been found by summing the moment projections. The results obtained have been verified using the virial theorem and the expression $$ \left(\overline{V}/\overline{T}=-2\right) $$ .

Solving the Schrödinger equation of atoms and molecules using one- and two-electron integrals only

A variational theory called free-complement (FC) ${s}_{ij}$ theory for solving the Schr\odinger equation of atoms and molecules using only one- and two-electron integrals over Slater or Gaussian functions is proposed. It is derived from the scaled Schr\odinger equation [Phys Rev. Lett. 93, 030403 (2004)] by replacing the two-electron part ${r}_{ij}$ of the scaling $g$ operator with ${s}_{ij}={r}_{ij}^{2}$, both of which solve the divergence difficulty inherent to the original Schr\odinger equation by avoiding the interelectron collisions. The ${s}_{ij}$ function can be rewritten with only one-electron functions so that when the initial wave function of the FC theory is composed of only one-electron functions, the FC wave function including ${s}_{ij}$ can be rewritten with only one-electron functions. Therefore, the variational calculations of the FC ${s}_{ij}$ theory can be performed with only one- and two-electron integrals. However, in comparison with the ${r}_{ij}$ function, the ${s}_{ij}$ function behaves less efficiently when two electrons come close to each other: the electron-electron cusp condition is not satisfied with the FC ${s}_{ij}$ theory, though the wave function has explicit ${r}_{ij}$ dependence. On the other hand, the electron-nuclear cusp condition is satisfied, even with the Gaussian functions, for the presence of the electron-nuclear function ${r}_{iA}$ in the $g$ operator. Test applications of the FC ${s}_{ij}$ theory were done to He, Li, and the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of carbon atom and to hydrogen molecule using the local-molecular orbital (MO)- and valence bond (VB)-type initial functions. We examined both Slater and Gaussian functions. As the order of the FC theory increased, the wave functions were improved and the energies approached from a few kcal/mol to less than 1 kcal/mol accuracies relative to the known exact values. Thus, with the FC ${s}_{ij}$ theory, the Schr\odinger equation was solved to less than 1 kcal/mol accuracy for all the systems examined. The costs for the ${s}_{ij}$ calculations were much smaller than those of the ${r}_{ij}$ theory. However, for the $^{5}S^{\mathrm{o}} {\mathit{sp}}^{3}$ state of the carbon atom, we observed that the convergence rate became slow when the calculation came close to $\ensuremath{\sim}1$ kcal/mol accuracy. This suggests that for the FC ${s}_{ij}$ theory, the calculations would become easier if the required accuracy is within a few kcal/mol for the absolute energy. Actually, what we need in chemical studies are mostly related to the difference energy, whose accuracy could be better than that of the absolute energy with the robust variational method. Two lines of future studies were suggested.

The study of Silver Nanoparticles in basis of Slater functions

The electronic structure of the silver nanoparticles were investigated by semi-empirical Wolfsberg – Helmholz method. It is avariant of the molecular orbitals method. Molecular orbitals are represented as a linear combination of valence atomic orbitals of the atoms of the nanoparticle. The atomic orbitals used 5s-, 5py-, 5pz- and 5px- Slateratomic orbitals of silver atoms. The exponential parameters of Slater functions were calculated and defined the analytic expression of the basis functions. The numerical values ​​of the unknown coefficients of the linear combination are found by solution of equations of molecular orbitals method. Calculations were carried out with computer program. The orbital energies, potential ionization, total electronic energy and the effective charge of atoms of silver nanoparticles were also calculated. The results indicate that the silver nanoparticles are tough, electrophile and stable dielectric material.

Кинетическая энергия электронов и потенциальная энергия атомов с открытой электронной оболочкой в базисе слейтеровских функций

Кинетическая энергия электронов и потенциальная энергия атомов с открытой электронной оболочкой в базисе слейтеровских функций

Calculation of Spectroscopic Parameters of Diatomic Molecules with an Open Electron Shell

The spectroscopic parameters of the BeH and CH molecules with an open electron shell are calculated in the basis of the Slater functions. An analytical expression for the total electron energy of molecules with an open electron shell is obtained. The total electron energy of the BeH and CH molecules was calculated by the Hartree–Fock–Roothaan (HFR) method for seven internuclear distances, including the equilibrium internuclear distance. A table of total energy values is constructed depending on the internuclear distance R. The total energy of a diatomic molecule is approximated as a third-degree polynomial in R. Applying the least squares method to energy values from the above-mentioned table, an analytical formula for the total energy of the BeH and CH molecules is obtained as a function of R. The analytical formula for E(R) has allowed the spectroscopic parameters ωe (harmonic vibrational frequency), ωexe (anharmonicity constant), Be (rotational constant), and αe (rovibrational interaction constant) to be calculated.

Calculation of the Energy of the Interelectron Interaction in Molecules in a Basis of Slater Functions

A calculation of the electron interaction energy for the CH molecule with an open electron shell is performed using a basis of Slater atomic orbitals. The molecular orbitals are represented in the form of a linear combination of the 1s, 2s, 2px, 2py, and 2pz atomic orbitals of the C atom and the 1s orbital of the H atom. As a result of solving the Hartree–Fock–Roothaan equation, numerical values of the coefficients in this linear combination have been found.

Shedding Light on the Basis Set Dependence of the Minnesota Functionals: Differences Between Plane Waves, Slater Functions, and Gaussians

The Minnesota family of exchange-correlation (xc) functionals are among the most popular, accurate, and abundantly used functionals available to date. However, their use in plane-wave based first-principles MD has been limited by their sparse availability. Here, we present an implementation of the M05, M06, and M11 families of xc functionals within a plane wave/pseudopotential framework allowing for a comprehensive analysis of their basis set dependence. While it has been reported that in Gaussian bases some members of the Minnesota family only converge slowly to the basis set limit, (1) we show that converged energies can be conveniently obtained from plane waves if sufficiently dense integration meshes are used. Based on the HC7/11 database, we assess the influence of basis set type on the calculation of reaction enthalpies and show that complete basis set values obtained in plane waves may occasionally differ notably from their atom-centered counterparts. We provide an analysis of the origin of these differences and discuss implications on practical usage.

Correlation energies for many-electron atoms with explicitly correlated Slater functions

In this work we propose a composite method for accurate calculation of the energies of many-electron atoms. The dominant contribution to the energy (pair energies) are calculated by using explicitly correlated factorizable coupled cluster theory. Instead of the usual Gaussian-type geminals for the expansion of the pair functions, we employ a two-electron Hylleraas basis set and discuss the advantages of the latter approach, e.g., a small number of nonlinear parameters that need to be optimized. The remaining contributions to the energy are calculated within the algebraic approximation by using large one-electron basis sets composed of Slater-type orbitals. The method is tested for the beryllium atom where an accuracy better than $1\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ is obtained. We discuss in detail possible sources of the error and estimate the uncertainty in each energy component. Finally, we consider possible strategies to improve the accuracy of the method by 1--2 orders of magnitude and apply it to larger atoms.

Numerical calculation of the Spherical Bessel Transform from Gaussian quadrature in the complex-plane

A numerical method for the calculation of spherical Bessel transforms via Gaussian quadrature in the complex plane is presented. The method is evaluated by transforming Slater and Gaussian-type spherical coordinate basis functions, used in electronic structure calculations, from position space to momentum space. The feasibility and efficiency of the method is explored for different regions of momenta and different orders of the spherical Bessel transform. The results illustrate that in general the method performs very well in the large p regions when applied to the Slater-type functions. On the other hand, a parameter has to be introduced for the application to Gaussian-type functions in order to avoid cancellation. In this case, the method performs well but accuracy is lost at larger p. Use of the parameter in application to Slater functions yields accurate results for all regions of p.

Simplified box orbitals for molecules containing atoms beyond Ar

ABSTRACTSimplified Box Orbitals (SBOs) are a kind of spatially restricted basis functions with a similar use to Slater functions, but fulfilling an exact version of the zero-differential overlap approximation. These functions also allow for a drastic reduction in the number of bielectronic integrals when dealing with huge systems, and can be adapted to study confined systems such as molecules in solution. In previous studies, the necessary SBOs parameters to be used for different elements were defined. However, the accuracy of those basis functions decreases with the atomic number of the atoms under study, and therefore their use was discouraged beyond the Ar atom. In the present study, we verify that slightly increasing the terms of SBOs for a better definition, is enough to correctly handling atoms beyond Ar. This, together with other improvements exposed in this work, allowed obtaining accurate SBOs for K–Kr atoms. To make possible the use of SBOs in standard quantum chemistry calculation software, Gaussian expansions to the proposed basis functions–were achieved. Then, simple formulas for directly obtaining those expansions were deduced. Finally, the results of an SZ basis set of the proposed SBOs are analysed and compared with a similar STO basis set.

Self-Friction Power Series and Their Use for Construction of One-Range Addition Theorems of Noninteger Slater Functions and Coulomb–Yukawa Like Potentials

Using complete orthogonal \( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-self-friction polynomials (\( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-SFPs) in standard convention introduced by the author, the formulas for transformation of noninteger type functions to the power series are presented, where \( p_{l}^{*} = 2l + 2 - \alpha^{*} \) and α * is the integer (\( \alpha^{*} = \alpha ,\; - \infty < \alpha \le 2 \)) or noninteger (\( \alpha^{*} \ne \alpha ,\; - \infty < \alpha^{*} < 3 \)) SF quantum number. With the help of these power series, the one-center one-range addition theorems for χ-Slater type orbitals and V-Coulomb–Yukawa like potentials with noninteger indices (χ-NISTOs and V-NICYPs) are established. As an application, the atomic nuclear attraction integrals of χ-NISTOs and V-NICYPs are investigated.

Study of chemical bonding in the interhalogen complexes based on density functional theory

The density functional theory analysis was used for a number XYL complexes (XY is a dihalogen molecule and L is a Lewis base), formed between molecules I2, ICl, IBr and pyridine. The calculated geometrical parameters, IR spectra and nuclear quadrupole interaction constants of iodine are consistent with the data of microwave spectroscopy and nuclear quadrupole resonance. The good correlation between the experimental and calculated binding energies of the inner electrons of iodine, chlorine and nitrogen atoms were found with the calculation using both Gaussian and Slater functions. The comparison of experimental and calculated changes in the electron density on the atoms upon complex formation suggested the choice of scheme for calculating the effective charge on the atoms, which allow us to interpret the experimental spectra. It is shown that the use of both calculated schemes allows us to predict the enthalpy of complex formation in close agreement with the experimental values. The energy analysis shows that in the complexes the electrostatic binding energy dominates that of covalent binding.

The Study of Silver Nanoparticles in Basis of Slater Functions

The Study of Silver Nanoparticles in Basis of Slater Functions

Use of Auxiliary Functions for Construction of Series Expansion Relations of Noninteger Slater Functions in Standard Convention

AbstractUsing complete orthonormal sets of ψ (α*) ‐self‐frictional exponential type orbitals (ψ (α*) ‐SFETOs) and Qq‐noninteger auxiliary functions (Qq‐NIAFs) introduced by the author, the combined formulas for the one‐ and two‐center one‐range addition theorems of χ‐noninteger Slater type orbitals (χ‐NISTOs) with arbitrary values of distances between centers Rab (for Rab = 0 and Rab ≠ 0), and of integer (for α* = α, –∞ &lt; α ≤ 2) and noninteger (for α* ≠ α, –∞ &lt; α* &lt; 3) self‐frictional (SF) quantum numbers are suggested. The presented relations for the one‐range addition theorems can be useful tools especially in the electronic structure studies of atoms, molecules and solids when χ‐NISTOs are employed as basis functions.

Unified Treatment of One-Range Addition Theorems for Complete Orthonormal Sets of Generalized Exponential-Type Orbitals and Noninteger n Slater Functions

Abstract By the use of L(pl*)-generalized Laguerre polynomials (L(pl*)-GLPs) introduced by the author, the combined formulas for the one- and two-center one-range addition theorems of complete orthonormal sets of φ (pl*)-generalized exponential-type orbitals (φ (pl*)-GETOs) and χ-noninteger n Slater type orbitals (χ-NISTOs) in terms of χ-integer STOs (χ-ISTOs) are suggested, where pl* = 2l + 2 − α* and α* are the integer (α* = α, −∞ &amp;lt; α ≤ 2) or noninteger (α* ≠ α, −∞ &amp;lt; α* &amp;lt; 3) self-frictional quantum numbers. The series expansion coefficients of these theorems are expressed through the overlap integrals over χ-NISTOs. As an application, the Combined Hartree–Fock–Roothaan (CHFR) total energy values for the ground states of some atoms obtained in the minimal basis set approximation are presented.

One‐range Addition Theorems for Complete Sets of Modified Exponential Type Orbitals and Noninteger n Slater Functions in Standard Convention

AbstractUsing the L ‐generalized Laguerre polynomials L ‐GLPs) and φ ‐generalized exponential type orbitals φ ‐GETOs) introduced by the author in standard convention, the one‐ and two‐center onerange addition theorems are established for the complete sets of Ψ(α*) ‐modified exponential type orbitals (Ψ(α*) ‐METOs) and noninteger n χ‐Slater type orbitals (χ‐NISTOs), where pl* = 2l + 2 ‐ α* and α* is the integer (α* = α, −∞ &lt; α ≤2) or noninteger (α* ≠ α, −∞ &lt; α* &lt; 3) self‐frictional quantum number. It should be noted that the origin of the L ‐GLPs, φ ‐GETOs and Ψ(α*) ‐METOs, therefore, of the one‐range addition theorems presented in this work is the Lorentz damping or self‐frictional field produced by the particle itself.

The Study of Gold Nanoparticles in basis of Slater Functions

The electronic structure of the gold nanoparticles were investigated by semi-empirical Wolfsberg - Helmholz method. This is a variant of the molecular orbitals method. Molecular orbitals are represented as a linear combination of valence atomic orbitals of the atoms of the nanoparticle. As the atomic orbitals used 6s-, 6p -, 6p - and 6p - y z x Slater atomic orbitals of gold atoms. The exponential parameters of Slater functions were calculated and defined the analytic expression of the basis functions. The numerical values of the unknown coefficients of the linear combination are found by solution of equations of molecular orbitals method. Calculations were carried out with own computer program. The orbital energies, potential ionization, the total electronic energy and effective charge of atoms of gold nanoparticles were calculated. The results indicate that the gold nanoparticles are soft, electrophile and conductive material.

Improved partition–expansion of two‐center distributions involving slater functions

The calculation of the electronic structure of large systems is facilitated by the substitution of the two-center distributions by their projections on auxiliary basis sets of one-center functions. An alternative is the partition-expansion method in which one first decides what part of the distribution is assigned to each center, and next expands each part in spherical harmonics times radial factors. The method is exact, requires neither auxiliary basis sets nor projections, and can be applied to Gaussian and Slater basis sets. Two improvements in the partition-expansion method for Slater functions are reported: general expressions valid for arbitrary quantum numbers are derived and the efficiency of the procedure is increased giving analytical solutions to integrals previously computed by numerical quadrature. The efficiency of the new version is assessed in several molecules and the advantages over the projection methods are pointed out.

On the fast evaluation of three-center nuclear attraction integrals using one-range addition theorems for Slater functions

Using one-range addition theorems, the three-center nuclear attraction integrals are expressed through the overlap integrals containing χ - and χα -Slater-type orbitals (χ -STOs and χα -STOs), where - ∞ < α ⩽ 2 and . For the fast calculation, the partial summation is utilized for some indices of series expansion relations which correspond to progressively increasing upper limits. The binomial coefficients are stored in the memory of the computer. The convergence and accuracy of series are tested by calculating concrete cases. The best values are obtained for α = 0 .