New method for the direct calculation of electron density in many-electron systems. I. Application to closed-shell atoms

Analytical properties of hydrogen-like atomic orbitals (HAO ) that are used in the MOLCAO approach to the quantum theory of molecules have been studied. Addition and expansion theorems for HAO have been proved, both in coordinate and momentum representations. A close relation has been established between HAO and the reduced Bessel functions of half-integer indices. New methods are suggested to calculate integrals for atomic and molecular form factors, and multicenter integrals, for the HAO basis in the MO LCAO theory.     

Bessel-zernike discrete variable representation basis

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermore, the Zernike polynomials may also be used to efficiently evaluate the Hankel transform for rapidly decaying functions or functions with finite support.     

Incompletely symmetric non-bloch electron states in perfect cubic crystals. I. Eigenproblem and its solution

In the studies on impure metals where the perturbation potential due to the impurity has symmetry of the point group of the crystal, subdivision of the electron density of the pure metal into irreducible representations of the point group is of significance. A method is presented for the calculation of wave functions of the perfect cubic crystal which transform according to the incompletely symmetric irreducible representations of the point group. The functions were evaluated in the LCAO approximation using s-type AOS for the face centered cubic lattice. These are in the form of standing waves, and their coefficient functions are linear combinations of the products of the cubic harmonics of a suitable type and the spherical Bessel functions. Properties of the solutions obtained were examined. Numerical calculations were made for four irreducible representations of the point group.     

A New Method of Calculating Multicenter Matrix Elements in MO LCAO Theory Using an Exponential Type Basis

Using integral representation of the product of reduced Bessel functions (RBF) specified on different centers and a new generalized integral identity for RBF one can prove that the 4-center integral of Coulomb repulsion in an exponential type AO basis may be expressed as a three-dimensional integral over the volume of a cube with an edge 1. A new method of calculating the multicenter matrix elements of quantum chemistry in an exponential AO basis is suggested based on this representation. Numerical calculations of a number of multicenter integrals using this algorithm illustrate the efficiency of the method.     

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrödinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrödinger equation and of coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known methods.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. II

A new method for calculating crystal orbitals in the Hartree-Fock-Slater approximation is proposed. The method makes use of x-ray crystallographic measurements of the deformation density, and uses transferable integrals to treat the neutral–atom potentials. Methods for evaluating matrix elements of neutral-atom potentials are discussed in detail, and in this connection, expansions of displaced Slater-type orbitals in terms of modified spherical Bessel functions and Legendre polynomials are presented. Tables of transferable integrals (moments of the neutral-atom potentials) are given for all the elements up to Z = 36, and tables of Fourier transforms of the neutral-atom potentials are also presented.     

Exact eigenvalues for the Coulomb potential with cut-off

The solution of the quantum-mechanical one-particle problem involving a Coulomb potential with a cut-off contains Whittaker functions, difficult to evaluate individually. However, the energy expression involves a ratio of two contiguous Whittaker functions, and the entire ratio is easily expressed as a convergent continued fraction. Representative energy eigenvalues are calculated, correcting the errors due to an earlier Bessel function approximation by Wannier.     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.     

Theoretical and Experimental Study of Radial Concentration Profiles Originating from a Non Punctual Injection Source

Abstract

Radial concentration profiles originating from non delta plug flow injection functions can be theoretically accounted for in terms of a Bessel function series. Results related to axial injections of various diameters relative to the column I.D. are described. They are in food agreement with the theory.     

The W algorithm and the D transformation for the numerical evaluation of three‐center nuclear attraction integrals

Three‐center nuclear attraction integrals over exponential‐type functions are required for ab initio molecular structure calculations and density functional theory (DFT). These integrals occur in many millions of terms, even for small molecules, and they require rapid and accurate numerical evaluation. The use of a basis set of B functions to represent atomic orbitals, combined with the Fourier transform method, led to the development of analytic expressions for these molecular integrals. Unfortunately, the numerical evaluation of the analytic expressions obtained turned out to be extremely difficult due to the presence of two‐dimensional integral representations, involving spherical Bessel integral functions. % The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of these spherical Bessel integrals. This algorithm, which is based on the nonlinear D transformation and the W algorithm of Sidi, can be computed recursively, allowing the control of the degree of accuracy. Numerical analysis tests were performed to further improve the efficiency of our algorithm. The numerical results section demonstrates the efficiency of this new algorithm for the numerical evaluation of three‐center nuclear attraction integrals. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

CHEMISORPTION ON ELECTRIFIED SUBSTRATES

The electronic structure of an electrified finite chain of atoms is investigated by means of a recursive Green-function (RGF) method, whose iterative nature results in a continued-fraction form of the RGF, which can be written as a ratio of Bessel functions. The chemisorption process is described by the self-consistent Anderson-Newns model. Using the Dyson equation, the adatom RGF is obtained from which the chemisorption functions are extracted. These provide access to the chemisorption energy and adatom charge transfer in terms of the system's parameters. Results are discussed in terms of the Wannier-Stark effect on magnetic versus non-magnetic chemisorption.     

Electrophoretic mobility of a charged cylindrical colloidal particle covered with an ion-penetrable uncharged polymer layer

Expressions are derived for the electrophoretic mobility of a cylindrical charged colloidal particle carrying a low zeta potential covered with an ion-penetrable uncharged polymer layer in an electrolyte solution. These expressions involve numerical integration of modified Bessel functions but are easily calculable with Mathematica. The obtained mobility expressions are a modification of Henry's mobility formula for a cylindrical particle taking into account the presence of the uncharged polymer layer.     

Yield stress analysis of 1D calcium and titanium precipitate-based giant electrorheological fluids

1D calcium and titanium composite nanorods synthesized were applied as electrorheological (ER) active materials with extremely high static yield stresses, i.e., giant ER effect. The yield stress of this giant ER fluid was analyzed using a new universal yield stress scaling equation in the form of the modified Bessel functions with two different limiting behaviors in a low and high electric field region. The universal yield stress equation collapsed the yield stress data onto a single curve.     

Wannier-Stark effect on surface states

A recursive Green-function (RGF) technique is used to study the surface states of an electrified solid. The concatenating nature of the method leads to a continued-fraction form of the RGF, which can be expressed as a ratio of Bessel functions. The surface density of states (SDOS), derived from the RGF, reveals a quasi-Stark-ladder distribution of the energy levels at the surface atom. The dependence of the SDOS on the applied field and surface perturbation parameters is discussed.     

Evaluation of bielectronic integrals for 1s Slater orbitals by using averages

A new approach for evaluating the four‐center bielectronic integrals (12|34), involving 1s Slater‐type orbitals, is presented. The method uses the multiplication theorem for Bessel functions. The bielectronic integral is expressed in terms of a finite sum of functions, and a scaling parameter is introduced. In the present work, the scaling parameter used is an average. The results show that the first term in the sum is always the principal contribution, and the remainder has a corrective character. The whole scheme and its numerical trend are understood on the basis of a theorem recently proved. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

On Kryachko's formula for the leaky aquifer function

Kryachko recently published in this journal a formula in which the incomplete Bessel function known as the leaky aquifer function is expanded in Legendre functions of the second kind. While Kryachko's analysis is ingenious, creative, and sound, it unfortunately contains errors that make its key result incorrect. The present contribution corrects and to some degree generalizes Kryachko's formula. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 332–334, 2001     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

Four‐center integrals for Gaussian and exponential functions

The shift operator technique is used for deriving, in a unified manner, the master formulas for the four‐center repulsion integrals involving Gaussian (GTO), Slater (STO), and Bessel (BTO) basis functions. Moreover, for the two classes of exponential‐type functions (ETO), i.e., STO and BTO, we give the expressions corresponding to both the Gauss and Fourier transforms. From the comparison of the master formulas of GTO and ETO, we conclude that ETO can perform more efficiently than GTO, and we remark the points where the effort must be focused to carry out this possibility. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 16–28, 2001     

A new method for approximate solution of one-dimensional Schrödinger equations

Summary A general method for approximate solution of one-dimensional Schrödinger equations with a wide range of square-integrable potentials is described. The potential is expanded in terms of either Jacobi or Bessel functions of argument exp(-r). This allows the Schrödinger equation to be solved by the Frobenius method. In the absence of super-computing power the input requirement of a large number of significant figures was handled by an algebraic computing package, for illustrative purposes. A sum of Gaussian wells and a Morse potential are treated as examples.     

Potential distribution in the solution interface of a scanning tunneling microscope

The linearized Poisson—Boltzmann equation is solved in the region between a sphere and a plane, which is modelling the electrolyte solution interface between the tip and the substrate in a scanning tunneling microscope. A series expansion in modified Bessel functions and Legendre polynomials, which are solutions to the linearized Poisson—Boltzmann equation, is used to fit the boundary conditions. Another numerical method of finite difference is also used with the domain transformed into bispherical coordinates. Results for cases of different potential values on the boundary surfaces and different distances of the sphere from the plane are presented.