arXiv subjects

P. E. Hoggan

Publications and source records attributed to P. E. Hoggan.

7 recordsLinked to original sources

arXiv2025

New Atomic Orbital Functions.Complete and Orthonormal Sets of ETOs with Non-integer Quantum Numbers.Results for He-like atoms

The Hartree-Fock-Rothaan equations are solved for He-like ions using the iterative self-consistent method. New complete and orthonormal sets of exponential-type orbitals are employed as the basis. These orbitals satisfy the orthonormality condition for quantum numbers with fractional power. They are solutions of a Schrodinger-like differential equation derived by the authors. In a recent study conducted for the calculation of the hydrogen atom energy levels, it has been demonstrated that the fractional formalism of the principal and the angular momentum quantum numbers converges to the 1s level of the ground state energy of hydrogen atom, obtained from the solution of the standard Schrodinger equation. This study examines the effect of fractional values of the quantum numbers for two-electron systems, which is the simplest system with electron correlation effects.

quant-ph↗

arXiv2023

Solution of the Dirac-Coulomb equation using the Rayleigh-Ritz method. Results for He-like atoms

The Dirac-Coulomb equation for helium-like ions is solved using the iterative self-consistent field method, with Slater-type spinor orbitals as the basis. These orbitals inherently satisfy the kinetic-balance condition due to their coupling for both large- and small-components. The $1/r_{12}$ Coulomb interaction is treated without constraints. Computations are carried out for total energies of atoms with nuclear charges up to $Z \leq 80$ using both minimal and extended basis sets. Variationally optimal values for orbital parameters are determined through the Rayleigh-Ritz variational principle. No manifestations related to the Brown-Ravenhall disease are found.

quant-ph↗

arXiv2022

I. Complete and orthonormal sets of exponential-type orbitals with noninteger principal quantum numbers

The definition for the Slater-type orbitals is generalized. Transformation between an orthonormal basis function and the Slater-type orbital with non-integer principal quantum numbers is investigated. Analytical expressions for the linear combination coefficients are derived. In order to test the accuracy of the formulas, the numerical Gram-Schmidt procedure is performed for the non-integer Slater-type orbitals. A closed form expression for the orthogonalized Slater-type orbitals is achieved. It is used to generalize complete orthonormal sets of exponential-type orbitals obtained by Guseinov in [Int. J. Quant. Chem. 90, 114 (2002)] to non-integer values of principal quantum numbers. Riemann-Liouville type fractional calculus operators are considered to be use in atomic and molecular physics. It is shown that the relativistic molecular auxiliary functions and their analytical solutions for positive real values of parameters on arbitrary range are the natural Riemann-Liouville type fractional operators.

physics.chem-ph↗

arXiv2020

Analytical evaluation of relativistic molecular integrals. III. Computation and results for molecular auxiliary functions

This work describes the fully analytical method for calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper [Phys. Rev. E 91, 023303 (2015)]. The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

physics.chem-ph↗

arXiv2017

Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

The auxiliary functions provide efficient computation of integrals arising at the self-consistent field (SCF) level for molecules using Slater-type bases. This applies both in relativistic and non-relativistic electronic structure theory. The relativistic molecular auxiliary functions derived in our previous paper [Phys. Rev. E 91, 023303 (2015)] are discussed here in detail. Two solution methods are proposed in the present study. The ill-conditioned binomial series representation formulae first, are replaced by convergent series representation for incomplete beta functions then, they are improved by inserting an extra parameter used to extend the domain of convergence. Highly accurate results can be achieved for integrals by the procedures discussed in the present study which also places no restrictions on quantum numbers in all ranges of orbital parameters. The difficulty of obtaining analytical relations associated with using non-integer Slater-type orbitals which are non-analytic in the sense of complex analysis at r=0 is therefore, eliminated.

physics.chem-ph↗

arXiv2015

Benchmark values for molecular three-center integrals arising in the Dirac equation

The authors in their previous papers obtained compact, arbitrarily accurate expressions for two-center one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In this present study, the accuracy limits of given expressions is examined for three-center nuclear attraction integrals, which are the first integral set do not have analytically closed form relations. They are expressed through new molecular auxiliary functions obtained via Neumann expansion of Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrarily values of orbital parameters, quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, Fourier transformation method, have been performed in order to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This is the first attempts to study the three-center integrals without any restrictions on quantum numbers and in all ranges of orbital parameters.

math-ph↗

arXiv2014

The use of Slater-type spinor orbitals in algebraic solution of two-center Dirac equation

The use of Slater-type spinor orbitals in algebraic solution of the Dirac equation is investigated. The one- and two-center integrals constitute the matrix elements arising in generalized eigenvalue equation for one-electron atoms and molecules are evaluated over Slater-type spinor orbitals via ellipsoidal coordinates. These integrals are calculated through numerical global-adaptive method with Gauss-Kronrod numerical integration extension. The calculations are performed for electronic structure of ground and excited states of one-electron atoms and diatomic molecules. The screening constants are allowed to be variationally optimum values for given nuclear separation. The obtained results are compaired with the results those found in the literature. The procedures discussed in this work are capable of yielding highly accurate relativistic two-center one-electron integrals for all ranges of orbital parameters. Besides provides an efficient way to overcome the problems that arise in relativistic calculations.

physics.atom-ph↗