arXiv subjects

A. Bagci

Publications and source records attributed to A. Bagci.

8 recordsLinked to original sources

arXiv2025

Multidimensional derivative-free optimization. A case study on minimization of Hartree-Fock-Roothaan energy functionals

This study presents an evaluation of derivative-free optimization algorithms for the direct minimization of Hartree-Fock-Roothaan energy functionals involving nonlinear orbital parameters and quantum numbers with noninteger order. The analysis focuses on atomic calculations employing noninteger Slater-type orbitals. Analytic derivatives of the energy functional are not readily available for these orbitals. Four methods are investigated under identical numerical conditions: Powell's conjugate-direction method, the Nelder-Mead simplex algorithm, coordinate-based pattern search, and a model-based algorithm utilizing radial basis functions for surrogate-model construction. Performance benchmarking is first performed using the Powell singular function, a well-established test case exhibiting challenging properties including Hessian singularity at the global minimum. The algorithms are then applied to Hartree-Fock-Roothaan self-consistent-field energy functionals, which define a highly non-convex optimization landscape due to the nonlinear coupling of orbital parameters. Illustrative examples are provided for closed$-$shell atomic configurations, specifically the He, Be isoelectronic series, with calculations performed for energy functionals involving up to eight nonlinear parameters. This work presents the first systematic investigation of derivative-free optimization methods for Hartree-Fock$-$Roothaan energy minimization with non-integer Slater orbitals.

quant-ph↗

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↗

arXiv2022

An efficient approximation for accelerating convergence of the numerical power series. Results for the 1D Schrödinger's equation

The numerical matrix Numerov algorithm is used to solve the stationary Schrödinger equation for central Coulomb potentials. An efficient approximation for accelerating the convergence is proposed. The Numerov method is error-prone if the magnitude of grid$-$size is not chosen properly. A number of rules so far, have been devised. The effectiveness of these rules decrease for more complicated equations. Efficiency of the technique used for accelerating the convergence is tested by allowing the grid-sizes to have variationally optimum values. The method presented in this study eliminates the increased margin of error while calculating the excited states. The results obtained for energy eigenvalues are compared with the literature. It is observed that, once the values of grid-sizes for hydrogen energy eigenvalues are obtained, they can simply be determined for the hydrogen iso-electronic series as, $h_{\varepsilon}(Z)=h_{\varepsilon}(1)/Z$.

quant-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↗

arXiv2019

On the use of Slater-type spinor orbitals in Dirac-Hartree-Fock method. Results for hydrogen-like atoms with super$-$critical nuclear charge

This work presents the formalism for evaluating molecular SCF equations, as adapted to four$-$component Dirac spinors, which in turn reduce to Slater$-$type orbitals with non$-$integer principal quantum numbers in the non$-$relativistic limit. The "catastrophe" which emerges for a charge numbers $Z>137$, in solving the Dirac equation with a potential corresponding to a point$-$charge is avoided through using Slater$-$type spinor orbitals in the algebraic approximation. It is observed that, ground$-$state energy of hydrogen$-$like atoms reaches the negative$-$energy continuum $\left(-mc^2 \right)$ while critical nuclear charge $Z_{c}$, about $Z_{c}=160$. The difficulty associated with finding relations for molecular integrals over Slater$-$type spinors which are not$-$analytic in the sense of complex analysis at $r = 0$, is eliminated. Unique numerical accuracy is provided by solving the molecular integrals through Laplace expansion of Coulomb interaction and prolate spheroidal coordinates. New convergent series representation formulae are derived. The technique draws on previous work by the author and the general formalism is presented in this paper.

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↗