Comment on Hartree Fock Roothaan Calculations for Ground States of Some Atoms Using Minimal Basis Sets of Integer and Noninteger n STOs

Recently published formulas for the one-center integrals arising in atomic Hartree-Fock-Roothaan (HFR) calculations with noninteger n STOs (S.Gumus, T. Ozdogan, Chin. J. Chem., 22 (2004) 1262) are critically analyzed. The purpose of this note is to point out that the presented in these work relations for the integer n -nuclear attraction and kinetic energy integrals which are available in the literature (C.C.J. Roothaan, J. Chem. Phys., 19 (1951) 1445) can not be used for noninteger STOs. In addition, the formulas for two-electron integrals can be obtained from the published in the literature (T. Koga, K. Kanayama, Chem.Phys.Let., 266(1997)123; I.I. Guseinov, B.A. Mamedov, Theor. Chem. Acc.,108 (2002) 21) relations by changing the indices. It should be noted that the accuracy of computer values for ground states energy of some closed and open shell atoms in the case of noninteger n STOs is not guaranteed since the calculations were performed by the use of integer n -one electron integrals. It is argued that the paper sheds no new light on the subject and that it is altogether misleading.Comment: 2 page

Response to Comment on "Combined open shell Hartree-Fock theory of atomic-molecular and nuclear systems" [I.I.Guseinov, J. Math. Chem., 42 (2007) 177] by B. N. Plakhutin and E. R. Davidson

This article is a thorough critique to the Plakhutin-Davidson's comments made to our paper published in the recent year. A detailed critical examination of the arguments that led to the suggested comments by Plakhutin and Davidson reveals some serious flaws. It is demonstrated that the principle of the indistinguishability of identical particles is not taking into account in Roothaan's open shell theory. This principle leads to the fact that the orbital-dependent energy functional and, therefore, the Hartree-Fock and Hartree-Fock-Roothaan equations for open shell systems presented by Roothaan and others are not, in general, invariant under unitary transformation of the combined closed-open shells orbitals. From a mathematical point of view this statement is fundamentally flawless. It is shown that the Plakhutin-Davidson's personal views about our assumptions concerning the insufficiencies of classic Roothaan's open-shell theory are undisputedly wrong.Comment: 3 page

New development in theory of Laguerre polynomials

The new complete orthonormal sets of -Laguerre type polynomials (-LTP,) are suggested. Using Schr\"odinger equation for complete orthonormal sets of -exponential type orbitals (-ETO) introduced by the author, it is shown that the origin of these polynomials is the centrally symmetric potential which contains the core attraction potential and the quantum frictional potential of the field produced by the particle itself. The quantum frictional forces are the analog of radiation damping or frictional forces suggested by Lorentz in classical electrodynamics. The new -LTP are complete without the inclusion of the continuum states of hydrogen like atoms. It is shown that the nonstandard and standard conventions of -LTP and their weight functions are the same. As an application, the sets of infinite expansion formulas in terms of -LTP and L-Generalized Laguerre polynomials (L-GLP) for atomic nuclear attraction integrals of Slater type orbitals (STO) and Coulomb-Yukawa like correlated interaction potentials (CIP) with integer and noninteger indices are obtained. The arrange and rearranged power series of a general power function are also investigated. The convergence of these series is tested by calculating concrete cases for arbitrary values of parameters of orbitals and power function.Comment: 10 pages, 2 figure

New complete orthonormal sets of exponential type orbitals in standard convention and their origin

In standard convention, the new complete orthonrmal sets of exponential type orbitals (ETOs) are introduced as functions of the complex or real spherical harmonics and modified and -generalized Laguerre polynomials (MPLs and GLPs), where, and is the noninteger or integer (for) frictional quantum number. It is shown that the origin of the ETOs, MLPs and GLPs is the self-frictional quantum forces which are analog of radiation damping or self-frictional forces introduced by Lorentz in classical electrodynamics. The relations for the quantum self-frictional potentials in terms of ETOs, MLPs and GLPs, respectively, are established. We note that, in the case of disappearing frictional forces, the ETOs are reduces to the oringers wave functions for the hydrogen-like atoms in standard convention and, therefore, become the noncomplete.Comment: 6 page

Nonrelativistic, Quasirelativistic and Relativistic Sets of Wave Functions, and Slater Orbitals of Particles with Arbitrary Spin

Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the -component relativistic tensor wave functions and tensor Slater orbitals in coordinate, momentum and four-dimensional spaces are derived, where. The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic tensor Slater orbitals with the same screening constants in coordinate space are also derived.Comment: 9 pages, 3 table

Comment on Unified treatment for the evaluation of arbitrary multielectron multicenter molecular integrals over Slater-type orbitals with noninteger principal quantum numbers

Ozdogan (Int. J. Quantum Chem., 92 (2003) 419) published formulas for evaluating the multielectron multicenter molecular integrals over Slater-type orbitals (STOs). It is demonstrated that the formulas presented in this work are not original and they can easily be derived by means of a simple algebra from the relationship of our published papers (I.I.Guseinov, J.Mol.Struct.(Theochem), 417(1997)117; J.Mol.Struct.(Theochem), 593 (2002) 65; I.I.Guseinov,B.A.Mamedov,F.Oner,S.Huseyin, J.Mol.Struct.(Theochem), 545(2001)265; I.I.Guseinov,B.A.Mamedov, J.Mol.Model., 8(2002)272; Theor.Chem.Acc., 108(2002)21).Comment: 5 page

Combined Theory of Basis Sets of Spinors for Particles with Arbitrary Spin in Position, Momentum and Four-Dimensional Spaces

The 2(2s+1)-component relativistic basis spinors for the arbitrary spin particles are established in position, momentum and four-dimensional spaces, where s=0,1 / 2,1, 3 / 2, 2, ... . These spinors for integral- and half-integral spins are reduced to the independent sets of one- and twocomponent spinors, respectively. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of generalized Dirac equation of arbitrary spin particles introduced by the author when the orthogonal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.Comment: 28 pages; 3 tables. arXiv admin note: substantial text overlap with arXiv:1008.526

Comment on Formulas and numerical table for the radial part of overlap integrals with the same screening parametres of Slater-type orbitals

Recently published formulas for the calculation of radial part of overlap integrals (E.Oztekin, M.Yavuz, S.Atalay, Theor.Cmem.Acc., (2001), 106, 264) are critically analyzed. It is demonstrated that the presented in this work formulas are not original and they can be easily be obtained from the formulas already established in our papers.Comment: 3 page

New Complete Orthonormal Basis Sets of Relativistic Exponential Type Spinor Orbitals and Slater Spinor Functions of Particles with Arbitrary Half-Integral Spin in Position, Momentum and Four-Dimensional Spaces

Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are reduced to the independent sets of two-component spinors defined as a product of complete orthonormal sets of radial parts of orbitals and twocomponent spinor type tensor spherical harmonics. We notice that the new idea presented in this work is the unified treatment of half-integral spin and scalar particles in position, momentum and four-dimensional spaces. Relations presented in this study can be useful in the linear combination of atomic orbitals approximation for the solution of different problems arising in the relativistic quantum mechanics when the orthonormal basis sets of relativistic exponential type spinor wave functions and Slater type spinor orbitals in position, momentum and four -dimensional spaces are employed.Comment: 27 pages, 4 figure

Unsymmetrical and symmetrical one-range addition theorems for Slater type orbitals and Coulomb-Yukawa like correlated interaction potentials of integer and noninteger indices

Using one-center expansion relations for the Slater type orbitals (STOs) of noninteger principal quantum numbers in terms of integer n STOs derived in this study with the help of - exponential type orbitals (-ETOs, the general formulas are established for the unsymmetrical and symmetrical one-range addition theorems of STOs and Coulomb-Yukawa like correlated interaction potentials (CIPs) with integer and noninteger indices. The final results are especially useful for computations of arbitrary multicenter multielectron integrals over STOs that arise in the Hartree-Fock-Roothaan (HFR) approximation and also in the correlated methods which play a significant role in theory and application to quantum mechanics of atoms, molecules, and solids.Comment: 5 page