On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

2004 ◽  
Vol 42 (5) ◽  
pp. 753-756 ◽  
Author(s):  
I.i. Guseinov ◽  
B.a. Mamedov
Keyword(s):  
Overlap Integrals ◽  
Quantum Numbers ◽  
Noninteger Principal Quantum Numbers

Unified treatment of overlap integrals with integer and noninteger n Slater-type orbitals using translational and rotational transformations for spherical harmonics

2004 ◽  
Vol 82 (3) ◽  
pp. 205-211 ◽  
Author(s):  
I I Guseinov ◽  
B A Mamedov
Keyword(s):  
Spherical Harmonics ◽  
Large Integer ◽  
Series Expansions ◽  
Slater Type Orbitals ◽  
Overlap Integrals ◽  
Addition Theorems ◽  
Slater Type ◽  
Quantum Numbers ◽  
Noninteger Principal Quantum Numbers ◽  
Analytical Relations

A unified treatment of two-center overlap integrals over Slater-type orbitals (STO) with integer and noninteger values of the principal quantum numbers is described. Using translation and rotation formulas for spherical harmonics, the overlap integrals with integer and noninteger n Slater-type orbitals are expressed through the basic overlap integrals and spherical harmonics. The basic overlap integrals are calculated using auxiliary functions Aσ and Bk. The analytical relations obtained in this work are especially useful for the calculation of overlap integrals for large integer and noninteger principal quantum numbers. The formulas established in this study for overlap integrals can be used for the construction of series expansions based on addition theorems. PACS Nos.: 31.15.–p, 31.20.Ej

COMPUTATION OF THREE-CENTER OVERLAP INTEGRALS OVER NONINTEGER n SLATER TYPE ORBITALS USING Ψα-ETO

2007 ◽  
Vol 06 (03) ◽  
pp. 641-646 ◽  
Author(s):  
I. I. GUSEINOV ◽  
B. A. MAMEDOV
Keyword(s):  
Exponential Type ◽  
Slater Type Orbitals ◽  
Overlap Integrals ◽  
Slater Type ◽  
Interaction Potentials ◽  
Quantum Numbers ◽  
Noninteger Principal Quantum Numbers ◽  
Multicenter Multielectron Integrals ◽  
Exponential Type Orbitals

Using complete orthonormal sets of Ψα-exponential type orbitals (Ψα- ETO , α = 1, 0, -1, -2,…), the three-center overlap integrals over noninteger n STO (NISTO) appearing in the evaluation of multicenter–multielectron integrals of central and noncentral interaction potentials are calculated. The final results are expressed in terms of one- or two-center overlap integrals between NISTO and integer n STO (ISTO). The formulas obtained are valid for arbitrary noninteger principal quantum numbers, screening parameters, and location of NSTO.

scholarly journals UNSYMMETRICAL AND SYMMETRICAL ONE-RANGE ADDITION THEOREMS FOR SLATER TYPE ORBITALS AND COULOMB–YUKAWA-LIKE CORRELATED INTERACTION POTENTIALS OF INTEGER AND NONINTEGER INDICES

2008 ◽  
Vol 07 (02) ◽  
pp. 257-262 ◽  
Author(s):  
I. I. GUSEINOV
Keyword(s):  
Basis Functions ◽  
Slater Type Orbitals ◽  
Addition Theorems ◽  
Slater Type ◽  
Interaction Potentials ◽  
Hartree Fock ◽  
Quantum Numbers ◽  
Noninteger Principal Quantum Numbers ◽  
Multicenter Multielectron Integrals ◽  
Exponential Type Orbitals

Using one-center expansion relations for the Slater type orbitals (STOs) of noninteger principal quantum numbers in terms of integer nSTOs derived in this study with the help of ψa-exponential type orbitals (ψa-ETOs, a = 1, 0, -1, -2,…), the general formulas through the integer nSTOs are established for the unsymmetrical and symmetrical one-range addition theorems for STOs and Coulomb–Yukawa-like correlated interaction potentials (CIPs) with integer and noninteger indices. The final results are especially useful for the computations of arbitrary multicenter multielectron integrals that arise in the Hartree–Fock–Roothaan (HFR) approximation and also in the correlated methods based upon the use of STOs as basis functions.

scholarly journals Systematic study of the muon-nucleus overlap integrals for various processes in muonic atoms

2019 ◽  
Vol 11 ◽  
Author(s):  
Nick Panagiotides ◽  
T. S. Kosmas
Keyword(s):  
Finite Size ◽  
Muonic Atom ◽  
Muon Decay ◽  
Muon Capture ◽  
Overlap Integrals ◽  
Muonic Atoms ◽  
Dirac Equations ◽  
Quantum Numbers ◽  
Electron Positron ◽  
The Standard Model

As it is known, the bound muon of a muonic atom can participate in many electroweak processes as the allowed channels of the ordinary ^""-capture by the atomic nucleus, μ~ •+- (A, Z) -» (A,Z — l) + νβ, and the muon decay in orbit, μ~ -» e~ + υμ ·+- ve, as well as the exotic channels of the muon-to-electron, μ~ + (A, Z) —> (.A, Z)* + e~, and muon-to-positron, μ~ + (A, Ζ) -» (A, Ζ — 2) + e+, conversions. The latter reactions have not been seen by experiments up to now, but they are predicted by various extensions of the standard model (they violate the flavor and/or lepton quantum numbers). For all the above muonic processes, the muon-nucleus overlap integrals are necessary in order to calculate the relevant rates. These integrals can be evaluated if, in addition to the nuclear states, the wave function of the bound muon (also that of the outgoinglepton) are known. In the present work, we perform precise calculations of the muon (and electron/positron) wave functions for both the Schrödinger and Dirac equations. We use modern neural network techniques to overcome the difficulties arising from the finite size of the nucléon and nuclear Coulomb potential. As some applications, the obtained muonnucleus integrals for various muonic atoms are going to be used for evaluating exclusive muon-capture rates and muon to electron/positron conversion branching ratios.

scholarly journals Evaluation of Self-Friction Three-Center Nuclear Attraction Integrals with Integer and Noninteger Principal Quantum Numbers n over Slater Type Orbitals

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Ebru Çopuroğlu
Keyword(s):  
Addition Theorem ◽  
Slater Type Orbitals ◽  
Slater Type ◽  
Arbitrary Integer ◽  
New Approach ◽  
Quantum Numbers ◽  
Noninteger Principal Quantum Numbers ◽  
Exponential Type Orbitals ◽  
Analytical Expressions ◽  
Nuclear Attraction

We have proposed a new approach to evaluate self-friction (SF) three-center nuclear attraction integrals over integer and noninteger Slater type orbitals (STOs) by using Guseinov one-range addition theorem in standard convention. A complete orthonormal set of Guseinov ψα exponential type orbitals (ψα-ETOs, α=2,1,0,-1,-2,…) has been used to obtain the analytical expressions. The overlap integrals with noninteger quantum numbers occurring in SF three-center nuclear attraction integrals have been evaluated using Qnsq auxiliary functions. The accuracy of obtained formulas is satisfactory for arbitrary integer and noninteger principal quantum numbers.

Sign in / Sign up

Export Citation Format

Share Document