Expansion of a function about a displaced center for multicenter integrals: A general and closed expression for the coefficients in the expansion of a Slater orbital and for overlap integrals

1976 ◽  
Vol 13 (2) ◽  
pp. 517-527 ◽  
Author(s):  
R. R. Sharma
Keyword(s):  
Overlap Integrals ◽  
Closed Expression ◽  
Slater Orbital ◽  
Multicenter Integrals

EVALUATION OF ONE- AND TWO-ELECTRON MULTICENTER INTEGRALS OF YUKAWA-LIKE SCREENED CENTRAL AND NONCENTRAL INTERACTION POTENTIALS OVER SLATER ORBITALS USING ADDITION THEOREMS

2005 ◽  
Vol 16 (06) ◽  
pp. 837-842 ◽  
Author(s):  
I. I. GUSEINOV ◽  
B. A. MAMEDOV
Keyword(s):  
Numerical Stability ◽  
Slater Type Orbitals ◽  
Overlap Integrals ◽  
Slater Type ◽  
Interaction Potentials ◽  
Multicenter Integrals ◽  
Expansion Formulae ◽  
Slater Orbitals ◽  
Exponential Type Orbitals ◽  
The One

By the use of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs, where α =1, 0, -1, -2, …), the series expansion formulae are established for the one- and two-electron multicenter integrals of arbitrary Yukawa-like screened central and noncentral interaction potentials (YSCPs and YSNCPs) in terms of two- and three-center overlap integrals of three Slater type orbitals (STOs). The convergence of the series is tested by the concrete cases of parameters. The formulae given in this study for the evaluation of one- and two-electron multicenter integrals of YSCPs and YSNCPs show good rate of convergence and numerical stability.

Multicenter integrals via Gaussian expansion of slater-orbital products

1969 ◽  
Vol 3 (7) ◽  
pp. 537-539 ◽  
Author(s):  
Hendrik J. Monkhorst ◽  
Frank E. Harris
Keyword(s):  
Slater Orbital ◽  
Multicenter Integrals

Unified treatment of electronic attraction, electric field, and electric-field gradient multicenter integrals of reened and nonscreened Coulomb potentials using overlap integrals for Slater orbitals

2004 ◽  
Vol 82 (10) ◽  
pp. 819-825 ◽  
Author(s):  
I I Guseinov
Keyword(s):  
Electric Field ◽  
Electric Field Gradient ◽  
Field Gradient ◽  
Slater Type Orbitals ◽  
Overlap Integrals ◽  
Slater Type ◽  
Multicenter Integrals ◽  
Slater Orbitals ◽  
Rotational Transformation ◽  
Coulomb Potentials

A unified treatment of multicenter electronic attraction (EA), electric field (EF), and electric-field gradient (EFG) integrals of Yukawa-like screened and nonscreened Coulomb potentials with Slater-type orbitals (STOs) is described. Using different sets of series expansion formulas of two-center distributions for STOs in terms of STOs at a displaced center the EA, EF, and EFG integrals over STOs are expressed through the overlap integrals between potentials or their derivatives and STOs. These two-center overlap integrals are evaluated by the use of rotational transformation for overlap integrals established by the author. The final results expressed through the overlap integrals of STOs with the same screening constants are valid for the arbitrary parameters of STOs and potentials. PACS No.: 31.15.–p

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

scholarly journals EXPANSION FORMULAE FOR ONE- AND TWO-CENTER CHARGE DENSITIES OVER COMPLETE ORTHONORMAL SETS OF EXPONENTIAL TYPE ORBITALS AND THEIR USE IN EVALUATION OF MULTICENTER–MULTIELECTRON INTEGRALS

2009 ◽  
Vol 08 (04) ◽  
pp. 597-602 ◽  
Author(s):  
I. I. GUSEINOV
Keyword(s):  
Exponential Type ◽  
Overlap Integrals ◽  
Charge Densities ◽  
Hartree Fock ◽  
Expansion Formulae ◽  
Multicenter Multielectron Integrals ◽  
Explicitly Correlated ◽  
Explicitly Correlated Methods ◽  
Exponential Type Orbitals ◽  
The One

The series expansion formulae are established for the one- and two-center charge densities over complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETO α = 1,0,-1,-2,…) introduced by the author. Three-center overlap integrals of Ψα appearing in these relations are expressed through the two-center overlap integrals between Ψα-orbitals. The general formulae obtained for the charge densities are utilized for the evaluation of arbitrary multicenter–multielectron integrals occurring when the complete orthonormal sets of Ψα-ETO are used as basis functions in the Hartree–Fock–Roothaan and explicitly correlated methods. The relationships for charge densities and multicenter–multielectron integrals obtained are valid for the arbitrary quantum numbers, screening constants, and location of Ψα-orbitals.

Sign in / Sign up

Export Citation Format

Share Document