Functions associated with the Burrau approximation for the hydrogen molecular ion and analogous molecules

Journal Article · Fri Dec 31 23:00:00 EST 1976 · J. Chem. Phys.; (United States)

DOI:https://doi.org/10.1063/1.433627· OSTI ID:7327893

McKinney, P C

The Schroedinger equation for the Burrau-like molecular species (A-B)/sub Z/A/sup +//sub Z/B/sup -1/ is (-h/sup 2//2m/sub e/) del/sup 2/ psi + (-Z/sub A/e/sup 2//r/sub A/-Z/sub B/e/sup 2//r/sub B/ +Z/sub A/Z/sub B/e/sup 2//R/sub A//sub B/)psi = Epsi. After separation using prolate spheroidal coordinates, the mathematical analysis of this equation can be reduced to the investigation of the ordinary differential equation (d/dx) ) (1-x/sup 2/)(dP (x)/dx)) + )(-m/sup 2//(1-x/sup 2/)) + 2..cap alpha beta..x + ..cap alpha../sup 2/x/sup 2/ + sigma) P (x) =0, where m is an integer, ..cap alpha.. and ..beta.. are molecular parameters expressed as elements of the real number set, and sigma is the eigenvalue of the equation which is dependent on two numbers kappa and m related to the quantum numbers found in the discussion of the hydrogen atom as well as ..cap alpha.. and ..beta... The solutions for the function P (x) where m is replaced by ..mu.. and ..mu.. is any number are expanded in power series of various kinds, in series of Legendre functions, and confluent hypergeometric functions to give three term recursion relations. The recursion relations are used to generate continued fractions which allow convenient numerical calculation of the eigenvalues and they are also helpful in the derivation of asymptotic expressions of the eigenvalues and eigenfunctions in terms of their dependent variables. Integral equation solutions are given for the two intervals xelement of(-1,1) and xelement of(1,infinity). For ..beta..=-(..nu..+..mu..+1) or ..beta..=..nu..+/sup +/+1, where ..nu..element of)0) UN, it is possible to generate polynomials solutions. (AIP)

• Research Organization: Department of Chemistry, Wabash College, Crawfordsville, Indiana 47933
• OSTI ID: 7327893
• Journal Information: J. Chem. Phys.; (United States), Journal Name: J. Chem. Phys.; (United States) Vol. 66:1; ISSN JCPSA
• Country of Publication: United States
• Language: English

Related Subjects

640302* -- Atomic
Molecular & Chemical Physics-- Atomic & Molecular Properties & Theory
640305 -- Atomic
Molecular & Chemical Physics-- Atomic & Molecular Theory-- (-1987)
74 ATOMIC AND MOLECULAR PHYSICS
ANALYTICAL SOLUTION
CRYOGENIC FLUIDS
DIFFERENTIAL EQUATIONS
ELEMENTS
EQUATIONS
FLUIDS
FUNCTIONS
HYDROGEN
MECHANICS
NONMETALS
QUANTUM MECHANICS
SCHROEDINGER EQUATION
WAVE EQUATIONS
WAVE FUNCTIONS