Short-range interaction energy for ground state H₂⁺

Abstract

Two of the Hermitian eigenvalue equations resulting from the separation of the three-dimensional Schroedinger equation for H 2 + in spheroidals are solved perturbatively for the ground state by expanding the action in positive powers of the internuclear distance R near the united atom He + . The dispersion relations between the separation constants A and E e are seen to have rigorous analytic solutions, the third-order equation leading to an exact expansion for the inner determinantal equation up to R 10 . The explicit form for the expansion coefficients is determined up to n = 10, and is seen to contain up to the third power of (γ + ln 4R) logarithmic terms. Even if the general range of validity of the short-range R n -expansion is expected to be smaller than the corresponding long-range R −n -expansion, it is important to stress that such higher expansion coefficients are calculated exactly for the first time. These formulae give extremely accurate numerical results up to R ∼ = 0.3a 0 .

Short-range interaction energy for ground-state ${{\bf{H}}}_{2}^{+}$

“…The separation of the 3-dimensional Schroedinger equation for + H 2 in confocal elliptic (spheroidal) coordinates x h j ( ) , , of the electron, with nuclei kept in fixed position [1][2][3][4][5][6][7][8][9][10][11][12][13], originates three one-dimensional differential equations, the outer ξ−equation, the inner η-equation and the j-equation. The subsequent variable transformation:…”

“…The regularity conditions upon the solutions of each separate equation in spheroidal co-ordinates yield two independent relations between the constants A and E e , that can be solved as a function of R. The inner ηequation has been satisfactorily solved by Hylleraas [1] by a determinantal method, whose zeroes yield the required functional relation. The outer ξ-equation has been solved perturbatively by us [8][9][10]13] in powers of the energy variation ΔE e which induces a variation in the 'potential' of the f-equation above. The perturbative solution of the Hamilton-Jacobi-Riccati (HJR) equation associated to this equation is expanded in powers of ΔE e and the regularity conditions imposed order by order so as to make the solution physically acceptable [10].…”

“…The outer ξ-equation has been solved perturbatively by us [8][9][10]13] in powers of the energy variation ΔE e which induces a variation in the 'potential' of the f-equation above. The perturbative solution of the Hamilton-Jacobi-Riccati (HJR) equation associated to this equation is expanded in powers of ΔE e and the regularity conditions imposed order by order so as to make the solution physically acceptable [10]. There results a relation between A and ΔE e which is independent of the distribution of the charge between the foci, being only dependent on the distance R and the sum of the charges.…”

“…The evaluation of the coefficients arising in the short-range expansion of the interaction in ground state + H 2 has been the object of many investigations [1][2][3][4][5][6][7][8][9][10][11]. In this paper we extend the calculations to the evaluation of the C 10 coefficient using our previous techniques.…”

“…The main principles and assumptions on which they are founded are examined and refined calculations are presented. In [7] perturbation theory was applied in conjunction with first-passage time techniques in order to solve both the so-called [4] inner and outer equations, while in subsequent work [8,10] it was found more suitable to make recourse to Hylleraas tridiagonal determinantal method [1] applied to the inner equation. This led straightforwardly to useful analytical results, although the physical picture of an electron stabilized by resonance in a double-well potential, which results from our transformed equations (2), (2ʹ) below, was somewhat hidden in that approach.…”