Short-range interaction energy for ground-state H 2 +

The analytical procedures previously developed by the authors (see references) in order to evaluate the short-range interaction energy of two protons bound by one electron are briefly reviewed. The Hamilton-Jacobi-Riccati outer equation is solved perturbatively up to fifth order in the perturbation parameter, where the zero-order function is the He+ ground-state function. The inner equation is solved by expanding Hylleraas determinant. The results are consistent with those of similar treatments and with accurate numerical calculations.


Introduction
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.

Methods
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: allows to write the two equations for L = 0 into the Hermitean form [13]: where A and E e are separation constants. In these equations, f is complex in the whole range of variation of ξ and g obviously real. The profile of the 'potential' in equation (2ʹ) changes from a symmetrical double well for real positive A to a single well for real negative A. Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. and the separation constant A. 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]. 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. Similarly, boundary conditions imposed upon the iterative solution of an integral equation yield the desired relationship between the constants, order by order [8], under the assumption that A may be expanded into power series of ΔE e .

The perturbative solution
The solution of the complete outer equation in X(ξ) is approximated through an expansion of the action 4 j(ξ) in powers of the energy difference ΔE e , the κth order solution being where it is recalled that all the j l , e λ=0, 1, 2 K κ are imaginary functions so that the ( j x l ( ) i e ) are real as a whole, for real ξ. Thus  [10,13]. The complete Riccati equation for the derivative of the action over f is: To get corrections up to O(R 10 ) it is necessary to calculate k p e up to D ( ) O E e 5 with the appropriate boundary conditions: These boundary conditions yield a kind of dispersion relation 5 between the constants A and E e : The solutions to the expanded HJR equation are linear in k A e and therefore it is possible to solve for this parameter which would depend in a complicated (non-analytic) way on R, although the solutions were analytic in R in the limit  Hamilton-Jacobi-Riccati equation. It is the quantum analogue of the classical 'reduced' action [14]. 5 Dispersion relations establish the law of variation of wave velocity with wavelength [15]. The role of squared wavenumber is played here by the constant A, which goes into squared angular momentum for vanishing R. A proof for the analyticity of dispersion relations is mentioned in [6].
The solutions to the expanded HJR equation up to κ=5 are reported in [10]. Equation (5) exhibits a singularity in the point ξ=1, consequently there are analytical and non-analytical solutions in the environment of this point. In [2] the analytical solution for X(ξ) was retained, the total charge being concentrated in one focus of the ellipse. The solution that was evaluated in [8][9][10] by perturbative series expansion of the HJR equation with boundary conditions 6 (6) is analytical in the environment of ξ=1.

Solution by integral equation
Omitting for short the index e, it is written [8,13] ò ò consequently, since p 0 (1)=0, it is obtained that the following condition should be necessarily satisfied [8]: In equation (9) the lower limit of integration has been put equal to 1, but it is at large extent arbitrary. In fact, using equation Consequently, the condition requiring the derivative over f to vanish [8] is independent of the lower extremum of integration, provided the factor multiplying the rhs of equation (12) is different from zero.

Expansion of coefficients a, b, c, d, e in powers of R
The coefficients a, b, c, d, e were evaluated in [8,10] 7 . The expanded form of these coefficients is the following, where it has been made use of appendix B: The first of these boundary conditions ensures that also x x -( ) ( ) / p 1 e 2 and consequently j(ξ), X(ξ) are analytical in the environment of ξ=1. 7 These coefficients were also evaluated in [17]. The present calculations of the expanded form are consistent with the results of that reference.    The nonexpanded form of coefficients may be found in [8,10].

Solution of the inner equation by Hylleraas determinantal method
The inner η-equation is solved by putting c . , 0 By direct substitution into the inner equation it is obtained a linear system whose infinite determinant must vanish. Battezzati and Magnasco [8] have shown that for = ℓ 0 There follows the expansion of the inner separation constant A k (see also [1,12,17]):  From equation (18) the A κ are evaluated recursively using the corresponding k κ , which are obtained from equations (7), (13)-(13″″), (17), (18) (see [10] for details).
Using the values of k κ which have been calculated in [9,10]   The wavefunctions may be obtained by a perturbative expansion [7] h y h y h y h µ + + + The solution (20) has been calculated up to κ=5 in [7] with the boundary condition -= ( ) q 1 0, e the result being: the result being an even function of R to this order. Retaining terms up to κ=3 there follows upon integration [7]: This function is consistent with the general form of solutions of a second-order ordinary differential equation whose lower order coefficients are singular in the point η=1 [18], which is therefore an algebraic critical point of the solution. The rhs. of equation (22) also exhibits a logarithmic singularity in the same point.
Dispersion relations may be derived by imposing appropriate boundary conditions to the solution modified by the integral equation method [7,8,13], which allows to represent the correctly symmetrized solution, while keeping into account variation of A k from the united atom value A . 0 e 7. Evaluation of coefficients C κ Upon substituting equation (17) into (7), there follows A k −A 0 expanded in powers of R [10]. Thus, using (18) and equating coefficients of equal powers of R, equal powers of R multiplied by ln4R, (ln4R) 2 , (ln4R) 3 the k κ are obtained recursively up to O(R 10 ). The result is  where the C κ =−k κ are the coefficients of the expansion of ΔE e in powers of the internuclear distance R. This result matches those obtained in [7][8][9][10] up to O(R 8 ), and that reported in [17] up to O(R 10 ) 8 .

Summary and conclusions
Previous calculations of short-range interactions in ground-state + H 2 are extended up to the evaluation of the C 10 coefficient.
Recently analytical methods for evaluation of + H 2 interaction energies were used extensively with the help of computer algebra and combined with highly accurate numerical calculations [12,17]. They contribute to shed light on the structure and properties of the solutions. In [17] a variant of the present approach was developed in order to check numerical data, from which exact values of h.o. coefficients of the expansion were inferred by numerical interpolation.
Equation (12) is proved by writing ò ò ò ò x a x a a = + - and then iterating. After every step the factor ò ò can be factorized out from the expansion of X k (ξ, β).

Appendix B. Expansion of integrals
The following integral identities have been proved with the help of [19] in order to expand the coefficients c, d in powers of R and lnR: there follows that a real wavefunction x ( ) X would certainly diverge at infinity if ΔE>4. More precisely: it can be ascertained by inspection of equations (9)-(13) of [10], or even proved to finite order by induction, that iρ(ξ) diverges at infinity x µ 2 with negative coefficient ε. ε is real and negative for ΔE<2, or −R for ΔE=2. ε assumes two complex-conjugate values if ΔE>2, that yield to X(ξ) the form of free travelling waves at large distances from the nuclei. ε may be evaluated from the perturbation expansion to fifth order reported in [10], which yields e = -.753 906 25 R for ΔE=2, and