Continued fractions and the hydrogen molecular ion H₂⁺

Abstract

The Jaffé continued fraction solution for the hydrogen molecular ion has been the basis for most H2+ energy calculations at physically relevant nuclear separation R. I show that its slow convergence at small R can be overcome so it can also be used to obtain the power series expansion of the energy about the united atom limit R = 0. I restrict the discussion to the H2+ ground state and show, by a generalization of Wynn's accelerated convergence technique and an analytical calculation, that the continued fraction yields the O(R5) series term which is the first term in which log(R) makes an appearance. For higher order terms I apply the convergence method numerically to obtain very accurate energies at small R and, by fitting, numerical coefficients in the R series. The analytic representations of the coefficients are identified using the integer relation algorithm PSLQ. All terms to O(R14) are obtained exactly and to O(R20) in a mixed analytic/numeric form. Comparisons with previous work based on wavefunction perturbation theory are made.

Experimental mathematics meets gravitational self-force

“…We used the simplification to aid this process, so we needed to include at most 3 transcendentals in the vector to which we apply PSLQ (for the 16.5PN linear logarithmic term). This procedure (of using PSLQ to iteratively improve a fit, aided by a conjecture for the form of certain leading logarithm-type terms) is very similar to the one used by Nickel to obtain highorder terms in the expansion of the ground state energy of H + 2 in [40]. We give the final results of this fit (both analytical and numerical) in the electronic Supplemental Material [60], including showing the remainder of the analytic terms after removing the portions given by the simplification.…”

“…This method should be contrasted with the more usual method of finding numerical values for all terms to some accuracy using a fit, then finding analytic forms for some coefficients, using these to improve the accuracy of the fit, and iterating. This fitting method was used in SFW and in Nickel's similar computation of high-order terms in the expansion of the ground state energy of H + 2 in powers of the distance [40]. 1 We also used this fitting method on the full ∆U to obtain even higher-order terms than we were able to obtain using the first method, though these terms were all obtained only in mixed numerical-analytic form.…”

“…Some other such high-accuracy computations in mathematical physics, including further applications of PSLQ, are discussed in [50]. Additionally, Nickel [40] also uses PSLQ to obtain analytic forms of high-order coefficients of a similar series for the ground-state energy of H + 2 . We also note, following Bini and Damour [24], that while one has to sum over all spherical harmonic (ℓ, m) modes to obtain ∆U to a given PN order (compared to, e.g., the energy flux, where one only has to sum a finite number of modes to obtain the expansion to a given PN order), this infinite sum is only necessary to obtain the nonlogarithmic integer-order PN terms.…”

Structure and dynamics of Xn-type clusters (n= 3, 4, 6) from spontaneous symmetry breaking theory

“…Here we will assume that the function E has derivatives at zero up to the second degree (this follows from [12][13][14][15]). …”

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

“…Commun. 2 (2018) 045018 M Battezzati and V Magnasco 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.…”