Towards the analytic theory of potential energy curves for diatomic molecules: studying He₂⁺ and LiH diatomics as illustration

Abstract

Following the first principles, the elements of the analytic theory of potential curves for diatomic molecules (diatomics) are presented. It is based on matching the perturbation theory at small internuclear distances R and multipole expansion at large distances, modified by exponentially small terms for homonuclear case, with the addition of the phenomenologically described equilibrium configuration, if it exists. It leads to a new class of (generalised) rational potentials (modified by exponentially small terms) with a difference in degrees of polynomials in numerator and denominator equal to 4 (6) for positively charged (neutral) diatomics. For example, the He${}_2^{+}$ and LiH diatomics in Born–Oppenheimer approximation are considered. For ${}^4$He${}_2^{+}$ (${}^3$He${}_2^{+}$) diatomics it is found the approximate analytic expression for the potential energy curves (analytic PEC) $V(R)$ for the ground state $X^2{\mathrm{\Sigma }}_u^{+}$ and the first excited state $A^2{\mathrm{\Sigma }}_g^{+}$. It provides 3–4 s.d. correctly for distances $R\in [1,10]$ a.u. with some irregularities for $A^2{\mathrm{\Sigma }}_g^{+}$ PEC at small distances (much are smaller than equilibrium distances) probably related to level (quasi)-crossings that may occur there. The analytic PEC for the ground state $X^2{\mathrm{\Sigma }}_u^{+}$ supports 829 (626) rovibrational states with 3–4 s.d. of accuracy in energy, which is only by 1 state less (more) than 830 (625) reported in the literature. The analytic PEC for the excited state $A^2{\mathrm{\Sigma }}_g^{+}$ supports all 9 reported weakly-bound rovibrational states. For ${}^7$LiH it is found the analytic expression for the ground state $X^1{\mathrm{\Sigma }}^{+}$ PEC in the form of a rational function, which supports 906 rovibrational states with 3–4 s.d. accuracy in energy, it is only of 5 states more than reported in the literature (901). For both diatomics, the difference in a number of rovibrational states is related to the non-existence/existence of weakly-bound states close to the threshold (to dissociation limit). Entire rovibrational spectra are found in a single calculation using the code based on the Lagrange mesh method.

GRAPHICAL ABSTRACT

Keywords:

• Perturbation theory
• multipole expansion
• one-instanton contribution
• matching small and large distances
• two-point Pade approximant
• Potential energy curve

Acknowledgments

A. V. T. thanks PASPA-UNAM for a support during his sabbatical stay at the University of Miami, where this work was finished. H. O. P. is grateful to the Instituto de Ciencias Nucleares, UNAM, Mexico for kind hospitality, where the present study was initiated and concluded. The authors thank anonymous referee for careful reading the paper and important insights. This work is dedicated to the memory of L. Wolniewicz whom both authors had no privilege to meet personally but always considered as the exemplary scientist. The authors thank J. Karwowski for interest to the present work and for the kind invitation to contribute to this volume.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Sometimes, in physics literature it is called the Born–Oppenheimer approximation of the zero order.

2 In this consideration it is always assumed that the nuclei are point-like, having no internal structure. We never go to ultra-small distances where strong interaction plays the role and nuclear forces should be considered.

3 Sometimes, it is called the super-ion.

4 The case $n=1/2$, which corresponds to the He${}_2^{+}$ molecular ion, will be presented later.

Additional information

Funding

The research is partially supported by CONACyT [grant number A1-S-17364] and DGAPA [grant numbers IN113819, IN113022] (Mexico).