The H${}_2^{+}$ molecular ion: low-lying states

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in {\it J. Phys. B44, 101002 (2011)}. The ten low-lying eigenstates of H${}_2^{+}$ of the quantum numbers $(n,m,\text{La},\pm )$, with $n=m=0$ at $\text{La}=0,1,2$, with $n=1$, $m=0$ and $n=0$, $m=1$ at $\text{La}=0$ of both parities are explored for all interproton distances $R$. For all these states this approximation provides the relative accuracy $\lesssim {10}^{-5}$ (not less than 5 s.d.) locally, for any real coordinate $x$ in eigenfunctions, when for total energy $E(R)$ it gives 10-11 s.d. for $R\in [0,50]$~a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions $E1$ is calculated with not less than 6~s.d. A dramatic dip in the $E1$ oscillator strength $f_{1s{\text{si}}_g-3p{\text{si}}_u}$ at $R\sim R_{eq}$ is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6~s.d. in oscillator strength. For two lowest states $(0,0,0,\pm )$ (or, equivalently, $1s{\text{si}}_g$ and $2p{\text{si}}_u$ states) the potential curves are checked and confirmed in the Lagrange mesh method within 12~s.d. Based on them the Energy Gap between $1s{\text{si}}_g$ and $2p{\text{si}}_u$ potential curves is approximated with modified Pade $R{e}^{-R}[Pade(8\mathrm{/}7)](R)$ with not less than 4-5 figures at $R\in [0,40]$,a.u. Sum of potential curves $E_{1s{\text{si}}_g}+E_{2p{\text{si}}_u}$ is approximated by Pade $1\mathrm{/}R[Pade(5\mathrm{/}8)](R)$ in $R\in [0,40]$,a.u. with not less than {3-4} figures.