  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 $(n,m,\La,\pm)$\, with $n=m=0$ at
$\La=0,1,2$, with $n=1$, $m=0$ and $n=0$, $m=1$ at $\La=0$ 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
$f_{1s\si_g-3p\si_u}$ 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, $1s\si_g$ and $2p\si_u$ states) the potential curves are checked
and confirmed in the Lagrange mesh method within 12~s.d. Based on them the
Energy Gap between $1s\si_g$ and $2p\si_u$ potential curves is approximated
with modified Pade $R e^{-R} [Pade(8/7)](R)$ with not less than 4-5 figures at
$R \in [0, 40]$\,a.u. Sum of potential curves $E_{1s\si_g} + E_{2p\si_u}$ is
approximated by Pade $1/R [Pade(5/8)](R)$ in $R \in [0, 40]$\,a.u. with not
less than {3-4} figures.
