{H${}_2^{+}$, HeH and H${}_2$}: approximating potential curves, calculating rovibrational states

Abstract

Analytic consideration of the Bohr-Oppenheimer (BO) approximation for diatomic molecules is proposed: accurate analytic interpolation for potential curve consistent with its rovibrational spectra is found. It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1s{\sigma }_g$ and $2p{\sigma }_u$, $2p{\pi }_u$ and $3d{\pi }_g$ of H${}_2^{+}$, the ground state X${}^2{\Sigma }^{+}$ of HeH and the two lowest states ${}^1{\Sigma }_g^{+}$ and ${}^3{\Sigma }_u^{+}$ of H${}_2$, the potential curves can be analytically interpolated in full range of internuclear distances $R$ with not less than {4-5-6} figures. Approximation based on matching the Taylor-type expansion at small $R$ and a combination of the multipole expansion with one-instanton type contribution at large distances $R$ is given by two-point Padé approximant. The position of minimum, when exists, is predicted within 1$\%$ or better. For the molecular ion H${}_2^{+}$ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states $(\nu ,L)$ associated with $1s{\sigma }_g$ and $2p{\sigma }_u$, $2p{\pi }_u$ and $3d{\pi }_g$ potential curves is calculated. In general, $1s{\sigma }_g$ electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state $2p{\sigma }_u$ the total number of rovibrational states (all with $\nu =0$) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state $2p{\pi }_u$ within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state $3d{\pi }_g$ is repulsive, no rovibrational state is found. The ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate analytical expression for the potential curves of the hydrogen molecule H${}_2$ for the states ${}^1{\Sigma }_g^{+}$ and ${}^3{\Sigma }_u^{+}$ is presented.

37 pages, 9 figures, 7 tables

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