H[formula omitted], HeH and H[formula omitted]: Approximating potential curves, calculating rovibrational states

Abstract

Analytic consideration of the Bohr–Oppenheimer (BO) potential curves for diatomic molecules is proposed: accurate analytic interpolation for a potential curve consistent with its rovibrational spectra is found.It is shown that in the BO approximation for four lowest electronic states 1sσg and 2pσu, 2pπu and 3dπg of H2+, the ground state X2Σ+ of HeH and the two lowest states 1Σg+ and 3Σu+ of H2, the potential curves can be analytically interpolated in full range of internuclear distances R with not less than 4–5–6 s.d. Approximation based on matching the Laurant-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 H2+ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states (ν,L) associated with 1sσg and 2pσu, 2pπu and 3dπg potential curves are calculated. In general, it coincides with spectra found via numerical solution of the Schrödinger equation (when available) within six s.d. It is shown that 1sσg curve contains 19 vibrational states (ν,0), while 2pσu curve contains a single one (0,0) and 2pπu state contains 12 vibrational states (ν,0). In general, 1sσg electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state 2pσu the total number of rovibrational states (all with ν=0) is equal to 3, within or beyond Bohr–Oppenheimer approximation. As for the state 2pπu within the Bohr–Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state 3dπg is repulsive, no rovibrational state is found.It is confirmed in Lagrange mesh formalism the statement that 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 H2 for the states 1Σg+ and 3Σu+ is presented. The ground state 1Σg+ contains 15 vibrational states (ν,0),ν=0–14. In general, this state supports 301 rovibrational states. The potential curve of the state 3Σu+ has a shallow minimum: it does not support any rovibrational state, it is repulsive.