H2+, HeH and H2: Approximating potential curves, calculating rovibrational states

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 1s sigma(g) and 2 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 s.d. Approximation based on matching the Laurant-type expansion at small R and a combination of the multipole expansion with oneinstanton type contribution at large distances R is given by two point Pade 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 (v, L) associated with lsag and 1s sigma(g) and 2 sigma(u), 2p pi(u), and 3d pi(g) potential curves are calculated. In general, it coincides with spectra found via numerical solution of the Schrodinger equation (when available) within six s.d. It is shown that 1s sigma(g) curve contains 19 vibrational states (v, 0), while 2p sigma(u), curve contains a single one (0, 0) and 2p pi(u) state contains 12 vibrational states (v, 0). 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 v = 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. 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 (1)Sigma(+)(g) and (3)Sigma(+)(u) is presented. The ground state (1)Sigma(+)(g) contains 15 vibrational states ( v, 0), v = 0-14. In general, this state supports 301 rovibrational states. The potential curve of the state (3)Sigma(+)(u) has a shallow minimum: it does not support any rovibrational state, it is repulsive. (C) 2018 Elsevier Inc. All rights reserved.