Adiabatic separations of stretching and bending vibrations: Application to H2O

Abstract

A detailed investigation is made into the use of adiabatic approximations for describing excited stretching and bending vibrations of the water molecule. The goal is to determine precisely how effective this approach can be in a fully quantum mechanical triatomic calculation which incorporates anharmonicities to all orders in each of the modes. Great care is taken to avoid introducing unnecessary limitations or approximations: (i) Curvilinear coordinates are used rather than the Cartesian coordinates which form the starting point for normal mode calculations; (ii) the exact quantum kinetic energy operator in these coordinates is used as the basis for both the adiabatic and full three-dimensional calculations; (iii) a Sorbie–Murrell-type potential energy surface is used, giving a reasonable representation of the ground electronic surface for large excursions from the equilibrium configuration. In addition to the bond and bond-angle variables of earlier local mode investigations, a slightly different set of fully curvilinear coordinates is also investigated. These coordinates are shown to provide a more nearly separable description in both the exact and adiabatic treatments of this specific problem. The conventional adiabatic approach, in which the slower bending mode experiences an effective force due to averaging over the faster stretching modes, is reaffirmed to be accurate for excited stretching states. For states with any appreciable bending excitation, however, it turns out that the adiabatic calculations quickly erode in reliability. In answer to this problem, the reverse adiabatic procedure (with the bend treated first) is also implemented here. While counterintuitive, this latter method is found to yield a significant improvement for the calculated bending overtones, as well as many of the combination bands. Thus, by thorough consideration of both the coordinates and order of averaging employed, the adiabatic method is shown to be very effective for either bending or stretching overtones in a realistic, fully anharmonic, triatomic vibrational problem. In addition, introduction of a new orthonormal set of basis functions for the bending angle overcomes some of the problems associated with use of the less flexible Legendre basis.

On adiabatic separation of stretching and bending vibrations

“…This is done in w 5 for two cases, first when the stretching frequency is higher than the bending frequency, and second, when the bending frequency is higher than the stretching frequency. In the first case, while the comparison between AS and AB approximations is erratic, in accordance with the observations made in [1] and [2], the MAB approximation is systematically superior to the AS + NAC approximation. In the second case, the AS approximation is found to be superior to the AB approximation, and the AS + NAC approximation superior to the MAB approximation.…”

“…As noted in [1] and [2], the comparison between AS and AB approximations is indeed erratic. Although AB approximation is superior for all states with 2 or more stretching quanta, it is clearly inferior for v = 0 and v = 1 states when the bending mode is excited.…”

“…In both these papers it was found that adiabatic stretch was often the better approximation, even when the stretching frequency was a few times the bending frequency, and particularly when the bending mode was excited. We quote from the abstract of [2] :…”

“…Johnson, Skodje and Reinhardt [1] have tested this assumption for a model hamiltonian with two degrees of freedom and Johnson and Reinhardt [2] have carried our a similar analysis of adiabatic approximations for vibrational spectrum of water molecule in states of zero angular momentum. (The model hamiltonian in the former paper is different from ours but has similar coupling between stretching and bending motion.)…”

Ab Initio Potential Energy Surfaces for Both the Ground (X̃1A′) and Excited (ALi1, Lin2, Xie3 2012Advances in Physical Chemistry33040View full textAdd to dashboardCite

“…where m 1 and m 2 are the atomic mass of H and Br, respectively. The transformation between the Radau coordinates (R 1 , R 2 , γ) and the internal coordinates (R HSi , R SiBr , θ) is well documented [29]. The vibrational wave functions were represented in a direct product discrete variable representation (DVR) [30] grid, and the vibrational energy levels were calculated using the Lanczos algorithm [31], which generates the eigenvalues of the Hamiltonian by a threeterm recursion.…”