On the use of optimal internal vibrational coordinates for symmetrical bent triatomic molecules

Abstract

The use of generalized internal coordinates for the variational calculation of excited vibrational states of symmetrical bent triatomic molecules is considered with applications to the SO2, O3, NO2, and H2O molecules. These coordinates depend on two external parameters which can be properly optimized. We propose a simple analytical method to determine the optimal internal coordinates for this kind of molecules based on the minimization with respect to the external parameters of the zero-point energy, assuming only quadratic terms in the Hamiltonian and no quadratic coupling between the optimal coordinates. The optimal values of the parameters thus obtained are shown to agree quite well with those that minimize the sum of a number of unconverged energies of the lowest vibrational states, computed variationally using a small basis function set. The unconverged variational calculation uses a basis set consisting of the eigenfunctions of the uncoupled anharmonic internal coordinate Hamiltonian. Variational calculations of the excited vibrational states for the four molecules considered carried out with an increasing number of basis functions, also evidence the excellent convergence properties of the optimal internal coordinates versus those provided by other normal and local coordinate systems.

Assigning quantum labels to variationally computed rotational-vibrational eigenstates of polyatomic molecules

“…However, assignment schemes can be very useful even if these requirements are not fully met. Among the techniques that have been employed in the analyses of variationally computed nuclear-motion wave functions are "node counting" along specified cuts of coordinate space, 17,18 the determina-tion of "optimally separable" coordinates, 10,[19][20][21][22][23][24][25][26][27][28] the use of natural modal representations, 18,29 and the evaluation of coordinate expectation values. 17 An alternative approach to assigning molecular eigenstates is provided by effective Hamiltonian methods, particularly in relatively low-energy regions.…”

Multimode wavelet basis calculations via the molecular self-consistent-field plus configuration-interaction method

“…Work on determination of optimal curvilinear coordinate systems ͑see Zúñiga et al 43 and references therein͒ is relevant here and can be carried out in conjunction with the computationally simple SCF stage. The accuracy of the molecular SCF method depends on the choice of coordinates in general, so a different set such as bond coordinates would be expected to produce larger errors.…”

Near- and long-term quantum algorithmic approaches for vibrational spectroscopy

“…Though various forms of perturbation theory and dimensionality reduction sometimes yield good results, one must often resort to exact diagonalization of the whole Hilbert space or similarly expensive methods [36][37][38][39][40][41]. We note that we are not constrained to use equation (3) but may choose any convenient coordinate system-it will often be the case that choosing a specialized coordinate system allows one to use a lower-order series expansion [42][43][44].…”

“…A fourth-order Hamiltonian has 8 types with the inclusion of q 4 i , q 3 i q j , q 2 i q 2 j , and q 2 i q j q k . Note that, depending on the choice of coordinate system, it often possible to exclude three-body terms q 2 i q j q k , while still obtaining sufficiently accurate results [42][43][44]68]. The Appendix gives Pauli operator counts for each of these 8 term types, for d = 4 (2 qubits) and d = 8 (3 qubits).…”