Increasing the accuracy of semiclassical energy quantization of multidimensional systems

We present a single-trajectory semiclassical method of spectroscopic accuracy for the calculation of molecular vibrational energies. The century old challenge of extending the Brillouin, Wentzel, and Kramers quantization rule for accurate predictions of energy levels in multidimensional systems is answered in two steps. One is ensuring that the semiclassical energy estimate agrees with vibrational perturbation theory. This is achieved by adding a constant energy shift of order ℏ2 to the quantization condition, which is readily estimated through knowledge of the third and fourth potential derivatives. The second is adapting the Fourier transform method to obtain an objective criterion for the validity of the adiabatic switching technique, which lies at the heart of the implementation of the Einstein-Brillouin-Keller semiclassical quantization method in multidimensional systems. The Fourier coefficients are calculated approximately by means of a single classical trajectory introduced to ascertain that indeed the actions were reasonably well conserved during the adiabatic switching phase. The result is a rather accurate single trajectory based quantization method. The theory is tested on bi-dimensional Hénon-Heiles models and on the non-rotating water molecule. When comparing with second-order perturbation theory and the current semiclassical accuracy, our approach is by far more accurate. This is remarkable, especially when considering that accurate energy evaluation on an absolute scale is important not only on the conceptual level of modification of the century old semiclassical quantization approximation but also in practice, as it affects different problems, such as non-adiabatic transitions, reaction rates, and isotopic effects.