Quantum calculation, vibrational energy

The vibrational relaxation of simple molecular ions M+ in the M+-M collision (where M = 02, N2, and CO) is studied using the method of distorted waves with the interaction potential constructed from the inverse power and the polarization energy. For M-M collisions the calculated values of the collision number required to de-excite a quantum of vibrational energy are consistently smaller than the observed data by a factor of 5 over a wide temperature range. For M+-M collisions, the vibrational relaxation times of M+ (r+) are estimated from 300° to 3000°K. In both N2 and CO, t + s are smaller than ts by 1-2 orders of magnitude whereas in O r + is smaller than t less than 1 order of magnitude except at low temperatures. 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

A vibrational basis set is chosen for the molecule and the quantum mechanical variational method (Eqs. (2.66)-(2.68)) is used to calculate vibrational energy levels from the analytic potential. The parameters/ -, byi i in the potential are varied until the experimental vibrational energy levels are fit. This variational method is only practical at low levels of excitation. 

J. L. Skinner and K. Park,/. Phys. Chem. B, 105,6716 (2001). Calculating Vibrational Energy Relaxation Rates from Classical Molecular Dynamics Simulations Quantum Correction Factors for Processes Involving Vibration-Vibration Energy Transfer. 

Skinner, ).L. and Park, K. (2001) Calculating vibrational energy relaxation rates from classical molecular dynamics simulations quantum correction factors for processes involving vibration-vibration energy transfer. J. Phys. Chem. B, 105 (28), 6716 6721. 

Fig. 5. The probability of energy transfer per unit energy P2(As) for the initial nj=2 vibrational level of D2, vevsus the energy transfer Ae, calculated with a Morse potential for D2 and hard-core repulsion for H-D, at initial kinetic energy E. The unit of energy is the first quantum of vibrational energy.

Quantum Calculations of Vibrational Energy Levels of Acetylene (HCCH) up to 13000 cm-1. 

A58, 727 (2002). Quantum Calculation of Highly Excited Vibrational Energy Levels of CS2(X) on a New Empirical Potential Energy Surface and Semiclassical Analysis of 1 2 Fermi Resonances. 

Equilibrium stable isotope fractionation is a quantum-mechanical phenomenon, driven mainly by differences in the vibrational energies of molecules and crystals containing atoms of differing masses (Urey 1947). In fact, a list of vibrational frequencies for two isotopic forms of each substance of interest—along with a few fundamental constants—is sufficient to calculate an equilibrium isotope fractionation with reasonable accuracy. A succinct derivation of Urey s formulation follows. This theory has been reviewed many times in the geochemical... 

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