Equipartition energy theorem, vibrational

A fundamental theorem of classical mechanics called the equipartition theorem (which we shall not derive here) states that the average energy of each degree of freedom of a molecule in a sample at a temperature T is equal to kT. In this simple expression, k is the Boltzmann constant, a fundamental constant with the value 1.380 66 X 10-21 J-K l. The Boltzmann constant is related to the gas constant by R = NAk, where NA is the Avogadro constant. The equipartition theorem is a result from classical mechanics, so we can use it for translational and rotational motion of molecules at room temperature and above, where quantization is unimportant, but we cannot use it safely for vibrational motion, except at high temperatures. The following remarks therefore apply only to translational and rotational motion. 

This is the classical equipartition theorem. It states that each rotation (which only contributes one term to the sum) adds RT/2 to the energy, whereas each vibration (which contributes two terms) adds RT to the energy. From Eq. (73), each of the... 

Because of the classic approach involved in a MD simulation, energy is equipartitioned among all the vibrational modes (equipartition theorem [24]). In a system with N elements, the total kinetic energy, E, is the sum ... 

If all translational and rotational modes are fully accessible in accordance with the equipartition theorem, then Utrans(T) and U ,t(T) are both equal to per molecule (except that U i T) equals k T for a linear molecule) k is Boltzmann s constant. However, the vibrational energy levels are often only partially excited at room temperature. The vibrational contribution to the internal energy at a temperature T thus requires knowledge of the actual vibrational frequencies. The vibrational contribution equals the difference in the vibrational enthalpy at the temperature T and at 0 K and is given by ... 

The Ed terms are obtained from the calculations and Eyib includes the zero point energy and temperature corrections to the vibrational enthalpy derived by computed harmonic vibrational frequencies. The corrections due to transhition, vibration and rotation are treated classically, using the equipartition theorem. The consideration that BH" " and B rotational contributions are quite similar, that the proton has only translational degrees of freedom and that the eventually different populations of the vibrational states that originate bringing the system from zero degrees to room temperature, are practically cancelled in the calculation of PA, yield tis to consider... 

Quantum mechanical vibrations do not obey a square-law potential. Quantum 1brational energies depend only linearly on the vibrational quantum number, Ev = (v + l/2)hv. Therefore the equipartition theorem for vibrations is different from Equation (11.53). Using Appendix D, Equation (D.9), the vibrational equipartition theorem is... 

---