Boltzmann-averaged energy

At the conclusion of the conformational search, the Boltzmann-averaged energy of the system was calcrilated using... 

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

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. 

The bond lengths were calculated as the Boltzmann average of the geometric properties of the individual vibronic energy levels. 

The denominator in this expression is the total population in all states the numerator is the total energy). At very low temperatures (kET <excited states, and the average energy is far less than k a T. For example, consider the case hv = 100kET. From the Boltzmann distribution, even the lowest excited state (with energy E = 100kE T) is almost empty ... 

After a proper Boltzmann averaging of the integral cross section over the kinetic energy in the center of mass system, ECM, one get the rate constants,... 

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