Molecular equipartition

The equipartition principle is a classic result which implies continuous energy states. Internal vibrations and to a lesser extent molecular rotations can only be understood in terms of quantized energy states. For the present discussion, this complication can be overlooked, since the sort of vibration a molecule experiences in a cage of other molecules is a sufficiently loose one (compared to internal vibrations) to be adequately approximated by the classic result. 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

In quantum statistical mechanics where a density operator replaces the classical phase density the statistics of the grand canonical ensemble becomes feasible. The problem with the classical formulation is not entirely unexpected in view of the fact that even the classical canonical ensemble that predicts equipartitioning of molecular energies, is not supported by observation. 

From a molecular viewpoint, we know that heat capacity is closely connected to internal modes of molecular vibration. According to the classical equipartition theorem (Sidebar 3.8), a nonlinear polyatomic molecule of Aat atoms has ftmodes = 3Aat — 6 independent internal modes of vibration, each of which would contribute equally to heat capacity... 

Boltzmann s Law. The law of the equipartition of energy to a molecular system. Stef an-Boltzmann Law states that the total energy radiated from a black body is proportional to its absolute temp raised to the fourth power. It is expressed by E= a (T4 - T ) where E - total energy in ergs,... 

Direct ab initio molecular dynamic simulations starting at the reactant with total Maxwell-Boltzmann equipartitioned thermal kinetic energy of 26kcalmol however, demonstrated that the reaction pathway did not follow the IRC (dotted line in Fig. 1) on the PES, but that it was rather... 

Harvey SC, Tan RK-Z, Cheatham TE III (1998) The flying ice cube velocity rescaling in molecular dynamics leads to violation of energy equipartition. J Comput Chem 19 726-740 Lundberg M, Nishimoto Y, Irle S (2012) Delocalization errors in a Hubbard-like model consequences for density-functional tight-binding calculations of molecular systems. Int J Quant Chem 112 1701-1711... 

The total kinetic energy of the multimolecular system is the sum of the kinetic energies of all atoms, and can be equated to the equipartition value, in the key link between molecular motion and temperature ... 

Molecular vibrations contribute to the heat capacity, but only when the temperature is high enough lor them to be significantly excited. For each vibrational mode, the equipartition mean energy is kT. so the maximum contribution to the molar heat capacity is R. However, it is very unusual for the vibrations to be so highly excited that equipartition is valid, and it is more appropriate to use the full expression lor the vibrational heat capacity which is obtained by differentiating eqn 17.28. The curve in Figure 17.12 of the... 

In this equation, p is the total momentum of particle i and m is its mass. According to the theorem of the equipartition of energy each degree of freedom contributes k T/2. If there are N particles, each with three degrees of freedom, then the kinetic energy should equal 3Nk T/2. Nc in Equation (6.14) is the number of constraints on the system. In a molecular dynamics simulation the total linear momentum of the system is often constrained to a value of zero, which has the effect of removing three degrees of freedom from the system and so would be equal to 3. Other types of constraint are also possible as we shall discuss... 

As will have been seen, the derivation of the equipartition law depends upon certain rather general, and perhaps somewhat abstract, assumptions about molecular dynamics. It is desirable, therefore, to bring the result as soon as possible into relation with experimentally measurable matters. The most striking method of doing this is afforded by the study of specific heats. 

From the formula for the pressure of a perfect gas, p = and the relation pV — NkT, it follows that the kinetic energy in the three translational degrees of freedom is pT. The allocation for each is thus kT. The equipartition law provides that where the energy is shared between s square terms, the average amount in a molecule is skT. The molecular heat, C , is therefore... 

There is no need to invoke assumptions regarding a molecular chaos or a statistical equivalence between average velocities for establishing this equipartition between capacitive subvarieties. 

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