The partition function Molecules

The partition function is the key quantity in the calculation of the Boltzmann equilibrium distribution of all molecular energies, and paves the way to a calculation of the total internal energy of any molecular system at equilibrium. Consider a system made of N molecules, each of which has a set of quantized energy levels Sj, known by the solution of some quantum chemical secular equation, as shown in Chapter 3. Energies are distributed over the accessible energy levels. With a modicum of mathematical derivation the following basic equations can be obtained  

Each overall energy level Si is in fact a combination of translational (Tr), rotational (Ro), vibrational (Vi), and electronic energy levels. As a first approximation, Tr, Ro, Vi, and electronic energies are considered as uncoupled, so that any overall molecular 

The moleeular partition funetion is then obtained by summing over all possible ej levels, whieh is equivalent to separately summing over all possible indices k, I, m, and n in equation 7.5  

Tr levels are so closely spaced that the summation can be replaced by an integral. 

Besides, n is always so large that rp - I = displacement are considered, one gets 

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