[CHM 330] Helpful Advice on Calculating Partition Functions

[Andrew J. Pounds]

As we noted in class the partition function Q is built by summing over 
the energy levels.  As the index for the sum increases these values get 
smaller and smaller.  As an example, in the problem dealing with the 
harmonic oscillator, the vibrational partition function is

Q=∑i=0Ne−iΔE/kbTQ=\sum_{i=0}^N e^{-i\Delta E/k_bT}because the argument 
of the exponential (other than i) is fixed, we can calculate it before 
taking the sum.  Let's imagine that value is 0.75.  Our new equation is 
then just

Q=∑i=0Ne−i0.75Q=\sum_{i=0}^N e^{-i 0.75}

Rather than explicitly calculating all of those terms individually and 
summing them up, you can use that $125 piece of tech most of you are 
toting around - your TI-8X calculator.

There are many YouTube videos on how to do summations on your 
calculators.  If you can't find one let me know and I can recommend 
one.  For any problems I give you the maximum number of terms should not 
exceed 100 -- which takes my HP calculator less than a second to solve 
(I don't have a TI-8X available for testing).  When I plug in the 
summation from above I get 1.8952551344.

You don't have to learn how to do this -- but it could definitely save 
you some time when doing problems that require you to calculate the 
fraction of molecules in a given vibrational level.


-- 
*/Andrew J. Pounds, Ph.D./*
/Professor of Chemistry and Computer Science/
/Director of the Computational Science Program/
/Mercer University, Macon, GA 31207 (478) 301-5627/
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