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<p><font face="serif">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</font></p>
<p><math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mi
mathvariant="normal">Δ</mi><mi>E</mi><mo>/</mo><msub><mi>k</mi><mi>b</mi></msub><mi>T</mi></mrow></msup></mrow><annotation
encoding="TeX">Q=\sum_{i=0}^N e^{-i\Delta E/k_bT}</annotation></semantics></math><font
face="serif">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</font></p>
<p><font face="serif"><math display="block"
xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mi>N</mi></munderover><msup><mi>e</mi><mrow><mo>−</mo><mi>i</mi><mn>0.75</mn></mrow></msup></mrow><annotation
encoding="TeX">Q=\sum_{i=0}^N e^{-i 0.75}</annotation></semantics></math></font></p>
<div class="moz-signature">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. <br>
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<div class="moz-signature">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.</div>
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<div class="moz-signature">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.</div>
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<div class="moz-signature">-- <br>
<b><i>Andrew J. Pounds, Ph.D.</i></b><br>
<i>Professor of Chemistry and Computer Science</i><br>
<i>Director of the Computational Science Program</i><br>
<i>Mercer University, Macon, GA 31207 (478) 301-5627</i></div>
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