Quantum Chemistry/Example 13

Write a question and its solution that shows the specific selection rule for a quantum harmonic oscillator

Calculate the energy for the vibrational transitions $v\rightarrow v+1$ and $v+6\rightarrow v+7$. If they have the same energy gaps comment on why?

$\Delta E_{v\rightarrow v+1}$$=h\nu _{0}{\Bigl (}(v+1)+{\tfrac {1}{2}}{\Bigr )}-hv_{0}{\Bigl (}v+{\tfrac {1}{2}}{\Bigr )}$

$=h\nu _{0}{\biggl (}(v+1)+{\tfrac {1}{2}}-{\Bigl (}v+{\tfrac {1}{2}}{\Bigr )}{\biggr )}$

$=h\nu _{0}{\biggl (}v+1+{\tfrac {1}{2}}-v-{\tfrac {1}{2}}{\biggr )}$

$=hv_{0}$

$\Delta E_{v+1\rightarrow v+2}$$=h\nu _{0}{\Bigl (}(v+7)+{\tfrac {1}{2}}{\Bigr )}-hv_{0}{\Bigl (}(v+6)+{\tfrac {1}{2}}{\Bigr )}$

$=h\nu _{0}{\biggl (}(v+7)+{\tfrac {1}{2}}-{\Bigl (}(v+6)+{\tfrac {1}{2}}{\Bigr )}{\biggr )}$

$=h\nu _{0}{\biggl (}v+7+{\tfrac {1}{2}}-v-6-{\tfrac {1}{2}}{\biggr )}$

$=h\nu _{0}$

The energy of the vibrational transitions from from $v\rightarrow v+1$ and $v+6\rightarrow v+7$ have the same energy gap $(h\nu _{0})$ . This is because of the specific selection rule for the quantum harmonic oscillator. The rule states that the molecule is only allowed to move up or down one vibrational energy level for the transition to occur. If the molecule diverges from the rule $\Delta v\pm 1$ then it is considered an overtone, and these are unlikely to occur.