Generalization to Arbitrary Transitions

We can take the second term in Equation (A6.39) and substitute in the general solutions for the harmonic oscillator discussed earlier (Equation (A6.33)). For a transition between two arbitrary states m and n the coupling matrix term would be 

We have stuck with the jc-direction to simplify the discussion but have dropped the subscript on the dipole moment, as now the molecular dipole for the polyatomic molecule need not be along X. There will also be transition dipole moments for y and z which should be taken into account in the same way. 

The Hermite polynomials have the following useful recurrence relationship  

This can actually be used to derive all the polynomials from Hq and Hi by using the n = 1 case to derive and then the n = 2 case to get Hi and so on. You may like to test that this expression holds for the polynomials in the first few wavefunctions given earlier in Equation (A6.34). 

In the consideration of selection rules. Equation (A6.41) allows us to transform the M, expression into integrals that only contain products of Hermite polynomials without the intervening x factor that is, because 

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