Lecture errata

I never defined the alpha used in the vibrational wavefunctions. It is not the same as the alpha (for the Morse potential) you have in the project instructions.

• For the harmonic oscillator alpha was simple a convenient short-hand for sqrt( reduced_mass * omega / hbar), not an independent parameter.
• For the Morse potential, I would say alpha and x_e are two ways to express the anharmonicity, related via equation (6).
• In both cases alpha has the dimension 1/length, but there is an extra factor in the Morse-alpha that makes it much smaller.

Near the end of the physics part of the lecture I mentioned some confusion about notation S or I for the Franck-Condon factor. There is some naming confusion also in the litterature, where S can have different meanings:

• On the lecture I used it (as P. Atkins does) for the overlap integral <final vibration n|initial vibration 0>=<n|0>, which should be squared to get a relative transition intensity I(n<-0) = |<n|0>|^2.
  We used the vibrational wavefunctions for specific vibrational quantum number n and the change Delta r in bond length to express I (which was some 80% for n=0 in N2 valence photoemission)
• In the project instructions (and most litterature, e.g. Cederbaum,Smedh,Mishima,Karabunarliev,Pullerits) S has a different meaning, more closely related to the change in bond length Delta r, given by equations (9)-(10) in the instructions. This S is not dependent on vibrational quantum number, instead the relation (eq. 7) I(n<-0)= S^n exp(-S) / n! is used to get the relative intensity. The benefit of this approximation is that it works for any final vibrational quantum number, n, and is much simpler than evaluating overlap between vibrational wavefunctions. This S is called the Huang-Rhys factor or sometimes poissonian parameter.

All sources except Atkins agree that it is I (the relative intensity) that should be called the Franck-Condon factor, so for the future I will update the project instructions (that used the name for S). You are still supposed to find a value for S in Exercise 3 and later, but it would be more correct to use the name "Huang-Rhys-factor" for it. Just to repeat, The Huang-Rhys factor then allows us to easily calculate any Franck-Condon factor I(n<-0) using equation (7). You will be excused if this naming confusion slips into your reports too

posted by Erik at Feb 16, 2012 9:20 PM