Oscillator strengths

What is the physical significance of the oscillator strength? Following Werner Kuhn's arguments (e.g. in this paper), it marks the number of electrons oscillating per spatial dimension during an electronic transition. The sum over the oscillator strengths of all the excited states amounts to the number electrons, which is the essence of the Thomas-Reiche-Kuhn sum rule. In other words, the oscillator strength counts how much of the total oscillating potential is used for a specific transition.

This interpretation explains for example the linear relationship of the oscillator strength of the lowest excited state with system size in the case of some conjugated organic polymers (see e.g. this paper): If there are more electrons available to oscillate, then the transition strength increases.

The oscillator strength f_ij between two non-degenerate states i and j is defined (in atomic units) as two thirds of the squared transition dipole moment multiplied by the energy gap

where the vector r contains all 3N spatial coordinates of the N electrons

The Thomas-Reiche-Kuhn sum rule now states that the sum over the oscillator strengths from one state i to all possible other states is equal to the number of electrons in the system, i.e.

In particular, if we consider excitations from the ground state, then all oscillator strengths are positive. Which means that the oscillator strengths can in fact be viewed as a partitioning of the number of electrons.

The derivation of this sum rule starts by realizing that the momentum operator with respect to any spatial coordinate x of any particle (e.g. x=y₂) is given as the commutator of the Hamiltonian with this coordinate

This follows whenever H is of the form

where clearly the derivatives with respect to x are the only part, which does not commute with x itself

By applying the product rule twice, the first term of this expression becomes

And in summary

The remaining proof follows what is shown here (sorry that I am switching the notation, but I copy-and-pasted a little bit ...). First one realizes that the commutator of x and p_x is equal to i

Then one expands the commutators and inserts a resolution of the identity over the eigenstates of the Hamiltonian

Insert the above expression for p_x

The commutators are evaluated by letting H act either on the bra or the ket, which results in a multiplication with the respective eigenvalue. And after summing together the equivalent terms one obtains

The actual r vector was composed of 3N individual electron coordinates. The above equation holds for each of these coordinates. Thus, in summary:

which is just what we wanted to show.

Edit: The discussion is also related to the relation of the length and velocity gauge for computing oscillator strengths. Some more information here.