In optical or core level spectroscopy, it is known that in dipolar electronic transitions between states (of well defined parity), one can have a dichroism or optical activity by properties of only the right symmetry, i.e. magnetic moments (axial vectors) but not electric polarization (polar true vectors). The situation reverses for mixed dipolar-quadrupolar transitions, dipolar electric-magnetic transitions etc, where the opposite symmetry is required for observables of properties. One can find in wikipedia, books or specialize articles tables of the symmetries.

My question is, is it possible to do in 2 or 3 lines, a very simple proof of this fact and with clear assumptions of the proof using Dirac notation and the symmetry or antisymmetry under P, T ? (and/or symmetry under parity or other of the electronic transition integral by using the fact that on "Fermi golden rule" the parity or symmetry of the integral is a product of symmetry properties of state 1 wavefunction, operator, state 2 wavefunction, i.e. like in addition of spherical harmonics...kind of the equivalent to 3D that the integral of an even function over a symmetric interval in 1D is obviously zero)

---- incorrect/non-rigorous outline of how I wish/pretend to do a proof -----

(It would be something like this but I fail to remember some of my faculty stuff from 15+ years ago, which is a bit of a shame and annoying...and I dont think it is correct not that I can write all the steps and assumptions out with minimum rigor/correction...sometimes I wish I could go back to University for 3 months or a semester but is not possible ...)

A transition between states 1 and 2 would be written as

< 2 | H+ | 1 > or < 2 | H- | 1 >

where I use H' for the perturbation or transition operator, using H+ and H- to denote the correspondence to transitions under C+ and C- light polarizations, and Bra and Ket Dirac notations for initial (1) and final states (2). This is Dirac formula, well know as Fermi golden rule #2.

If I consider a circular dichroism CD experiment, I can build the contrast as

CD= < 2 | H+ | 1 > - < 2 | AH- | 1 > . This still has not used the dipolar approximation.

1) proof reversing light polarization (using time reversal)

One could introduce the fact that light polarization reverses under time reversal, T, i.e. C+ transforms in C- under time reversal symmetry ,

CD_time= < 2 | H+ | 1> - < 2| TH+ | 1 >

And now I move the T operator onto the bra. This I dont recall very well but I have to use I think the adjoint operator or conjugate,

If the bra state < 2 |   is magnetic (i.e., it has well defined axial symmetry), it will change sign under time reversal to - < 2 | , and then I have a CD that is non zero; in contrast for states or properties symmetric under time reversal, it will not change sign, hence they will cancel and one gets null CD.

So probably not rigorously but if not fully incorrect gives a flavor...

2) proof reversing the state (using parity or time reversal)

Instead of changing the light or x-ray polarization, I try to use the change of the magnetic moment or the polarization vector, i.e. the property on the material. The first one I can use T, whereas for the second one I can use inversion symmetry, P.

Lets consider a polar state such as a state with an electric polarization moment

If the states are of well defined parity, P| 1>=-| 1> or P| 1>=| 1>, one would have:

P = +1 if the states correspond to a system described by a property that is axial like magnetism, that does not change sign under inversion, or

P = -1 if the system has polar character, like electronic polarization.

Lets assume we used C+ polarization, and we denote < 2' | and | 1'> the states where a property is reversed (magnetization, polarization,...)

CD_reversal = < 2 | H+ | 1> - < 2'| H+ | 1' >= < 2 | H+ | 1> - < 2P| H+ | P1 > =

If the states have well defined parity (assumption 1),

we get either P|1>=-|1> and P|2>=-|2> (or =|2>, but in any case the two signs factor whether positive or negative,

and one has if I correctly get the P operator out of bra and kets,

= < 2 | H+ | 1> - < 2| P' H+ P | 1 >

This clarifies that this is how the operator is transformed under parity (or any operation)

P' | H+| P

a) So if H+ commutes with P we use PH+ = H+P and P'P=Identity, so

= < 2 | H+ | 1> - < 2| P' H+ P | 1 >=< 2 | H+ | 1> - < 2| H+ P'P | 1 >=

=< 2 | H+ | 1> - < 2| H+| 1 > = 0

b) if H+ anticommutes with P, we have

= < 2 | H+ | 1> - < 2| P' H+ P | 1 >=< 2 | H+ | 1> - < 2| -H+ P'P | 1 >=

=< 2 | H+ | 1> +< 2| H+| 1 >=2< 2| H+| 1 >, generally different from zero

If one considers dipolar transitions, in the dipolar term, the operator has L=1, so it transforms under parity such as (-1)ˆL = -1, so we are on the second case b).

We seem to get the opposite result to the know result. So there is a mistake. The mistake relates to the fact that states 1 and 2 have different orbital moment in electronic dipolar transitions, where H+=E1, otherwise < 2 | E1 | 1>=0 ; but if states |1> and |2> have different orbital moment, i.e. one is a p state and the other a d state, they have different parity, so one reverses and the other not under parity, which goes as (-1)ˆL . So what has to be wrong is to consider that reversing the magnetic state represented by | 1'> and | 2'> can be simply written as P| 1> and P| 2>.

Anyone wants to have fun writing this in the correct way and note the assumptions? I would expect takes 3 lines and less than 5 min to a good university professor or chemist...

Greetings,

Manuel