Addition of orbital angular momentum in valence 4f^2 electronic configuration

[rlantha]

Main Question or Discussion Point

I am trying to self-learn quantum mechanics pertaining to Lanthanide ions.

For a given set of J and MJ quantum numbers in a valence 4f^2 electronic configuration, J=0,2,4,6 and MJ=0,6,-6. The |J,MJ> basis functions are |0,0>, |2,0>, |4,0>, 1/2[|6,6>+|6,-6>]+sqrt(1/2)|6,0>, and 1/2[|6,6>+|6,-6>]-sqrt(1/2)|6,0>. I understand that the MJ quantum number goes from 0,1,2,... to J. I am guessing that the coefficients 1/2 and sqrt(1/2) are Clebsch-Gordon coefficient?

Can someone explain how the |J,MJ> basis functions are obtained for J=6 and MJ=0,6,-6? Please refer me to a book or a website where I could understand better on this subject. Thank you.

 

[DrDu]

The last two functions are obviously not eigenfunctions of MJ. I guess that they are symmetry adapted functions pertaining to some special coordination geometry, e.g. cubic or the like?

 

[rlantha]

Thank you DrDu for your reply.

Yes. This is for a Pr3+ ion that resides in a D3h site symmetry (the crystal system is hexagonal).

I got this from a book, V.S. Sastri, J.-C. Bunzli, V. R. Rao, G.V.S. Rayudu, and J.R. Perumareddi, “Modern Aspects of Rare Earths and Their Complexes,” Elsevier 2003. This is a specialized book, and it assumes that the reader has prior knowledge on the subject.

I am able to understand how the J and the MJ values are derived, but I don't understand how the |J, MJ> basis functions are derived.

I did look into symmetry adapted linear combination (SALC), which I am foreign to. If I understand correctly, SALC requires that each of the |J,MJ> that are to be combined, i.e., |6,6>, |6,0>, and |6,-6>, is to be assigned to each of the symmetry operations in the site group. The symmetry operations for D3h site group are E, C3, C'2, sigma-h, S3, and sigma-v.

Could I know how one knows which of the symmetry operations should each of the |J,MJ> be assigned to?

 

[DrDu]

Are you familiar with group theory?
D3h is a subgroup of SO3, so you could work out the irreps working out the characters of the |J MJ> under D2h.
Obviously this can be looked up somewhere.
But I still don't see what precisely you want to learn.

 

[rlantha]

I have not had a formal course on group theory. Could you please suggest some introductory books to read that may help me apply on this topic?

I am trying to learn how to calculate the energy level splitting of Ln3+ in a crystalline environment. The crystalline environment is specified by one of the 32 point groups. The energy level is found by diagonalizing the energy matrix Hamiltonian. The Hamiltonian consists of the free-ion and the crystal-field terms.
For this example on Pr3+ ion, it consists of 7 2S+1L electronic states, 3(H,F,P) and 1(I,G,D,S), and in turns, it has 13 2S+1LJ terms.

Here, I am trying to understand how the crystal-field energy matrix Hamiltonian terms are derived. From the full-rotational symmetry group compatibility table ( G. F. Koster, J. O. Dimmock, R. G. Wheeler, and H. Statz, Properties of 32 Point Groups (M.I.T. Press, Cambridge, Mass., 1963), the irreps associated with the J quantum numbers (J=0,1,2...) for the specific site group can be determined.

The site group is also associated with crystal quantum number, Greek symbol mu, this assigns the irreps to the quantum number MJ's. In this case, the irreducible representation Gamma-1 (using Koster notation) is associated with J=0,2,4,6 and MJ=0,6,-6.

In order to derive the crystal field energy matrix elements <4f^2, 2S+1LJ,MJ|HCF|4f^2, 2S+1LJ,MJ > I need to understand how to derive the |4f^2, 2S+1LJ,MJ> basis sets.

 

[DrDu]

This will be difficult without some background in group theory. I think Cotton's book "Chemical applications of group theory" which is quite an easy read may be helpful for you to catch up the main concepts.

It should be clear that the functions |J MJ> span representations of the group D3h.
You have to work out the characters of the representations spanned by these functions.
For SO3 they are given as $\sin((M_J+1/2)\phi)/sin(\phi/2)$, where $\phi$ is the angle of rotation around an arbitrary axis
see. e.g.,
http://faculty.ksu.edu.sa/Kayed/eBooksLectureNotes/Group%20Theory%20%28Applications%20in%20Quantum%20Mechanics%29%20By%20Dimitry%20Vvedensky/GTChapter8.pdf [Broken]
You will need this for values π and 2π/3. Furthermore you have to figure out how the functions transform under inversions.
Then you can set up the character table for these representations in D3h and determine the irreps they carry by application of the orthogonality theorem.
In your case various short cuts are possible, e.g. most information should be obtainable considering only the sub groups SO2 and C3 as the action of the mirror and C2 rotations are almost trivial to take into account.

 

[rlantha]

Thank you DrDu for the link and for the book title. Could I also know where I could find basic information on the derivation of symmetry adapted functions?

 

[DrDu]

There is plenty of stuff on the net, e.g.:


http://www.google.de/url?sa=t&rct=j&q=characters projection operators salcs&source=web&cd=2&ved=0CCsQFjAB&url=http://goicoechea.chem.ox.ac.uk/teaching/s2/s2_lecture1.pdf&ei=77c4ULruEKTY4QTc9YCICw&usg=AFQjCNF7bL2II1TYXrIAtSUIqdZoS9hBTQ&cad=rja

 

[rlantha]

Thank you DrDu. I will take it from here. Regards.

 

[M Quack]

A classic on group theory is Tinkham

https://www.amazon.com/dp/0486432475/?tag=pfamazon01-20

For your case learning group theory will be helpful in the long run.

To understand the physics quickly you should consider crystal fields (sometimes called crystal electric fields but the main interaction is exchange). In chemistry they are called molecular fields.

The upshot is that the interaction of your local ion with the crystal lattice may be represented as a series of operators constructed from to momentum operator J.

Which forms of the operator are allowed or forbidden depends on the site symmetry. Group theory is very helpful for treating this systematically. This goes back to Hans Bethe who apparently worked this out as an exercise to better understand group theory. But you can just take the operator from literature and run with that.

The most important term is the one of lowest order in J. For uniaxial symmetries like hexagonal or tetragonal this term is proportional to Jz²-1/3 J(J+1)

So as an exercise you can try to work out eigenstates of J² and this operator - or just see if the states you cite are eigenvectors

 

[M Quack]

A classic paper on crystal fields in cubic symmetry is Lea Leask and Wolf J chem phys vol 23 pp 1381

Reading and understanding this paper and then reproducing their results is very highly recommended.

Cubic symmetry is the easiest case because there are only two CF parameters, one of which is a scaling factor.