How Can I write the Hamiltonian for Helium Atom including spin orbit interaction term ?

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Pedro L. Contreras E.

I really don't know much about electronic structure calculations, Prof. Nityananda Das, but:

This is what books suggest, you can try this methodology in one of the free software available without much computational cost since the LCAO atomic orbitals implementation are much cheaper computationally than any grid solutions:

1. The HSO contains three terms (see *) The first contributes to the energy of the system, while the second and third (they are usually called "spin-other orbital interactions ") make much smaller contributions.

* https://xumuk.ru/encyklopedia/2/4171.html

2. For spin-orbit interactions, you can neglect the second and third terms in the hamiltonian.

3. Then with the first term, you can implement the LCAO-approximation.

https://xumuk.ru/encyklopedia/2361.html

https://chem.libretexts.org/Courses/New_York_University/CHEM-UA_127%3A_Advanced_General_Chemistry_I/14%3A_Linear_combination_of_atomic_orbitals

The best way to do all this is with a robust free electronic package calculation.

Chinaza Godswill Awuchi

Kindly check https://www.physik.fu-berlin.de/einrichtungen/ag/ag-heyne/docs/teaching/SoSe08/Amol1/AMol_Teil2.pdf

Also check http://www.umich.edu/~chem461/QMChap8.pdf

Nityananda Das

Thanks a lot Dear Chinaza Godswill Awuchi.

I am trying to get it solved.

Thanks

N Das

I found what I was looking for since the spin-orbit coupling has to be treated on a different basis, Prof. Nityananda Das

Hso= sum over i of ( ai [ lzi szi + 1/2 (li+ si− + li− si+)] )

Notice the helicity of the new spin operator s with + and - in the XY plane, in the spin-orbit coupling, spin is not a good quantum number, so it changes to a more general helicity meaning. The decoupled transformation of the basis.

This is a hard question, and it is not usually explained in QM books.

http://www2.chem.umd.edu/groups/alexander/teaching/

http://www2.chem.umd.edu/groups/alexander/teaching/spin-orbit_coupling.pdf

Tanvir Singh

Agreed Chinaza Godswill Awuchi and Pedro L. Contreras E.

Aref Wazwaz

Hi Dr Nityananda Das . I hope the following link could help . Best wishes. Kindly see the link:

https://www.physik.fu-berlin.de/einrichtungen/ag/ag-heyne/docs/teaching/SoSe08/Amol1/AMol_Teil2.pdf