By W. G. Harter, C. W. Patterson
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Extra info for A unitary calculus for electronic orbitals
I ~Ic~:~l~ -,>. ~---~ : c~ . ~ I . . • I . ,, ~ ... . - :: ~ "~ ~',-i ~- ~ • ~-~ . ~i~i~---~. SSs L~L~I~ ~ ..... ~ ..... ,--i II ~-4 L",I O,l . 25. 26a) and the desired interaction matrix (Eq26b). ,,31T> o -9IT> - (26c) v3. ) to label the repeated states. V ~ are multiples in the ([It , LII> ) representation,(M 1 of the unit matrix is the U s invariant, 1 while V -V is proportional to L 2) so eigenvectors of V3. 28 must also be eigenvectors ~I~'~ of the pairing operator.
4) L ~ ~v~ = 2E21 + (gE32+ ~ E 4 3 + 2E54 For any atomic configuration the highest tableau always corresponds to the highest L term in the multiplet with z-component of orbit M=L. For the example 121~= 12H L=5, M=5~. 6 and an orthogonalization which is described below, gives all the other terms. " I 41 For the example we obtain the 12G 4,4~ orthogonalization with the 12H 5,4) state (Eq. 8. 8c. lO. The sum there is over the equal M vectors previously derived such as those in Eq°9. s=i- L>M M fixed MI (io) Certain properties of the S operator minimize the computational labor.
Nevertheless, we expect that one will generally prefer to use computing machines to perform calculations, that applications and so it is important of the calculus be explained in a straight- forward way that can be coded. The application lar factor analysis below in Sec. of the unitary formulation of complex atomic spectra is described i. Here we find one set of formulas and rules that are the same for all configurations~ and no matter how many sub-shells or angular momentum conservation with independent pure or mixed, or electrons We explain a way to take advantage are involved.
A unitary calculus for electronic orbitals by W. G. Harter, C. W. Patterson