By W. G. Harter, C. W. Patterson
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Additional info for A unitary calculus for electronic orbitals
S=i- L>M M fixed MI (io) Certain properties of the S operator minimize the computational labor. First, < S > is a symmetric projector so its rows or its columns are the components of the desired eigenvectors of L 2. S = S) so the scalar product (ilj) of the i-th row vector (ilwith the j-th column vector matrix
For (d) 3doublets 33 ' ,. Then all the U m (m=2~+1) "lowering operators" E21, E32'E43 . . . Em m-1 are applied once to the initial tableau. The new tableaus that result are transcribed along with the matrix elements (Recall Fig. 9) for the change. Only tableaus that have the numbers equal or increasing toward the right in their rows, and increasing but not equal down their columns, are transcribed, For (d) 3 doublets we obtain 11 the tableaus ~2 and 3 =1 a n d < ~ l l with matrix elements -2 <121 E21 211~ / E321 ~1~ =i respectively.
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.