By W. G. Harter, C. W. Patterson

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**Extra resources for A unitary calculus for electronic orbitals**

**Example text**

However, the explicit Pauliantisymmetric states of orbit and spin (viz. 45. 46. The first term has an orbit factor with the "first tableau"~, (In~, and the spin factor with the "last tableau"~ i. ) Each following term has in the orbit factor, an allowed which is a permutation of numbers i n ~ 1, and this term is positive (negative) if this permutation is even (odd). (45) % • • I ; ~j =~i • I~ 56 ~~ 12 7 , 4' 2 - ~/~ 4 + I/~r~- 2 3 2 , 6 12Lr , 7 3 123 , I' ,I",I, 7 I 4 @ , ~' I ' 4 , 112. 33 67 @@ = i/~ 34 , 6 24 , @ @ - i/~ 24 , 67 34 , @ @ It is undeed fortunate ity does not have to be brought that most of this complexout again when this math- ematics is used for physical problems.

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. > (28) ~> [%

33 67 @@ = i/~ 34 , 6 24 , @ @ - i/~ 24 , 67 34 , @ @ It is undeed fortunate ity does not have to be brought that most of this complexout again when this math- ematics is used for physical problems. notation and accompanying However, correspondences calculus In this respect the is most efficient. the proofs of tableau formulas and the 26 between Gelfand and tableau approaches have been quite difficult former is algebraic to make, possibly because the while the latter is not. The nature of this dilemma is seen in the proofs and discussion here, in 27 Part II, and in works by others.