Yamanouchi-Kotani spin

The c a are the spin-coupling coefficients of the a-th configuration. One should mention that in the earlier MCSC version of the method [1] all orbital configurations shared the same linear combination of the f(Ne,S) Yamanouchi-Kotani spin functions there was a single set of f(Ne,S) spin-coupling coefficients, denoted simply csk, just as in single-configuration spin-coupled theory. 

The spin functions obtained this way are usually referred to as Yamanouchi-Kotani functions. 

Finally, s,M a is the linear combination of Yamanouchi-Kotani (YK) spin functions associated with the a-th configuration... 

When the orbitals are ordered so that the first two are the inner orbitals and, if a valence orbital is even-numbered (odd-numbered), its symmetry-equivalent counterparts also are even-numbered (odd-numbered), then the spin part of the SC wavefunction is dominated by the perfect-pairing Yamanouchi-Kotani (YK) spin function, with a coefficient exceeding 0.99. The coefficients of the other 13 YK functions are all smaller than 0.01. 

The Yamanouchi-Kotani basis in the 77-electron --adapted spin space is closely related to the standard Young tableaux used in characterizing irreps of the symmetric group [50] and is conveniently represented by Van Vleck s branching diagram [18, 42]. To a basis function QfM we assign an array... 

Figure 1 Van-Vleck s branching diagram (A) and the reversed branching diagram (B) for S = 1, N = 8. Either 6 (full lines) or 8 (full and broken lines) spins are coupled in all ways allowed by the Yamanouchi-Kotani scheme. At vertices and arcs their weights (if different from 0) are shown.

The Yamanouchi-Kotani basis is best suited if we want to solve the Heisenberg problem in the complete spin space. However, the number of spins that can be handled this way, soon reaches an end due to the rapid growth of the spin space dimension f(S,N). Even with the present day computers, the maximum number of spins that can be treated clusters around N = 30. For larger values of N one must resort to approximate treatments, one of which, as described hereafter, is based on the idea of resonating valence bonds (RVB) coming from the classical VB model developed by Pauling and Wheland back in the early 1930 s [37, 51]. In essence,... 

Fig. 3 Yamanouchi-Kotani branching diagram governing the spin-levels of d for N = Nsvin = 1, 2, 3, 4, 5 (see Ref. 19, p. 44)...

Spin functions constructed in this way are known as Kotani-Yamanouchi or simply Kotani spin functions after those who introduced them. They are orthonormal (see equation 14). 

Serber[12] and Van Vleck and Sherman[13] continued the analysis and introduced symmetric group arguments to aid in dealing with spin. About the same time the Japanese school involving Yamanouchi and Kotani[14] published analyses of the problem using symmetric group methods. 

Serber[15] has contributed to the analysis of symmetric group methods as an aid in dealing with the twin problems of antisymmetrization and spin state. In addition, Van Vleck espoused the use of the Dirac vector model[16] to deal with permutations. [17] Unfortunately, this becomes more difficult rapidly if permutations past binary interchanges are incorporated into the theory. Somewhat later the Japanese school involving Yamanouchi[18] and Kotani et al.[19] also published analyses of this problem using symmetric group methods. 

The set of spin functions constructed in this way is commonly termed the standard or Kotani basis—though a more correct name would be the Young-Yamanouchi basis . There are, however, many other possible bases of spin functions (in general, there is an infinite number of choices) and we... 

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