Kotani basis

The spin-coupling pattern go for the reordered orbital set is largely dominated by its perfect-pairing component [ qq-s = in the Kotani basis, see Eq. (5)], in... 

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... 

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,... 

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... 

Of course for equilibrium intemuclear distances, HCN is better described in terms of electron-pair bonds, i.e., in the Serber or Rumer basis of spin functions. This brings us to the problem of transforming from one set of spin functions to another. In practical terms the problem is as follows. A calculation has been carried out using, e.g., the Kotani basis (which may be the most convenient) and we have... 

For (a), of course, the choice of spin basis may be very important for highlighting different features of the spin coupling, with our most common choices being the Rumer, Kotani, Serber, or projected spin function bases. Transformation between these (complete) bases is, in any case, very straightforward [36,66,67]. 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

In the case of the 1,3-dipolar cycloaddition offulminic acid to ethyne, none of the three common spin bases leads to any particular interpretational advantages. For this reaction, we report the composition of the optimal spin-coupling pattern (2) in the Kotani spin basis, which is orthonormal as a result, the weights of the individual spin functions making up oo are given simply by the squares of the corresponding spincoupling coefficients,... 

For a further description of this basis and how the US(P) matrices in it are constructed, the reader is referred to the article by Kotani et al.s... 

SC theory does not assume any orthogonality between the orbitals ij/ which, just as in the GVB-PP-SO case, are expanded in the AO basis for the whole molecule Xp P 1,2,..., M. The use of the full spin space and the absence of orthogonality requirements allow the SC wavefunction to accommodate resonance which is particularly easy to identify if 0 sm is expressed within the Rumer spin basis. In addition to the Rumer spin basis, the SC approach makes use of the Kotani spin basis, as well as of the less common Serber spin basis. When analysing the nature of the overall spin function in the SC wavefunction (3.9), it is often convenient to switch between different spin bases. The transformations between the representations of 5M in the Kotani, Rumer and Serber spin bases can be carried out in a straightforward manner with the use of a specialised code for symbolic generation and manipulation of spin eigenfunctions (SPINS, see ref. 51). 

While the concentration dependence of the experimental fields are reproduced rather well by the theoretical fields (a phase transition to the BCC structure occurs around 65% Fe), the later ones are obviously too small. This finding has been ascribed in the past to a shortcoming of plain spin density functional theory in dealing with the core polarization mechanism (Ebert et al. 1988a). Recent work done on the basis of the optimized potential method (OPM) gave results for the pure elements Fe, Co and Ni in very good agreement with experiment (Akai and Kotani 1999). 

Kotani, A., S. Kojima, Y. Hayashi, R. Matsuda, and F. Kusu. 2008. Optimization of capillary liquid chromatography with electrochemical detection for determining femtogram levels of baicalin and baicalein on the basis of the FUMI theory. J. Pharm. Biomed. Anal. 48 780-787. 

Projected spin functions have recently been reintroduced by Friis-Jensen and Rettrup (see also Refs. 12, 13, and 18). These spin eigenfunctions are linearly independent, but are nonorthogonal. They have been introduced into some versions of the spin-coupled codes and, using modem technology, provide a useful shortening of execution times, sometimes by as much as a factor of four. The main drawback of the projected spin functions is their lack of physical interpretability. However, the resulting SC coefficients can always be transformed into a more familiar representation such as the Kotani, Rumer, or Serber basis. Used in this way, the projected spin functions provide a useful addition to the SC codes. 

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