Kotani spin function

The Kotani spin functions for a six-electron singlet i constmcted by... 

The Kotani spin functions for a six-electron singlet 0oo yt be constmcted by successive coupling of six one-electron spin functions (a or p) to an overall singlet according to the mles for addition of angular momenta. Each spin function is uniquely defined by the series of partial resultant spins of the consecutive groups of 1, 2,. .., 5 electrons, which can be used as an extended label for the spin function... 

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. 

Just as in the case of the Kotani spin functions, the intermediate values of the total spin can be used in order to introduce a compact definition of a Serber spin function ... 

Corresponding to this we couple together Kotani spin functions from each subsystem to form complete functions of the form... 

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). 

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]. 

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,... 

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. 

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). 

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... 

Because of this, only one spin function is allowed that corresponding to a series of pairs of spins coupled to give singlets. In the Kotani scheme of ordering, this is the last function in the set, k = /s, as shown. If all the electrons are accommodated in doubly occupied orbitals, function (20) is nothing else but a single Slater determinant. 

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. 

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... 

This is not merely a question of theoretical interest, since it is intimately connected to important physical and chemical information which a spin-coupled calculation can yield about a given system. Certainly such a transformation is always possible and it remains to develop an efficient algorithm for carrying it out. This has been accomplished by means of the code SPINS, which transforms a set of spin-coupling coefficients between the Kotani, Rumer, and Serber bases of spin functions. It is also possible to combine any of these transformations with a reordering of the active orbitals in a SC wave function in any manner. 

Several simple models exist5 that approximately describe the temperature dependence of x for transition metal cations that do not represent spin-only centers. As one example that is applicable to coordination complexes at low temperatures, the Kotani theory6 incorporates the effects of spin-orbit coupling into the Van Vleck equation and describes y(T) as a function of the spin-orbit coupling energy C,. 

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... 

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). 

The methods that we consider fall into two types synthetic (see e.g. Kotani et ai, 1955), in which the full set of linearly independent eigenfunctions of given 5 and M is built up by some systematic procedure and analytic (Ldwdin, 1955, 1956a), in which a spin eigenfunction with the required values of 5 and M is extracted from an arbitrary function (i.e. a mixture of spin eigenfunctions) by means of a suitable projection operator. There are many possible procedures of both types, all exhaustively treated in the book by Pauncz (1979). 

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