Kotani theory

Table AlO.l Set of energy levels corresponding to Kotani theory...

These expressions were first derived by Kotani (by a less circuitous route ) and the simple theory which neglects distortion and the orbital reduction factor is called Kotani theory. Note that levels which are split apart in the more detailed analysis are degenerate in the Kotani treatment. 

Kotani, M., Proc. Shelter Island Conference on Quantum Mechanical Methods in Valence Theory, p. 139. Best orbital for the hydrogen molecule."... 

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 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. 

All this history is somewhat more fully discussed (from several points of view) in a couple review articles [2] as well as the first two chapters of [9] Valence-Bond Theory and Chemical Structure. These reviews describe through the period of this eclipse VB-theoretic work which was continued by a (prestigious or perhaps stubborn) minority of researchers (including Daudel, Hartmann, Simpson, Kotani, McWeeny, Lowdin,... 

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). 

The only magnetic data for the oxidation state II are those reported by Nyholm (337) on an unanalyzed sample. The susceptibility was low, only 150 x 10 c.g.s. units/mole, equivalent to 0.6 B.M., compared to the spin-only value of 2.88. Since Kotani s theory predicts only a reduction of 2.6 B.M. owing to spin-orbit coupling, antiferromagnetic interaction via direct metal—metal bonds is assumed to be operative here also. 

In the spirit of the Kotani/Toyozawa/Gunnarsson/Schonhammer approach, which was shown to work well for the calculation of the core levels of La203, the core level spectra of CeOj have been calculated (Jo and Kotani 1985 see also Schneider et al. 1985, Fujimori 1985). Figure 9 shows the calculated spectrum for one of the 3d3/2,s/2 doublets and the inset gives a measured spectrum (Kotani and Jo 1986). The calculation clearly reproduces the three final states which can be read off the diagram in fig. 7. The inset shows a measured 3d spectrum of Ce02 (similar to the one in fig. 8) and one can realize that the theory represents the experimental data quite well (Kotani and Jo 1986). 

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. 

A direct test of this birefringence variation has been carried out by Kotani and Sternstein by observation of the polarization pattern between crossed polaroids using a microscope. They have found a distribution of birefringence which quantitatively agrees with the prediction of their theory. 

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