Nuclear spin additional functions

If any atoms have nuclear spin this part of the total wave function can be factorized and the energy treated additively. ft is for these reasons that we can treat electronic, vibrational, rotational and NMR spectroscopy separately. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

The impurity interacts with the band structure of the host crystal, modifying it, and often introducing new levels. An analysis of the band structure provides information about the electronic states of the system. Charge densities, and spin densities in the case of spin-polarized calculations, provide additional insight into the electronic structure of the defect, bonding mechansims, the degree of localization, etc. Spin densities also provide a direct link with quantities measured in EPR or pSR, which probe the interaction between electronic wavefunctions and nuclear spins. First-principles spin-density-functional calculations have recently been shown to yield reliable values for isotropic and anisotropic hyperfine parameters for hydrogen or muonium in Si (Van de Walle, 1990) results will be discussed in Section IV.2. 

Fig. 9 Nuclear spin-spin coupling constants J(195Pt-205Tl) for complexes I-V (see Fig. 8), from ZORA DFT computations. Data taken from Autschbach and Le Guennic [126]. Different computational models were applied Model A includes explicit water molecules. In Model B, a continuum model (conductor-like screening model, COSMO) is applied in addition to the explicit solvent molecules of model A. Model C differs from model B in that instead of the VWN functional the statistical averaging of orbital potentials (SAOP) XC potential was used, which allows more accurate computations of NMR parameters [32]. The NMR measurements were carried out in aqueous solution [99,130]...

The analysis of the spectrum was accomplished using a case (a) basis, with the addition of two nuclear spins, I and h, for 63Cu and 19F respectively. The basis functions therefore take the form A S, E J, 72, I, F, h- F, Mp), and leaving aside nuclear spin interactions, the theory follows closely the same path as that already described for 3n CO in chapters 9 and 10. The effective Hamiltonian is the sum of terms representing the spin orbit, rotational, spin-rotation, spin-spin and centrifugal distortion contributions and is written [56] ... 

An additional constant factor enters into these expressions if the nuclei have nonzero spins. If i and are the two nuclear spins of a diatomic molecule, the complete rotational function is written as... 

Finally, nuclear spin also produces an additional constant factor in the partition function a nucleus with spin quantum number s contributes a factor 2s + 1 to... 

For a macroscopic sample it is necessary to define a different set of wave functions because the spins are in a mixed state. The mixed state indicates that the wave functions of a particular nuclear spin are subject to additional molecular contributions that might differ over the whole sample. The expectation value of a mixed state now uses the averaged coefficients and is... 

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