Spherical polar coordinates inverse

For this integral to be non-zero, the integrand must be invariant to the inversion operation. We have shown above that the product under inversion, and the changes in the spherical polar coordinates under this operation are... 

Show that the spherical harmonic function Poo eigenfunction of the inversion operator with eigenvalue 1, while the spherical harmonic function Y is an eigenfunction with eigenvalue — 1. In spherical polar coordinates the inversion operator replaces 0by tt — 0 and replaces by n + (j). Show that rotation of 180° around an axis perpendicular to the bond axis gives the same result. 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

It is known, that axial crystalline field constant D in EPR spectra is proportional to polarization P or for paramagnetic centers without or with inversion center respectively. The coordinate dependence of polarization inside the spherical nanoparticle can be obtained by corresponding Euler-Lagrange equation solution, as it was discussed earlier in the Chap. 3, Sect. 3.2.2.3. The dependence of polarization on r and R can be written as follows [93] ... 

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