Spherical harmonics, tensor properties

In this book, we adopt a form in which the function is expressed as a linear combination of spherical harmonics. This form is particularly appropriate for systems with near-spherical symmetry (such as Rydberg states or molecules which conform to Van Vleck s pure precession hypothesis [68, 69]) and is also consistent with the spirit of spherical tensors, which have the same transformation properties under rotations as spherical harmonics. The functional form of the ket rj, A) is written... 

Transition metal clusters also have Axy and atomic orbitals, which are classified as 5-type in TSH theory. To represent the transformation properties of these orbitals, we use second derivatives of the spherical harmonics, that is, tensor spherical harmonics - hence the name of the theory. As for the vector surface harmonics, there are again both odd and even 5 cluster orbitals, denoted by L and L, respectively. Usually, both sets are completely filled in transition metal clusters, and we will not consider their properties in any detail in this review. However, the cases of partial occupation are important and have been described in previous articles. ... 

Since the transformation properties of spherical harmonics are well known, the spherical-tensor notation has some advantages, particularly in the derivation of general theorems however, the reality of cartesian tensors also has its attractions, especially for small values of /. Normally, the moments of a particular three-dimensional molecule are most conveniently given in an x,y,z frame. 

To study the bonding in transition metal cluster compounds, a new type of Spherical Harmonic, with tensor properties, is required. This is because the metal d orbitals have (in addition to 1 o and 2 jt components) 2 8 components (d and dx2 y2) which are doubly noded in the plane perpendicular to the radial vector (see Fig. 16c) and which, therefore, behave as tensors. Two Tensor Surface Harmonic functions may be obtained from each Scalar Spherical Harmonic as follows146 ... 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

These techniques of electromagnetic absorption spectrometry readily give the orientation parameter (P2(cosar)) with respect to external axes. Combined with the appropriate form of microscopy the methods can, in principle, give orientational information on the scale of the microstructure. However, the absorption property is a second-rank tensor, which means that it is only able to give the orientational information contained in the first spherical harmonic component, Pzicos or)). The higher-order components and hence the full orientation function are inaccessible. The techniques do have the advantage of speed of measurement and provide a real-time evaluation procedure. 

The spherical harmonics y (i) can be replaced by the tensor operators C (i), which have the same transformation properties as the spherical harmonics ... 

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