Spherical harmonics tensor operators

There is no paradox [112] in the use of e(3) as an operator as well as a unit vector. In the same sense [112], there is no paradox in the use of the scalar spherical harmonics as operators. The rotation operators in space are first-rank Toperators, which are irreducible tensor operators, and under rotations, transform into linear combinations of each other. The Toperators are directly proportional to the scalar spherical harmonic operators. The rotation operators, J, of the full rotation group are related to the T operators as follows... 

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

In these expressions the quantity Cj,11 is proportional to the spherical harmonics ylp(f) and represents a tensor operator of rank 1 and component p. Results for both linear and circular polarizations can then be comprised in... 

The crystal field interaction can be treated approximately as a point charge perturbation on the free-ion energy states, which have eigenfunctions constructed with the spherical harmonic functions, therefore, the effective operators of crystal field interaction may be defined with the tensor operators of the spherical harmonics Ck). Following Wyboume s formalism (Wyboume, 1965), the crystal field potential may be defined by ... 

Y x) denotes a spherical harmonic. The scalar product between two tensor operators Cl(y) Cl(x) is given by... 

The simplest example of the Wigner—Eckart theorem is given by the Gaunt integral over three spherical harmonics, which is the matrix element for the transition between eigenstates m) and fm ) of a single orbital angular momentum observable due to a tensor operator Tj. We prefer to use the renormalised tensor operator C, which simplifies the expression. 

Similar to the ODF for texture, SODF can be subjected to a Fourier analysis by using generalized spherical harmonics. However, there are three important differences. The first is that in place of one distribution (ODF), six SODFs are analyzed simultaneously. The components of the strain, or the stress tensor can be used for analysis in the sample or in the crystal reference system. The second difference concerns the invariance to the crystal and the sample symmetry operations. The ODF is invariant to both crystal and sample symmetry operations. By contrast, the six SODFs in the sample reference system are invariant to the crystal symmetry operations but they transform similarly to Equation (65) if the sample reference system is replaced by an equivalent one. Inversely, the SODFs in the crystal reference system transform like Equation (65) if an equivalent one replaces this system and remain invariant to any rotation of the sample reference system. Consequently, for the spherical harmonics coefficients of the SODF one expects selection rules different from those of the ODF. As the third difference, the average over the crystallites in reflection (83) is structurally different from Equations (5)+ (11). In Equation (83) the products of the SODFs with the ODF are integrated, which, in comparison with Equation (5), entails a supplementary difficulty. 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

We have already seen that the crystal field potential can be rather simply expressed through a set of radial parameters and the irreducible tensor operators Tk m which refer to a certain, symmetry-predetermined combination of spherical harmonic functions (see Table 8.12). Such an expression can be written in several equivalent forms... 

Here y m are the spherical harmonic functions Q m = yj47r/(2k + 1) y, m is the Racah tensor operator = rk Ykm is the irreducible tensor operator Pk m (not to be confused with the Legendre polynomials) are unnormalised homogeneous polynomials of Cartesian coordinates proportional to the function rk Ykm + Yk m) Ok are referred to as equivalent operators which are constructed of only the angular momentum operators. 

Table 1. Tensor Operators in the Rotating Frame and Modified Spherical Harmonics for the Dipolar and CSA Interaction. The calibration has been chosen such that (j>) 0) d (cos 0)d is independent of q. ...

Tensor Operators for the Dipolar Interaction Tensor Operators for the CSA Interaction Modified Spherical Harmonics Frequency... 

The various mechanisms of mixing are thoroughly discussed by Wybourne (2). One of the most important mechanisms responsible for the mixing is the coupling of states of opposite parity by way of the odd terms in the crystal field expansion of the perturbation potential V, provided by the crystal environment about the ion of interest. The expansion is done in terms of spherical harmonics or tensor operators that transform like spherical harmonics. This can be formulated in a general Eq. (1)... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

The spin and orbital gyromagnetic ratios g, and gj are the free nucleon values Sg, and 8gi are caused by both core polarization and meson exchange. The last term, arising from the dipole-dipole interaction, is a rank-one tensor product of the spherical harmonic of order 2, Y, and the spin operator. The separate contributions of Sg, 8g(, and gp can be extracted from experimental data that include high-spin and low-spin states and can also be calculated by theory. For the rather extensive evaluation of the neighborhood of 2o pjj/i57) gystematjcg hag en closed... 

Philip and Kuchel described a graphical representation of the spin states of quadrupolar nuclei in NMR experiments. The spherical harmonics represent the irreducible spherical tensor operator basis set, and therefore any density matrix can be illustrated with computer graphics by converting it to a sum of... 

There is a lot compressed in this expression so it is weU that we spend some time unravelling it First of aU, the subscripts t and u label the angular momenta of the real spherical harmonics and take the Im values 00,10,11c,11s,20,21c,21s,22c,22s,--- (the labels c and s stand for cosine and sine respectively). Qf is the real form of the multipole moment operator of rank t centered on A and expressed in the local-axis system of A, while T/J is the so-called T-tensor that carries the distance and angular dependence. The T-function of ranks h and I2 has a distance dependence where R is the separation between the centers of A and B. 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

A is the measured differential hfs anomaly, determination is that it provides a new operator ent from the magnetic moment (] = g + gjW, with which to test nuclear wave functions. The square Bracket is a tensor product of order 1 between the spin operator and the spherical harmonic Y. It is a measure of the angular asymmetry of the nuclear spin distribution. 

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