Spherical harmonic representation

In the literature three different approaches were reported based on the spherical harmonics representation of the SODFs by Wang et by... 

In fact. Section 4.1 is geared to the spherical harmonic representation emphasizing that the functions involved are eigenfunctions of L and L, for which the unit vectors i /f matching the eigenfunctions are appropriate. Now the functions involved are common eigenfunctions of L , H, and the parity operators, calling for fhe representation... 

The distributed multipole analysis method of Stone and co-workers is similar in concept but is based on nonredundant spherical harmonic representation of the multipoles (recall that whereas there are six second moments, only five are independent). He initially places numerous site multipoles at centers of orbital overlap. The individual monopoles are spread out along the molecular axis, and are thought to represent the distribution of charge the site dipoles are also spread out along the bond axis. This very detailed description is simplified into a three-site model, which includes a site in the F—H bond. However, the multipole expansion does not converge well, especially for the bond center site. 

Potential coefficients coefficients in the spherical harmonic representation of the gravitational potential U by the equation... 

Figure 10. The coefficients g(R) in a spherical harmonics representation of orientation correlation in liquid chloroform CHCil3...

In practice the evaluation of the ERIs in the real spherical harmonic representation runs over the Cartesian representation and a transformation occurs between the Cartesian and the real spherical harmonic representation (see Table 2). This is not due to the lack of explicit formulae for ERI generation in the real spherical harmonic Gaussian representation but rather a matter of implementation. The transformation is expressed as... 

Cartesian Versus Real Spherical Harmonic Representation... 

Figure 15 Spherical harmonic representations " of the mosaic virus Ivtm viral structure. Viral RNA is included in the helix. Structure was constructed with the symmetry server library within the AVS data-flow visualization environment. (Image courtesy of Bruce Duncan, The Scripps Research Institute, La Jolla, CA)...

Figure 22 Construction of the icosahedral capsid of poliovirus using a low-order spherical harmonic representation of the viral protomers (one copy each of the proteins VPl, VP2, VP3, and VP4). " The individual boundaries of the four protein chains are texture mapped onto the surface. One pentameric assembly intermediate has been translated away from the capsid along a five-fold axis. The model was constructed, and can be manipulated interactively in real time using the symmetry server library developed in the Olson group, within the AVS data-flow visualization environment. (Image courtesy of Arthur J. Olson, The Scripps Research Institute, La Jolla, CA)...

The representation of foe angular part of foe two-body problem in spherical harmonics, as developed in Section 6.4, is applicable to any system composed... 

To find the irreducible representations of 0(3) it is necessary to find a set of basis functions which transform into their linear combinations on operating with the elements of 0(3). The set of 21 + 1 spherical harmonics Y[m(d, ), where l = 0,1, 2... and —l[Pg.91]

A problem that arises in connection with the construction of the basis is that of finding what are the allowed values of the quantum numbers of the subalgebra G contained in a given representation of G. For example, what are the allowed values of Mj for a given J in Eq. (2.12). In this particular case, the answer is well known from the solution of the differential (Schrodinger) equation satisfied by the spherical harmonics (see Section 1.4), that is,... 

The applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

For each nonnegative integer f, the space of spherical harmonics of degree f (see Dehnition 2.6) is the vector space for a representation of 50(3). These representations appear explicitly in our analysis of the hydrogen atom in Chapter 7. Recall the complex scalar product space L (S ) from Definition 3.3. 

In this section we use the results of Section 7.1 and our knowledge of irreducible representations to show that the spherical harmonic functions span the space of square-integrable functions on the two-sphere. In other... 

We choose the following representation for the four-dimensional spherical harmonics. We set... 

While we have chosen to proceed here by reducing representations for the full group D3h, it would have been simpler to take advantage of the fact that D3h is the direct product of C3u and C where the plane in the latter is perpendicular to the principal axis of the former. The behaviour of any atomic basis functions with respect to the C3 subgroup is trivial to determine, and there are only two classes of non-trivial operations in C3v. In more general cases, it is often worthwhile to look for such simplifications. It is seldom useful, for instance, to employ the full character table for a group that contains the inversion, or a unique horizontal plane, since the symmetry with respect to these operations can be determined by inspection. With these observations and the transformation properties of spherical harmonics given in the Supplementary Notes, it should be possible to determine the symmetries spanned by sets of atomic basis functions for any molecular system. Finally, with access to the appropriate literature the labour can be eliminated entirely for some cases, since... 

We list here full matrix representations for several groups. Abelian groups are omitted, as their irreps are one-dimensional and hence all the necessary information is contained in the character table. We give C3v (isomorphic with D3) and C4u (isomorphic with D4 and D2d). By employing higher 1 value spherical harmonics as basis functions it is straightforward to extend these to Cnv for any n, even or odd. We note that the even n Cnv case has four nondegenerate irreps while the odd n Cnv case has only two. 

Fig. 3.14 Representation of the parfde-on-a-sphere wavefunctions (spherical harmonics) with 1=0,1, and 2, given in Table 3.1.

On the U(l) level, the transverse components of eM are physical but the longitudinal component corresponding to M = 0 is unphysical. This asserts two states of transverse polarization in the vacuum left and right circular. However, this assertion amounts to Cq = e[i = 0, meaning the incorrect disappearance of some vector spherical harmonics that are nonzero from fundamental group theory because some irreducible representations are incorrectly set to zero. 

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