Spherical basis vectors

To determine the matrix, let us introduce the basis vectors and ey with respect to the plane throngh the vectors n,. and. The vector is perpendicnlar to the reference planes, whereas the vector Cy is parallel to them. Transforming these vectors into spherical basis vectors yields the contravariant spherical basis vectors [30] e"(nj ) and e (n J, which are rotated with respect to the vectors e (n,) and (n ) through the angles

[Pg.225]

To bring out the symmetry character, however, it is much more convenient to introduce spherical basis vectors that transform under rotation like multipole moments, that is monopole, dipole, quadupole, etc. The spherical basis vectors are defined by... 

M= 0 because there is no < -dependence owing to the assumed axial symmetry of the problem with respect to the Z axis. Thus, the spherical basis vectors transform under rotation as the Legendre polynomials Pl(cosO), see Eq. (10.23) below, where classically 9 is the angle between J and the Z axis, see Figure 10.7. The spherical basis vectors also satisfy the orthonormaUty condition, T Ty = Sy. Then N may be expanded in terms of the spherical basis set as... 

The factor C k of B(R) is often called the /th component of the Racah spherical tensor of rank k the three tensor components of rank 1 may be considered unit basis vectors spanning the (spherical) space. 

We will use the basis vectors (1) where > i2 and apply equation (4) when needed. For two tlu electrons, our basis (1) consists of 15 different state vectors I/) (for two holes, the fivefold hu degeneracy leads to 45 states). In the following we will study the intramolecular correlations of electrons (holes) within a multipole expansion of the two-body Coulomb potential V(r, f) = 1/lr —1 (charge e = unity). In terms of real spherical harmonics YJ, where r stands for m = 0,... 

In the discussion of light polarization so far the Cartesian basis and spherical basis have been considered. Because the linear polarization might be tilted with respect to the (ex, e -basis, a third basis system has to be introduced against which such a tilted polarization state can be measured via its non-vanishing components. This coordinate system is called (e e and its axes are rotated by +45° with respect to the previous ones. This leads to a third representation of the arbitrary vector b ... 

In (16) and (18) an expression of the type (30) has been used also for the spherical basis due to a confusion of the transformation properties of basis vectors and components. However, for Me, called simply M in (16, Eq. 4.43) the standard order of functions is ex Yez so that an inversion of the matrix elements in the center of the matrix connects our Me and M of (16). The relation between our D0>> of Table 2 and itf of (16) is therefore a reflection of the matrices in the secondary diagonal. In both cases the translation should be accomplished by the transformation (a/ y) = ([Pg.79]

This transformation is in accordance with the Condon-Shortley phase conventions for the spherical basis functions [7]. In fact, our initial Hamiltonian matrix in Eq. (7.21) was constructed in this way. The resulting vector corresponds to the triplet spin functions, which we used in Sect. 6.4. The total spinor product space has dimension 4. The remainder after extraction of the three triplet functions corresponds to the spin singlet, which is invariant and transforms as a scalar. Spinors are thus the fundamental building blocks of 3D space. Their transformation properties were known to Rodrigues as early as 1840. It was some ninety years before Pauli realized that elementary particles, such as electrons, had properties that could be described... 

In the following, vectors are boldface, scalars are not. Basis vectors are i, j, k in Cartesian coordinates and f, 6, in spherical coordinates. Alternative notations associated with the vector A and scalar A are listed in Table Al.l. 

Steric effects are vector quantities. This is easily shown by considering, for example, the pentyl and the 1,1-dimethylpropyl groups which have the same volume but a different steric effect. That of the former is less than half that of the latter. In order to account for this let us examine what happens when a nonsymmetric substituent is in contact with an active site. Consider, for example, the simple case of a spherical active site Y, in contact with a carbon substituent, CZLZMZS, where the superscripts, L, M and S represent the largest, the medium-sized and the smallest Z groups, respectively. There are three possible conformations of this system which are shown in topviews in Figure 1. As all steric interactions raise the energy of the system, the preferred conformation will be the one that results in the lowest energy increase. This is the conformation which presents the smallest face to the active site, conformation C. This is the basis of the minimum... 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

Let us now consider the overlap between the spherical and the Stark basis. For the latter, the momentum space eigenfimctions, which in configuration space correspond to variable separation in parabolic coordinates, are similarly related to alternative hyperspherical harmonics [2]. The connecting coefficient between spherical and 5 torA basis is formally identical to a usual vector coupling coefficient (from now on n is omitted from the notation) ... 

The H 1 form a basis set of functions of the orientations of three molecules which transform as a vector. The i/-th spherical component of the dipole moment n may be expanded in terms of this basis, according to... 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new... 

This matrix I 1 (A) differs from that in eq. (11.6.19) which describes the transformation of the basis x y z). The first term in the symmetrized basis in eq. (25) is the spherical vector... 

The components C2qi (9, spherical harmonics, with the angles 9 and < /> defined in figure 8.52, shown in appendix 8.1. Equation (8.229) is similar to (8.10), except that we have chosen to couple the vectors differently because of the basis set used in the present problem. Clearly the components of the cartesian tensor T are related to those of the spherical tensor T2(C) these relationships are derived in appendix 8.2. 

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