The Spherical-Symmetry Group

7 Application Spin Hamiltonian for the Octahedral Quartet State. 184 

Consider the row vector of coordinate functions y) z in 3D space. A rotation 

Ceutemans, Group Theory Applied to Chemistry, Theoretical Chemistry and Computational Modelling, DOl 10.1007/978-94-007-6863-5 7, 

The transformations of the standard vector form the fundamental irrep of spherical symmetry. All other irreps can be constructed by taking direct products of this vector. In particular, the spherical harmonic functions can be constructed by taking fully symmetrized powers of the vector. The symmetrized direct square of the / -functions yields a six-dimensional function space with components z, x, yz, xz, xy. This space is not orthonormal the components are not normalized, and the first 

A rotation C about the z-direction affects this function in the following way  

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The basis functions in a one-centre calculation belong to representations of the spherical symmetry group. There is, therefore, a higher degree of symmetry in the integrals over the basis functions than in the integrals over the molecular orbitals. This can be exploited in self-consistent field calculations and in orbital transformations. 

Thus, the problem of enumeration and construction of projective fullerenes reduces simply to that for centrally symmetric conventional spherical fullerenes. The point symmetry groups that contain the inversion operation are Q, C, h, (m even), Dmh (m even), Dmd (m odd), 7, Oh, and 7. A spherical fullerene may belong to one of 28 point groups ([FoMa95]) of which eight appear in the previous list C,-, C2h, Dm, Da, D3d, Du, 7, and /. Clearly, a fullerene with v vertices can be centrally symmetric only if v is divisible by four as p6 must be even. After the minimal case v = 20, the first centrally symmetric fullerenes are at v = 32 (Dm) and v = 36 (Dm). 

Ion distribution around a spherical micelle can also be described with models that consider that polarizable and not very hydrophilic species (such as Br ) interact both coulombically and by a specific, noncoulombic, interaction [26]. This latter interaction allows the ion to intercalate at the micellar surface and to neutrahze an equivalent number of head groups. Ion distribution around a micelle is then calculated by solving the Poisson-Boltzmann equation (PBE) in the spherical symmetry with allowance for specific interactions via a Langmuir or Volmer isotherm [31]. The original kinetic treatment for a micelle of radius a, aggregation number iV in a cell of radius R yields [31] ... 

For example the dipole polarizability given in (2.33) has spherical tensor components OQQdl) and a2K(11) The dipole-quadrupole polarizability (A .gY in Cartesian notation), which describes the quadrupole moment induced by an electric field or the dipole moment induced by an electric field gradient, has components 0 (12), 021(02) and 02k(12). The polarizabilities are even (g) or odd (u) under inversion according as 1+1 is even or odd. This information is then sufficient, with the help of Table 3, to determine the transformation properties in the molecular symmetry group. Any component which transforms according to the totally symmetric representation may have a non-zero value. 

A FIGURE 7.10 Energies of the transition metal atom d orbitals as a function of ligand distance, drawn for CoCls. The five d orbitals, degenerate at large distance R, split into two groups of orbitals, and tig, asthe ions approach the metal and distort the spherical symmetry of the potential energy. 

The chief source of trouble in molecular problems is the absence of the spherical symmetry of the isolated atom. The operators and its components, which played such an important part in our treatment of atomic structure, no longer commute with the Hamiltonian, and so they lose their usefulness. It is true that many molecules have some elements of symmetry, when the problem can be simplified with the aid of group theory, but these symmetry elements are properties of individual molecules and cannot be used in the general theory of molecular structure. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

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