Full rotation group

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

It is noteworthy that dq(e,t) does not satisfy this relation, as equality [J,x, dq] = 2 C q dq+ll (the definition of an irreducible tensor operator) does not hold for it [23]. Integration in (7.18), performed over the spherical angles of vector e, may be completed up to an integral over the full rotational group due to the axial symmetry of the Hamiltonian relative to the field. This, together with (7.19), yields... 

EXAMPLE 7.3 The construction of representations the full rotation group. 

In Table 7.5, we show the character (defined as the set of character elements of a representation) of different representations (from / = 1 to 6) of the 0 group. The character elements were obtained from Equation (7.7). These representations, which were irreducible in the full rotation group, are in general reducible in 0, as can be seen by inspecting the character table of 0 (in Table 7.4). Thus, the next step is to decompose them into irreducible representations of 0, as we did in Example 7.1. Table 7.5 also includes this reduction in other words, the irreducible representations of group O into which each representation is decomposed. We will use this table when treating relevant examples in Section 7.6. 

In the ideal case of free Eu + ions, we first must observe that the components of the electric dipole moment, e x, y, z), belong to the irreducible representation in the full rotation group. This can be seen, for instance, from the character table of group 0 (Table 7.4), where the dipole moment operator transforms as the T representation, which corresponds to in the full rotation group (Table 7.5). Since Z)° x Z) = Z) only the Dq -> Fi transition would be allowed at electric dipole order. This is, of course, the well known selection rule A.I = 0, 1 (except for / = 0 / = 0) from quantum mechanics. Thus, the emission spectrum of free Eu + ions would consist of a single Dq Ei transition, as indicated by an arrow in Figure 7.7 and sketched in Figure 7.8. 

Under the action of the electric field of the 0 environment, the state of the ( ) 3d configuration will split up into two states. The orbital degeneracy of a D state is 2 X 2 -f- 1 = 5. From the above discussion each of the resulting states must belong to one of the irreducible representations of Oh given in Table I. The state W corresponds to an irreducible representation of the group of symmetry operations of a sphere, i.e., the full rotation group R(3). 

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... 

This implies that the fields B(V), B 2, and B 2 are also operators of the full rotation group, and are therefore irreducible representations of the full rotation group. Specifically... 

The set of properly orthogonal transformations R1 forms the group SO(3), the reflexion Z1 at the origin of the LS likewise leaves A symmetric, since the eulerian angles remain unaffected by Z1. Therefore, H is symmetric w.r.t. the full rotation group 0(3/. However, in agreement with the usual conventions we will omit the elements Z R1 0(3). As a consequence we will consider hence -forward the group... 

The full rotation group is infinite and its operators are continuous. In this context, it is useftd to define an operator. / which produces an infinitesimally small rotation... 

All rotations through a finite angle a form a single class of the full rotation group. Thus... 

We next seek the irreducible representations of the full rotation group, formed by the infinite number of finite rotations R(ait). Because all such rotations can be expressed in terms of the infinitesimal rotation operators Jx, Jy and Jz (or equivalently J+, J and Jz), we start from these. 

Even and odd linear combinations of these spin states may be built up. As there is no spin-orbit coupling, the full rotation group operates in each spin space with the representation <5Di/2 for a spin 1/2 associated with each electron. We have ... 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

The J and M selection rules for the nonvanishing matrix elements, <(/,T,M ett y, T, M ), may be obtained from the transformation properties of the wavefunctions and rotational operators under the full rotation group. We will not discuss this point in detail and simply refer to the table of matrix elements given in Appendix II. In general, there may be nonvanishing off-diagonal elements with / =/, / l,/ 2 and with M = M, Af 1. The operators Jg rot, and commute with Jz and are diagonal in the quantum number M,... 

The matiix forms an irreducible representation of the full rotation group. Because the basis vectors j,m) are orthonormal and remain so on rotation, the matrices are unitary ... 

Any rotation in x, y, z space can be described in terms of the infinitesimal rotation operators /, ly, and 7 whose matrix representations were defined in Eq. 3.31. These operators satisfy certain commutation relationships Eq. 3,32, and they add like ordinary unit vectors. Any small rotation may be expressed as a linear combination of 7i, ly, I,. As before, the basic problem is to find the reps of the full rotation group. 

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