## Spherical tensor operators. Tables of matrix elements and symmetries

So the dependence on 3 parameters - corresponding to the 3 rotations - is eliminated. And if you are interested in "where" it might come useful: Some of the selection rules for optical transitions can be obtained from it, and I faintly recall that it helps rewriting the Hamiltonian for Spin-Orbit coupling into a much more convenient form.

The common function here is r itself. This radial integral is basically the reduced matrix element or at least very closely related to this reduced matrix elements as some definitions have extra factors in the numerator or denominator. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What is the usefulness of the Wigner-Eckart theorem?

Ask Question. Asked 8 years, 7 months ago. Active 4 years, 5 months ago. Viewed 11k times. Cogitator Cogitator 2 2 gold badges 8 8 silver badges 15 15 bronze badges. Kostya Kostya Jul 1 '14 at You said So instead of doing all of them we got to exploit their symmetry. What symmetry are we talking about? Sperical symmetry? Mar 8 '17 at Marek Marek Though I am a little shaky on the rest of the group theory and the notation the first part , this answer is helpful, particularly the last three paragraphs.

A representation is said to be irreducible if it is not possible to find a basis in which all the matrix representatives of the group elements have the same block diagonal form. The explicit matrix representations Mr G are dependent on the choice of the basis vectors. However, for a given representation r, the characters, defined as the traces of the matrix representations. All elements in the same class have the same character for all representations r, i. Suppose now that the group G contains a finite subgroup g containing h g elements. A representation r of the group G is also a representation of the subgroup 0.

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The finite group orthogonality theorem  implies that the number of independent linear combinations of basis vectors spanning the representation r which are invariant under all of the subgroup operations G e 0 is given by. The last two formulations on the right-hand side of 11 are equivalent since all elements in the same class have the same character. This equation leads to the following property:. The sum of matrices in the representation r vanishes if the characters sum to zero over all classes of g, taking into account the number of subgroup elements he g in each class.

For each group element G, there exists a corresponding operator G acting on functions of the coordinate vectors f x to generate new functions f ' x , defined as follows:. The definition above corresponds to an active transformation of the object f . The average function over a finite subgroup g of G is defined by. Any operator G is then represented by an m x m matrix Mr G acting on the set of basis functions from the right :.

The right-hand side corresponds to Equation 15, evaluated at any point x1 in the set. Equation 20 is the central result of this section. The point symmetry group of an n-dimensional regular polytope is a finite group which acts transitively on the polytope vertices. It is a subgroup of the infinite orthogonal group O n , which is the group of all the n-dimensional space transformations in with a single fixed point and which preserve distance between transformed points.

Using Equation 20, the averaging properties of a function over the vertices of a polytope may be deduced from the characters of the symmetry elements and the classes of its symmetry point group. This result is now applied to the spherical moments of the regular solids. Three dimensional polytopes are known as polyhedra. In this section we discuss the averaging properties of the regular polyhedra with respect to spherical harmonics. Although this topic has been treated before in Reference, a recapitulation is useful for framing the discussion of four-dimensional symmetries.

In addition, the treatment in Reference did not exploit all the available symmetries, as discussed below. The proper rotations in three dimensions may be defined in various ways. The identity operation R 0 does not need any specification of the rotation axis. The improper rotations in three-dimensional space may be expressed in various ways. By definition, the inversion operation corresponds to the improper rotation R 0 , where the rotation axis does not need to be specified in this case.

The action of any O 3 operation G on these functions defines an operator G which is represented by a. In the case of a proper rotation R, the matrix representative is given by the rank-i Wigner matrix:. The character of a proper rotation for the rank-i representation is equal to the trace of the corresponding Wigner matrix, xD? The character of an improper rotation is the same as for the corresponding proper rotation, but with a change in sign for odd values of i:.

The five regular convex polyhedra have been known since the Greeks. Their names and properties are listed in Figure 1. The five Platonic solids therefore belong to only three symmetry point groups: i Td, represented by the tetrahedron; ii Oh, populated by the cube and the octahedron; and iii Ih, populated by the icosahedron and the dodecahedron. The symmetry point groups of the regular polyhedra are given explicitly in Table 1.

Figure 1. The 3D regular convex polyhedra organised according to their symmetry group.

Here N0 is the number of vertices, N is the number of edges and N2 is the number of faces constituting the solid. The theorem in Equation 20 may be used with Table 1 and the characters given in Equations 26 and 28 to deduce the vanishing spherical moments of the regular polyhedra. In general, both improper and proper rotations must be taken into account.

The treatment in Reference  uses only the proper rotations, and gives slightly different results for the groups Oh and Ih see below. As a first example, consider the tetrahedron. This proves the well-known fact that a tetrahedron averages second-rank spherical harmonics to zero:. The point symmetry groups of the octahedron and icosahedron contain the inversion element.

Each proper rotation is therefore accompanied by an improper rotation through the same angle, as shown in Table 1. It follows that all odd-rank spherical moments harmonics vanish when summed over the vertices of polyhedra with symmetries Oh and Ih:. The treatment of Reference  does not predict this result, since only proper rotations were taken into account.

There are 4 regular non-convex polyhedra star-polyhedra , which all fall in the group Ih .

Four of them have the same vertices of the icosahedron while one has the same vertices as the dodecahedron. All have the same spherical moment characteristics as the icosahedron. Table 1. The three symmetry point groups of the regular polyhedra. The last column shows the number of elements in each class in square parentheses , followed by a single symmetry element of the class, for a polyhedron in standard orientation.

R 0 is the identity operation and R 0 is the inversion operation. In this section we derive the spherical averaging properties of the regular 4-polytopes. In the discussion below, we make extensive use of quaternions . As shown in Equation 7, quaternions provide a correspondence between points on a unit sphere in four-dimensional space, and the group of three-dimensional rotations. Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector.

Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors q 1 and q 2 according to.

Figure 2. Spherical rank profiles for the regular convex 3D polyhedra. Closed circles indicate that there is at least one non-zero spherical moment of rank t. The inverse q-1 is defined for any non zero quaternion q as the unique quaternion that satisfies:. From Equation 7, a unit quaternion and its inverse represent a pair of rotations through opposite angles about the same axis. The group of unit quaternions Q is homomorphic with the group of proper three-dimensional rotations SO 3 .

The relationship between the product of quaternions and the product of proper 3D rotations is expressed by. The explicit correspondence between the Euler angles and the unit quaternion components is as follows:. Isometries in 4D space are classed as either proper preserving the handedness of the four-dimensional axis system or improper changing the handedness of the axis system. The group of all isometries with one fixed point in four dimensions is called O 4.

Proper operations will be denoted by Rqi,qr and improper operations by Rqi,qr respectively. The action of a proper rotation Rqi,qr on a point in 4D space q may be written as follows:. In this section we give the explicit matrix representations of the O 4 operators and their characters in the basis of the Wigner matrices. These results will then be used to establish the spherical averaging properties of the regular 4-polytopes.

According to Equations 14, 44, 38 and 39, a proper transformation in O 4 defines an operator Rqi,qr which acts as follows on the Wigner matrix elements evaluated at any unit quaternion q:. Similarly according to Equations 14, 45, 38 and 39 an improper transformation in O 4 defines an operator Rqi,qr which acts as follows:. This proves that the Wigner matrices are a basis for the representation of the group O 4. The matrix representations are given by. For proper rotations this leads to the following result:. For improper rotations, on the other hand, we get.

The six regular convex polytopes are summarized in Figure 3. The six regular convex 4-polytopes therefore belong to only four symmetry groups. These are i the group A4 isomorphic to the permutation group of 5 elements, S5 , populated by the 5-cell hypertetrahedron ; ii the group B4, populated by the mutually reciprocal 8-cell hypercube and cell hyperoctahedron ; iii the group F4, populated by the cell; and iv the group H4, populated by the mutually reciprocal cell hyperdodecahedron and cell hypericosahedron. Figure 3. A list of the 4D regular convex polytopes organized according to their symmetry group.

Here N0 is the number of vertices, N1 is the number of edges, N2 is the number of faces and N3 is the number of three dimensional cells. The two dimensional graphs indicate the vertex connections. Table 2 reports the four symmetry groups of the six regular polytopes and their symmetry elements, given in the quaternion form. The numbers of operations in each class are provided, together with one representative operation, using the notation Rqi,qr for proper transformations and Rqi,qr for improper transformations.

In the case of the group H4, the symmetry classes and representative operations are given directly in quaternion form in Reference. For the other groups, the information given in the literature [] is not directly suitable for this type of analysis. In these cases, the quaternion form of the representative operations and the class structure were obtained by using the information provided in Reference with the help of the symbolic software platform Mathematica .

Table 2. The four symmetry groups of the 4D regular polytopes. The last column shows the number of elements in each class in square parentheses , followed by a single symmetry element of the class, for a polytope in standard orientation. The symmetry elements are denoted Rqi,qr for a proper rotation and Rqi,qr for an improper rotation, see Equations 42 and The spherical averaging properties of the regular 4-polytopes may be deduced by using Equation 20 together with the sets of symmetry operations Table 2 , and the characters of the 4D rotations, given in Equations 51 and As an example, consider the 5-cell, which has symmetry group A4.

From Table 2, there are seven symmetry classes. The four classes of proper operations have 1,15, 20,24 elements respectively. The remaining three classes of improper operations have 10, 30,20 elements respectively. As in the 3D case, even the cell and cell, which have the highest symmetry, fail to average out the rank-6 Wigner matrices. This figure is slightly misleading since only integer ranks t are shown. The group A4, on the other hand, lacks the inversion, so the spherical moments of half-integer rank do not vanish in this case.

The fact that A4 and B4 appear to have the same rank profiles in Figure 4 is therefore due to the omission of half-integer ranks. Most applications of orientational averaging only require integer ranks, in which case the properties shown in Figure 4 are appropriate. There are 10 regular non-convex polytopes star-polytopes in four dimensions, which all fall in the group H4 . Nine of them have the same vertices as the cell, while one has the same vertices as the cell. All have the same spherical moment characteristics as the cell.

Under the reviewing of this paper, an anonymous referee pointed out that the pattern of empty and filled circles in Figure 4 may also be derived using the theory of spherical designs . Figure 4. Spherical rank profiles of the regular convex 4-polytopes. In order to facilitate exploitation of these results, we provide explicit tables of Euler angles derived from the vertices of the regular 4-polytopes. The z — y — z convention for the Euler angles is used throughout. All Euler angle sets are derived from 4-polytopes in their standard orientations, as defined. All angles are reduced to the interval 0 to 2n by a modulo-2n operation. Table 3. The coordinates of the six convex regular 4-polytopes vertices in standard orientation, as reported in Reference .

The double round parentheses indicate that all even permutations of the quartet are taken. All the polytopes are centred at the origin of the coordinate system, with the vertices lying on the hypersphere of radius 1. Different Euler angle sets with the same spherical averaging properties may be constructed by applying an equal but arbitrary 4D isometry to all the quaternions underlying the set.

The set of Euler angles corresponding to the 5 vertices of the 5-cell is provided in Table 4. As shown in Figure 4, all first-rank spherical moments vanish for this set of Euler angles. Since the 5-cell lacks an inversion operation, the number of orientations is the same as the number of vertices in this case.

The sets of Euler angles corresponding to the 8 vertices of the cell, and the 16 vertices of the 8-cell are provided in Table 5 and 6. As shown in Figure 4, all first-rank spherical moments vanish for these sets of Euler angles. The symmetry groups of both polytopes include an inversion operation, so the number of distinguishable orientations is therefore one-half the number of the vertices.

Clearly the four rotations specified in Table 5 comprise the most economical way to set all first-rank spherical moments to zero. The set of 12 Euler angles corresponding to the 24 vertices of the cell is provided in Table 7. As shown in Figure 4, all first and second-rank spherical moments vanish for this Euler angle set. The sets of 60 and Euler angles corresponding to the vertices of the cell and the cell are provided in Table 8 and 9. Figure 4 shows that all spherical moments up to and including rank 5 vanish for these Euler angle sets.

The most economical way of annihilating spherical ranks up to and including rank 5 is therefore the angle set in Table 8.

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This rotation set was previously described in Reference , where it was presented without any supporting theory. Table 4. The set of Euler angles in degrees corresponding to the 5 vertices S of the 5-cell whose cartesian coordinates are given in Table 3. Table 5. The set of Euler angles in degrees corresponding to the 8 vertices V of the cell whose cartesian coordinates are given in Table 3. Table 6. The set of Euler angles in degrees corresponding to the 16 vertices W of the 8-cell whose cartesian coordinates are given in Table 3.

Table 7. The set of Euler angles in degrees corresponding to the 24 vertices T of the cell whose cartesian coordinates are given in Table 3. It is worth pointing out that the 3D rotations discussed above for the cell and the cell have more inutitive descriptions. The set of Euler angles obtained from the vertices of the cell generates exactly the 12 rotational symmetries of the tetrahedron, compare Table 7 with the last column for the group Td in Table 1.

Similarly the set of Euler angles obtained from the vertices of the cell generates exactly the 60 rotational symmetries of the icosahedron, compare Table 8 with the last column for the group in Table 1. The 24 rotational symmetries of the cube Oh group do not corresond to any regular 4-polytope. In fact they are not well distributed in the sense of particle repulsion over the hypersphere in 4D as the other polytopic cases. Regarding this last point, it has been rigorously proven that that some of the regular 4-polytopes the 5-cell, the cell and the cell minimize a full class of repulsive potentials over the 4D sphere .

Table 8. The set of Euler angles in degrees corresponding to the vertices I of the cell whose cartesian coordinates are given in Table 3. Table 9.

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