Matrices symmetry operations

When we carry out mathematically a symmetry operation on a ligand vector, we perform a matrix multiplication between the symmetry operation matrix and the vector composed of its directional numbers. For instance, when we operate C3 (1) on xi, we do the following ... 

The relationships among symmetry operations, matrix representations, reducible and irreducible representations, and character tables are conveniently illustrated in a flowchart, as shown for C2v symmetry in Table 4.8. 

Here, Xr(R) is the eharaeter belonging to symmetry E for the symmetry operation R. Applying this projeetor to a determinental flinetion of the form ( )i( )j generates a sum of determinants with eoeffieients determined by the matrix representations Ri ... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

We ean likewise write matrix representations for eaeh of the symmetry operations of the C3v point group ... 

For any symmetry operator T = T 0) (rewritten r when operating on the domain of basis functions x)) for instance, the rotation-reflexion about the z-axis, with matrix representation... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

In the examples presented in the previous section, the vectors % of displace meat coordinates [Eqs. (12) and (19)] were used as a basis. It should not be surprising that the matrices employed to represent the symmetry operations have different forms depending on the basis coordinates. In effect, there is an infinite number of matrices that can serve as representations of a given symmetry operation. Nevertheless, there is one quantity that is characteristic of the operation - the trace of the matrix - as it is invariant under a change of basis coordinates. In group theory it is known as the character. 

Since the Hartree-Fock wavefunction 0 belongs to the totally symmetric representation of the symmetry group of the molecule, it is readily seen that the density matrix of Eq. (10) is invariant under all symmetry operations of that group, and the same holds, therefore, for the Hartree-Fock operator 7. 

It is worthwhile now to introduce a mechanism devoted to representing the symmetry operations of our ABe center. The symmetry operation (rotation) of Figure 7.1 transforms the coordinates (x, y, z)into(y, -x, z). This transformation can be written as a matrix equation ... 

It is then possible to represent the above-mentioned symmetry operation by the 3x3 matrix of Equation (7.1). In a more general way, we can associate a matrix M with each specific symmetry operation R, acting over the basic functions x, y, and z of the vector (x,y, z). Thus, we can represent the effect of the 48 symmetry operations of group Oh (ABe center) over the functions (x, y, z) by 48 matrices. This set of 48 matrices constitutes a representation, and the basic functions x, y, and z are called basis functions. 

An inspection of this scheme shows that the character (+1) of the transformation matrix given in Equation (7.3) has some physical sense in this case, it indicates that only one orbital, the Pz, is unchanged by the symmetry operation C4(001) (see Figure 7.3). 

An important conclusion envisaged from the previous paragraph is that all of the information needed for a symmetry operation is contained in the character of the matrix associated with this operation. This leads to the first great simplification we do not need to write the full matrix associated with any transformation - its character is sufficient. 

Each of the symmetry operations we have defined geometrically can be represented by a matrix. The elements of the matrices depend on the choice of coordinate system. Consider a water molecule and a coordinate system so oriented that the three atoms lie in the x-z plane, with the z—axis passing through the oxygen atom and bisecting the H-O-H angle, as shown in Figure 5.1. 

Although every symmetry operation can be represented by a matrix, many matrices correspond to linear transformations that do not have the properties of symmetry operations. For example, every symmetry operation has the property that the distance between any two points and the angles between any two lines are not altered by the operation. Such a geometric transformation, that does not distort any object that it acts on, is called an orthogonal transformation. A matrix that corresponds to such a transformation is called an orthogonal matrix. 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

Because the traces contain sufficient information to decompose Ftot into irreducible representations, it is necessary to compute only the diagonal elements of the matrices of the representation. If a particular atom changes position under a symmetry operation, its displacements can contribute no diagonal elements to the matrix therefore, for that symmetry operation, such an atom may be ignored. For example, the displacements of the hydrogen atoms in water do not contribute to the character of C2 in Ftot- The displacement of Hi means that the elements (1,1), (2,2) and (3,3) of the matrix are zero. 

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

We have about exhausted possible scattering matrix symmetries obtained by appealing to approximate or exact solutions to specific scattering problems. But more can be said about particles, regardless of their shape, size, and composition, without explicit solutions in hand. The scattering matrix for a given particle implies those for particles obtained from this particle by the symmetry operations of rotation and reflection. We shall consider each of these symmetry operations in turn. 

We shall need one more matrix, that for a particle obtained by the symmetry operations of reflection and rotation this follows readily from (13.13) and (13.18) ... 

It is also possible to construct matrix representations by considering the effeot that the symmetry operations of a point group have on one or more sets of base veotors. We will consider two cases, both using the point group as an example (1) the set of base veotors eu e, and e introduced in 5-2 (2) three sets of mutually perpendioular base vectors, each located at the foot of a symmetric tripod. 

The matrix Dfl y(R) that corresponds to the symmetry operation R is a square matrix of order and its elements are the (n n,)1 possible... 

Sylow, 6. symbol list, xv. symmetric matrix, 59. symmetry element, 7. symmetry operations, 7, 10 algebra of, 15 for symmetric tripod, 27. symmetry operator, 10. symmetry orbitals, 203, 206, 207, 210, 212, 246. 

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