Symmetry operations, matrix representations

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. 

Each of the 9X9 matrices in equation 13.2 is called a representation of the corresponding symmetry operation. These representations are complete, but cumbersome. And this just for a molecule that has three atoms. For N atoms, the 3N X 3N matrix contains 9N terms. Therefore the complete representation for dimethyl ether, ( 113)20, which also has 2 symmetry, can be defined by four 27 X 27 matrices with each having 27 = 729 numbers in it To be sure, most of them are zero (as they are above), but determining which are exactly zero is a chore. Dimethyl ether is still a rather small molecule. We need a simpler representation. 

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 ... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

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... 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

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 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. 

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. 

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. 

Hence we have the matrix equation RS = W, so that the matrices (9.44) multiply the same way the symmetry operations do and form a representation of the point group. The functions Fl,F2,...,Fn are said to form a basis for the representation (9.44), which consists of the matrices that describe how these functions transform upon application of the symmetry operators. Any member of the set Fv...,Fn is said to belong to the representation (9.44). We denote the representation (9.44) by TF it may be reducible or irreducible. 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

It is relatively easy to deduce the four one-dimensional representations. As in every group, there must be the so-called totally symmetric representation, in which every symmetry operation is represented by the one-dimensional matrix 1. At this point, we have in hand the following part of the character table ... 

Inspection of this result shows that it is not equivalent to the partner function Eq. 3.15, and that although it and the result of Eq. 3.17 are linearly independent they are not orthogonad. Explicit Schmidt orthogonadization of the result of Eq. 3.18 to that of Eq. 3.17 yields the same functions as those obtained from the full matrix projection and shift operators. However, without knowledge of the full matrix representations we cannot identify these character projection results with specific rows of the e irrep. In fact, in generad the results of character projection will not yield basis functions that can be identified with symmetry species. 

In 1989, however, a practical resolution of this problem was derived independently by Haser and by Almlof [16]. They obtain the necessary matrix representation information by utilizing the reducible representation matrices obtained from symmetry transformations on the AO basis as in Eq. 5.1. Full details of their procedure is beyond the scope of this course, but, as would be expected, it has many similarities to the non-totally symmetric operators discussed in the previous section. 

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