Representation matrices

The matrix Rij,kl = Rik Rjl represents the effeet of R on the orbital produets in the same way Rik represents the effeet of R on the orbitals. One says that the orbital produets also form a basis for a representation of the point group. The eharaeter (i.e., the traee) of the representation matrix Rij,id appropriate to the orbital produet basis is seen to equal the produet of the eharaeters of the matrix Rik appropriate to the orbital basis Xe (R) = Xe(R)Xe(R) whieh is, of eourse, why the term "direet produet" is used to deseribe this relationship. 

The coordinate representation matrix of this operator has the typical element ... 

As indicated above there may be many equivalent matrix representations for a given operation in a point group. Although the form depends on the choice of basis coordinates, the character is Independent of such a choice. However, for each application there exists a particular set of basis coordinates in terms of which the representation matrix is reduced to block-diagonal form. This result is shown symbolically in Fig. 4. ft can be expressed mathematically by the relation... 

Rfl. 4 Block-diagonal form of a representation matrix The tedueed representation. 

Generate the corresponding spin representation matrix, V (P), combine it with two-electron integrals, and apply the resulting components of the Hamiltonian to the Cl vector. 

NB In the inverse we have interchanged the index labels of the irreducihle representation matrix. 

A full proof of this can be found in Taylor [15]. The factor p is +1 if the operator is Hermitian and —1 if it is anti-Hermitian. We can see an immediate complication relative to our earlier formula Eq. 5.11 in that a full representation matrix for irrep T is required. This is considered in more detail below. Additional redundancies in the P2 list that arise from the form of particular operators axe also treated by Taylor. 

Heisenberg representation (matrix mechanics) the position and momentum are represented by matrices which satisfy this commutation relation, and ilr by a constant vector in Hilbert space, the eigenvalues E being the same in two cases,... 

If each of the blocks in the matrices comprising the matrix system A cannot be reduced ftirther, the matrix system has been reduced completely and each of the matrix systems A1, A2, . .. in the direct sum is said to be irreducible. Matrix systems that are isomorphous to a group G are called matrix representations (Chapter 4). Irreducible representations (IRs) are of great importance in applications of group theory in physics and chemistry. A matrix representation in which the matrices are unitary matrices is called a unitary representation. Matrix representations are not necessarily unitary, but any representation of a finite group that consists of non-singular matrices is equivalent to a unitary representation, as will be demonstrated in Section A1.5. 

Inserting resolutions of the identity written in terms of tensor product states RjAajA) (with 0 < j < [n — 1)) in the coordinate and diabatic state representation, matrix elements of the time dependent density operator are conveniently written as... 

For an infinitesimal transformation, the irreducible representation matrix is Dab = Sab + SDab, and the coordinate transformation returning to the original coordinate value is (A 1) x,v = x 1 - )Jdx v = x 1. Defining the transformation with this reverse step makes hx1 = 0, while the functional form of the field changes according to... 

For the Dirac bispinor, the irreducible representation matrix Dab for each helicity component is a Pauli spin matrix a multiplied by ti/2. Then... 

These three rotations do in general not commute, that is, the product depends on the order in which the matrices are multiplied. The representation matrix of the complete transformation can be composed from these individual rotation matrices as... 

One must agree that the precise recipe implied by Van Vleck s and Sherman s language is daunting. The use of characters of the irreducible representations in dealing with spin state-antisymmetrization problems does not appear to lead to any very useful results. Prom today s perspective, however, it is known that some irreducible representation matrix elements (not just the characters) are fairly simple, and when applications are written for large computers, the systematization provided by the group methods is useful. 

It is evident that the matrix elements of the Hamiltonian and overlap are independent of the index r of BT in Eq. (13) and only the first diagonal element of the irreducible representation matrix, D P), is required, which has been well discussed [31,33,42,43], and is easily determined. It is worthwhile to emphasize that Eqs. (21) and (22) are the unique formulas of the matrix elements in the spin-free approach, even though one can take some other forms of VB functions. For example, it is possible to construct VB functions by Young operator [2], but the forms of the matrix elements are identical to Eqs. (21) and (22) [44],... 

The constitution group Kt is identical with the racemate group Rx it is stored as a representation matrix ... 

The numbers c, n and a are connected through the representation matrix GyM(S) and the following equations... 

The converse implication is that if a transformation preserves orthonormality, the coiTesponding representation matrix will be unitary. Here, the starting point is the assumption that the basis remains orthonormal after transformation ... 

---