Representation symmetric direct product

The symmetric direct product of a one-dimensional representation with itself is clearly the same as the ordinary direct product of the representation with itself if TF is one-dimensional, then its matrices are the same as its characters, so that... 

Since the functions (9.144) are transformed into linear combinations of one another by the symmetry operators of the group, they form a basis for a representation VK of dimension + 1) this representation is called the symmetric direct product of TF with itself. We write... 

We can take the symmetric direct product of a representation with itself any number of times. For example, if Fl,...,Fm form a basis for T, then the functions... 

Equation (9.155) is a recursion relation for the characters of the symmetric direct product of a representation with itself n times note that it is consistent with (9.147). 

The antisymmetrized direct product of the two-dimensional representations with themselves is A for C and A2 for independent of whether n is even or odd. The basic rule (p. 210) for the choice of standard basis functions for the irreducible representations naturally proposes the fimction as a standard function for this representation. It is noted that Cn in this way requires two standard basis functions for A, namely further the function representing the symmetrized direct product of the most reduced representations with themselves. This is due to the fact that the group Cn with a real basis is not a simply reducible group since in... 

In systems with two- and fourfold rotation axes, the e JT effect, discussed so far, is not operative. In the corresponding (sometimes called tetragonal) point groups, the decomposition of the symmetrized direct product E)" of an irreducible representation E with itself reads... 

Table 3. Symmetrized direct products of irreducible representations and vibronic activity...

The alternative method of specifying distortion coordinates is used in the application of the JT effect. The rule of the JT instability states [8] that the JT active chstortion coordinates of a degenerate electronic state of symmetry type T must span representations A belonging to the non-totally-symmetric part of the symmetrized direct product ofT. 

If the system contains symmetry, there are additional Cl matrix elements which become zero. The symmetry of a determinant is given as the direct product of the symmetries of the MOs. The Hamilton operator always belongs to the totally symmetric representation, thus if two determinants belong to different irreducible representations, the Cl matrix element is zero. This is again fairly obvious if the interest is in a state of a specific symmetry, only those determinants which have the correct symmetry can contribute. 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

Although the resulting direct product may not be reduced, it can be made so by application of the magic formula, or often by inspection. The nonvanishing of the integral is then determined by the existence of the totally symmetric representation in the resulting direct sum. This procedure will be illustrated by the development of spectroscopic selection rules in Section 12.3.3. 

As indicated in Section 3.4, the integral of an odd function, taken between symmetric limits, is equal to zero. More generally, the integral of a function that is not symmetric with respect to all operations of the appropriate point group will vanish. Thus, if the integrand is composed of a product of functions, each of which belongs to a particular irreducible representation, the overall symmetry is given by the direct product of these irreducible representations. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

A tableau may be used to define certain subgroups of which are themselves direct products of smaller permutation groups the symmetrizing and antisymmetrizing operators for these subgroups lead, as we shall see, to projection operators on irreducible representations of... 

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. 

Simplification of secular equations. Because the Hamiltonian is totally symmetric - that is, for a molecule of C2v symmetry such as H2O, of symmetry species Ai - the matrix elements Hij = ipi, Ti. ipj) as well as the overlap integrals Sij = (tpi, ipj) will be equal to zero unless the direct product representation r. contains Ai. This is the basis for the assertion that states of different symmetry do not mix. ... 

The polarizability tensor of a molecule related the components of the induced dipole moment of the molecule to the components of the electric field doing the inducing. It therefore has 9 components, axx, ctxy, etc., only 6 of which are independent. The theory of the Raman effect shows that a vibrational transition, from the totally symmetric ground state to an excited state of symmetry species F, will he Raman active if at least one of the following direct products contains the totally symmetric representation ... 

The Jahn-Teller theorem was proved by showing that for all symmetry groups except and there was at least one normal mode of vibration which belonged to a non symmetric representation f,- such that the direct product of F/ with the representation Fy of the degenerate electronic state contained the representation Fy. 

If we consider in the direct product representation rH P then since Hitf belong to P, rH P = P and therefore TH — P. Hence, any operator which commutes with all 0M of a point group can be said to belong to the totally symmetric irreducible representation P. 

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