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 shall not need the complete matrices of the symmetric direct product only their traces are required. The character of R in the sym-... 

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... 

We shall abbreviate (9.151) as [T p and shall use [TJp to indicate the symmetric direct product of TF with itself n times. The basis functions for [TJP are... 

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). 

According to the Jahn-Teller theorem the active modes for an orbital multiplet are given by the non-totally symmetric part of the symmetrized direct product of the electronic degeneracy ... 

The existence of these structures may be explained by the JT distortion of a parent Dih structure in E or E" electronic states. The symmetry of JT active coordinates may be obtained from the symmetric direct product within Djh group... 

The symmetric direct product within D h symmetry group... 

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... 

The symmetric dire< t product finds its most extensive use in vibrational spectroscopy. When more than one quantum of u degenerate vibration is absorbed, the symmetry of the resulting states (overtone) are obtained from the symmetric direct product. In general, the species deriving from the pth overtone of a vibrational state of symmetry r(o() is... 

Since (a) and (d) are identically zero, the matrix elements of the (a) and (d) rows of the direct product will vanish. The function (b) is the negative of (c), so the set of functions of E

antisymmetric product is only one-dimensional. If equals the symmetric product yields three independent functions and the antisymmetric product yields none. It is important to use symmetric direct products when examining the symmetry of products of partner functions. We now outline the general method of obtaining the characters of these direct product matrices. Once the characters are known the reduction to constituent reps can be carried out in the usual way. 

According to this result the integral is zero if does not belong to Eig, A ig, or A ig. The Ei functions are the same so we require the symmetric direct product (Ei X Ei ). This is given by... 

Problems involving degenerate levels are not always as simple. 4.15 shows double occupancy of an e orbital. 4.16 and 4.17 show that because of the restrictions of the Pauli principle, there can only be three singlets and one triplet and not (as in 4.14) four of each. In this case wc need to define a different sort of direct product. The usual simple product leads to the characters defined in equation 4.33. But the symmetric direct product which we use to generate the singlet levels of 4.15 is... 

The JT problem is determined by the symmetrized direct product of Tiu. As we have seen in the previous problem, this product contains A g- -Eg- - T2g. Since Aig modes do not break the symmetry, the JT problem is of type T x (e+f2).In the linear problem only two force elements are required. The distortion matrix is thus as follows ... 

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... 

Benzene has four e2g vibrational modes — z/ig in Herzberg numbering) which are linearly JT active according to the decomposition of the symmetrized direct products... 

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

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