The Direct Product

Suppose that R is an operation in the symmetry group of a molecule and Xx, X2,. . . , X , and Yu Y2,. . . , Y are two sets of functions (perhaps eigenfunctions of the wave equation for the molecule), which are bases for representations of the group. As shown earlier, we may write 

Thus the set of functions Xyk, called the direct product of A, and Yk, also forms a basis for a representation of the group. Tlie zjUk are the elements of a matrix X of order (mn) x (mn). 

We now have a very important theorem about the characters of theX matrices for the various operations in the group  

The characters of the representation of a direct product are equal to the products of the characters of the representations based on the individual sets of functions. 

if we want to know the characters x(R) of a representation that is the direct product of two other representations with characters Xi(R) and X2W, these are given by 

In this section we discuss the direct product of two irreducible representations. The concept of direct product may seem abstract initially. But it should become easily acceptable once its applications are discussed in subsequent sections. 

When we obtain the direct product /), of two irreducible representations TJ and TJ, i.e., 

Now let us apply these formulas to some examples. All the direct products formed by the three irreducible representations of the Civ group are summarized in Table 6.4.1. These results show that the direct product of two irreducible representations is sometimes also an irreducible representation. When it is not, its constitution can be readily determined using eq. (6.4.3). 

When eq. (6.4.3) is examined more closely, it is seen that, the division by h, the group order, makes the equation untenable for groups with an infinite order, namely, Coov and 7 lCOh. Fortunately, in most cases the decomposition of a reducible representation for such groups can be achieved by inspection. Also, a work-around method is available in the literature (see references 19 and 20 listed at the end of the chapter). In any event, examples will be given later. 

Based on the rather limited examples of direct products involving the representations of the C3v group, we can draw the following conclusions (or guidelines) regarding the totally symmetric representation Tts  

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The CNP group of a molecule containing / identical nuclei of one type, m of another, n of another and so on is the direct product group... 

The complete nuclear pemnitation inversion (CNPI) group of the PH molecule is the direct product of the complete nuclear pemnitation (CNP) group 3 (see (equation Al.4.19)) and the inversion group P= E, E ]. This is a group of 12 elements that we call... 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

In the Bom-Oppenlieimer approxunation the vibronic wavefrmction is a product of an electronic wavefimction and a vibrational wavefunction, and its syimnetry is the direct product of the synuuetries of the two components. We have just discussed the synuuetries of the electronic states. We now consider the syimnetry of a vibrational state. In the hanuonic approximation vibrations are described as independent motions along nonual modes Q- and the total vibrational wavefrmction is a product of frmctions, one wavefunction for each nonual mode ... 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

At 321 mn there is a vibronic origin marked This has one quantum of v, the antisynnnetric C-H stretching mode, in the upper state. Its intensity is induced by a distortion along This state has B2 vibrational symmetry. The direct product of B2 and A2 is B, so it has B vibronic syimnetry and absorbs x-polarized light. One can also see a 4 6,, vibronic origin which has the same syimnetry and intensity induced by... 

Now, consider the subgroup C3 of D3 , (since no out-of-plane bending is possible for a triatomic system, and hence the subgroup C3 may be used for the discussion). Then, Eq. (E. 15) contains only four symmetry types of electronic operators Iia, Iia, h, and hy. The direct product decompositions for C3 may then be shown (see Table 57 of [28]) to assume the fomi... 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

From the direct-product characters X(S) belonging to a particular electronic configuration (e.g., ai a2 e ), one must still decompose this list of characters into a sum... 

In order to obtain the direct product of two species we multiply the characters under each symmetry element using the mles... 

Assign the allene molecule to a point group and use the character table to form the direct products A-2 x5i,5i X 82,82 xE and E X E. Show how the symmetry species of the point group to which 1,1-dilluoroallene belongs correlate with those of allene. 

It was explained in Section 4.3.2 that the direct product of two identical degenerate symmetry species contains a symmetric part and an antisymmetric part. The antisymmetric part is an A (or 2") species and, where possible, not the totally symmetric species. Therefore, in the product in Equation (7.79), Ig is the antisymmetric and Ig + Ag the symmetric part. 

Manufacture is either by reaction of molten sodium with methyl alcohol or by the reaction of methyl alcohol with sodium amalgam obtained from the electrolysis of brine in a Castner mercury cell (78). Both these methods produce a solution of sodium methylate in methanol and the product is offered in two forms a 30% solution in methanol, and a soHd, which is a dry, free-flowing white powder obtained by evaporating the methanol. The direct production of dry sodium methylate has been carried out by the introduction of methanol vapors to molten sodium in a heavy duty agitating reactor. The sohd is supphed in polyethylene bags contained in airtight dmms filled in a nitrogen atmosphere. 

There are many variations on crystalline cane sugar from refineries, in addition to the direct production and noncentrifugal sugars described above. 

Manufacture. Most chlorate is manufactured by the electrolysis of sodium chloride solution in electrochemical cells without diaphragms. Potassium chloride can be electroly2ed for the direct production of potassium chlorate (35,36), but because sodium chlorate is so much more soluble (see Fig. 2), the production of the sodium salt is generally preferred. Potassium chlorate may be obtained from the sodium chlorate by a metathesis reaction with potassium chloride (37). 

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. 

The direct product G H x k x. .. of arbitrarily many permutation groups is defined similarly. With this definition of a direct product the degrees are added and the cycle indices multiplied. 

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