Product direct

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... 

There is no component of ai symmetry in the triple direct product, so the integral... 

There will be many occasions when we shall need to multiply symmetry species or, in the language of group theory, to obtain their direct product. For example, if H2O is vibrationally excited simultaneously with one quantum each of Vj and V3, the symmetry species of the wave function for this vibrational combination state is... 

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

Using the rules for forming direct products it can be shown also that, in the 2 point group,... 

Except for the multiplication of by we follow the rules for forming direct products used in non-degenerate point groups the characters under the various symmetry operations are obtained by multiplying the characters of the species being multiplied, giving... 

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. 

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