D2h point group

For a molecule belonging to the D2h point group deduce whether the following vibrational transitions, all from the zero-point level, are allowed in the infrared spectmm and/or Raman spectmm, stating the direction of the transition moment and/or the component of the polarizability involved ... 

A 3 — Ag electronic transition in a molecule belonging to the D2h point group is forbidden. at are the possible symmetry species of a vibration X which would result in the Xq and X transitions being allowed ... 

Pyrazine belongs to the D2h point group, and Table A.32 in Appendix A shows that since B u = r(Tj) the Oq band is polarized along the x axis, which is perpendicular to the molecular plane. 

Direct Product Table for C2v Point Group in Coordinate Space with / to Give the D2h Point Group in Momentum Space... 

Naphthalene and anthracene are archetypes of the even and odd members of the polyacene series. In each subseries, one can start by classifying the classical Kekule structures by using the symmetry operations i, C2, and point group. Then one can form symmetry-adapted linear combinations of the mutually transformable Kekule structures and deduce their bonding characteristics. Finally, these 1 Ag and 1 B2u symmetry-adapted combinations are allowed to mix and form the states of interest, the ground and first covalent excited states (16). 

The point group D]h is the same as C2v. Ethylene (Figure 4.1a) and naphthalene (Figure 4.3c) belong to the D2h point group in which, because of the equivalence of the three mutually perpendicular C2 axes, no subscripts are used for the planes of symmetry. 

The character table of a point group defines the symmetry properties of a (wave) function as either 1 for symmetric or —1 for antisymmetric with respect to each symmetry operation/ The first row lists all the symmetry operations of the point group and the first column lists the Mulliken symbols of all possible irreducible representations, the symmetry transformation properties that are allowed for wavefunctions. As an example, the character table for the D2h point group is given in Table 4.1. The character tables of all relevant point groups are given in many textbooks.134,273-275 The last column shows the transformation properties of the axes x, y and z, which are used to determine electronic dipole and transition moments (Section 4.5). 

Point Group CO2 has Do, , symmetry so the D2h point group will be used. 

Matching Orbitals on the Central Atom To determine which atomic orbitals of carbon are of correct symmetry to interact with the group orbitals, we will consider each group orbital separately. The carbon atomic orbitals are shown in Figure 5.19 with their symmetry labels within the D2h point group. 

Ethylene has E, three independent C2 axes, three independent planes of symmetry, and a center of inversion. This total of eight symmetry elements composes the D2h point group. 

Symmetry operators and their products can be represented conveniently by matrices. Suppose an atom in the D2h point group has coordinates r= x,y,z) relative to the molecule s center of mass. If we represent the coordinates by a column vector and use the expression for multiplication of a vector by a matrix (Eq. A2.5 in Appendix A.2), the actions of the symmetry operators on the atom s location can be written ... 

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