D h point group

The linear C02 and XeF2 molecules belong to the D h point group and, for purposes of classifying their atomic orbitals, the principal quantum numbers (2 for C, 5 for Xe) of their valence electrons are omitted in the following discussion. 

For the D ,h point group, we can use the projection operators from the subgroup D4h. Applying the project operators Peg and PEu to the above 4 primary BTs, we obtain... 

The D h point group is derived from C, by the inclusion of oh therefore, all linear molecules with a plane of symmetry perpendicular to the axis belong to Di0ch. Acetylene... 

For BF3, there is a C3 axis as shown in the structure and the molecule lies in a horizontal plane, designated as <7/,. There are also three vertical planes. Each B-F bond also defines a C2 axis where the 07, intersects each av. Therefore, the trigonal planar BF3 molecule has one C3 axis, one ah, three av, and three C2 axes. A molecule that has a C axis and n C2 axes perpendicular to it resulting from the horizontal plane intersecting the three vertical planes belongs to the D h point group. Therefore, D3h is the appropriate point group for BF3. Note that a structure such as... 

The 2s and 2pz orbitals of the same atom belong to the same irreducible representation of the D h point group. Their energies are also similar so they cannot be separated completely. Another way of making linear combinations is to first combine the 2s and 2pz orbitals of the same atom... 

The next step is to determine how these group orbitals transform in the D(,h point group. The D6h character table is given in Table 6-7. Since most of the AOs in the suggested group orbitals are transformed into another AO by most of the symmetry operations, the representations will be quite simple, though still reducible ... 

If the molecule possesses several rotation axes (the principal axis being of order n), there are three possibilities (t) there are no other symmetry elements besides these axes (D point-groups) (it) there are naj planes that bisect the Cz axes (D j point groups) (tit) there is also a CTfc plane, perpendicular to the principal axis (D h point groups). 

The reduction formula (6.5), in combination with the characters of the irreducible representations of the D h point group found in Table 6.21, leads to ... 

We shall limit ourselves here to the determination of the symmetry-adapted A" and E" orbitals in the Fj,x representation. Table 6.25 shows the action of all the symmetry operations of the D h point group on the generating functions pxi and (px2 — Pxs), as well as the characters of the A" and E" irreducible representations. 

FIGURE 4.4. Assembly of the molecular orbital diagram for the planar AII3 molecule D h point group) using symmetry-adapted orbitals on the ligands and the central s andp orbitals. 

The energies are thus seen to form N discrete levels, which are points on a cosine curve, as shown in Fig. 4.7. Except for k = 0, and in the case of N even, k = N/2, all levels Ek and are twofold-degenerate. Closed-shell structures thus will be realized for N = 4n +2, which is the famous Hiickel condition for aromaticity. These cyclic labels can easily be expanded to the full irrep designations of the D h symmetry group for benzene. The atomic p -orbitals transform as bi in the C2v site group. In accord with the conventions for the D h point group symmetry, as pictured in Fig. 3.10, this site group is based on operators of type and a. The induced irrep of the six atomic orbitals then becomes... 

A trans disubstituted six-coordinate complex of the type A4MB2 has D4 , symmetry. Some of the degeneracies in the three stretching vibrations of the Oh point group are removed to give five stretching vibrations in the D h point group, of which two are infrared active. 

The orientations of the principal axes are easily obtained by inspection in molecules with sufficient symmetry. Any C symmetry axis (n > 2) coincides with a principal axis. In the D2h ethylene molecule, for example, the CjCx), C2(y), and 2(2) twofold axes are principal axes. All of the principal moments of inertia in ethylene are different in magnitude, so this molecule is an asymmetric top. All molecules with one C3 or higher-order axis are symmetric tops, because the principal moments about two axes normal to a C axis with>i 3 are necessarily equal. Such molecules include BF3 (which belongs to the D h point group), NH3 and CH3I (C3p), and benzene (Dg ). Molecules with two or more C3 or higher... 

Show that the five M-L a bonding orbitals in a MLj trigonal bipyramidal complex have 2A l + A 2 + E symmetries. For this problem the character table of the D h point group is... 

Hybridization and symmetry are intimately connected because of the relationships seen above Atomic centers in molecules that have certain hybridization have specific shape. Atoms that make sp hybrid orbitals impart a linear shape about that atomic center (which might suggest Coov or D h point groups, for simple molecules). Atoms that have sp hybrid orbitals make bonds in a threefold, or trigonal, shape. 

Figure 3.22 Example molecules belonging to D h point groups (a) ethene, Djh/ (b) the trans-isomer of [CoINH I Cb], 04. ...

Figure 3.23 shows the two point groups that arise for linear molecules. If the two ends of the molecule are different, then the only symmetry elements are and the vertical mirror planes, so the point group is Coov by analogy with C2v, Csv, etc. If the molecule has equivalent points at either end of the axis then it will also have a horizontal mirror plane

A molecule which has an -fold axis of symmetry C , n twofold perpendicular axes, and n vertical planes of symmetry and which also has a horizontal plane of symmetry perpendicular to C belongs to the D h point group. 

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