D4h point group

Figure 3. The common distortions of a porphyrin, written in terms of the irreducible representations of the D4h point group. 66-127[Phys. Chem. Chem. Phys. 2002, 4, 5878] - Reproduced with permission of the PCCP Owner Societies.

Here, S is the total spin of the cluster, J is the parameter of the HDVV exchange interaction, t is the parameter of double exchange, Sa and 5 are the spins of the cluster ions, and is the minimal of these spins. For the sake of definiteness the orbitals are assumed to be spherical, and the symmetry labels of D4h point group are employed. The presence of these levels in the electronic spectrum of MV dimers and their mixing by the out-of-phase mode q (q = (Qa - Qb)lV2 Qa and Qb are the full symmetric vibrational modes of the cluster fragments (Fig. 29a))... 

Scheme 4.2 Frontier orbitals of the porphyrins according to the Gouterman four-orbital model for D4h point group (top). In the presence of a perturbation (substituents or ligand distortion) the degeneracy might be lifted (bottom). With permission from Wiley, ref 7a.

Fig. 7. A. Schematic illustration of an antisymmetric vibrational mode of/l2g symmetry in the D4h point group. The diagram can represent a much simplified view of a porphyrin-type molecule. B. Illustration of the variation in the transition moments of E y (04, ) symmetry during an 2g vibration. The full arrows represent the transition moments x°, y° corresponding to the equilibrium configuration Q,42g = 0. The dashed arrows ate the moments x, y corresponding to a turning point of the vibration. In the limit Q 0, the only non-vanishing variations are shown by the white arrows...

In the case being examined, we therefore come to the same conclusion as that established in the preceding section from a limited number of symmetry elements the only interactions that occur concern the (tt, TV ) orbitals on the Ugands, and (xz, yz) on the metal centre, which constitute two degenerate pairs of orbitals with e symmetry in the D4h point group. 

Tables 6.18 and 6.19, which give the character table for the D4h point group and the full set of the symmetry operations of this group, should he helpful, as should Scheme 6-24 in which the different symmetry elements are shown.

Similarly for cyclo-butadiene (D4h point group) Fa transforms according to the hig representation and the a2 and l>2u bonding and antibonding molecular orbitals shown in Fig. 13 are related by the following direct product ... 

Problem 3.2 Show that the product of a Q" axis and the cf plane in the D4h point group implies that there are also mirror planes. 

During the study of inorganic chemistry, the structures for a large number of molecules and ions will be encountered. Try to visualize the structures and think of them in terms of their symmetry. In that way, when you see that Pt2+ is found in the complex PtCl42 in an environment described as D4h, you will know immediately what the structure of the complex is. This "shorthand" nomenclature is used to convey precise structural information in an efficient manner. Table 5.1 shows many common structural types for molecules along with the symmetry elements and point groups of those structures. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

The cross terms (df)0(d,), with i j in Eq. (10.3) do not appear in the case of the isolated atom for which the electron density is the sum of the square of the atomic orbitals. In the molecular case, the cross terms will only be nonzero for orbitals belonging to the same representation of the point group of the molecule, like the eg orbitals in the case of trigonal site symmetry discussed above. In the square-planar point group D4h(4/m mm), the orbitals have alg, blg, b2g, and eg symmetry, and no such mixing occurs. 

The M 1 matrices specific for higher point groups are obtained by omission of symmetry-forbidden columns in the full 15 x 15 matrix. This leads to rows with zero elements for the nonallowed cross products between d orbitals, which are subsequently omitted to recover a reduced matrix. The matrix for the point group D4h = 4/m mm is shown as an example in Table 10.2. 

For the point group D4h, the atomic d-orbital functions belong to four different group-theoretical representations. When the same representation occurs more than once, as it does, for example, in trigonal point groups, M-1 will contain cross terms between orbitals of the same symmetry, as shown in Table 10.3(a). In this case, we are interested in the population of the symmetry-adapted orbitals ysk, such as defined for the trigonal case by expression (10.2). The symmetry-adapted orbitals are linear combinations of the original functions, that is,... 

Find the IRs of the point group D4h for which the following Cartesian tensors form bases ... 

For point group invariant systems, the intrinsic group is G = Point group. The construction of the basis for these systems is a standard group theoretical problem. For the groups D4h, D6h and 0 it was done by Hamermesh many years ago [13]. I report here only the case of G D4i,. For positive parity one has Table 3. For negative parity one has Table 4. 

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