P orbital transformation

Table 5.5 Central Atom s and p Orbital Transformations under Different Symmetries. ...

In the point-group Td, p-orbitals transform in the same way as dxy, dyt, and dtx, owing to the absence of a center of symmetry. This has the important consequence that all six orbitals are mixed together and the electronic transitions have some d-p character they are therefore not forbidden and have higher intensities than the nearly pure d-d transitions of octahedral complexes. 

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

The central atom p orbitals transform as Oj + and thus the molecular... 

Generation of a set of 1 x 1 matrix representations of the symmetry operations in the Cj point group using the (a) orbital, (b) orbital, and (c) Pj, orbital as a basis set. The shaded lobe of the orbitals indicates a positive sign for the wave function, whereas the hollow lobe indicates a negative sign for the wave function. The three p orbitals transform with the same representations as the corresponding coordinate axes. 

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... 

The entry under a iyz) is of different sign from the corresponding one in equation 4.32 since this symmetry operation sends each F orbital to —p. Reduction of equation 4.34 either by inspection or using equation 4.30 gives us the result that these two p orbitals transform as 02 + b. Although equation 4.30 allows access to the results of interest, with a little practice, the reducible representation may be reduced by inspection of the character table. 

Having considered the case of the H20 molecule, we would like to be able to use the same procedures to construct the qualitative molecular orbital diagrams for molecules having other structures. To do this requires that we know how the orbitals of the central atom transform when the symmetry is different. Table 5.5 shows how the s and p orbitals are transformed, and more extensive tables can be found in the comprehensive books listed at the end of this chapter. 

Thus, the orbitals uk and vk satisfy Hartree-Fock equations which are identical in form and differ only in the numerical values of the constants X/Jt and Ajk respectively. But since the latter are unknowns in the equation, and since 7(p) is itself invariant as shown in Eq. (21), we can say that the Hartree-Fock self-consistent-field equations are invariant under the orbital transformation given by Eqs. (5) and (6). This means in effect, that the energy integral ( H "X11/0 is minimized by the vk s as well as by the uk s — a circumstance which is in agreement with the invariance of and ( 1 under the transformation (5). 

Diphosphaallene derivatives ArP=C=PAr are peculiar compounds because of the presence of the two orthogonal carbon-phosphorus double bonds. The compounds were transformed into cation-radicals on electrochemical or chemical one-electron oxidation. As found, the unpaired electron is located on an MO constituted mainly by a p orbital of each phosphorus atom and a p orbital of the carbon atom (Chentit et al. 1997, Alberti et al. 1999). 

Here, the p-space natural momentals 4>i(p) Fourier transforms of the r-space natural orbitals ... 

Fig. 7.17 Successive transformations of linear combinations of atomic orbitals, beginning with atomic s and p orbitals, proceeding to sp3 hybrids, forming bond orbitals and antibond orbitals, then coupling to form the valence band and conduction band respectively. (After Harrison (1980).).

No doubt the j orbit of the H-atom in a molecule will not correspond to the p-orbit of a free atom. It will be some transformed p-orbit responsible for the actual electron energy level in a molecule,... 

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