Symmetry orbital transformations

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

The first line indicates that the syrrumetry could not be determined for this state (the symmetry itself is given as Sym). We will need to determine it ourselves. Molecular symmetry in excited states is related to how the orbitals transform with respect to the ground state. From group theory, we know that the overall symmetry is a function of symmetry products for the orbitals, and that only singly-occupied orbitals are... 

C2 and change signs under C2 E, axz, or ayz operations). Although it may not be readily apparent, the py orbital transforms as B2. Using the four symmetry operations for the C2 point group, the valence shell orbitals of oxygen behave as follows ... 

If we now consider a planar molecule like BF3 (D3f, symmetry), the z-axis is defined as the C3 axis. One of the B-F bonds lies along the x-axis as shown in Figure 5.9. The symmetry elements present for this molecule include the C3 axis, three C2 axes (coincident with the B-F bonds and perpendicular to the C3 axis), three mirror planes each containing a C2 axis and the C3 axis, and the identity. Thus, there are 12 symmetry operations that can be performed with this molecule. It can be shown that the px and py orbitals both transform as E and the pz orbital transforms as A, ". The s orbital is A/ (the prime indicating symmetry with respect to ah). Similarly, we could find that the fluorine pz orbitals are Av Ev and E1. The qualitative molecular orbital diagram can then be constructed as shown in Figure 5.10. 

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

Having seen the development of the molecular orbital diagram for AB2 and AB3 molecules, we will now consider tetrahedral molecules such as CH4, SiH4, or SiF4. In this symmetry, the valence shell s orbital on the central atom transforms as A, whereas the px, py, and pz orbitals transform as T2 (see Table 5.5). For methane, the combination of hydrogen orbitals that transforms as A1 is... 

If there is a molecular symmetry group whose elements leave the hamiltonian 36 invariant, then the closed-shell wavefunction belongs to the totally symmetric representation of both the spin and symmetry groups.8 It is further true that under these symmetry operations the molecular orbitals transform among each other by means of an orthogonal transformation, such as mentioned in Eq. (5) 9) and, therefore, span a representation of the molecular symmetry group. In general, this representation is reducible. 

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 electrons which are important for the bonding in the water molecule are those in the valence shell of the oxygen atom 2s22p4. It is essential to explore the character of the 2s and 2p orbitals, and this is done by deciding how each orbital transforms with respect to the operations associated with each of the symmetry elements possessed by the water molecule. 

In the example under discussion the 2s and 2p, orbitals transform as a, the 2px orbital transforms as b, and the 2p, orbital transforms as b2. In this context transform refers to the behaviour of the orbitals with respect to the symmetry operations associated with the symmetry elements of the particular group. 

The classification of the 2s and 2p atomic orbitals of the central oxygen atom. The 2s(0) orbital transforms as Gg, the 2px and 2p orbitals transform as the doubly degenerate nu representation, and the 2pz orbital transforms as om+. In some texts the + and superscripts are omitted, but throughout this one there is strict adherence to the use of the full symbols for all orbital symmetry representations. Unlike the 90° case, the 2s and 2p7 orbitals have different symmetry properties so there is no question of their mixing. 

C2u character table because when x2 and y2 are of the same symmetry, any linear combination of the two will also have that symmetry. Note that although both the and d - orbitals transform as at in this point group, they are not degenerate because they do not transform together, ft would be a worthwhile exercise to confirm that the s, p, and d orbitals do have the symmetry properties indicated in a Cu molecule. Keep in mind, in attempting such an exercise, that the signs of orbital lobes are important... 

Reduction of fr (Eq. 3.1) shows that it is composed of ee, and bin. The character table reveals that no orbitals transform as but that p. belongs to while df. and dn belong to eg. That these three orbitals on platinum are allowed by symmetry to participate in out-of-plane it bonding is reasonable since they are all oriented perpendicular to the plane of the ion (the xy plane). Selection of orbitals on platinum suitable for in-plane it bonds is left as an exercise. (Hint In choosing vectors to represent the suitable atomic orbitals, remember that the in-plane and out-of-p[ane ir bonds will be perpendicular to each other and that the regions of overlap lor the former will be on eadi side of a hording axis. Thus the in-plane vectors should be positioned perpendicular to the bonding axes.)30... 

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. 

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