S orbital transformation

To complete this symmetry analysis of the surface states in the L2/ — Li band gap, we consider the upper edge of the gap formed by the Li band. The orbital composition of Li is 1 (s state) and l/Vs (%y + yz + zx) (d state). The s orbital transforms totally symmetric under all symmetry operations and therefore only Pai s = sis different from zero. Following the arguments in the previous section, one concludes that the character of the actual surface states stems from a hybridization of the states at the upper and lower gap edge, respectively. Accordingly, one of the... 

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... 

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). 

As a result of the fact that the polynomial subscript to an orbital symbol tells us that the orbital transforms in the same way as the subscript, we can immediately determine the transformation properties of any orbital on an atom lying at the center of the coordinate system by looking up its subscript in the appropriate column on the right of a character table, if it is a p or d orbital. An s orbital always transforms according to the totally symmetric... 

It is easy to apply the same treatment to electrons in other types of orbitals than d orbitals. The results obtained are collected in Table 9.2. It will be seen that an s orbital is totally symmetric in the Oh environment. The set of p orbitals remains unsplit, transforming as r, this same conclusion could have... 

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... 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new... 

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