Orbitals with nonzero overlap

The rules just given let us decide which atomic orbitals may have nonzero overlap in a molecule. We have seen that sN may have nonzero overlap with s1 (the combination s_A + s_B + s_C), so bonding and antibonding molecular orbitals can form from (sN, s1) overlap (Fig. 12.28). The general rule is that only orbitals of the same symmetry species may have nonzero overlap, so only orbitals of the same symmetry species form bonding and antibonding combinations. It should be recalled from Chapter 11 that the selection of atomic orbitals that had mutual nonzero overlap is the central and initial step in the construction of molecular orbitals by the LCAO procedure. We are therefore at the point of contact between group theory and the material introduced in that chapter. The molecular orbitals formed from a particular set of atomic orbitals with nonzero overlap are labelled with the lowercase letter corresponding to the symmetry species. Thus, the (sN, s1)-overlap orbitals are called a1 orbitals (or a1*, if we wish to emphasize that they are antibonding). The linear combinations s2 = 2s_a− s_b − s_c and s₃ = s_b − s_c have symmetry species E. Does the N atom have orbitals that have nonzero overlap with them (and give rise to e molecular orbitals)? Intuition (as supported by Figs. 12.28b and c) suggests that N2p_x and N2p_y should be suitable. We can confirm this conclusion by noting that the character table shows that, in C3v, the functions x and y jointly belong to the symmetry species E. Therefore, N2p_x and N2p_y also belong to E, so may have nonzero overlap with s₂ and s₃. This conclusion can be verified by multiplying the characters and finding that the product of characters can be expressed as the decomposition E ×E=A1+A2+E. The two e orbitals that result are shown in Fig. 12.28 (there are also two antibonding e orbitals). We can see the power of the method by exploring whether any d orbitals on the central atom can take part in bonding. As explained earlier, reference to the C_3v character table shows that dz2 has A1 symmetry and that the pairs (d_x2−y2, dxy) and (dyz,dzx) each transform as E. It follows that molecular orbitals may be formed by (s₁,d_z2) overlap and by overlap of the s₂,s₃ combinations with the E d orbitals. Whether or not the d orbitals are in fact important is a question group theory cannot answer because the extent of their involvement depends on energy considerations, not symmetry.

Fig. 12.28 Orbitals of the same symmetry species may have non-vanishing overlap. This diagram illustrates the three bonding orbitals that may be constructed from (N2s, H1s) and (N2p, H1s) overlap in a C_3v molecule. (a) a1; (b) and (c) the two components of the doubly degenerate e orbitals. (There are also three antibonding orbitals of the same species.)

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