2.5. The AH₂ Walsh Diagram

As mentioned above, the MOs derived in Sect. 2.4 for methylene are applicable to any AH₂ fragment where A is a main group element. So-called Walsh diagrams consider the changes in the shapes and energies of individual MOs as the geometry is changed, in this case the dependency of the MOs of AH₂ fragments on the central H-A-H angle. Figure 2.15 shows the Walsh diagram that allows us to estimate the central angle in AH₂.

Figure 2.15 Walsh diagram for AH₂ molecules with a bent (left) and a linear (right) H-A-H angle.

The s_CH-orbital consists essentially of an s-orbital on carbon. s-Orbitals are spherical and therefore overlap equally well with ligand orbitals in any direction. The energy of the s_CH is therefore largely unaffected by the change from a bent to a linear geometry. The p_CH2-MO, on he other hand, is quite sensitive to the H-A-H angle. As shown in Sect. 2.3, overlap with p-orbitals is at a maximum when the ligand lies on the principal axis of the p-AO. By moving the hydrogen ligands from a bent to a linear geometry, we move them into the principal axis of the carbon-p, and therefore to the position of maximum overlap. The p_CH2 is therefore more stable in the linear geometry than in the bent. The energy of the n- (lone pair) orbital depends strongly on the H-A-H angle for another reason. In the bent geometry, the s- and p-AOs on carbon can mix with each other. This mixing can be considered to stabilise the p-orbital. At the linear geometry, however, the s- and p-AOs are orthogonal, so that the lone pair becomes a pure p-orbital that is degenerate with the original p-MO, which remains unaffected by the change in geometry because it is localised on the carbon. Thus, the energy of the n-orbital rises steeply on changing the geometry from bent to linear.

These changes in orbital energy can be used to rationalise the observed geometries of AH₂ fragments. Figure 2.16 shows the energy levels given in Figure 2.15 with schematic orbital occupations for singlet and triplet CH₂, H₂O and H₂S, BH₂⁺ and BeH₂.

Figure 2.16 Walsh diagram for AH₂ showing orbital occupation diagrams for 4-, 8- and singlet and triplet 6-electron species.

Singlet methylene, the molecule used to derive the MOs in Sect. 2.4.1, has six valence electrons. These occupy the lowest three MOs, the s_CH, the p_CH2 and the n. As described above, the s_CH has no preference for a bent or linear geometry, whereas the p_CH2 prefers linearity and the n-lone pair a bent geometry. The latter two orbitals therefore oppose each other in their geometric preferences and we must decide which of the two has the stronger preference (i.e. which slope of the two orbital correlation lines is larger). In this case, the n-lone pair dominates and singlet methylene is strongly bent. Adding two electrons to obtain the 8-electron species water or H₂S has little or no effect on the preferred H-A-H angle because the two additional electrons occupy the p-MO, which is independent of the angle.

Triplet methylene has a larger H-C-H angle than the singlet because the n-orbital is only singly occupied and therefore half as important energetically as in the singlet. The geometric preference of the p_CH2 is therefore more important for the triplet and the bond angle is larger. Finally, the four-electron species BH₂⁺ and BeH₂ have only the s_CH and p_CH2 orbitals occupied and therefore prefer linear geometries. Table 2.1 gives the bond angles found for the species shown in Figure 2.16.

Table 2.1 Bond angles found for the species shown in Figure 2.16.

Note that H₂S has a smaller bond angle than ¹CH₂ or H₂O, although they are all have predicted to have similar geometries by the Walsh diagram. This is a general trend between molecules involving main group elements of the first and higher long periods.

Walsh diagrams are a powerful tool for explaining the geometries of covalent molecular fragments, but can fail for highly ionic species. One well know example is Li₂O, which ought to have a geometry similar to that of water according to the Walsh approach, but is in fact found to be linear (Figure 2.17). The reason for this discrepancy is that the bonding in Li₂O is not described well by a covalent treatment such as that implicit in the Walsh diagram shown above and that the bond angle is determined by the electrostatic repulsion between the two positively charged lithium ions:

Figure 2.17 The 180° bond angle in Li₂O minimises the electrostatic repulsion.

H₂ and He₂ -The Simplest Examples
The Effect of Electronegativity
p-Orbitals and p-Overlap
Combining AOs to Build MOs
The AH₂ Walsh Diagram