Chapter
4

Electrons in molecules: polyatomics

Chapter 4: Electrons in molecules: polyatomics

4.1 The simplest triatomic: H₃⁺

MOs can be constructed for simple polyatomic molecules, such as H₃⁺ and H₃. The principles are quite general, and they can be used to describe more complex examples.

4.1.1 Constructing the MOs using symmetry

For linear H₃⁺ the three H 1s AOs can be divided into a central atom and two identical end atoms as shown in Figure 4.1. This Figure also shows the key mirror plane.

As a consequence the AO for the central atom is symmetric and the AOs for the two identical end atoms have no mirror symmetry (see Figure 4.2).

However, symmetry orbitals (SOs) can be created by combining the AOs, as shown in Figure 4.3. Adding the two end AOs gives a symmetric in-phase SO (i.e., SO2), while subtracting gives an antisymmetric out-of-phase SO (i.e., SO3). SO2 is expected to be lower in energy than the 1s AO (SO1), while SO3 is expected to be higher.

Figure 4.4 shows the resultant MO diagram for linear H₃⁺. Interaction between the symmetric SO1 and SO2 gives bonding MO1 and antibonding MO3, while antisymmetric SO3 has no other orbitals to overlap with, and forms a weakly antibonding MO2.

Figure 4.5 shows contour plots of the MOs. All three orbitals have cylindrical symmetry (there are no other possibilities for this linear arrangement of atoms!), and are hence σ orbitals. MO1 is bonding with no nodes, while MO2 and MO3 are antibonding with one and two nodes respectively. Notice that the energy of these MOs increases with increasing numbers of nodes

Both electrons in H₃⁺ are allocated to the bonding MO1 and are delocalized over the three atoms.

4.1.2 Constructing the MOs for triangular H₃⁺

Experiment has shown that triangular H₃⁺ is the most stable geometry. All vertices are equivalent in an equilateral triangle and so all atoms have the same symmetry. By choosing one mirror plan, the ion has one symmetric 1s AO (SO1) and two AOs with no symmetry (see Figure 4.6).

Figure 4.7 shows that by combining the two AOs with no symmetry an in-phase symmetric orbital (SO2) results, as well as an out-of-phase antisymmetric SO (SO3).

The MO diagram for triangular H₃⁺ is shown in Figure 4.8. The actual linear combinations of atomic orbitals are the same as for the linear arrangement, and they can be pictured as in Figure 4.9. However, the label σ is no longer appropriate and MO2 and MO3 are degenerate. The MO2 has a much higher energy than the antibonding MO2 in the linear ion.

4.1.3 The optimum geometry of H₃⁺

For the triangular H₃⁺ tion, the two electrons will again occupy the lowest energy orbital, and the MO1 is lower than the MO1 for linear H₃⁺. As a consequence, H₃⁺ is expected to be triangular. This orbital is delocalised over all three atoms; three nuclei are bound by a single electron pair.

Figure 4.10 shows the stability of H₃⁺ as a function of H-H-H angle, and the most stable is when this angle is 60^o (i.e., equilateral triangle).

Geometry of H₃

For neutral H₃ the third electron is placed in the antibonding MO2, which is more weakly antibonding for linear H₃ than for the triangular geometry. Hence, linear H₃ is the preferred geometry.

4.2 More complex linear triatomics

The approach of constructing SOs and then MOs can be extended to more complex molecules. An example is linear FHF^–, which has the valence AO electron configuration 1s¹ for H and 2s²2p⁵ for each F plus one extra electron. The 2s AOs are nonbonding σ-type MOs on the F. Overlap occurs between the 1s and 2p_z AOs, to form symmetric SO1 and SO2, and antisymmetric SO3. This is represented in Figure 4.11.

The MO diagram, which combines the symmetric SO1 and SO2 into a lower energy bonding MO1 and higher energy antibonding MO3 is shown in Figure 4.12. MO2 is, as for linear H₃^–, weakly antibonding. The form of MO1 and MO2 are plotted in Figure 4.13.

The pair of electrons in the bonding MO1 are shared across all three atoms. The bonding is again delocalized.



4.3 MOs of water and methane

4.3.1 Using symmetry to construct an MO diagram of water

Water is bent, with an H-O-H bond angle of 104.5^o. The valence AO electron configuration is 1s¹ for each H and 2s²2p⁴ for each O. For this geometry overlap will occur between the 2s, the 2p_z and the 2p_y on O giving symmetric SO1 and SO2 and antisymmetric SO3. Overlap will also occur between the two 1s AOs on the two H atoms to give symmetric SO4 and antisymmetric SO5. This is shown in Figure 4.14.

The approximate MO diagram for water is represented in Figure 4.15. There are a total of six MOs, with oxygen 2p_x leading to a nonbonding orbital, MO4. As there are eight valence electrons, all the MOs up to MO4 are fully occupied by pairs of electrons. MO1 and MO2 are bonding, MO3 is weakly bonding and MO4 nonbonding and hence a lone pair on the O. The different representations for the MOs are shown in Figure 4.16 on page 148, but have not been reproduced in these notes.

4.3.2 MOs of methane

MO diagrams with molecules with more valence electrons and AOs than H₂O, become too difficult and complex to construct. For example, OF₂ will have 12 MOs; four AOs on each atom. When compared with OF₂, methane has more atoms, fewer AOs (8) and higher symmetry. As a consequence group theory needs to be used. This introduces a complication, and so the MOs for such molecules are best and most accurately determined using a computer program and are shown in Figure 4.17 for CH₄. MO1 is the lowest energy orbital, while MO2, MO3 and MO4 are degenerate. The MOs are delocalised over the 4 bonds.

Hybrid atomic orbitals have been introduced in order to interpret the properties of more complex molecules, without resorting to computer programs. Hybrid AOs are discussed in the next section

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4.4 Hybrid atomic orbitals

Simple overlap between s and p atomic orbitals is unable to account for the geometries of simple molecules such as H₂O, NH₃ and CH₄. In all cases the bond angles are nowhere near 90^o, the value expected from the use of only p orbitals on the central atoms. This dilemma was solved in first-year by introducing hybridisation or hybrid atomic orbitals (HAOs). HAOs can be useful in MO theory to help visualise certain concepts.

The common textbook explanation of the hybridisation process is to first promotion an electron (which requires considerable energy), then mixing of orbitals to form directed hybrids (which requires no energy). This first step is required for the C atom; without it the electron configuration would be 1s²2s²2p_x¹2p_y¹, and only two singly-occupied p orbitals would be available for overlap, and hence bonding. But little mention is made of the energy required for this process. It turns out to be considerable, and of the order of 800 kJ mol^-1. Moreover, promotion is not required to form sp³ hybrid orbitals for N and O atoms (or many others). Promotion of electrons for this purpose is unnecessary and is a complete fiction!

Hybridisation is the formation of linear combinations of s, p and sometimes d orbitals. It is part of the language of chemistry learned by every student, and it does indeed rationalise the shapes of many, many complex molecules, especially organic molecules.

For our purposes in this unit it is sufficient to note the following, which appear beyond dispute:

1. Certain mathematical linear combinations of s, p and d orbitals can be constructed which yield a set of identical hybrid orbitals which possess marked directional character, and point in directions that accord with the basic VSEPR shapes.
2. The number of hybrid orbitals formed is the same as the number of atomic orbitals used.
3. The most common hybrid orbitals are sp (digonal, or linear), sp² (trigonal planar) and sp³ (tetrahedral).
4. Less common are sp²d (trigonal bipyramidal) and sp²d² (octahedral).
5. Chemical bonding can be described by overlap of appropriate singly-occupied hybrid orbitals with singly-occupied orbitals on neighbouring atoms. Multiple bonds arise from additional overlap of unhybridised singly-occupied p orbitals; for an sp hybridised atom there are two sets of unhybridised p orbitals, and for an sp² hybridised atom there is one set of unhybridised p orbitals.

4.4.1 sp³ hybrids

For methane sp³ HAOs, the 2s and three 2p AOs on carbon combine. Four HAOs result and are represented in Figure 4.18. The direction of each HAO varies (109.5^o between any two hybrids), but the energies and shapes are identical. The energy of each HAO lies between that of the 2s and 2p AOs, but closer to the 2pbecause sp³ is 75% 2p character. Each HAO has both a major (blue) and a minor (pink) lobe.

Overlap then occurs between each HAO and the 1s on each H, to give bonding and antibonding σ and σ^* MOs, as shown in Figure 4.20. This means that the bonding is localized between each C-H, rather than delocalized as in the full and accurate MO description.

Figure 4.20 also shows the shapes of the four bonding and four antibonding MOs for C-H, while Figure 4.22 shows the plots for the MOs for C-C hybridised bonding and antibonding.

4.4.2 Doubly-bonded carbon: sp² hybrids

For double-bonded ethene, sp² HAOs are assumed. These are a combination of 2s and two 2p AOs on carbon. The shapes of the resultant three HAOs and the unhybridized 2pz AO are represented in Figure 4.24. The direction of each HAO varies (120^o between any two hybrids), but the energies and shapes are identical. The energy of each HAO lies between that of the2s and 2p AOs, but closer to the 2p because sp³ is 67% 2p character. Each HAO has both a major (blue) and a minor (pink) lobe.

Overlap then occurs between each HAO and the 1s on each H and the sp² on the second carbon, to give bonding and antibonding σ and σ^* MOs, as shown in Figure 4.26. Also the two out-of plane 2p_z AOs overlap to form bonding and antibonding π and π^* MOs. Again the bonding is localized between each C-H and C-C, rather than delocalized as in the full and accurate MO description.

Rotation of the C-C double bond is difficult because it requires the breaking of the π bond between the two carbons. See Figure 4.27.

4.4.3 Triply-bonded carbon: sp hybrids

For triple-bonded ethene, sp HAOs are assumed. These are a combination of 2s and 2p AOs on carbon. The shapes of the resultant two HAOs and the unhybridized 2p_x and 2p_y AOs are represented in Figure 4.28. The direction of each HAO varies (180_o between the two hybrids), but the energies and shapes are identical. The energy of each HAO lies midway between that of the 2s and 2p AOs. Each HAO has both a major (blue) and a minor (pink) lobe.

Overlap then occurs between each HAO and the 1s on each H and the sp on the second carbon, to give bonding and antibonding σ and σ^* MOs, as shown in Figure 4.30. Also the out-of plane 2p_x and 2p_y AOs overlap to form bonding and antibonding π and π^* MOs.

4.4.4 Summary

Each molecular shape is a result of, or indeed is a cause of, hybridisations involving various s, p and d orbitals.

The common geometric shapes and the atomic orbitals, which make up these hybrids are as follows:

Coordination number	Arrangement	Composition
2	Linear	sp, pd, sd
	Angular	sd
3	Trigonal planar	sp2, p2d
	Unsymmetrical planar	spd
	Trigonal planar	pd2
4	Tetrahedral	sp3, p3d, pd3
	Irregular tetrahedral	spd2, p3d, pd3
	Square planar	p2d2, sp2d
5	Trigonal bipyramidal	sp3d, spd3
	Tetragonal bipyramidal	sp2d, sd4, pd4, p3d2
	Pentagonal bipyramidal	p2d3
6	Octahedral	sp3d2
	Trigonal prismatic	spd4, pd5
	Trigonal antiprismatic	p3d3

Note that trigonal bipyramidal or square pyramidal structures are not symmetrical, and the hybridisation can be considered a combination of different subsets of hybrid or unhybridised orbitals.

4.5 Comparing the hybrid and full MO approaches

Nitrogen has the electronic configuration 1s² 2s² 2p³. As N₂ is a linear molecule the hybridization approach would be to assume two sp HAOs on each N (energy midway between 2s and 2p). Overlap would lead to bonding and antibonding σ MOs, and two non-bonding orbitals (sp_c and sp_d) would form. In addition, the 2p_x,y AOs overlap to form bonding and antibonding π MOs. The hybridization MO diagram is shown in Figure 4.31(a). The 10 valence electrons would fill the bonding σ, π MOs and the non-bonding sp_c and sp_d MOs.

In comparison, Figure 4.31 (b) is the full MO diagram of N₂. Here the occupied orbitals are 2σ_g (bonding), 2σ_u (weakly antibonding), 1π_u (bonding) and 3σ_g (weakly bonding, HOMO).

Similarities between two models are that both predict two π bonds and both predict that the LUMO is π^*. Differences between two models are that the HAO approach predicts two lone pairs, whereas the full MO model predicts two weakly bonding and antibonding σ MOs (supported by the photoelectron spectrum, Figure 3.37, p. 126). Also the full MO model predicts delocalization, which is an accurate description of the bonding, while for the HAO model, MOs are localized at bonds.

4.5.1 More about lone pairs

Lone pairs are commonly identified by a lack of symmetry between AOs.

Lone pairs could also be indentified in the full MO model, in the case when the bonding and corresponding antibonding MOs are occupied. For example, for N₂ both 2σ_g and 2σ_u are filled, and so these could be considered to represent one lone pair on each nitrogen. This would allow the prediction of two lone pairs on each oxygen for O₂, and three lone pairs of each fluorine for F₂.



4.6 Extending the hybrid concept

The bond angles for hydrides are shown in the following table:

Molecule	H-X-H angle	Molecule	H-X-H angle
NH3	107.8o	H2O	104.5o
PH3	93.5o	H2S	92.1o
AsH3	92.0o	H2Se	91.1o
SbH3	91.5o	H2Te	89.5o

These structures can be described using the hybrid approach by varying the ratio of s to p, as shown in Figure 4.32. For 109.5^o (e.g., methane) sp³, is the best description, while for 120^°, sp² and for 180_°, sp is used. These represent 25%, 33% and 50%, respectively.

It is not necessary for all HAOs on an atom to be equivalent, but the overall percentage s character must be equal to 25^°, for sp³-type structures or 33% for sp²-type etc.

4.6.1 Water

For water, the H-O-H angle can be reduced from 109.5^o to 21% (from 25%) and increasing the s character of the other two HAOs to ensure 25% s character overall. The iso-surfaces of the four HAOs are shown in Figure 4.33. HAO1 and HAO2 have reduced s character and overlap with the 1s on H lead to MOs with a H-O-H bond angle of 104.5^o. HAO3 and HAO4 have increased s character and are associated with equivalent the non-bonding MOs (lone pairs).

Alternatively, the starting point could be sp², and the s character of two of these HAOs increased to match the H-O-H bond angle of 104.5^o. This would lead to one lone pair based on sp² and one lone pair based on the 2p AO.

The photoelectron spectrum of water, reproduced in Figure 4.34, indicates one lone pair (MO4), while MO1 (not shown), MO2 and MO3 are bonding orbitals, delocalised over the two O-H bonds. This is in agreement with the full MO diagram (Figure 4.15, page 147), but not with the two hybrid models (using either sp³ or sp² hybrids), which predict two non-bonding lone pairs.