Chapter
1

Introduction

Chapter 3: Electrons in molecules: diatomics

MO theory is capable of providing the most reliable quantitative predictions of structure and bonding in molecules, and it is used almost routinely these days to complement research in all areas of chemistry. In that guise it is called computational chemistry. Underlying these calculations are sophisticated mathematics and computer programming, which cannot be conveyed in this unit. In these chapters, the basic ideas are covered with application to simple diatomic molecules and diatomic fragments.

3.1 Introducing molecular orbitals

Why is H₂+ is lower in energy than separated H+ and H?

Figure 3.1 shows that in H₂⁺ there are two attractive contributions to the potential energy between the electron and two nuclei and one repulsive contribution between the two nuclei.

The attractive contribution decreases as the distance between the nucleus and electron decreases. This is shown in Figure 3.2 (blue line). Repulsions between the nuclei dominate as the distance between the nuclei decreases (red line in Figure 3.2). The kinetic energy (green line) also increases with decreasing distances. The sum of these three energy contributions gives a minimum at Re, where the stable H₂⁺ occurs.

3.1.1 Linear combination of atomic orbitals

The linear combination of atomic orbitals (LCAO) approximation is an excellent method to form each molecular orbitals.

The best way to appreciate these statements is by example, and the simplest example is the H₂⁺ ion (one electrons, two nuclei). In the LCAO approximation, the MOs are formed from combinations of the hydrogen 1s AOs, one on centre A - ψ_A - and one on centre B - ψ_B.

where the c_A and c_B are the orbital coefficients. Two linear combinations are possible from these two orbitals

ψ₊ = N₊[ψ_A +ψ_B] and ψ_- = N_- [&psi_A -ψ_B]

Here, N₊ and N_– are normalizing factors, and ψ₊ is the in-phase (bonding) interaction, and ψ_- is the out-of-phase (antibonding) interaction. The energies of these two molecular orbitals are plotted in Figure 3.3. The minimum in E+ occurs where R = 2.5a₀ (a₀ is the Bohr radius).

In general, combinations of n AOs gives rise to the same number - n - of MOs. As for atoms, the Pauli exclusion principle applies, so that each MO is occupied by at most two electrons, with opposite (paired) spins.

Bonding orbitals are lower in energy than the AOs from which they are formed. Antibonding orbitals are always higher in energy than the AOs from which they are formed. Representations of the H₂⁺ ion for the bonding and antibonding MOs are shown in Figure 3.4 (Figures 3.5 and 3.6, pp. 96-97 in the textbook show contour and shaded plots of the MOs). The bonding orbital has enhanced electron density between the two nuclei, arising from addition of 1s orbitals on the two atoms, while the antibonding orbital, has a node - zero electron density - between the two nuclei, arising from subtraction of 1s orbitals on the two atoms.

3.1.2 What makes an orbital bonding or antibonding?

Bonding MOs have a minimum energy and electron density between the nuclei. Antibonding MOs do not have a minimum when the AOs overlap and have nodes between the nuclei.

3.1.3 Symmetry labels

A simple MO description can be summarised as follows:

1. The MOs are classified by symmetry label: σ orbitals are cylindrically symmetric about the bond axis, with no nodal planes. For one or two nodal planes the labels are either π orbitals or δ orbitals, respectively.
2. 1σ MOs are formed from the sum of 1s orbitals on the two atoms (bonding) and the difference of 1s orbitals (antibonding).

The notation for MO theory uses the labels σ, π and δ to describe the orbital type, and subscripts g and u to identify their symmetry with respect to inversion through the bond midpoint. g denotes an even orbital, one which is unchanged under inversion (gerade in German), while u denotes an odd orbital, one whose sign changes under the inversion operation (ungerade in German). Orbitals of each type and symmetry are numbered consecutively with increasing energy. Figure 3.9 illustrates the symmetry for 1σ MOs.

3.1.4 Molecular orbital diagrams

Figure 3.10 shows the MO diagrams for the overlap of two 1s AOs as a function of distance between nuclei. With decreasing distance the energy difference between MOs increases.

3.1.5 Significance of the sign of the wavefunction

In its simplest form, MO theory assumes that we can write the wavefunction for a system of N electrons as the product of n one-electron wavefunctions - the molecular orbitals:

This is known as the orbital approximation

, and these MOs are interpreted in the same way as atomic orbitals - their square is the probability distribution for the electron in that orbital.

The following figures from Shriver and Atkins, Inorganic Chemistry, 1999, show the effect on the sign during constructive and destructive interference.

3.1.6 Overlap and the overlap integral

Overlap integrals provide good estimates of bond strengths. The overlap integral for two 1s AOs is shown in Figure 3.11 as a function of separation distance. Nuclear repulsions need to be taken into account as the nuclei move closer together.

Overlap integrals are dependent on the size of the AOs, as shown in Figure 3.12.

3.1.7 Summary

This section of Chapter 3 looked at MOs for the one electron H₂⁺. Key points are:

• MOs are linear combinations of AOs.
• Two 1s AOs gives two MOs: one bonding and one antibonding.
• MO energies depend on internuclear separation: bonding MO is higher and antibonding MO is lower than AO.
• Bonding MO due to constructive interference of AOs. Antibonding MO due to destructive interference of AOs.
• Overlap integrals provide information on bond strengths.

3.2 H2, He2 and their ions

The same energy level diagram to describe H₂⁺, H₂, He₂ and H₂^– or He₂⁺. From these diagrams (shown in Figure 3.13) we can predict:

1. H₂ will have a stronger bond than H2+, because of the extra bonding electron.
2. H₂^– will be stable, but does have one antibonding electron, giving instability.
3. He₂ will not form, due to the two antibonding electrons.

The following table (Table 3.1, p. 105) confirms these conclusions:

Molecule	Configuration	Dissociation energy / kJ mol-1	Bond length / pm
H2+	1σg1	256	106
H2	1σg2	432	74
He2+	1σg21σu1	241	108
He2	1σg21σu2	not observed

3.3 Homonuclear diatomics of the second period

3.3.1 Rules for forming MOs

MO theory is applied to a given arrangement of atoms (i.e., we must know the geometry - bond lengths etc - of the molecule) and it determines the various molecular orbitals that can form by combining all available orbitals on all of the atoms. The basic ideas of MO theory can be summarised:

• The number of MOs produced is always equal to the total number of AOs from the atoms involved.
• A bond arises from overlap of AOs on adjacent atoms. Thus the symmetry of orbitals is important.
• AOs combine to form MOs most effectively when the atomic orbitals are of similar energy. This means that 1s orbitals will not combine with 2s or 2p orbitals to form MOs; but 1s and 1s will, and 2s and 2p orbitals will combine well with 2s and 2p orbitals etc.
• An AO that is closest in energy to a MO contributes most to the characteristics of that MO.
• AOs that are of a similar size contribute most to the MOs and hence the properties of the molecules.

Details of these basic ideas for determining the key overlaps of AOs to form MOs in each molecule are presented in this section of the textbook.

3.3.2 Types of MOs from 2s and 2p AOs

For the homonuclear diatomics Li₂ through F₂ we assume that an adequate description of bonding can be obtained by using just the valence 2s and 2p orbitals. In this fashion we use 4 AOs per atom, or 8 AOs to construct 8 MOs for each diatomic molecule. As already seen in the first-year course, σ orbitals can arise by overlap of 2s–2s, 2s–2p and 2p–2p atomic orbitals. The 2s–2p overlap is also not discussed in this section because the energy difference is considered too large.

MOs from 2s-2s overlap

The MO iso-surfaces for the 2s-2s overlap to form 2σ_g (bonding) and 2σ_u (antibonding) are shown in Figure 3.20. The shapes and the labels are equivalent to MOs calculated for the 1s–1s overlap.

MOs from 2p-2p overlap s overlap

The MO iso-surfaces from overlap of two 2p_z orbitals to form 3σ_g (bonding) and 3σ_u (antibonding) are shown in Figure 3.22.

MOs from 2p-2p overlap π overlap

The MO iso-surfaces from overlap of two 2p_x orbitals to form 1&pi'_u (bonding) and 1π_g (antibonding) are shown in Figures 3.25(a) and (b) respectively.Figures 3.25(c) and (d) show the iso-surfaces from overlap of two 2p_y orbitals to also form 1π_u and 1π_g MOs

3.3.3 Idealized MO diagram

Figure 3.28 shows the idealized MO diagram for the homonuclear diatomics Li₂ through F₂. 2s and 2p orbitals in these atoms are separated by a considerable energy difference, which increases with increasing atomic number. Mixing wavefunctions (i.e., orbitals in this case) is strongest for those with similar energies, and hence in this idealized MO we can often ignore the contribution of s,p overlap to the σorbitals.

Bonding in O2, F2 and Ne2

In exactly the same way as for atoms, we build up electron configurations for molecules, starting from the lowest energy levels, and placing two spin-paired electrons in each MO. As for p orbitals in atoms, where each is singly occupied before pairing occurs (Hund's rule), the π MOs are singly occupied before pairing occurs (eg see Figure 3.29).

Valence electron configurations for molecules are written in the same form as for atoms:

Li2	1σg22σg2	Be2	1σg21σu22σg22σu2
B2	1σg21σu22σg22σu21πu2	C2	1σg21σu22σg22σu21πu4
N2	1σg21σu22σg22σu21πu43σg2	O2	1σg21σu22σg22σu23σg21πu41πg2
F2	1σg21σu22σg22σu23σg21πu41πg4	Ne2	1σg21σu22σg22σu23σg21πu41πg43σu2

Bond order (BO) provides a connection between formal Lewis structures, where the number of shared electron pairs is used to indicate single, double or triple bonds, and hence imply something about their strength and length. BO is defined as

BO = ½ (no. of bonding electrons – no. of antibonding electrons)

ie. half the difference between the number of bonding and antibonding electrons. This amounts to counting an electron pair in a bonding orbital as a "bond", and an electron pair in an antibonding orbital as an "antibond". BO is the excess number of bonds over antibonds. The following table compares BO with bond dissociation energies.

Diatomic molecule	Bonding electrons	Antibonding electrons	Bond order	Bond dissociation energy (kJ mol-1)
O2	8	4	2	494
F2	8	6	1	154
Ne2	8	8	0	unstable

Paramagnetism

Paramagnetic materials have permanent dipole moments even in the absence of a magnetic field. For homonuclear diatomics, paramagnetic molecules contain one or more unpaired electrons. The MO diagram in Figure 3.29 shows that O₂ has unpaired electrons and so is paramagnetic. This property is demonstrated in Figure 3.30.

For diamagnetic materials show magnetism only in the presence of an externally applied magnetic field. For homonuclear diamagnetic diatomics (e.g., F₂) all electrons are spin paired.

Ions of O2

The following table (Table 3.2, p. 118) shows that MO theory can accurately predict trends in bond lengths and dissociation energies, as well as paramagnetism or diamagnetism of all the common O₂ ions.

Species	Bond length (pm)	Dissociation energy (kJ mol-1)	Bond order	Paramagnetic?
O2+	112	643	2.5	yes
O2	121	494	2	yes
O2-	135	395	1.5	yes
O22-	149	204	1	no

3.3.4 Allowing for s-p mixing

Interactions or mixing between 2s and 2p_z AOs cause a reordering of the MOs as shown in Figure 3.31. This mixing results in a decrease in energy for MO₁ and MO₂, while MO₃ and MO₆ increase. Hence 2σ_u and 1π_u switch positions (see Figure 3.31(b)). This switch is the order that is observed for Li₂ … N₂, while O₂ and F₂ show the order in (a).

3.3.5 Properties of the homonuclear diatomics

The following table (Table 3.3, p. 121) compares electron configurations from MO theory with bond length, dissociation energy and paramagnetism:

Species	Configuration	Bond length (pm)>	Dissociation energy (kJ mol-1)	Paramagnetic?
Li2	1σg21&sigmau22&sigmag2	267	105	no
Be2	1σg21σu22σg22σu2	245	~ 9	no
B2	1σg21σu22σg22σu21πu2	159	289	yes
C2	1σg21σu22σg22σu21πu4	124	599	no
N2	1σg21σu22σg22σu21πu43σg2	110	942	no
O2	1σg21σu22σg22σu23σg21πu41πg2	121	494	yes
F2	1σg21σu22σg22σu23σg21πu41πg4	141	154	no
Ne2	1σg21σu22σg22σu23σg21πu41πg43σu2	310	<1	no

Several correlations with bond order are apparent, in particular

1. bond enthalpy increases as bond order increases;
2. bond length decreases as bond order increases;
3. hence, we would expect bond enthalpy to increase as bond length decreases.

Figure 3.34 shows the decrease in orbital energy of the MOs across the period from Li₂ to F₂. This is caused by the increasing nuclear charge. Also note the switch in the 3σ_g and 1π_u between N₂ and O₂ caused by s–p mixing.

3.3.6 Limitations of qualitative MO diagrams

It is difficult to ‘guess’ the details of the MO diagram or the key AO interactions or to predict properties such as bond length and bond dissociation energy, without detailed computational calculations.

3.4 Photoelectron spectra

UV photoelectron spectroscopy is one elegant experimental method, which yields almost direct information on the energies of molecular orbitals. It yields a spectrum of ionisation energies of the higher energy electrons for molecules, and these can usually correlated directly with MO energy levels, and hence verify the predictions made by the MO model.

The orbital energy can be estimated from the energy of the incident UV photon and the measured energy of the ejected electron:

– orbital energy ≈ ionization energy = energy of photon – energy of ejected electron

Figure 3.36 shows the photoelectron spectrum of H₂. This has a single band that corresponds to the ionization of a 1σ_g electron. The multiple peaks are due to electrons ejecting from a range of stimulated vibrational energy levels.

The UV photoelectron spectrum of N₂, shown in Figure 3.37, has three bands corresponding to 3σ_g, 1π_u and 2σ_u MOs. Note that extensive vibrational structure (e.g., for 1π_u) indicates that the removal of an electron from this MO causes a significant change in the bonding. Both 3σ_g and 2σ_u are weakly bonding and antibonding.

3.5 Heteronuclear diatomics

Heteronuclear diatomic molecules can be treated in much the same way as homonuclear diatomics, but now of course the MOs will contain unequal contributions from the two atoms. Atomic orbitals are no longer at the same level and they may not have the same symmetry. If symmetry does not match, then the non-bonding MO orbitals are localized on one atom.

As a general rule, in heteronuclear diatomics, the more electronegative element makes the larger contribution to bonding orbitals, and the less electronegative element contributes more to the antibonding orbitals. The heteronuclear bonds formed are always polar, sometimes quite strongly, with bonding electrons usually nearer the more electronegative atom and antibonding electrons nearer the less electronegative atom.

3.5.1 LiH and HF: two typical hydrides

In this section, and the following one on NO, CO and LiF, the text examines the consequences flowing from MO energy level diagrams. Appreciate that you do not need to memorise these diagrams, nor will you need to create them from scratch. But you should learn how to "read" them, at least partly understand why they look like they do (especially the relative ordering of the contributing AOs), and be able to deduce bonding characteristics and infer molecular properties from them.

HF

The MO energy level diagram for HF is reproduced in Figure 3.39.

The most important points are:

• The H 1s energy lies well above the F 2s and 2p AOs;
• The valence electron configuration can be written 3σ²1π⁴;
• The H 1s orbital contributes only to the σ MOs, as does one of the F 2p orbitals (hence the lines in Figure 3.39 connecting these AOs and the 3σ and 4σ MOs);
• The remaining F 2p orbitals (ie those perpendicular to the bond axis) are unaffected by bonding, and these form the 1π MOs;
• The 1π orbitals are nonbonding - they are not affected energetically by the interaction between the atoms, and are hence neither bonding nor antibonding;
• The 3σ orbital is weakly bonding, and largely F 2p;
• The 4σ orbital is antibonding, and largely H 1s;
• Since most of the occupied orbitals concentrate electron density largely on the F atom, the molecule is expected to be polar, with negative charge (ie excess electrons) on the F atom. These can be seen by the more dense lines around F in the 3σ contour map in Figure 3.39.

Figure 3.40 shows the analogous MO diagram and photoelectron spectrum for HCl. The spectrum has two bands corresponding to non-bonding 1p MOs (negligible vibrational structure) and the 3s bonding MO (vibrational structure).

3.5.2 NO, CO and LiF

CO

CO is considerably more complicated than HF, as now both atoms contribute 2s and 2p AOs to the molecular orbitals.

The most important points are:

• The C 2s and 2p AOs lie above the respective O 2s and 2p AOs;
• The valence electron configuration can be written 3σ²4σ²1π⁴5σ²;
• The degenerate 1π MOs are bonding and have a larger contribution from the oxygen;
• The highest occupied MO (HOMO) is the 5σ orbital, a combination of C 2s and O 2p, is weakly bonding (perhaps a nonbonding lone pair on the Figure 3.43;
• The lowest occupied MO (LUMO) are the pair of 2π orbitals, and these orbitals are expected to be mainly C 2p in character;
• Much of the reactivity of CO can be explained and rationalised in terms of these HOMO and LUMO orbitals (see next section);
• The experimental dipole moment for CO is small, and actually ^δ-C=O^&delta+, rather than the more obvious ^δ+C=O^δ-. Dipole moments represent a balance between the distribution of (positive) nuclear charge and (negative) electronic charge, and in many instances this is a very subtle property, one which is not well predicted by simple MO theory;

The isoelectronic principle states that molecules with the same number of electrons and atoms will have similar structures and chemical properties. Most generally, the valence electrons of molecules are considered isoelectronic. However, the principle can also apply only to the inner electrons of molecules, or both. CO and N₂ are isoelectronic molecules and their properties are remarkably similar.

3.5.3 The HOMO and the LUMO

The concept of frontier orbitals is often useful, especially in discussing reactivity of various species. The orbitals that are important are the HOMO - highest occupied molecular orbital - and LUMO - lowest unoccupied molecular orbital.

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