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CHEM 1004 | Inorganic Chemistry MOLECULAR ORBITAL THEORY Background

• For covalent bonding to occur, ionic bonding must be unfavourable. The energies of the electrons to be shared must be similar on both atoms, thereby forming A–B instead of A+B- or A-B+

• Atomic orbitals of the atoms must overlap or share the same region of space (symmetry)

• In molecules, as in atoms, electrons reside in orbitals. The shapes are different and they are associated with more than one atomic nuclei

• Atomic orbitals are those that are associated with atoms while molecular orbitals (MO) are those associated with molecules

• Valence bond theory has been superseded by MO theory

MO Theory

• Assumption: The 2 or more nuclei are placed at equilibrium distance and electrons are added to molecular orbitals in much the same way as electrons are added to atomic orbitals in atoms

• The 3 principles (Pauli exclusion principle, aufbau principle and Hundʼs rule) also apply

• For a given molecule, a wave equation can be written and the solutions of which are molecular wavefunctions or MO (cf SE where solutions give the wavefunction

Potential energy curve for H2 Energy falls below that of 2 separated H atoms as the 2 atoms are brought within bonding distance The 2 electrons are free to migrate to the other atom

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of atomic orbitals)

• Certain quantum nos will be associated with the wave equation and there will be certain allowed solutions. In atoms, there are s, p, d etc. orbitals whereas in molecules, these are called σ, π, δ etc.

• Note: Wave functions for the molecular orbitals are represented by combinations of the wavefunctions for individual atomic orbitals

Overlap

• Considering 2 H atoms which are far apart, when they are brought closer together, overlap occurs. Interference occurs between the 2 wavefunctions.

• For bonding orbitals, constructive interference between the H 1s orbitals

• Enhanced electron density in internuclear distance

• Energy of molecule < Energy of 2 separate atoms as electron density shields nuclei and system is stabilised and energy is lowered

• For antibonding orbitals, destructive interference between the H 1s orbitals (cancels out amplitude, giving rise to nodal plane)

• Electrons are excluded from the internuclear region and occupy energetically less favourable regions

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• Nuclei is poorly shielded due to nodal plane, system is destabilised and energy is increased

• To determine probability, consider (ψA + ψB)2

• For bonding orbitals, overlap is positive and electron density between the nuclei is increased. However, for antibonding orbitals, overlap is negative and electron density between the nuclei is decreased

• No of molecular orbitals = no of atomic orbitals from which they are formed e.g. two 1s orbitals result in a bonding and antibonding orbital

• A nodal plane occurs in antibonding orbitals and is perpendicular to the internuclear axis

• MO diagram:

• Note: In general, ΔE2 > ΔE1 by a small amount

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• For H2, the orbitals involved are sigma (σ) orbitals which are cylindrically symmetrical wrt internuclear axis. Antibonding orbitals are denoted by σ*

• 2 electrons go into σ and none into σ*. Hence, overlap is very good and ΔE1 is large, causing H2 to be stable and exists rather than H atoms as the normal form of this element

• However, for He2, both σ and σ* are filled. Since ΔE2 > ΔE1, He2 is not stable and helium exists as He atoms rather than He2 molecules

Bond Order = ½ (no of electrons in σ – no of electrons in σ*) (i.e. When BO = 1, 2, 3, it represents single, double and triple bonds respectively) Example Li2 and Be2

• Be2 is not stable for the same reasons as He2

• However, Li2 might be stable (present in lithium vapour), but as a solid or liquid it is a metal. The overlap is not so good for Li and another arrangement of electrons (metallic bonding) is more stable.

• The σ* orbital in Li2 is low in energy and not much higher than the σ orbital. This situation usually makes single diatomic species less stable wrt other types of bonding

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Symmetry, Overlap and Diatomic Molecules

• The main difference between bonding and antibonding orbitals lies in the region between the nuclei

• This is where the overlap integral, S, has an appreciable non-zero value

• When S > 0, there is bonding. When S < 0, there is antibonding. When S = 0, there is non-bonding

• In general, bond strength is roughly proportional to the overlap and bonds form in a way which maximises overlap

• No problems with s orbitals with regard to orientation in achieving maximum overlap as they are spherical

• For p and d orbitals, orientation and symmetry is all-important

• σ orbitals are cylindrically symmetrical about the internuclear axis and are analogous to the s orbitals and look like them end on

• π bonds have a similar relationship to p orbitals and have a nodal plane along the internuclear axis

2 nodal planes retained (from atomic orbitals)

Extra nodal plane

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Note:

• Any nodal plane present in atomic orbitals will be present in molecular orbitals

• All antibonding orbitals have an additional nodal plane between the nuclei and perpendicular to the internuclear axis

• Delta (δ) bonds can also be formed – they have four main lobes, 2 orthogonal nodal planes along the internuclear axis and resemble d orbitals

Diatomic Molecules

• Need to position the atomic orbitals according to energy, where 2p > 2s but the same for each atom since they are identical atoms (for O2, F2)

• Ignore 1s orbitals since they are very low in energy and that they are core electrons so antibonding orbitals cancel out bonding orbitals

• Note: The π orbitals derived from px and py have the same energy (degenerate) and π* also forms a degenerate pair at higher energy. In the case of the σ bond, formed from the overlap of pz, the overlap is better since orbitals point directly towards each other

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• Hence, the σ bond is lower in energy than π bond and σ* is higher in energy than π*

Answering Technique

1. Consider the electronic configuration of the atom

2. Draw out the 2s and 2p orbitals, label the atoms on either ends, and the respective antibonding and bonding orbitals

3. Start filling electrons with the lowest energy first, noting Hundʼs rule, Pauli exclusion principle and Aufbau principle

4. Write electronic configuration as (bond (*) orbital)# electrons (e.g. (σ*2s)2

5. Calculate bond order

• Inferences: paramagnetic (unpaired electron hence will be attracted in magnetic field) or diamagnetic

• In the case of B2, C2 and N2, there is a slight difference in the ordering of orbital energies where σp and π orbitals are swapped (s-p mixing), which results in B2 being paramagnetic and C2 being diamagnetic

• Reason: Overlap or mixing will take place between all orbitals which have the right symmetry but with the condition that overlap or mixing will be reduced if energy difference is large (allows us to ignore 1s orbitals)

• Example: s orbital on 1 atom overlapping with pz on another atom

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• The energy difference between 2s and 2p is not very large although it varies across the row. ENC from B to F increases the energy separation between the 2s and 2p increases considerably

• The relative energies of the σ2pz orbital compared to the π2p orbitals depend on the Z value of the atoms. If Z is > 8, the σ2pz orbital is lower in energy.

• Hybridisation can be used to tackle this problem. The s and pz orbitals hybridise

to give 2 sp hybrid orbitals on each atom

• These overlap so as to lower the energy of the 2 σ orbitals and to raise the energy of the 2 highest

• Note: The π orbitals are not changed in energy as they are not involved in hybridisation

Heteronuclear Diatomics

• When 2 atomic orbitals with different energies overlap, the lower MO is primarily composed of the lower atomic orbital and vice versa

Example HF

• Since F has a higher ENC and is more electronegative than H, it has lower energy than H

• The F 2s orbital is low in energy and largely non-bonding

• The px and py are completely non-bonding by symmetry

Pz

Pz

Expected (F2, O2) sp mixing (C2, N2, B2)

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Example CO

• CO is isoelectronic to N2 and MO diagram for both should be similar

• Difference: orbital energies for the 2 atoms are different. Since O has higher ENC than C, the valence orbitals on O will be at lower energy and the energy difference between 2s and 2p will be greater

• The bonding orbital lies closer in energy to the lowest energy atomic orbital and

• The antibonding orbital is closer to the highest energy atomic orbital

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• The π bonding orbitals are closer in energy to O 2p than to C 2p and are therefore polarised towards O (i.e. the π electrons spend more time closer to the O atom)

• π* orbitals are closer in energy to C 2p than to O 2p and are correspondingly polarised towards C

• Important when considering electron acceptor properties of CO in TM carbonyl complexes (e.g. W(CO)6) where CO bonds to TM through the C atom instead of O atom

Important:

• The lower energy orbitals, 1σ and 2σ are centred more on O than on C and the higher energy orbitals 3σ and 4σ are centred more on C than O

• 1σ and 4σ are strongly bonding and antibonding respectively

• 2σ and 3σ are weakly bonding and antibonding respectively OR non-bonding or lone pair orbitals with 2σ centred largely on O and 3σ on C

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• Since 3σ > 2σ in energy, this accounts for CO acting as a 2 electron donor or Lewis base through C rather than O, forming M—CO rather than M—OC

• Hence, BO of CO = 3 (since 1σ and the 2 π orbitals are strongly bonding, and 2σ and 3σ which are weakly bonding and antibonding respectively cancel out)

• Considering NO which has 1 more electron than CO and it goes into a π* orbital, BO = 2.5

• The electron is readily lost to give NO+ which is isoelectronic with CO (BO = 3) Triatomic Molecules

• Triatomic molecules are either linear, bent or cyclic Example H3

+ and H3-

• For H3

+, H3- and diborane (B2H6), they involve 3-centre, 2-electron and 3-centre,

4-electron bonding

• The principal shapes for H3+ and H3

- are triangular (D3h) and linear (D∞h) respectively

• The symmetry-adapted linear combinations (SALC) of triangular H3+ are easily

derived from the 3 H 1s AOs which combine to give 3 MOs

• H3+ is described as a 3-centre (3 H atoms) and 2-electron (2 available electrons)

[3C-2e] bond

• H3+ is thermodynamically stable wrt H+ (g) and H2 (g) due to the delocalisation of

electron pairs which leads to stability of molecules

• H3

- is made up of 2 outer H atoms combining with a central H atom, giving a 3-centre 4-electron bond

• The ʻsurplusʼ 2 electrons are accommodated in a H—H non-bonding MO which is localised on the terminal H atoms

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Walsh Diagram

• Refers to a correlation diagram portraying the variation in orbital energies of a molecule as its shape is changed

• By selecting the geometry that results in the lowest total energy (approximately the sum of the orbital energies), it is possible to predict likely shape of the molecule from the occupation of its orbitals and also from their relative energies

• Walshʼs rule: A molecule adopts a structure that best stabilises the HOMO. If the HOMO is unperturbed by the structural change under consideration, then the unoccupied MO lying closest to it governs the geometric preference

• Note: Molecules tend to adopt the geometry that maximises the HOMO-LUMO gap

H3

- H3+

Larger gap

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Carbon Dioxide

• The σ bonding and anti-bonding orbitals will arise from overlap of the 2s and 2pz orbitals, but assumption is that O 2s orbitals are too low in energy to take much part in σ bonding

• Hence, the σ interactions are therefore derived from the overlap of C 2s and 2pz and O 2pz

• Note: all orbitals are delocalised over all 3 atoms. The π system in CO2 is made up from the overlap of C and O px and py orbitals which form degenerate pairs

• The bonding and antibonding orbitals are in phase and out of phase respectively while the non-bonding orbital has a nodal plane on the central atom (C) and is a lone pair type of orbital (which is delocalised over both O)

Note: s-p mixing not considered here