Chemistry 2000 Lecture 2: LCAO-MO theory for homonuclear diatomic molecules

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1 Chemistry 2000 Lecture 2: LCAO-MO theory for homonuclear diatomic molecules Marc R. Roussel January 5, 2018 Marc R. Roussel Homonuclear diatomics January 5, / 17

2 MO theory for homonuclear diatomic molecules We want to develop MO theory for diatomic molecules involving atoms beyond H and He. Initially, we will focus on homonuclear diatomic molecules, i.e. those in which both atoms are of the same element. Convention: The bond axis is the z axis. Marc R. Roussel Homonuclear diatomics January 5, / 17

3 Review: LCAO-MO theory of first-row diatomics In the last lecture, we combined 1s AOs to obtain sigma bonding and antibonding molecular orbitals for the first row of the periodic table: σ 1s σ 1s Marc R. Roussel Homonuclear diatomics January 5, / 17

4 LCAO-MOs of higher orbitals We can combine higher AOs in a similar way. σ and σ orbitals obtained from the 2s AOs: σ 2s Note that these look very similar to the corresponding MOs obtained by LCAO of first-row 1s AOs. The change in sign of the AO is far from the bonding region and has relatively little effect on bonding. Marc R. Roussel Homonuclear diatomics January 5, / 17 σ 2s

5 2p orbitals can be combined in two different ways: 1 The 2p z orbitals are oriented toward each other so adding them gives sigma (cylindrically symmetric) orbitals: add σ 2p subtract σ 2p Marc R. Roussel Homonuclear diatomics January 5, / 17

6 Note: Subtracting the 2p z orbitals is equivalent to adding them out of phase: add σ 2p Marc R. Roussel Homonuclear diatomics January 5, / 17

7 3 The 2p x and 2p y orbitals are perpendicular to the bond axis. Adding them gives π orbitals: add π 2p add π 2p Marc R. Roussel Homonuclear diatomics January 5, / 17

π orbitals because we can combine p x or p y orbitals in this

8 Pi orbitals have a nodal plane through the bond axis. Note: There are two degenerate π orbitals and two degenerate π orbitals because we can combine p x or p y orbitals in this manner: Marc R. Roussel Homonuclear diatomics January 5, / 17

9 Core orbitals Bonding involves the outer (valence) electrons. We can therefore ignore the core electrons. If we do an ab initio (from first principles) calculation, which includes all the electrons, we can compute MOs that derive from the core electrons. However, these MOs are essentially identical to the corresponding AOs. Marc R. Roussel Homonuclear diatomics January 5, / 17

Core orbitals (continued) Example: Here are the core MOs of Li 2, arising from the two possible linear combinations of the 1s orbitals: σ core σ core Although formally bonding and antibonding,

10 Core orbitals (continued) Example: Here are the core MOs of Li 2, arising from the two possible linear combinations of the 1s orbitals: σ core σ core Although formally bonding and antibonding, these two MOs have essentially identical energies. The lack of overlap means that these core orbitals do not participate in bonding. Marc R. Roussel Homonuclear diatomics January 5, / 17

11 AO energies and LCAO theory Principle: AOs of similar energies and appropriate symmetries can mix to form MOs. 2s and 2p atomic orbital energies: s 2p ε i /R H Li Be B C N O F Ne For higher-z elements in period 2, there won t be much mixing between the 2s and 2p orbitals, and the MOs will be those described above. Marc R. Roussel Homonuclear diatomics January 5, / 17

12 The normal MO diagram E 4σ 2p A 2π 1π 3σ 2p B 2σ 2s A 2s B 1σ Applies to O 2, F 2 and Ne 2 Marc R. Roussel Homonuclear diatomics January 5, / 17

13 The normal MO diagram E 4σ 2p A 2π 1π 3σ 2p B 2σ 2s A 2s B 1σ Note that 1 We only consider the valence MOs and number these MOs starting from 1 (e.g. lowest valence MO is 1σ). 2 The σ and π orbitals are numbered independently. Marc R. Roussel Homonuclear diatomics January 5, / 17

14 s-p mixing The closer the 2s and 2p orbitals are in energy, the more these mix to form MOs. s-p mixing lowers the energies of the 1σ and 2σ MOs, and raises the energies of the 3σ and 4σ MOs. If this mixing is strong enough, the 3σ can end up higher in energy than the 1π orbitals. Marc R. Roussel Homonuclear diatomics January 5, / 17

15 MO diagram with strong s-p mixing E 4σ 2p A 2π 3σ 1π 2p B 2σ 2s A 2s B 1σ Applies to Li 2 N 2 Marc R. Roussel Homonuclear diatomics January 5, / 17

16 Ground-state electronic configurations of the 2nd period homonuclear diatomics Bond order Correlation with Lewis diagram Paramagnetic or diamagnetic? Spoiler: Oxygen is paramagnetic. Marc R. Roussel Homonuclear diatomics January 5, / 17

17 Lessons learned from the homonuclear diatomics Atomic orbitals of the same symmetry can mix to form MOs. For diatomics, we mean symmetry with respect to the bond axis. σ symmetry: s, p z π symmetry: p x, p y In order for mixing to be significant, the atomic orbitals must have similar energies. Rule of thumb: Mixing is (relatively) insignificant if the difference in energy between the AOs is much greater than 1 Ry (1 Rydberg = R H ). Marc R. Roussel Homonuclear diatomics January 5, / 17

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