Bonding in Molecules Prof John McGrady Michaelmas Term 2009

Transcription

1 Bonding in Molecules Prof John McGrady Michaelmas Term lectures building on material presented in Introduction to Molecular Orbitals (HT Year 1). Provides a basis for analysing the shapes, properties, spectra and reactivity of a wide range of molecules and transition metal compounds.

2 Topics The essentials of molecular orbital theory Symmetry and molecular orbital diagrams for the first row hydrides AH n The use of Walsh diagrams in exploring molecular shapes Photoelectron spectroscopy and experimental energy levels Molecular orbital diagrams for hypervalent molecules Complexes of the transition metals: octahedral, tetrahedral and square planar.

3 Bibliography DeKock and Gray: Chemical Structure and Bonding (Benjamin) Jean, Volatron and Burdett; An Introduction to Molecular Orbitals (OUP) Murrell, Kettle and Tedder: The Chemical Bond (Wiley) McWeeny: Coulson's Valence (OUP) More specialized reading Albright, Burdett and Whangbo: Orbital Interactions in Chemistry (Wiley) Balhausen and Gray: Molecular Electronic Structures (Benjamin-Cummings) Streitweiser: Molecular orbital theory for Organic Chemists (Wiley) Albright and Burdett: Problems in Molecular Orbital Theory (OUP) Figgis: Introduction to Ligand Fields (Interscience) Larsen and La Mar The angular overlap model J. Chem. Ed.1974,

4 A good theory of bonding should be consistent with quantum theory, in particular the Schrödinger wave equation HΨ = EΨ provide a basis for good numerical calculations provide simple pictures for the interpretation of trends be widely applicable to a range of molecular types

5 Some basics The orbital approximation assume that the wavefunction can be expressed as an antisymmetrised product of wavefunctions that describe the individual electrons Ψ = (1,2,... N) φ1φ 2... φ N 2 dτ ρ Ψ = A molecular orbital, φ i, is the wave function of an electron in a molecule moving under the influence of the nuclear attraction and the average repulsion of all other electrons. An orbital is associated with a specific energy, ε i

6 Some basics The Linear Combination of Atomic Orbitals (LCAO) approximation assume that a molecular orbital can be expressed as a linear combination of the atomic orbitals of the component atoms. φ = c χ + c χ + c χ +... i A A B B C C Justification: in the region of space near A, the electron will behave much like an electron in atom A in isolation Simplest version only valence orbitals of atoms are included More sophisticated versions use additional functions in the expansion, which is known as a basis set

7 Some basics The MOs form an orthonormal set: φ φ dτ = i i 1 φ φ dτ = 0 i j

8 Overlap In order for bonding to take place, atomic orbitals must overlap χ χ dτ = A B S AB 0 Note: in diatomics, it is easy to identify non-zero overlaps by inspection. in more complex molecules, we will need to use group theory.

9 Overlap Constructive overlap leads to a build up of electron density in the region between the nuclei ψ ψ Destructive overlap leads to a removal of electron density from the region between the nuclei ψ destructive interference ψ node

10 Molecular orbital diagrams: H 2 +

11 An example: H 2 +

12 An example: H 2 +

13 An example: H 2 +

14 An example: H 2 +

15 An example: H 2 + Obtained by solving secular equations see Dr Vallance s notes pp KS 1 S KS 1+ S AB 2 AB AB 2 AB KS ε ( σ u ) ε (1 s) 1 S AB 2 AB interaction S AB KS ε ( σ g ) ε (1 s) S AB 2 AB Note: antibonding orbital is destabilised more than bonding orbital is stabilised

16 An example: H 2 +

17 In anticipation of group theory: In the case of H 2+, it is fairly obvious that we can have only two linear combinations of 1s orbitals symmetric (σ g ) and antisymmetric (σ u ). What do the labels mean? χ(1s A ) χ(1s B ) Γ red = Γ = σ g + σ u

18 Some quantitative results for H 2 + Simple LCAO approximation using atomic orbitals of H (Z = 1) as basis 1 Z φ(1 s) = π a 0 3/2 e Zr a Yields only 64% of experimentally measured bond energy and overestimates bond length by ~0.26 Å 0 Can be improved by: Varying Z as well as coefficients (optimum Z turns out to be 1.24) Including p z orbitals in the basis set (improves overlap in bonding orbital) Note s/p mixing is symmetry allowed because Γ(2 x 2p z ) = σ g + σ u

19 Molecular orbital diagrams for multi-electron molecules: We can build up MO diagrams for many-electron atoms using i) The Aufbau principle ii) The Pauli exclusion principle iii) Hund s rule Note: the orbital approximation lies at the heart of this process. Bond length /pm diss energy /kcalmol -1 H

20 Molecular orbital diagrams: We can build up MO diagrams for many-electron atoms using i) The Aufbau principle ii) The Pauli exclusion principle iii) Hund s rule Note: the orbital approximation lies at the heart of this process. Bond length /pm diss energy /kcalmol -1 H H

21 Molecular orbital diagrams: We can build up MO diagrams for many-electron atoms using i) The Aufbau principle ii) The Pauli exclusion principle iii) Hund s rule Note: the orbital approximation lies at the heart of this process. Bond length /pm diss energy /kcalmol -1 H H He

22 Molecular orbital diagrams: Similar processes for: B 2 C 2 N 2 O 2 F 2 O 2+ O 2 O 2 O 2 2 Note: subtle changes in s/p mixing can sometimes cause a re-ordering of levels (e.g σ g /π u in N 2 /O 2 ) See Introduction to Molecular orbitals (Year 1)

23 Heteronuclear diatomics: What happens when the two ends of the molecule are no longer identical? interaction S E 2 AB χ A χ B Corollaries: orbitals do not interact (i) if they do not overlap (ii) if they have very different energies Molecular orbital resembles the atomic orbital to which it lies closest in energy

24 Heteronuclear diatomics: What happens when the two ends of the molecule are no longer identical? χ B χ B χ A χ A

25 Heteronuclear diatomics, AH: (LiH FH) More complex atomic basis: on A 2s, 2p x,y,z on H 1s Symmetry analysis: Γ(H 1s ) = = σ Γ(A 2s ) = Γ(A 2p z ) = Γ(A 2px,y ) = 2-2 2cosφ 1 = σ 0 = π p y p x cosφ sinφ = sinφ cosφ ( px ' py ') ( px py ) p y p x

26 Heteronuclear diatomics, AH: (LiH FH) More complex atomic basis: on A 1s, 2s, 2p x,y,z on H 1s Symmetry analysis: Γ(H 1s ) = Γ(A 1s ) = Γ(A 2p z ) = Γ(A 2px,y ) = 2-2 2cosφ 1 = σ 1 = σ 0 = π (if atom lies on an invariant point) φ( nσ ) = c χ + c χ + c χ φ( π ) = χ F px, y 1 H (1 s) 2 F (2 s) 3 F (2 p z ) (2 ) 3 orbital interactions: one orbital will be bonding, one anti-bonding and one non-bonding

27 Energy

28 Energy (σ) (σ + π) (σ)

29 Energy for non-degenerate atomic orbitals (σ + π) E(H(1s)-F(2p z )) ( ) 2 (σ) int S AB E E(H(1s)-F(2s)) (σ)

30 Energy (σ) (σ + π) (σ) φ( nσ ) = c χ + c χ + c χ 1 H (1 s) 2 F (2 s) 3 F (2 p z )

31 Energy (σ) (σ + π) (σ) φ( nσ ) = c χ + c χ + c χ 1 H (1 s) 2 F (2 s) 3 F (2 p z )

32 Energy (σ) (σ + π) (σ)

33 Energy (σ) (σ + π) (σ)