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Chemical bonding in molecules. Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 I - What is a molecule?

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Presentation on theme: "Chemical bonding in molecules. Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 I - What is a molecule?"— Presentation transcript:

1 Chemical bonding in molecules

2 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 I - What is a molecule? Modeling ? Electronic level Vibrational levels Rotational levels V=0 V=1 V=2 K M M Electronic energy Vibrational energy Rotational energy e-e-

3 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 I - What is a molecules? Modeling ? Total Hamiltonian for a diatomic molecule A B e1e1 e2e2 e3e3 O where N-electron atom wave function with obeys the Pauli exclusion principle Aim: Solve the time independent Schrödinger equation

4 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Step 1: Electronic part – no spin Electronic Hamiltonian with We solve the time independent Schrödinger equation at fixed R with the electronic wave function which forms a basis set The exact molecular wave function can be expanded such as

5 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Step 2: The total Hamiltonian – no spin We bravely solve by projecting this equation on all electronic wavefunctions We obtain coupled equations for electron and nuclear wavefunctions

6 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Step 3: Born-Oppenheimer approximation - adiabatic approximation - We find Introducing spherical coordinate for T N

7 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Step 3: Born-Oppenheimer approximation - adiabatic approximation - In the case that the motion of the nucleus is slow with respect to the motion of the electrons Assuming We just “need” to solve the nuclear wave function in a potential made by the electrons

8 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Step 3: Born-Oppenheimer approximation - adiabatic approximation - General form of the electronic energy is the electronic dissociation energy Limit of validity: -Coupling between states -Collision experiments -Rydberg states -…

9 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – symmetries L z commutes with H e Spectroscopic notation Electron configurationElectronic state Value Value  Letter  Letter  1 ) Cylindrical symmetry x y z e-e- +  - 

10 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – symmetries x y z e-e- 2) Symmetry plane Reflection R y,, commutes with H e Electronic states -Two symmetries when -Doubly degenerated when Reflection R y does not commute with L z

11 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – symmetries x y z e-e- For homonuclear molecules (N 2, O 2,…) Inversion I r,, commutes with H e and L z Electronic states -Symmetric gerade (g) -Anti-symmetric ungerade (u)  g,u  g,u 

12 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – symmetries x y z e-e- For homonuclear molecules (N 2, O 2,…) Inversion I R,, commutes with H e and L z Electronic states - unaffected by I R - affected by I R

13 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Total wavefunction – Hund’s coupling cases Hund’s case a: L and S precess about R with well-defined componets  and , along R. N couples with  R to form J, where  ^ Electronic interaction is much larger than spin orbit coupling interaction which in turn is much larger than the rotational energy

14 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – term manifold Hetero-nuclear molecules From separated atoms A (L 1, S 1 ) B (L 2, S 2 ) Molecular State Parity = (Parity atom A) * (Parity Atom B) Example 1: Molecular states made from two atoms with L 1 =L 2 =1

15 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – term manifold Hetero-nuclear molecules From separated atoms A (L 1, S 1 ) B (L 2, S 2 ) Molecular State Multilicity = (2S+1) Molecular State Parity = (Parity atom A) * (Parity Atom B)

16 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – term manifold Hetero-nuclear molecules Example 2: NH molecule N:1s 2 2s 2 2p 3 ( 2 P, 2 D, 4 S) – odd (u) H:1s – even (g) O:1s 2 2s 2 3p 4 ( 1 S, 1 D, 3 P) – even (g) …to unified atom

17 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – term manifold Homo-nuclear molecules Example 3: Determination of the dissociation limit of O 2 molecule

18 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – molecular orbital United atomMolecule statel MOOccupation ns00 ns  2 np z 10 np  2 np x,np y 11 np  4 nd z 2 20 nd  2 nd xz,nd yz 21 nd  4 22 nd  4 From the orbital of the united atoms to to one electron molecuar orbital (state)

19 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – molecular orbital Many electrons molecular states Non equivalent electrons Equivalent electrons Example 4: Determination of the molecular state of the BH molecule

20 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – molecular orbital Correlation diagrams from united atom to separated atoms Conservation laws: The quantum number =|m l | is independent of R. The principal quantum number n and the angular quantum number l can change. Wave function parity does not depend on the inter-nuclear separation. If two states in the united atom have the same symmetry, quantum number , and multiplicity (2S+1), they can not cross for any inter-nuclear distance.

21 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – molecular orbital Hetero-nuclear molecules

22 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Electronic wavefunction – molecular orbital Homo-nuclear molecules

23 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Chemical bounding – molecular orbital Homo-nuclear molecules

24 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Chemical bounding – molecular orbital How to fill molecular orbitals (MO’s): i. MO’s with the lowest energy are filled first (Aufbau principle) ii. There is a maximum of two electrons per MO with opposite spins (Pauli exclusion principle ) iii. When there are several MO's with equal energy, the electrons fill into the MO's one at a time before filling two electrons into any (Hund's rule) The chemical bound is stable if the bond order is positive The filled MO highest in energy is called the Highest Occupied Molecular Orbital (HOMO) The empty MO just above it, is the Lowest Unoccupied Molecular Orbital (LUMO)

25 Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011 Chemical bounding – molecular orbital B 2 - diboron O 2 - dioxygen B: 1s 2 2s 2 2pO: 1s 2 2s 2 2p 4


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