1. Chemical bonding in molecules

2. I - What is a molecule? Modeling ?
Electronic energy
Vibrational energy
Rotational energy M M e- 1 2
V=2 1 2
V=1 1 2
V=0
Electronic level
Vibrational
levels
Rotational
levels K

3. I - What is a molecules? Modeling ?B e1 e2 e3 O
Total Hamiltonian for a diatomic molecule
Aim: Solve the time independent Schrödinger equation
where N-electron atom wave function with obeys the Pauli exclusion principle

4. 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. 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. Step 3: Born-Oppenheimer approximation - adiabatic approximation -
Introducing spherical coordinate for TN
We find

7. 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. Step 3: Born-Oppenheimer approximation - adiabatic approximation -
General form of the electronic energy
Limit of validity:
Coupling between states
Collision experiments
Rydberg states …
is the electronic dissociation energy

9. Electronic wavefunction – symmetriesx y z e- + L - j
1 ) Cylindrical symmetry
Lz commutes with He
Spectroscopic notation
Electron configuration
Electronic state
Value l
Value L
Letter S P D F
Letter s p d f

10. Electronic wavefunction – symmetriesz
2) Symmetry plane e-
Reflection Ry, , commutes with He y x
Reflection Ry does not commute with Lz
Electronic states
Two symmetries when
Doubly degenerated when

11. Electronic wavefunction – symmetriesz
For homonuclear molecules (N2, O2,…) e-
Inversion Ir, , commutes with He and Lz y x
Electronic states
Symmetric gerade (g)
Anti-symmetric ungerade (u)
Sg,u Pg,u …

12. Electronic wavefunction – symmetriesz
For homonuclear molecules (N2, O2,…) e-
Inversion IR, , commutes with He and Lz y x
Electronic states
unaffected by IR
affected by IR

13. Total wavefunction – Hund’s coupling cases
Hund’s case a: L and S precess about R with well-defined componets L and S, along R. N couples with WR to form J, where W=L+S ^
Electronic interaction is much larger than spin orbit coupling interaction which in turn is much larger than the rotational energy

14. Electronic wavefunction – term manifold
Hetero-nuclear molecules
A (L1, S1)
B (L2, S2)
From separated atoms
Example 1: Molecular states made from two atoms with L1=L2=1
Molecular State Parity =
(Parity atom A) * (Parity Atom B)

15. Electronic wavefunction – term manifold
Hetero-nuclear molecules
A (L1, S1)
B (L2, S2)
From separated atoms
Molecular State Multilicity =
(2S+1)
Molecular State Parity =
(Parity atom A) * (Parity Atom B)

16. Electronic wavefunction – term manifold
Hetero-nuclear molecules
Example 2: NH molecule
N:1s22s22p3 (2P,2D,4S) – odd (u)
H:1s – even (g)
…to unified atom
O:1s22s23p4 (1S,1D,3P) – even (g)

17. Electronic wavefunction – term manifold
Homo-nuclear molecules
Example 3: Determination of the dissociation limit of O2 molecule

18. Electronic wavefunction – molecular orbital
From the orbital of the united atoms to to one electron molecuar orbital (state)
United atom
Molecule
state l MO
Occupation ns
nss 2
npz 1
nps
npx,npy
npp 4
ndz2
nds
ndxz,ndyz
ndp
ndd

19. 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. Electronic wavefunction – molecular orbital
Correlation diagrams from united atom to separated atoms
Conservation laws:
The quantum number l=|ml| 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 L, and multiplicity (2S+1), they can not cross for any inter-nuclear distance.

21. Electronic wavefunction – molecular orbital
Hetero-nuclear molecules

22. Electronic wavefunction – molecular orbital
Homo-nuclear molecules

23. Chemical bounding – molecular orbital
Homo-nuclear molecules

24. Chemical bounding – molecular orbital
How to fill molecular orbitals (MO’s):
MO’s with the lowest energy are filled first (Aufbau principle)
There is a maximum of two electrons per MO with opposite spins (Pauli exclusion principle )
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. Chemical bounding – molecular orbital
B2- diboron
O2- dioxygen
B: 1s22s22p
O: 1s22s22p4