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Chemical bonding in molecules
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
I - What is a molecule? Modeling ?
Electronic level
Vibrational levels
Rotationallevels
V=0
V=1
V=2
K
012
012
012
M
M
• Electronic energy
• Vibrational energy
• Rotational energy
e-
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
I - What is a molecules? Modeling ?
• Total Hamiltonian for a diatomic moleculeA
B
e1
e2
e3
O
where N-electron atom wave function withobeys the Pauli exclusion principle
Aim: Solve the time independent Schrödinger equation
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
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
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Step 3: Born-Oppenheimer approximation- adiabatic approximation -
• We find
• Introducing spherical coordinate for TN
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 potentialmade by the electrons
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-…
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – symmetries
Lz commutes with He
Spectroscopic notation
Electron configuration Electronic state
Value 0 1 2 3 Value 0 1 2 3
Letter Letter
1 ) Cylindrical symmetry
x
y
z
e-
+
-
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – symmetries
x
y
z
e- 2) Symmetry plane
Reflection Ry, , commutes with He
Electronic states
-Two symmetries when
-Doubly degenerated when
Reflection Ry does not commute with Lz
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – symmetries
x
y
z
e-For homonuclear molecules (N2, O2,…)
Inversion Ir, , commutes with He and Lz
Electronic states
-Symmetric gerade (g)
-Anti-symmetric ungerade (u)g,ug,u…
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – symmetries
x
y
z
e-For homonuclear molecules (N2, O2,…)
Inversion IR, , commutes with He and Lz
Electronic states
- unaffected by IR
- affected by IR
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
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – term manifold
Hetero-nuclear molecules
• From separated atomsA (L1, S1) B (L2, S2)
Molecular State Parity =
(Parity atom A) * (Parity Atom B)
Example 1: Molecular states made from two atoms with L1=L2=1
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – term manifold
Hetero-nuclear molecules
• From separated atomsA (L1, S1) B (L2, S2)
Molecular State Multilicity=
(2S+1)Molecular State Parity
= (Parity atom A) * (Parity Atom B)
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – term manifold
Hetero-nuclear molecules
Example 2: NH molecule
N:1s22s22p3 (2P,2D,4S) – odd (u)
H:1s – even (g)
O:1s22s23p4 (1S,1D,3P) – even (g)
• …to unified atom
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 O2 molecule
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – molecular orbital
United atom Molecule
state l MO Occupation
ns 0 0 ns 2
npz 1 0 np 2
npx,npy 1 1 np 4
ndz2 2 0 nd 2
ndxz,ndyz 2 1 nd 4
2 2 nd 4
From the orbital of the united atoms toto one electron molecuar orbital (state)
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
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 =|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 , and multiplicity (2S+1), they can not cross for any inter-nuclear distance.
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – molecular orbital
Hetero-nuclear molecules
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Electronic wavefunction – molecular orbital
Homo-nuclear molecules
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Chemical bounding – molecular orbital
Homo-nuclear molecules
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)
Lunds universitet / Fysiska institutionen / Avdelningen för synkrotronljusfysik FYST20 VT 2011
Chemical bounding – molecular orbital
B2- diboron O2- dioxygen
B: 1s22s22p O: 1s22s22p4