chemical bonding in molecules. lunds universitet / fysiska institutionen / avdelningen för...

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