Vibrations of polyatomic molecules

1. Vibrationsofpolyatomic molecules MJ, Feb 7

2. Outline MJ, Feb 7 * Normal modes* Selection rules* Group theory (Tjohooo!)* Anharmonicity

3. Describing the vibrations MJ, Feb 7 Molecule with N atoms has 3N-6 vibrational modes, 3N-5 if linear. Find expression for potential energy. Taylor expansion around equilibrium positions.

4. Total energy ; where ; where Kinetic energy Introduce mass weighted coordinates: MJ, Feb 7

5. Total energy Nasty cross-terms when What we want is to find set of coordinates where the cross-terms disappear. Is this at all possible? MJ, Feb 7 We can now write the total vibrational energy as:

6. Can be broken down as linear combinations of: These modes do not change the centre of mass,and they are independent. MJ, Feb 7 A look at CO2 Vibrations of the individual atoms

7. MJ, Feb 7 Normal coordinates So, we can write the energy as: where Q are the so called normal coordinates. They can be a bit tricky to find, but at least we know they are there. Before we see how can use this, lets have a look at the normal modes for our CO2.

8. MJ, Feb 7 Normal modes of CO2 3 x 3 - 5 = 4 vibrational modes Symmetric stretch Anti-symmetric stretch Orthogonal bending

9. Also the vibrational wavefunction separates into a product of single mode wavefunctions: MJ, Feb 7 QM Since the total energy is just a sum of terms, so is the Hamiltonian of the vibrations. We write it as:

10. MJ, Feb 7 Schrödinger equation The Scrödinger equation then becomes: … and this we recognise, right? Harmonic oscillator with unit mass and force constant k.

11. Harmonic oscillator Energy levels: Wavefunctions: Total vibrational energy: MJ, Feb 7

12. Ground state energy: Ground state wavefunction: MJ, Feb 7 Harm. Osc. … We know the ground state: All normal modes appear symmetrically, and as squares The ground state is symmetric with respect to all symmetry operations of the molecule.

13. Selection rules Dipole transition matrix element of a particular mode: 0 if , due to orthogonality if MJ, Feb 7 Molecular dipole moment depends on displacements of the atoms in the molecule: Taylor expand...

14. Selection rules Similarly, by observing that we get selection rules for Raman activity: MJ, Feb 7 ; Selection rules for IR absortion: It can be hard to see which vibrations are IR/Raman active, but, as we have seen before, Group Theory can come to rescue.

15. Group theory and vibrations z3 z1 First find a basis for the molecule. Let’s take the cartesian coordinates for each atom. y3 x3 z2 y1 x1 y2 x2 Water belongs to the C2v group which contains the operationsE, C2, sv(xz) and sv’(yz). The representation becomes E C2 sv(xz) sv’(yz) Gred 9 -1 1 3 MJ, Feb 7 The details of a normal mode depend on the strength of the chemical bonds and the mass of the atoms. However the symmetries are just a function of geometry. Example: H2O (the following stolen from Hedén)

16. Continued water example MJ, Feb 7 Character table for C2v. Now reduce Gred to a sum of irreducible representations. Use inspection or the formula.

17. MJ, Feb 7 Continued water example The representation reduces to Gred=3A1+A2+2B1+3B2 Gtrans= A1+B1+B2 Grot=A2+B1+B2 Gvib=2A1+B2 Modes left for vibrations

18. What to use this for? So for , and must span the same irreducible representation for their product to be in A1. transform as translations, so: MJ, Feb 7 We know that that the ground state is totally symmetric: (A1) First excited state of a normal mode belongs to the same irred. repr. as that mode because For a transition to be IR active, the normal mode must be parallel to the polarisation of the radiation.

19. This also leads to the exclusion rule: In a molecule with a centre of inversion, a mode cannot be both IR and Raman active. MJ, Feb 7 What more to use this for? By the same argument one can come the the conclution that For a transition to be Raman active, the normal mode must belong to the same symmetry species as the components of the polarisability These scale as the quadratic forms x2, y2, xy etc.

20. Water again... MJ, Feb 7 A1 Gvib=2A1+B2 A1 All three modes are both IR and Raman active, no centre of inversion. (a) and (b) are excited by z-polarised light, and (c) by y-polarised. B2

21. At this point overtones like can be allowed, since the matrix element containing the quadratic term Qi2not necessarily vanishes. This can not be determined from group theory, but must be calculated for every molecule. MJ, Feb 7 Anharmonicity Electric anharmonicity occurs when our expansion of the dipole moment to first order is not valid.

22. MJ, Feb 7 Anharmonicity We also see from the presence of QiQj cross-terms can cause a mixing of normal modes. In a perfectly harmonic molecule, energy put into one normal mode stays there. Anharmonicity causes the molecule to thermalise.

23. 2a 1b 1a 0a 0b MJ, Feb 7 Anharmonicity Also mechanical anharmonicity can lead to mixing of levels if one needs to add cubic and further terms in the expression for the potential.

24. Inversion doubling Consider ammonia: pyramidal molecule with two sets of vibrational levels: Coupling between the levels lead to mixing of up and down wavefunctions which lifts the degeneracy of the levels

25. Summary • Harmonic approximation of energy gives transition rules for IR and Raman activity. • Group theory can help us figure out which transition are active. • However, anharmonic terms can come in play and mess everything up.