Normal Coordinate Analysis

Salvatore Cardamone

• Normal modes are a mathematical formalism used to describe the coupled oscillations of a chemical system

• For a system which is translationally and rotationally invariant, there are 3N-6 such normal modes, with N being the number of atoms present in the system

• Very important concept in spectroscopy; both Raman and IR spectroscopy are heavily reliant on normal modes

Introduction

‘Model System’

• Expand as power series

• Choose E = 0 at V0 ; eliminate 1st 2 terms• Assume small amplitude of vibration; eliminate 4th and higher

terms• is the ijth element of the Hessian of force constants

Potential Energy

• We write the potential energy in matrix notation

• And define

as the ‘F-Matrix’

Potential Energy

• If we analyse our model system once again and define several new vectors

Kinetic Energy

• Internal coordinates in terms of vector displacements are

Kinetic Energy

• This may be written in matrix form

• Where

is termed the ‘S-Matrix’

Kinetic Energy

• We go on to define the ‘G-Matrix’

where M-1 gives the inverse masses of the atoms, so that

• Multiplying this out gives

Kinetic Energy

• This can be significantly simplified [thank God], by use of several simple relationships

Kinetic Energy

Dot product of perpendicular vectors

• Leading to

• We finally state the formula for kinetic energy

• As such, we have obtained statements which map both the potential and kinetic energies

Now what?

Kinetic Energy

• Normal modes are obtained from the solution to the eigenvalue problem

where λ is a function of the frequency of a normal mode

• We have 3 internal coordinates, which means we obtain 3 eigenvalues in the above relation

Normal Modes

• The symmetry coordinate matrix, Ω, is used to transform our F and G matrices, which facilitates the determination of eigenvalues in the previous relationship

• We define

Symmetry Coordinates

• Resulting in

Symmetry Coordinates

• For atom 3...

• For atoms 1 and 2...

Which we solve by factorisation to obtain the 2 values for λ

Symmetry Coordinates

• As such, we obtain 3 distinct eigenvalue solutions, which correspond to the 3 distinct normal modes of a 3 atom molecule

Normal Modes