normal coordinate analysis
TRANSCRIPT
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