Chapter 6: Rotational and Rovibrational Spectra
Different Approximations
Spectrum for Harmonic Oscillator + Rigid Rotator
Polyatomic Molecules
Harmonic Oscillator + Rigid Rotor Model to Obtain Quantities from Statistical Mechanics
More Conventional Discussion of Rotations for Diatomics
Chapter 6: Rotational and Rovibrational Spectra
A)General discussion of two-body problem with central potential
Examples:
a) Diatomic
b)Hydrogen atom with charge
Let us consider 2 particles with mass ,
Coordinates: ,
,
Define center of mass coordinate
,
Also define the relative coordinate
Then we will show that
Hence the Hamiltonian can be written as
Now we can apply a separation of variables:
, translational energy The energy we are really interested in
If we put the system in a (very large) box, the center of mass problem just yields the particle in the box solutions
This describes translational kinetic energy of the center of mass motion [used in stat mech, Chem 350, Chem356]
Our interest is the relative motion:
Because only depends on , it is very convenient to use spherical coordinates
To develop this further we need to obtain
In spherical coordinates, this is a very tedious exercise using the chain rule.
The basic step in the derivation is like this:
and
express everything in spherical coordinates
You might appreciate that deriving the operator is a lot of work (use MathCad?!)
Let me give you the result for the kinetic energy operator
Let me also give you
So is precisely the operator corresponding to the total angular momentum
This kinetic energy operator can also be written as
Let us note that only depends on the angular coordinates ad so does
In a later lecture I will derive the eigenfunctions and eigenvalues of the and operators. We will use the commutation relations to do this (compare harmonic oscillator)
For now I will just list the solutions
These functions are the angular parts of the familiar hydrogen orbitals
, s-orbitals
, p-orbitals (3)
, d-orbitals (5)
, f-orbitals (7)
Always -type orbitals
These functions are normalized as
for ,
otherwise
The extra factor is related to the surface element
More general for spherical coordinates
Volume element
To solve in spherical coordinates we try the solution
This yields a radial differential equation:
Together with normalization condition
This radial equation, depending on , is a completely general result, valid for any2-body problem, with a central potential
When discussing chapter 7 (Hydrogen atom), I will say more about the spherical solutions, and also discuss the radial problem for
At this point, I want to return to diatomics. The solutions to the S.E. for a vibrating and rotating molecule are
The angular functions are the exact solutions related to rotations. The radial equation is “the ‘harmonic oscillator’ in disguise.”
Let us bring the equation to a more familiar form:
Define
Then
Hence
Or using :
This starts to look like a harmonic oscillator equation
normalization:
If we have harmonic oscillator except for the term
Different Approximations
a)Harmonic oscillator + rigid rotor:
Set
Replace : common variable in rotational spectroscopy
Simplest solution; often used
b)Solve for anharmonic wavefunction
Then evaluate
Rotational constant depends on vibrational level (smaller , larger , as increases)
We only need to calculate few vibrational wavefunctions
c)Solve equation exactly for each; degeneracy is always ,
Spectrum for Harmonic Oscillator + Rigid Rotator
,
Define
Selection rules (diatomic, H.O./R.R.)
a)Pure rotational transitions
rotational constant (Hertz)
(cm-1) (use in cm s-1)
Rotational transitions (selection rule; discussed later)
Set of Equidistant lines
At room temperature both ground and excited rotational levels are occupied, and we get absorption for all . This leads to a set of equidistant lines in the spectrum.
Let us now consider transitions to different vibrational state
,
If we go beyond H.O., then
; and “centrifugal distortions”
Working it out:
Also the lines in pure rotational spectrum are not exactly equally spread.
Why: solve equation exactly!
Polyatomic Molecules(briefly)
Assume we know equilibrium geometry, a quadratic force constant matrix, then we can define a center of mass motion.
And a wavefunction (particle in the box) associated with
Another 3 coordinates are associated with overall rotation and we can associate classical rigid body moment of inertia:
This symmetric matrix can be diagonalized yielding eigenvalues and a corresponding set of axis. These correspond to displacement of all the nuclei using a rigid rotation.
The remaining or coordinates define the normal modes
The rotational problem for any can be solved using matrix diagonalization: numerically straight forward
This separation of variables is valid in the H.O./R.R. approximation. These approximations are quite good for spectroscopy; Very good for stat mech.
Harmonic Oscillator + Rigid Rotor Model to Obtain Quantities from Statistical Mechanics
I want to recall the basic formulas from Statistical Mechanics here, and show the immense usefulness of H.O./R.R. for thermochemistry.
a)Exact formulation of Stat Mech
Define system partition function (e.g. gas of molecules):
energy levels
Connection to thermodynamics:
From the absolute value of the Helmholtz free energy we can get any thermodynamics function.
How?
, ,
etc…
Now proceed to independent molecules in the gas phase:
translation, vibration, rotation, nuclear, electronic
: from particle in the box quantum solutions
Vibrational:
From Harmonic oscillator Q.M. sum each level
Rotational + Nuclear spin (complication leads to symmetry factor ):
: Moments of inertia from rigid Rotor quantum mechanics
separated atom limit
binding energies, bottom of well
One can calculate for each molecular species in the gas phase
Using a slight extension one can also calculate and from this one can obtain reaction rates
Simple Quantum models
-Particle in the box
-Harmonic oscillator
-Rigid rotor
-Electronic energies ( no good simple models)
Accurate thermochemistry for gas phase reactions
More Conventional Discussion of Rotations for Diatomics
Two bodies at fixed distance rotating around center of mass:
Moment of Inertia:
Angular momentum: (see Bohr atom)
General for rigid body rotation of linear molecules:
(more general for polyatomic)
Relative motion in quantum mechanics
,
Solutions:
Harmonic Oscillator + Rotational rigid body motion.
Selection rules for H.O/R.R.
Molecules needs permanent dipole to observe rotational transitions
Note: It is hard to see where the approximation comes in or how to improve on it.
Chapter 6: Rotational and Rovibrational Spectra / 1