Molecules

Lecture notes for the exercises class Struttura della Materia Dec. 14, 2001.

The following exercises study general properties of diatomic molecules.

The general framework that is assumed here is the Born-Oppenheimer (adiabatic) separation of the electronic and ionic motions.

The standard approximations involved include the separation of translational, rotational and vibrational motions.

The trivial translational motion is treated exactly.

The rotational motion is interpreted in terms of the rigid linear rotor approximation.

The vibrational motion is typically studied in the harmonic approximation, unless dissociation properties are investigated. For those a slightly better model is the Morse potential.

The description of an ensemble of identical molecules in a diluted gas is studied according to standard Boltzmann statistics. See also this document for ro-vibrational states and this and this other document for chemical bonding.

Exercise 1

The vibrational transition of the Br2 molecule (mol. weight=160) occurs at ν[bar]=322 cm-1.
  1. Determine the zero-point vibrational energy.
  2. Compute the inter-atomic force constant.
  3. If the ground-state equilibrium distance is 2.3 Å, compute the rotational energy in the level L=10, and the wavelength for the dipole-allowed transitions involving this level.
  4. Determine the ratio between the number of molecules in the L=10 state and those in the ground state at T=300 K.
  5. Determine the most populated rotational multiplet.
  6. Compute the energy correction to the L=10 rotational level due to the centrifugal distortion.

RESULT:
  1. 20.0 meV
  2. 244.36 N/m
  3. E11->10=0.2173 meV, E10->9=0.198 meV, and the wavelengths are fractions of cm.
  4. n10/n0=21 exp[-(E10-E0)/(kBT)] = 20.1
  5. L(nmax)=36
  6. ΔE/E = 5.2×10-5

Exercise 2

Compute the energy necessary to dissociate the the 7Li19F molecule into the composing ions, assuming that the bond length is 1.4 Å and that the ion-ion potential is of the kind:

U(r) = - qe2 / (4 π ε0) / r + b/r9

Take into account the zero-point energy. Compute the same dissociation energy for the 6Li19F molecule.

Exercise 3

A gas of HCl molecules is confined to a plane. In this plane, each molecule can vibrate and rotate. Assuming a vibrational frequency of 3000 cm-1 and a bond length of 1.47 Å, discuss the absorption spectrum and the observed characteristic wavelengths. At T=300 K, which fraction of the molecules can emit vibrational quanta, and which fraction of the molecules can emit rotationally?
RESULT: Rotationally excited fraction = 96%. Vibrationally excited fraction = 5.64×10-7.
Note: the spectrum of a 2-dimensional quantum rotor of inertial moment I involves states labeled by the (positive, null, and negative) integer K at energies: ℏ2K2/(2 I)

Exercise 4

Redo the previous exercise at T=3000 K (assuming that the molecules do not dissociate).
Note: the approximate summation implied by the partition function must be carried out in a completely different fashion, for example by replacing it by an integration.

Exercise 5

Let the adiabatic potential energy of the H35Cl molecule be expressed by the relation:

U(R) = - a R-3 + b R-9

where a and b are positive constants, and R is the internuclear distance.
  1. Fit a and b on the observed vibrational frequency ν[bar] = 2890 cm-1 and rotational energy spacing B = ℏ²/I = 21.3 cm-1.
  2. Determine the dissociation energy of the molecule, accounting for zero-point motion.
  3. Based on the harmonic approximation, estimate the number of bound vibrational states.
  4. What is the most populated rotational level at 300 K and at 1103 K?
  5. What fraction of molecules is in the L=2 rotational state at 100 K?
  6. What fraction of molecules is in some excited vibrational state at 300 K and at 1103 K?
RESULT:
  1. RM=(3b/a)1/6 = = 1.276·10-10 m; U''(RM)=6·31/6a11/6b-5/6 = k = 512.1 Kg s-2; whence a = k RM5/18; b = k RM11/54.
  2. Ediss = 1.74 eV.
  3. about 4 or 5.
  4. 300 K: L=3; 1103 K: L=6 (not 5!)
  5. 29 %
  6. exp(-ℏω/kBT). 300 K: 5.9·10-7; 1103 K: 0.0202.
Find a more detailed solution of this problem in this pdf document. See also these extra exercises on molecules.

Try and use this nice applet to simulate the solutions of the Schrödinger equation in a model molecular potential of your choice. For example simulate a Lennard-Jones potential with: 

hbar^2/2m	= 0.5
Xmin		= 0.65
Xmax		= 5
Grid Points     = 1000
Nodes           = 0, then 1, 2 ecc.
V scale		= 200
V(x)		= -1000*x**(-6)+1000*x**(-12)

Comments and debugging are welcome!

created: 07 Jan 2002 last modified: 19 Nov 2020 by Nicola Manini