VIBRATIONS OF DIATOMIC MOLECULES
CLASSICAL APPROACH
The classical one-dimensional harmonic oscillator is a particle which is bound to an equilibrium position by a force which is proportional to the displacement from that position and is expressed as
and (1)
mass of particle, m
displacement of particle, x
elastic restoring force, Fe
force constant (elastic spring constant), ke
potential energy of system, U
The solution to this classical problem is
(2)
amplitude, A
initial phase angle, a
classical oscillation frequency, (3)
classical total vibrational energy of the oscillator,
(4)
The total vibrational energy of the particle can have any value and is proportional to the square of the amplitude and the square of the oscillation frequency.
QUANTUM APPROACH
The motion of a vibrating system is governed by the Schrodinger equation [Atkins]. We will consider the solutions of the Schrodinger equation for the vibrations of the centre of mass of a diatomic molecule as shown in figure 1. Analytic and numerical solutions of the Schrodinger equation will be compared.
Fig. 1.
Vibrations of a diatomic molecule and
its centre of mass.
For small amplitude vibrations about the equilibrium separation distance Req of the two atoms, the potential energy U(R) as a function of the separation distance R can be approximated by a parabola
(5)
Assuming the two atoms with masses m1 and m2 are at locations x1 and x2 and are free to move parallel to the X-axis, then the Schrodinger equation for the motion for the atoms can be written as
(6)
We can simplify equation (6) using a number of transformation of the variables
total mass of molecule, m = m1 + m2
reduced mass, 1/m = 1/m1 + 1/m2
separation of atoms, R = x1 – x2
location of centre of mass, x = (m1/m)x1 + (m2/m) x2
wavefunction,
The Schrodinger equation can now be written as two separate equations:
(1) The free translational motion for a particle of mass m along the X-axis
(7)
(2) The internal vibrational (harmonic) motion for a particle of mass m
(8)
We can ignore the free motion of the molecule and concentrate on the internal vibrational molecular motion as described by equation (8). Equation (8) can be solved analytically and its solutions can be expressed in terms of Hermite polynomials [McKelvey].
The particle executes harmonic motion due to the restoring force acting between the two atoms. The motion of the atoms are restricted because as R increases the potential energy U(R) rises to large values and the wavefunction yvib(R) will tend towards zero. Since the vibrations are restricted and for physically acceptable solutions of the wavefunction, the total vibrational energy Evib = En is quantized and the discrete set of energy level eigenvalues are
n = 0, 1, 2, 3, … (9)
The equal spacing between adjacent energy levels
(10)
is an unique property of the parabolic potential well.
The ground state (lowest energy state, n = 1) of the harmonic oscillator is characterized by a minimum zero-point vibrational energy, . The distance between the particles is fluctuating incessantly around the equilibrium separation whereas from a classical mechanics point of view, the particles would be perfectly stationary.
The wavefunction for each state is related to a Hermite polynomials and the normalized wavefunctions are
(11)
normalization constant,
Gausian function,
Hermite polynomial, Hn(b)
The Hermite polynomials satisfy the differential equation
(12)
and the first few polynomials are
n / Hn(b)0 / 1
1 / 2b
2 / 4 b2 - 2
3 / 8 b3 – 12 b
4 / 16 b4 - 48 b2 +12
5 / 32 b5 -160 b3 + 120 b
6 / 64 b6 – 480 b4 + 720 b2 -120
However, the best way to calculate the Hermite polynomials is by the use of the recursion relationship
(13)
MATLAB EXPLORATIONS
(1) Classical vibrations: shm.m
For a classical oscillator, executing simple harmonic motion (SHM) calculate the following quantities for a macroscopic particle attached to a spring and set vibrating (amplitude of oscillation, A = 1 m, mass of object, m = 1.00 kg and force constant, ke= 25.0 N.m-1)
· angular frequency (rad.s-1)
· natural frequency (Hz)
· period (s)
· separation between energy levels (eV)
Animate the motion of the particle and plot the following graphs
· x/ t, v / t, a / t and v / x
· K / t, U / t and E / t
· U / x and Fe / x
For our bound classical particle, the probability of locating it in an interval dx is proportional to the time dt spent by the particle in that interval. The classical probability density function DP is derived from equation (2)
(14)
(15)
(16)
The probability of finding the particle is 1, therefore
(17)
from which the normalizing constant, N can be found. The derivative dt/dx can be found numerically using the Matlab function gradient and the value of N found by numerically by performing the integration given in equation (17) by Simpson’s rule with the function simpson1d.
Plot the probability density function DP and calculate the probability of locating the particle within ± A/10 of the origin and ± A/10 of the extreme positions.
Comment on the significance of each result.
Matlab outputshm.m
CLASSICAL OSCILLATOR - SIMPLE HARMONIC MOTION
Angular frequency (rad/s,) wo = 2.19e+001
Natural frequency (Hz), fo = 3.49e+000
Period (s), T = 2.87e-001
Total Energy (J), Etot = 2.40e+000
Separation of energy levels (eV), dE = 1.39´10-15
Comments
At any time t, the particle’s trajectory, velocity, acceleration and energy can be predicted. The energy separation DE = for any macroscopic system is negligible, for practical purposes the total energy varies continuously and not in discrete jumps.
Classical probability distribution function
Comments
(2) Quantum vibration: vibrations02.m
The force constants for typical diatomic molecules are in the range between 400 to 2000 N.m-1.
Molecule / HF / HCl / HBr / HI / CO / NOForce constant, ke (N.m-1) / 970 / 480 / 410 / 320 / 1860 / 1530
Atom / H / F / Cl / Br / C / O
mass (amu) / 1.008 / 19.00 / 35.45 / 79.91 / 12.01 / 16.00
1 amu = 1.66056´10-27 kg
For the diatomic molecules listed above, calculate the following:
· angular frequency (rad.s-1)
· natural frequency (Hz)
· period (s)
· separation between energy levels (J and eV)
· wavelength l of the electromagnetic radiation emitted in the transition
n + 1 à n.
Comment on the significance of your calculations
Matlab outputQuantum oscillator - diatomic molecule
HCl
Angular frequency (rad/s,) wo = 5.45e+014
Natural frequency (Hz), fo = 8.68e+013
Period (s), T = 1.15e-014
Spacing between energy levels (J), dE = 5.75e-021
Spacing between energy levels (eV), dE = 3.59e-002
Wavelength (m) lambda = 3.46e-005
Wavelength (mm) lambda = 3.46e+001
Comments
The radiation associated with the vibration of molecules is mainly in the infrared part of the spectrum. Diatomic molecules will be strong absorbers and emitters in the infrared part of the spectrum due to molecular vibrations.
For molecules such as HCl, the mass of chlorine atom is much greater than the mass of the proton. The chlorine atom can be regarded as stationary and the proton vibrating towards and away from the more massive atom.
(3) Hermite polynomials:
function Hn = hermite(n,Hx)
Hn is the value of the Hermite polynomial of order n evaluated at Hx (º b)
hermitePlot.m
Evaluate the Hermite polynomials Hn(b) using the recursion relationship given by equation (13) and plot them for the range -5 £ b £ +5
Matlab output(3) Hermite polynomials:
9
dell/b05/qmbook/doc/vibrations.doc 7:32:46 AM 30-Nov-05