LINEAR MOLECULE ROTATIONAL TRANSITIONS:
J = 4
J = 3
J = 2
J = 1
J = 0
LINEAR MOLECULE ROTATIONAL SPECTRUM:
Intensity
J = 4←3
J = 1←0


2B
Absorption Frequencies →
EFFECTS OF MASS AND MOLECULAR SIZE:
The slide that follows gives B values for a
number of diatomic molecules with different
reduced masses and bond distances. What is
the physical significance of the very different
B values seen for H35Cl and D35Cl? All data
are taken from the NIST site.
http://www.nist.gov/pml/data/molspec.cfm
REDUCED MASSES AND BOND DISTANCES:
Molecule
Bond
Distance (Å)
B Value
(MHz)
H35Cl
H37Cl
D35Cl
H79Br
H81Br
D79Br
24Mg16O
107Ag35Cl
1.275
1.275
1.275
1.414
1.414
1.414
1.748
2.281
312989.3
312519.1
161656.2
250360.8
250282.9
127358.1
17149.4
3678.04
REAL LIFE – WORKING BACKWARDS?
In the real world spectroscopic experiments
provide frequency (and intensity) data. It is
necessary to assign quantum numbers for
the transitions before molecular (chemically
useful) information can be determined.
Sometimes “all of the data” are not
available!
SPECTRUM TO MOLECULAR STRUCTURE:
Class Example: A scan of the microwave
(millimeter wave!) spectrum of 6LiF over the
range 350 → 550 GHz shows lines at
358856.2 MHz, 448491.1 MHz and 538072.7
MHz. Assign rotational quantum numbers
for these transitions. Determine a B value
and the bond distance for 6LiF. Are the
“lines” identically spaced?
HIGHER ORDER ENERGY TERMS:
The slightly unequal spacing of lines in the
6LiF
spectrum occurs because very rapidly
rotating diatomic molecules distort. A
“higher order” energy expression accounts
for this effect
EJ = hBJ(J+1) – hDJJ2(J+1)2
DJ is the (quartic) centrifugal distortion
constant.
HIGHER ORDER FREQUENCY EXPRESSION:
The energy expression on the previous slide
can be used with the selection rule ΔJ = +1
(for absorption) and ΔE = hν to give:
ν = 2B(J+1) - 4DJ(J+1)3
This expression will be used in the lab
(HCl/DCl spectrum). A typical frequency
calculation is shown on the next slide.
NON-RIGID ROTOR CALCULATION, 7LIF:
Here B = 40,026.883 MHz & DJ = 0.3505 MHz
Transition
2B(J+1)
4DJ(J+1)3
Freq. Calc.
Freq. Obs.
J=1←0
80053.766
1.402
80052.36
No Data
J=2←1
160107.53
11.216
160096.32
160096.33
J=3←2
240161.30
37.854
240123.45
240123.47
J=4←3
320215.07
89.728
320125.34
320125.36
J=5←4
400268.83
175.25
40093.58
400093.62
J=6←5
480322.60
302.83
480019.77
480019.73
NON-RIGID MOLECULES:
Aside: Every spectroscopic constant tells us
something. A “small” DJ value suggests that
a molecule does not distort easily.
Comparisons can be made for inertially
similar molecules. Explanation?
Molecule
B (GHz)
DJ (kHz)
6LiF
45.23
443
13C18O
52.36
151
SPECTRA OF NONLINEAR MOLECULES:
With the particle in the box energy expressions
grew more complex as we moved from one to
three dimensions.
PIAB one dimension: Energy (eigenvalues!)
expression has one term and one quantum
number.
PIAB three dimensions: Energy expression has
(up to!) three terms and three quantum
numbers.
ROTATIONS IN THREE DIMENSIONS:
For nonlinear molecules the number of
quantum numbers and rotational constant
needed to describe rotational energies is
greater than one. We also have more than
one I value. In general, we have a (3x3)
matrix (moment of inertia tensor) that cane
be diagonalized to simplify the mathematics.
NONLINEAR RIGID ROTORS:
After diagonalization the moment of inertia
tensor has three elements with Ia ≤ Ib ≤ Ic.
Types of Rotors:
1. Spherical tops: Ia = Ib = Ic. We need just
one quantum number (J again) to describe
rotational energies. Examples: CH4, SF6 and
C60.
NONLINEAR RIGID ROTORS:
2. Symmetric tops: Ia = Ib ≤ Ic (oblate top)
and Ia ≤ Ib = Ic (prolate top). Examples:
Oblate top: CHF3, HSi79Br3.
Prolate top: CH3F, CH3-CN.
For symmetric tops we need two quantum
numbers, J and K, to describe rotational
energies.
NONLINEAR RIGID ROTORS:

DEGENERACY:
In organic chemistry courses you have
discussed NMR spectra – and removal of
spin degeneracy using a magnetic field. For
the spin case, I = ½, there is a two-fold
energy degeneracy in the absence of a
magnetic field. In rotational spectroscopy
there is, similarly, a (2J+1) fold degeneracy.
DIPOLES AND ELECTRIC FIELDS.
From physics, the energy of a linear rod with
an electric dipole moment (μ) placed in an
electric field can be found as μEcosθ. Similar
to NMR, the degeneracy of rotational energy
levels can be removed by applying an
electric filed to a gas. This enables the size of
a molecule’s dipole moment to be
determined.
DEGENERACY AND DIPOLE MOMENTS:
We will not use the degeneracy of rotational
levels for several weeks. It will be useful in
calculating the relative intensities of spectral
lines (after Boltzmann!). By experiment, it is
found that a molecule must have a
permanent non-zero electric dipole moment
to have a pure rotational spectrum.
MOLECULAR STRUCTURE/DIPOLE MOMENTS:
From first year chemistry courses you should
be able to take a simple molecular formula
and (a) draw a Lewis structure for the
molecule, (b) determine a molecule’s shape
and (c) predict whether a molecule has net
polarity. Review examples on the next slide.
MOLECULAR SHAPES AND POLARITY:
Molecular Formula
H2
CO
HCN
CO2
CH2Cl2
SF6
O=C=C=C=S
CHF=CHF (cis)
CHF=CHF (trans)
Electrically Polar
Pure Rotational
Spectrum?