Rotational states and introduction to molecular alignment
• Rotational states
• Molecular alignment is suitable tool to exert strong-field control over
molecular properties.
• Some of research fields in which molecular alignment plays a key role
• High harmonics generation
• Molecular phase modulators
• Control of fragmentation of molecules by molecular alignment
• Selective rotational manipulations of close molecular species
EnerMaterials and acknowledgments:
Gamze Kaya and Sunil Anumula, TAMU
Cohen-Tannoudji C., Diu B., Laloe F. Quantum mechanics,
vol. 1,2
Tom Ziegler , Department of Chemistry ,
University of Calgary
Rigid body angular momentum
L  r p
If we split the whole body into
small pieces, then each contribution
with magnitude:
l  l sin  p  r p
iz
i
i
i
i
Direction: li  perpendicular to ri and pi
 n
2 
Lz   liz   mi vi ri   mi (  ri )  ri     mi ri 
i 1
i 1
i 1
 i 1

Lz  I z
angular momentum
n
n
n
L  I
kinetic energy
K  I / 2
2
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Quantum mechanical angular momentum
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Courtesy of Tom Ziegler , Department of Chemistry , University of Calgary
Shapes of spherical
harmonic functions
l
0
1
m
2
3
3
2
1
0
-1
-2
First Sixteen Spherical Harmonic Functions
-3
Rotational energies of a molecule in a
particular vibrational state
J is the total orbital angular momentum of the whole molecule
B is called rotational constant
D is a centrifugal distortion constant (a correction
due to molecular stretching)
Rotational molecular states: random alignment
Energy corresponding to a rotational level (with angular quantum number J) is given by:
E= B J (J+1)
where J =1,2,3,…….
Difference between two energy states:
ΔE= EJ-EJ-1 = 2BJ
which is very small and can be
archived at room temperature, i.e. kT~ ΔE
In general, an ensemble of molecules is in a thermal distribution of multiple J states.
Molecules can be thought of as randomly aligned at normal room temperature,
i.e. their the directions of their axes are isotropically distributed.
Effects of the laser field on molecular state
Laser field
If the laser field frequency is far from resonance, the Hamiltonian has the
following contributions
H(t) = BJ2 + V µ(θ) + V 𝜶(θ)
Corresponds to field free
rotational energy.
Corresponds to
permanent dipole
moment
Corresponds to induced
dipole moment
Induced dipole momet
Time period of IR field at 800 nm (2.66 fs) < typical rotational period of molecules
Effect of a short laser pulse on molecular alignment: adiabatic
and non adiabatic regimes
Trev  1/ (2 Bc)
Rotational time period of molecule can be written as
This value ranges from few femto seconds to pico seconds
Different types of interactions with the laser field:
1. Adiabatic: Trot < pulse width
Dipole is induced due to interaction between laser field and molecules, which causes the molecules
to align along the laser field. Molecules follow laser fields, as if it were static fields.
2. Non adiabatic ( field free, or impulsive): Trot > pulse width
An ensemble of Rotational wave packets of molecules are created by applying short
intense laser filed. These molecules can dynamically rotate their molecular axes after the laser pulse. And
these rotating molecules repeatedly come to a phase and diphase at a period of certain revival time in a field
free environment.
Molecular rotational constants
B
1
2Trev c
Table. 1 Our experimental data and comparison to theoretical molecular
rotational constants from the literature.
N₂
O₂
CO₂
CO
C₂H₂
Our Experimental data (cm-1)
2.0102±0.011
1.4611±0.022
0.3971±0.018
1.9393±0.004
1.1801±0.003
Theoretical (cm-1)
1.9896a
1.4297a
0.3902a
1.9313a
1.1766b
a W.
M. Haynes, CRC Handbook of Chemistry and Physics: A Ready-Reference Book
of Chemical and Physical Data. Boca Raton, FL.: CRC Press, 2011.
b M. Herman, A. Campargue, M. I. El Idrissi, and J. Vander Auwera, "Vibrational
Spectroscopic Database on Acetylene," Journal of Physical and Chemical Reference
Data 32, 921-1361 (2003).
Courtesy of Gamze Kaya
Molecules in external laser field
When an electric dipole with a dipole moment ‘P’ is placed in an electric field, E,
The net torque about an axis through “O” is given as Τ=PxE
Then, internal energy of the dipole is given as
U = -P.E
In case of induced polarization in molecules, we can write P= α. E ,
where, α is the polarizability tensor of molecule.
Internal energy of molecule becomes
U= - α. E. E
Polarizability tensor of a linear molecule
In case of linear molecules:
Details of derivation of the potential energy in a laser field
So,
Molecules in external laser field
The degree of alignment of a molecular sample is characterized by the
expectation value of
To find the wave function one needs to solve the Schroedinger equation
Table. 1 Relevant parameters for the molecules investigated in the experiment
Molecule
Trev(ps)
Ip(eV)
HOMO symmetry
N₂
8.4
15.6
σg
O₂
11.6
12.7
u
CO₂
42.7
13.8
g
CO
8.64
14.01
σg
C₂H₂
14.2
12.9
u
O₂
CO
Highest occupied molecular orbital (HOMO) of the molecules investigated.
Courtesy of Gamze Kaya
Diagram of Molecular orbitals for N2
σ*u
LUMO
HOMO
π*g
σg
πu
σ*u
σg
HOMO : highest occupied molecular orbital
LUMO : lowest unoccupied molecular orbital
N2 has 10
valence
electrons.
The degree of alignment of molecules is characterized by
cos 2 .
The rotational wavepacket evolution in time  ( ,   cJ m J , m exp( i  J ( J  1) )
J ,m
 the time is given in units of Trev
The alignment factor:
cos 2  ( )   ( , cos 2   ( ,

*
2
c
c
J
',
m
'
cos
 J , m exp( i   J '( J ' 1)  J ( J  1)  )
  J ' m' J m
J ', m ' J , m






J
,
J
'


J , J '2 m,m '
 JJ '
Infrared spectroscopy does not involve electric dipole transitions. Thus, no
electric dipole moment is required; the principal selection rule for linear
molecules here is J  0, 2
J  J ' the time-dependent phase disappears
J  J ' 2
 J , J  2  exp( i  (4 J  6))
Zon (1976),
Friedrich +
Herschbach (1995),
Seideman (1995)
24
Isotropic case
Experimental results of N2 (for 2:1 ratio of even and odd J states)
 tot   el  vib  rot  ns Weight:
14
N2
14
N is a Boson (I=1), so  tot SYM
SYM S
S
 el =SYM, ( 3  g  ), hence:
Trev
S
AS AS
Trev
4
S
2
3Trev
4
2
1
2:1
Expected ratio
of contributions
Trev
Molecular revivals of N2 molecules by linearly polarized probe pulse I0=7.2 10^13 W/cm2;
measured by detecting the ionization yield.
Courtesy of Gamze Kaya
Finding excited rotational wave packet
Ortigoso et al. J. Chem. Phys., Vol. 110, No. 8, 3874, 1999
Markus Gühr, SLAC National Accelerator Laboratory
Calculations of the rotational wavepacket at maximal
alignment for different temperatures and intensities
Calculated with the code of Markus Gühr, SLAC National
Accelerator Laboratory
Conclusions: effects due to alignment
The alignment effect manifests itself in such processes as
ionization, high harmonic generation; even configuration of
molecular orbitals can be tested.
Fragmentation of molecules also changes due to alignment.
Alignment introduces changes of the refractive index, introduces
anisotropy and birefringence.
The alignment effect is reducing with temperature, but
increasing with the intensity, though the intensity still should be
below the values when significant ionization occurs.