1. Rotational states and introduction to molecular alignment
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 

2. 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 

3. Rigid body angular momentum
If we split the whole body into
small pieces, then each contribution
with magnitude:
Direction: li  perpendicular to ri and pi
angular momentum
kinetic energy

4. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

5. Quantum mechanical angular momentum
Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

6. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

7. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

8. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

9. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

10. Courtesy of Tom Ziegler , Department of Chemistry  , University of Calgary 

11. Shapes of spherical harmonic functions 3 2 1 -1 -2 -3
First Sixteen Spherical Harmonic Functions

12. 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)

13. 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.

14. Effects of the laser field on molecular state
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 induced
dipole moment
Corresponds to permanent dipole moment
Induced dipole momet
Corresponds to field free
rotational energy.
Time period of IR field at 800 nm (2.66 fs) < typical rotational period of molecules

15. Effect of a short laser pulse on molecular alignment: adiabatic
and non adiabatic regimes
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:
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.

16. Molecular rotational constants
Table. 1 Our experimental data and comparison to theoretical molecular rotational constants from the literature.
Our Experimental data (cm-1)
Theoretical (cm-1) N₂
2.0102±0.011
1.9896a O₂
1.4611±0.022
1.4297a
CO₂
0.3971±0.018
0.3902a CO
1.9393±0.004
1.9313a
C₂H₂
1.1801±0.003
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, (2003).
Courtesy of Gamze Kaya

17. 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

18. Polarizability tensor of a linear molecule
In case of linear molecules:

19. Details of derivation of the potential energy in a laser field

20. 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

21. Raman mechanism of the rotational wave packet excitation
The interaction of a short laser pulse with a molecule produces excitation of rotational states. This excitation can be due to Stokes Raman scattering: the scattered photon has a smaller energy than the incident photon with the difference transferred to the molecule. Since the pulse duration is typically shorter than the rotational period of the molecule, it can be considered as an instantaneous “kick” that creates rotational states, which continue to exist after the pulse is gone (this is why it is called field-free alignment, since it happens after the pulse).
The molecular system is left in a coherent superposition of rotational states. This superposition is called
rotational wavepacket, which evolves in time. The rotational states dephase and rephrase again after the rotational period
and also partially at half and quarter of the rotational period. This causes the molecular ensemble to acquire
|< > -1/3 |>0, which indicates some degree of alignment or antialignment.

22. Table. 1 Relevant parameters for the molecules investigated in the experiment
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
C₂H₂
14.2
12.9 O₂ CO
Highest occupied molecular orbital (HOMO) of the molecules investigated.
Courtesy of Gamze Kaya

23. Diagram of Molecular orbitals for N2
N2 has 10 valence electrons.
LUMO
HOMO
HOMO : highest occupied molecular orbital
LUMO : lowest unoccupied molecular orbital

24. The degree of alignment of molecules is characterized by .
The rotational wavepacket evolution in time
the time is given in units of
The alignment factor:
Infrared spectroscopy does not involve electric dipole transitions. Thus, no electric dipole moment is required; the principal selection rule for linear molecules here is
Zon (1976), Friedrich + Herschbach (1995), Seideman (1995)
the time-dependent phase disappears

25. Isotropic case

26. Expected ratio of contributions
Experimental results of N2 (for 2:1 ratio of even and odd J states)
2:1
Expected ratio of contributions
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

27. Finding excited rotational wave packet
Ortigoso et al. J. Chem. Phys., Vol. 110, No. 8, 3874, 1999
Markus Gühr, SLAC National Accelerator  Laboratory

28. Calculations of the rotational wavepacket at maximal alignment for different temperatures and intensities
Calculated with the code of Markus Gühr, SLAC National Accelerator  Laboratory

29. 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.