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 

Cohen-Tannoudji C., Diu B., Laloe F. Quantum mechanics, vol. 1,2

Tom Ziegler , Department of Chemistry , University of Calgary

Gamze Kaya and Sunil Anumula, TAMU

EnerMaterials and acknowledgments:

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

siniz i i i il l p r p

L r p

zz

i

n

iiii

n

iiii

n

ii

n

iizz

IL

rmrrmrvmlL

2

1111

)(

L Iangular momentum

kinetic energy 2 / 2K I

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

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

Shapes of spherical harmonic functions

First Sixteen Spherical Harmonic Functions

l m0 1 2 3

3

2

1

0

-1

-2

-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)

Energy corresponding to a rotational level (with angular quantum number J) is given by:

E= B J (J+1)

ΔE= EJ-EJ-1 = 2BJ

Difference between two energy states:

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.

Rotational molecular states: random alignment

where J =1,2,3,…….

which is very small and can be archived at room temperature, i.e. kT~ ΔE

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 (θ)

Corresponds to permanent dipole moment

Corresponds to induced dipole moment

Corresponds to field free rotational energy.

Laser field

Time period of IR field at 800 nm (2.66 fs) < typical rotational period of molecules

Induced dipole momet

Rotational time period of molecule can be written asThis value ranges from few femto seconds to pico seconds

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.

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

Different types of interactions with the laser field:

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.

1/ (2 )revT Bc

Molecular rotational constants

1

2 rev

BT c

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, 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:

So,

Details of derivation of the potential energy in a laser field

The degree of alignment of a molecular sample is characterized by the expectation value of

Molecules in external laser field

To find the wave function one needs to solve the Schroedinger equation

Table. 1 Relevant parameters for the molecules investigated in the experiment

Highest occupied molecular orbital (HOMO) of the molecules investigated.

O₂ CO

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

Courtesy of Gamze Kaya

Diagram of Molecular orbitals for N2

N2 has 10 valence electrons.

σg

σg

σ*u

σ*u

πu

π*g

HOMO

LUMO

LUMO : lowest unoccupied molecular orbital

HOMO : highest occupied molecular orbital

24

,

( , , exp( i ( 1) )J mJ m

c J m J J

2 2

* 2' '

', ' ,

', '

, ', ' 2

cos ( ) ( , cos ( ,

', ' cos , exp( i '( ' 1) ( 1) )J m J mJ m J m

JJm m

J JJ J

c c J m J m J J J J

Infrared spectroscopy does not involve electric dipole transitions. Thus, no electric dipole moment is required; the principal selection rule for linear molecules here is 0, 2J

the time-dependent phase disappears

' 2J J

'J J

, 2 exp( i (4 6))J J J

The rotational wavepacket evolution in time

revT the time is given in units of The alignment factor:

2cos The degree of alignment of molecules is characterized by .

Zon (1976), Friedrich + Herschbach (1995), Seideman (1995)

Isotropic case

Molecular revivals of N2 molecules by linearly polarized probe pulse I0=7.2 10^13 W/cm2;

measured by detecting the ionization yield.

4revT

2revT 3

4revT revT

Experimental results of N2 (for 2:1 ratio of even and odd J states)14

2 tot el vib rot ns

14tot

3el

N Weight:

N is a Bo SYM S S S S son (I=1), so 2

SYM,

SYM

= ( ), hence: g

AS AS 1

2:1Expected ratio

of contributions

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.