Non-adiabatic Collision Dynamics in the Ion- and the …€¦ · · 2009-10-05Non-adiabatic...
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Non-adiabatic Collision Dynamics in the Ion- and the Electron-Molecule Reactions
Sanjay KumarDepartment of Chemistry
Indian Institute of Technology MadrasChennai 600 036 India
email: [email protected]
3
Solar flare:
Solar flare is the sudden eruption ofhydrogen gas on the sun’s surface.
Picture of solar flare reportedby NASA on 28-10-2003
Solar flare primarily consists of acceleratedprotons, electrons and other heavy ions withmore than 1 MeV energy.
P. J. Crutzen et al. Science 189 457 (1975)W. Klemperor, Nature 227 1230 (1970)
H+ molecule collision are important to study chemistry of upper atmosphere.
H3+, N2H+, [HCO]+, HO2
+ are detected in the interstellar medium.
5Proton Energy Loss Spectrum
Schematic diagram of a Molecular Crossed Beam-scattering apparatus
Steinfeld, Francisco and Hase, Chemical Kinetics and Dynamics, Prentice Hall,1998.
H+
source
Target M SourceDetector
6Molecular beam experiments
G. Niedner-Schatteburg and J. P. Toennies, Adv. Chem. Phys. LXXXII 553 (1992))
+
7
Endoergic and Endoergic and ExoergicExoergic
H+ + M
H + M+Ene
rgy
Intermolecular distance
V1
V2
H+ + M
H + M+
Ene
rgy
Intermolecular distance
V2
V1
Exoergic Endoergic
I.P (M) < I.P (H) I.P (M) > I.P (H)
M Molecular targetH Hydrogen
G. Niedner-Schatteburg and J. P. Toennies, Adv. Chem. Phys. LXXXII 553 (1992)
88
to t e n m p
e e n e ee n n
H H T H
H T V V V
= + +
= + + +
2
1
12
N
mp iitot
HM =
⎛ ⎞=− ∇⎜ ⎟
⎝ ⎠∑
*
( ) ( , ) ( ) ( , ), 1, 2,..
( , ) ( , )e i i i
i j ij
H R R r E R R r i
R r R r dr
ψ ψ
ψ ψ δ
= = ∞
=∫
1
1 1
( , ) ( ) ( , )
( ) ( ) ( , ) ( ) ( , )
tot ni ii
n e mp ni i tot ni ii i
R r R R r
T H H R R r E R R r
ψ ψ ψ
ψ ψ ψ ψ
∞
=
∞ ∞
= =
=
+ + =
∑
∑ ∑
ΗΨ = ΕΨ
99
( )
{ }
2
1 1
2
1 1
1 1
2 2
( )
[( ) ( )]
( ) 2( )( ) ( )
n e mp ni i tot ni ii i
n ni i e ni i mp ni i tot ni ii i
n i n ni ni n itot ni i
i ini e i ni mp i
i n ni n i n ni ni n i
ni i i ni
H H E
H H E
EH H
E
ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψ ψ ψ
ψ ψ ψ ψψ ψ
ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψψ ψ ψ
∞ ∞
= =
∞ ∞
= =
∞ ∞
= =
∇ + + =
∇ + + =
∇ ∇ + ∇⎧ ⎫⎪ ⎪ =⎨ ⎬+ +⎪ ⎪⎩ ⎭
∇ + ∇ ∇ + ∇+ +
∑ ∑
∑ ∑
∑ ∑
1 1tot ni i
i imp i
EH
ψ ψψ
∞ ∞
= =
⎧ ⎫⎪ ⎪ =⎨ ⎬⎪ ⎪⎩ ⎭
∑ ∑*
*
| |
|
|i j ij
H d H
d
ψ ψ τ ψ ψ
ψ ψ τ ψ ψ
ψ ψ δ
=
=
=
∫∫
1010
( )
2 2
1
2 | |
| |
| |
j n i n ni
n ni i ni j n i ni tot nii
j mp i ni
E E
H
ψ ψ ψ
ψ ψ ψ ψ ψ ψ
ψ ψ ψ
∞
=
⎧ ⎫∇ ∇⎪ ⎪⎪ ⎪⎪ ⎪
∇ + + + ∇ =⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪+⎩ ⎭
∑
R
E
R1
R2E
Avoided crossing Conical intersection
NACME
Mass polarization
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Diabatization
2 211 1 1 11 22
ˆ = d d delV H U Cos U Sinψ ψ α α= +
12 1 2 11 22ˆ = ( ) d d d
elV H U U Sin Cosψ ψ α α= −
{ } 1 1 111 122
21 222 2 2
1 ˆ ˆ1 12
d d dd dn n n
Q d dd d dn n n
V VE
M V Vψ ψ ψ
ψ ψ ψ
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞− ∇ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠
2 222 2 2 11 22
ˆ = d d delV H U Sin U Cosψ ψ α α= +
D. G. Truhlar et al., Faraday Discuss. 127 59 (2004)
1 1
2 2
cos sinsin cos
d a
d a
ψ ψα αα αψ ψ
⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
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F. T. Smith, Phys. Rev. 179, 111(1969).A. Mead and D. J. Truhlar, J. Chem. Phys. 77, 6090 (1982).V. Sidis, Adv. Chem. Phys. 82, 73 (1992).T. Pacher, L. S. Cederbaum, and H. KÖppel, Adv. Chem. Phys.
84, 293 (1993).D. Simah, B. Hartke and W. J. Werner, J. Chem. Phys.
111,4523 (1999).M. Baer, Adv. Chem. Phys. 124,39 (2002).M. Baer, Phys. Rep.,358, 75 (2002).M.S. Child, Adv. Chem. Phys. 12, 1 (2002).S. Adhikari and G. Billing, Adv. Chem. Phys. 124, 3355 (2002).W. Jasper et al, Faraday Discuss. 127, 1 (2004).H. KÖppel, Faraday Discuss. 127, 35 (2004).T. Vertesi et al, J. Phys. Chem. A 109, 34476 (2005).
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0 01 ( ) | ( )
2a a a am n m nR R R R
R Rψ ψ ψ ψ∂
= + Δ − Δ∂ Δ
1 2 0d d
Rψ ψ∂
=∂
1 2a a d
R dRαψ ψ∂
=∂
1 2( )ref
ref
Ra a
RR
dR dRdR
α α ψ ψ ′= +′∫
1 1
2 2
cos sinsin cos
d a
d a
ψ ψα αα αψ ψ
⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
0 0
1 2 1 2( , ; ) ( )R r
a a a a
R r
R r dR drR r
α γ α γ ψ ψ ψ ψ∂ ∂= + +
∂ ∂∫ ∫
Diabatization
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Mixing Angle directly from the adiabatic CI wavefunctions
R
E1 1
1a
k kk
cψ ξ=∑ 2 22a
k kk
cψ ξ=∑
( )( )
12 21
'1 '
21sin
kk
kk
c
cα −
⎛ ⎞⎜ ⎟
= ⎜ ⎟⎜ ⎟⎝ ⎠
∑
∑1aψ
2aψ
Mixing Angle directly from the (quasi)diabatic CI wavefunctions
2 2( ') ( ) ( ') ( ) ....i j j jq q q qφ φ φ φ+ + Jacobi rotation or unitary
Transformation1
† 2( )T S S S−
=N
d dm lm l
ldψ χ=∑ d cA= 1
† 2( )A B BB−
=† refB c d=
T. Pacher, L. S. Cederbaum, and H. Köppel, Adv. Chem. Phys. 84, 293 (1993). W. Domcke and C. Woywod, Chem. Phys. Lett. 216, 362 (1993).D. Simah, B. Hartke and W. J. Werner, J. Chem. Phys. 111,4523 (1999).
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Ab initio diabatic potential energy surfaces
The H+ + H2 system
Ab initio diabatic potential energy surfacesH3
+
R+
r
H(2S) + H2+ (1 2Σg)
+
H+ + H2 (1 1Σg)+
NACME (a.u.) Mixing angle α (deg)
Rr
Rr
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M. Baer, G. Niedner-Schatteburg and J. P. Toennies, J. Chem. Phys. 91 4169 (1989)A. Saieswari and Sanjay Kumar, J. Chem. Phys. 127, 214304 (2007).
Inelastic vibrational excitation(IVE)
Rainbow maximum
EXPT ≈ 9.8°
Present work: 10.5°
Baer et al. (1989): 12.5°
State-resolved rotationally-summed differential cross section
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M. Baer, G. Niedner-Schatteburg and J. P. Toennies, J. Chem. Phys. 91 4169 (1989)A. Saieswari and Sanjay Kumar, J. Chem. Phys. 127, 214304 (2007).
Vibrational charge transfer (VCT)
Rainbow maximum
EXPT ≈ 10.5°
Present work: 11.5°
Baer et al. (1989): 13.5°
State-resolved rotationally-summed differential cross section
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Transition probability
R. Schinke, M. Dupuis, W. A. Lester, Jr, J. Chem. Phys. 72 3909 (1980)A. Saieswari and Sanjay Kumar, J. Chem. Phys. 128, 064301 (2008)
The Present calculations:
Reproduces the experimentalpattern and agrees well with earlier theoretical study at Ec.m. = 10 eV.
In addition the camparisionat Ec.m. = 7.3 eV also reproduces the experiments.
Inelastic vibrational excitation(IVE)
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32 ( )gH O+ −+ ∑
2 22( ) ( )gH S O++ ∏
Rr
Mixing angle (α) in degrees
2 42( ) ( )uH S O++ ∏
Diabatic
Adiabatic
The H+ + O2 system
2121
2H O+ +
Adiabatic PESs Quasidiabatic PESs
F. George, A. Saieswari and Sanjay Kumar – to be published
2323
GS, first and second ES Adiabatic PESs
11 +Σ
12 +Σ
13 +Σ
12 +Σ
13 +Σ
11 +Σ
H+ →COγ = 180o
H+ → OCγ = 0o
R (ao)
R (ao)
r (a o)
r (a o)
E (a
.u.)
E (a
.u.)
H+ + CO
T J Dhilip Kumar and Sanjay Kumar, J. Chem. Phys. 121, 191 (2004)
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H+
γ = 15º
Or
R
N
Adiabatic
Diabatic
A. Saieswari,Sanjay Kumar, and H. KÖppel, J. Chem. Phys. 128, 124305-1/11 (2008).
The H+ + NO system
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+R
r
Ab initio diabatic potential energysurfaces ([HNO]+)
H+ + NO (1 2Π)
H(2S) + NO+ (1 1Σ )+
V22d
V11d
1 2A′
2 2A′
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• Enhanced cross section around certain incident energy (Eres)
• [AB-]* is a metastable anion formed by occupation to an MO
e- + AB (v')[AB-]*e- ( Eres) + AB (v) A + B- or
A- + B
Resonant scattering
Dissociative Electron Attachment (DEA)
Vibrational Excitation (VE)
Non-resonant (direct) scattering
• In non-resonant (direct) scattering the VE and DEA cross sectionsvary smoothly with incident energy and are called background crosssection
• Cause: energy transfer is efficient in resonant than in non-resonant
• Elastic scattering is dominated by the direct process
Electron-molecule Interactions
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Relevance• Technological
• Environmental
• Biological
• Astrophysical
─ Plasma chemistry─ Semiconductor industry to produce itchant gases─ Lasers
─ Atmospheric chemistry
─ Radiation damage to DNA molecule
─ Studying planetary atmosphere and interstellar medium
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R
V (R
) DEA
A- + B
A + B
Complex energy curve
A BR
Aut
o-de
tach
men
t
Formation of Resonant Negative ions and its decay pathways
3030Construction of discrete and continuum components by
R-matrix theory
The technique
Separation of the space of the scattered electron into two regions by a sphere Ω .
Solve a variational problem for the Hamiltonian(HΩ) of electron + target system inside the sphere.
AB + e-
( ) 0k kH E ψΩ − =
HΩ is ensured to be Hermitian
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Time-dependent formulations – only a few studies reported.Time-independent quantum mechanical formulations – well studied
Time-Dependent Quantum Mechanical Formulations
Diabatic electronic basis set
Simplest case of an isolated single-channel resonance
The electronic Hilbert space:
Projection operators
Nuclear Dynamics – electronic decay, vibrational excitations, DEA in a complex, energy dependent and non-local effective potential!
Difficult to solve!
dφ and kφ
Discrete electronic state
continuum stateScattering
d dQ φ φ= k k kP K dK d φ φ= Ω∫1, 0P Q PQ+ = =
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dφ kφ
Discrete electronic state
: exact (energy nondiagonal)background scattering states
: solution of fixed-nuclei e- - molecule scattering problem in P-space
Assumptions :
(1) Action of TN on the wavefunctions of the diabatic electronicstates , is negligible
(2) Neglect direct continuum-continuum coupling via TN
Well-justified for low-energy resonant e- - molecule collisions
dφ kφ
[ ]{ }
2( ) ( ) / 2 +
+
d N d d k k N o k
k d dk k
H T V R K K dK d T V R k
K dK d V Hermition conjugate
φ φ φ φ
φ φ
⎡ ⎤= + Ω + +⎣ ⎦
Ω +
∫∫
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( )tψ
( ) ( )i t H ttψ ψ∂
=∂
( ) ( ) ,d dt tψ ψ ψ= ( ) ( )k kt tψ ψ ψ=
[ ]( ) ( ) ( ) ( )d N d d k dk ki t T V R t K dK d V ttψ ψ ψ∂
= + + Ω∂ ∫
2( ) ( ) / 2 ( ) ( )k N o k kd di t T V R k t V ttψ ψ φ∂ ⎡ ⎤= + + +⎣ ⎦∂
( ) ( )2 2ˆ ˆ/2 /2 '1
0
( ) (0) ' ( ')o ot
i H k t i H k tk k kd dit e dt e V tψ ψ ψ− + +⎡ ⎤
= +⎢ ⎥⎣ ⎦
∫
: Full time-dependent state vector of the collision system
Defining the electronically projected states,
Results in coupled equations.
Vib. for the discrete stateˆdH
Vib. for the targetˆoH
Time-Dependent Schrödinger equation
34ˆ( ) ( ) ( ) ( )d d d k dk ki t H R t K dK d V t
tψ ψ ψ∂
= + Ω∂ ∫
( ) ( )2 2ˆ ˆ/ 2 / 2 '1
0
( ) e (0) e ( ')o ot
i H k t i H k t
k k kd dit V tψ ψ ψ⎡ ⎤ ⎡ ⎤− + +⎣ ⎦ ⎣ ⎦
⎧ ⎫= +⎨ ⎬
⎩ ⎭∫
1
0
( ')( ) ( ) ' ) ( )(t
d d d dii t H t dt SF t t tt
tψ ψ ψ∂= +−+
∂ ∫
( ) F t
( ) S t : “Source term”, depends on the initial condition
: Accounts for discrete-continuum coupling on the nuclear motion in the discrete state
For resonant electron-molecule collisions the appropriate initial condition is
(0)id dkVψ ν= (0) 0kψ =;
: momentum of the incoming electron
: vibrational state of the target molecule
: “entrance amplitude” for the electron
ikν
dkV
( )2ˆ / 2( ) = oi H k tk dk kdK dK dF t V e V− +
Ω∫( )2ˆ / 2 )) (( 0= oi H k t
k dk kK dK dS t V e ψ− +Ω∫
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R
V (R
) DEA
A- + B
A + B
Complex energy curve
A BR
Aut
o-de
tach
men
t
Formation of Resonant Negative ions and its decay pathways
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e- + O3 [O3]- O- + O2
O + O2-
Allan et al (1996) - DCS (elastic, vibrational inelastic and DEA- Vibrational excitation peaks (4.2 eV and 6.6 eV)
- Very small KER at 1.3 eV.
Assignment 2A12B2
e- + O3 system
Curran (1961)
Peterson et al (1990)
Koch et al (1993)
- Beam experiment (0-3 eV)- two dissociation channels O-/O3 and O2
-/O3
- CASSCF calculations for X 2B1 state of O3-
- Vertical excitation energies (eg. geometry of X 2B1)- To explain experimental absorption bands of Ozonides in solution
Shape resonance
- DEA peaks at 0.4, 1.3, 3.2 and 7.5Tail of X 2B2of O3
-
Valenceexcited
resonance
2A1 Shape resonance
Feshbachresonance
eV
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Sarpal, Nestmann and Peyerimhoff (1997)
- Calculated integral and differential cross section using fixed-nuclei R-matrix method
Morokuma et al (1998) - ab initio calculations, mechanism in the experiments of Continetti et al
- Found 1 2B2 as a bound state and also the lowest state at bond angle below 90o
Senn et al (1998)
- reported of a large peak for O- production at zero incident energy!
- O- ion production at ~0.0 eV is four times that of at 1.3 eV
38
Rangwala, SVK Kumar and Krishna Kumar (1999), TIFR experiment
- Accurate measurements of absolute cross-section
39
X 1A1 (O3) : (core) + (6a1)2(4b2)2(1b1)2(1a2)2 (2b1)0(7a1)0(5b2)0
X 2B1 (O3-) : (core) + (6a1)2(4b2)2(1b1)2(1a2)2 (2b1)1
EA = 2.103 eV (Novick et al, 1979)
Shape resonances (O3-)
X 1A1 (O3) + e- [X 1A1] (2b1)0(7a1)1 2A1 4.8 eVX 1A1 (O3) + e- [X 1A1] (2b1)0(5b2)1 2B2 8.5 eV
Feshbach resonance (O3-)
O3- metastable states O3 parent states
Resonance pos. (Eres) (eV)*
[X 1A1] 4b2-1 2b1
2 (2B2)
[X 1A1] 1a2-1 2b1
2 (2A2)
[X 1A1] 6a1-1 2b1
2 (2A1)
[X 1A1] 1b1-1 2b1
2 (2B1)
[X 1A1] 4b2-12b1
1 (1/3A2)
[X 1A1] 1a2-1 2b1
1 (1/3B2)
[X 1A1] 6a1-1 2b1
1 (1/3B1)
[X 1A1] 1b1-1 2b1
1 (1/3A1)
0.87
1.47
1.54
7.30
Low-lying resonant states of O3-
*) From MRCI /pVDZ results by Bonn group (Hanrath et al)
40Formation probability and contribution to DEA
*) R-matrix results: Nestmann, SVK Kumar and Peyerimhoff, Phys. Rev. A, 71, 012705 (2005)
The partial cross section for formation of resonant ions2 2
( ) | ( , ) |EE f E R vEπσ = Γ
22(R )
( ) |eqEE f v
Eπ
σΓ
≈
1( )Lifetime τ =Γ
With Condon-type approx. for Γ
Symmetry Γ(Eres)/2 Lifetime(fs) Character
2B22A12A22A12B12B2
1.1 x 10-4
1.6 x 10-4
1.3 x 10-2
0.543.1 x 10-2
8.5 x 10-2
3000210025
0.61114.0
FeshbachFeshbachFeshbach
ShapeFeshbach
Shape
A period of vibration≈ 12 fs
≈ 100% dissociation
( )N d E ET V f E f+ =
41
Calculation of DEA cross section from 2A2 Feshbach resonance
1. In C2v symmetry (γ = 90 deg.)
2. In Cs symmetry ( i.e. in full dimension)
• Franck-Condon reflection principle (TN = 0)
• Time-dependent WP calculation
The 2A2 state in C2v symmetry corresponds to 2 2A'' state in Cs symmetry
• Generation of 3-dimensional PES (by MRCI/pVTZ method)
42Calculation of DEA cross section by FC reflection principle
X 1A1
2A2
( PESs close to FC region)
(O3)
(O3-)
• The KE of nuclear motion not takeninto account ( i.e. TN = 0)
• Harmonic approx. for PESs near FC region
2
( , )
2
( ) ( ( , ), , )
( , )
resresE r R E
v
dE E r R r RE dE
r R dr dR
πσ
χ
≤= Γ
×
∫
• The total resonant cross section formula†
by FC principle:
†
res targ
targ 0 1 2
with ( , ) ( , )
( , ) 0.5( ) 0.11 eV
andd
e e
E r R V r R E
E V r R ν ν
= −
= + + =
B. M. Nestmann, V. Brems, A. Dora and S. Kumar, J. Phys. B., 38, 75 (2005)
43Total cross section corresponding to the 2A2 Feshbach resonance, calculated at different levels of accuracy by FC reflection principle
B. M. Nestmann, V. Brems, A. Dora and S. Kumar, J. Phys. B., 38, 75 (2005)
+++
Energy- and geometry- dependent Γ
Energy- and geometry-independent Γ
Total Cross-sections of Rangwala et al
Energy-dependent but geometry-independent Γ with linearized 2A2 PES
Energy-dependent but geometry-independent Γ
45Wave packet evolution by split operator method
Technical parameters:spatial grid size 1024 x 1024total time = 27.5 fsΔt = 0.54 fs
2( , ) ( ) ( ) ( ) ( ', ')id N d L L di R t T V R R R R t
tψ ψ∂
= + + Δ − Γ⎡ ⎤⎣ ⎦∂
4646
The Fourier transform of auto-correlation function yields the spectrum of the molecule
( ) ( )iEtC E e C t dt∞
−∞
= ∫
20( ) |
2 EC E f vπ=
22
0( ) |EE f vEπσ ≈ Γ
2 ( )C EEπ
≈ Γ
In time-independent way C(E) is related as
The formation cross section of resonant ions( )N d E ET V f E f+ =
4747Total cross section corresponding to the 2A2 Feshbach resonance, calculated at different levels of accuracy by WP dynamics calculation
(a) Franck-Condon reflection with PES linearization
(b) Franck-Condon reflection
(c) TDQM
(d) TDQM with a constant complex absorbing potentiali.e. Γ(Eres) = 0.026 eV)
(e) TDQM with constant complex absorbing potential and explicitenergy dependence for entry amplitude Γ(E,re,Re)
A. Dora, S. Kumar, V. Brems, and Nestmann,, J. Chem. Phys. (to be submitted)]
48
Acknowledgements
Professor S. D. Peyerimhoff and Dr. B. M. NestmannInstitute of Theoretical & Physical ChemistryUniversity of Bonn, Germany
Professor S V K KumarAtomic & Molecular PhysicsTata Institute of Fundamental researchMumbai India
Dr. A. SaieswariPost Doctoral FellowThe Fritz Haber Research Centre andThe Department of Physical ChemistryHebrew University of Jerusalem Jerusalem
Dr. Amar DoraPost-doctoral Research FellowDepartment of Physics & AstronomyUniversity College London UK