1

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: sanjay@iitm.ac.in

22

Chemical Reaction Dynamics Beyond TheBorn-Oppenheimer Approximation

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.

4

EarthEarth’’s Atmospheres Atmosphere

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

11

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

⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

12

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).

13

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

14

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).

15

Ab initio diabatic potential energy surfacesH3

+ +

R

r

H(2S) + H2+ (1 2Σg)

+

H+ + H2 (1 1Σg)+

16

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

17

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

18

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

19

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)

20

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

22

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)

24

25

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

26

+R

r

Ab initio diabatic potential energysurfaces ([HNO]+)

H+ + NO (1 2Π)

H(2S) + NO+ (1 1Σ )+

V22d

V11d

1 2A′

2 2A′

2727

• 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

2828

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

29

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

31

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+ = =

32

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 +

∫∫

33

( )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∫

35

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

36

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

37

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

4444

O3/O3- PESX1A1 O3

2A2 O3-

Energy Difference between 2A2 and X1A1

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

49