Theoretical Study of Charge Transfer in Ion-Molecule Collisions

Emese Rozsályi University of Debrecen2012.09.19. Department of Theoretical Physics

C2+ + OH → C+ + OH+

C2+ + HF → C+ + HF+

C2+ + HCl → C+ + HCl+

By the Wigner and von Neumann non-crossing rule, adiabatic potential energy curves for states of the same symmetry cannot cross.

Potential energy curves for states of the same symmetry can approach each other in a narrow region avoided crossing

The charge-transfer process is driven by means of the nonadiabatic interactions in the vicinity of avoided crossings.

Comparison of depth-dose profiles

The dose to normal healthy tissue is the least by using carbon-ion therapy. This depth-dose profile is the closest to the desire profile in diagram (a) in terms of tumour coverage, critical organ avoidance and minimised entry channel dosage.

Ion-diatomic collisional system

B R A

ρ ϴ

C

The projectile follows straight-line trajectories:

X

v

b R ϑ

θ ρ Z Y

The electronic motion is described by the eikonal equation:

Semiclassical treatment

Sudden approximation

No appreciable change in the ro-vibrational wavefunction is effected in the time interval in which the electronic transition takes place.

Molecular close-coupling treatment:

Semiclassical formalism

For a given nuclear trajectory and fixed :

The coefficients are subject to the initial condition:

X

Z’ v

Dynamical couplings: R

ρ Z

X’

Cross sections

The probability for transition to the final state is:

The cross section for transition to state , for each value of ρ is:

The total cross section is a sum of the partial cross sections:

Franck-Condon approximation The coefficients are slowly varying functions of ρ

it is possible to substitute them with values at the equilibrium distance of the diatomic molecule ρ0

F0ν is the Franck-Condon factor between the BC and BC+ vibration wave functions at equilibrium geometry for the vibrational level ν=0 and ν, respectively.

EIKONXSR.J. Allan, C. Courbin, P. Salas, P. Wahnon, J. Phys. B 23, L461 (1990).

LEVEL 7.7 R.J. Le Roy [http://leroy.uwaterloo.ca]

Dissociation limits and their atomic terms

States of HF+

Corresponding symmetry of states within the C2v point group

Asymptotic energies (a.u.)CASSCF/aug-cc-pVTZ

Asymptotic energies (a.u.)MRCI/aug-cc-pVTZ

H++F(2P) 2Σ+

2Π2A12B1, 2B2

-99.4658-99.4482

-99.6230-99.6236

H(2S)+F+(3P) 2,4Σ-

2,4Π2B1, 2B22A24B1, 4B24A2

-99.3536-99.3472-99.3350-99.3355

-99.4908-99.4902-99.4896-99.4897

H(2S)+F+(1D) 2Σ+

2Π 2Δ

2A12B1, 2B22A22A1

-99.2661-99.2531-99.2427-99.2391

-99.3940-99.3939-99.3932-99.3933

Dissociation limits and their atomic terms

Energies obtained from NIST database (in eV)

CASSCF/aug-cc-pVTZ energies (in eV)

MRCI/aug-cc-pVTZ energies(in eV)

H++F(2P) 0 0 0

H(2S)+F+(3P) 3.8452 3.1075 3.6246

H(2S)+F+(1D) 6.4123 5.6273 6.2505

States of HF+

NIST H+F+ , H++F

MOLPROH.J. Werner, P. Knowles, MOLPRO (version 2009.1) package of ab initio programs

The quasimolecule CHF2+

E∞(eV) C+-state HF+-state CHF2+-state

1. 0 2P◦ 12Π 1,3Σ+, 1,3Π, 1,3Δ

2. 3.9093 2P◦ 12Σ+ 1,3Σ+, 1,3Π

3. 5.1775 4P 12Π 3,5Σ+, 3,5,Π, 3,5Δ

4. 9.2453 4P 12Σ+ 3,5Σ+, 3,5Π

5. 9.3017 2D 12Π 1,3Σ+, 1,3Π, 1,3Δ, 1,3φ

Comparison of asymptotic energies (in eV):

Configuration This calculation Separated species

C2+(1S) + HF(1+) 8.368 8.279

C+(2P) + HF+(2+) 3.824 3.909

C+(2P) + HF+(2Π) 0 0

Three 1+ states and two 1Π states are considered in the process:

C2+(1s22s2)1S + HF(1+) 1+

C+(1s22s22p)2P + HF+(2+)1+, 1Π

C+(1s22s22p)2P + HF+(2Π)1+, 1Π

C2++HF

1. 1. C+(1s22s22p)2P + HF+(2Π) 1+, 1Π

2. 2. C+(1s22s22p)2P + HF+(2+) 1+, 1Π

3. 3. C2+(1s22s2)1S + HF(1+) 1+

Potential energy curves, θ=0◦, ρHF=eq., 1Σ+, 1Π.

Radial coupling matrix elements between 1+ states, θ=0◦, ρHF=eq.

Rotational coupling matrix elements between 1+ and 1Π states, θ=0◦, ρHF=eq.

C2++HF

Total and partial charge transfer cross sections at equilibrium, ϴ=0° ; full line: with translation factors; broken line: without translation factors.

Total and partial charge transfer cross-sections for the vibration coordinate rHF=1.5 a.u., θ=0°.

Radial coupling matrix elements between 1+ states, θ=0°, Dotted line, rHF=2.0 a.u.; full line, rHF=1.73836832 a.u. (equilibrium); dashed line, rHF=1.5 a.u.

Total and partial charge transfer cross-sections for the vibration coordinate rHF=2.0 a.u., θ=0°.

Total charge transfer cross-sections, θ=0°, for different values of the vibration coordinate rHF.

C2++HF/ C2++OH

ν V=0.15 V=0.15 V=0.15 V=0.15 V=0.15

0 14.260 14.424 15.379 15.339 17.071

1 5.007 5.064 5.400 5.385 5.994

2 1.279 1.293 1.379 1.375 1.531

3 0.305 0.308 0.329 0.328 0.365

4 0.073 0.075 0.080 0.079 0.088

Total charge transfer cross-sections for the C2+- HF system in the linear approach, θ=0°, for different values of the vibration coordinate rHF.

Total charge transfer cross-sections for the C2+- OH system in the linear approach, θ=180°, for different geometries of the OH radical.

Total cross sections for the C2+ + HF(=0) →C+ + HF+

() charge transfer process (in 10-16 cm2) for different velocities v (in a.u.).

— ϴ = 0o — ϴ = 20o— ϴ = 45o— ϴ = 90o ····· ϴ = 135o ····· ϴ = 160o ····· ϴ = 180o

Potential energy curves,ρHF=eq., 1Σ+, 1Π.

θ=90◦ θ=180◦

Evolution of the radial couplings for different orientations.

rad23rad12

C2++HF

Total charge transfer cross-sections at equilibrium, for different orientations θ from 0° to 180°.

Radial coupling matrix elements between 1+ states for different orientations θ from 0° to 180°. Dotted line, θ=90°; dotted-dashed line, θ=45°; dashed line, θ=135°; thin full line, θ=0°; full line, θ=180°.

C2++HF

Velocity(a.u.)

Elab

(keV)

sec3231Σ+ - 21Σ+

secpi3231Σ+ - 21Π

sec3131Σ+ - 11Σ+

secpi3131Σ+ - 11Π

sectot

0.05 0.75 7.04 2.68 0.48 0.34 10.54

0.1 3 8.41 3.99 0.86 0.96 14.22

0.15 6.75 8.78 4.79 1.04 1.48 16.10

0.2 12 8.75 5.31 1.24 1.33 16.64

0.25 18.75 8.56 6.04 1.76 1.53 17.89

0.3 27 8.13 6.97 2.19 1.81 19.10

0.35 36.75 8.18 7.68 2.52 2.10 20.51

0.4 48 8.25 8.10 2.89 2.25 21.49

0.45 60.75 8.16 8.22 3.38 2.24 22.01

0.5 75 7.94 8.15 3.96 2.15 22.20

0.6 108 7.37 7.64 5.25 1.96 22.23

Charge transfer cross sections averaged over the different orientations.

C2++HCl

Potential energy curves for the 1+ (full line) and 1Π (broken line) states of the C2+-HCl molecular system at equilibrium, θ=0°.

Four 1+ states and three 1Π states are considered in the process:

1.C+(1s22s22p)2P° + HCl+(2Π) 1+, 1Π

2. C+(1s22s22p)2P° + HCl+(2+) 1+, 1Π

3. C+(1s22s22p)2D + HCl+(2Π) 1+, 1Π

4. C2+(1s22s2)1S + HCl(1+) 1+

C2++HCl

Total and partial charge transfer cross sections at equilibrium, ϴ=0° ;

Radial coupling matrix elements between 1+ states, θ=0◦, ρHCl=eq.

Rotational coupling matrix elements between 1+ and 1Π states, θ=0◦, ρHCl=eq.

C2++HCl Velocity

(a.u.)

Elab

(keV)

sec43

41Σ+31Σ+

secpi43

41Σ+31

Π

sec42

41Σ+21Σ+

secpi42

41Σ+21

Π

sec41

41Σ+11Σ+

secpi41

41Σ+11

Π

Sectot

C 2++HCl

Sectot

C 2++HF

0.05 0.75 6.99 1.78 0.14 0.22 0.17 0.09 9.38 10.54

0.1 3 12.22 2.80 0.39 0.77 0.14 0.27 16.58 14.22

0.15 6.75 7.48 4.78 0.36 1.14 0.27 0.23 14.26 16.10

0.2 12 3.73 4.99 1.80 1.38 0.73 0.42 13.05 16.64

0.25 18.75 2.84 3.86 3.50 0.99 0.57 0.82 12.59 17.89

0.3 27 2.36 2.91 3.91 0.95 0.77 0.81 11.70 19.10

0.35 36.75 2.14 2.34 3.43 1.07 0.88 0.66 10.52 20.51

0.4 48 2.08 2.03 2.72 1.16 0.77 0.55 9.32 21.49

0.45 60.75 2.07 1.86 2.14 1.21 0.68 0.47 8.42 22.01

0.5 75 2.06 1.74 1.77 1.21 0.65 0.43 7.87 22.20

0.6 108 2.07 1.57 1.44 1.15 0.67 0.48 7.37 22.23

The comparative results show that the charge-transfer mechanism is fundamentally dependent of the specific nonadiabatic interactions involved in each system.

Charge transfer cross sections for the C2+ + HCl collision system (in 10-16 cm2). Comparison with the C2+ + HF collision system.

Publication list The presentation is based on the following papers: 1. E. Bene, E. Rozsályi, Á. Vibók, G. J. Halász, M. C. Bacchus-Montabonel: Theoretical treatment of direct

and indirect processes in ion-biomolecule collisions, AIP Conf. Proc. 1080, 59-70 (2008). 2. E. Rozsályi, E. Bene, G. J. Halász, Á. Vibók, M. C. Bacchus-Montabonel: Theoretical treatment of charge

transfer in collisions of C2+ ions with HF: Anisotropic and vibrational effect, Phys. Rev. A 81, 062711 (2010). 3. E. Rozsályi, E. Bene, G. J. Halász, Á. Vibók, M. C. Bacchus-Montabonel: Ab initio molecular treatment of

C2+ + HF collision system, Acta Physica Debrecina, XLIV, 118 (2010). 4. E. Rozsályi, E. Bene, G. J. Halász, Á. Vibók, M. C. Bacchus-Montabonel: Ab initio study of charge-transfer

dynamics in collisions of C2+ ions with hydrogen chloride, Phys. Rev. A 83, 052713 (2011). 5. E. Rozsályi: Charge transfer in collisions of C2+ ions with HCl molecule, Acta Physica Debrecina, XLV, 166

(2011). 6. E. Rozsályi, E. Bene, G. J. Halász, Á. Vibók, M. C. Bacchus-Montabonel: Analysis of the charge transfer

mechanism in ion-molecule collisions. Advances in the Theory of Quantum Systems in Chemistry and Physics; Progress in Theoretical Chemistry and Physics; 22, (355-368), 2012, ISBN 978-94-007-2075-6, Springer.

Further publication: 7. E. Rozsályi, L. F. Errea, L. Méndez, I. Rabadán: Ab initio calculation of charge transfer in proton collisions

with N2, Phys. Rev. A 85, 042701 (2012).

Thanks to...

Dr. Ágnes Vibók, Dr. Marie-Christine Bacchus-Montabonel, Dr. Erika Bene and Dr. Gábor Halász for their support, inspiring comments during the research.

The presentation is supported by the TÁMOP-4.2.2/B-10/1-2010-0024 project.The project is co-financed by the European Union and the European Social Fund.