1. Theoretical Study of Charge Transfer in Ion- Molecule Collisions Emese Rozsályi University of Debrecen 2012.09.19. Department of Theoretical Physics

2. C 2+ + OH → C + + OH + C 2+ + HF → C + + HF + C 2+ + 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.

3. 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.

4. Ion-diatomic collisional system B R A ρ C

5. 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

6. 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:

7. 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’

8. 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:

9. 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 ρ 0  F 0ν 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. EIKONXS R.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]

10. 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( 2 P) 2 Σ + 2 Π 2 A 1 2 B 1, 2 B 2 -99.4658 -99.4482 -99.6230 -99.6236 H( 2 S)+F + ( 3 P) 2,4 Σ - 2,4 Π 2 B 1, 2 B 2 2 A 2 4 B 1, 4 B 2 4 A 2 -99.3536 -99.3472 -99.3350 -99.3355 -99.4908 -99.4902 -99.4896 -99.4897 H( 2 S)+F + ( 1 D) 2 Σ + 2 Π 2 Δ 2 A 1 2 B 1, 2 B 2 2 A 2 2 A 1 -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( 2 P)000 H( 2 S)+F + ( 3 P)3.84523.10753.6246 H( 2 S)+F + ( 1 D)6.41235.62736.2505 States of HF + NIST H+F +, H + +F MOLPRO H.J. Werner, P. Knowles, MOLPRO (version 2009.1) package of ab initio programs

11. The quasimolecule CHF 2+ E ∞ (eV) C + -stateHF + -stateCHF 2+ -state 1.0 2P◦2P◦ 12Π12Π 1,3 Σ +, 1,3 Π, 1,3 Δ 2.3.9093 2P◦2P◦ 12Σ+12Σ+1,3 Σ +, 1,3 Π 3.5.1775 4P4P12Π12Π 3,5 Σ +, 3,5,Π, 3,5 Δ 4.9.2453 4P4P12Σ+12Σ+3,5 Σ +, 3,5 Π 5.9.3017 2D2D12Π12Π 1,3 Σ +, 1,3 Π, 1,3 Δ, 1,3 φ Comparison of asymptotic energies (in eV): ConfigurationThis calculationSeparated species C 2+ ( 1 S) + HF( 1  + ) 8.3688.279 C + ( 2 P) + HF + ( 2  + ) 3.8243.909 C + ( 2 P) + HF + ( 2 Π)00 Three 1  + states and two 1 Π states are considered in the process: C 2+ (1s 2 2s 2 ) 1 S + HF( 1  + ) 1  + C + (1s 2 2s 2 2p) 2 P + HF + ( 2  + ) 1  +, 1 Π C + (1s 2 2s 2 2p) 2 P + HF + ( 2 Π) 1  +, 1 Π

12. C 2+ +HF 1. 1. C + (1s 2 2s 2 2p) 2 P + HF + ( 2 Π) 1  +, 1 Π 2. 2. C + (1s 2 2s 2 2p) 2 P + HF + ( 2  + ) 1  +, 1 Π 3. 3. C 2+ (1s 2 2s 2 ) 1 S + HF( 1  + ) 1  + Potential energy curves, θ=0 ◦, ρ HF =eq., 1 Σ +, 1 Π. Radial coupling matrix elements between 1  + states, θ=0 ◦, ρ HF =eq. Rotational coupling matrix elements between 1  + and 1 Π states, θ=0 ◦, ρ HF =eq.

13. C 2+ +HF Total and partial charge transfer cross sections at equilibrium, ϴ=0° ; full line: with translation factors; broken line: without translation factors.

14. Total and partial charge transfer cross-sections for the vibration coordinate r HF =1.5 a.u., θ=0°. Radial coupling matrix elements between 1  + states, θ=0°, Dotted line, r HF =2.0 a.u.; full line, r HF =1.73836832 a.u. (equilibrium); dashed line, r HF =1.5 a.u. Total and partial charge transfer cross-sections for the vibration coordinate r HF =2.0 a.u., θ=0°. Total charge transfer cross-sections, θ=0°, for different values of the vibration coordinate r HF.

15. C 2+ +HF/ C 2+ +OH νV=0.15 014.26014.42415.37915.33917.071 15.0075.0645.4005.3855.994 21.2791.2931.3791.3751.531 30.3050.3080.3290.3280.365 40.0730.0750.0800.0790.088 Total charge transfer cross-sections for the C 2+ - HF system in the linear approach, θ=0°, for different values of the vibration coordinate r HF. Total charge transfer cross-sections for the C 2+ - OH system in the linear approach, θ=180°, for different geometries of the OH radical. Total cross sections for the C 2+ + HF( =0) →C + + HF + ( ) charge transfer process (in 10 -16 cm 2 ) for different velocities v (in a.u.).

16. — = 0 o — = 20 o — = 45 o — = 90 o ····· = 135 o ····· = 160 o ····· = 180 o Potential energy curves, ρ HF =eq., 1 Σ +, 1 Π. θ=90 ◦ θ=180 ◦ Evolution of the radial couplings for different orientations. rad23 rad12

17. C 2+ +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°.

18. C 2+ +HF Velocity (a.u.) E lab (keV) sec32 3 1 Σ + - 2 1 Σ + secpi32 3 1 Σ + - 2 1 Π sec31 3 1 Σ + - 1 1 Σ + secpi31 3 1 Σ + - 1 1 Π sectot 0.050.757.042.680.480.3410.54 0.138.413.990.860.9614.22 0.156.758.784.791.041.4816.10 0.2128.755.311.241.3316.64 0.2518.758.566.041.761.5317.89 0.3278.136.972.191.8119.10 0.3536.758.187.682.522.1020.51 0.4488.258.102.892.2521.49 0.4560.758.168.223.382.2422.01 0.5757.948.153.962.1522.20 0.61087.377.645.251.9622.23 Charge transfer cross sections averaged over the different orientations.

19. C 2+ +HCl Potential energy curves for the 1  + (full line) and 1 Π (broken line) states of the C 2+ -HCl molecular system at equilibrium, θ=0°. Four 1  + states and three 1 Π states are considered in the process: 1.C + (1s 2 2s 2 2p) 2 P° + HCl + ( 2 Π) 1  +, 1 Π 2. C + (1s 2 2s 2 2p) 2 P° + HCl + ( 2  + ) 1  +, 1 Π 3. C + (1s 2 2s 2 2p) 2 D + HCl + ( 2 Π) 1  +, 1 Π 4. C 2+ (1s 2 2s 2 ) 1 S + HCl( 1  + ) 1  +

20. C 2+ +HCl Total and partial charge transfer cross sections at equilibrium, ϴ=0° ; Radial coupling matrix elements between 1  + states, θ=0 ◦, ρ HCl =eq. Rotational coupling matrix elements between 1  + and 1 Π states, θ=0 ◦, ρ HCl =eq.

21. C 2+ +HCl Velocity (a.u.) E lab (keV) sec43 4 1 Σ + 3 1 Σ + secpi43 4 1 Σ + 3 1 Π sec42 4 1 Σ + 2 1 Σ + secpi42 4 1 Σ + 2 1 Π sec41 4 1 Σ + 1 1 Σ + secpi41 4 1 Σ + 1 1 Π Sectot C 2+ +HCl Sectot C 2+ +HF 0.050.756.991.780.140.220.170.099.38 10.54 0.1 312.222.800.390.770.140.2716.58 14.22 0.156.757.484.780.361.140.270.2314.26 16.10 0.2123.734.991.801.380.730.4213.05 16.64 0.2518.752.843.863.500.990.570.8212.59 17.89 0.3272.362.913.910.950.770.8111.70 19.10 0.3536.752.142.343.431.070.880.6610.52 20.51 0.4482.082.032.721.160.770.559.32 21.49 0.4560.752.071.862.141.210.680.478.42 22.01 0.5752.061.741.771.210.650.437.87 22.20 0.61082.071.571.441.150.670.487.37 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 C 2+ + HCl collision system (in 10 -16 cm 2 ). Comparison with the C 2+ + HF collision system.

22. 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 C 2+ + 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 C 2+ ions with hydrogen chloride, Phys. Rev. A 83, 052713 (2011).  5. E. Rozsályi: Charge transfer in collisions of C 2+ 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 N 2, Phys. Rev. A 85, 042701 (2012).

23. 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.