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

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.

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 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 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 [

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 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 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 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) H( 2 S)+F + ( 1 D) States of HF + NIST H+F +, H + +F MOLPRO H.J. Werner, P. Knowles, MOLPRO (version ) package of ab initio programs

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 P◦2P◦ 12Σ+12Σ+1,3 Σ +, 1,3 Π P4P12Π12Π 3,5 Σ +, 3,5,Π, 3,5 Δ P4P12Σ+12Σ+3,5 Σ +, 3,5 Π D2D12Π12Π 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  + ) C + ( 2 P) + HF + ( 2  + ) 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

C 2+ +HF C + (1s 2 2s 2 2p) 2 P + HF + ( 2 Π) 1  +, 1 Π C + (1s 2 2s 2 2p) 2 P + HF + ( 2  + ) 1  +, 1 Π 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.

C 2+ +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 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 = 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.

C 2+ +HF/ C 2+ +OH νV= 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 cm 2 ) for different velocities v (in a.u.).

— = 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

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°.

C 2+ +HF Velocity (a.u.) E lab (keV) sec Σ Σ + secpi Σ Π sec Σ Σ + secpi Σ Π sectot Charge transfer cross sections averaged over the different orientations.

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  +

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.

C 2+ +HCl Velocity (a.u.) E lab (keV) sec Σ Σ + secpi Σ Π sec Σ Σ + secpi Σ Π sec Σ Σ + secpi Σ Π Sectot C 2+ +HCl Sectot C 2+ +HF 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 cm 2 ). Comparison with the C 2+ + 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, (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, (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, (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, ( ), 2012, ISBN , 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, (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/ project. The project is co-financed by the European Union and the European Social Fund.