Chemical Physics 410 (2013) 9–18

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Chemical Physics

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Dynamics of proton collisions with acetylene, ethylene and ethane at 30 eV

Cong-Zhang Gao a,b, Jing Wang a,b, Feng-Shou Zhang a,b,c,⇑a The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University,Beijing 100875, Chinab Beijing Radiation Center, Beijing 100875, Chinac Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China

a r t i c l e i n f o

Article history:Received 31 July 2012In final form 17 October 2012Available online 26 October 2012

Keywords:Time-dependent local densityapproximationMolecular dynamicsElectron captureReactive channelIonization cross section

0301-0104/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.chemphys.2012.10.007

⇑ Corresponding author at: The Key Laboratory of BModification of Ministry of Education, College of NucBeijing Normal University, Beijing 100875, China. Fax

E-mail address: fszhang@bnu.edu.cn (F.-S. Zhang).

a b s t r a c t

The collision processes, H+ + C2Hm (m = 2,4,6), are studied at 30 eV by using time-dependent densityfunctional theory combined with molecular dynamics approach. The reaction channel, electronic densityevolution, scattering pattern, total ionization cross sections are presented based on three projectile inci-dent orientations. We find that the primary mechanism of target ionization at 30 eV is electron capture,and the bond dilution effect is prevalent in present collisions. The structure of the scattering angle is dif-ferent in distinct orientations, which is investigated by means of the force which proton is subjected to.The rainbow angle is in reasonable agreement with experimental and other theoretical results. Calculatedionization cross sections increase with increasing C–C bond length except for the orientation parallel tothe C–C bond, and the reason is explained by the C–C bond effect.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the past decades, study on proton-molecule collisions has re-ceived a great attention in a wide range of areas, such as accelera-tors, material science, plasmas, astrophysical phenomena, andupper atmosphere reactions. As a consequence, a number of simplemolecules have been studied by the crossed proton beam in exper-iments [1–9], like H2, O2, N2, H2O, CO, CO2, N2O, CH4, C2H2, andC2H4. However, the detailed understanding related to collision pro-cesses is still obscure, especially for the low energy region. When aprojectile with low energies (1 eV 6 ELab 6 30 eV) attacks a target,some phenomena will occur in the collision process, such as vibra-tional excitation, dissociation, charge transfer, nuclear exchange,etc. Because of the unattainable precision in time scale and instru-mental errors in experiments, the phenomena are hardly to be ob-served or detected directly. However, these difficulties could beovercome by powerful theoretical methods in short time scales.

Considering the extensive existence of proton-hydrocarbon col-lisions and its important role in fusion devices [10], we devote tostudy proton colliding with C2Hm (m = 2,4,6). In experiments,there are a series of work relevant to proton-molecule collisionswhich have been done by Toennies group [1,2,4–6,8,9]. They em-ployed crossed proton beam over the energy range from 10 to

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46 eV, especially for C2H2 [8] and C2H4 [9] molecules with the col-lision energy of 30 eV. They measured the differential cross sectionas well as time-of-flight spectra, and analyzed collision-inducedvibration modes, charge transfer, energy loss, and fragmentation.These desirable results had enriched our knowledge of collisionprocesses and were used to appraise the model as a standard. How-ever, there was no experiment available for H+ colliding with C2H6

at low energies. For theory aspect, Malinovskaya et al. [11] studieddynamics of proton-acetylene collisions at 30 eV within the elec-tron nuclear dynamics (END) method [12]. Recently, the same col-lision process was also performed by Morales et al. [13] combiningthe coherent state dynamics (CSD) [14] with END method. The useof CSD allows for all the molecular degrees of freedom: transla-tional, rotational, vibrational and electronic state, which are ex-pressed by a set of full quantum dynamical equations. Bothsimulations displayed valuable mechanistic information whichwere not accessible by experimental means at low energies. Kim-ura and coworkers [15,16] and Cabrera-Trujillo et al. [17] didH+ + C2H2 collisions in keV energy range. Electron capture and tar-get excitation are the main processes. One should note that themethod used by Kimura made an assumption of nuclei being fro-zen in collision process. Compared with acetylene, studies onC2H4 scattering by H+ are fewer at low-collision energies. At keVenergy, Suzuki et al. [18] and Getahun et al. [19] undertook theproton-ethylene collisions with averaging on three and six projec-tile incident orientations, respectively. The former employed afully quantum–mechanical approach and the latter used theclassical trajectory Monte Carlo method. For C2H6, Kimura and

10 C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18

coworkers [16,20] investigated the collision process, i.e., the vibra-tional effect, capture cross section, fragmentation, and steric effect.Cabrera-Trujillo [17] predicted the energy dependence of C2H6

molecular fragmentation cross sections at proton energy from 10to 10 keV, which lacks of experiments to compare with. Thesimulations above could reproduce results compatible with exper-iments, and helped to revise our previous views of molecularcollision dynamics. However, for H+ + C2Hm (m = 2,4,6) collisions,few simulations involved in the mechanism of target ionization,and the comparison of scattering properties at low energy. Suchproblems need to be solved.

In this work, we study the collisions of H+ + C2Hm (m = 2,4,6) at30 eV. The calculations are based on the time-dependent densityfunctional theory (TDDFT) [21] combined with molecular dynam-ics (MD) approach, using the time-dependent local density approx-imation (LDA) [22]. The TDLDA-MD code package which we use isdeveloped by the Toulouse-Erlangen group [23]. The TDLDA-MD isa non-adiabatic approach which employs classical mechanics todescribe the motion of the nuclei and uses quantum mechanicsto treat the dynamics of electrons. It has been applied to investi-gate the stopping power of proton and antiproton in metals[24–26], the excitation and ionization of molecules (i.e., water[27], ethylene [28–30], Li4 [31]) by atom or proton scattering,and electronic excitation induced by proton collision with graph-ene [32,33]. These studies provide a reasonable qualitative andsemi-qualitative picture. Obviously, the use of local densityapproximation (LDA) for the collision problems has been acknowl-edged to be a good first approximation, and LDA is believed tocapture the essentials of the collision process. Encouraged by thesesuccessful study with TDLDA-MD method, we are devoted toinvestigate the mechanism of target ionization in H+ + CHm

(m = 2,4,6) collisions at low energies and to compare the scatteringproperties between the three molecules.

The article is organized as follows. In Section 2, we explain thetheoretical model TDLDA-MD. Followed by Section 3, we presentcomputational details. The results and discussion are presentedin Section 4. Finally, the conclusions are drawn in Section 5.

Atomic units are employed throughout this work, except wherespecifically mentioned.

2. Theoretical model

2.1. The total energy

Detailed reviews of TDLDA-MD method have been given else-where [23,34–36]. Hence, only a brief summary is presented here.In the framework of density functional theory (DFT), any observa-ble could be written uniquely as a functional of the density of anon-interacting reference system [37]. The total energy of systemconsists of three parts: ion-ion, electron–electron, and ion–electron coupling interaction, which is given as below,

Etot ¼ Eion þ Eel þ Ecoup

¼ Ekin;ionðf _RIgÞ þ Epot;ionðfRIgÞ þ Ekin;elðfuigÞ þ ECðqÞ

þ ExcðqÞ þ ESICðfjuij2gÞ þ EPsPðfuig; fRIgÞ þ Eext: ð1Þ

The different contributions to the total energy, one by one, arethe ion kinetic energy Ekin;ion, the ion potential Epot;ion, the electronkinetic energy Ekin;el, the direct Coulomb energy EC , the exchange–correlation energy Exc , the self-interaction correction term ESIC ,the pseudo-potential EPsP , and the external energy Eext .

In the present work, for the exchange and correlation term Exc ,we employ the widely used exchange–correlation functional ofPerdew and Wang [38]. Self-interaction correction (SIC) is intro-duced by the self-Coulomb energy contained in ECðqÞ and is not

eliminated by the LDA exchange term. It will lead to deviationsfrom experiments in handling dynamic process without SIC, forexample, electron emission [39,40]. Actually, one can employ thesimple average-density SIC (ADSIC) [41],

EADSIC ¼ �Nel;" ECq"

Nel;"

� �þ ELDA

xc

q"Nel;"

;0� �� �

� Nel;# ECq#

Nel;#

� �þ ELDA

xc 0;q#

Nel;#

� �� �; ð2Þ

where Nel;r is the number of electrons with spin orientationr; r 2 f"; #g.

We only consider the valence electrons of molecule for presentcollision energy, since the core electrons are far below the valenceelectron energy levels. The coupling between electrons and ioniccores is described via Goedecker–Teter–Hutter pseudopotentials[42].

The term Eext stands for the external energy derived from theby-passing ion. Here, we consider the simply external Coulomb po-tential Uext .

2.2. Ground state of molecule

The stationary equations are derived from minimization of thetotal energy by means of the variational conditions,

duyi

Etot �X

n

�iðuijuiÞ" #

¼ 0; dRI Etot ¼ 0; ð3Þ

where one should note that the external potential Eext is set to zerobecause of no excitation.

For a given ionic geometry, the electronic structure is obtainedby solving the static Kohn–Sham (KS) equation,

hKSui ¼ �iui ð4Þ

and the Hamiltonian is,

hKS ¼p2

2þ UKS ¼

p2

2þ VC þ Vxc þ VPsP; ð5Þ

where p2

2 denotes the kinetic energy of electron and UKS is Kohn–Sham effective potential.

2.3. Dynamical equation

For the motion of ions, the classical molecular dynamics (MD)equations are obtained by variation with regard to RI and PI ,

@

@tRI ¼

PI

MI; ð6Þ

@

@tPI ¼�5RI

12

XI–J

ZIZJ

jRI�RJjþXNel

i¼1

huijVPsPðr�RIÞjuiiþVext;ionðRI;tÞ" #

; ð7Þ

where MI and ZI represents the mass and the charge of ion numberI, and Vext;ion is the interaction between ions and the external field.

For describing the dynamics of electrons, the single-particlewave function uiðr; tÞ should satisfy the time-dependent Kohn–Sham (TDKS) equation

i@

@tuiðr; tÞ ¼ �1

2r2 þ UKS½q�ðr; tÞ

� �uiðr; tÞ; ð8Þ

where the electronic density is

qðr; tÞ ¼XNel

i¼1

juiðr; tÞj2: ð9Þ

C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18 11

2.4. Numerical methods and observables

For ground-state calculation, KS equation (4) is efficientlysolved by an iterative method. The damped gradient method isused [34],

uðnþ1Þj ¼ bO uðnÞj � bDðhðnÞ� < uðnÞj jh

ðnÞjuðnÞj >ÞuðnÞj

n o; ð10Þ

where the operator bO represents ortho-normalization of the newset fuðnþ1Þ

j g, the superscript (n) counts the iteration, and bD is a con-vergence generating damping operator. The speed of convergencecan be reached by tuning bD, and here we employ

bD ¼ gbT þ �0

; ð11Þ

where bT is the operator of kinetic energy, and g and �0 is adjustableparameters.

For calculating the ionic geometry, one way is to follow the pathalong the steepest downhill gradient RI RI � dIrRI Etot , where dI

is an appropriate step size. In order to avoid that simple downhillmethod gets stuck in local minimum, the simulated annealingmethod [34,35] is used to optimize the ionic configuration.

The MD equations (6) and (7) are solved by using Verlet algo-rithm [43].

The dynamical equations for electrons and ions should be com-puted and solved simultaneously, and both electrons and ions arefully propagated in time. In fact, propagation schemes of electronicwave-function over a time interval Dt is,

ujðt þ DtÞ ¼ Texp �iZ tþDt

tdt0h½qðt0Þ�

� �ujðtÞ; ð12Þ

where T is the time ordering operator and h ¼ bT þ U½qðr; tÞ; t�.Eq. (12) is solved by the time-splitting method [44,45]. The es-

sence of this method lies in splitting the propagation between ki-netic and potential terms, and can be written as,

ujðr;tþDtÞ¼exp �iDt2

UKSðr;tþDtÞ� �

exp �iDtbT� exp �i

Dt2

UKSðr;tÞ� �

ujðr;tÞ; ð13Þ

where the local density qðt þ DtÞ is built after applying kinetic term,and then the new KS potential is obtained for the second half-po-tential step. The kinetic step is performed by Fourier transformingthe wave function into momentum space, applying the kineticpropagator, and transforming back to real space.

In order to describe electronic excitation, the basic relation

NðtÞ ¼Z

Vdrqðr; tÞ; ð14Þ

which associates the number of bound electrons NðtÞwith the time-dependent electronic density qðr; tÞ within the finite volume V (alarge enough computational box). Then, one could calculate the to-tal number of escaped electrons (Nesc) as,

NescðtÞ ¼ Nðt ¼ 0Þ � NðtÞ; ð15Þ

note that the absorbing boundary conditions (ABS) [23,35,46] is em-ployed, and its advantage is to avoid ejected electrons being re-flected back to the numerical box when they reach the boundary.

The probabilities PðkþÞðtÞ which denote for a certain time t theexcited molecule is in one of possible charge state k to which itcan ionize. First, we write out the bound state probability NjðtÞand continuum one NjðtÞ [46] for each KS single-particle orbital,

NjðtÞ ¼Z

Vdrnjðr; tÞ ¼

ZV

drjujðr; tÞj2; ðj ¼ 1; . . . ;NelÞ; ð16Þ

NjðtÞ ¼Z

Vdrnjðr; tÞ ¼

ZV

drjujðr; tÞj2; ðj ¼ 1; . . . ;NelÞ; ð17Þ

where V denotes the volume which is outside the computationalbox.

Deduced from NjðtÞ, the expressions of the approximate ionprobabilities [46] PðkþÞðtÞ read,

P0ðtÞ ¼ N1ðtÞ . . . NNðtÞ; ð18Þ

Pð1þÞðtÞ ¼XN

i¼1

N1ðtÞ . . . NiðtÞNNðtÞ; ð19Þ

Pð2þÞðtÞ ¼XN�1

i¼1

XN

j¼2;j>i

N1ðtÞ . . . NiðtÞ . . . NjðtÞ . . . NNðtÞ; ð20Þ

..

.

PððN�1ÞþÞðtÞ ¼XN

i¼1

N1ðtÞ . . . NnðtÞNNðtÞ; ð21Þ

PðNþÞðtÞ ¼ N1ðtÞ . . . NNðtÞ; ð22Þ

although the KS single-particle orbitals have no really physicalmeaning, one should consider the obtained PðkþÞðtÞ as a reasonableapproximation to the exact ion probabilities, which have been ap-plied in some calculations [23,46–49].

3. Computational details

In this work, the coordinate space grid method is utilized, and3-dimensional Cartesian coordinate system is represented by72�72�64 a3 grid points with a mesh size a = 0.412 a0. In Fig. 1,the pictorial representation of C2H4 case is displayed. The C–C bondis placed along the y axis with its midpoint in the origin. The C–Hbonds are along the y axis for C2H2, distribute symmetrically in they-z plane for C2H4, and stretch along y axis in spatial space for C2H6.For studying the collision dynamics in details, three different pro-jectile incident orientations are considered (see in Fig. 1): (I) theproton (b, �30 a0, 0) approaches the target parallel to the C–C bondin the x-y plane, (II) the proton (30 a0, 0, b) attacks the target per-pendicular to the C–C bond along the x axis in the x-z plane, (III) theproton (b, 0, 30 a0) moves to the target perpendicular to x-y planealong the z axis. The impact parameter b belongs to 0 � 5 a0 withstep Db = 0.1 a0 and 5.0 � 10.0 a0 with Db = 0.5 a0. The proton withcollision energy of 30 eV corresponds to an initial velocity of1.43 a0/fs. The target molecules are initially at rest. The evolutiontime of every trajectory is 40 fs with a time stepDt ¼ 6:05� 10�4 fs, and the longer time is required by the individ-ual events.

4. Results and discussion

4.1. Ground state structure

The dynamical collision process is based on molecular ground-state. Therefore, the first step in the simulation is to calculate theground state of C2Hm (m = 2,4,6) molecules: both the lowest en-ergy ionic geometry and the corresponding electronic configura-tion. The calculated equilibrium geometric structure parametersand occupied orbital energies with experimental results [52,53]are listed in Tables 1 and 2, respectively.

Acetylene (C2H2) belongs to D1h symmetry group. The outervalence region consists of five molecular orbitals: ð2rgÞ2

ð2ruÞ2ð3rgÞ2ð1puÞ4. The highest occupied molecular orbital(HOMO) is 1pu, which is important to electron ionization. The tri-ple bond (C�C) in acetylene is composed of one r (ð3rgÞ2) and twop (ð1puÞ4) bonds. The distance between two carbon atoms in

Fig. 1. The three initial configurations of H+ + C2H4 system are shown: (I) the proton (b, �30 a0, 0) approaches the target parallel to the C–C bond in the x-y plane, (II) theproton (30 a0, 0, b) attacks the target perpendicular to the C–C bond along the x axis in the x-z plane, (III) the proton (b, 0, 30 a0) moves to the target perpendicular to x-y planealong the z axis. h denotes scattering angle. The center of mass of targets is in the origin. Note that black, blue, and red ball represent carbon, hydrogen, and proton,respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2Molecular point group, molecular orbitals, occupation number of electrons. Orbitalenergies in eV from present calculations are compared with these from experiments[53].

Molecule and symmetry MO N Exp. [53] This work

C2H2 (D1h) 1pu 4 �11.49 �12.073rg 2 �16.7 �16.442ru 2 �18.7 �17.992rg 2 �23.5 �23.2

C2H4 (D2h) 1b3u 2 �10.68 �11.51b3g 2 �12.8 �12.393ag 2 �14.8 �14.41b2u 2 �16.0 �16.512b1u 2 �19.1 �18.52ag 2 �23.6 �23.3

C2H6 (D3d) 1eg 4 �12.0 �11.753a1g 2 �12.7 �12.521eu 4 �15.0 �15.082a2u 2 �20.4 �19.742a1g 2 �23.9 �23.18

Table 1Calculated and experimental equilibrium geometries for acetylene, ethylene andethane in Cartesian coordinate, distances (r) in Å, angles (\) in degree.

Molecule Label Exp. [52] This work

C2H2 rCC 1.203 1.164rCH 1.063 1.058

C2H4 rCC 1.339 1.292rCH 1.086 1.07\HCC 121.2 124.5\HCH 117.6 117.7

C2H6 rCC 1.536 1.461rCH 1.091 1.083\HCC 110.91 110.6\HCH 108.0 108.0

12 C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18

experiment is slightly larger than that in present calculation, andsimilar trend is observed in rCH . Acetylene is a well-known stan-dard linear structure with HCC angles 180�. Occupied orbital ener-gies of present method are larger than experimental values exceptthe HOMO.

Ethylene (C2H4) belongs to D2h point group with six outermolecular orbitals. The electronic configuration is: ð2agÞ2ð2b1uÞ2

ð1b2uÞ2ð3agÞ2ð1b3gÞ2ð1b3uÞ2. The HOMO is 1b3u, and it belongs tothe p bond. 3ag exists between C–C bond along y axis characterizedby the r bond. This special distribution results in ethylene a planarstructure with a C@C double bond. Bond lengths from experimentsare slightly larger than these from present method, but experimen-tal bond angles are smaller than our calculations. Occupied orbitalenergies compare well with experimental results.

Ethane (C2H6) belongs to D3d point group with fourteen valenceelectrons occupying seven molecular orbitals, namelyð2a1gÞ2ð2a2uÞ2ð1euÞ4ð3a1gÞ2ð1egÞ4. There is no p type molecularorbital. Compared with experiments, the calculated bond lengthsare somewhat shorter and bond angles are more or less the same.Like other two molecules, the calculated occupied orbital energiesare slightly larger than these from experiment.

In general, present results show a more compact geometricstructure than the experimental one, this is because LDA overesti-mates the binding energy of molecules [50,51]. However, the cal-culated bond lengths and bond angles are in reasonableagreement with experimental results within a 5% discrepancy.The occupied orbital energies are well reproduced by presentmethod. These make us confidence that our calculations giveacceptable results for the ground state of molecules.

4.2. Dynamical process

The collision process is characterized by chemical reactivechannels, especially in low energy regions. Here, the collisions ofH+ with C2Hm (m = 2,4,6) are studied at 30 eV by TDLDA-MDmethod. When the projectile moves far enough away from thetarget, the simulation is terminated, and then we can analyze theresidual products by means of charge state and the variation inbond length.

Theoretically predicted product channels include:

(a) Non-charge transfer scattering (NCT),

Hþ þ C2Hm ! Hþ þ C2Hm; ð23Þ

(b) Electron capture scattering (EC),

Hþ þ C2Hm ! Hq þ ½C2Hm�q0þ ðNescÞe�;

qþ q0 ¼ þð1þ NescÞ: ð24Þ

(c) Proton exchange process (E),

pþ C2Hm ! Hq þ ½C2Hm�1p�q0þ ðNescÞe�;

qþ q0 ¼ þð1þ NescÞ: ð25Þ

(d) Collision-induced dissociation (D),

Hþ þ C2Hm ! Hq þHq0 þ ½C2Hm�1�q00þ ðNescÞe�;

qþ q0 þ q00 ¼ þð1þ NescÞ: ð26Þ

Where q; q0 and q00 denote the charge state of ions, and Nesc

stands for the number of escaped electrons.Processes (a)–(d) show distinct reactive properties, and the key

results are summarized in Table 3. In present study, non-chargetransfer scattering occurs where the impact parameters are large

Table 3Reactive process for different initial conditions (target, projectile orientation, andimpact parameter): Non-charge transfer (NCT), electron capture (EC), protonexchange (E), dissociation (D). Impact parameter in a0.

Target Projectile orientation Impact parameter Reactive process

C2H2 0.0,0.5–6.0 ECI 0.1–0.2 E

0.3–0.4 D6.1–10.0 NCT

II 0.0–6.0 EC6.1–10.0 NCT

III 0.0–6.0 EC6.1–10.0 NCT

C2H4 I 0.0–7.0 EC7.1–10.0 NCT

II 0.0–7.5 EC7.6–10.0 NCT

III 0.0–6.5 EC6.6–10.0 NCT

C2H6 0.0–1.4 ECI 1.5–2.1 D

2.2–7.5 EC7.6–10.0 NCT

II 0.0–6.5 EC6.6–10.0 NCT

III 0.0–6.5 EC6.6–10.0 NCT

Fig. 2. The electronic density qðxÞ, integrated over y and z, is shown in a series oftime from 14.31 to 30.60 fs for H+ + C2H4 collision at orientation II from b ¼ 0:8 a0.The red ball denotes projectile proton. The logarithmic coordinate is used. Note thatat t = 28.79 fs, the blue dash line represents the boundary. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 3. The relative distances between atoms for the same event as Fig. 2. Theschematic structure of C2H4 is displayed in the inset.

C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18 13

enough (i.e., b > 7:5 a0 for C2H4 for orientation II), which corre-sponds to the ‘‘blast-through’’ of the unscattered proton in exper-iments [6]. In this case, the energy of proton rarely transfers to thetarget, thus the dynamics of electron and ion may be negligible. Forsmall impact parameters, there are diverse reaction processes,such as proton exchange process (E) and collision-induced dissoci-ation (D). Therefore, we aim at studying collisions at small impactparameters.

There are two mechanisms [54] in collision process for targetionization: direct electron emission (DEE) and electron capture(EC). Thus, we should make clear the nature of target ionizationat 30 eV. We show an example of H+ + C2H4 collision for orientationII at b ¼ 0:8 a0 (see in Fig. 1). In Fig. 2, the electronic density qðxÞ,which is integrated over y and z, is shown in a series of time from14.31 to 30.60 fs. The proton has entered the computational box at

14.31 fs. A bump appears around the proton at 16.12 fs, and itcould be considered as the electrons trapped by the Coulomb po-tential of the proton. With the time evolution (t = 17.93 fs), thebump moves following with the proton. The bump disappearswhen the proton passes into the charge density center area ofC2H4 molecule at 21.55 fs. This is because the number of trappedelectrons is much smaller than the total electrons in system, thebump could not be discerned. The bump appears once again at23.36 fs when the proton moves along x direction. The proton is al-ways capable of capturing some amounts of electrons (see 25.17 fsand 26.98 fs). As is expected, the trapped electrons are absorbed bydegrees when proton enters the boundary (at 28.79 fs). At 30.60 fs,the bump around the proton becomes smaller, and finally disap-pears. Eventually, the total number of captured electrons is 0.45,which could be calculated from Eq. (15) or integrated over x inFig. 2. It should be noted that the fractional electron number hasto be understood statistically [31,49,55,56], so the value 0.45 isthe average electron number captured by the proton. Furthermore,if we consider one hundred identical collisions, the total capturedelectron number 0.45 means that, on the average, 45 electrons willbe captured in all. As one can invariably find a small density peakaround the proton, we conclude that electron capture is the pri-mary mechanism of target ionization at 30 eV.

Besides the electron capture, one could observe an inter-ionvibration behavior. Fig. 3 displays the distance between atomsfor the same collision as Fig. 2. Within 20 fs, all bond lengths arein equilibrium state, and C1-proton distance decreases monotoni-cally as time increases. When the proton approaches closest tothe target at 20.8 fs, C–C and C–H bonds are motivated to be an in-ter-ion oscillation. Note that due to the symmetry, the curve of C1–H1 (H2) and C2-H3 (H4) is overlapping. Bond lengths are induced toincrease, this phenomenon is so-called ‘‘bond dilution’’ effect. Theeffect is first introduced by 10 eV H+ + H2 calculation [57], whichresults from the temporary withdrawal of electron density fromthe molecular orbitals. The bond dilution effect is also found inexperiments [1] and other calculations [58]. In the present study,similar dilution effect is also found in C2H2 and C2H6. After20.8 fs, it is observed as the distance between C1 and proton in-creases monotonically with time increases, which suggests thatthe proton is scattered away. The entire collision is an electron cap-ture scattering.

In order to visualize the collision process, the six snapshots ofH+ + C2H4 collision for orientation I at b ¼ 1:2 a0 are shown inFig. 4. From these snapshots, the collision is displayed vividly.

Fig. 4. Six snapshots of TDLDA-MD simulation at 30 eV for H+ + C2H4 collision for orientation I at b ¼ 1:2 a0. Blue, red, and green ball denote carbon, hydrogen, and proton,respectively. The electronic density scales down from red to blue. (For interpretation of the references to color in this figure legend, the reader is referred to the web version ofthis article.)

Fig. 5. (a) The trajectory in x-y plane is plotted for H+ + C2H2 collision at orientationI at b ¼ 0:1 a0. H2 hydrogen is scattered away, and the proton makes bond with C2carbon. This is a proton exchange process. Note that the initial and final structure isdisplayed in the upper-left and lower-left corner, respectively. (b) The timeevolution of the atomic distance is depicted. (c) The total number of escapedelectrons as a function of time is shown.

14 C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18

The proton disturbs the electronic density and captures some elec-trons temporarily when it enters the electron cloud area (see20.57 fs). Finally, the proton is scattered away with capturing someamounts of electrons (see 25.41 fs). Obviously, this is an electroncapture scattering process, and is in accordance with the mecha-nism of target ionization discussed above.

To get a profound understanding of reactive dynamics and frag-mentation, we should study some representative reactions. Theproton exchange process (E) for H+ + C2H2 collision for orientationI at b ¼ 0:1 a0, where the proton approaches parallel to the C–Cbond in the x-y plane, is interpreted as the trajectory of atoms, var-iation in bond length and total number of escaped electrons inFig. 5. From Fig. 5(a), the collision process is well reproduced bythe trajectory, and could be divided into three steps (i) the protonattacks the target which is initially a linear structure shown in theinset, (ii) when it approaches the closest to the target, the H2hydrogen obtains 22.4 eV energy from the proton, and then H2 isscattered away, (iii) the proton is decelerated and replaces theH2. The collision leads to a distorted structure of ½C2H2�q

0shown

in the inset, where q0 is 0.25. This process is again confirmed bythe distance between atoms in Fig. 5(b). The distance of C2–H2 in-creases monotonically after collision (t > 20 fs), while the distanceof C2-proton reduces firstly and afterward exhibits a weak oscilla-tion. This indicates there occurs C2–H2 bond breaking and C2-pro-ton bond making. Apart from the H2 atom, other atoms oscillateweakly around the equilibrium position. In Fig. 5(c), no ejectedelectron is found before 25 fs, and the rise in Nesc corresponds tothe electrons captured by H2 whose number is 0.25, as one canread from the final plateau. If consider the stability of the newstructure, one should evolve much longer time (i.e., 200 fs), whichis not the concern of present study.

Another intriguing process (c.f. (d)) is the collision-induced dis-sociation (D) for H+ + C2H6 collision for orientation I at b ¼ 1:6 a0.In Fig. 6(a), the ionic trajectory clearly shows the dissociation pro-cess. There’s no difference with proton exchange process exceptthat the proton is scattered away instead of replacing the H2hydrogen. This is attributed to the amount of energy transferredfrom the proton to H2 atom. In present study, the energy trans-ferred to atoms (except H2) is usually 0.1 � 2 eV, if the loss energyof proton is more than 95%, the proton is inclined to be deceleratedto replace the H2 hydrogen, and new structure is formed, like thecase proton exchange process discussed above. However, for the

collision-induced dissociation, the proton loses 12.3 eV energy(41%), and meanwhile the H2 atom obtains an energy of 11.0 eV.Thus, it’s impossible for proton to be decelerated and to be attained

Fig. 6. (a) The collision-induced dissociation for H+ + C2H6 collision in orientation Iat b ¼ 1:6 a0 is interpreted as trajectory in x-y plane. (b) The relevant atomicdistance is displayed as a function of time. (c) The time evolution of the totalnumber of escaped electrons is depicted.

Fig. 7. Scattering angle h as a function of impact parameter b for acetylene, ethyleneand ethane is shown.

C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18 15

by the target molecule. Proton is scattered away, and soon after H2hydrogen runs away. In Fig. 6(b), the distance of C1–H2 and C1-proton increases monotonically with time after 23 fs, which meansthe bond breaking indeed happens and no new bond making oc-curs. In collision, other bond lengths oscillate weakly except C2–H6 bond, because H6 is closer to the proton than other atoms. InFig. 6(c), the two rises in total number of escaped electrons corre-spond to the boundary absorption of electrons captured by the pro-ton and by H2, respectively. From that figure, total number ofescaped electrons is 1.48, the electrons captured by proton are0.73 (first peak at 35.61 fs), and electrons captured by H2 hydrogenis 0.75 (second peak at 45.52 fs).

4.3. Scattering pattern

The scattering angle h, which is used to describe the interactionbetween the projectile and target, is defined as the projectile de-flected from initial momentum (see Fig. 1). The expression canbe given by,

cosh ¼ PjPj ; ð27Þ

where P is the final momentum component of the outgoing particlealong the initial incident direction, and P is the final total momen-tum of the outgoing one.

First of all, we introduce three kinds of special scattering angles:(i) the maximum scattering angle (hmax) refers to the strongest

repulsive interaction, (ii) the glory angle (hg) appears at the impactparameter bg where hgðbÞjb¼bg

¼ 0�, which signifies the projectilehas no deflection, (iii) the rainbow angle (hr) occurs at the impactparameter br where dh

db jb¼br¼ 0, which represents the most obvious

attractive interaction.The scattering angle h as a function of the impact parameter b is

plotted in Fig. 7. For the orientation I (upper panel), the largestscattering angle occurs at bmax ¼ 0 a0 for the three molecules. Withincreasing the impact parameter, the scattering angle decreases.Note that for C2H2 at b ¼ 0:1–0:4 a0 and C2H6 at b ¼ 1:2–2:1 a0,there appears local peaks. This is because the proton exchange ordissociation reactions happen in this impact parameter range,and the energy transferred from the proton to the target moleculeresults in the redistribution of the momentum on proton. Aroundb ¼ 2:0 a0, the glory angle hg occurs for C2H2 and C2H4. The gloryangle is considered to be associated with the symmetry of the inci-dent orientation [57]. The general idea is that the glory angle goesto zero for the symmetric orientation, and it becomes the rainbowangle (non-zero) for the nonsymmetric orientation. However, thisidea seems to fail to be applicable to the present study, since theorientation I is not symmetric but the glory angle still appearsfor C2H2 and C2H4. This would be investigated by means of theforce which the proton is subjected to in the following. Forb > bg , the scattering angle increases gradually to the maximumcalled rainbow angle, where hrðC2H2Þ ¼ 9:5�; hrðC2H4Þ ¼ 12:2�,and hrðC2H6Þ ¼ 8:9�. For b > br , the scattering angle decreases asthe impact parameter increases, since the interaction between pro-ton and target is weaker as b increases. For the orientation II (mid-dle panel), the most striking difference with I is that the position ofhmax shifts from bmax ¼ 0 a0 to bmax ¼ 0:2 a0. In case II, at bmax ¼ 0 a0,the proton just suffers the force along the incident direction due tothe symmetry of molecule. While, at bmax ¼ 0:2 a0, the forces actedon proton are not only restricted to the incident direction, and itexhibits the strongest repulsive behavior with the maximum scat-tering angle 52�; 30:6� and 18:6� for C2H2, C2H4 and C2H6, respec-

Fig. 8. (a) The variation of scattering angle with time is shown at some impactparameters. (b) the force along z-direction that proton is subjected to is depicted,and each is translational upwards one unit for comparison. The data are fromH+ + C2H2 collision in orientation III.

16 C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18

tively. The structure of curve for the orientation III (lower panel) issimilar to that of curve for II, and the relevant angles with the cor-responding impact parameters are collected in Table 4.

In order to better understand the structure of the scattering an-gle in Fig. 7, it is instructive to examine the variation of scatteringangle with time and the corresponding force which the proton issubjected to. In Fig. 8, the time evolution of the scattering angleand the force along incident direction that the proton is subjectedto are shown as an example from H+ + C2H2 collision for orientationIII. The result shows that for b < bg ¼ 1:72 a0 the scattering angle inFig. 8(a) first reaches to the maximum under the Coulomb repul-sion (see Fig. 8(b)), and then decreases to an asymptotic value un-der the Coulomb attraction. We find that the narrow maximumpeak is determined by the strong repulsion. The peaks,(22.6,53.5) for b ¼ 0:2 a0, (22.0,26.0) for b ¼ 0:9 a0, (21.6,3.1) forb ¼ 1:72 a0, have a slightly shift to the left with increasing b, whichis consistent with the left shift of the maximum force in Fig. 8(b).The peaks disappear for b > bg , since the Coulomb repulsion be-comes weak as b increases, and the final scattering angle ishðb ¼ 2:9Þ ¼ 8:3� > hðb ¼ 2:3Þ ¼ 6:6� > hðb ¼ 4:0Þ ¼ 3:7�.

The scattering angle in Eq. (27) results from the integral of theforce in time, Dp ¼

RFðtÞdt, as shown in Fig. 8(b). From b ¼ 0:2 a0

to b ¼ 1:72 a0, the Coulomb repulsion reduces which leads to thedecrease of the integral, so the scattering angle decreases fromb ¼ 0:2 a0 to b ¼ 1:72 a0. At b ¼ 1:72 a0, the final angle is h ¼ 0�,this is called the glory angle. For clarifying this angle, we integratethe force over time, and find that the contribution of the repulsionand attraction to the change of the momentum is equal, but oppo-site in sign. In other words, the net effect of the force is zero. Forb > bg , the Coulomb attraction is dominant from the beginning,when the proton approaches close enough to the target, the repul-sion becomes prevailing. In this impact parameter range, the obvi-ous attractive interaction appears at b ¼ 2:9 a0 corresponding tothe rainbow angle. For b > br , the small force leads to the small an-gle scattering. It is worth noting that the glory angle never resultsfrom the force balance, but arises from the net effect of the Cou-lomb repulsion and attraction. For the case which no glory angleoccurs, it indicates that the net effect of the force is non-zero, suchas H+ + H [12] and H + He [59] collision. Overall, the glory anglecould be considered as the transition point for the Coulomb repul-sion and attraction, for b < bg , Coulomb repulsion is dominant andthe proton is repulsed by target, while for b > bg , Coulomb attrac-tion is leading and the proton is attracted by target. We concludethat the force which the projectile is subjected to may be moreappropriate than the symmetry of incident orientation to explainthe structure of the scattering angle.

In experiment, Aristov et al. [8] employed a crossed protonbeam to collide with C2H2 gas, and detected the rainbow peakextending from about 3 to 8�. The present calculated rainbow an-gles for C2H2 are 8.3� for II and III, and 9.5� for I. This disagreementwith experimental results lies in the orientation effect. In fact, the

Table 4The maximum repulsive angle, the glory angle and the rainbow angle with corresponding idegree.

Incident orientation (Impact parameter,angle)

I (bmax; hmax)(bg ; hg)(br ; hr)

II (bmax; hmax)(bg ; hg)(br ; hr)

III (bmax; hmax)(bg ; hg)(br ; hr)

rainbow angle from various incident orientations is usually differ-ent, e.g., I and II, because the proton received distinct interactionfrom the repulsion and attraction. The incident orientations are di-verse in experiment, and the rainbow angle is a result averaging onall possible orientations. Therefore, it is not surprising that resultsof present three orientations are not completely consistent withexperimental ones. Adding more orientations in simulation is ex-pected to narrow the gap with experiments. First calculation forC2H2 at 30 eV is performed by Malinovskaya et al. [11]. The resultsof the scattering angle are in good agreement with our calculations,for example, the maximum angle h ¼ 56� at b ¼ 0:3 a0, which cor-responds to present orientation III h ¼ 52:5� at b ¼ 0:2 a0. The goodagreement is attributed to the classical treatment for nuclei andquantum mechanic description for electrons in our TDLDA-MDmethod, which is similar to the treatment in minimal END ap-proach used by Malinovskaya et al. A more exhaustive study forC2H2 is carried out by Morales et al. [13]. Their calculated rainbowangles were 13.8 and 3.8� corresponding to present orientation I

mpact parameter b for C2H2, C2H4, and C2H6. The impact parameter in a0 and angles in

C2H2 C2H4 C2H6

(0,180) (0,80) (0,163)(2.1,0) (2.0,0) –(3.0,9.5) (3.2,12.2) (3.8,8.9)(0.2,52) (0.3,30.6) (0.5,18.6)(1.7,0) (1.9,0) –(2.9,8.3) (2.7,5.4) (3.8,5.1)(0.2,52.5) (0.3,35.2) (0.4,19.0)(1.72,0) (2.02,0) (1.78,0)(2.9,8.3) (3.3,5.4) (3.4,5.4)

C.-Z. Gao et al. / Chemical Physics 410 (2013) 9–18 17

with hr ¼ 9:5� and II with hr ¼ 8:3�, respectively. Theoretical meth-od (e.g., the description of wave packet for nuclei, the employmentof coherent-state dynamics) may result in the discrepancy with ourresults. Although, no clear evidence for the rainbow angle wasshown in 30 eV proton colliding with C2H4 gas experiments [9]due to a poor signal-to-noise ratio, we could give the rainbow an-gle 12.2 and 5.4� for I and II (III), respectively. In addition, there isno available experiment and theoretical calculation which are in-volved in H+ + C2H6 at 30 eV.

4.4. Ionization cross section

In order to describe the probability of ionization, we calculatethe ionization cross section (ICS), which is defined as

rICS ¼ 2pZ bmax

0bPðbÞdb; ð28Þ

where PðbÞ is the ionization probability for the target and the upperlimit bmax represents the impact parameter where extremely weakinteraction occurs between projectile and target.

Up to now, the data of experimental ionization cross sectionsare absent for C2Hm (m = 2,4,6) scattering by 30 eV proton. Forthe sake of comparison, the data of 30 eV electron [60] and450 eV proton [16,61] collision with the studied molecules are col-lected. Besides, we also performed the calculations at the energy of450 eV. The results are shown in Fig. 9. The ionization cross sec-tions of 30 eV calculations increase as the order of C2H2, C2H4

and C2H6 except for orientation I. A possible explanation may be gi-ven by the C–C bond effect. For orientation II and III (see Fig. 1), theproton moves perpendicular to the C–C bond, ionization in C–Cbond is dominant. There are a triple, double, and single C–C bondfor C2H2, C2H4 and C2H6, respectively. For the three molecules,the electronic density distribution concentrated in C–C bond in-creases with bond order increasing [62]. The more electrons in-volve in chemical bond, the stronger binding of the molecule. Forthis reason, the order of rICSðC2H2Þ < rICSðC2H4Þ < rICSðC2H6Þ is ob-served for orientation II and III. However, for orientation I, the pro-ton approaches the target parallel to the C–C bond, the protonenters C–H bond region firstly, and then is scattered away beforereaching the C–C bond region. The ionization in C–C bond couldnot play a key role for I, which is different from II and III. The sim-ilar order for the electron-capture cross section [20] was found forC2H2, C2H4 and C2H6 in the energy range from 0.2 to 3 keV. It wassupported by the fact that the smaller geometrical size leads to re-duce the effective scattering region, which resulted from the nat-ure of C–C bonding.

Fig. 9. Comparison of the ionization cross section for present calculations, 30 eVelectron [60] and 450 eV proton [16,61]. Note that for 30 eV electron case a factor ofthree is multiplied.

Compared with experimental results of 450 eV proton collisionwith C2Hm, the average results of 450 eV calculations for C2H4 andC2H6 are much smaller. In experiment, diverse reactions (or orien-tations) will happen in collision cell, even the secondary collisionoccurs, and the results obtained from the detector are reasonableto be larger than present three orientation simulations. Note that450 eV calculated results for C2H2 are slightly larger than 450 eVexperimental ones. For 30 eV, the calculated cross sections ofC2H2 approximate to the experimental results (450 eV), but thecross sections for C2H4 and C2H6 are much smaller than experi-mental ones. The reason may be that the more binding in C2H2

makes it nearly unrelated with energy in the certain energy range,and the more flexible structure of C2H4 (planar) and C2H6 (spatial)results in the cross section dependence on the impact energy. Forclear comparison, we multiply a factor of three in ionization crosssection in the case of 30 eV electron collision with C2Hm [60]. Themass of electron is much smaller than that of proton, for the samecollision energy (30 eV), the velocity of electron is much largerthan that of proton. Swift electron results in a shorter time interac-tion with the target. As a result, the ionization cross section of30 eV electron is roughly an order of magnitude lower than thatof present calculation results.

To sum up, in present simulation, the ionization cross sectionwhich increases with increasing C–C bond length is only observedin certain orientations (e.g. II and III), this results from the C–Cbond effect. In experiments, although similar dependence withC–C bond length occurs, it is associated with the geometrical sizeof the molecules.

5. Conclusions

In the framework of time-dependent local density approxima-tion, applied to valence electrons, and coupled non-adiabaticallyto molecular dynamics of ions, we have investigated proton colli-sions with C2Hm (m = 2,4,6) at 30 eV. The optimized ground-statestructures by present method are in good agreement with experi-mental results with a 5% discrepancy. The simulations based onthree orientations obtain various reactions. By means of electronicdensity evolution, we find that the electron capture is the primarymechanism of target ionization at 30 eV. Ionic trajectory diagramsuggests that the bond dilution effect is prevalent in Hþ + C2Hm col-lisions. The complex structure of the scattering angle results fromthe force which the projectile is subjected to. The glory angle neverresults from the force balance, but arises from the net effect of theCoulomb repulsion and attraction. Therefore, it could be consid-ered as the transition point for Coulomb repulsion and attraction.Calculated ionization cross sections increase with C–C bond lengthincreasing for orientation II and III, and the difference with orien-tation I is discussed. In addition, we also study the proton exchangeprocess and collision-induced dissociation reactions in details forbetter understanding the reactive dynamics and fragmentation.

Acknowledgments

This work was supported by the National Natural Science Foun-dation of China under Grants Nos. 11025524 and 11161130520,and the National Basic Research Program of China under GrantNo. 2010CB832903.

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