A screened independent atom model for the description of ion collisions from atomic and molecular clusters
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r A screened independent atom model for the description of ion collisions from atomicand molecular clusters
Hans Jürgen Lüdde ∗ Institut für Theoretische Physik, Goethe-Universität, D-60438 Frankfurt, Germany
Marko Horbatsch † and Tom Kirchner ‡ Department of Physics and Astronomy, York University, Toronto, Ontario M3J 1P3, Canada (Dated: April 10, 2018)We apply a recently introduced model for an independent-atom-like calculation of ion-impact elec-tron transfer and ionization cross sections to proton collisions from water, neon, and carbon clusters.The model is based on a geometrical interpretation of the cluster cross section as an effective areacomposed of overlapping circular disks that are representative of the atomic contributions. The lat-ter are calculated using a time-dependent density-functional-theory-based single-particle descriptionwith accurate exchange-only ground-state potentials. We find that the net capture and ionizationcross sections in p-X n collisions are proportional to n α with / ≤ α ≤ . For capture from waterclusters at 100 keV impact energy α is close to one, which is substantially different from the value α = 2 / predicted by a previous theoretical work based on the simplest-level electron nuclear dy-namics method. For ionization at 100 keV and for capture at lower energies we find smaller α valuesthan for capture at 100 keV. This can be understood by considering the magnitude of the atomiccross sections and the resulting overlaps of the circular disks that make up the cluster cross sectionin our model. Results for neon and carbon clusters confirm these trends. Simple parametrizationsare found which fit the cross sections remarkably well and suggest that they depend on the relevantbond lengths. PACS numbers: 34.10.+x, 34.50.Gb, 34.70.+e, 36.40.-c
I. INTRODUCTION
Ionization in charged-particle matter interactions is aprocess of relevance to both fundamental and more ap-plied research areas, but is difficult to describe in quan-titative terms if the objects under study are sufficientlycomplex. On the experimental side, challenges associatedwith preparation and control of the projectile and targetspecies as well as the detection of multiple reaction prod-ucts, possibly in coincidence, have to be addressed. Onthe theoretical side, the challenge resides in the descrip-tion of an interacting few- or many-body system far awayfrom its ground state, a problem that is straightforwardto formulate for (nonrelativistic) Coulomb systems, buthard to solve even with present-day supercomputers [1].Time-dependent density functional theory (TDDFT)was conceived by Erich Runge and Hardy Gross [2]with this problem in mind and the objective to developa time-dependent description of scattering experimentsthat would circumvent the calculation of the many-bodywave function [3]. However, applications of the time-dependent Kohn-Sham (TDKS) scheme to collision prob-lems have remained relatively sparse. This is differentfrom the situation for the somewhat related problem ofionization in strong laser fields (see, e.g., the books [4, 5]and references therein). There are no obvious symmetries ∗ [email protected] † [email protected] ‡ [email protected] in collisional ionization and furthermore for positivelycharged projectile ions direct target ionization competeswith electron transfer to bound projectile states. In theframework of the semiclassical approximation, in whichthe projectile is assumed to move on a classical (straight-line) path, projectile-centered states must be augmentedby so-called electron translation factors (ETFs) to ac-count for the relative motion and preserve Galilean in-variance.These issues have been analyzed in some detail forthe two-center ion-atom case, often for prototypical one-electron problems such as the proton-hydrogen collisionsystem [6, 7]. Collisions involving helium are perhapsthe next best studied systems, but the vast majorityof calculations have been based on simplified descrip-tions since explicit solutions of the two-electron time-dependent Schrödinger equation are exceedingly difficultand computationally costly (see, e.g., reference [8] andreferences therein). For atoms with more than two elec-trons, let alone for the atomic and molecular clusters ad-dressed in the present work, they are out of reach.A popular framework for a simplified treatment ofmany-electron collision systems is the independent elec-tron model (IEM). However, the most sophisticatedIEM variant, the time-dependent Hartree-Fock (TDHF)scheme, has been applied to only a handful of cases.This is due to difficulties associated with the nonlocalexchange interaction and, more fundamentally, with thenonlinearity of the TDHF equations, which manifests it-self in the occurrence of fluctuating transition probabil-ities when analyzing the TDHF wave function with re-spect to eigenstates of a static asymptotic Hamiltonian[9–11]. The latter problem is known as the TDHF crosschannel correlation or projection problem and has alsobeen discussed in the context of nuclear reactions [12, 13].About 20 years ago we started to look into atomic col-lisions involving many-electron targets such as neon andargon atoms using a TDDFT-inspired single-particle de-scription based on atomic ground-state DFT potentials[14, 15]. The time dependence of these potentials overthe course of the collision was neglected and the projec-tion problem avoided. Orbital propagation was achievedusing the basis generator method (BGM), a basis set ex-pansion technique built on atomic orbitals and dynami-cally adapted pseudostates [16]. The BGM and the morerecent two-center version TC-BGM [17] proved capableof accounting for target excitation, electron transfer, andionization channels in ion-atom collision problems overa wide range of collision energies (see, e.g., references[18, 19] and references therein).Subsequently, we amended the no-response approxima-tion of frozen ground-state potentials by simple responsemodels which did not increase the computational bur-den significantly and allowed us to analyze the projec-tion problem and study dynamical potential effects ina semi-quantitative way [20]. We found that responsehas the tendency to lower probabilities for total electronremoval (the sum of electron transfer to the projectileand direct target ionization) at low to intermediate col-lision energies. As soon as the projectile speed is sig-nificantly larger than the average orbital velocity of agiven target electron, that electron becomes insensitiveto time-dependent changes in the interelectronic poten-tial simply because ionization happens too rapidly. Asa consequence, response model cross sections approachno-response results towards high collision energies. Also,response effects turned out ot be generally small for pro-ton impact on first-row elements. This is so becausemultiple-electron removal is a weak process in these colli-sions and our model is designed in such a way that dynam-ical screening becomes appreciable only after one electronis removed on average. This choice was motivated by thesuccess of so-called frozen TDHF calculations in studiesconcerned with the (photo-) ionization of a single electron[21].Collisions of projectile ions (with or without projectileelectrons) from small molecules, such as H O [22, 23], or CH [24] were treated within a framework where sim-ple self-consistent field wave functions were projectedonto atomic orbitals calculated in DFT. These orbitalswere then evolved using the TC-BGM and transitionamplitudes were calculated on the basis of interpretingKohn-Sham determinants. For biomolecules and clustersthis approach is not suitable. Direct implementations ofTDDFT equations for ion collisions with small moleculeswere reported by other groups [25, 26]. For larger systemsa few calculations based on first-principles approacheshave been carried out [27–29], but most of the availablecross section results for electron transfer and target ion- ization have been obtained using simplified and classi-cal models (see, e.g., references [30–34] and referencestherein).Given this situation we recently introduced an indepen-dent-atom-model (IAM) description to deal with complexmulticenter collision systems on the basis of atomic no-response TC-BGM calculations [35]. The simplest real-ization of the IAM is Bragg’s additivity rule (IAM-AR)according to which a net cross section for a complex tar-get such as a molecule or cluster is obtained from addingup atomic net cross sections for all atoms that make upthe system. Our model goes beyond the IAM-AR by as-sociating the atomic cross sections in the AR sum withweight factors. The latter are determined from a geo-metrical interpretation of a cross section as an effectivearea using the following procedure. First, each atom ina given target is surrounded by a sphere of a radius rep-resentative of the atomic cross section for either net elec-tron transfer to the projectile or to the continuum (forbrevity referred to as net capture and net ionization inthe following). Secondly, the resulting three-dimensionalstructure of overlapping spheres is projected on a planewhich is perpendicular to the projectile beam axis. In thelast step, the effective area in that plane is taken as thecross section for net capture or net ionization of the sys-tem in that particular geometry. An orientation averageis calculated to make contact with experimental data forrandomly oriented molecules or clusters. We refer to themodel as IAM-PCM, since the effective cross-sectionalarea, and by extension the weight factors attached tothe atomic contributions in a given orientation, are cal-culated using a pixel counting method (PCM).The IAM-PCM was successfully applied to a number ofcollision systems involving proton projectiles and molec-ular targets such as CO, H O, and C H N O (uracil).It was demonstrated that IAM-AR cross sections fornet capture and ionization are reduced substantially andagreement with experimental data is improved in regionsin which the atomic cross section contributions are largeand the overlap effects significant [35].In this work, we use the IAM-PCM to calculate netcapture and ionization cross sections in proton collisionswith water, neon, and carbon clusters comprising sys-tems with hydrogen bonds, van der Waals bonds, andcovalent bonds. We begin in section II with a discussionof the atomic ingredients, i.e., the solution of the (approx-imate) ion-atom TDKS equations using the TC-BGM(section II A), the calculation of cross sections for netcapture and ionization (section II B), and results for thep-H, p-C, p-O, and p-Ne systems (section II C). This isfollowed by a description of the IAM-PCM in section III.Results for the proton-cluster collision systems are pre-sented in section IV and the paper ends with a few con-cluding remarks in section V. Atomic units, characterizedby ~ = m e = e = 4 πǫ = 1 , are used unless otherwisestated. II. THE BASIS GENERATOR METHOD FORION-ATOM COLLISIONSA. Solution of the single-particle equations
The TDKS scheme was anticipated by Runge andGross in their original 1984 work [2] and put on firmgrounds by van Leeuwen in 1999 [36]. For a thoroughdiscussion of the foundational theorems of TDDFT werefer the reader to the books [4, 5] and references therein.For an N -electron ion-atom collision problem in thesemiclassical approximation the TDKS equations can bewritten in the form i∂ t ψ i ( r , t ) = (cid:18) − ∇ − Z T r T − Z P r P + v ee [ n ]( r , t ) (cid:19) ψ i ( r , t ) ,i = 1 , . . . , N, (1)where r , r T , and r P denote the electronic position vec-tor with respect to the center of mass (CM), the target,and the projectile, respectively, and Z T and Z P are thecharge numbers of the nuclei. We assume the projectileto follow a straight-line path R ( t ) = r T − r P = ( b, , vt ) characterized by the impact parameter b and the constantspeed v .The effective electron-electron potential v ee in equa-tion (1) is a functional of the density n and can besplit into the usual Hartree, exchange, and correlationcontributions. In the no-response approximation for theproblem of a bare projectile ion impinging on an atomictarget, v ee is given by a (spherically-symmetric) ground-state DFT potential. More specifically, we use Hartree-exchange potentials obtained from the exchange-only ver-sion of the optimized potential method (OPM) [37, 38] forthe neutral ( N = Z T ) carbon, oxygen, and neon atomsof interest in the present study and neglect correlationeffects. An important feature of the OPM potentials istheir complete cancellation of self-interaction contribu-tions contained in the Hartree potential such that thecorrect asymptotic behaviour v OPM ee ( r T ) r T →∞ → N − r T (2)is ensured. This is crucial for a proper description oftarget electron removal [14, 15].The approximate TDKS equations (1) with the ground-state potential v OPM ee are propagated using the TC-BGM,which, like any basis-expansion technique, assumes thatthe solutions can be represented in terms of a finite setof states. The TC-BGM set includes N T atomic stateson the target center, i.e., bound eigenstates of ˆ h T = − ∇ − Z T r T + v OPM ee ( r T ) (3) ≡ − ∇ + V T , (4)a set of N P eigenstates of the projectile Hamiltonian ˆ h P = − ∇ − Z P r P ≡ − ∇ + V P (5) to describe capture, and a set of pseudostates which over-lap with the continuum. It is the specific choice of the lat-ter that distinguishes the TC-BGM from other coupled-channel methods for atomic collisions. The guiding ideais to span a subspace of Hilbert space which dynamicallyadapts to the time evolution of the system in such a waythat couplings to the complementary space are small andcan be neglected without introducing significant errors.The benefit of using time-dependent basis states is thatone can hope for reasonable convergence without havingto include a very large number of states.It was shown on theoretical grounds [16] and demon-strated in a number of practical applications (see, e.g.,references [18, 19] and references therein) that good con-vergence can be achieved by using basis functions of theform χ Jj ( r , t ) = [ W P ( r P )] J φ j ( r , t ) (6) W P ( r P ) = 1 r P (cid:0) − e − r P (cid:1) (7) φ j ( r , t ) = (cid:26) φ j ( r T ) exp( i v T · r ) if j ≤ N T φ j ( r P ) exp( i v P · r ) if N T < j ≤ N T + N P , (8)where v T and v P denote the (constant) velocities of thetarget and projectile with respect to the CM, and thefunctions φ j on the right hand side of equation (8) satisfystationary eigenvalue equations for ˆ h T ( j ≤ N T ) and ˆ h P ( j > N T ) in the (moving) target and projectile referenceframes, respectively. The phase factors are ETFs whichensure Galilean invariance. Acting with spatial and timederivative operators on them leads to the modified eigen-value equations (ˆ h T,P − i∂ t ) | φ j i = g j | φ j i (9)with g j = ε j + v T,P (10)and atomic energy eigenvalues ε j for the target and pro-jectile orbitals φ j ( r , t ) = h r | φ j i in the CM referenceframe.Expanding the TDKS orbitals in this non-orthogonal,time-dependent TC-BGM basis ψ i ( r , t ) = X j,J c ij,J ( t ) χ Jj ( r , t ) (11)turns the single-particle equations into a set of coupledequations for the expansion coefficients i X j,J S KJkj ( t ) ˙ c ij,J ( t ) = X j,J M KJkj ( t ) c ij,J ( t ) (12)with overlap S KJkj = h kK | jJ i (13)and interaction M KJkj = h kK | ˆ h T + V P − i∂ t | jJ i (14)matrix elements. In equations (13) and (14) we have usedthe short-hand notation | jJ i = W JP | j i (15)for the BGM basis states, i.e., the functions χ Jj ( r , t ) = h r | jJ i .The calculation of the matrix elements proceeds in sev-eral steps. First, the interaction matrix elements (14) arerewritten by using similar arguments as in references [39]and [23] to arrive at M KJkj = h kK | KJ (cid:18) ∇ W P W P (cid:19) + KK + J V ¯ j + JK + J V ¯ k | jJ i− JK + J i∂ t h kK | jJ i + Kg j + Jg k K + J h kK | jJ i , (16)where for j ≤ N T we set V ¯ j = V P and V j = V T , whilefor j > N T we set V ¯ j = V T and V j = V P . In contrast toequation (14) the equivalent form (16) does not involvederivatives of basis functions.In a second step, the set of TC-BGM pseudostates {| jJ i , J > } is orthogonalized to the generating two-center basis {| j i} to separate the ionized and boundparts of the TDKS orbitals. Finally, an LU decomposi-tion is carried out to turn the basis into a completely or-thonomalized set of states and the coupled-channel equa-tions (12) into the form i ˙ d i = ˜ M d i , (17)in which ˜ M is the transformed interaction matrix and d i the transformed expansion coefficient vector of the i -thTDKS orbital. The set of matrix equations (17) is solvedusing standard methods [40]. B. Calculation of net cross sections
The atomic contributions used in the IAM-AR andIAM-PCM are cross sections for net capture and net ion-ization. They are calculated, exploiting cylindrical sym-metry, via σ net x = 2 π Z ∞ bP net x ( b ) db, (18)where x denotes capture ( x = cap) or ionization ( x = ion) and P net x is the corresponding (impact-parameter-dependent) net electron number. Provided that at anasymptotic time t f after the collision the one-particledensity n can be split into non-overlapping contributionsassociated with electrons captured by the projectile ( P ),promoted to the continuum ( C ) and retained by the tar-get ( T ), one can write for the total electron number [48] N = Z P n ( r , t f ) d r + Z C n ( r , t f ) d r + Z T n ( r , t f ) d r, (19) where the integrals are over (non-overlapping) P , C , and T subspaces, and identify P net cap = Z P n ( r , t f ) d r, (20) P net ion = Z C n ( r , t f ) d r. (21)Equations (20) and (21) show that net electron numbers,and as a consequence of equation (18) net cross sectionsas well, are explicit density functionals. This makes themconvenient observables in TDDFT-based studies: Theonly fundamental approximation involved in a TDDFTnet cross section calculation is the choice made for theTDKS potential. If one wishes to calculate a cross sectionthat corresponds to a coincident measurement of singleor multiple capture and ionization, one faces the addi-tional challenge that the exact density dependence of theobservables is not known and additional approximationsare required [8, 48].We conclude this section by noting that instead of in-tegrating the electron density over subspaces of R weuse the TC-BGM basis representation to calculate netcapture and ionization directly from the asymptotic ex-pansion coeffients of equation (17) P net cap = N X i =1 P X k | d ik ( t f ) | , (22) P net ion = N − P net cap − N X i =1 T X k | d ik ( t f ) | . (23)If the sums over k include all appreciably populatedbound projectile ( P ) and target ( T ) states and providedthe above-mentioned condition of non-overlapping P , T ,and C components is fulfilled, the channel and real-spacerepresentations of P net cap and P net ion are equivalent. C. Sample results
In Figure 1 we show no-response TC-BGM net ioniza-tion and net capture cross section results for the proton-atom collision systems of interest in this work: p-H, p-C, p-O, and p-Ne. The p-H system in particular hasbeen studied extensively over the years and many setsof theoretical results have been reported. Figure 1 doesnot provide comparisons with those previous calculations,since a review of the current status of atomic cross sec-tion calculations is outside the scope of this article. Thepurpose of Figure 1 is limited to an illustration of thelevel of accuracy and the asymptotic behaviour obtainedin the (no-response) TC-BGM framework. To this end,experimental data for p-H and p-Ne, the only systems forwhich direct measurements of net ionization and capturecross sections are available, and fits of the asymptoticBethe-Born ionization cross section formula [49, 50] σ Bethe = A ln E + BE , (24)
HCONe
Bethe = (A ln E + B)/Ep+H: Shah81p+H: Shah87p+Ne: Rudd85
HCONep+H: McClure66p+H: Bayfield69p+H: Wittkower66p+Ne: Rudd83
FIG. 1. Total cross sections for net ionization (left panel) and net capture (right panel) in p-H, p-C, p-O, and p-Ne collisionsas functions of impact energy. Experiments: Shah81 [41], Shah87 [42], Rudd85 [43], McClure66 [44], Bayfield69 [45], Wit-tkower66 [46], Rudd83 [47]. For the p-H system the reported experimental uncertainties are below 10% and the error bars aresmaller than the size of the symbols. in which E is the projectile energy and A and B aretreated as fit parameters, are included. For a broaderdiscussion of p-H cross section results we refer the readerto the recent work [51]. The p-O and p-Ne systems werestudied in our previous papers [52] and [53], respectively.For the various atomic targets we included in the basisall atomic orbitals of the KLM N shells of both projectileand target plus sets of 73–111 pseudostates constructedaccording to equations (6) and (7). The Bethe-Born crosssections were obtained by fitting the parameters A and B of equation (24) to the current TC-BGM results at highenergies using the Fano representation, in which Eσ net ion is plotted against ln( E ) (using appropriate units) [50].For the p-H system the fitted parameters are consistentwith the values that can be deduced from Bethe’s originalwork [49].Obviously, the agreement of the TC-BGM results withthe experimental data and the Bethe-Born prediction athigh E is very good. It is interesting to see that thenet ionization cross sections for p-C and p-O do not onlyagree in shape, but also in magnitude in this region. Forthe oxygen case we found excellent agreement with exper-imental data for equivelocity electron impact correspond-ing to E ≥ keV [52], which confirms the validityof first-order perturbation theory. Furthermore, within10% accuracy the high-energy p-C and p-O results arefour times larger than the p-H ionization cross sectionand, as found in additional calculations (not included inFigure 1), they also coincide (within 10%) with resultsfor p-N collisions. This implies that for a large classof biomolecules consisting of H, C, N, and O atoms the IAM-AR will predict very simple scaling relations. Wefound, somewhat surprisingly, that the same relationshold within the IAM-PCM described in the next section.An analysis of these scaling relations will be presentedin a future publication focusing on ion-biomolecule colli-sions. III. A PIXEL COUNTING METHOD FORSCREENED INDEPENDENT ATOM MODELCALCULATIONS
The IAM-PCM is best explained by way of an example.Consider net capture in p-H O collisions at relatively lowimpact energy E . The ingredients of the IAM are the netcapture cross sections for p-H and p-O collisions. Thesecross sections are assigned radii according to r j = [ σ net capj /π ] / , (25)where j = 1 , , enumerates the atoms. We place the L = 3 atomic nuclei at their equilibrium positions inground-state H O and surround each of them by a sphereof radius r j . The impinging projectile then encounters anobject made up of overlapping spheres and an effectivecross-sectional area is determined by projecting that ob-ject on the plane that is perpendicular to the projectilebeam.Figure 2a displays the overlapping spheres for captureat E = 10 keV. It is important to keep in mind that theobject shown is not a model of the water molecule, but athree-dimensional representation of net capture. A pro-jectile approaching the molecule from a given direction -303 X [ A ] -3 0 3 Y [ A ] -303 Z [ A ] (a) (b) FIG. 2. Net capture in p-H O collisions at E = 10 keV: (a)three-dimensional image and (b) projection on the x - y plane.The radii of the spheres and circular disks are determinedacoording to equation (25). -303 X [ A ] -3 0 3 Y [ A ] -303 Z [ A ] (a) (b) FIG. 3. Net capture in p-H O collisions at E = 100 keV: (a)three-dimensional image and (b) projection on the x - y plane.The radii of the spheres and circular disks are determinedacoording to equation (25). will ’see’ the projected cross-sectional area as in classicalscattering from superimposed hard spheres. Figure 2bshows this projection for projectile impact along the z -direction of the coordinate system used. The effectivearea, i.e., the molecular net cross section, can be repre-sented as a weighted sum of atomic cross sections σ net x mol ( E, α, β, γ ) = L X j =1 s xj ( E, α, β, γ ) σ net xj ( E ) (26)with weight factors ≤ s xj ≤ and the Euler angles α, β, γ which characterize the orientation of the molecule.The notation used in equation (26) shows the dependen-cies of the various quantities and indicates that we usethe prescription for both capture and ionization. Wenote in passing that the screening corrected additivityrule (SCAR) for electron-molecule scattering is basedon similar ideas and uses a similar equation, but withorientation-independent weight factors that are obtainedfrom a heuristic recurrence relation [54].Figure 3 shows net capture at the higher energy E =100 keV. At this energy, the atomic net capture cross sec-tions are small and the spheres do not overlap. The pro-jection on the x - y plane is simply the sum of the atomic cross sections, i.e., the weight factors are equal to oneand the IAM-AR result is recovered.In practice, we calculate the cross-sectional area ofoverlapping circular disks in the following way. The x - y plane is represented by a (pixel) matrix of dimension × with square elements (pixels) whose size isdetermined by choosing a resolution (we typically use . × . Å pixels). The circular atomic cross sectiondisks are ’colored’ according to their atomic identifier j and the pixel matrix is filled with the identifiers corre-sponding to the atomic cross sections from backgroundto foreground as seen by the impinging projectile. Foreach j the area that is exposed to the projectile is de-termined by counting the visible pixels of that color andthe screening coefficients in equation (26) are obtainedby normalizing the area to the total (unscreened) atomiccross section s xj ( E, α, β, γ ) = σ vis xj ( E, α, β, γ ) σ net xj ( E ) . (27)It was noted in reference [35] that the procedure canbe criticized for overemphasizing the contribution of anatom located at the front, while possibly completely ne-glecting the contribution of an atom at the back of themolecule (cf. Figure 2b). However, as long as one isinterested in net cross sections only, this is a minor con-cern, since there is no need to attach physical significanceto the individual screening coefficients and partial crosssection areas. One may view them as purely auxiliaryquantities to calculate the total projected area accordingto equation (26). Obviously, the area can be decomposedin different, but equivalent ways.To make contact with experimental data for randomlyoriented molecules, IAM-PCM calculations are carriedout for a number of orientations and are averaged overthe Euler angles. For all results shown in this work weexploit the fact that a rotation about the z -axis does notchange the size of the visible area and vary only two outof three Euler angles on fine grids for a total of 40 × O system inFigure 4. We compare IAM-PCM net capture and ion-ization cross sections with experimental data and withprevious TC-BGM calculations obtained in the molecu-lar framework mentioned in the Introduction, in whichsimple self-consistent field wave functions were projectedonto atomic orbitals calculated in DFT [23].For net ionization (Figure 4a) the IAM-PCM outper-forms the molecular TC-BGM: The cross section maxi-mum appears at the correct position and the agreementwith the measurements of Rudd and coworkers [55] isvery good, except at energies below 20 keV where thesedata are underestimated. By contrast, the molecular TC-BGM cross section curve peaks at too low an energyand underestimates the experimental data above 100 keV.The IAM-AR results show the same overall behaviouras IAM-PCM, except that the cross section values are
IAM-ARIAM-PCMMurakami12Rudd85Bolorizadeh86 IAM-ARIAM-PCMMurakami12Rudd85Toburen68
FIG. 4. Total cross sections for (a) net ionization and (b) net capture in p-H O collisions as functions of impact energy.Murakami12 refers to the molecular TC-BGM calculation of reference [23]. Experiments: Rudd85 [55], Bolorizadeh86 [56],Toburen68 [57]. somewhat larger around the maximum, in seemingly ex-cellent agreement with the measurements of Bolorizadehand Rudd [56]. However, these cross section data haverelatively large error bars. They were obtained fromintegrating absolute differential measurements and aredeemed less accurate than those of reference [55], whichwere obtained from a more direct parallel-plate-capacitormethod. Overall, the comparison indicates that the inclu-sion of geometric screening corrections via the IAM-PCMrepresents an improvement.This becomes more obvious in the case of net cap-ture. The linear plot in the inset of Figure 4b showsthat the simple IAM-AR results in a strong overestima-tion towards low energies where the atomic capture crosssections are large (cf. Figure 1). The overlap effect issignificant (cf. Figure 2) and leads to a substantial re-duction of the molecular cross section. Still, the IAM-PCM results overestimate the experimental data at en-ergies below 30 keV. It was argued in reference [35] thatthis overestimation is a consequence of the strong (reso-nant) p-H contributions in the IAM, which are unphys-ical given that there is no resonant capture channel inthe p-H O collision system. The comparison with themolecular TC-BGM calculations confirms this. Down tothe lowest energy of 20 keV for which these calculationswere carried out they are in excellent agreement with theexperimental data.The situation is different at energies above 100 keVwhere the overlap effect in the IAM is negligible (cf. Fig-ure 3). IAM-PCM and IAM-AR results coincide and arein excellent agreement with the measurements of Tobu-ren et al. [57]. The molecular TC-BGM cross section is higher by about a factor of two in this region. Noexplanation for this discrepancy has been found yet [58].
IV. RESULTS FOR PROTON-CLUSTERCOLLISIONS
Motivated by the goal to aid the microscopic under-standing of proton cancer therapy a recent theoreticalwork looked into proton collisions from water clusters at E = 100 keV [59]. This is the region of the so-calledBragg peak, which marks the point of maximum energydeposition near the end of the path of an ion travelingthrough matter [60].The calculation of reference [59] was based on the sim-plest-level electron nuclear dynamics (SLEND) method(see also reference [27]), in which classically moving nu-clei are nonadiabatically coupled to electrons representedin terms of an unrestricted Hartree-Fock (UHF) determi-nantal wave function. Based on calculations for (H O) n with n = 1 , . . . , it was found that the total (one-electron) capture cross section σ ( n ) scaled as n / . Thiswas rationalized by associating each cluster with a sphereof volume V ( n ) , assuming V ( n ) ∝ n and arguing thatthe capture cross section should be proportional to theeffective area of the sphere exposed to the incident ion.Ionization was not considered in reference [59], since thebasis sets used did not allow for a representation of thecontinuum part of the spectrum. In addition, ETFs (cf.equation (8)) were neglected.We have applied the IAM-PCM to test the predictionof reference [59] and to explore the scaling of both net FIG. 5. Net capture cross sections in p-(H O) n collisions at E = 100 keV for n = 2 , , , , , from top left to bottomright. The representation is analogous to those of Figures 2b and 3b with radii determined acoording to equation (25). Thecorresponding plots of net ionization are similar, but show larger disks and more significant overlap, since the atomic ionizationcross sections are larger. capture and net ionization cross sections in p-(H O) n collisions in the impact energy range from 10 to 1000keV and for cluster sizes up to n = 20 . Specifically, weused the set of isomers included in the Cambridge Clus-ter Database [61] , whose structures were calculated atthe restricted Hartree-Fock/6-31G ( d, p ) level [62]. Fig-ure 5 shows the IAM-PCM net capture cross sectionsat E = 100 keV for a subset of these clusters and arbi-trary geometries in a similar representation as used inFigures 2b and 3b. As a consequence of the relativelylarge distances between the monomers in the clusters andthe relative weakness of electron capture at 100 keV (cf.Figure 4b) the overlap effect is small. This suggests thecross section scaling σ net cap ( n ) ∝ n α with a value of α close to one. Indeed, as Figure 6 shows, the IAM-PCMcapture results for n = 1 , . . . , are almost perfectly fit-ted by σ net x ( n ) = an α (28) For (H O) we chose the prism structure and omitted the cagestructure. with a = 1 . Å and α = 0 . . Here a represents aneffective capture (or ionization) cross section (in Å ) forthe case n = 1 , but is treated as a fit parameter in ordernot to give too much weight to the monomer.Ionization is stronger than capture at E = 100 keV (cf.Figure 4) and, accordingly, the overlap effect is larger.This translates into the optimal fit parameter α = 0 . ,which is still substantially larger than the value α = 0 . found by Privett et al . [59]. The different scaling be-haviour between our calculations (which treat ionizationproperly) and those of reference [59] may have variousreasons. Our calculations are based on a model, whereasPrivett et al. considered the molecular structure of thewater clusters in the UHF framework. As mentionedabove, ETFs and ionization channels were neglected intheir calculations. Also, they did not consider net cap-ture, but one-electron capture. The latter is probably aminor concern given that both quantities should be sim-ilar in a calculation in which the only other contributionto net capture is two-electron capture.Figure 7 shows IAM-PCM results for net capture andnet ionization at E = 10 keV. For capture the overlapeffect is large at low impact energy and the best fit of p+(H O) n : E=100 keVnet ionnet capPrivett17nn n n FIG. 6. Total cross sections for net ionization and net capturein p-(H O) n collisions at E = 100 keV as functions of clustersize n . The straight lines are obtained from equation (28) fordifferent parameter choices and are included to guide the eye.Privett17: SLEND calculation for one-electron capture fromreference [59]. p+(H O) n : E=10 keVnet ionnet capnn n FIG. 7. Total cross sections for net ionization and net capturein p-(H O) n collisions at E = 10 keV as functions of clustersize n . The straight lines are obtained from equation (28) fordifferent parameter choices and are included to guide the eye. the calculations is obtained with α = 0 . . By contrast,ionization is weak and σ net ion ( n ) scales almost linearlywith n . Linear scaling is also obtained at high energieswhere the IAM-PCM cross sections for net ionization andnet capture approach the IAM-AR predictions.We tabulated the optimal parameters α and a for bothnet ionization and net capture at all impact energy valuesin the ≤ E ≤ keV range for which we carried outcalculations and found that the IAM-PCM cross sectionresults can be parametrized by using equation (28) andassuming α ( a ) = (cid:26) − a/ . . if a ≤ . / otherwise (29)for the exponent. This is demonstrated in Figure 8. Eachpoint on the graph corresponds to the best fit of the IAM-PCM σ net x ( n ) results for a given E to equation (28), i.e.,to slope and intercept of that straight line that fits thecross section results for capture or ionization on a double-logarithmic plot as used in Figures 6 and 7 for E = 100 and E = 10 keV, respectively.The only deviation from the almost perfect linear de-pendence of α on a is observed for net capture at thelowest energy E = 10 keV (i.e., the point at a = 13 . Å ), suggesting that α cannot fall below 0.67. This lowerlimit is implemented explicitly in the parametrization bythe piecewise definition of α ( a ) and seems plausible giventhe arguments provided by Privett et al. [59] and thegeometrical construction of the IAM-PCM cross section.In other words, the IAM-PCM appears to be consistentwith those arguments in the limit of strong overlap. Inthe limit of weak overlap, the IAM-PCM approaches theIAM-AR prediction of a linear cross section scaling withcluster size n . Given the energy dependence of the atomiccross section magnitudes and overlaps the n -dependenceis not universally determined by the geometry of the clus-ter as the arguments provided by Privett et al. mightsuggest.To further test these observations we carried out IAM-PCM calculations for proton impact on neon clusters.The relevant structure information is also taken from theCambridge Cluster Database using d = 3 . a.u. as the in-ternuclear distance of the dimer [61]. For p-Ne n collisionswith n = 1 , . . . , we find α ≥ . for both net captureand net ionization in the entire impact energy range from10 to 1000 keV. Figure 9 shows the IAM-PCM cross sec-tions and the fits according to equation (28) at E = 10 keV. The cross sections are smaller than for p-(H O) n collisions, since the Ne electrons are more tightly bound.Given that the average distance between the monomersis similar in neon and water clusters, the atomic cross sec-tion overlaps are smaller and α is larger for the former.Remarkably, the p-Ne n results over the entire impactenergy range can also be parametrized by equation (29).This is shown in Figure 10, which is analogous to Figure 8for p-(H O) n collisions. The range of α ( a ) points for neonclusters is compressed compared to Figure 8 reflecting thesmaller atomic cross sections and overlaps.0 E=10keVE=20keVE=50keVE=100keVE=50keVE=200keV E=20keVE=500keVE=10keVE=1000keVE=100keVE>200keV net ionnet cap(a)=-a/36+1, 0...12; 2/3, otherwise
FIG. 8. The exponent α in equation (28) for net ionizationand net capture in p-(H O) n collisions plotted versus the pa-rameter a . Each data point corresponds to the best fit ofthe IAM-PCM results for σ net x ( n ) by equation (28) at theindicated impact energy. The full line corresponds to theparametrization (29). p+(Ne) n : E=10 keVnet ionnet capnn n n FIG. 9. Total cross sections for net ionization and net capturein p-Ne n collisions at E = 10 keV as functions of cluster size n .The straight lines are obtained from equation (28) for differentparameter choices and are included to guide the eye. p+Ne n net ionizationp+Ne n net capture(a)=-a/36+1, 0...12;2/3, otherwisep+C n net ionizationp+C n net capture(a)=-a/12+1, 0...4;2/3, otherwise FIG. 10. The exponent α in equation (28) for net ionizationand net capture in p-Ne n and p-C n collisions plotted versusthe parameter a . Each data point corresponds to the bestfit of the IAM-PCM results for σ net x ( n ) by equation (28) ata given impact energy. The yellow line corresponds to theparametrization (29) and the light-blue line to (30). Finally, we consider proton collisions from a selectionof carbon clusters C n with ≤ n ≤ . The relevantstructure information is taken from reference [63]. Simi-larly to the (H O) n case we find that net capture scales as n / at low energy, while α approaches unity more slowlythan for water clusters towards higher energies. In thecase of net ionization we also obtain somewhat smaller α values for C n than for (H O) n signaling larger overlaps.Figure 11 illustrates these observations for E = 100 keV.For capture the optimal α value is 0.95, while for ion-ization α = 0 . provides the best fit of the IAM-PCMcalculations. This is to be contrasted with α = 0 . and α = 0 . for p-(H O) n collisions, respectively (cf. Fig-ure 6).The parametrization (29) does not work for fullerenes,but we found that the ansatz (28) together with α ( a ) = (cid:26) − a/ . . if a ≤ . / otherwise (30)provides a good fit of the results in the 10 to 1000 keVimpact energy range. These results are included in Fig-ure 10. One may argue that the slope of α ( a ) for a givencluster species is reflective of the average distance be-tween monomers in the clusters. Additional calculationsfor p-Ar n support this and all data taken together suggestthat the slope is approximately inversely proportional tothat distance. Systematic measurements for a set of clus-ters over a range of impact energies would be highly de-sirable to test these predictions.1 p+C n : E=100 keVnet ionnet capnn n n FIG. 11. Total cross sections for net ionization and net cap-ture in p-C n collisions at E = 100 keV as functions of clustersize n . The straight lines are obtained from equation (28) fordifferent parameter choices and are included to guide the eye. Experimental data are available for net ionization ofC at high impact energies [64]. In Figure 12 we com-pare these measurements with IAM-PCM and IAM-ARcalculations. The overlap effect is significant, reducingthe net ionization cross section by more than a factor oftwo for most of the impact energy interval shown. Theexperimental data are even lower than the IAM-PCM re-sults with the latter just lying outside of the error bars.One can regard the agreement as fair. Clearly, data atlower energies (and for net capture as well) would beneeded for a better assessment of the quality of the IAM-PCM results. V. CONCLUDING REMARKS
34 years after the publication of the Runge-Gross theo-rem full-fledged TDDFT calculations for ion-impact colli-sions have remained a rarity compared to the widespreadapplication of TDDFT to laser-matter interaction prob-lems. As yet, simplified approaches and models are indis-pensable for a semi-quantitative understanding of elec-tron removal processes in collisions involving complexmulticenter Coulomb systems. The IAM-PCM is onesuch model. It is based on a geometrical interpretation ofthe cross section as an effective area composed of overlap-ping circular disks whose areas represent the atomic crosssections that contribute to net capture or net ionizationin the system of interest. The atomic cross sections arecalculated based on a TDDFT-inspired single-particle de-scription using atomic ground-state DFT potentials and
IAM-ARIAM-PCMTsuchida98
FIG. 12. Total cross section for net ionization in p-C collisions as function of impact energy. Experimental data:Tsuchida98 [64]. the two-center basis generator method for orbital propa-gation. The effective area calculation is carried out usinga pixel counting method.The IAM-PCM is flexible and efficient. Once theatomic cross sections have been calculated and the re-quired information on the geometric structure of the tar-get, i.e., the equilibrium positions of the nuclei, is avail-able it takes about three minutes on a single-core desktopor laptop computer to calculate the net ionization or netcapture cross section at a given impact energy for a sys-tem as complex as C .To date, we have applied the IAM-PCM to protoncollisions from a variety of targets: covalently boundmolecules in reference [39] and, in this work, clusters withhydrogen bonds ((H O) n ), van der Waals clusters (Ne n ),and covalently-bound fullerenes (C n ). One major objec-tive of this work has been to test and generalize a scalinglaw found by Privett et al. [59] in capture from waterclusters at E = 100 keV to capture and ionization over awide range of energies.Our results can be summarized as follows: Both netcapture and net ionization cross sections at a given im-pact energy scale as n α , but α varies as a function of E and reaches the value of 2/3 found by Privett et al. for one-electron capture only in situations in which theatomic cross section overlaps are large. This is the casefor capture at low impact energy ( E = 10 keV) in p-(H O) n and p-C n collisions, but not in p-Ne n collisionsand never for ionization. For capture from water clustersat E = 100 keV we find α = 0 . in stark contrast to theresult of Privett et al. Furthermore, we showed that the variations of α can be2modeled by the ansatz σ net x = an α ( a ) and a linear func-tional dependence of α on a . Our results suggest thatthe slope of this linear function is largely determined bythe average distance between the monomers in a givencluster. It will be interesting to see how general this re-sult is and where its limitations are. Further calculationsfor other systems and, more importantly, systematic ex- perimental measurements will be required to answer thisquestion.This work was supported by the Natural Sciences andEngineering Research Council of Canada (NSERC). Oneof us (H. J. L.) would like to thank the Center for Scien-tific Computing, University of Frankfurt for making theirHigh Performance Computing facilities available. [1] I.K. Gainullin and M.A. Sonkin, Computer Physics Com-munications , 68 (2015).[2] E. Runge and E. K. U. Gross, Phys. Rev. Lett. , 997(1984).[3] M. A. L. Marques and E. K. U. Gross, Annu. Rev. Phys.Chem. , 427 (2004).[4] M. A. L. Marques, N. T. Maitra, F. M. S. Nogueira,E. K. U. Gross, and A. Rubio, eds., Fundamentalsof Time-Dependent Density Functional Theory , LectureNotes in Physics, Vol. 837 (Springer, Berlin, 2012).[5] C. A. Ullrich,
Time-Dependent Density-Functional The-ory: Concepts and Applications (Oxford University Press,Oxford, 2012).[6] W. Fritsch and C. D. Lin, Phys. Rep. , 1 (1991).[7] B. H. Bransden and M. R. C. McDowell,
Charge Ex-change and the Theory of Ion-Atom Collisions (Claren-don Press, Oxford, 1992).[8] M. Baxter and T. Kirchner, Phys. Rev. A , 012502(2016).[9] K. C. Kulander, K. R. Sandhya Devi, and S. E. Koonin,Phys. Rev. A , 2968 (1982).[10] W. Stich, H. J. Lüdde, and R. M. Dreizler, Phys. Lett. , 41 (1983).[11] K. Gramlich, N. Grün, and W. Scheid, J. Phys. B ,1457 (1986).[12] J. J. Griffin, P. C. Lichtner, and M. Dworzecka, Phys.Rev. C , 1351 (1980).[13] Y. Alhassid and S. E. Koonin, Phys. Rev. C , 1590(1981).[14] T. Kirchner, L. Gulyás, H. J. Lüdde, A. Henne, E. Engel,and R. M. Dreizler, Phys. Rev. Lett. , 1658 (1997).[15] T. Kirchner, L. Gulyás, H. J. Lüdde, E. Engel, and R. M.Dreizler, Phys. Rev. A , 2063 (1998).[16] O. J. Kroneisen, H. J. Lüdde, T. Kirchner, and R. M.Dreizler, J. Phys. A , 2141 (1999).[17] M. Zapukhlyak, T. Kirchner, H. J. Lüdde, S. Knoop,R. Morgenstern, and R. Hoekstra, J. Phys. B , 2353(2005).[18] T. Kirchner, H. J. Lüdde, and M. Horbatsch, RecentRes. Dev. Phys. , 433 (2004).[19] A. C. K. Leung and T. Kirchner, Phys. Rev. A , 042703(2017).[20] T. Kirchner, M. Horbatsch, H. J. Lüdde, and R. M. Drei-zler, Phys. Rev. A , 042704 (2000).[21] K. C. Kulander, Phys. Rev. A , 778 (1988).[22] H. J. Lüdde, T. Spranger, M. Horbatsch, and T. Kirch-ner, Phys. Rev. A , 060702(R) (2009).[23] M. Murakami, T. Kirchner, M. Horbatsch, and H. J.Lüdde, Phys. Rev. A , 052704 (2012).[24] A. Salehzadeh and T. Kirchner, Eur. Phys. J. D , 66(2017). [25] X. Hong, F. Wang, Y. Wu, B. Gou, and J. Wang, Phys.Rev. A , 062706 (2016).[26] E. E. Quashie, B. C. Saha, X. Andrade, and A. A. Cor-rea, Phys. Rev. A , 042517 (2017).[27] A. J. Privett and J. A. Morales, Chem. Phys. Lett ,82 (2014).[28] M. C. Bacchus-Montabonel, Chem. Phys. Lett. , 173(2016).[29] C. Covington, K. Hartig, A. Russakoff, R. Kulpins, andK. Varga, Phys. Rev. A , 052701 (2017).[30] C. Dal Cappello, P. A. Hervieux, I. Charpentier, andF. Ruiz-Lopez, Phys. Rev. A , 042702 (2008).[31] H. Lekadir, I. Abbas, C. Champion, O. Fojón, R. D. Ri-varola, and J. Hanssen, Phys. Rev. A , 062710 (2009).[32] C. Champion, P. F. Weck, H. Lekadir, M. E. Galassi,O. A. Fojón, P. Abufager, R. D. Rivarola, andJ. Hanssen, Phys. Med. Biol. , 3039 (2012).[33] Pablo de Vera, Rafael Garcia-Molina, Isabel Abril, andAndrey V. Solov’yov, Phys. Rev. Lett. , 148104(2013).[34] L. Sarkadi, Phys. Rev. A , 062704 (2015).[35] H. J. Lüdde, A. Achenbach, T. Kalkbrenner, H.-C.Jankowiak, and T. Kirchner, Eur. Phys. J. D , 82(2016).[36] R. van Leeuwen, Phys. Rev. Lett. , 3863 (1999).[37] J. D. Talman and W. F. Shadwick, Phys. Rev. A , 36(1976).[38] E. Engel and S. H. Vosko, Phys. Rev. A , 2800 (1993).[39] H. J. Lüdde, A. Henne, T. Kirchner, and R. M. Dreizler,J. Phys. B , 4423 (1996).[40] A. Hindemarsh, in Scientific Computing , edited byR. Stepleman (Elsevier, 1983).[41] M. B. Shah and H. B. Gilbody, J. Phys. B , 2361(1981).[42] M. B. Shah, D. S. Elliott, and H. B. Gilbody, J. Phys.B , 2481 (1987).[43] M. E. Rudd, Y. K. Kim, D. H. Madison, and J. W.Gallagher, Rev. Mod. Phys. , 965 (1985).[44] G. W. McClure, Phys. Rev. , 47 (1966).[45] J. E. Bayfield, Phys. Rev. , 105 (1969).[46] A. B. Wittkower, G. Ryding, and H. B. Gilbody, Proc.Phys. Soc. , 541 (1966).[47] M. E. Rudd, R. D. DuBois, L. H. Toburen, C. A. Ratcliffe,and T. V. Goffe, Phys. Rev. A , 3244 (1983).[48] H. J. Lüdde, in Many-Particle Quantum Dynamics inAtomic and Molecular Fragmentation , edited by J. Ullrichand V. P. Shevelko (Springer, Heidelberg, 2003) p. 205.[49] H. Bethe, Ann. Physik , 325 (1930).[50] M. Inokuti, Rev. Mod. Phys. , 297 (1971).[51] I. B. Abdurakhmanov, A. S. Kadyrov, S. K. Avazbaev,and I. Bray, J. Phys. B , 115203 (2016). [52] T. Kirchner, H. J. Lüdde, M. Horbatsch, and R. M. Drei-zler, Phys. Rev. A , 052710 (2000).[53] T. Kirchner, H. J. Lüdde, and R. M. Dreizler, Phys. Rev.A , 012705 (2000).[54] F. Blanco and G. García, Phys. Lett. A , 458 (2003).[55] M. E. Rudd, T. V. Goffe, R. D. DuBois, and L. H. To-buren, Phys. Rev. A , 492 (1985).[56] M. A. Bolorizadeh and M. E. Rudd, Phys. Rev. A ,888–892 (1986).[57] L. H. Toburen, M. Y. Nakai, and R. A. Langley, Phys.Rev. , 114 (1968).[58] T. Kirchner, M. Murakami, M. Horbatsch, and H. J.Lüdde, Adv. Quant. Chem. , 315 (2013).[59] A. J. Privett, E. S. Teixeira, C. Stopera, and J. A.Morales, PLoS ONE , e0174456 (2017).[60] H. Bichsel, Adv. Quant. Chem.31