Proton-impact-induced electron emission from biologically relevant molecules studied with a screened independent atom model
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Proton-impact-induced electron emission from biologically relevantmolecules studied with a screened independent atom model
Hans Jürgen Lüdde ∗ Frankfurt Institute for Advanced Studies (FIAS), D-60438 Frankfurt, Germany
Marko Horbatsch † and Tom Kirchner ‡ Department of Physics and Astronomy,York University, Toronto, Ontario M3J 1P3, Canada (Dated: May 8, 2019)
Abstract
We use the recently introduced independent-atom-model pixel counting method to calculateproton-impact net ionization cross sections for a large class of biologically relevant systems includingpyrimidines, purines, amino acids, and nucleotides from 10 keV to 10 MeV impact energy. Overallgood agreement with experimental data, where available, is found. A scaling prescription thatinvolves coefficients derived from the independent atom model is shown to represent the crosssection results better than scalings based on the number of (bonding) valence electrons of thetarget molecules. It is shown that the scaled net ionization cross sections of the proton-nucleotidecollision systems can be represented in terms of a simple analytical formula with four parametersto within 3% accuracy.
PACS numbers: 34.10.+x, 34.50.Gb, 34.70.+e, 36.40.-c ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION There has been growing interest in collisions involving organic compounds and biomoleculesin recent years. That interest and the ensuing research activity are largely driven by dataneeds in areas ranging from astrochemistry to ion-beam cancer therapy. Ideally, the crosssection data required, e.g., for a detailed understanding of the radiation damage of biologicaltissue [1], would be obtained from systematic measurements and first-principles calculations.While progress has been made on both experimental (see [2–9] for proton-impact collisions)and theoretical [10–12] fronts, the complexity and multitude of molecules of interest suggestthat there is a role to be played by simplified models which are easily applicable to a widerange of collision systems.Classical arguments or quantum-mechanical first-order perturbation theory represent nat-ural starting points for modeling electron removal in ion-molecule collisions. Making use ofthese ingredients, a number of analytical cross section formulae have been proposed andapplied (see, e.g., [13–15] and the discussion in [9]). Their advantage is simplicity—typicallythey only require the binding energies of the target electrons as physical input—but theirsuccess depends on (additional) parameters which are determined (semi-) empirically. As aconsequence, these analytical models do not have sufficient predictive power, and are thusof limited value for problems for which measurements or more sophisticated calculations arenot available.Most attempts at constructing more sophisticated models are based on the idea that theionization or capture cross section of a complex molecule may be linearly combined fromsmaller parts. In the complete neglect of differental overlap (CNDO) approach a Mulliken-likepopulation analysis is applied to an electronic structure calculation of the target molecule.The molecular cross section is then written as a linear combination of contributions fromall atomic orbitals involved, with the Mulliken populations as weight factors [16]. A relatedclass of models is based on the additivity rule (AR) according to which the ionization orcapture cross section for a specifc target system is obtained as a sum of cross sections of itsbuilding blocks. Normally, atomic building blocks are used and the description is referredto as independent-atom-model (IAM)-AR. In a recent work it was argued that one can alsostart from small molecular constituents to assemble the ionization cross section of a largermolecule [17]. A caveat of this independent-molecule-model (IMM)-AR is that there are2sually many different ways to decompose a given molecule into molecular building blocksand depending on which ansatz is used the resulting cross section may vary. It shouldalso be noted that the IMM-AR work of [17] used experimental cross section data for thebuilding blocks as input, while the atom-like contributions in the CNDO approach are usuallyobtained from perturbative collision calculations [16, 18–20].Motivated by the somewhat limited scope of these models we recently introduced anIAM-based description of ion-molecule collisions which uses first-principles-based atomiccross section calculations and goes beyond the simple AR [21]. The main assumption of ourmodel is that the net ionization and capture cross sections can be represented as weighted sums of atomic net cross sections for all the atoms that make up the molecule. The weightfactors account for the geometrical overlap which occurs when one projects the loci of theatomic centers of the molecule onto a plane perpendicular to the projectile beam directionand pictures the atomic cross sections as circular disks in that plane. The “visible” effectivecross sectional area is calculated using a pixel counting method (PCM) and, accordingly, werefer to the model as IAM-PCM. In the limit of small overlaps the IAM-PCM cross sectionapproaches the result of the IAM-AR. This is a desirable property since the IAM-AR isknown to give fairly accurate results in the high-impact-energy regime in which captureand ionization cross sections are small and geometrical overlap is negligible. Toward lowerenergies, capture and ionization become stronger and the IAM-AR tends to overestimateexperimental data [17, 21]. In applications to proton collisions from medium-sized moleculessuch as H O and from larger water, neon, and carbon clusters we have found that geometricaloverlap does occur at these lower energies and leads to significant cross-section reductions [21,22]. Compared to the IAM-AR the agreement with experimental data, where available, isimproved.Given these promising results and the relative simplicity of the model, the IAM-PCMseems ideally suited to study collisions from complex biomolecules for which neither ab-initio calculations nor measured cross section data are available. This is the main objectiveof the present work. In particular, we examine scaling relations obtained from IAM-PCMcalculations for proton-impact net ionization cross sections of different groups of systemssuch as amino acids and nucleotides, and suggest a parametrization of our results in termsof a simple analytical formula.The layout of the paper is as follows. We begin in section II with a short summary3f the IAM-PCM. For a more detailed description the reader is referred to [22]. In sec-tion III we seek to further validate the model, beyond the results presented in [21, 22], bycomparing calculated net ionization cross sections with available measurements and previoustheoretical and semi-empirical model data for 10 keV to 10 MeV proton collisions from pyrim-idine (C H N ), purine (C H N ), tetrahydrofuran (THF – C H O) and trimethyl phosphate(TMP – (CH ) PO ). Different scaling prescriptions for the net ionization cross section areexamined and applied to a large class of systems including amino acids and nucleotides insection IV and a simple parametrization of these results is suggested in section V. The paperends with concluding remarks in section VI. II. THEORETICAL MODEL
The ingredients of the IAM-PCM are atomic cross sections and molecular ground-stategeometries. The latter are taken from the literature using a Cartesian coordinate repre-sentation [23], which is commonly referred to as xyz-file format. More specifically, for thebiomolecules studied in this work we use data provided through [24]. The proton-atom netionization cross sections are calculated in a density functional theory (DFT) framework onthe level of the independent electron model. We use a well-tested no-response model inwhich the Kohn-Sham potential is approximated by an accurate exchange-only ground-statepotential [25] and time-dependent screening and exchange effects are neglected. The nonper-turbative two-center basis generator method (TC-BGM) is used for orbital propagation [26].It yields transition probabilities and cross sections for target excitation and electron transferto the projectile (capture) and the continuum (ionization). In the present work we only lookat the ionization channel, since it usually exhibits simpler scaling properties and there areno capture data available for comparison for most of the molecules studied here.The net ionization cross sections σ net j for the j = 1 , . . . , N atoms that make up themolecule under study are combined according to σ netmol ( E, α, β, γ ) = N X j =1 s j ( E, α, β, γ ) σ net j ( E ) (1)to yield the molecular cross section at projectile energy E . The weight factors ≤ s j ≤ account for the overlap of the atomic cross sections and depend on the relative orientationof the molecule with respect to the ion beam direction. This dependence is captured by the4uler angles α, β, γ .To calculate the weights we picture the atomic cross sections as circular disks with radii r j ( E ) = [ σ net j ( E ) /π ] / in a plane perpendicular to the ion beam axis. The combined area ofoverlapping circles is broken up into pixels and calculated by counting those pixels that arevisible to the impinging projectile. Accordingly, the screening coefficients can be determinedby s j ( E, α, β, γ ) = σ vis j ( E, α, β, γ ) σ net j ( E ) , (2)where σ vis j is the visible part of the j th atomic cross section.The IAM-PCM is similar in spirit to the so-called screening-corrected additivity rule(SCAR) developed and used for electron-molecule collisions [27]. SCAR cross sections, how-ever, are based on a heuristic recurrence relation for the determination of the screeningcoefficients in an orientation-independent version of (1), whereas the IAM-PCM procedureto calculate them for any given orientation is numerically exact. In order to compare IAM-PCM calculations with experimental data for randomly oriented molecules we repeat thepixel count for an ensemble of Euler angle triples and average the cross-section results ap-propriately.Once the atomic cross sections have been calculated, the IAM-PCM procedure is not atall resource intensive. It takes just a few minutes on a single-core computer to carry outan orientation average at a given energy for a system consisting of dozens of nuclei andhundreds of electrons. The reader is referred to [22] for more details. III. VALIDATION OF THE MODEL
The molecules pyrimidine, purine, THF, and TMP are structural analogues of DNAconstituents and have been studied in recent collision experiments [8, 9, 28], since they weredeemed more amenable to gas-phase cross-section measurements than actual DNA buildingblocks . More specifically, pyrimidine is a single carbon-nitrogen-ring molecule from whichthe nucleobases cytosine and thymine (and the RNA nucleobase uracil) are derived, whilethe double-ring molecule purine is a precursor of the nucleobases adenine and guanine. Both While it is difficult to prepare well-characterized gas targets of large neutral DNA components, the tech-nique of electrospray ionization offers a pathway to bringing charged complex biomolecules into the gasphase and use them in collision experiments [29]. IG. 1. Net ionization in E = 500 keV proton collisions with (from left to right) pyrimidine(C H N ), purine (C H N ), THF (C H O), TMP ((CH ) PO ) for particular orientations. Theradii of the circular disks are given by r j = [ σ net j /π ] / using TC-BGM proton-atom net ionizationcross sections σ net j . pyrimidine and purine are also used as generic names for wider classes of similar one-ringand two-ring molecules, respectively, which include the DNA and RNA nucleobases (seesection IV). THF serves as a model for the monosaccharide deoxyribose in DNA, and TMPrepresents the phosphate residue which together with the sugar molecule forms the DNAbackbone [30, 31].Figure 1 displays effective cross sectional areas for proton collisions from these fourmolecules for arbitrary orientations with respect to the projectile beam axis. The atomicnuclei are placed at their equilibrium ground-state positions and the plots are obtained fromnet ionization calculations at E = 500 keV impact energy. For pyrimidine and purine thereis no overlap of the atomic cross sections for the chosen geometries, while a modest overlapeffect occurs for THF and a somewhat larger one for TMP. One can imagine how the mag-nitude of the overlap effect varies as a function of orientation and collision energy and thateven for pyrimidine, the smallest of the four molecules, the orientation-averaged IAM-PCMcross section will be smaller than the zero-overlap limit corresponding to the IAM-AR.Figure 2 shows the orientation-averaged net ionization cross section for proton-pyrimidinecollisions as a function of impact energy. In the left panel [figure 2(a)], we compare severalmodel calculations on a double-logarithmic scale which emphasizes the fall-off of the ioniza-tion cross section toward high impact energy. This shows nicely how the present IAM-ARand IAM-PCM results merge in the E ≥ keV range in which the atomic cross sectionsare so small that no significant overlap occurs for any orientation. This is very different at6 +C H N IAM-PCMIAM-ARIMM-ARmodel 1model 2 p+C H N IAM-PCMCB1Bug14Wolff14Rudek16
FIG. 2. Total net ionization cross section for proton-pyrimidine (C H N ) collisions as a functionof impact energy on (a) a double-logarithmic scale and (b) a single-logarithmic scale. IAM-PCM,IAM-AR: present calculations, IMM-AR [9], model 1 and model 2 denote model calculations basedon equations (3) and (4) as described in the text; CB1: first Born approximation with correctedboundary conditions within the CNDO approach [9]; experiments: Bug14 [32] (see also [28]) forelectron impact using equivelocity conversion, Wolff14 [8], Rudek16 [9] for proton impact. lower energies: At the cross section maximum around E = 60 keV the IAM-AR cross sectionis about a factor of two larger than its IAM-PCM counterpart.The two IAM calculations bracket the IMM-AR results reported in [9] in most of theimpact energy range shown. This is expected based on geometrical considerations. On theone hand the (experimental) molecular cross sections used in the IMM-AR to assemble thecross section for pyrimidine should be smaller than the zero-overlap IAM-AR predictions, buton the other hand their sum should be larger than the IAM-PCM cross section for pyrimidine,since overlaps of the contributing molecular cross sections are neglected in the IMM-AR.The fact that the IMM-AR results become slightly smaller than the IAM-PCM calculationsabove E = 1000 keV cannot be explained on geometrical grounds. It points either to anunderestimation of the overlap effect by the IAM-PCM or to a small inconsistency betweenthe (atomic) TC-BGM results and the experimental cross section data used in the IMM-AR7alculation of [9]. Given that no details about the latter were provided, we cannot offer amore definitive explanation for the (small) discrepancy.The two model calculations included in figure 2(a) are based on a semi-empirical scalingrelation proposed by Montenegro and co-workers [15], and used for proton-pyrimidine colli-sions in [8]. According to that model the net ionization cross section of an atom or moleculecan be written as σ net ( E ) = X k σ k ( E ) = X k Z k δ k I k F (cid:18) E/MI k (cid:19) (3)with the universal function F ( x ) = A ln(1 + Bx ) x − AB (1 + Cx ) . (4)The sum in (3) is over the contributing atomic or molecular orbitals, Z k is the occupationnumber and I k the ionization potential (in atomic units) of the k th orbital, δ k a parameter,and E/M the collision energy in keV/amu. The universal parameters in (4) are A = 6 . × , B = 7 . × − , and C = 1 . × − to yield cross sections measured in units of − cm [15].The model 1 calculation is identical to that reported in [8]. It uses the ionization potentialsof the eleven most loosely bound, doubly-occupied ( Z k = 2 ) orbitals of pyrimidine quotedin [8] and δ k = 0 . for all k . For model 2 we make the same choices for Z k and δ k , butuse the ionization potentials of the fifteen valence orbitals provided in [9], which are slightlydifferent from those quoted in [8] for the outermost eleven. Both model variants lead tosimilarly shaped cross section curves with the model 2 results being larger than the model 1results by 12–14 % in the E = 300 keV to E = 10 MeV range in which the curves appear asalmost perfect straight lines on the double-logarithmic plot. The difference in magnitude ismostly due to the fact that four additional orbitals are taken into account in model 2.We also carried out a model calculation based on the fifteen most loosely bound atomic orbitals using the exchange-only DFT orbital energy eigenvalues on which the TC-BGMcalculations are based. Given that the high-energy behavior of the model cross section iscontrolled by the first, Bethe-Born-like, term in equation (4) and the TC-BGM has beenshown to give results which agree very well with Bethe-Born predictions at high energies (seesection V and [22]), one would expect that this model variant would agree with the IAM-PCM calculations at least in the high-energy limit. However, we found that the results ofthe atomic model are very similar to those of model 2 (which is why they are not included8n the figure) and, as a consequence, somewhat smaller than the IAM-PCM cross sectionat E ≥ keV. We have checked that the high-energy discrepancy would essentially beeliminated if one would use δ k = 1 . instead of δ k = 0 . in the model. Such a choice wouldbe consistent with the findings of [15]: In that paper δ = 0 . was used to describe p-Hcollisions, while δ = 1 . was found to give excellent fits of the experimental ionization crosssections in the p-N and p-CH systems. But even such an amended model would not agreewith the IAM-PCM results at lower impact energies, i.e., the latter are incompatible withequations (3) and (4). This suggests that in general one cannot expect very accurate resultswhen applying these equations to collisions involving complex biomolecules.Figure. 2(b) displays on a single-logarithmic plot the IAM-PCM results together withexperimental data for proton [8, 9] and equivelocity electron impact [32] and the resultsof a first Born calculation with corrected boundary conditions (CB1) performed withinthe CNDO approach [9]. The CB1 calculation gives smaller cross section values than theIAM-PCM at energies above E = 200 keV, in apparent agreement with most of the ex-perimental data points of [8]. However, the CB1 and experimental results differ in energydependence above E = 1000 keV. This is better seen on the double-logarithmic plot providedin figure 5(a) of [9]. The IAM-PCM results agree very well with the electron-impact mea-surements of [32] in the region above the cross section maximum in which electron-impactdata are expected to approach the proton-impact cross section. Indeed, within combined er-ror bars the electron-impact data are in marginal agreement with the proton measurementsof [8], although it appears as if the latter fall somewhat below the former. New experimentaldata with smaller error bars would be needed to draw more definitive conclusions about thehigh-energy behavior of the cross section.The three experimental proton-impact data points of [9] at intermediate energies havetoo large uncertainties to help shed light on the increasing deviations between the CB1 andIAM-PCM calculations in this region. Given the first-order nature of the CB1 one would notexpect this model to be valid below E = 100 keV, but in the absence of more experimentaldata the overall situation remains unclear.Figure 3 shows IAM-PCM net ionization cross section results for proton-purine collisions.We compare them with equivelocity electron-impact measurements from [32] only, since weare not aware of other theoretical calculations or measurements for proton impact. Con-sistent with the pyrimidine case, the agreement is excellent in the energy region above the9 +C H N IAM-PCMBug14
FIG. 3. Total net ionization cross section for proton-purine (C H N ) collisions as a function ofimpact energy. IAM-PCM: present calculation; experiments: Bug14 [32] (see also [28]) for electronimpact using equivelocity conversion. experimental cross section maximum in which the sign of the projectile charge is deemed tobe unimportant.Similar observations are made for the THF and TMP target molecules, as shown infigures 4 and 5, respectively. For these two cases, in addition to the electron data from [32],proton-impact cross section measurements and CB1 calculations from [9] are available forcomparison. As for the proton-pyrimidine case, the CB1 results are somewhat lower than theIAM-PCM cross sections at relatively high impact energies where first-order perturbationtheory is expected to be valid. They cross the IAM-PCM curve between E = 100 and E = 200 keV and are probably too high at lower energies where the perturbation is too strongfor a first-order theory to be reliable. The measured data points of [9] appear somewhatunsystematic and have error bars that are too large to differentiate between the CB1 andIAM-PCM calculations. At high energies in particular, new measurements with smalleruncertainties are highly desirable to clarify the situation. This caveat notwithstanding, theIAM-PCM results appear to agree with most experimental data points for the four collisionsystems studied in this section. We thus feel encouraged to expand the application of themodel to a larger class of systems for which experimental data are sparse and theoreticalpredictions are required. 10 +C H OIAM-PCMCB1Bug14Rudek16
FIG. 4. Total net ionization cross section for p-THF (C H O) collisions as a function of impactenergy. IAM-PCM: present calculation, CB1: first Born approximation with corrected boundaryconditions within CNDO approach [9]; experiments: Bug14 [32] (see also [28]) for electron impactusing equivelocity conversion, Rudek16 [9] for proton impact. p+(CH ) PO IAM-PCMCB1Bug14Rudek16
FIG. 5. Total net ionization cross section for p-TMP ((CH ) PO ) collisions as a function of impactenergy. IAM-PCM: present calculation, CB1: first Born approximation with corrected boundaryconditions within CNDO approach [9]; experiments: Bug14 [32] (see also [28]) for electron impactusing equivelocity conversion, Rudek16 [9] for proton impact. V. SCALING PROPERTIES
Previous experimental and theoretical work provided some evidence that the (total)proton-impact electron emission cross sections of large biomolecules scale with the num-ber of valence electrons [5, 9, 17, 28]. However, only a relatively small set of molecules hasbeen investigated so far and it is not clear whether the observed approximate scaling appliesto a larger class of systems and can be used to predict cross sections for experimentallyinaccessible compounds. In this section we investigate this question by using the IAM-PCMto calculate proton-impact net ionization cross sections for four groups of biologically rel-evant systems: pyrimidines, purines, amino acids, and nucleotides, the latter constitutingthe monomeric units of RNA and DNA [30, 31]. To our knowledge, for most of the studiedspecies the results presented here are the first cross section data obtained from system-atic, parameter-free calculations. Exceptions are the five DNA/RNA nucleobases cytosine,thymine, uracil, adenine and guanine. As mentioned in section III the first three fall into thecategory of pyrimidines, while the latter two are purines. For all of them classical [33, 34] andperturbative [35] cross section calculations have been carried out, the latter in some caseswithin the CNDO approach [16, 18, 20]. A comparison with those earlier calculations, theIMM predictions of [17], and the limited experimental data available [4–7] will be presentedelsewhere [36].In the following, we consider three different scaling prescriptions. The first one usesthe standard textbook definition of valence electrons according to which all electrons inthe outermost n -shells of the atoms that form the molecule under study are included inthe valence-electron count [37]. In the second variant we only count the bonding valenceelectrons, i.e., those electrons that form lone pairs are excluded. The scaled cross section isobtained by dividing the orientation-averaged IAM-PCM result for a molecule with formulaC n H n N n O n P n by the numbers N VE = 4 n + n + 5 n + 6 n + 5 n (5)and N BVE = 4 n + n + 3 n + 2 n + 5 n , (6)respectively, assuming in the latter case that all L -shell electrons of carbon participate inbonds (through hybridization), while in nitrogen one and in oxygen two electron pairs do12 +Hp+Cp+Np+Op+P Bethe = (A ln E + B)/E
Bethe (1 - a exp(- E)) p+H: Shah81p+H: Shah87 FIG. 6. Total net ionization cross sections for proton collisions with atomic hydrogen (H), carbon(C), nitrogen (N), oxygen (O), and phosphorus (P) as functions of impact energy. Full lines: presentTC-BGM calculations, dashed lines: Bethe-Born ionization cross sections σ Bethe = ( A ln E + B ) /E with fit parameters A and B in appropriate units (see section V), ( • ) : parametrizations discussedin section V. Experiments for p-H: Shah81 [38], Shah87 [39]. not.The third prescription is based on the IAM and the observation that at high impactenergies the net ionization cross sections for p-C, p-N, and p-O collisions are very similar(i.e., σ net X ≡ σ net C ≈ σ net N ≈ σ net O ) and to a good approximation four times larger than thep-H cross section [22]. This is demonstrated in figure 6, where the corresponding TC-BGMresults are shown. Since the nucleotides studied further below contain phosphorus, the p-Pcollision system is included in figure 6 as well. In this case, we find that the net ionizationcross section is approximately 1.5 times as large as that for carbon, nitrogen, and oxygen.Accordingly, the high-energy IAM prediction for the net ionization cross section of themolecule C n H n N n O n P n is to a good approximation ( n + n / n + n + 3 n / σ net X ,and we scale the (orientation-averaged) IAM-PCM cross sections in this variant by dividingthem by the (fractional) coefficients N IAM = n + n n + n + 3 n . (7)Figure 7 displays the scaled and unscaled cross section results for all systems studied in13his section. The unscaled IAM-PCM cross sections are also provided in tabulated form inthe Appendix. For the pyrimidines [figure 7(a)] and purines [figure 7(b)] the scaling withrespect to the number of bonding valence electrons yields better results than that withrespect to the number of all valence electrons. For the amino acids [figure 7(c)] the situationis reversed, most visibly so in the region around the cross section maximum, while for thenucleotides [figure 7(d)] both scaling procedures appear to work equally well.A more conclusive picture emerges when the cross sections are scaled by dividing them bythe fractional IAM coefficients (7). In this case, we obtain for each group of systems a nearlyuniversal curve. Given the atomic results shown in figure 6 this is to be expected at highenergies where the IAM-PCM cross sections approach the IAM-AR limit. However, it is notobvious that the scaling should also hold at lower energies where significant atomic crosssection overlaps occur and the IAM-PCM calculations are not orientation independent; e.g.,we found that at E = 100 keV the net ionization cross section of pyrimidine varies withina factor of two as a function of orientation. It is one of the main results of this work thatdespite these caveats the scaling holds if one accepts tolerances on the order of 10%.Figure 8 provides a more differentiated view of the approximate universality of the IAM-based scaling. It shows on a linear scale averages of the IAM-normalized cross sections andthe deviations from these averages as error bars. For the pyrimidines these deviations are aslarge as ∼ ± %, while for the DNA nucleotides they are smaller than ± %. The negligiblespread of the latter can be traced back to the sugar-phosphate backbone, which is the samefor all DNA nucleotides, and, due to the large p-P cross section (cf. figure 6), is the maincontributor to the total cross section. Our results then indicate that the differences in thecross sections of the pyrimidines and purines are not relevant for the ionization of a DNAmolecule. Rather, they suggest that one can understand the latter as the ionization of oneor another of its nucleotide monomers, all of which fulfill the IAM-based scaling relationvery accurately, i.e., it does not matter much which of them is actually ionized. V. PARAMETRIZATION
The finding that the IAM-based scaling works very well for a large class of systems raisesthe question whether the proton-impact electron emission cross sections of biomolecules canbe parametrized in a convenient way. To address this question we re-inspect figure 6 which in14
AM-PCMscaled with (7)scaled with (5)scaled with (6) x 10p+barbituric acid C H N O p+cytosine C H N O p+orotic acid C H N O p+pyrimidine C H N p+thymine C H N O p+uracil C H N O IAM-PCMscaled with (7)scaled with (5)scaled with (6) x 10p+adenine C H N p+caffeine C H N O p+guanine C H N Op+isoguanine C H N Op+purine C H N p+theobromine C H N O p+uric acid C H N O p+xanthine C H N O p+glycine: C H NO p+alanine: C H NO p+threonine: C H NO p+valine: C H NO p+leucine: C H NO p+proline: C H NO p+asparagine: C H N O IAM-PCMscaled with (7)scaled with (5)scaled with (6) x 10 IAM-PCMscaled with (7)scaled with (5)scaled with (6) x 10p+dAMP: C H N O Pp+dCMP: C H N O Pp+dGMP: C H N O Pp+dTMP: C H N O Pp+UMP: C H N O P FIG. 7. Total net ionization cross sections for proton collisions with (a) the pyrimidines barbi-turic acid, cytosine, orotic acid, pyrimidine, thymine, uracil; (b) the purines adenine, caffeine,guanine, isoguanine, purine, theobromine, uric acid, xanthine; (c) the amino acids glycine, alanine,threonine, valine, leucine, proline, asparagine; (d) the DNA nucleotides deoxyadenosine monophos-phate (dAMP), deoxycytidine monophosphate (dCMP), deoxyguanosine monophosphate (dGMP),deoxythymidine monophosphate (dTMP), and the RNA nucleotide uridine monophosphate (UMP)as functions of impact energy. Full lines: present IAM-PCM calculations, long-dashed lines: presentIAM-PCM calculations divided by the number of valence electrons (5), short-dashed lines: presentIAM-PCM calculations divided by the number of bonding valence electrons (6) and further dividedby ten for visibility, dotted lines: present IAM-PCM calculations divided by the IAM coefficients (7). onisation of biomoleculesp + amino acidsp + pyrimidinesp + purinesp + deoxinucleotides FIG. 8. Average (IAM-based) scaled net ionization cross sections using (7) for proton collisionswith pyrimidines, purines, amino acids, and DNA nucleotides as functions of impact energy. Thedeviations of the actual results displayed in figure 7 from the averages are shown as error bars. addition to the atomic net ionization cross sections obtained from the TC-BGM calculationsdisplays Bethe-Born results in which the parameters A and B in the cross section formula σ Bethe ( E ) = A ln EE + BE (8)were determined by a fitting procedure using the Fano representation [22]. The agreementis excellent for impact energies above E ≈ keV. At lower energies, the TC-BGM crosssections are smaller than the Bethe-Born predictions, mostly because electron capture (whichis described by the TC-BGM) gains importance and ultimately takes over as the dominantelectron removal channel. One may argue that the occurrence of capture effectively decreasesthe projectile charge Q P , thereby reducing the amount of direct ionization to the continuum,which in the Bethe-Born approximation is proportional to the square of Q P [40]. Thissuggests the ansatz σ ion ( E ) = [ Q P, eff ( E )] σ Bethe ( E ) (9)with an effective charge of the form Q P, eff ( E ) = 1 − a exp( − αE ) . (10)We note that similar parametrizations have been proposed in the past to model the electronloss from neutral projectiles (see [41] and references therein). The full circles in figure 6 show16 ABLE I. Parameters α (in keV − ) and a used in (10) to model the effective charges for the atomictargets hydrogen (H), carbon (C), nitrogen (N), oxygen (O), and phosphorus (P).H C N O P α the results obtained from equations (9) and (10) with the parameters listed in Table I. Theagreement with the TC-BGM cross section curves is almost perfect. A similar parametriza-tion should then work for biomolecular targets. However, in this case we found that theoverlap effects taken into account in the IAM-PCM result in a flatter shape of the crosssection curves compared to their atomic counterparts and the modified formula σ ionmol ( E ) = [1 − a exp( − α √ E )] σ Bethe ( E ) (11)provides better fits. This is demonstrated for the nucleotides in figure 9. Formula (11) agreeswith the orientation-averaged IAM-PCM calculations in the entire impact energy range from E = 10 keV to E = 10 MeV to within ± %. VI. CONCLUDING REMARKS
We have used the IAM-PCM, introduced in recent work, to calculate proton-impact netionization cross sections for a large class of biologically relevant molecules from E = 10 keV to E = 10 MeV impact energy. We have found overall good agreement with thelimited experimental data available for pyrimidine, purine, THF, and TMP and have madepredictions for a number of larger systems including amino acids and nucleotides. To ourknowledge, the results reported for the latter are the first cross sections obtained from aparameter-free theoretical model. It is shown that they follow a scaling rule which is basedon IAM-derived fractional coefficients and can be represented by a simple analytical formulato within 3% accuracy. Scaling prescriptions based on the number of (bonding) valenceelectrons which were advocated in previous works yield less conclusive results.The IAM-PCM is based on a geometrical interpretation of a molecular total cross sectionas the combined area of overlapping circles which represent atomic cross sections. The latterare calculated from first principles using accurate atomic potentials and the nonperturbative17 caled with (7)fit based on (11)p+dAMP: C H N O Pp+dCMP: C H N O Pp+dGMP: C H N O Pp+dTMP: C H N O Pp+UMP: C H N O P FIG. 9. IAM-based scaled net ionization cross sections using (7) for proton collisions with theDNA nucleotides deoxyadenosine monophosphate (dAMP), deoxycytidine monophosphate (dCMP),deoxyguanosine monophosphate (dGMP), deoxythymidine monophosphate (dTMP), and the RNAnucleotide uridine monophosphate (UMP) as functions of impact energy. Crosses: fit based on (11)using the parameters A = 135 . Å keV , B = − . Å keV, a = 0 . and α = 0 . keV − / . TC-BGM for orbital propagation. Once these cross sections have been computed, the IAM-PCM procedure to assemble the molecular cross section is computationally inexpensive andnumerically accurate.We envision that the method can be used to describe collisions involving long-chainmolecules and polymers, such as peptides and large DNA sections, in terms of cross sectioncalculations for (clusters of) amino acids and nucleotides. Further work in this direction isin progress. Future studies will also be concerned with the electron capture channel andwith the extraction of charge-state-correlated multiple ionization data. The latter will beparticularly relevant for collision systems involving multiply-charged projectile ions.
ACKNOWLEDGMENTS
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TABLE II. Orientation-averaged IAM-PCM net ionization cross sections for proton collisions withpyrimidines (in Å ).E [keV] Barbituric acid Cytosine Orotic acid Pyrimidine Thymine UracilC H N O C H N O C H N O C H N C H N O C H N O
10 9.62 9.34 11.19 7.67 10.62 8.9220 15.62 15.42 17.60 12.66 17.41 14.8350 20.12 19.44 22.60 15.98 21.93 18.90100 19.59 18.75 22.14 15.05 21.17 18.23200 16.50 15.49 18.64 12.11 17.63 15.13500 10.95 10.28 12.64 7.96 11.72 9.991000 6.86 6.46 8.07 4.96 7.35 6.212000 4.03 3.76 4.82 2.87 4.26 3.655000 1.94 1.80 2.30 1.38 2.03 1.7610000 1.08 1.02 1.32 0.77 1.14 0.98 ABLE III. Orientation-averaged IAM-PCM net ionization cross sections for proton collisions withpurines (in Å ).E [keV] Adenine Caffeine Guanine Purine Theobromine Uric Acid XanthineC H N C H N O C H N O C H N C H N O C H N O C H N O
10 11.32 16.80 12.04 10.36 14.86 12.13 11.4020 18.53 26.78 19.70 16.55 23.77 19.47 18.0050 22.96 32.82 24.52 20.61 29.42 24.63 22.64100 22.17 31.56 23.68 19.67 28.49 24.11 22.03200 18.41 26.52 19.91 16.32 23.94 20.32 18.53500 12.40 18.04 13.33 10.96 16.18 13.94 12.681000 7.86 11.25 8.47 6.88 10.35 8.86 8.102000 4.58 6.66 4.96 4.07 6.03 5.20 4.785000 2.18 3.20 2.39 1.93 2.95 2.52 2.3210000 1.23 1.82 1.33 1.12 1.62 1.40 1.28TABLE IV. Orientation-averaged IAM-PCM net ionization cross sections for proton collisions withamino acids (in Å ).E [keV] Alanine Asparagine Glycine Leucine Proline Threonine ValineC H NO C H N O C H NO C H NO C H NO C H NO C H NO