Ionization of biological molecules by multicharged ions using the stoichiometric model
aa r X i v : . [ phy s i c s . a t m - c l u s ] S e p Ionization of biological molecules by multicharged ions using the stoichiometric model
A. M. P. Mendez, C. C. Montanari, J. E. Miraglia
Instituto de Astronom´ıa y F´ısica del Espacio (CONICET–UBA),Buenos Aires, Argentina.
In the present work, we investigate the ionization of molecules of biological interest by the impactof multicharged ions in the intermediate to high energy range. We performed full non–perturbativedistorted–wave calculations (CDW) for thirty–six collisional systems composed by six atomic targets:H, C, N, O, F and S –which are the constituents of most of the DNA and biological molecules– andsix charged projectiles (antiprotons, H, He, B, C, and O). On account of the radiation damagecaused by secondary electrons, we inspect the energy and angular distributions of the emittedelectrons from the atomic targets. We examine seventeen molecules: DNA and RNA bases, DNAbackbone, pyrimidines, tetrahydrofuran (THF), and C n H n compounds. We show that the simplestoichiometric model (SSM), which approximates the molecular ionization cross sections as a linearcombination of the atomic ones, gives reasonably good results for complex molecules. We alsoinspect the extensively used Toburen scaling of the total ionization cross sections of molecules withthe number of weakly bound electrons. Based on the atomic CDW results, we propose new activeelectron numbers, which improves significantly scaling for all the targets and ions studied here in theintermediate to the high energy region. The new scaling describes well the available experimentaldata for proton impact, including small molecules. We perform full molecular calculations for fivenucleobases and test a modified stoichiometric formula based on the Mulliken charge of the compositeatoms. The difference introduced by the new stoichiometric formula is less than 3%, which indicatesthe reliability of the SSM to deal with this type of molecules. The results of the extensive ion–targetexamination included in the present study allow us to assert that the SSM and the CDW–basedscaling will be useful tools in this area. PACS numbers: 34.50Gb, 34.80Gs, 34.80Dp
I. INTRODUCTION
The damage caused by the impact of multichargedheavy projectiles on biological targets has become a fieldof interest due to its recent implementation in ion–beamcancer therapy. The effectiveness of the radiation de-pends on the choice of the ions. In particular, theoreticaland experimental studies with different projectiles haveconcluded that charged carbon ions could be the mostsuitable ions to be used. Nonetheless, the study of suchsystems represents a challenge from the theoretical pointof view.The ionization of biological molecules by multichargedions constitutes the primary damage mechanism. Themost widely used method to predict such processes isthe first Born approximation. At high energies, this per-turbative method warrants the Z laws, where Z is theprojectile charge. However, the damage is concentratedin the vicinities of the Bragg peak –at energies of hun-dreds of keV/amu–, precisely where the Born approxi-mation starts to fail. Another theoretical issue arises dueto the targets themselves; we are dealing with complexmolecules, and the description of such targets representsa hard task for ab initio calculations. The objective ofthis article is to deal with these two aspects; first, weperform more appropriate calculations on the primarydamage mechanism, which can replace the Born results.Second, we inspect and test a stoichiometric model todescribe the ionization of molecular targets.To overcome the first perturbative approximation lim- itations, and since the projectiles are multicharged ions,we resort to the Continuum Distorted Wave–Eikonal Ini-tial State (CDW) [1–3], which includes higher pertur-bative corrections. Details on the CDW calculation aregiven in Section II. We start from the premise that theionization process is the mechanism that deposits themost significant amount of primary energy. Moreover,the residual electrons from the ionization are known tobe a source of significant local biological damage. Thesecondary electrons are included in Monte Carlo simula-tions, and hence their behavior requires further investi-gation. In Section II A and II B, we calculate the meanenergy and angular distributions of the ejected electrons.Surprisingly, we found a substantial dependence on theprojectile charge, which is unexpected in the first Bornapproximation.In Section III A, we deal with the molecular structurecomplexity of the targets by implementing the simpleststoichiometric model (SSM): the molecules are assumedto be composed of isolated independent atoms, and thetotal cross section by a linear combination of stoichiomet-ric weighted atomic calculations. By implementing theCDW and the SSM, we calculate ionization cross sectionof several molecules of biological interest, including DNAand RNA molecules, such as adenine, cytosine, guanine,thymine, uracil, tetrahydrofuran (THF), pyrimidine, andDNA backbone, by the impact of antiprotons, H + , He + ,Be + , C + , and O + . In Section III B, we test the Tobu-ren scaling rule [4, 5], which states that the ratio betweenthe ionization cross section and the number of weaklybound electrons can be arranged in a narrow universalband in terms of the projectile velocity. We applied thisrule to several hydrocarbons and nucleobases and notedthat the width of the resulting universal band could besignificantly reduced if we consider the number of activeelectrons in the collision based on the CDW results forthe different atoms. The new scaling was then testedtheoretically and by comparison with experimental dataavailable.The approach SSM considers the atoms in the moleculeas neutral, which is not correct. In Section III C, weused the molecular electronic structure code gamess [6]to calculate the excess or defect of electron density on theatoms composing the molecules. Then, we modified theSSM to account for the departure from the neutrality ofthe atoms. We find that the modified SSM for the DNAmolecules does not introduce substantial changes in thecross sections. II. THEORY: IONIZATION OF ATOMS
In the present study, we consider six atoms, α = H,C, N, O, P, and S, and six projectiles, antiprotons ¯ p , H + ,He + , Be + , C + , and O + . Most of the organic moleculesare composed of these atoms. Some particular moleculesalso include halogen atoms such as fluor and bromine;ionization cross sections of these elements have been pre-viously published [2].The total ionization cross sections of these atoms werecalculated using the CDW. The initial bound and fi-nal continuum radial wave functions were obtained byusing the radialf code, developed by Salvat and co–workers [7], and a Hartree–Fock potential obtained fromthe Depurated Inversion Method [8, 9]. We used a fewthousand pivot points to solve the Schr¨odinger equation,depending on the number of oscillations of the contin-uum state. The radial integration was performed usingthe cubic spline technique. We expand our final contin-uum wave function as usual, ψ −Ð→ k (Ð → r ) = l max ∑ l = l ∑ m = − l R − kl ( r ) Y ml (̂ r ) Y m ∗ l (̂ k ) . (1)The number of angular momenta considered varied from8, at very low ejected–electron energies, up to l max ∼ Z .The impact energies considered range between 0.1 to 10MeV/amu, where the CDW is supposed to hold. In fact,for the highest projectile charges the minimum impactenergy where the CDW is expected to be valid could behigher than 100 keV. We also performed similar calcula-tions with the first Born approximation, and we corrobo-rated that it provides quite reliable results only for ener-gies higher than a couple of MeV/amu. We use the sameline color to indicate the projectile charge throughout allthe figures of this work: dashed–red, solid–red, blue, ma-genta, olive and orange for antiprotons, H + , He + , Be + ,C + , and O + , respectively. Notably, there is no completetabulation of ionization of atoms by the impact of mul-ticharged ions. We hope that the ones presented in thisarticle will be of help for future works.Simultaneously, we will be reporting state to state ion-ization cross sections for the 36 ion–target systems con-sidered in the present work [11]. A great numerical effortwas paid to obtain these results, and we expect that theywill be useful to estimate molecule fragmentation. A. Emitted electron energies
In a given biological medium, direct ionization by ionimpact accounts for just a fraction of the overall damage.Secondary electrons, as well as recoil target ions, alsocontribute substantially to the total damage. We canconsider the single differential cross section of the shell nl of the atom α , dσ αnl / dE , to be a function of the ejectedelectron energy E as a simple distribution function [12].Then, we can define the mean value E α as in Ref. [13], E α = ⟨ E α ⟩⟨ ⟩ = σ α ∑ nl ∫ dE E dσ α,nl dE , (2) ⟨ ⟩ = σ α = ∑ nl ∫ dE dσ α,nl dE , (3)where Σ nl takes into account the sum of the differentsub–shell contributions of the element α .The mean emitted electron energies E α for H, C, N,O, P and S are shown in Fig. 2. The range of impactvelocities was shortened to v =
10 a.u. due to numer-ical limitations in the spherical harmonics expansion ofEq. (1). As the impact velocity v increases, so do ⟨ E α ⟩ and l max , which results in the inclusion of very oscillatoryfunctions in the integrand. Furthermore, the integrandof ⟨ E α ⟩ includes the kinetic energy E (see Eq. (2)), whichcancels the small energy region and reinforces the largevalues, making the result more sensible to large angu-lar momenta. Regardless, for v >
10 a.u., the first Bornapproximation holds.In Fig. 2, we estimate E α of the emitted electron inthe 10–70 eV energy range, for all the targets. Our re-sults agree with the experimental findings [12]. As can I o n i z a t i o n C r o ss S ec t i o n / Z ( − c m ) H × +1 − H + ¯ p He Be C O C +1 − N +1 − O +1 − Impact energy (MeV/amu)P +1 − S +1 − FIG. 1: Reduced CDW total ionization cross section of six atomic targets. The curves are labeled with the charge statecorresponding to the six multicharged projectiles. E ( e V ) H − H + ¯ p He Be C O C N O Impact energy (MeV/amu)P S Born
FIG. 2: Mean emitted energy distribution for ionization by the impact of multicharged ions, given by Eq. (2). Dashed lines for¯ p , solid lines for ion charges +
1, 2 + , 4 + , 6 + and 8 + , as indicated. be noted in the figure, the mean energy value is surpris-ingly sensitive to the projectile charge Z , which can du-plicate the proton results in the intermediate region, i.e.,100–400 keV/amu. The effect observed can be attributedto the depletion caused by the multicharged ions to theyields of low energy electrons. This behavior cannot befound in the first Born approximation, where the Z lawcancels the Z dependence in Eq. (2). At high energies, E α tends to a universal value for all ions, as can be seenin Fig. 2. B. Emitted electron angles
As mentioned before, secondary electrons contribute tothe total damage. Then, not only the ejection energy is θ ( a n g ) H − H + ¯ p He Be C O C N O Impact energy (MeV/amu)P S Born
FIG. 3: Mean emitted angle distribution for ionization by impact of multicharged ions. essential but also the angle of emission. Once again, wecan consider the single differential cross section in termsof the ejected electron solid angle Ω, dσ α,nl / d Ω, to beexpressed as a distribution function, and the mean angle θ α can be defined as θ α = ⟨ θ α ⟩⟨ ⟩ = σ α ∑ nl ∫ d Ω θ dσ α,nl d Ω (4)The mean emitted electron angles θ α for the six atomsand six ions of interest are shown in Fig. 3. An significantdependence of θ α with Z is noticed for all the cases. Onceagain, this effect could not be observed in the first Bornapproximation (dotted line).It is a general belief [14] that the angular dispersionof emitted electrons is nearly isotropic. This behav-ior is thought to be caused by the insignificant angularanisotropy of sub–50–eV yield. A typical value for theejection angle considered in the literature is θ α ∼ ° [12],and it is quite correct in the range of validity of the firstBorn approximation for any target. However, when adistorted wave approximation is used, θ α decreases sub-stantially with Z in the intermediate energy region. Thiseffect is evident in Fig. 3; for example, at 0.3 MeV/amu–where the Bragg peak for C + impact occurs– the CDW θ α is half of the first Born value. This correction closesthe damage to the forward direction. We can attributethis forward direction correction to the capture to thecontinuum effect; the higher the charge Z , the smaller θ will be. Of course, this effect only holds at intermediateenergies and not at high impact energies.Furthermore, Fig. 3 provides an illustrative descriptionof the behavior of antiprotons: the projectile repels theelectrons, being θ α ∼ ° . Note the opposite effect of proton and antiprotons respect to the first Born approx-imation; this phenomenon constitutes an angular Barkaseffect. III. IONIZATION OF MOLECULESA. The stoichiometric model
Lets us consider a molecule M composed by n α atomsof the element α , the SSM approaches the total ionizationcross section of the molecule σ M as a sum of ionizationcross sections of the isolated atoms σ α weighted by n α , σ M = ∑ α n α σ α . (5)We classified the molecular targets of our interest in threefamilies: CH, CHN, and DNA, as in Table I.In Fig. 4, we report the total ionization cross sectionsby the impact of multicharged ions for adenine, cyto-sine, guanine, and thymine, by combining the SSM andCDW results in Eq. 5. For adenine, the agreement withthe experimental data available for proton impact [15]is excellent. To the best of our knowledge, there are noexperimental data on ion–collision ionization for the restof the molecules. We have also included in this figureelectron impact measurements [16] with the correspond-ing equivelocity conversion for electron incident energieshigher than 300 eV. In this region, the proton and elec-tron cross section should converge. Although the elec-tron impact measurements are above our findings for allthe molecular targets, it is worth stating that our resultsagree very well with other electron impact theoreticalpredictions [17, 18]. CH CH (methane), C H (acetylene),C H (ethene), C H (ethane),C H (benzene)CHN C H N (pyridine), C H N (pyrimidine),C H N (dimenthylamine),CH N (monomethylamine)DNA C H N (adenine), C H N O (cytosine),C H N O (guanine), C H N O (thymine),C H N O (uracil), C H O (THF),C H O P (DNA backbone),C H N O P (dry DNA)TABLE I: Molecular targets studied in this work, classified inthree families. Our results for uracil, DNA backbone, pyrimidine, andTHF are displayed in Fig. 5. For uracil, the agreementwith the experimental proton impact measurements byItoh et al. [19] is good. However, for the same tar-get, our theory is a factor of two above the experimen-tal ionization measurements by Tribedi and collabora-tors [20, 21] by the impact of multicharged ions. Nonethe-less, it should be stated that our theoretical results coin-cide with calculations by Champion, Rivarola, and col-laborators [20, 22], which may indicate a possible misstepof the experiments.For pyrimidine, we show a comparison of our resultswith experimental data for proton impact by Wolff [23]and also for electron impact ionization [24] at high ener-gies. The electron impact measurements agree with ourcalculations for energies higher than 500keV. Unexpect-edly, the proton impact cross sections are significantlylower than our findings. Much more experiments areavailable for ionization of THF molecule by proton [25]and by electron [24, 26, 27] impact. Our SSM with CDWresults show overall good agreement with these data.
B. Scaling rules
1. Toburen rule
The first attempt to develop a comprehensive butstraightforward phenomenological model for electronejection from large molecules was proposed by Toburenand coworkers [4, 5]. The authors found it convenient toscale the experimental ionization cross section in terms ofthe number of weakly–bound electrons, n e . For instance,for C, N, O, P, and S, this number is the total numberof electrons minus the K–shell. Following Toburen, thescaled ionization cross section per weakly bound electron σ Te is σ Te = σ M n e , (6) where n e = ∑ α n α ν Tα , and ν Tα are the Toburen numbersgiven by ν Tα = ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ , for H,4 , for C,5 , for N and P,6 , for O and S . (7)The Toburen rule can be stated by saying that σ e is a universal parameter independent on the molecule, whichdepends solely on the impact velocity, and holds for highimpact energies (i.e., 0.25–5 MeV/amu). These ν Tα canbe interpreted as the number of active electrons in thecollision. At very high energies, the K–shell electronswill also be ionized, and these numbers will be different.A similar dependence with the number of weakly boundelectrons was found in Ref. [19] for proton impact onuracil and adenine.Following the Toburen scaling, we computed the scaledCDW cross sections σ Te for the molecular targets of Ta-ble I. Our results are shown in Fig. 6a as a function ofthe impact energy for different projectile charges. Al-though the Toburen scaling holds for high energies, itsperformance is still not satisfactory: the universal bandis quite broad, as can be noted in this figure.
2. CDW–based scaling
The departure of our theoretical results from the To-buren rule can be easily understood by inspecting Fig. 1.It can be noted that the rule σ α / ν Tα ∼ σ Te , approximatelyconstant, is not well satisfied by the CDW. For example,Fig. 1 shows that the cross sections for O are very similarto the cross sections for C, suggesting 4 active electronsin O instead of 6. In the same way, the number of activeelectrons for N, P, and S obtained with the CDW are alsodifferent from the ν Tα of Eq. (7).Based on the CDW results, we propose a new scaling, σ ′ e = σ M n ′ e , (8)where n ′ e = ∑ α n α ν CDW α , and ν CDW α are the numbers ofactive electrons per atom obtained from the CDW ion-ization cross sections for different ions in H, C, N, O, P,and S targets, given as follows, ν CDW α ∼ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ , for H,4 , for C, N, and O,4 . , for P and S . (9)The new scaled cross sections σ ′ e are plotted in Fig. 6b.The experimental data for ionization of adenine [15],uracil [19], pyrimidine [23], and THF [25] by proton im-pact in Fig. 6b seems to corroborate the new scaling.We also included the electron impact ionization measure-ments with equivelocity conversion on pyrimidine [24] I o n i z a t i o n C r o ss S ec t i o n / Z ( − c m ) AdenineC H N +1 − H + ¯ p He +2 Be +4 C +6 O +8 H + e − CitosineC H N O +1 − GuanineC H N O +1 − Impact Energy (MeV/amu) ThymineC H N O +1 − FIG. 4: Reduced CDW ionization cross section by the impact of multicharged ions. Experiments: ○ [15] for proton impactand ◻ [16] for electron impact with equivelocity conversion. I o n i z a t i o n C r o ss S ec t i o n / Z ( − c m ) UracilC H N O +1 − H + ¯ p He +2 Be +4 C +6 O +8 H + C C , O , F O , F PyrimidineC H N +1 − + e − THFC H O +1 − − e − e − Impact Energy (MeV/amu) DNA backboneC HO P +1 − FIG. 5: Reduced CDW ionization cross section by the impact of multicharged ions. Experiments: proton impact on △ uracil [19], ▽ pyrimidine [23] and ◇ THF [25]. Impact of ⊖ C + , ⊕ C + , O + , F + , and ⊗ O + , F + on uracil [20, 21]. Symbols ⊳ [24], ⊲ [26], and ☆ [27] for electron impact with equivelocity conversion. and THF [24, 26, 27]. It will be interesting to cross–check with future experiments, mainly for higher projec-tile charge states.By using Eq. (9), we define new active electron num-bers n ′ e for some molecules of interest in Table II. Thesevalues are different from the ones proposed by Toburenand used by other authors [19]. Moreover, an alternativeway of showing the scaling can be attained by plotting the ionization cross sections of molecules as a function ofthe number of active electrons from Table II. Our find-ings are displayed in Fig. 7 for impact energies 0.5, 1,and 2 MeV. As can be noted, the computed CDW ion-ization cross sections for all the molecules show a lineardependence with the number of electrons from Table II.We obtain similar results, even for E =
10 MeV. Thecomparison with the experimental data available shows
Impact Energy (MeV/amu) σ e ( − c m ) (a)Toburen rule +1+2+4+6+8 H + He +2 Be +4 C +6 O +8 Present work (b) +1+2+4+6+8
FIG. 6: Scaled ionization cross section per weakly bound electron using (a) the Toburen numbers ν Tα , and (b) our proposednumbers ν CDW α . Experiments: proton impact on ○ adenine [15], △ uracil [19], ▽ pyrimidine [23] and ◇ THF [25]; electronimpact on ⊳ pyrimidine [24], and ⊲ , ☆ [26, 27] THF. n ′ e I o n i z a t i o n C r o ss S ec t i o n ( − c m ) H H ONH CH CH N C H N THF C H Uracil ThyminePyrimidine Cytosine AdenineGuanine
FIG. 7: Ionization cross sections by the impact of protons at 0.5, 1, and 2 MeV in terms of the number of active electrons givenby Table II. Experiments: ○ adenine [15], △ uracil [19], ▽ pyrimidine [23], ▲ C2H N, CH N, methane and ammonia [28], ☀ ammonia and H [29], and ● water [30]. overall good agreement, for the smallest molecules, H ,H O, and CH , up to the most complex ones, like ade-nine. For electron impact data, the experimental datawas interpolated between close neighbors.While finishing the present work, we became aware ofan accepted manuscript by Luedde et al. [31] on totalionization of biological molecules by proton impact, us-ing an independent–atom–model pixel counting method.The authors also raised a scaling with ν α = ν α = C. Molecular structure of targets
To test the range of validity of the SSM, we performed ab initio molecular calculation of five nucleobases by em-
Molecule n ′ e Molecule n ′ e Molecule n ′ e H H N 19 C H N O 37H O 6 C H O 28 C H N O H N
28 C H N H
30 C H N O 49CH N 13 C H N O
36 C H O P 54.5TABLE II: New scaling numbers for some molecular targetsof biological interest. ploying the gamess code. The geometry optimizationand single point energy calculations were performed im-plementing the restricted Hartree–Fock method and the3-21G basis set.The molecular binding energies of the valence electronsfor adenine, cytosine, guanine, thymine, and uracil areshown in Fig. 8. The binding energy of the highest molec-ular orbital (HOMO) agrees with the experimental val-ues [32–34] within 2% for all the DNA bases considered.On the left side of Fig. 8, we show the atomic Hartree–Fock energies of the constituent elements, which gives aninsight into the distribution of the weakly bound elec-trons in the molecules. A dashed line around −
26 eV isdrawn to separate the molecular band in two. We canconsider the atomic energy levels above this line as theones corresponding to the weakly bound electrons fromEq. (9). For example, the 2 s and 2 p electrons of carbonare placed above the separating line, which correspondsto the 4 electrons given by CDW–scaling. In the caseof O, only the 4 electrons of the 2 p orbitals are locatedabove the separating line, which corresponds to the num-ber of weakly bound electron given by our new scaling.The N case is not as straightforward; the ν CDW N = wouldsuggest that one out of the two 2 s electrons contributeto the molecular scheme.
1. A modified stoichiometric model
The SSM considers the molecule to be assembled byisolated neutral atoms, which is definitively unrealistic.A first improvement can be suggested by assuming thatthe atoms are not neutral and that they have an unevendistribution of electrons within the molecule, which canbe expressed as an effective charge q α per atom. TheMulliken charge gives a possible value for q α ; however,there are a wide variety of charge distributions [35].To take this effect into account, we can consider thatthe total amount of electrons Q α on the element α isequally distributed on all the α atoms. Therefore, eachelement α will have an additional charge, q α = Q α / n α ,which can be positive or negative. This amount will de-pend on the relative electronegativity respect to the otheratoms [36]. Following this idea, we can estimate a new number of atoms per molecule n ′ α , given by n ′ α = n α − q α ν CDW α (10)In the case of neutral atoms, q α = n ′ α = n α , as itshould be. In Table III, we display the average effectivecharge per atom q α of C, H, N, and O, for five DNAmolecules, obtained from the full molecular calculationdescribed above.By implementing Eq. (10), it is possible to determinea new stoichiometric formula (last column of Table III).Now, instead of having an integer number of atoms n α ,we obtain a fractional number n ′ α . New molecular crosssections σ ′ M = ∑ α n ′ α σ α can be computed consideringsuch values. Relative errors for the ionization cross sec-tions were computed for the DNA bases from Table III.The differences obtained were less than 3%, which in-dicates that the modified SSM is a quite robust modelto handle these type of molecules within the range errorexpected for this model. IV. CONCLUSIONS
In this work, we have dealt with the calculation of ion-ization cross sections of seventeen biological moleculescontaining H, C, N, O, P, and S by the impact of an-tiprotons, H + , He + , Be + , C + , and O + . To that end,we have employed the full CDW method and the simplestoichiometric model. The mean energy and angle of theemitted electrons, of importance in post–collisional radi-ation damage, has also been calculated, showing a cleardependence with the ion charge Z . For a given target as Z increases, E α increases, but θ α decreases, showing aclear tendency to the forward direction. At impact ener-gies greater than 2 MeV/amu, these values converge tothe Born approximation, which embodies the simple Z law.Total ionization cross sections for adenine, cytosine,thymine, guanine, uracil, DNA backbone, pyrimidine,and THF are presented and compared with the scarcelyavailable experiments. We explore the rule of Toburen,which scales all the molecular ionization cross sectionnormalizing with a certain number of weakly bound orvalence electrons. We found that the ionization crosssections scales much better when normalizing with thenumber of active electrons in the collision obtained fromthe CDW results for atoms. This new scaling was testedwith good results for the six projectiles and seventeenmolecules studied here. The comparison with the experi-mental data reinforce these results. Furthermore, we alsotested the scaling by including experimental data of ion-ization of H , water, methane, and ammonia by protonimpact showing good agreement at intermediate to highenergies.Finally, we performed full molecular calculations forthe DNA basis. By inspecting the molecular binding en-ergy from quantum mechanical structure calculations, we C H N O C H N C H N O C H N O C H N O C H N O − − − − − − − − B i nd i n g e n e r g y ( e V ) A C G T U
FIG. 8: Theoretical molecular binding energies for adenine, cytosine, guanine, thymine, and uracil compared to those of atomicconstituents. Element C H N O New stoichiometryAdenine +0.32 +0.23 –0.55 C . H . N . Cytosine +0.28 +0.21 –0.56 –0.53 C . H . N . O . Guanine +0.46 +0.20 –0.58 –0.36 C . H . N . O . Thymine +0.20 +0.19 –0.54 –0.52 C . H . N . O . Uracil +0.31 +0.22 –0.59 –0.47 C . H . N . O . TABLE III: Average effective Mulliken charge per atom q α , and new stoichiometric formula defined by Eq. (10) for five DNAmolecules. were able to understand the number of electrons proposedin our new CDW-based scaling. We attempt to improvethe stoichiometric model by using the Mulliken charge toget fractional rather than integer proportions. We foundno substantial correction, which indicates that the SSMworks quite well.In conclusion, the present results reinforce the relia-bility of the SSM to deal with complex molecules in the intermediate to high energy range. Moreover, the sim-ple stoichiometric model and the CDW cross sections inRef. [11] opens the possibility to describe a wide range ofmolecules containing H, C, N, O, P, and S, by the impactof multicharged ions. [1] Fainstein P.D., Ponce V. H. and Rivarola R. D. J. Phys.B: At. Mol. Opt. Phys.
287 (1988).[2] J. E. Miraglia and M. S. Gravielle. Ionization of the He,Ne, Ar, Kr, and Xe isoelectronic series by proton impact.Phys Rev A , 052705 (2008)[3] J. E. Miraglia, Ionization of He, Ne, Ar, Kr, and Xeby proton impact: Single differential distributions. Phys.Rev. A , 022708 (2009).[4] W. E. Wilson and L. H. Toburen. Electron emission fromproton –hydrocarbon-molecule collisions at 0.3–2.0 MeV.Phys. Rev. A , 1303 (1975).[5] D. J. Lynch, L. H. Toburen, and W. E. Wilson. Electronemission from methane, ammonia, monomethylamine, and dimethylamine by 0.25 to 2.0 MeV protons. J. Chem.Phys. , 2616 (1976).[6] M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. El-bert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Mat-sunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M.Dupuis, J. A. Montgomery J. Comput. Chem. , 1347-1363 (1993).[7] Salvat, F., Fern´andez-Varea, J.M., Williamson, W. Com-put. Phys. Commun. , 151–168 (1995)[8] A.M.P. Mendez, D.M. Mitnik, and J.E. Miraglia. Depu-rated inversion method for orbital-specific exchange po-tentials. Int. J. Quantum Chem. ,116 (2016).[9] A.M.P. Mendez, D.M. Mitnik, and J.E. Miraglia. Local Effective Hartree–Fock Potentials Obtained by the Depu-rated Inversion Method, Adv. Quant. Chem. , 117–132(2018).[10] C. C. Montanari, J. E. Miraglia, Ionization probabilitiesof Ne, Ar, Kr, and Xe by proton impact for different ini-tial states and impact energies. Nucl. Instr. Meth. Phys.Res. B , 236–243 (2017).[11] J. E. Miraglia. Shell-to-shell ionization cross sections ofantiprotons, H + , He + , Be + , C + and O + on H, C, N,O, P, and S atoms (to be published).[12] Multiscale approach to the physics of radiation dam-age with ions. E. Surdutovich and A. V. Solov’yov,arXiv:1312.0897v, (2013)[13] P. de Vera1, I. Abril, R. Garcia-Molina andA.V.Solov’yov, Ionization of biomolecular targetsby ion impact: input data for radiobiological applica-tions. Journal of Physics: Conference Series , 012015(2013).[14] M. E. Rudd, Y.-K. Kim,, D. H. Madison and T. J. Gay.Electron production in proton collisions with atoms andmolecules: energy distributions. Rev. Mod. Phys. , 44–490 (1992).[15] Y. Iriki, Y. Kikuchi, M. Imai, and A. Itoh Phys. Rev. A , 052719 (2011).[16] M. A. Rahman and E. Krishnakumar, Electron ionizationof DNA bases, J. Chem. Phys. , 161102 (2016).[17] P. Mozejko and L. Sanche, Cross section calculations forelectron scattering from DNA and RNA bases. RadiatEnviron. Biophys , 201 (2003).[18] H. Q. Tan, Z. Mi, and A. A. Bettiol, Simple and uni-versal model for electron-impact ionization of complexbiomolecules, Phys. Rev. E , 032403 (2018)[19] A. Itoh, Y. Iriki, M. Imai, C. Champion, and R. D. Ri-varola, Cross sections for ionization of uracil by MeV-energy-proton impact, Phys. Rev. A , 052711 (2013).[20] A. N. Agnihotri, S. Kasthurirangan, S. Nandi, A. Kumar,M. E. Galassi, R. D. Rivarola, O. Foj´on, C. Champion,J. Hanssen, H. Lekadir, P. F. Weck, and L. C. Tribedi.Ionization of uracil in collisions with highly charged car-bon and oxygen ions of energy 100 keV to 78 MeV. Phys.Rev. A , 032711 (2012).[21] A N Agnihotri, S Kasthurirangan, S Nandi, A Kumar,C Champion,, H Lekadir, J Hanssen, P FWeck, M EGalassi, R D Rivarola, O Fojon and L C Tribedi, Abso-lute total ionization cross sections of uracil (C H N O )in collisions with MeV energy highly charged carbon,oxygen and fluorine ions J. Phys. B , 185201 (2013).[22] C Champion, M E Galassi, O Foj´on, H Lekadir, JHanssen, R D Rivarola, P F Weck, A N Agnihotri, SNandi, and L C Tribedi. Ionization of RNA-uracil byhighly charged carbon ions. J. Phys.: Conf. Ser. ,012004 (2012).[23] W. Wolff, H. Luna, L. Sigaud, A. C. Tavares, and E. C.Montenegro Absolute total and partial dissociative cross sections of pyrimidine at electron and proton intermedi-ate impact velocities J. Chem. Phys. , 064309 (2014).[24] M. U. Bug, W. Y. Baek, H. Rabus, C. Villagrasa,S. Meylan, A. B. Rosenfeld, An electron-impact crosssection data set (10 eV–1 keV) of DNA constituentsbased on consistent experimental data: A requisite forMonte Carlo simulations, Rad. Phys. Chem. , 459–479 (2017).[25] M. Wang, B. Rudek, D. Bennett, P. de Vera, M. Bug,T. Buhr, W. Y. Baek, G. Hilgers, H. Rabus, Cross sec-tions for ionization of tetrahydrofuran by protons at ener-gies between 300 and 3000 keV Phys. Rev. A , 052711(2016).[26] W. Wolff, B. Rudek, L. A. da Silva, G. Hilgers, E.C. Montenegro, M. G. P. Homem, Absolute ioniza-tion and dissociation cross sections of tetrahydrofuran:Fragmentation–ion production mechanisms J. Chem.Phys. , 064304 (2019).[27] M. Fuss, A. Muoz, J. C. Oller, F. Blanco, D. Almeida,P. Limo-Vieira, T. P. D. Do, M. J. Brunger, G.Garc´ıa, Electron-scattering cross sections for collisionswith tetrahydrofuran from 50 to 5000 eV Phys. Rev. A , 052709 (2009).[28] D. J. Lynch, L. H. Toburen, and W. E. Wilson, Electronemission from methane, ammonia, monomethylamine,and dimethylamine by 0.25 to 2.0 MeV protons J. Chem.Phys. , 2616 (1976).[29] M.E. Rudd, Y.-K. Kim, D.H. Madison, J.W. Gallagher,Electron production in proton collisions: total cross sec-tions, Review of Modern Physics, , 965–994 (1985).[30] H. Luna, A. L. F. de Barros, J. A. Wyer, S. W. J. Scully,J. Lecointre, P. M. Y. Garcia, G. M. Sigaud, A. C. F.Santos, V. Senthil, M. B. Shah, C. J. Latimer, and E. C.Montenegro, Water-molecule dissociation by proton andhydrogen impact, Phys. Rev. A , 042711 (2007).[31] H J Luedde et al. , 11 (1975).[33] Verkin, B.I.; Sukodub, L.F.; Yanson, I.K., Ionization po-tentials of nitrogenous bases of of nucleic acids, Dokl.Akad. Nauk SSSR, , 1452 (1976).[34] Dougherty, D.; Younathan, E.S.; Voll, R.; Abdulnur, S.;McGlynn, S.P., Photoelectron spectroscopy of some bio-logical molecules, J. Electron Spectrosc. Relat. Phenom., , 379 (1978).[35] Jung-Goo Lee, Ho Young Jeong, and Hosull Lee, Chargesof Large Molecules Using Reassociation of Fragments.Bull. Korean Chem. Soc. , 369 (2003).[36] A. K. Rappe, A. K.and W. A. Goddard III,. J. Phys.Chem.95