Structure, Ionization, and Fragmentation of Neutral and Positively

Article pubs.acs.org/JPCA

Structure, Ionization, and Fragmentation of Neutral and Positively Charged Hydrogenated Carbon Clusters: CnHq+ m (n = 1−5, m = 1−4, q = 0−3) Published as part of The Journal of Physical Chemistry A virtual special issue “Spectroscopy and Dynamics of Medium-Sized Molecules and Clusters: Theory, Experiment, and Applications”. Juan P. Sánchez,† Néstor F. Aguirre,*,† Sergio Díaz-Tendero,†,‡ Fernando Martín,†,¶,‡ and Manuel Alcamí†,¶ †

Departamento de Química, Módulo 13 and ‡Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain ¶ Instituto Madrileño de Estudios Avanzados en Nanociencias (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain S Supporting Information *

ABSTRACT: In this work we present a systematic theoretical study of neutral and positively charged hydrogenated carbon clusters (CnHq+ m with n = 1−5, m = 1−4, and q = 0−3). A large number of isomers and spin states (1490 in total) was investigated. For all of them, we optimized the geometry and computed the vibrational frequencies at the B3LYP/6-311++G(3df,2dp) level of theory; more accurate values of the electronic energy were obtained at the CCSD(T)/6-311++G(3df,2dp) level over the geometry previously obtained. From these simulations we evaluated several properties such as relative energies between isomers, adiabatic and vertical ionization potentials, and dissociation energies of several fragmentation channels. A new analysis technique is proposed to evaluate a large number of fragmentation channels in a wide energy range.

1. INTRODUCTION Astrochemistry is a discipline that has received a great deal of attention in the last decades. A special area of interest is the study of molecular gas clouds where many reactions involving heavy elements (C, N, O, ...) and their combination with H can occur, leading to species of increasing complexity. Today ∼180 molecules have been detected in the interstellar media (ISM) or Circumstellar Shells.1 Among them, hydrocarbons constitute a great proportion of the ISM chemistry: from the simplest CH, detected in the early 1940s,2 to more complex CnHm,3 a large variety of this kind of molecules have been detected in several environments in the ISM such as diffuse clouds, cold dense molecular clouds, star forming regions, photodissociation regions, and circumstellar envelopes.4−6 Also their presence could be responsible for some detected spectroscopic features still unidentified, known as diffuse interstellar bands.7 Hydrogenated carbonaceous molecules play a key role in the chemical evolution of interstellar species5,8 and in combustion processes.9 A very important part of the ISM chemistry is triggered after the interaction of energetic particles, such as electrons, ions, or photons that flow outward from stars (solar wind) with such molecules, leading to highly reactive charged and radical species.10−12 Thus, the study of this kind of interaction has grown in the last years. New experimental © XXXX American Chemical Society

techniques have been developed allowing storage and measurement of properties of multicharged cations.13−16 The use of heavy-ion accelerators and synchrotron radiation sources for excitation, followed by detection through multicoincidence spectra,17,18 provide very useful information on the stability of charged and excited molecular systems.19,20 For example, triple coincidence techniques have allowed the determination of the momentum of all fragments produced from highly excited C3 molecules.21 Furthermore, the use of time-dependent photoionization techniques allows the experimental study of their evolution and permit to investigate their dynamics (see, e.g.,22 where ultrafast isomerization of acetylene cations [HCCH]+ is detected). Experimental and theoretical efforts have been done to provide data that help in the understanding of interstellar composition. Theoretical modeling is essential for the correct identification of molecules or ions already detected in ISM, to characterize their properties and to model the chemical processes in which they are involved. In particular, geometries, simulated spectra, energies of their different species, and the Received: October 16, 2015 Revised: December 18, 2015

A

DOI: 10.1021/acs.jpca.5b10143 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

Coupled Cluster calculations65,66 using single and double substitutions and including triple excitations noniteratively.67 In particular we use single-point CCSD(T)/6-311++G(3df,2p) calculations for all compounds using the B3LYP/6-311+ +G(3df,2p) optimized geometries. These calculations were performed using the Gaussian09 computational package.68 This level of theory has been shown to give very successful results to describe the properties of bare carbon clusters,69,70 even though pathological behaviors of the wave function can arise when using single-reference based methods. In those cases where spin contamination is high, one can obtain spurious results at DFT level.71 Therefore, in all DFT calculations of this study, we evaluated the mean value of S2, and we performed a wave function stability test to identify the results that are stable and those that present different kinds of instabilities (internal, restricted−unrestricted (R-U), or combined (internal + RU)).72−74 In a previous work on linear C2H and C4H the unrestricted−restricted difference is used to quantify the spin contamination.74 Additionally the monoreferential character in the CCSD(T) calculations was evaluated using the T1 diagnostic75 and the norm of the coupled cluster wave function. To test the reliability of the method used we chose some benchmark systems to perform calculations using 12 different DFT functionals and two post Hartree−Fock methods. Equilibrium geometries, vibrational frequencies, and ionization potentials were evaluated and compared with the corresponding experimental data available in the literature. The assessed DFT methods cover the range from meta-GGA, hybrid, and long-range corrected hybrid models (M06L,76 VSXC,77 B3LYP,78−80 PW91PW91,81−85 PBE1PBE,86−88 mPW3PBE,89 BHandHLYP, 68 LC-wPBE, 90−92 CAM-B3LYP, 93 LCBLYP,79,80,94,95 wB97XD96). Post Hartree−Fock methods, Coupled Cluster, and CASPT297−103 were also used to evaluate qualitatively the dynamic and static correlation contributions. In the CASPT2 calculations all valence electrons and orbitals were included in the active space. CASPT2 calculations were done using the Molpro2009 Software Package.104 To test the convergence in the basis set size, calculations with a larger basis set, 6-311++G(3df,3pd), were also performed. In addition the adiabatic and vertical ionization potentials (IP) were calculated. The adiabatic IPs were computed as the energy difference between the most stable structure for a given charge q, CnHq+ m , and the most stable structure after the extraction of one electron, CnH(q+1)+ . These values include ZPE m corrections for products and reactants. The vertical IPs were computed as the difference between the energy of the most stable neutral isomer and the energy obtained keeping fixed the geometry and extracting 1, 2, or 3 electrons, that is, considering a fast ionization where the molecule does not have time to relax its geometry (Franck−Condon type transition). All evaluated IPs are given in Tables 1 and 2 of the Supporting Information. As it was pointed out by Dixon et al.,105 the electronic energy converges more rapidly for the IP than for the electron affinity, where the diffuse nature of the anion’s electron density requires extra diffuse basis functions. In the systems that we studied, by extending the basis set we estimated an error of ∼0.006 eV in the ZPE values and ∼0.005 eV in the electronic energy, yielding a final calculated error in IP of ∼0.011 eV.

rate of the possible reactions are crucial quantities in astrochemical modeling. In the last years there has been a large effort to provide such information for carbon clusters and hydrocarbon molecules, both in neutral and ionic states.19,23−29 All these efforts converge into different databases such as the Kinetic Database for Astrochemistry (KIDA).30 Recent works have focused on understanding the formation and growth of hydrogenated carbon clusters in the ISM with different theoretical methodologies.31−33 Some studies have been also performed on the anharmonicities present in this kind of systems, such as C3H+3 ,34,35 an important case due to its aromatic stability. In the present article we studied in a systematic way the properties of neutral and positively charged hydrogenated carbon clusters following our previous theoretical studies for carbon clusters.36−38 Previous theoretical studies have been performed for some of the neutral CnHm28,39−50 and singly charged systems47,51−55 considered in this work. For dications only very specific cases, such as acetylene dication,56 have been theoretically studied before, and one previous work for trications focuses on pure carbon clusters.38 Theoretical studies on anionic species have been also reported.51,57,58 In the case of neutral species the information is quite fragmentary, as different authors have only considered specific isomers at different levels of theory. For singly charged systems one previous systematic study performed by Fantuzzi et al.51 (n = 1−4; m = 1−4) considers, as in the present work, a very large number of isomeric forms. Our final goal is to extend our fragmentation models,37 previously reported for carbon clusters,59 to hydrogenated species for the interpretation of recent experimental measurements.23,60,61 To study the different ionization and fragmentation channels of clusters it is necessary to have a consistent description of all species at the same level of theory. Since the previous works were performed using different levels of theory, the sparse and heterogeneous available information cannot be used for our purposes; properties computed with different methods can lead to biased results, providing an unsatisfactory comparison between them. In addition to providing theoretical information for sizes, isomers, and charged states not considered so far, the present work aims at homogenizing the sparse available theoretical data for the hydrogenated carbon clusters. We focus on neutral and positively charged CnHq+ m (n = 1−5, m = 1−4, q = 0−3). We present a systematic theoretical study of the structure, frequencies, ionization potentials, and dissociation energies of these molecules. The data reported in this work should be useful to infer the chemical species present in the ISM, to model the chemical processes taking place in the ISM, and to extend databases such as KIDA.30

2. METHODOLOGY For all systems under study density functional theory (DFT) calculations were performed by using the hybrid B3LYP functional combined with a 6-311++G(3df,2p) basis set.62 The B3LYP functional combines the Becke three-parameter nonlocal hybrid exchange potential with the nonlocal correlation functional of Lee, Yang, and Parr.63,64 Geometry optimization was performed at this level for the different molecules under consideration. Harmonic vibrational frequencies were obtained at the same level to classify the stationary points of the potential energy surface as local minima or transition states and to estimate the corresponding zero point energies (ZPE). To improve the energies obtained at DFT level we also performed

3. RESULTS AND DISCUSSION This section is divided as follows: we will first analyze in Section 3.1 the results of the methodology assessment for some B


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The Journal of Physical Chemistry A Table 1. Comparison between Calculated and Experimental Resultsa for CH(2Π) and CH+(1Σ+) Molecules CH(2Π)

basis set 6-311++G(3df,2p) experimental CASPT2 CCSD(T) B3LYP CCSD(T)//B3LYP PW91PW91 PBE1PBE mPW3PBE TPSSTPSS BHandHLYP M06L VSXC CAM-B3LYP LC-wPBE LC-BLYP wB97XD a

CH+(1Σ+)

re

νe

(Å)

(cm−1)

109

2858.382

⟨S ⟩ 2

109

1.118 1.1319 1.1221(−3) 1.1229(−5)

2851(+15) 2825(+15)

1.1341(−6) 1.1250(−5) 1.1253(0) 1.1298(−6) 1.1105(−4) 1.1249(+3) 1.1295(−9) 1.1199(−5) 1.1199(−6) 1.1166(−5) 1.1225(−4)

2741(+20) 2845(+11) 2831(+8) 2782(+8) 2951(+6) 2810(−35) 2795(+14) 2871(+16) 2914(+20) 2934(+27) 2847(+9)

re

νe

(Å)

(cm−1)

110

ionization potential (eV) ⟨S2⟩ adiabatic

2857.56110

vertical

10.64111,112

0.750 0.760 0.753

1.1309 1.1110 1.1317(−4) 1.1373(−8)

2858(+12) 2798(+15)

0.000 0.000

0.753 0.754 0.754 0.755 0.754 0.755 0.754 0.753 0.754 0.753 0.753

1.1514(−10) 1.1401(−8) 1.1410(−9) 1.1415(−8) 1.1235(−7) 1.1530(+8) 1.1488(−16) 1.1359(−8) 1.1374(−4) 1.1362(−6) 1.1363(−5)

2692(+19) 2811(+11) 2792(+13) 2775.0(+9) 2932.0(+8) 2128(−184) 2701(+13) 2828(+16) 2859(+16) 2862(+24) 2827(+7)

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

10.54 11.02 10.55 11.07 10.99 11.04 10.93 10.84 10.91 10.98 11.03 11.13 11.08 10.84

10.55 11.02 10.55 11.07 11.00 11.05 10.94 10.85 10.96 10.99 11.04 11.14 11.09 10.85

See text for details.

Table 2. Calculated Values for the C−C Bond, C−H Bond, and Angles of the Cyclic C3H+3 for All the DFT Functionals and CCSD(T), Compared to Experiment c-C3H+3 (1A1)a

basis set 6-311++G(3df,2p)

a

rC−C

rC−H

ν8

ν10

(Å)

(Å)

(cm−1)

(cm−1)

1290113

3138113

1315(−5) 1318(0) 1292(−1) 1344(−1) 1332(0) 1306(0) 1372(−1) 1337(−5) 1319(−2) 1344(−1) 1376(−1) 1383(0) 1342(−1)

3256(−6) 3235(+6) 3172(+9) 3253(+4) 3240(+6) 3201(+3) 3330(+2) 3241(−9) 3231(+4) 3254(+6) 3279(+10) 3272(+12) 3261(+5)

experimental

1.363−1.373106

CCSD(T) B3LYP PW91PW91 PBE1PBE mPW3PBE TPSSTPSS BHandHLYP M06L VSXC CAM-B3LYP LC-wPBE LC-BLYP wB97XD

1.3667(+1) 1.3574(+1) 1.3653(+2) 1.3561(+1) 1.3569(+1) 1.3648(0) 1.3469(0) 1.3547(−1) 1.3630(+2) 1.3520(+1) 1.3505(+2) 1.3431(+1) 1.3546(+1)

1.0815(−7) 1.0803(−5) 1.0875(−4) 1.0819(−4) 1.0821(−5) 1.0839(−5) 1.0725(−5) 1.0805(−12) 1.0810(−4) 1.0800(−5) 1.0814(−4) 1.0809(−5) 1.0803(−6)

For all cases the angles C−C−C and H−C−C are 60.0° and 150.0°, respectively. These structures are close to the D3h symmetry.

C2H, C3H, and C4H, where we may expect to have strong static electronic correlation as in their non-hydrogenated counterparts.36 In most cases B3LYP and CCSD(T) calculations are compared with CASPT2 ones. Results are summarized in Table 1 for CH and CH+, in Table 2 for the cyclic C3H+3 cation, and in Table 3 for CnH systems. In Tables 1 and 2 the underlined values correspond to the DFT results in better agreement with the experiment. In the three tables, DFT and CCSD(T) values were computed with the 6-311++G(3df,2p) and 6-311++G(3df,3pd) basis set. The values shown correspond to the 6-311++G(3df,2p) basis and the difference with respect to the larger 6-311++G(3df,3pd) basis is given in parentheses, as the numbers corresponding to the last digits of the actual value (i.e., re = 1.1229(−5) Å means a value of 1.1229 using the 6-311++G(3df,2p) basis set and a value of 1.1224 when using the 6-311++G(3df,3pd) basis set) . The first observation in Table 1 is that the differences using a 6-311++G(3df,2p) or a

small/prototype molecules to determine the effect of the basis sets and the method to evaluate some properties where experimental results are available. In Section 3.2 we will discuss the strategy used for a systematic search of all possible isomers. One of the main difficulties of the present work is that hundreds of different structures need to be calculated and analyzed; the technique used to do a global analysis of the results will be discussed in detail in Section 3.3. The vibration spectra are discussed in Section 3.4; the relative stability of the isomers are in Section 3.5; and the most relevant energetic quantities for use in astrochemical modeling are shown in Section 3.6IPsand Section 3.7dissociation energies. 3.1. Methodology Assessment. We chose three sets of systems: the smallest ones CH and CH+, where 12 different functionals as well as CCSD(T) and CASPT2 were tested; the particular case of cyclic C3H3+ using 12 different DFT functionals and CCSD(T); and the most stable isomers of C


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are in agreement with both CCSD(T) and experimental results.106 For larger clusters Table 3 shows that the geometry does not vary significantly between the B3LYP and CASPT2 optimizations, being the larger differences of the order of 0.05 Å, and both give results in good agreement with the experimental values. In the case of C2H the adiabatic ionization potential at the CCSD(T) level differs in 0.15 eV from the CASPT2 one and 0.31 eV from the experimental value. Coupled cluster and configuration interaction results were examined to assess the reliability of single reference quantum methods for these kinds of systems. We found that, even when T1 diagnostic is high, the wave function shows a monoreferential character since the norm of the CCSD(T) wave function is close to 1.0.107 In this case the norm is 1.1347; that is, the reference configuration is predominant, and therefore the system can be studied with single-reference based methods such as DFT. Besides, there is a large improvement of results when moving from CASSCF to CASPT2 implying that in this kind of system the dynamic correlation has a much larger weight than the static one. The deviation of the order of 0.01 Å in the equilibrium distances and 0.3 eV in ionization energies at the CCSD(T)//B3LYP level with respect to the experimental value confirm the validity of this approach (see Table 3). This is in agreement with previous results from Mostafenejad et al.,108 where they also found a similar contribution of the most dominant configuration evaluated at the equilibrium geometry. From these results, the spin-unrestricted B3LYP functional turns out to be a trustworthy method, which gives reliable and accurate results with a reasonable computational cost, mainly regarding geometrical parameters and frequencies. Therefore, in the rest of the article we will use the B3LYP/6-311+ +G(3df,2p) level for computing the geometry and frequencies, and single-point calculations at the CCSD(T)/6-311++G(3df,2p) level using as reference the stable UHF solution to obtain total energy values. This level of theory is denoted hereafter as CCSD(T)//B3LYP. 3.2. Search for Isomers. A systematic geometry search was performed on each stoichiometric combination of the 80 CnHq+ m molecules studied (n = 1−5, m = 1−4, and q = 0−3). This geometry search was performed optimizing several initial geometries generated systematically: ∼700 initial structures were considered in total (for all n and m combinations) for each charge (q = 0−3) and considering different multiplicity values (singlet and triplet or doublet and quadruplet for even or odd number of electrons, respectively). After the optimization, duplicated geometries and nonminimum structures (characterized by imaginary vibrational frequencies) were removed. Thus, the obtained geometries are likely to cover most of the configurational space. A total of 255 different geometries were found for all charge states considered, and they are depicted in Table 3 of the Supporting Information. Table 4 indicates the number of calculated geometries (hereafter called isomers) and the associated structures (the latter distinguish between the different spin states) found as a minimum in the potential energy surface for each CnHmq+ molecule (hereafter two structures are considered different if they differ on geometry and/or spin). For charges +2 and +3 some molecules experience Coulomb explosions (specially the small ones); that is, in the geometry optimization, we do not found a minima, but instead we observe disociation with charge split among the species produced. Thus, the number of isomers and structures found decreases as charge increases.

Table 3. Theoretical Results for C2H, C3H, and C4H Compared with Experimental Data C2H(2Σ+) rC−C[Å] rC−H[Å] IPadiabatic[eV] T1 coef CCSD(T) norm ⟨S2⟩ C3H(2B2) rC1−H[Å] rC1−C2[Å] rC2−C3[Å] T1 coef CCSD(T) norm ⟨S2⟩ C4H (2Σ+) rC1−H[Å] rC1−C2[Å] rC2−C3[Å] rC3−C4[Å] T1 coef CCSD(T) norm ⟨S2⟩

experimental

CCSD(T)//B3LYP

CASPT2

1.217114 1.047114 11.610114

1.1994(0) 1.0636(−4) 12.31 0.061 1.096 1.1019

1.07640

1.0797(−5)

1.091948

1.374

40

1.3608(+1)

1.391448

1.377

40

1.3682(0)

1.394548

1.221 1.076 11.451

0.054 1.0876 1.070 1.055115

1.0633(−7)

1.053116

1.215

115

1.2228(−1)

1.206116

1.359

115

1.3274(0)

1.378116

1.224

115

1.2816(0)

1.211116

0.057 1.1684 1.200

6-311++G(3df,3pd) are always smaller than 1 × 10−3 Å for the distances and 20 cm−1 for the frequencies (except for the M06L in CH+). Therefore, the basis set 6-311++G(3df,2p) is adequate to describe these clusters. Results for CH and CH+ also show that the internuclear distance is above the experimental value for all the functionals except for the BHandLYP functional. The best results for the distances are obtained with the long-range modifications in the B3LYP functional (CAM-B3LYP and LC-BLYP), but the improvement of the results with respect to B3LYP is only of ∼1−6 × 10−3 Å, which does not compensate the increase in computational time. The frequencies are important information for this kind of systems since they give a fingerprint of the molecular structure, and they serve for calculation of the ZPE and other thermochemical properties. The frequencies obtained span a range of 200 cm−1 for the neutral CH and of 240 cm−1 for CH+. PBE1PBE functional for neutral and the LC-wPBE one for the singly charged give the better agreement with the experimental values. CCSD(T) calculations for geometries and frequencies are in a very good agreement with experiments, thus being an expensive but reliable method for benchmarking. Spin contamination is a very good test to evaluate different DFT functionals, as large spin contamination can lead to unreliable results. Values of ⟨S2⟩ are also given in Table 1. In the case of CH+(1Σ+), we performed spin-unrestricted calculations (allowing the HOMO−LUMO orbitals to mix) as a way to test if the wave function is stable or leads to large spin contamination. Overall B3LYP is one of the functionals giving the best results. Regarding IPs, Table 1 shows that all DFT methods have errors of ∼0.3 eV with respect to the experiment. CCSD(T) (either optimizing the geometry or using the B3LYP one) improves the results giving a difference of only 0.1 eV. The results for the cyclic C3H+3 cation in Table 2 also show that, even for this more complex system, DFT values D


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2

1 1 4 6 16 30 42 75 122 214 185 326 5 5 8 15 17 29 45 83 141 232 216 364 5 10 10 17 17 28 46 79 155 256 233 390

1 0

5 9 10 18 17 29 48 82 162 272 242 410 1 1 1 2 6 11 19 34 57 89 84 137

3 2

2 2 2 3 6 11 17 37 60 91 87 144

total (CnHm for n constant)

A total number of 1490 structures were found. A full compilation of all these values is given in Table 4 of the Supporting Information; geometries and frequencies are available from the authors upon request. Previous theoretical works were performed by Fantuzzi et al.51 for the CnHm+ family (n = 1−4, m = 0−(2n + 2)). In their systematic search, they computed a lower number of isomers and spin states. Our results agree with the most stable structures that they proposed. Other stoichiometric combinations were previously studied and also give rise to the same ground-state geometries found in this paper.28,48−50,54,55,117 The C3H+3 cation has also been widely studied because it is the simplest hydrocarbon with Hü c kel aromaticity. According to previous theoretical works118,119 it has four different local minima, in agreement with the number of structures and energetic ordering found in this work. Bare carbon clusters with charges 0 to 3 were previously studied in our group36,37 using the same approach. Studies on multiple charged molecules are more scarce; in Zyubina’s et al. work56 for C2H2+ 2 they found three singlet states and two triplet states, whereas in our calculations we located two singlet states and two triplet states. In Zyubina’s work two of the singlet molecules present very similar geometry, and in our work only one of them is taken into account. 3.3. Molecular Structure. As indicated above, sketches of the 255 different geometries that we found are available in Table 3 of the Supporting Information. These representations are approximate since, for example, with increasing charge the angles tend to be wider and bond distances larger. The geometries of the most stable CnHq+ m cluster for each n, m, and q are summarized in Figures 1 and 2. An overall inspection of this figure shows that in general linear forms are the most stable ones when charge is increased, since this is the best way to minimize Coulombic repulsions. C−C bonds generally enlarge too for the same reason. This is the case for molecules with one hydrogen atom: the most stable structures are the linear ones with the hydrogen on one end. The only exception is neutral C3H, which is cyclic. This radical has been widely studied,28,48−50,118,120,121 and it has been shown that linear and cyclic isomers are very close in energy. Most theoretical and experimental studies reveal that the most stable isomer is the cyclic 2B2 one, very close in energy to the linear 2Π, which is the same result obtained in our CCSD(T)//B3LYP approach, giving a difference in energy between them of only 0.14 eV. A similar trend is found in the molecules containing two H atoms: linear isomers with the hydrogens on each end of the molecule are preferred except for neutral and singly charged C3H2 and neutral C5H2 molecules that show a three-membered cycle in its geometry. The appearance of these cycles for structures with odd number of carbon atoms can be understood if one considers the possible Lewis structures (see Figure 3). In the case of molecules with an even number of carbons, the terminal hydrogens force a polyynic structure as the most stable one, while for an odd number of carbons a partial cumulenic structure can be formed, but implies to keep two unpaired electrons in a triplet electronic state. A more stable structure can be obtained if an internal cyclization is produced closing the electronic shell. The neutral C3H2 linear isomer is more stable in the triplet state but less stable than the cyclic singlet one by 0.68 eV. In the case of doubly charged structures the unpaired electrons are removed; the most stable form is linear, and the spin state is singlet.

1 2 2 4 6 11 19 35 67 115 95 167 total CnHm

C5

C4

C3

C2

1

1 2(2) 3 5(4) 3 4(4) 5 8(8) 14 24 26 43 1 2 3 5 3 4 6 9 15 25 28 45

0 charge

isomers structures isomers structures isomers structures isomers structures isomers structures isomers structures C

In parentheses the number of structures found by Fantuzzi et al.51 is included. a

1 0 3

0 0 2 3 3 6 11 21 29 64 45 94 1 2 2 4 3 6 15 24 43 78 64 114

2 1

1 2(1) 2 4(4) 3 6(7) 14 23(19) 45 72 65 107 1 1 2 4 3 6 14 25 50 85 70 121

0 3

0 0 1 1 4 8 7 12 24 41 36 62 2 1 2 4 5 8 7 12 25 39 41 64

2 1

2 4(2) 3 5(4) 5 7(6) 8 13(12) 29 48 47 77 2 4 3 5 5 8 9 13 30 47 49 77

0 3

0 0 0 0 3 5 5 8 12 20 20 33 0 0 2 4 3 4 6 10 13 24 24 42

2

1 2(1) 2 3(4) 6 11(7) 19 35(18) 67 112 95 163

H4 H3 H2 H

Table 4. Numbera of Considered Geometries for Each CnHm Combination and Number of Optimized Structures Including Different Spin Multiplicities

3


E


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Figure 3. Different electron distributions according to the parity of the number of carbon atoms.

For the molecules with three hydrogens the preference for linear structures is still observed, with the hydrogens on both sides. Some exceptions appear, in this case, for molecules with an odd number of C atoms: C3H+3 and C5H+3 . They present three-membered cyclic structures, which are particularly more stable than the rest. These cases have been previously studied,118,119 and their stabilities have been explained in terms of Hückel aromaticity: C3H+3 satisfies the 4n + 2 relation for Hückel aromaticity, and C5H+3 presents a local Hückel aromatic character (see Figure 4).

Figure 1. Most stable geometries for the CnHq+ m molecules with m = 1,2.

Figure 4. Structure with Hückel aromatic character (4n + 2 π electrons, n = 0).

Finally, molecules with four hydrogen atoms present the same general trend of favoring linear structures. The hydrogen atoms also prefer to be at the ends of the molecule except for C4H2+ 4 , where the particular formation of a tetrahedron is favored. This polyhedral-like structure is the most stable one, although it should be noted that the energy difference with the linear structure (H2CCCCH2) is very small (0.016 eV). Formation of a stable three-membered cycle structure is also observed in the most stable isomer of C5Hq+ 4 cations. 3.4. Harmonic Frequencies. To obtain a general idea of the infrared absorption of these compounds, all the frequencies from 0 to 4000 cm−1 of all the considered molecules (including all 1490 structures) were put together on a single histogram as shown in Figure 5. As a general trend two regions with large probability are observed: one at ∼3000 cm−1 and a broader one from almost 0 to 2500 cm−1. The region of frequencies ∼3000 cm−1 coincides with what is known as the C−H stretching modes, whereas in the broad band there is a huge mixing of several vibrational modes, such as C−C−H and C−C−C bending, C−C stretching, and several others. In particular, a more populated region is seen at ∼1800 cm−1, which could correspond to double bond CC stretching and would justify the expected highly unsaturated hydrocarbon behavior of this family of (both neutral and positively charged) molecules. 3.5. Relative Isomer Energies. The large number of studied molecules forces us to introduce a proper labeling. We will refer to a particular molecule CnHq+ m with a single number, Molecule Identifier (Mol-ID). This follows a linear indexing strategy based on the relation Mol−ID[Cn Hq+ m ] = (q + 1) + 4(m − 1) + 16(n − 1), which sorts all the elements in

Figure 2. Most stable geometries for the CnHq+ m molecules with m = 3,4.

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Figure 5. Histogram of vibrational frequencies from all the isomers studied (all possible geometries, charges, and spins were included).

Figure 6. Relative isomer energies. (a) Histogram counting the number of structures as a function of their Mol-ID. (b) Bivariate histogram that represents the number of structures for each Mol-ID as a function of the relative energy with respect the most stable one.The color code indicates the number of structures existing in a range of 0.2 eV. Purple lines indicate the energy difference between the two most stable structures for each value of Mol-ID. (c) Number of structures as a function of the relative energy (all possibilities of geometry and spin are included).

ascending order of mass and charge (CH, CH+, CH2+, CH3+, 3+ + CH2, CH+2 , CH2+ 2 , CH2 , CH3, CH3 , ...). A detailed analysis of the geometries of the 1490 molecules studied is not possible, but some general trends can be observed when analyzing the relative structure stability. Figure 6a shows a histogram counting the number of structures as a function of their Mol-ID. As can be seen a drastic increase of the number of structures is observed when the size of the molecule increases. Figure 6b shows a bivariate histogram of the number of structures as a function of Mol-ID and the energy relative to that of the most stable isomer. We have fixed bins of width 0.2 eV; thus, there are regions where it is possible to find up to six structures for a given Mol-ID in this range. The Figure 6b shows that the density of stable structures increases as a function of the size of the molecules. Figure 6c shows that the density of possible structures for all molecules reaches a maximum at ∼3 eV; that is, the largest number of structures

appear at a relative energy of 3 eV with respect the most stable structure. These trends are related to the typical C−C and H− C bond dissociation energies.122−125 First the strongest bond in this kind of system corresponds to the triple C−C bond, which for acetylene is 10.0 eV; thus, the upper limit for dissociations should be around this value. Second the most common C−C and H−C bond breaking energies are ∼3.5 and 4.5 eV, respectively; therefore, at these excitation energies more isomers are available, because new structures could be built through the breaking and forming of these bonds. The purple lines in Figure 6b connect the two most stable structures for a given Mol-ID, and therefore, their length tells us the relative energy between them. According to the latter, the structures can be classified in three main groups as a function of the required energy: • High energies (above 4 eV): these are the most energetic excitations/isomerizations and generally involve elecG


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The Journal of Physical Chemistry A tronic changes within the same or similar geometry; these compounds are somewhat rigid and reflect changes in the spin over structural changes. The most clear example of this is CH4 (Mol-ID = 13), which has only two structures separated by 8.44 eV corresponding to the singlet−triplet excitation but keeping essentially the same geometry. • Middle energies (between ∼1 and 4 eV approximately): here we found the combination of two effects, changes in the geometry and in the spin. The energy jumps are smaller than those observed in the changes implying only spin (high energies) since in these cases the change in geometry allows a better accommodation of the electrons in a new spin configuration. Neutral C2H (Mol-ID = 17) is an example in this energy range, where changes in the spin (from doublet to quartet) and in the geometry (from linear to angular) occur at 3.46 eV. • Low energies (below ∼1 eV): here we found changes in geometry (sometimes experimentally unresolvable) maintaining the electronic configuration (spin). Thus, molecules in this energy range present geometrical isomers at lower excitation energies. An example of this type is CH+2 (Mol-ID = 6), which goes from a bent doublet to a linear doublet isomer at 0.233 eV. This result has been previously reported in ref 126, which shows that this method is accurate enough even for small differences in energy. 3.6. Ionization Potentials. Figure 7 summarizes the vertical and adiabatic IPs for the most stable structures found for each molecule. As a general trend both IPs become smaller with the number of atoms, specially in the case of second and third IP; this is because the bigger the molecule the better the extra positive charge can be accommodated. We also observe oscillations: every four points the IPs increase. These points correspond to the CnH4 molecule. With no exceptions, the C n H 4 molecule always has a larger IP than the C n H 3 counterpart. This effect is clearer for the first IP, but it is also observed in the second and third IP (note that the energy scales are different). The most likely reason for this behavior is that CnH3 compounds (and many times CnH and CnH2) have a radical character in their most stable neutral form, being the electron easier to detach, while in all cases CnH4 structures are singlets in their ground neutral state (i.e., they present an electronic closed shell, and thus higher energy is required for ionization). 3.7. Dissociation Energies and Fragmentation Channels. One of the most relevant informations that can be extracted from the present study is the dissociation energy for each possible fragmentation channel. The fact that all systems have been treated at the same level of theory allows us to obtain this information. For these particular systems, there are many dissociation pathways that must be taken into account, even if one only considers two-fragments dissociations CnHqm+ → C(n − x)H((qm−−zy))+ + Cx Hzy+

(1)

Figure 7. Adiabatic (red) and vertical (green) IPs as a function of the Mol-ID for the neutral systems. To compare similar values for adiabatic and vertical IP, curves labeled as 1st, 2nd, and 3rd IP 3+ correspond to CnHm → CnHm+, CnHm → CnH2+ m , and CnHm → CnHm , respectively.

where x is the number of carbon atoms that are lost, y is the number of hydrogen atoms lost, and z is the charge lost. Some possibilities are given in Figure 1 of the Supporting Information. To summarize the most important results we present in Table 5 the lowest dissociation energy for the twofragment channels in the 80 CnHq+ m molecules considered in this work.

For neutral systems, the channels with lower dissociation energy correspond to the loss of H, H2, C2H, or C2H2. H


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The Journal of Physical Chemistry A Table 5. Dissociation Energies (D) of the Lowest Dissociation Two-Fragments Channelsa for the CnHq+ m Families channel

D (eV)

CH → C + H C2H → C2 + H C3H → C3 + H C4H → C4 + H C5H → C5 + H CH+ → C+ + H C2H+ → C+2 + H C3H+ → C+3 +H C4H+ →C+4 + H C5H+ → C+5 + H C2H2+ → CH+ + C+ C3H2+ → C+3 + H+ C4H2+ → C3H+ + C+ C5H2+ → C3H+ + C+2

3.344 5.364 3.026 4.437 3.444 3.981 4.859 5.617 5.159 6.302 −3.167 0.112 −0.809 1.971

+ C3H3+ → C2+ 3 + H C4H3+ → C3H2+ + C+ + C5H3+ → C2+ 5 + H

−9.584 −5.957 −5.024

CH3 → CH + H2 C2H3 → C2H2 + H C3H3 → C3H + H2 C4H3 → C4H2 + H C5H3 → C5H + H2 C5H3 → C3H + C2H2

4.407 1.465 3.576 1.637 3.338 3.706

CH+3 → CH+2 + H C2H+3 → C2H+ + H C3H+3 → C3H+ + H2 C4H+3 → C+4 H2 + H C5H+3 → C5H+ + H2 + + CH2+ 3 → CH2 + H 2+ + C2H3 → C2H2 + H+ + + C3H2+ 3 → C3H2 + H + + C4H2+ → C H + C 3 3 3 + + C5H2+ 3 → C3H + C2H2 + 3+ 2+ C2H3 → C2H2 + H + 2+ C3H3+ 3 → C3H2 + H + 2+ C4H3+ → C H + H 3 4 2 3+ 2+ C5H3 → C5H2 + H+ a

channel CH2 → C + H2 C2H2 → C2H + H C3H2 → C3 + H2 C4H2 → C2H + C2H C5H2 → C3 + C2H2 CH+2 → C+ + H2 C2H+2 → C2H+ + H C3H+2 → C3H+ + H C4H+2 → C4H+ + H C5H+2 → C5H+ + H + + CH2+ 2 → CH + H + + C2H2+ → CH + C 2 2 + + C3H2+ 2 → C2H + CH + 2+ + C4H2 → C3H2 + C + + C5H2+ 2 → C4H2 + C + 2+ C2H3+ → CH + C 2 2 2+ C3H3+ + H+ 2 → C3H + 2+ C4H3+ 2 → C3H2 + C + 2+ C5H3+ → C H + C 2 4 2 CH4 → CH3 + H C2H4 → C2H2 + H2 C3H4 → C3H2 + H2 C4H4 → C4H2 + H2 C4H4 → C2H2 + C2H2 C5H4 → C5H2 + H2 C5H4 → C3 + C2H2 CH+4 → CH+3 + H C2H+4 → C2H+2 + H2 C3H+4 → C3H+3 + H C4H+4 → C4H+2 + H2 C5H+4 → C5H+3 + H + + CH2+ 4 → CH3 + H 2+ + C2H4 → C2H3 + H+ + + C3H2+ 4 → C3H3 + H + + C4H2+ → C H + C 4 2 2 2H2 2+ + C5H4 → C3H2 + C2H+2 + 2+ C1H3+ 4 → C1H3 + H + 2+ C2H3+ → C H + H 4 2 3 + 2+ C3H3+ → C H + H 4 3 3 3+ 2+ C4H4 → C3H2 + CH+2 2+ + C5H3+ 4 → C3H2 + C2H2

5.179 4.107 5.038 3.930 4.090 −4.902 −1.970 −0.238 −0.037 0.390 −10.727 −9.788 −5.715 −5.596

D (eV) 3.186 4.876 2.952 5.461 2.991 4.058 5.821 4.286 5.569 3.925 −3.162 −1.298 0.931 0.976 2.834 −10.173 −7.738 −6.653 −3.860 4.388 1.740 3.238 1.463 1.557 3.177 3.216 1.283 2.597 1.947 2.557 1.968 −4.719 −2.471 −0.521 0.104 0.356 −14.591 −12.128 −7.863 −6.227 −6.014

Channels with dissociation energy close to the lowest one are also shown in some relevant cases. ZPE corrections are taken into account.

Loss of C2H2 is also commonly observed in the fragmentation of polyciclyc aromatic hydrocarbons.127 We can observe some trends in the loss of the simplest fragments H, H+, C, and C+. For neutral and singly charged molecules, the most favorable single-atom fragmentation corresponds to the emission of neutral H, showing oscillations with the number of carbon atoms. With the exception of CnH3, neutral molecules with an even number of carbons present higher energy dissociations for the loss of H than molecules with an odd number of carbons. In the case of doubly and triply charged clusters the most likely emission of single fragments is the loss of H+ and C+. In both cases dissociation energies increase with the number of C atoms. 3.7.1. Fragmentation Channel Distributions. The previous analysis of the dissociation energies provides only a partial picture of the complex fragmentation processes of CnHmq+ molecules. Appendix A describes a method to calculate the

However, for singly charged ones, loss of H and, in a few cases, H2 are the most stable channels (see Table 5). This is in agreement with the experiments,61 which show that the channel for emission of H is the one with the highest fragmentation branching ratio. For the doubly charged systems, fragmentation products tend to separate the charge. H+ or C+ are the most common fragments, but in some cases more complex fragments such as CH+ or C2H+2 are observed; note that the formation of complex final fragments would imply in some cases atom reorganization inside the molecule and therefore complex reaction mechanisms and large energy barriers. For triply charged molecules, the charge is also splitted, and the higher charge is always contained in the bigger fragment. Loss of H+ or C+ dominate except for the largest molecules, where the C2H+2 fragment is again produced. The loss of C2H2 and its cation is expected since it is a well-known stable molecule: acetylene. I


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The Journal of Physical Chemistry A Table 6. Possible Fragmentation Channelsa for C3H2 and C3H+2 Molecules 1f (8) C3H2 (8)

1f (7) C3H+2

a

(7)

2f (36) H+C3H H2+C3 C+C2H2 CH+C2H CH2+C2

(4) (4) (10) (10) (8)

2f (73) C+C2H+2 CH2+C+2 H2+C+3 H+C3H+ CH+2 +C2 CH+C2H+ H+2 +C3 + C +C2H2 H++C3H CH++C2H

(10) (8) (5) (4) (8) (10) (4) (10) (4) (10)

3f (46) 2H+C3 H+C+C2H H+CH+C2 H2+C+C2 2C+CH2 C+2CH 3f (111) +

C+CH+CH H++CH+C2 H2+C+C+2 H+C++C2H H+2 +C+C2 2H+C+3 H2+C++C2 H+H++C3 C++2CH H++C+C2H C+C++CH2 H+C+C2H+ H+CH+C2+ H+CH++C2 2C+CH+2

(4) (10) (4) (4) (16) (8)

4f (20)

5f (8)

2H+C+C2 (4) H+2C+CH (8) H2+3C (8)

2H+3C (8)

4f (52)

(8) (4) (4) (10) (4) (5) (4) (4) (8) (10) (16) (10) (4) (4) (16)

+

H +2C+CH H+2C+CH+ 2H+C+C+2 H+H++C+C2 H+2 +3C H2+2C+C+ 2H+C++C2 H+C+C++CH

5f (16) (8) (8) (4) (4) (8) (8) (4) (8)

H+H++3C (8) 2H+2C+C+ (8)

The number of different considered structures is given in parentheses. Channels in bold present the larger number of possible combinations.

Fragmentation Channels Distributions (FCDs) avoiding the generation of all possible fragmentation channels. This method takes advantage of the convolution theorem, producing a function where each peak represents a possible fragmentation channel, and thus, a single visualization allows to identify energetic regions with larger probability of fragmentation. To clarify the concept of FCDs we focus on the molecules C3H2 and C3H+2 . Table 6 shows all the possible fragmentation channels for these systems. Fragmentation of C3H2 and C3H+2 involves 16 and 36 channels, respectively. These numbers reveal that when the charge increases the number of possible channels rapidly grows. Actually, by considering the possible spin configurations, the number of channels increase to 1104 and 3395, respectively, which obscures a possible pattern in this process making a detailed analysis unfeasible. Table 6 also shows the number of possible structures classified by channel and number of fragments. Figure 8 presents a graphical illustration of the complex fragmentation and how the FCDs help in its analysis, taking the C3H2 as an example. Panel (a) in the figure shows an energetic scheme with the values associated with all possible fragmentation channels (including all possible spin multiplicities and geometries of the fragments in each channel). The first line in each group of lines represents the energy window for a given number of fragments; we observe that it is necessary to increase the internal energy of the system up to 2.94 and 7.33 eV to reach the two-fragments and three-fragments windows, respectively. Panel (b) in the same figure represents the FCD for a given number of fragments, which is the density for each group of lines in panel (a), and appear in the corresponding energy window. As discussed in Appendix A, this density is proportional to the combinatorial entropy, and thus, it can be understood as the probability driven by some entropic considerations. Thus, with the FCDs we have a simple picture of the energetic and entropic contributions to the fragmentation.

Figure 8. Energy levels diagram for all possible fragmentation channels of C3H2 molecule grouped by number of fragments in different colors. (a) Energy levels diagram and (b) density of this energy levels or FCD. See text for details.

When the size of the molecules increases the fragmentation becomes much more complex, and a larger number of possible channels appear. Thus, it is much harder to generate all possible fragmentation channels as we did for C3H2 (Figure 8a). To overcome this problem, we propose a simple method for computing the FCDs, explained in detail in Appendix A. The first step is to build the FCDs, corresponding to the generation of the Structures Distribution Plot (SDP) for each considered molecule. The SDP represents the relative energy of the different structures (considering geometries and spin states) with respect to the most stable one. The height of each line is the multiplicity of the corresponding structure. We generated the 80 different SDPs for the CnHq+ m molecules considered here. J


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Figure 9. SDP for C3H2. See text for details.

Figure 10. Fragmentation Channels (top) and number-of-fragments distributions (bottom) for the neutral C3H2 molecule. Only low energy dissociation part is shown.

Figure 11. Fragmentation Channels (top) and number-of-fragments distributions (bottom) for the singly charged C3H+2 molecule. Only low energy dissociation part is shown.

corresponds to the linear form and has a relative energy of 0.588 eV. The other structures are plotted in the same way using the energies given in Table 4 of the Supporting Information. In general, the most probable structures are those with the larger spin-multiplicity (larger values of SDP). As explained in Appendix A, all possible fragmentation channels for a given molecule (i.e., the FCD) can be represented in a single plot by convoluting the corresponding SDPs. The FCD of both C3H2 and C3H+2 systems are shown in

In particular, Figure 9 shows the SDP for neutral C3H2 (blue lines) including a Gaussian broadening profile (red line) with a width (standard deviation) of 0.1 eV. This profile mimics the distribution of energy among the internal degrees of freedom of the molecule. The most stable isomer (line at zero energy) corresponds to the one denoted as 5 in Table 4 of the Supporting Information and has a singlet spin multiplicity; isomerization to obtain isomer 2 in a singlet state (second line in Figure 9) implies only 0.444 eV; the first stable triplet K


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The Journal of Physical Chemistry A Figures 10 and 11, respectively, below 12 eV. (see Table 6). At lower energies we observe the SDP for the parent molecule (compare top panels in Figures 10 and 11), and at ∼3 eV the first fragmentation channel appears (C3 + H2), overlapping with the peaks corresponding to the less stable unfragmented isomers; a second peak for the same fragment corresponding to a different structure appears at 4 eV (note that the FCD does not identify specific structures). At higher energies (above 4 eV and below 8 eV), competition between four different channels appears: H + C3H, C2 + CH2, C+C2H2, and a third peak of the C3 + H2 channel corresponding to different C3 structures. In the range of 3−6 eV and based on the height of the peaks, it is clear that the most favored channel corresponds to the loss of a carbon atom and the most probable channel is the loss of molecular hydrogen, but loss of atomic H is also possible. For C3H+2 (see Figure 11) in the range of 4−8 eV, the most probable channels are the loss of atomic and molecular hydrogen, being the first one the most favored energetically. In both cases the charge is localized on the heaviest fragment. Above 8 eV the FCD gets a more complex structure since the charge can be located at different fragments, and the number of channels increases (see other examples in Figures 6 to 25 of the Supporting Information). We can compare these results with the available experiments,23,60 where the two-fragments channel with higher branching ratio (BR) corresponds to the production of a hydrogen atom (BR close to 50% for both systems), followed by the loss of H2 and the loss of C (both channels with BR close to 25%). We found that for singly charged C3H+2 the loss of atomic hydrogen is indeed the most favored channel energetically. In the FCD it can be seen that in the region between 6 and 8 eV other two channels start to compete, the loss of H2 and the loss of C, also in agreement with the experiments. In the case of neutral C3H2 this agreement is worse since the theory favors the loss of H2. Loss of hydrogen atom and loss of one C atom also appear but at slightly higher energies and with lower probability. This disagreement with the experimental results suggests that additional factors should be taken into account, not just the energetics and the degeneracy due to spin multiplicity. Indeed it has been already shown that a full statistical analysis considering vibrational and rotational effects and the energy deposited in the experiments37 must be done for a correct interpretation of the experimental BRs for fragmentation of excited carbon clusters.59 Nevertheless the FCD provides in many cases a qualitative picture of which channels are expected to appear in fragmentation experiments at different internal energies of the molecule. The FCD for the 80 molecules considered in this work, extended to a complete energy range, are given in the Supporting Information. The bottom panels of Figures 10 and 11 show the FCDs grouped by the number of fragments for both C3H2 and C3H+2 . From these figures it is possible to characterize some regions or energetic windows that are dominated by a given number of fragments. For C3H2 the one-fragment window extends from zero to 3.0 eV; however, for C3H+2 it extends to almost 5 eV. In both cases, the windows for two and three fragments extend above 12 eV. Then, to obtain a global view of these fragmentation processes, we collected all energetic windows for a given number of fragments for all molecules grouped by charge. Figure 12 shows for charges 0 and +3 (the extreme cases) the energetic windows for one to four fragments by considering all studied molecules. Results including other charges and with more fragments can be found in the

Figure 12. Energetic windows for one to four number of fragments. Total FCD by considering contributions from all systems is also shown. (upper) Neutral systems. (lower) Triply positively charged systems. A complete version of this figure including the complete number of fragments and charges considered is given in the Supporting Information.

Supporting Information. For the neutral clusters, it can be clearly seen that the peaks in the distributions are wellL


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vertical and adiabatic IPs stays almost constant showing a similar geometrical relaxation energy after ionization. Dissociation energies show the existence of preferential fragmentation channels: mainly the loss of H and H2 for neutral and singly charged clusters, and the loss of H+ for doubly and triply charged ones. The study of the CnHq+ m fragmentation is extremely complex due to the huge number of possible dissociation channels and possible isomers. To solve this problem we have proposed a new method that consists in calculating all the possible stable geometries and spin states to build the energetic distribution considering all possible dissociation channels. This method allows a qualitative analysis of the fragmentation of such complex clusters.

separated, that is, well-defined energetic windows with a given number of fragments appear. This is compatible with the results of Tuna et al.,61 who showed that the number of emitted fragments reflects the energy deposited in the system and may be very different from one experiment to another. Branching ratios for a given number of fragments are less sensitive to the excitation energy distribution and reflect more intrinsic properties of the parent molecule. In the case of charge +3, separation of these energetic windows is no longer clear. In fact, these regions overlap almost in the whole energy range, and start at negative values of energy (i.e., these are exothermic processes). However, most of these fragmentation processes involve charge separation (Coulomb explosion) and need first to overcome an energy barrier; thus, experimental branching rations will be mainly affected by the height and width of the barrier instead of the dissociation energies. By increasing the charge of the system, the number of dissociation channels increases due to the large number of possible ways to locate the charge, and some of them present very different energies. The singly charged molecules require a similar fragment distribution compared to the neutral ones with well-separated peaks for each number of fragments (see Supporting Information). For the triply charged ones, the behavior changes qualitatively: much broader energy distributions of fragments are obtained, and overlapping energy windows are observed. For instance, for dissociation into two fragments, there are favorable situations in which the charges split between the two fragments, others in which the total charge remains in the larger fragment and the energy of the products is higher, and cases in which the charge remains in the smaller fragment thus leading to a very unfavorable situation from the energetic point of view.

■

APPENDIX A: FRAGMENTATION CHANNELS DISTRIBUTION CALCULATION In this appendix we derive the Fragmentation Channel Distributions (FCDs), which are discussed in Section 3.7.1. An important result from the probability theory128 is the statistical theorem of convolution, which establishes that if X and Y are two independent random variables with probability density functions f and g, respectively, then the probability density function of their sum τ = X + Y is given by the convolution f⊗g, defined as (f ⊗ g )(τ ) =

∫τ=X+Y dXf (X )g(Y )

(2)

This shows, in an explicit way, the constrain between the two variables, performing the integration in the whole domain. Additionally, by taking into account that the convolution of a function is commutative and associative, we can introduce the convolution of multiple functions as

4. CONCLUSIONS We have shown that the CCSD(T)/6-311++G(3df,2p)// B3LYP/6-311++G(3df,2p) level of theory is a reliable and feasible computational approach for the study of geometric and energetic properties of CnHq+ m clusters, giving qualitative and quantitative information for the interpretation of current experiments. From the structural point of view, a systematic search for possible isomers shows that linear carbon chains are energetically favored with the hydrogen atoms on the ends of the chain. In some clusters, such as those containing an odd number of carbon atoms, the formation of three-membered cycles (in some cases aromatic ones) leads to more stable structures. A noticeable exception is C4H2+ 4 , which adopts a tetrahedral structure. The harmonic frequencies distribution analysis shows that all computed frequencies are found around two regions in the spectrum corresponding to the C−H stretching modes (∼3000 cm−1) and C−C−C, C−C−H, H−C−H bending modes and C−C stretching mode (0−2500 cm−1). The number of isomers increases exponentially with the number of atoms. The energy distribution shows that, with increasing the size of the system the most probable relative energy is centered around 3 eV, which corresponds to the average C−C and H−C bond cleavage energy. Also there is an upper limit around 10 eV, which is close to the triple CC bond breaking energy, the strongest bond in these systems. Ionization potentials decrease with the size of the system as expected: the larger the molecule the easier to accommodate the additional charge. In addition, the difference between

N

⊗ fi (τ ) = f1 (X1) ⊗ f2 (X 2) ⊗ ⋯ ⊗ fN (XN )

i=1

N

=

∫τ=X +⋯+X 1

N

dX1⋯dXN − 1 ∏ fi (Xi) i=1

(3)

We use this expression for computing the FCDs. To unify the notations, we first describe the expressions we are going to use. To this we consider the fragmentation of the CH2 molecule as an example. This molecule can break leading to the following fragments S = {S1, S2 , S3 , S4 , S5} = {CH 2 , CH, H 2 , H, C}

(4)

Each one of them may be present in the fragmentation showing several electronic states or geometries, which we are going to call structures. Some structures reported in literature for these fragments (S1,129 S2,130 S3,131 S4,132 S5132) are S1 = {CH 2(X 3B1), CH 2(a1A1), CH 2(b1B1), CH 2(c1A1), ···} S2 = {CH(X 2Π), CH(a 4Σ−), CH(A 2Σ−), ···} S3 = {H 2(X1Σ+g ), H 2(a 3Σ+u ), H 2(A 1Σ+u ), ···} S4 = {H(1 2S), H(2 2P), H(2 2S), ···} S5 = {C(2 3P), C(21D), C(21S), ···}

(5)

The molecule CH2/S1 may be broken through several fragmentation channels (Ci): M


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fragments. In general, the FCD for a given N-fragments channel (S1 + S2 +···) will be given by

C = {C1, C2 , C3, C4 } = {CH 2 , H 2 + C, CH + H, H + H + C}

(6)

N

Ω∑i Si(E − E0) = ⊗ ΩSi(E − E0)

Each channel is characterized uniquely by including the geometry and the specific electronic states of each fragment as follows: A ⊕ B = {{a + b} | a∈A and b∈B }. Thus,

i=1

In practice, we do not use a Dirac delta distribution but a normalized Gaussian function with a full width at half maximum of 0.1 eV. These FCDs can be understood in terms of probabilities or entropy. Basically if a system is isolated and in statistical equilibrium, it is possible to show that the entropy of this system is given by S = k ln Ω(E). In fact, the fragmentation of carbon clusters is mainly dictated by regions in the phase space with largest density of states (DOS) or equivalently those regions where the entropy is maximum.37 It has been shown that the DOS can be expressed as a product of different contributions, which can be divided into two different groups: the first one is associated with the phase space of each fragment, which involves its vibrational, rotational, and translational degrees of freedom, and the second one describes the number of ways to arrange all possible fragments (combinations) and the degeneration of their electronic states. Thus, the FCDs calculated in this work are completely equivalent to the second contribution to the DOS.

C1 = S1 = {CH 2(X 3B1), ···, CH 2(c1A1), ···} C2 = S3 ⊕ S4 = {H 2(X1Σ+g ) + C(2 3P)···, H 2(a 2Σ+u ) + C(21D)···} C3 = S2 ⊕ S5 = {CH(X 2Π) + H(2 3P)···, CH(A 2Σ−) + H(2 2S), ···} C4 = S4 ⊕ S4 ⊕ S5 = {H(1 2S) + H(1 2S) + C(2 3P), ···, H(1 2S) + H(2 2S) + C(21D), ···}

(7)

Now, computing the FCDs is reduced to calculating how many fragmentation channels exist between the energy interval E and E + dE, where E corresponds to the electronic energy. One way to proceed is by generating all combinations of all possible structures by including the mass conservation constraint. However, in practice, this is computationally expensive. The strategy we adopted here is to take advantage of the convolution theorem described above. Therefore, in a first step we generate the Structures Distribution Plot (SDP) for each considered molecule, that is, S, by using Equation 8 ΩS(E) =

1 wS

■

∑ w(Si)δ[E − E(Si)] ∴ wS ≔

∑ w(Si)

i=1

i=1

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b10143. Adiabatic and vertical ionization potentials, dissociation energy for H, H+, C, and C+ loss, dissociation energy windows classified by the number of fragments, fragmentation channel distribution plots, geometries of all stable isomers, and complete set of energies for all studied structures. (PDF)

(8)

where E(Si) and wi represents the electronic energy and the weight of the i-th structure, respectively, and δ[x] indicates the Dirac delta function. Equation 8 is proportional to the probability to found one structure of the molecule S between the interval of energy E and E + dE. In our case, predefined weights wi are given by the number of possible microstates associated with the electronic configuration, which is given by the spin electronic degeneracy wi = 2Si + 1. In a second step, we obtain the probability density distribution for a given channel by the convolution of their SDP associated with the related fragments. If we consider the fragmentation of the initial molecule S0 in to two fragments: S0 → S1+ S3, then, according to the convolution theorem (3), the probability to find a combination of structures from these molecules between the interval of energy E and E + dE is given by

■

1 S1 S3 w w

N1

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We acknowledge the allocation of computer time at the Centro de Computación Cientı ́fica at the Universidad Autónoma de Madrid (CCC-UAM). This work was supported by Project Nos. FIS2013-42002-R and CTQ2013-42698-P (MINECO) and NANOFRONTMAG S2013/MIT-2850 (CAM). J.P.S. acknowledges the FPU doctorate grant FPU12/03050 of the Spanish MECD. S.D.-T. gratefully acknowledges the “Ramón y Cajal” program of the Spanish MINECO ref RYC-2010-07019.

N3

∑ ∑ wiS wjS δ[E − E(S1i) − E(S3j)] 1

3

■

i=1 j=1

(9)

REFERENCES

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Thus, it is clear that Ω ⊗Ω (E + E0) will represent the combinatorial probability density as a function of the energy to break down the molecule S0, by taking as energy reference the most stable isomer for the initial molecule (E0). Note that the peaks of this distribution will correspond to the dissociation energies for specific fragmentation channels; that is, ΩS1⊗ΩS3(E + E0) will give the probability to break the molecule into two S1

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Corresponding Author

ΩS1+ S3(E) = ΩS1 ⊗ ΩS3(E) =

ASSOCIATED CONTENT

S Supporting Information *

NS

NS

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S3

N


Article

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The Journal of Physical Chemistry A (132) Kramida, A.; Ralchenko, Y.; Reader, J. NIST Atomic Spectra Database (version 5.2); http://physics.nist.gov/asd, 01 Oct 2015; National Institute of Standards and Technology: Gaithersburg, MD, 2014.

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