Structures and energetics of hydrocarbon molecules in a wide hydrogen chemical potential range
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Structures and energetics of hydrocarbon molecules in a widehydrogen chemical potential range
Y. X. Yao
Ames Laboratory-U.S. DOE and Department of Physics and Astronomy,Iowa State University, Ames, Iowa IA 50011
C. Rareshide
Physics department, Stetson University, DeLand, FL 32723
T. L. Chan, C. Z. Wang, and K. M. Ho
Ames Laboratory-U.S. DOE and Department of Physics and Astronomy,Iowa State University, Ames, Iowa IA 50011
Abstract
We report a collection of lowest-energy structures of hydrocarbon molecules C m H n (m=1-18;n=0-2m+2). The structures are examined within a wide hydrogen chemical potential range. Thegenetic algorithm combined with Brenner’s empirical potential is applied for the search. Theresultant low-energy structures are further studied by ab initio quantum chemical calculations. Thelowest-energy structures are presented with several additional low-energy structures for comparison.The results are expected to provide useful information for some unresolved astronomical spectraand the nucleation of growth of nano-diamond film. NTRODUCTION
Hydrocarbon molecules, especially polycyclic aromatic hydrocarbons (PAHs), have re-ceived much attention for the broad astronomical interests. PAHs were proposed to be thepossible carriers of the unidentified infrared (IR) emission bands[1], the interstellar ultra-violet (UV) extinction curve[2, 3] and the diffuse interstellar bands[4–6]. Though quite afew achievements have been made in recent years[7, 8], the PAH hypotheses are not fullyaddressed. Possible different charged states of PAHs may increase its complexity, however,one intrinsic difficulty could be stemmed from the very complicated phase space of hydrocar-bons due to the enormous bonding ability of carbon. Furthermore, the phase space is alsodependent of the hydrogen chemical potential, as the observed IR spectra may be differentin different interstellar environments[8].In this paper, we focus on providing a collection of lowest-energy structures for C m H n (m=1-18; n=0-2m+2) in a wide hydrogen chemical potential range with the aid of a seriesof unbiased global searches. Our results are expected to be helpful for addressing the abovePAH hypotheses. Moreover, since the most abundant reactive elements in the cosmos arecarbon and hydrogen, our database should be helpful in elucidating the evolution of carbonfrom its birthsite in circumstellar shells through interstellar medium, which may be related toastrobiology ultimately[8]. Useful information for a better understanding of the mechanismof nucleation and growth for nano-diamond films may also be extracted from our results[9,10]. COMPUTATIONAL METHODS
The global structure optimization is based on our early developed genetic algorithm(GA)[11]. An illustrative online version of the code for the optimization of a two-dimensionalmap has been published in nanohub[12]. Each GA run is started with a randomly gener-ated pool with typically 100 structures which are relaxed to local minimum with Brenner’sempirical potential[13]. The size of pool may vary with different investigated systems. Therandom generation of the pool may be guided by some physical intuition. The evolution ofthe pool is realized by performing mating operations. Each operation includes randomly se-lecting two structures (parents) from the pool, cutting them with a common plane randomly2elected, then combining the opposite parts relative to the cutting plane from the parentsto create a new structure (child). The child structure is relaxed within Brenner’s model[13].The decision of replacing the highest-energy structure in the pool with the child structureis based on energy criterion f defined as f = ( E − n H µ H ) /n C (1)where E is the total energy of the cluster with n H hydrogen atoms and n C carbon atoms. n C is fixed for each GA run, while n H may be varied in a physically reasonable range. µ H is thehydrogen chemical potential, which may be fixed to some particular value(s)[14] or a rangeof interest. In this work, we are interested in a wide hydrogen chemical potential range,which is taken as (-6eV, 0) in Brenner’s model, i.e., the lowest (highest)-hydrogen chemicalpotential value guarantees that the optimum molecules are pure carbon clusters (alkanes).In practice µ H is uniformly sampled in this chemical potential range by step size of 0.05eV. The mating operations performed with parents of different number of hydrogen atomsmay provide a superior sampling of the potential energy landscape[14]. Each GA run hastypically 10000 generations with 15 random pairs of molecules mated for each generation. Atthe end of each GA run, the candidate structures are further relaxed by ab initio quantumchemistry method.Recently it has been pointed out that density functional theory (DFT) is unreliablefor computing hydrocarbon isomer energy differences[15, 16]. Fig.1 shows the correlationbetween experimental isomer energy differences and those calculated at second order Møller–Plesset perturbation theory (MP2), DFT-B3LYP, DFT-PBEOP levels with basis set of 6-31(d) for 9 C H , 7 C H , 5 C H , 4 C H and 9 C H molecules. MP2 gives a bestcorrelation with the experiments, especially for the low-energy molecules. Hence all thecandidate hydrocarbon structures in the final pool from GA are relaxed at level of MP2/6-31G(d) using GAMESS package[17]. RESULTS
We systematically searched for the lowest energy structures of hydrocarbon molecules C m H n (m=1-18; n=0-2m+2) in a wide hydrogen chemical potential range of (-6eV, 0).Fig.2-14 show MP2-relaxed lowest-energy structures from our unbiased global searches. Sev-3 .0 0.1 0.2-0.050.000.050.100.150.200.25 0.0 0.1 0.2 MP2 E ( e V / C a t o m ) E exp (eV/C atom) B3LYP
PBEOP
FIG. 1: (Color online) Computed isomer energy differences at MP2 (black square), DFT-B3LYP(red circle) and DFT-PBEOP (blue triangle) level versus experimental values for 34 hydrocarbonmolecules. eral low-energy isomers were also shown for comparison. The ground state geometries ofpure carbon clusters have been extensively studied and reviewed[18]. It usually requireshigher level of quantum chemistry calculations to determine the fine structures and energydifferences between the isomers. In this paper we focus on providing the lowest-energyhydrocarbon clusters in various hydrogen chemical potentials and the lowest-energy purecarbon clusters are intended to be chosen as reference systems. Nevertheless, the generalfeatures (i.e., chain or ring structure) of the ground state geometries of pure carbon clustersare consistent with the results in the literature[18, 19]. C H m Fig. 2 shows the lowest energy structures with a backbone of 6 carbon atoms in the fullhydrogen chemical potential range. The relative energy E associated with each structureis defined as E = E C n H m − E C n − mE H (2)where E C n H m , E C n and E H are the total energies of the hydrocarbon C n H m , pure car-bon cluster C n and hydrogen atom H , with E H = − . eV. The energies are eval-uated at MP2/6-31G(d) level. The pure C cluster is chosen as the reference point.In this C6 group, the lowest-energy hydrocarbon structures in the hydrogen chemical4 FIG. 2: 11 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV.11 lowest energy structures of C H m in full hydrogen chemical potential range. The number below each structure is the relativeenergy E in unit of eV. potential range of ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV,
0) are C (chain), C H (Benzyne), C H (Benzene), C H (a,Cyclohexane) and C H (a,2,2-Dimethylbutane), respectively. Several low energyisomers in C H and C H group are also shown for reference. Cyclohexane with boatconformation C H (c) is 0 . eV higher in energy than Cyclohexane with chair conforma-tion C H (a), close to the experimental result 0.238 eV[20]. Methylcyclopentane C H (b)is in between and 0.197 eV higher than C H (a), close to experimental value 0.177 eV[21]. C H (d) is rather high in energy because of its diradical structure. The energy orders of thefour alkane isomers are correctly predicted by MP2 as compared with experimental resultsalthough the energy difference can be quite small[21].5 FIG. 3: 7 lowest energy structures of C H m in full hydrogen chemical potential range. The numberbelow each structure is the relative energy E in unit of eV. C H m Fig. 3 shows the lowest energy structures with a backbone of 7 carbon atoms in the fullhydrogen chemical potential range. In this C7 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range of ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV,
0) are C (chain) , C H (Methylbenzene) , C H (Methylcyclohexane) and C H (a,2,2-Dimethylpentane), respectively. From C6 to C7group, the system mainly choose to add one methyl to the dominant lowest energy hydro-carbon molecules (i.e., Benzene and Cyclohexane) and exhibit four different lowest energychemical compositions in the whole hydrogen chemical potential range. The alkane subgroupis different and still the one with one end methyl pair has lowest energy, in consistence withexperiment[21]. C H m Fig. 4 shows the lowest energy structures with a backbone of 8 carbon atomsin the full hydrogen chemical potential range. In this C8 group, the lowest-energyhydrocarbon structures in hydrogen chemical potential range ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . − . eV,
0) are C (ring), C H (a,Ethynylbenzene), C H (a,1,2-Dimethylbenzene), C H (a,cis-1,3-6
8 (a) (b)C8H10 (a) (b) (c) (d)C8H16 (a) (b) (c) (d)C8H18 (b) (a) (c ) (d)0.000 C8H6−37.077 −37.067 −37.065 −36.960−51.833 −51.831 −51.784 −51.753−56.365 −56.326 −56.323 −56.309−24.739−25.459
FIG. 4: 15 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. Dimethylcyclohexane) and C H (a,2,2-Dimethyhexane), respectively. Ethynylbenzene haslower energy than C H (b,Pentalene) because of its conjugated bond configuration. Thefirst three low energy Benzene-based isomers in C8H10 subgroup are very close in energy.MP2-level calculations can not correctly predict the energy orders between them and fa-vor 1,2-Dimethylbenzene. In contrast, gas phase thermochemistry measurement shows that C H (c,1,3-Dimethylbenzene) is 0.018 eV lower in energy than 1,2-Dimethylbenzene[21].On the other hand, MP2 predicts that C H (d,Ethylbenzene) is higher in energy than theDimethylbenzenes in agreement with experiment[21]. For the C8H16 subgroup, MP2 alsopredicts that Dimethylcyclohexanes have lower energies than Ethylcyclohexane and the en-ergy differences between Dimethylcyclohexanes can be very small. In the alkane subgroup,the one with one end methyl pair is still the lowest energy isomer.7 FIG. 5: 15 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. C H m Fig. 5 shows the lowest energy structures with a backbone of 9 carbon atoms in the fullhydrogen chemical potential range. In this C9 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (chain) , C H , C H (indene), C H (a,1,2,5-Trimethylbenzene), C H (1,3,5-Trimethylcyclohexane) and C H (a,2,2,5-Trimethyhexane), respectively. The group showsmore ground state chemical compositions in the full hydrogen chemical potential range.MP2 predicts that the isomer with 1,3,5-Trimethyl has the lowest energy in both C9H128 FIG. 6: 7 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. and C9H18 subgroups due to maximal reduction of the repulsion energy between the hydro-gen atoms. The ground state geometry of the alkane subgroup is predicted to have threemethyls in accordance with experiment[21]. C H m Fig. 6 shows the lowest energy structures with a backbone of 10 carbon atoms in the fullhydrogen chemical potential range. In this C10 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H (Naph-thalene), C H (Adamantane), C H (a,trans-1,2,4,5-Tetramethylcyclohexane) and C H (a,2,2,5,5-Tetramethyhexane), respectively. Interestingly, the smallest 3D fragmentof diamond (i.e., Adamantane) appears as a ground state geometry in the group. The cor-responding chemical potential range is quite narrow though. The 2D cyclohexane-basedstructure is still present. The isomer with methy pair at both ends is predicted to have theground state geometry in the alkane subgroup.9
11 C11H7−28.3440.000 −27.630 (b) (a) (a)C11H18 C11H24 (a) (b) (b) −71.828 (a)C11H24−71.742 (b) −71.726 (c) −71.706 (d)−58.243 −58.074 C11H10C11H10 −38.208−38.193
FIG. 7: 11 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. C H m Fig. 7 shows the lowest energy structures with a backbone of 11 carbon atomsin the full hydrogen chemical potential range. In this C11 group, the lowest-energyhydrocarbon structures in hydrogen chemical potential range ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H , C H (a,2-Methylnaphthalene), C H (a,1-Methyladamantane) and C H (a,2,2,5,5-Tetramethyheptane), respectively. The energy order of the isomers inC11H10 subgroup is correctly predicted by MP2 compared with experiment although theenergy difference is tiny[21]. MP2 favors the growth of one methyl at the monohydrogensite and predicts that 1-Methyladamantane is 0.169 eV lower in energy than C H (b,2-Methyladamantane), close to the experimental value of 0.206 eV[21]. One may also observethat the 2D cyclohexane-based structure disappears in the ground state geometries of thehydrocarbons with carbon number larger than ten. Thus we observe a crossover between10
12 C12H8(a) 0.000 C12H12 (a) (b) (c) (d)C12H20 C12H26 (a) (a)C12H26 (d) (c) (b) (a) −32.579 (b) −31.749−43.824 −43.814 −43.810−43.792−63.908 −77.284−77.211 −77.209 −77.185−63.742 (b)
FIG. 8: 13 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. populations of 2D cyclohexane-based structure and 3D diamond fragment with increasingsize of hydrocarbon. The ground state isomer in the alkane subgroup is still predicted tothe one with methyl pair at both ends. C H m Fig. 8 shows the lowest energy structures with a backbone of 12 carbon atoms in the fullhydrogen chemical potential range. In this C12 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range ( − . eV, − . eV ), ( − . eV, − . eV ),11 FIG. 9: 10 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV.10 lowest energy structures of C H m in full hydrogen chemical potential range. The number below each structure is the relativeenergy E in unit of eV. ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H (Ace-naphthylene), C H (a,2,6-Dimethylnaphthalene), C H (a,1,3-Dimethyladamantane)and C H (a,2,2,7,7-Tetramethyloctane), respectively. Similar rules as mentioned abovecan be applied here: 2,6-Dimethylnaphthalene is the lowest energy isomer because of themaximal reduction of the repulsion energy between the hydrogen atoms; monohydrogen siteis the preferred site to grow methyl on adamantane; the isomer with methyl pair at bothends is still lowest in energy in the alkane subgroup.12 H m Fig. 9 shows the lowest energy structures with a backbone of 13 carbon atoms in the fullhydrogen chemical potential range. In this C13 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H (Phenalene), C H (a,1,3,5-Trimethylnaphthalene) and C H (a,2,2,8,8-Tetramethylnonane), respectively. Phenaleneis the lowest energy isomer in C13H9 subgroup due to its closely packed aromatic rings. 1,3,5-Trimethylnaphthalene is predicted to be much lower in energy than C H (b,dodecahydro-1H-Phenalene). The ground state structure in the alkane subgroup still keeps the singlecarbon chain with methyl pair at both ends. C H m Fig. 10 shows the lowest energy structures with a backbone of 14 carbon atoms in the fullhydrogen chemical potential range. In this C14 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range of ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H (Cyclopent[fg]acenaphthylene), C H (a,Phenanthrene), C H (a,Diamantane), C H (a,1,3,5,7-Tetramethyladamantane) and C H (a,2,2,5,8,8-penmethylnonane), respectively. Contrary to the rule of maximal reduction of repulsionenergy between hydrogen atoms, Phenanthrene is correctly predicted to be 0.283 eV lowerin energy than C H (b,Anthracene), comparable to the experimental result of 0.216 eV.A bigger diamond fragment (Diamantane) appears as ground state structure in a narrowhydrogen chemical potential range. The fifth methyl appears in the middle of the carbonchain besides methyl pair at both ends of the lowest energy isomer in the alkane subgroup. C H m Fig. 11 shows the lowest energy structures with a backbone of 15 carbon atomsin the full hydrogen chemical potential range. In this C15 group, the lowest-energyhydrocarbon structures in hydrogen chemical potential range of ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are13 .000 (a) (b)C14 C14H8−32.899 −32.185−39.182−39.465 −38.227C14H10 (a) C14H24C14H20 (c)−64.614 C14H30−87.187 −87.171 −87.142 (a) (b) (c) (b) (a) (b)−74.208 −73.685
FIG. 10: 12 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. C (ring), C H (a), C H (3-Methylphenanthrene), C H (a,3-Methyldiadamantane)and C H (a,2,2,5,5,8,8-Hexamethylnonane), respectively. C H (a) is predicted to be thelowest energy isomer due to its compact structure and minimal influence of the five fold ring.Methyl is still preferred to grow at the monohydrogen site of diadamantane. Three pairs ofmethyls are distributed symmetrically on the carbon chain in the ground state structure ofalkane subgroup. 14 .000 (a)C15C15H9 (c) −38.153−37.135C15H22 (a)−71.534 (b)−71.433 −93.953 −93.887 (a) (b)C15H32−46.341C15H12 (b)−37.722C15H9−37.001 (d) FIG. 11: 10 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. C160.000C16H24 C16H10(a) −40.507 (b) −38.975−74.928 (a)−97.153C16H34(b)−97.079
FIG. 12: 6 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. C H m Fig. 12 shows the lowest energy structures with a backbone of 16 carbon atoms in the fullhydrogen chemical potential range. In this C16 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range ( − . eV, − . eV ), ( − . eV, − . eV ),15 .000C17C17H26 C17H36C17H12C17H11−82.376 −48.001−44.505(a)−104.501 (b)−104.490 FIG. 13: 6 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H (a,Pyrene), C H , and C H (a,2,2,5,5,9,9-Hexamethyloctane), respectively. C H m Fig. 13 shows the lowest energy structures with a backbone of 17 carbon atoms in the fullhydrogen chemical potential range. In this C17 group, the lowest-energy hydrocarbon struc-tures in hydrogen chemical potential range of ( − . eV, − . eV ), ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H , C H (2-Methylpyrene), C H , and C H (a,2,2,6,6,10,10-Hexamethyludecane), respectively. C H m Fig. 14 shows the lowest energy structures with a backbone of 18 carbon atomsin the full hydrogen chemical potential range. In this C18 group, the lowest-energyhydrocarbon structures in hydrogen chemical potential range ( − . eV, − . eV ),( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV, − . eV ), ( − . eV,
0) are C (ring), C H ( a ), C H (2,7-Dimethylpyrene), C H (Diamantane) and C H (a,2,2,5,5,11,11-Hexamethyldodecane), respectively. Similar empirical rules are observedin C16-C18 groups: Poly-aromatic rings tend to form compact structure; Methyl favors16 .000C18C18H24 C18H38−79.087 C18H10 C18H14(a)−43.769 −54.026(b)−110.284(a)−110.361(b)−43.606 FIG. 14: 7 lowest energy structures of C H m in full hydrogen chemical potential range. Thenumber below each structure is the relative energy E in unit of eV. the monohydrogen site in diamond fragment structure; ground state structure of alkanesubgroup favors methyl pairs spaced by at least two carbon atoms over the main carbonchain. CONCLUSION
An unbiased evolution-based optimization method combined with Brenner’s empiricalpotential is used to search for ground state structures of hydrocarbon molecules in a widehydrogen chemical potential range. The resultant structures are further sorted by quantumchemical calculations at MP2 level. The collection of lowest energy structures of hydrocarbonmolecules C m H n (m=1-18; n=0-2m+2) is presented. A crossover between populations of 2Dcyclohexane-based structure and 3D diamond fragment with increasing number of carbonatoms of the hydrocarbon molecules is demonstrated. Besides the PAH compounds, wealso show that PAH with methyls can also be important in the interstellar medium. Thespectra of the PAH compounds and how are they affected by the various configurations ofthe radicals (e.g., methyl) are worthy of further studying. ACKNOWLEDGMENTS
We want to thank M.S. Tang for many useful discussions. Work at the Ames laboratorywas supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of17aterials Science and Engineering including a grant of computer time at the National EnergyResearch Supercomputing Center (NERSC) at the Lawrence Berkeley National Laboratoryunder Contract No. DE-AC02-07CH11358. C. Rareshide acknowledges the support fromNSF sponsored Research Experience for Undergraduates (REU) program at Iowa StateUniversity. [1] A. Leger, and J. L. Puget, A&A, , L5 (1984).[2] C. Joblin, A. Leger and P. Martin, Astrophys. J.
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