Abstract
A very recently developed optimization algorithm for carbon clusters (C n s) (Yen and Lai J Chem Phys 142:084313, 2015) is combined separately with different empirical bond-order potentials which were proposed also for carbon materials, and they are applied to calculate the lowest energy structures of C n s studying their structural changes at different size n. Based on predicted structures, we evaluate the practicality of four analytic bond-order empirical potentials, namely the Tersoff, Tersoff–Erhart–Albe, first-generation Brenner and second-generation Brenner (SGB) potentials. Generally, we found that the cluster C n (n = 3–60) obtained by the SGB potential undergoes a series of dramatic structural transitions, i.e., from a linear → a single ring → a multi-ring/quasi-two-dimensional bowl-like → three-dimensional fullerene-like shape; such variability of structural forms was not seen in the other three potentials. On closer examination of the C n s calculated using this potential and further comparing them with those obtained by the semiempirical density functional tight-binding theory calculations, we found that these C n are more realistic than similar works reported in the literature. In this respect, due to its potential applications in the study of chemically complex systems of different atoms especially chemical reactions (Che et al. Theor Chem Acc 102:346, 1999), the SGB potential can, moreover, be used to investigate larger size C n , and calculated structural results by this potential are naturally input configurations for higher-level density functional theory calculations. Another most remarkable finding in the present work is the C n results calculated by Tersoff–Erhart–Albe empirical potential. It predicts a two-dimensional development of graphene structure, exhibiting always a zigzag edge in the optimized clusters. This empirical potential can thus be applied to study graphene-related materials such as that shown in a recent paper (Yoon et al. J Chem Phys 139:204702, 2013).
Similar content being viewed by others
Notes
Zhou et al. [33] developed their quantum-based bond-order potential from a series of works by Pettifor DG and coworkers (see this reference for references cited therein).
There are several calculations in the literature also employing the FGB potential to study Cn clusters either for a particular size n or in a specific range of n. Among them, Halicioglu [42] considered Cn < 6 using exactly the same Brenner potential expressions as us. The binding energies of the few Cn obtained by him are generally higher than ours due to his crude means of energy minimization. Others such as Wang et al. [43] applying a time-going-backward quasi-dynamics method were less systematic in structural studies of Cn or Zhang et al. (Ref. [44] below) who used genetic algorithm associated with simulated annealing method, and Yamaguchi and Maruyama [45, 46] performing molecular dynamics simulations; all these authors employed a modified FGB potential in which \({\pi}^{RC}_{ij}\) (see Eq. (10)) was omitted and their calculated Cns were therefore inappropriate for direct comparison with our results.
References
Gupta RP (1981) Phys Rev B 23:6265
Daw MS, Baskes MI (1984) Phys Rev B 29:6443
Daw MS, Foiles SM, Baskes MI (1993) Mater Sci Rep 9:251
Finnis MW, Sinclair JE (1984) Philos Mag A 50:45
Finnis MW, Sinclair JE (1986) Philos Mag A 53:161 (for erratum)
Sutton AP, Chen J (1990) Philos Mag Lett 61:139
Ercolessi F, Parrinello M, Tosatti E (1998) Philos Mag A 58:213
Calvo F, Spiegelmann F (2000) J Chem Phys 112:2888
Assadollahzadeh B, Schwerdtfeger P (2009) J Chem Phys 131:064306
Abell GC (1985) Phys Rev B 31:6184
Ferrante J, Smith JR, Rose JH (1983) Phys Rev Lett 50:1385
Rose JH, Smith JR, Ferrante J (1983) Phys Rev B 28:1835
Austin BJ, Heine V, Sham LJ (1962) Phys Rev 127:276
Anderson PW (1968) Phys Rev Lett 21:13
Weeks JD, Anderson PW, Davidson AGH (1973) J Chem Phys 58:1388
Tersoff J (1986) Phys Rev Lett 56:632
Tersoff J (1988) Phys Rev B 37:6991
Tersoff J (1988) Phys Rev B 38:9902
Tersoff J (1988) Rev Lett 61:2879
Tersoff J (1989) Phys Rev B 39:5566
Brenner DW (1990) Phys Rev B 42:9458
Brenner DW (1992) Phys Rev B 46:1948 (for erratum)
Robertson DH, Brenner DW, White CT (1991) Phys Rev Lett 67:3132
Brenner DW, Robertson DH, Elert ML, White CT (1993) Phys Rev Lett 70:2174
Che J, Cagin T, Goddard III WA (1999) Theor Chem Acc 102:346
Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB (2002) J Phys Condens Matter 14:783
Erhart P, Albe K (2005) Phys Rev B 71:035211
Jakse N, Arifin R, Lai SK (2011) Condens Matter Phys 14:43802
Yoon TL, Lim TL, Min TK, Hung SH, Jakse N, Lai SK (2013) J Chem Phys 139:204702
Schall JD, Gao G, Harrison JH (2008) Phys Rev B 77:115209
Kumagai T, Izumi S, Hara S, Sakai S (2007) Comput Mater Sci 39:457
Kumagai T, Hara S, Choi J, Izumi S, Kato T (2009) J Appl Phys 105:064310 (See also references cited in)
Zhou XW, Ward DK, Foster ME (2015) J Comput Chem 36:1719
Drautz R, Hammerschmidt T, Čak TM, Pettifor DG (2015) Model Simul Mater Sci Eng 23:074004
Hur J, Stuart SJ (2012) J Chem Phys 137:054102
Yen TW, Lai SK (2015) J Chem Phys 142:084313
Harrison JA, Fallet M, Ryan KE, Mooney BL, Knippenberg WT, Shall JD (2015) Model Simul Mater Sci Eng 23:074003
van Duin ACT, Dasgupta S, Lorant F, Goddart WA III (2001) J Phys Chem A 105:9396
Hobday S, Smith R (1997) J Chem Soc Faraday Trans 93:3919
Hobday S, Smith R (2000) Mol Simul 25:93
Cai W, Shao N, Shao X, Pan Z (2004) J Mol Struct (Theochem) 678:113
Halicioglu T (1991) Chem Phys Lett 179:159
Wang Y et al (2008) Phys Rev B 78:026708
Zhang C, Xu X, Wu H, Zhang Q (2002) Chem Phys Lett 364:213
Yamaguchi Y, Maruyama S (1998) Chem Phys Lett 286:336
Yamaguchi Y, Maruyama S (1998) Chem Phys Lett 286:343
Kosimov DP, Dzhurakhalov AA, Peeters FM (2008) Phys Rev B 78:235433
Kosimov DP, Dzhurakhalov AA, Peeters FM (2010) Phys Rev B 81:195414
Backman M, Juslin N, Nordlund K (2012) Eur Phys J B 85:317
Stuart SJ, Tutein AB, Harrison JA (2000) J Chem Phys 112:6472
Monteverde U, Migliorato MA, Pal J, Powell D (2013) J Phys Condens Matter 25:425801
Wales DJ, Doye JP (1997) J Phys Chem A 101:5111
Li Z, Scheraga HA (1987) Proc Natl Acad Sci USA 84:6611
Liu D, Nocedal J (1989) Math Progr B 45:503
Lai SK, Hsu PJ, Wu KL, Liu WK, Iwamatsu M (2002) J Chem Phys 117:10715
Hsu PJ, Lai SK (2006) J Chem Phys 124:044711
Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE (1985) Nature 318:162
Slanina Z, Ishimura K, Kobayashi K, Nagase S (2004) Chem Phys Lett 384:114
Drowart J, Burns RP, De Maria G, Inghram MG (1959) J Chem Phys 31:1131
Menon M, Subbaswamy KR, Sawtarie M (1993) Phys Rev B 48:8398
Mylvaganam K, Zhang LC (2004) Carbon 42:2025
Kutana A, Giapis KP (2008) J Chem Phys 128:234706
Acknowledgements
This work is supported by the Ministry of Science and Technology (MOST103-2112-M-008-015-MY3), Taiwan. We thank Prof. R. Smith for sending us the code of the first-generation Brenner potential.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Empirical bond-order potentials
In this appendix, we briefly summarize some of the essential features of the four selected empirical potentials to be used in this work. They are presented all with reference to Sect. 2 above.
1.1 Tersoff empirical potential
Starting from the general formula of Eq. (1) and the equations that follow, the form proposed by Tersoff [20] is that he sets \( r_{ij}^{\left( 0 \right)} = 0 \) in Eq. (3), γ ik = 1 in Eq. (6) and also assumes b ij = b ji which implies that the parameters γ ik , c ik , d ik and h ik in Eq. (6) are independent of the atom at site k, i.e., one may write (γ ik , c ik , d ik , h ik ) simply as (γ i , c i , d i , h i ). With these approximate replacements, the author determined the parameters in the remaining formulas by fitting them to bulk properties. Two further comments are in order. Firstly, the bond order considered in this potential only works well in the graphite and diamond environments (through fitting parameters to them), and it is a lack of the non-local environment (conjugation and radical effect) [21, 22]. Secondly, the bond angle θ jik is fitted only to optimal bulk bond angle that satisfies θ jik < 180o which means that only bent structures of sp 2 and sp 3 bonds are allowed.
1.2 Tersoff–Erhart–Albe empirical potential
The empirical potential of Erhart and Albe [27] is basically of Abell–Tersoff-type described by the same Eq. (1). Referring to Eq. (3), here the parameter \( r_{ij}^{\left( 0 \right)} \) is not zero in the TEA potential and it is fitted to the dimer bond length, a term omitted in the Tersoff potential. The authors also assume γ ik ≠ 1, b ij ≠ b ji and quantities A, B, λ and μ in Eq. (3) are now expressed as
where D 0 is the dimer energy, ω is associated with the ground-state oscillation frequency of the dimer, and Z is a parameter adjusted to the slope of the Pauling plot [62]. The TEA potential is thus constructed through fitting D 0 to the measured dimer energy and \( r_{ij}^{\left( 0 \right)} \) to dimer bond length as well as the cohesive energy of cubic (simple cubic, body-centered and face-centered lattices) and diamond structures. By this numerical procedure, the TEA potential [27] improved on the approximations that Tersoff potential made on parameters γ ik , c ik , d ik and h ik mentioned above since more realistic angular bonding factor such as b ij ≠ b ji has now been explicitly taken into account in these parameters by fitting to cohesive energies and bond lengths of several high-symmetry structures as well as to the elastic constants of ground-state structures. Finally, as in Tersoff potential, the optimal bond angle h ik used in Eq. (6) corresponds to optimal angle of the bulk system implying that only sp 2 and sp 3 bonds with θ jik < 180o are included. Further detailed comparison of the parameters in the Tersoff and TEA empirical potentials are compiled in Table I of Ref. [29].
1.3 Brenner potential: first generation
The FGB empirical potential [21, 22] has essentially the same mathematical structure as the Tersoff [20] and TEA [27] potentials (see Appendix “Tersoff empirical potential” and Tersoff–Erhart–Albe empirical potential). The major modification is the author’s observation that the bond-order interaction in either Tersoff or TEA potential is mainly fitted to bulk graphite and diamond solids where only single- and double-bond characters appear, notably in graphite. These latter two empirical potentials do not therefore work well when the radical effects have to be taken into proper account and in circumstances for conjugated systems. To include these features, he wrote
where the function \( \pi_{ij}^{RC} \) is introduced to rectify the unphysical situation of overbinding of radicals and the apparently contradictory feature that could happen between conjugated and nonconjugated double bonds in the absence of non-local effects. In other words, \( \pi_{ij}^{RC} \) depends on whether the bond between atoms i and j that each with its own total number of C neighbors has a radical character and is part of a conjugated system. This term therefore represents the influence of radical energetics and π-bond conjugation on the bond energies, and it describes radical structures correctly as well as including also for non-local conjugation effects such as those governing different properties of the carbon–carbon bonds. In this FGB potential, the first term [cf. Eq. (4)] in Eq. (10) reads
where fC(r ik ) is a cutoff function similar to Eq. (7) and b ji is defined by writing i → j. The b ij or b ji depends on the bond angle through g i (g j ) between bond ij (ji) of atom i (j) and ik (jk), and the local coordination through the Δ in {…} of atom i (j). Both Δ and α ijk are identically zero in this FGB potential, and values of δ i and δ j are both set equal to 0.80469 [21, 22]. Note that the function g i (θ jik ) has the same form as Eq. (6), but the value of h ik has been now chosen to be cos180o. For other parameters that appear in Eq. (11), the author determined them from the binding energies, lattice constants of graphite, diamond, simple cubic and face-centered cubic lattices of pure carbon, and also from the vacancy and formation energy of diamond and graphite. All of these numerical values are given in Ref. [21, 22].
Before proceeding further, a relevant comment is perhaps relevant. It is that this FGB potential does not take into account properly the angular dependence of the optimal angle for the planar ring structure. Neither does it include the rotations about carbon–carbon double bonds and non-bonded interaction.
1.4 Brenner potential: second generation
One major modification in the SGB empirical potential [26] is to add to Eq. (10) on the right-hand side a term
where the function T ij is a tricubic spline, cosΘ kijℓ = e jik · e ijℓ with e jik a unit vector in the direction of the product R ji × R ik , R ji being the vector connecting atoms j and i. The f C(r ij ) has the same meaning as that defined by Eq. (7). Equation (12) describes the dihedral angle for carbon–carbon double bonds. We should point out that in this SGB potential values of δ i and δ j in Eq. (11) are both 0.5, and g i (θ jik ) or g j (θ ijk ) does not have the same functional form as Eq. (6) but is calculated instead as sixth-order polynomial splines [26] in cosΘ kijℓ . The latter change in functional form was carried out to rectify the above-mentioned deficiency in the FGB potential and it considered more realistic dependences on the angular interaction. In this work, we shall examine also this SGB potential which is well coded and readily for use in the LAMMPS software (See http://lammps.sandia.gov for LAMMPS code). A similar potential which was generalized to include oxygen has been reported earlier by Kutana and Giapis [62], and Harrison et al. [37] revisited this potential very recently and gave a brief overview.
Appendix 2: modified basin hopping method
The numerical procedure of the BH method and its modifications is summarized as follows. For a cluster of n atoms, we randomly generate an atomic configuration and confine the n atoms inside a sphere of radius \( R_{d}^{*} \) (see Sect. 2) whose origin is positioned at the center of mass of the cluster. The searching of the lowest energy value follows the BH procedure originally proposed for metallic and nonmetallic clusters. We depict schematically in Fig. 7 a flowchart that describes in greater detail how the BH method is modified. The definition of all variables in the chart is the same as in Lai et al. [55], and readers interested in technical details are referred to this reference.
Rights and permissions
About this article
Cite this article
Lai, S.K., Setiyawati, I., Yen, T.W. et al. Studying lowest energy structures of carbon clusters by bond-order empirical potentials. Theor Chem Acc 136, 20 (2017). https://doi.org/10.1007/s00214-016-2042-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00214-016-2042-2