^{1}

^{*}

^{2}

^{1}

^{2}

Edited by: Jorge M. C. Marques, University of Coimbra, Portugal

Reviewed by: Stefan Goedecker, University of Basel, Switzerland; Jun Zhang, Pacific Northwest National Laboratory, United States

This article was submitted to Theoretical and Computational Chemistry, a section of the journal Frontiers in Chemistry

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Candidate structures for the global minima of adamantane clusters, (C_{10}H_{16})_{N}, are presented. Based on a rigid model for individual molecules with atom-atom pairwise interactions that include Lennard-Jones and Coulomb contributions, low-energy structures were obtained up to

Global optimization is an important topic in the physical and chemical sciences, whether we want to refine a force field, predict the native structure of a protein or the crystal structure of some condensed material, or find a practical solution to machine learning problems (Andricioaei and Straub,

The difficulties in practically solving the global optimization problem for atomic and molecular systems are at least 2-fold. Firstly, the number of available local minima is thought to increase exponentially with size, making systematic enumeration virtually impossible already above a few tens of particles (Hartke et al., ^{60} minima (Tsai and Jordan,

One additional difficulty arises in molecular systems, even described as rigid bodies, because of the interplay between translational and orientational degrees of freedom. In some cases, the molecules themselves are such that they impose drastic constraints on the collective arrangements that can be adopted by the clusters, starting with the dimer. This occurs, e.g., for planar polycyclic aromatic hydrocarbons (PAHs), which tend to assemble into columnar motifs (Rapacioli et al., _{2} (Maillet et al.,

In the present work we are interested in clusters of the adamantane molecule (C_{10}H_{16}). Adamantane is a small hydrocarbon molecule with pure sp^{3} hybridized carbon atoms arranged in a tetrahedral point group, often referred to as a diamondoid. It has a very high thermal stability, and could be found in deep petroleum sources (Dahl et al.,

Adamantane clusters were recently synthesized in the cryogenic environment of helium nanodroplets, in which they could be size-selected after ionization by an electron gun (Goulart et al.,

So far, the structure of adamantane clusters has not been characterized at the atomistic level of details, but indirect structural information could be drawn from the experimental mass spectra, which show special abundances at the sizes of 13, 19, and 38 molecules. While the former two magic numbers are compatible with icosahedral arrangements, the latter is strongly indicative of a close-packed face-centered cubic structure, suggesting a size-induced structural transition taking place above only a few tens of molecules. Icosahedral-to-cubic transitions are common in atomic and molecular clusters, as they convey the increasing energetic penalty that the highly connected icosahedral structures have to sustain, eventually in favor of less connected but also less strained close-packed structures (Doye et al.,

However, the experimental magic numbers do not provide any insight into the orientational ordering within the clusters, and in particular whether the tetrahedral symmetry plays any role on the structures. In order to shed some light onto the relative importance of the translational and orientational degrees of freedom and their interplay, and more generally to confirm whether adamantane clusters do indeed correspond to the speculated structures, we have carried out a systematic global optimization investigation in the size range up to 42 molecules, using the basin-hoping method as our main tool. Two complementary strategies have been employed, namely an all-atom (AA) approach based on a rigid body description, and a highly simplified, coarse-grained (CG) model averaging over all possible orientations.

At the all-atom level, our calculations predict that adamantane clusters are most stable as icosahedra until 14 molecules are reached, and above which the structural arrangement becomes close packed. The special stabilities in the mass spectra are reproduced by the second-energy difference in our all-atom model. At the coarse-grained level, differences appear already above six molecules, although both the icosahedral and cubic motifs at sizes 13 and 38 are correctly reproduced. Comparison between the two models confirms the important role played by the orientational degrees of freedom, despite adamantane being of a rather high symmetry, and shows that the close-packed structures are ideally composed of planes with alternating molecular orientations, a feature that the coarse-grained model is obviously unable to capture.

This paper is organized as follow. We present the potential energy surfaces in section 2 and the methodology employed in the global optimization in section 3. The results are discussed in section 4, and we summarize our conclusions in section 5.

Complete global optimization using an explicit description of electronic structure is unfeasible for systems containing hundreds or thousands of atoms, which furthermore can adopt many nearly degenerate local minima. For the present system, and using the model described just below, more than 20 local minima are found just for the adamantane dimer within 2 kJ/mol of the putative global minimum. Moreover, the interactions between neutral adamantane molecules are essentially non-covalent in nature, a notorious issue in quantum chemistry dealing with large molecules. However, the closed-shell electronic structure of the adamantane molecule makes classical force fields particularly attractive for modeling the potential energy surface. A primary assumption usually made at low temperatures relevant for cryogenic environments is to treat the molecules as rigid bodies, with all vibrations frozen. In this work, two models were considered for the interactions between adamantane molecules.

Following the traditional approach of classical force fields, we assume that adamantane molecules interact with each other through a sum of pairwise forces comprising repulsion-dispersion and Coulomb contributions. The interaction _{ab} between two rigid adamantane molecules

where _{i} and _{j} are the partial charges on site _{ij} is the Cartesian separation between the two sites. In the above expression, all sums were implicitly assumed to be between atoms from different molecules: no intramolecular potential acts for such rigid molecules.

The LJ parameters between sp^{3} carbon and hydrogen atoms are taken from the popular OPLS force field (Jorgensen et al., _{CC} = 0.458 kJ/mol, ε_{HH} = 0.066 kJ/mol, σ_{CC} = 3.4 Å, σ_{HH} = 2.649 Å, Lorentz-Berthelot combination rules providing the complementary values for C-H interactions. The partial charges on individual atoms and the equilibrium geometry of isolated adamantane were obtained from a quantum chemical calculation at the DFT/M06-2X/6-311G(d,p) level of theory. They read _{C} = −0.71 and _{H} = +0.14 for carbon and hydrogen atoms in CH_{2} groups, and _{C} = +0.70 and _{H} = −0.055 for carbon and hydrogen atoms in CH groups, in units of the electron charge magnitude.

To assess the accuracy and relevance of our simple force field, we have performed dedicated quantum chemical calculations for the adamantane dimer using various levels of theory. Density-functional theory (DFT) is probably the most practical method to deal with such molecules, and here we have chosen the modern functionals PBE0 (Adamo and Barone,

From the resulting geometries, the basic geometric properties of distance _{int} was also determined from the total energies of the optimized monomer and dimer using the standard equation

and the resulting values for _{int} are given in

Interaction energy and geometric properties of the adamantane dimer, as predicted by different quantum chemical methods and by the present empirical potential.

_{int} (kJ/mol) |
|||
---|---|---|---|

DFT/PBE0/6-311G(d,p) | −5.19 | 6.83 | −0.31 |

DFT/PBE0/aug-cc-pvDZ | −7.70 | 6.59 | −0.33 |

DFT/wB97xD/6-311G(d,p) | −21.33 | 6.05 | −0.25 |

DFT/wb97xD/aug-cc-pvDZ | −24.72 | 6.02 | −0.26 |

DFT/M06-2X/6-311G(d,p) | −12.97 | 6.14 | −0.27 |

DFT/M06-2X/aug-cc-pvDZ | −17.66 | 6.07 | −0.27 |

SCS-MP2/6-311G(d,p) | −9.97 | 6.26 | −0.31 |

SCS-MP2/aug-cc-pvDZ | −8.81 | 6.66 | −0.26 |

Force field | −14.92 | 6.22 | −0.36 |

Unsurprisingly, we find a significant spreading among the DFT results, with a marked dependence of the interaction energy on the functional used, and notably a factor >4 between PBE0 and wB97xD results, SCS-MP2 and M06-2X data lying in-between those extremes. The weaker binding predicted by PBE0 is consistent with this functional not properly accounting for dispersion interactions. Basis set effects further contribute to some variations, although with one magnitude lower. The strong differences between the predictions of PBE0 and wB97xD are comparable to those obtained earlier in other intermolecular interactions problem involving fullerenes and hydrogen (Kaiser et al.,

The force field based on OPLS with multipolar contributions obtained from partial charges derived from DFT performs very satisfactorily against the not-so-extreme quantum chemical predictions from M06-2X and SCS-MP2 both in terms of energy and geometry. The good performance of the force field against the Minnesotta functional M06-2X is also consistent with an earlier study on microhydrated RNA precursors (Bacchus-Montabonel and Calvo,

The high symmetry of adamantane encourages us to attempt a simplified description based on a coarse-grained model of the previous all-atom potential. Such an approach has been highly successful in the past for isotropic molecules such as C_{60}, for which simple analytical expressions can be obtained for the integrals (Girifalco, ^{6} independent orientational configurations.

In

Spherically averaged potential energy curve of the adamantane dimer (red circles) and its best fit giving the coarse-grained potential (black line). The minimum energy in the all-atom model is shown as a blue circle, and the inset highlights the distance range where the potential is minimum. The tetrahedral symmetry of adamantane molecules in the dimer geometry is also shown.

The energy and equilibrium position in the AA model, also highlighted in the figure, show that the CG model underestimates the binding energy by about one order of magnitude, owing to the strong repulsion between peripheral hydrogen atoms, and presents an equilibrium position at a larger distance. The effective potential is very steep, as also expected for an interaction between sizeable molecules. It has thus an effectively short range, which should favor close packing (Doye et al.,

The CG potential can be fitted into a simple expression only dependent on the interparticle distance

with a short-range cut-off function _{cut} that reads

The optimal parameters of the CG potential were found to be ^{−1}, _{0} = 9.405 Å, _{6} = 423040.5 kJ· mol^{−1}·Å^{6}, _{8} = 56522581.1 kJ· mol^{−1}·Å^{8},

The global energy minima were located using the basin-hopping (BH) or Monte Carlo plus minimization method (Li and Scheraga,

Basin-hopping is a stochastic algorithm that transforms the PES into a collection of basins of attraction and explore them by random large amplitude, collective moves between minima. This transformed PES preserves all local minima, including the global minima, and the search proceeds by successive applications of the Monte Carlo Metropolis acceptance rule to the locally minimized energies. The BH method has been successfully applied to a plethora of atomic and molecular clusters in the past (Wales et al.,

For the CG potential, only translational moves have to be considered, and several series of 10^{5} local minimizations were carried out for each cluster size, the fictitious temperature parameter being set such as _{B}

For the AA model, the translational and rotational moves can be either managed on a similar footing, or distinguished from one another. In the most general version, a random move thus consists of perturbing all positions of the centers of mass and rotating the molecules, both displacements being performed simultaneously before local minimization is carried out. Here we have chosen to represent the orientational degrees of freedom using angle-axis coordinates

Test runs performed for the 12-molecule cluster and employing 5 × 10^{4} BH steps allowed us to evaluate suitable parameters for the basin-hopping optimizations with the AA model, namely _{B}^{4} BH collective steps were performed for each cluster size.

The results reported below are thus the results of three combined approaches relying on basin-hopping but altering the entire set of degrees of freedom, only the orientations, or exploring the random removal of one molecule followed by further local search. The orientational minimization was also used to produce all-atom clusters with a specific translational ordering but lying in a different funnel as the global minimum. In practice it allowed us to generate icosahedral and cubic clusters in a broader size range, providing further insight into the related structural transition. In all our BH searches, the geometry was reset to the local minimum before a random perturbation was attempted again.

For the analysis of cluster minima, different order parameters and structural indicators were considered to probe the extent of translational and orientational orderings. The bond-orientational order parameter _{6} involves the relative positions of the molecular centers of mass and is useful to discriminate icosahedral and cubic packings (Calvo et al.,

where

_{b} being the number of bonds defined when the distance between of center of masses of two adamantane molecules is lower than 7.5 Å. _{6m}(θ_{ij}, ϕ_{ij}) is the spherical harmonic function of degree 6 and order _{6} parameter can be evaluated for both the AA and CG structures.

An orientational order parameter respecting the tetrahedral symmetry of adamantane was constructed to measure the extent of alignment within the clusters. More precisely, and following Fel (Fel,

For a set of molecules, an orientational order parameter κ that is tetrahedrally invariant is defined by considering the pairs of nearest-neighbor molecules as

where _{nn} is the number of nearest-neighbor molecules. The prefactor ensures that κ = 1 if all molecules are tetrahedrally aligned.

In addition to purely geometric indicators, energetic parameters were also evaluated to measure the relative stability of the clusters, and quantify the role of orientational strain (

The putative global minima of adamantane clusters were obtained with full atomistic details up to size 42. All structures are available in the

To estimate the relative stability of different cluster sizes, we evaluated the second-energy derivative of the PES, _{N} is the energy of the global minimum for (C_{10}H_{16})_{N}. Maxima in Δ^{2} correspond to clusters with enhanced stability, and are thus closely related to special abundances experimentally measured by mass spectrometry.

The variations of Δ^{2}

Second-energy derivative vs. cluster size for the all-atom and coarse-grained models.

From this figure it is clear that the two models do not predict the same special stabilities in the entire size range considered, except at

The energetic data obtained with atomistic details are essentially consistent with experimental data, indicating that our modeling of adamantane clusters is realistic. The contrasted behaviors between the two models suggest that the mutual orientations of the adamantane molecules play a significant role on the cluster structures.

Selected structures obtained with the AA and CG models are presented in

Remarkable structures obtained for selected adamantane clusters with 13–15, 26, and 38 molecules, in the all-atom and coarse-grained models.

For _{h} point group, the symmetry being lowered due to the important strain within the cluster. In the AA description the molecules manage to adopt appropriate orientations that bring the translational structure closer to the perfect icosahedron.

At size 14 both models predict a qualitatively different structural motif, as a capped icosahedron with all atoms, but showing a decahedral arrangement after coarse-graining. Decahedral motifs are known to occur as an intermediate packing scheme on the way from the highly coordinated, but highly strained icosahedra to the low coordinated and weakly strained close packed structures (Doye et al.,

At size 15 the all-atom model now predicts a cubic motif while the isotropic potential still yields a (doubly capped) decahedron. The cubic translational arrangement is preserved at sizes 16 and beyond, while the coarse-grained model further experiences some structural changes. At sizes 26 and above, both models favor close-packed cubic structures, leading to the perfect truncated octahedron at

To shed more light onto the respective roles of translational and orientational orderings on the stable structures of adamantane clusters, and to clarify the effects of coarse-graining, we now consider the structural order parameters introduced in section 3.2 in comparison between the two models. Near size 14 where the icosahedral-cubic transition takes place, additional but metastable structures were generated as belonging to the icosahedral and cubic families, by performing basin-hopping global optimization with orientational moves only.

The bond-orientational order parameter _{6} is shown against increasing cluster size for both models in

Bond-orientational order parameter _{6} obtained from the relative positions of the centers of mass of adamantane clusters in the all-atom and coarse-grained models. The values obtained for metastable icosahedral and cubic conformations near the corresponding transition are also shown.

Within the all-atom description, _{6} exhibits irregular, essentially decreasing variations during the completion of icosahedral packing at _{6} reaches about 0.58 and stays constant at this value, indicating that the face-centered cubic structure is robust and regular with no point defect or stacking fault.

In the CG model, _{6} displays the same value as in the AA description up to size 7, indicating that translational structures are identical. Differences above the critical size of _{6} reaches the same value as in the AA model. As confirmed by visual inspection along the lines of _{6} being lower than ~0.4, except for _{6} being also higher.

The differences between the AA and CG models further support that orientational ordering plays a role in establishing the close-packed translational ordering itself. To further explore this aspect, the order parameter κ was evaluated for atomistic global minima, the results of which are depicted in

Tetrahedral order parameter κ between nearest-neighbor molecules in adamantane clusters, as obtained from the all-atom model. The values obtained for metastable icosahedral and cubic conformations near the corresponding transition are also shown.

Similar to _{6}, the orientational order parameter displays irregular variations during the completion of the icosahedron at

This stability further indicates that the clusters adopt a constant growth scheme. However, the negative value of κ also shows that the tetrahedral molecules do not display a single, common orientation within the cluster, as otherwise κ would be closer to unity. To illustrate the specific orientational ordering, we have represented in

Selected adamantane clusters for which the second energy derivative shows peaks in the 19–29 size range. In the all-atom model, molecules were replaced by their equivalent tetrahedra to emphasize orientational ordering.

In this size range, the coarse-grained potential predicts both decahedral (19 and 29) and cubic (24) motifs. The structures obtained with the AA model show the same close-packed motif, with clusters of a given size that are subparts of larger global minima. More interestingly, and as suggested by the indicators previously discussed, the molecules show two different possible orientations that alternate between planes in the largest clusters. While molecules with the same orientation within a same plane are next nearest neighbors, nearest-neighbor molecules precisely belong to different planes and present parallel contact faces.

However, in the dimer at equilibrium (see

We have quantified the importance of strain in adamantane clusters by removing from their total potential energy the contribution between nearest-neighbor pairs, as if these pairs were at equilibrium (including their orientational degrees of freedom) (Doye et al., _{strain} reads

where _{min} denotes the minimum energy in the dimer at equilibrium.

In order to compare the two models, we have deemed more suitable to further normalize the strain energy by the magnitude of the dimer binding energy, considering thus a strain factor _{strain}/|_{min}| instead of the absolute strain energy. The variations of the strain factor with increasing size are represented in

Strain energy normalized by dimer energy, as a function of cluster size and for the all-atom and coarse-grained models. The equivalent tetrahedra in the equilibrium dimer geometry are depicted. The values obtained for metastable icosahedral and cubic conformations near the corresponding transition are also shown.

With the coarse-grained description, which ignores orientational degrees of freedom, most structures are either icosahedral or decahedral and thus exhibit moderate strain (Doye et al., _{6} previously considered.

In contrast, the all-atom model shows strongly increasing strain as the cluster size increases, with a peak at

The remarkable thermodynamical and chemical stability of adamantane makes it a valuable building block of supramolecular materials, including non-covalent molecular clusters. Recent mass spectrometry measurements under the cryogenic conditions of helium droplets have found magic numbers for cationic adamantane clusters at the sizes of 13, 19, and 38 molecules, as well as others higher suggesting close packed geometries (Goulart et al.,

Using the basin-hopping algorithm, the putative global minima of (C_{10}H_{16})_{N} clusters were found to follow icosahedral packing up to

Comparison between the all-atom and coarse-grained models highlights and explains the importance of orientational strain in the structure of adamantane clusters, in particular the sharp transition toward the cubic motif which arises due to a combination between the short range of the potential and the optimal orientations presented by the nearest-neighbor molecules with tetrahedral facets parallel to one another.

Here we have neglected the cationic nature of the adamantane clusters in the mass spectrometry experiments, but in a first approximation it could be accounted for by adding a polarization contribution and assuming the

The raw data supporting the conclusions of this manuscript will be made available by the authors, without undue reservation, to any qualified researcher.

JH-R and FC conceived the project, performed the electronic structure calculations and prepared, wrote, and discussed the manuscript. FC built and conducted the coarse-grained calculations. JH-R conducted the all-atom basin-hopping calculations.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

_{2})

_{n}with 13 ≤

^{+}(Ar)

_{n}clusters

_{2}on fullerenes and the solvation of C

_{60}by hydrogen clusters at finite temperature: a theoretical assessment

_{60}(H

_{2}O)

_{n}clusters

_{2}O)

_{n},

_{2}, N

_{2}, and SF

_{6}clusters

_{2}O)

_{n}with 2 ≤

_{13}and (H

_{2}O)

_{8}clusters

_{2}O)

_{n},