Lukas N. Wirz

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207

Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes

Fullerene (Version 4.4) is a general purpose open‐source program that can generate any fullerene isomer, perform topological and graph theoretical analysis, as well as calculate a number of physical and chemical properties. The program creates symmetric planar drawings of the fullerene graph and generates accurate molecular 3D geometries by way of force‐field optimization, serving as a good starting point for further quantum theoretical treatments. It includes a number of fullerene‐to‐fullerene transformations, such as Goldberg–Coxeter transforms, Stone–Wales transforms, Endo–Kroto, Yoshida–Fowler, and Brinkmann–Fowler vertex insertions. The program is written in standard Fortran and C++ and can easily be installed in a Linux or UNIX environment.

From small fullerenes to the graphene limit: A harmonic force-field method for fullerenes and a comparison to density functional calculations for Goldberg-Coxeter fullerenes up to C980

We introduce a simple but computationally very efficient harmonic force field, which works for all fullerene structures and includes bond stretching, bending, and torsional motions as implemented into our open‐source code Fullerene. This gives accurate geometries and reasonably accurate vibrational frequencies with root mean square deviations of up to 0.05 Å for bond distances and 45.5 cm−1 for vibrational frequencies compared with more elaborate density functional calculations. The structures obtained were used for density functional calculations of Goldberg–Coxeter fullerenes up to C980. This gives a rather large range of fullerenes making it possible to extrapolate to the graphene limit. Periodic boundary condition calculations using density functional theory (DFT) within the projector augmented wave method gave an energy difference between −8.6 and −8.8 kcal/mol at various levels of DFT for the reaction C60→graphene (per carbon atom) in excellent agreement with the linear extrapolation to the graphene limit (−8.6 kcal/mol at the Perdew–Burke–Ernzerhof level of theory).

Structure and Properties of the Nonface-Spiral Fullerenes T -C 380 , D 3 -C 384 , D 3 -C 440 , and D 3 -C 672 and Their Halma and Leapfrog Transforms.

The structure and properties of the three smallest nonface-spiral (NS) fullerenes NS-T-C₃₈₀, NS-D₃-C₃₈₄, NS-D₃-C₄₄₀, and the first isolated pentagon NS-fullerene, NS-D₃-C₆₇₂, are investigated in detail. They are constructed by either a generalized face-spiral algorithm or by vertex insertions followed by a force-field optimization using the recently introduced program Fullerene. The obtained structures were then further optimized at the density functional level of theory and their stability analyzed with reference to Ih-C₆₀. The large number of hexagons results in a higher stability of the NS-fullerenes compared to C60, but, as expected, in a lower stability than most stable isomers. None of the many investigated halma transforms on nonspiral fullerenes, NS-T-C₃₈₀, NS-D₃-C₃₈₄, NS-D₃-C₄₄₀, and NS-D₃-C₆₇₂, admit any spirals, and we conjecture that all halma transforms of NS-fullerenes belong to the class of NS-fullerenes. A similar result was found to not hold for the related leapfrog transformation. We also show that the first known NS-fullerene with isolated pentagons, NS-D₃-C₆₇₂, is a halma transform of D3-C₁₆₈.

Naming Polyhedra by General Face-Spirals - Theory and Applications to Fullerenes and other Polyhedral Molecules*

Abstract We present a general face-spiral algorithm for cubic polyhedral graphs (including fullerenes and fulleroids), and extend it to the full class of all polyhedral graphs by way of the leapfrog transform. This yields compact canonical representations of polyhedra with a simple and intuitive geometrical interpretation, well suited for use by both computers and humans. Based on the algorithm, we suggest a unique, unambiguous, and simple notation for canonical naming of polyhedral graphs, up to automorphism, from which the graph is easily reconstructed. From this, we propose a practical nomenclature for all polyhedral molecules, and an especially compact form for the special class of fullerenes. A unique numbering of vertices is obtained as a byproduct of the spiral algorithm. This is required to denote modifications of the parent cage in IUPAC naming schemes. Similarly, the symmetry group of the molecule can be found together with the canonical general spiral at negligible cost. The algorithm is fully compatible with the classical spiral algorithm developed by Manolopoulos for fullerenes, i.e., classical spirals are accepted as input, and spiralable graphs lead to identical output. We prove that the algorithm is correct and complete. The worst case runtime complexity is for general N-vertex polyhedral graphs, with J the sum of all jump lengths. When the number of faces of any particular size is bounded by a constant, such as the case for fullerenes, this reduces to . We have calculated canonical general spirals for all 2,157,751,423 fullerene isomers from C20 to C200, as well as for all fullerene graphs that require jumps up to C400. Further, we have calculated canonical general spirals for large fullerenes with few or no classical spirals: all the Goldberg-Coxeter transforms up to C50,000 of the the non-spiralable chiral T-C380, D3-C384, D3-C440, and D3-C672 fullerenes, and for assorted fullerenes with no pentagon spiral starts. We verify exhaustively that the algorithm is linear for all the 2.7 × 1012 fullerene isomers up to C400, and show that this holds also for 11,413 large GC-transform fullerenes up to C50,000. On the used hardware, each single general spiral took about N × 200ns to produce for a CN fullerene, and the canonical general spiral was found in N × 22μs–32μs. Hence, we claim the algorithm to be efficient even for very large polyhedra. The algorithm is implemented in our program package Fullerene. In addition, the source code for a reference implementation of our proposed nomenclature for polyhedral molecules can be downloaded from http://erda.ku.dk/vgrid/Polyhedra/spirals/.

Fitting alignment tensor components to experimental RDCs, CSAs and RQCs

AbstractResidual dipolar couplings, chemical shift anisotropies and quadrupolar couplings provide information about the orientation of inter-spin vectors and the anisotropic contribution of the local environment to the chemical shifts of nuclei, respectively. Structural interpretation of these observables requires parameterization of their angular dependence in terms of an alignment tensor. We compare and evaluate two algorithms for generating the optimal alignment tensor for a given molecular structure and set of experimental data, namely SVD (Losonczi et al. in J Magn Reson 138(2):334–342, 1999), which scales as

Magnetically Induced Ring-Current Strengths in Möbius Twisted Annulenes

{{\mathcal {O}}(n^2)}

Comment on “A Tensor-Free Method for the Structural and Dynamic Refinement of Proteins using Residual Dipolar Couplings”

O(n2), and the linear least squares algorithm (Press et al. in Numerical recipes in C. The art of scientific computing, 2nd edn. Cambridge University Press, Cambridge, 1997), which scales as