Molecular Structure Refinement Based on Residual Dipolar Couplings: A Comparison of the Molecular Rotational-Sampling Method with the Alignment-Tensor Approach.

In NMR experiments, residual dipolar couplings (RDCs) in a molecule can be measured by averaging the dipolar couplings (DCs) over the rotational motion of a molecule in an environment that induces a slight anisotropic orientation distribution of the molecule. Since the shape of the anisotropic distribution cannot be measured, it is standard practice to use a particular orientation distribution of the molecule with respect to the magnetic field, in the form of a so-called alignment tensor (AT), to calculate RDC-values for the molecule. Since the same alignment tensor is commonly used to calculate the different RDCs of a molecule, this approach rests on the assumption that the rotational motion of the molecule is decoupled from its internal motions and that the molecule is rigid. The validity of these two assumptions is investigated for a small, simple molecule, using a relatively rigid atomic interaction function or force field and a more flexible one. By simulating the molecule using an orientation-biasing force an anisotropic rotational distribution can be generated, for which RDCs can be obtained. Using these RDCs as target RDCs when applying one of the two approaches of structure refinement based on RDCs, it can be investigated how well the target RDCs are approximated in the RDC restraining and whether the corresponding nonuniform orientation distribution is reproduced. For the relatively rigid version of the molecule, the AT approach reproduces the target RDC-values, although the nonuniform orientation distribution of the angle θab,H between the vector r⃗ab connecting two atoms a and b in the molecule and the vector representing the direction of the magnetic field H⃗ as generated in the orientation-biasing simulation cannot be reproduced in the AT RDC-restraining simulation. For the relatively flexible version of the molecule, the AT approach fails to reproduce both the target RDC values and the nonuniform orientation distribution. For biomolecules with flexible parts, the application of the AT approach is thus not recommended. Instead, a method based on sampling of the rotational and internal degrees of freedom of the molecule should be applied in molecular structure determination or refinement based on measured RDCs.

Magnetically Induced Current Densities in π-Conjugated Porphyrin Nanoballs.

Magnetically induced current densities (MICDs) of Zn-porphyrinoid nanostructures have been studied at the density functional theory level using the B3LYP functional and the def2-SVP basis set. Six of the studied Zn-porphyrinoid nanostructures consist of two crossing porphyrinoid belts, and one is a porphyrinoid nanoball belonging to the octahedral (O) point group. The Zn-porphyrin units are connected to each other via butadiyne linkers as in a recently synthesized porphyrinoid structure resembling two crossed belts. The MICDs are calculated using the gauge-including magnetically induced current method. Current-density pathways and their strengths were determined by numerically integrating the MICD passing through selected planes that cross chemical bonds or molecular rings. The current-density calculations show that the studied neutral molecules are globally nonaromatic but locally aromatic sustaining ring currents only in the individual porphyrin rings or around two neighboring porphyrins. The ring-current strengths of the individual porphyrin rings are 20% weaker than in Zn-porphyrin, whereas oxidation leads to globally aromatic cations sustaining ring currents that are somewhat stronger than for Zn-porphyrin.

Magnetically Induced Current Densities in Zinc Porphyrin Nanoshells.

The molecular structures of porphyrinoid cages were obtained by constructing small polyhedral graphs whose vertices have degree-4. The initial structures were then fully optimized at the density functional theory (DFT) level using the generalized gradient approximation. Some of polyhedral vertices were replaced with Zn-porphyrin units and their edges were replaced with ethyne or butadiyne bridges or connected by fusing two neighboring Zn-porphyrin units. Molecule 1 is an ethyne-bridge porphyrinoid nanotube, whose ends are sealed with a Zn-porphyrin. Molecule 2 is the corresponding open porphyrinoid nanotube. Molecule 3 is a clam-like porphyrinoid cage, whose shells consist of fused Zn-porphyrins, and the two halves are connected via butadiyne bridges. Molecule 4 is a cross-belt of fused Zn-porphyrins, and molecule 5 is a cross-belt of Zn-porphyrins connected with butadiyne bridges. The magnetically induced current density of the optimized porphyrinoid cages was calculated for determining the aromatic character, the degree of aromaticity and the current-density pathways. The current-density calculations were performed at the DFT level with the gauge─including magnetically induced currents (GIMIC) method using the B3LYP hybrid functional and def2-SVP basis sets. Calculations of the current densities show that molecule 2 sustains a paratropic ring current around the nanotube, whereas sealing the ends as in molecule 1 leads to an almost nonaromatic nanotube. Fusing porphyrinoids as in molecules 3 and 4 results in complicated current-density pathways that differ from the ones usually appearing in porphyrinoids. The aromatic character of molecules 4 and 5 changes upon oxidation. The neutral molecule 4 is antiaromatic, whereas the dication is nonaromatic. Molecule 5 is nonaromatic, and its dication is aromatic.

A method for designing a novel class of gold-containing molecules.

We propose a novel class of gold-containing molecules, which have been designed using conjugated carbon structures as templates. The sp-hybridized carbons of C2 moieties are replaced with a gold atom and one of the adjacent carbons is replaced by nitrogen. Applying the procedure to hexadehydro[12]annulene yields the well-known cyclic trinuclear gold(i) carbeniate complex. Planar, tubular and cage-shaped complexes can be obtained by taking similar sp-hybridized carbon structures as the starting point.

Magnetically Induced Ring-Current Strengths in Möbius Twisted Annulenes.

The topology of twisted molecular rings is characterized by the linking number, which is equal to the sum of the twist-a local property of the molecular frame-and the writhe-a global parameter, which represents the bending of the molecular ring. In this work, we investigate a number of cyclic all- trans C40H40 annulenes with varying twisting numbers for a given linking number and their dications. The aromatic character is assessed by calculating ring-current strength susceptibilities using the gauge-including magnetically induced currents (GIMIC) method, which makes it possible to conduct a systematic study of the relation between the topology and aromaticity of twisted molecules. We found that the aromatic properties of the investigated Möbius twisted molecules are not only dependent on the linking number as previously suggested but also depend strongly on the partitioning of the linking number into the twist and writhe contributions.

On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability.

The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.

Block-restraining of residual dipolar couplings to allow fluctuating relative alignments of molecular subdomains.

Residual dipolar couplings (RDCs), unlike most other types of NMR observables, provide orientational information, reporting on the alignment of inter-spin vectors (ISVs) relative to the magnetic field. A great challenge in using experimental RDCs to restrain molecular dynamics (MD) simulations is how to represent this alignment. An alignment tensor is often used to parameterise the contribution of molecular alignment to the angular dependence of RDCs. All ISVs that share the same tensor have fixed relative alignment, i.e. if just one tensor is used, the molecule is internally rigid. Here we propose and illustrate a method for subdividing molecules into individually aligned blocks during MD simulations restrained to fit RDCs. This allows the relative orientation of each block to vary during the simulation, which in turn ensures that the internal structure of each block is more realistically reproduced.

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 Šfor 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).

Novel hollow all-carbon structures.

A new family of cavernous all-carbon structures is proposed. These molecular cage structures are constructed by edge subdivisions and leapfrog transformations from cubic polyhedra or their duals. The obtained structures were then optimized at the density functional level. These hollow carbon structures represent a new class of carbon allotropes which could lead to many interesting applications.

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 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.

Fitting alignment tensor components to experimental RDCs, CSAs and RQCs.

Residual 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 [Formula: see text], 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 [Formula: see text].

Structure and properties of the nonface-spiral fullerenes T-C380, D3-C384, D3-C440, and D3-C672 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.