Tuning (Anti)Aromaticity: Variations on the [8]-Circulene Framework.

Optoelectronic properties of organic molecules are underpinned by delocalisation and delocalisability of π-electrons. These properties are sensitive to small changes in electron count, whether achieved by heteroatom substitution or redox chemistry. One measure of the delocalisability of π-electrons is the current induced by an external magnetic field, which is diagnostic of (anti)aromaticity. The ab initio ipsocentric method is used here to model diverse ring-current patterns in the family of [8]-circulenes based on tetracyclopenta[def,jkl,pqr,vwx]tetraphenylene (TCPTP), in different charge states, with disjoint hetero-atom substitution, and with CC units systematically replaced by BN pairs. Maps calculated at the CHF/CTOCD-DZ2/6-31G** level reveal that these modifications of the TCPTP framework access the full range of possibilities for current from concentric global circulations (typically counter rotating) to full (non-aromatic) localisation. In the ipsocentric approach, induced current density is partitioned into robust orbital contributions that obey selection rules based on orbital symmetry, energy and nodal character. The selection rules are applied here to interpret current-density and exploit insights gained from simpler models to suggest design strategies for fine-tuning of π-delocalisability (aromaticity and antiaromaticity) in macrocyclic frameworks.

Periodoannulenes: A Generalized Annulene-within-an-Annulene Paradigm for Combined σ and π Ring Currents.

Periodoannulene molecules and ions CxIxq in planar geometry offer examples of systems with the potential for outer σ and inner π ring-current double aromaticity, given a sufficient overlap of tangential pσ-orbital manifolds on the large atoms of the outer cycle. Previous theoretical work indicated concentric diatropic currents in the dication C6I62+. Ab initio ipsocentric calculations support an account in terms of frontier-orbital selection rules for current contributions in C6I62+ (and radical C6I6+, implicated in recent experimental work on the oxidation of periodobenzene). A σ/π analogue of the annulene-within-an-annulene model is applied here to periodo systems based on cyclooctatetraene. Model species C8I8q with charges q = 0, +1, +2, +4, -2 and structures constrained to a planar D4h symmetry exhibit maps with all combinations of σ/π con- and counter-rotation, comprising global σ ring currents on the iodine perimeter and central π ring currents on the carbocycle. All can be rationalized by the separate application of the tropicity selection rules to the two subsystems, whether in singlet or triplet states.

A Correlated Source-Sink-Potential Model Consistent with the Meir-Wingreen Formula.

We model a molecular device as a molecule attached to a set of leads treated at the tight-binding level, with the central molecule described to any desired level of electronic structure theory. Within this model, in the absence of electron-phonon interactions, the Landauer-Büttiker part of the Meir-Wingreen formula is shown to be sufficient to describe the transmission factor of the correlated device. The key to this demonstration is to ensure that the correlation self-energy has the same functional form as the exact correlation self-energy. This form implies that nonsymmetric contributions to the Meir-Wingreen formula vanish, and hence, conservation of current is achieved without the need for Green's Function self-consistency. An extension of the Source-Sink-Potential (SSP) approach gives a computational route to the calculation and interpretation of electron transmission in correlated systems. In this picture, current passes through internal molecular channels via resonance states with complex-valued energies. Each resonance state arises from one of the states in the Lehmann expansion of the one-electron Green's function, hole conduction derived from ionized states, and particle conduction from attached states. In the correlated device, the dependence of transmission on electron energy is determined by four structural polynomials, as it was in the tight-binding (Hückel) version of the SSP method. Hence, there are active and inert conduction channels (in the correlated case, linked to Dyson orbitals) governed by a set of selection rules that map smoothly onto the simplest picture.

Modelling aromatisation of (BN)nH2n azabora-annulenes.

Despite a long tradition of descriptions of borazine as an 'inorganic benzene', this molecule is a non-aromatic species according to the magnetic (ring-current) criterion. Borazine, borazocine, and the larger neutral (BN)nH2n azabora-annulene heterocycles in planar conformations, although π-isoelectronic with [2n]annulenes, support only localized induced currents in perpendicular magnetic fields. The π-current maps of these systems comprise superpositions of separate 'lone-pair' circulations on all nitrogen centres. For the systems with n > 4, planarity must be enforced by a constraint. Qualitative orbital analysis based on the ipsocentric approach to calculation of induced current density suggests that global induced currents could be produced through strategic changes to the π electron count. In ab initio calculations, azabora-annulenes with rings of size [8]- and larger were indeed found to support global diatropic ring currents in both anionic and cationic forms with (4N + 2) π electron counts. The planar conformation of the charged ring typically occupies a stationary point of higher order on the potential energy surface, rather than a minimum. However, the borazocine dianion, [B4N4H8]2-, occupies a planar minimum, supports a diatropic ring current of strength comparable to that in benzene, and is predicted to participate in sandwich compounds; it is therefore a good candidate for an aromatised azabora-annulene.

Ring-Current Maps for Benzenoids: Comparisons, Contradictions, and a Versatile Combinatorial Model.

As a key diagnostic property of benzenoids and other polycyclic hydrocarbons, induced ring current has inspired diverse approaches for calculation, modeling, and interpretation. Grid-based methods include the ipsocentric ab initio calculation of current maps, and its surrogate, the pseudo-π model. Graph-based models include a family of conjugated-circuit (CC) models and the molecular-orbital Hückel-London (HL) model. To assess competing claims for physical relevance of derived current maps for benzenoids, a protocol for graph-reduction and comparison was devised. Graph reduction of pseudo-π grid maps highlights their overall similarity to HL maps, but also reveals systematic differences. These are ascribed to unavoidable pseudo-π proximity limitations for benzenoids with short nonbonded distances, and to poor continuity of pseudo-π current for classes of benzenoids with fixed bonds, where single-reference methods can be unreliable. Comparison between graph-based approaches shows that the published CC models all shadow HL maps reasonably well for most benzenoids (as judged by L1-, L2-, and L∞-error norms on scaled bond currents), though all exhibit physically implausible currents for systems with fixed bonds. These comparisons inspire a new combinatorial model (Model W) based on cycle decomposition of current, taking into account the two terms of lowest order that occur in the characteristic polynomial. This improves on all pure-CC models within their range of applicability, giving excellent adherence to HL maps for all Kekulean benzenoids, including those with fixed bonds (halving the rms discrepancy against scaled HL bond currents, from 11% in the best CC model, to 5% for the set of 18 360 Kekulean benzenoids on up to 10 hexagonal rings). Model W also has excellent performance for open-shell systems, where currents cannot be described at all by pure CC models (4% rms discrepancy against scaled HL bond currents for the 20112 non-Kekulean benzenoids on up to 10 hexagonal rings). Consideration of largest and next-to-largest matchings is a useful strategy for modeling and interpretation of currents in Kekulean and non-Kekulean benzenoids (nanographenes).

Design of annulene-within-an-annulene systems by the altanisation approach. A study of altan-[n]annulenes.

The altanisation strategy, devised to design molecules with large and paratropic perimeter circulations, is applied to the family of [n]annulenes to give, altan-[n]annulenes, i.e. [n,5]coronenes. Analytical expressions are obtained for the eigenvalues of the Hückel Hamiltonian for altan-[n]annulenes, and used in conjunction with selection rules derived from the ipsocentric approach to predict patterns of global ring current in these systems. Density-functional calculations performed on seven altan-[n]annulenes, three neutral and four charged, give current-density maps in essential agreement with the predictions obtained at the unperturbed Hückel level. All but one of the systems show patterns with the tropicities expected for isolated annulenes, in line with the altanisation concept. The apparent exception is altan-[11]annulene-, the only singlet system with a well defined open-shell character in the studied set. The key role of open-shell character can be accommodated by appropriate choice of the occupation numbers of the initial Hückel molecular orbitals, where the anion altan-[11]annulene- is considered as an [11]annulene inside the [22]annulene anion.

Molecular graphs and molecular conduction: the d-omni-conductors.

Ernzerhof's source-and-sink-potential (SSP) model for ballistic conduction in conjugated π systems predicts transmission of electrons through a two-wire device in terms of characteristic polynomials of the molecular graph and subgraphs based on the pattern of connections. We present here a complete classification of conduction properties of all molecular graphs within the SSP model. An omni-conductor/omni-insulator is a molecular graph that conducts/insulates at the Fermi level (zero of energy) for all connection patterns. In the new scheme, we define d-omni-conduction/insulation in terms of Fermi-level conduction/insulation for all devices with graph distance d between connections. This gives a natural generalisation to all graphs of the concept of near-omni-conduction/insulation previously defined for bipartite graphs only. Every molecular graph can be assigned to a nullity class and a compact code defining conduction behaviour; each graph has 0, 1, >1 zero eigenvalues (non-bonding molecular orbitals), and three letters drawn from {C, I, X} indicate conducting, insulating or mixed behaviour within the sets of devices with connection vertices at odd, even and zero distances d. Examples of graphs (in 28 cases chemical) are given for 35 of the 81 possible combinations of nullity and letter codes, and proofs of non-existence are given for 42 others, leaving only four cases open.

In search of Coulson's lost theorem.

In Hückel theory, the bond number is the sum of the orders of the π bonds incident on a given carbon center. From the work of Coulson and his school, it has been believed for over 70 years that the bond number has a maximum of 3 and that this bound is realized by exactly one conjugated framework, that of the trimethylenemethane radical. Search of published literature and archived correspondence failed to find any formal proof of these two statements. Here, we provide a new formula for bond number that leads to an easily checked proof of both. The bond number of graphene is 1.574 597… (90.9% of the mathematical limit), and this value appears to act as a separator for the classes of metallic and semiconducting single-walled nanotubes, as defined within Hückel theory.

Can percolation theory explain the gelation behavior of diblock copolymer worms?

It is well known that polymerization-induced self-assembly (PISA) offers an efficient synthetic route for the production of highly anisotropic diblock copolymer worms. When prepared in aqueous media, such worms form thermoresponsive free-standing hydrogels that are (i) readily sterilizable, (ii) can act as a 3D matrix for the culture of normal mammalian cells and (iii) can induce stasis in human stem cell colonies. Herein we critically examine the gelation behavior of two types of diblock copolymer worms in terms of recent advances in percolation theory for rigid rods, which explicitly account for the effect of rod length polydispersity. More specifically, we use small-angle X-ray scattering (SAXS) to determine the weight-average worm contour length, L w, and the mean worm cross-sectional radius, R. This approach enables a direct comparison to be made between the theoretical critical worm volume fraction, φ c, required for gelation and the experimental values indicated by rheological measurements and tube inversion experiments. Given that these diblock copolymer worms are relatively flexible rather than truly rod-like, reasonably good agreement between these two parameters is observed, particularly for shorter, relatively stiff worms. For longer, more flexible worms a proportionality constant of approximately two is required to reconcile theory with experimental values for φ c. These findings are expected to have important implications for the aqueous gelation behavior exhibited by various other anisotropic nanoparticles, such as cellulose nanocrystals and semicrystalline block copolymer rods, and also fibril-forming small molecule (e.g. dipeptide) gelators.

Near omni-conductors and insulators: Alternant hydrocarbons in the SSP model of ballistic conduction.

Within the source-and-sink-potential model, a complete characterisation is obtained for the conduction behaviour of alternant π-conjugated hydrocarbons (conjugated hydrocarbons without odd cycles). In this model, an omni-conductor has a molecular graph that conducts at the Fermi level irrespective of the choice of connection vertices. Likewise, an omni-insulator is a molecular graph that fails to conduct for any choice of connections. We give a comprehensive classification of possible combinations of omni-conducting and omni-insulating behaviour for molecular graphs, ranked by nullity (number of non-bonding orbitals). Alternant hydrocarbons are those that have bipartite molecular graphs; they cannot be full omni-conductors or full omni-insulators but may conduct or insulate within well-defined subsets of vertices (unsaturated carbon centres). This leads to the definition of "near omni-conductors" and "near omni-insulators." Of 81 conceivable classes of conduction behaviour for alternants, only 14 are realisable. Of these, nine are realised by more than one chemical graph. For example, conduction of all Kekulean benzenoids (nanographenes) is described by just two classes. In particular, the catafused benzenoids (benzenoids in which no carbon atom belongs to three hexagons) conduct when connected to leads via one starred and one unstarred atom, and otherwise insulate, corresponding to conduction type CII in the near-omni classification scheme.

Raman Deuterium Isotope Probing Reveals Microbial Metabolism at the Single-Cell Level.

We illustrate that single-cell Raman microspectroscopy, coupled with deuterium isotope probing (Raman-DIP), provides a culture-independent and nondestructive approach to probe metabolic pathways of carbon substrates at the single-cell level. We found a distinguishable C-D vibration band at 2070-2300 cm-1 in single-cell Raman spectra (SCRS) when Escherichia coli used deuterated glucose and Pseudomonas sp. used deuterated naphthalene as sole carbon sources. The intensity of the C-D band is proportional to the extent of deuteration in the carbon source, and as little as 5% deuteration can be distinguished by analysis of SCRS. It suggests that Raman-DIP could be used to semiquantitatively and sensitively indicate the metabolism of deuterated carbon source in microbes. A lower lipid conversion rate of deuterated naphthalene compared to that of deuterated glucose was observed, presumably owing to different anabolic pathways and membrane alteration. Apart from the C-D band shift from C-H, SCRS also reveal several isotopic shifts of the phenylalanine band, of which the positions correlate well with a computational model. A reduction in phenylalanine deuteration in Pseudomonas sp. compared to that in E. coli is due to the dilution effect of different pathways of phenylalanine biosynthesis in Pseudomonas sp. Collectively, we demonstrate that Raman-DIP can not only indicate metabolic activity using deuterated carbon sources but also reveal different metabolic pathways by analyzing SCRS. By harnessing such low-cost and versatile deuterated substrates, Raman-DIP has the potential to probe a wide range of metabolic pathways and functions at the single-cell level.

Bespoke Diblock Copolymer Nanoparticles Enable the Production of Relatively Stable Oil-in-Water Pickering Nanoemulsions.

Sterically stabilized diblock copolymer nanoparticles with an intensity-average diameter of 25 nm are prepared in the form of a concentrated aqueous dispersion using polymerization-induced self-assembly (PISA). The addition of n-dodecane followed by high-shear homogenization produces n-dodecane-in-water Pickering macroemulsions of 22-46 µm diameter. If the nanoparticles are present in sufficient excess, then subsequent processing using a high-pressure microfluidizer leads to the formation of Pickering nanoemulsions with a mean oil droplet diameter below 200 nm. The size of these Pickering nanoemulsions can be tuned by systematically varying the nanoparticle concentration, applied pressure, number of passes, and oil volume fraction. High-internal-phase emulsions can also be achieved by increasing the n-dodecane volume fraction up to 0.80. TEM studies of (dried) n-dodecane droplets confirm the presence of intact nanoparticles and suggest a relatively high surface coverage, which is consistent with model packing calculations based on radius ratios. Such Pickering nanoemulsions proved to be surprisingly stable with respect to Ostwald ripening, with no significant change in the mean DLS droplet diameter after storage for approximately 4 months at 20 °C.

From C(58) to C(62) and back: Stability, structural similarity, and ring current.

An increasing number of observations show that non-classical isomers may play an important role in the formation of fullerenes and their exo- and endo-derivatives. A quantum-mechanical study of all classical isomers of C58 , C60 , and C62 , and all non-classical isomers with at most one square or heptagonal face, was carried out. Calculations at the B3LYP/6-31G* level show that the favored isomers of C58 , C60 , and C62 have closely related structures and suggest plausible inter-conversion and growth pathways among low-energy isomers. Similarity of the favored structures is reinforced by comparison of calculated ring currents induced on faces of these polyhedral cages by radial external magnetic fields, implying patterns of magnetic response similar to those of the stable, isolated-pentagon C60 molecule. © 2016 Wiley Periodicals, Inc.

An atlas of endohedral Sc2S cluster fullerenes.

Structural identification is a difficult task in the study of metallofullerenes, but understanding of the mechanism of formation of these structures is a pre-requisite for new high-yield synthetic methods. Here, systematic density functional theory calculations demonstrate that metal sulfide fullerenes Sc2S@Cn have similar cage geometries from C70 to C84 and form a close-knit family of structures related by Endo-Kroto insertion/extrusion of C2 units and Stone-Wales isomerization transformations. The stabilities predicted for favoured isomers by DFT calculations are in good agreement with available experimental observations, have implications for the formation of metallofullerenes, and will aid structural identification from within the combinatorially vast pool of conceivable isomers.

A Hückel source-sink-potential theory of Pauli spin blockade in molecular electronic devices.

This paper shows how to include Pauli (exclusion principle) effects within a treatment of ballistic molecular conduction that uses the tight-binding Hückel Hamiltonian and the source-sink-potential (SSP) method. We take into account the many-electron ground-state of the molecule and show that we can discuss ballistic conduction for a specific molecular device in terms of four structural polynomials. In the standard one-electron picture, these are characteristic polynomials of vertex-deleted graphs, with spectral representations in terms of molecular-orbital eigenvectors and eigenvalues. In a more realistic many-electron picture, the spectral representation of each polynomial is retained but projected into the manifold of unoccupied spin-orbitals. Crucially, this projection preserves interlacing properties. With this simple reformulation, selection rules for device transmission, expressions for overall transmission, and partition of transmission into bond currents can all be mapped onto the formalism previously developed. Inclusion of Pauli spin blockade, in the absence of external perturbations, has a generic effect (suppression of transmission at energies below the Fermi level) and specific effects at anti-bonding energies, which can be understood using our previous classification of inert and active shells. The theory predicts the intriguing phenomenon of Pauli perfect reflection whereby, once a critical electron count is reached, some electronic states of devices can give total reflection of electrons at all energies.

Effect of Ring Size and Migratory Groups on [1,n] Suprafacial Shift Reactions. Confirmation of Aromatic and Antiaromatic Transition-State Character by Ring-Current Analysis.

Suprafacial sigmatropic shift reactions of 5-substituted cyclopentadienes, 3-substituted cyclopropenes, and 7-substituted cycloheptatrienes have been studied computationally at the MP2/6-31+G* level for structures and energetics and with the ipsocentric method at the CHF/6-31G** level to calculate current-density maps. The hydrogen shifts in cyclopentadienes have a diatropic ring current indicating aromatic, cyclopentadienide anion character. This result stands in contrast to the fluorine shift in 5-fluorocyclopentadiene which requires much more energy and has a paratropic ring current in the TS pointing to antiaromatic, cyclopentadienyl cation character. [1,3] hydrogen shifts in cyclopropenes are very difficult, passing through transition states that have an extended C-C bond. For 3-fluorocyclopropene, the [1,3] fluorine shift is much easier than the hydrogen shift. For 7-fluorocycloheptatriene, the [1,7] hydrogen shift is predicted but requires very high energy and has a paratropic ring current and antiaromatic character. The [1,7] suprafacial fluorine shift is relatively easy, having a TS with cycloheptatrienyl cation character. Patterns of currents, and the reversal for H and F migration, are rationalized by orbital analysis based on the ipsocentric method. Calculated charges and structural features for reactants and transition states support these conclusions.

Determining the Effective Density and Stabilizer Layer Thickness of Sterically Stabilized Nanoparticles.

A series of model sterically stabilized diblock copolymer nanoparticles has been designed to aid the development of analytical protocols in order to determine two key parameters: the effective particle density and the steric stabilizer layer thickness. The former parameter is essential for high resolution particle size analysis based on analytical (ultra)centrifugation techniques (e.g., disk centrifuge photosedimentometry, DCP), whereas the latter parameter is of fundamental importance in determining the effectiveness of steric stabilization as a colloid stability mechanism. The diblock copolymer nanoparticles were prepared via polymerization-induced self-assembly (PISA) using RAFT aqueous emulsion polymerization: this approach affords relatively narrow particle size distributions and enables the mean particle diameter and the stabilizer layer thickness to be adjusted independently via systematic variation of the mean degree of polymerization of the hydrophobic and hydrophilic blocks, respectively. The hydrophobic core-forming block was poly(2,2,2-trifluoroethyl methacrylate) [PTFEMA], which was selected for its relatively high density. The hydrophilic stabilizer block was poly(glycerol monomethacrylate) [PGMA], which is a well-known non-ionic polymer that remains water-soluble over a wide range of temperatures. Four series of PGMA x -PTFEMA y nanoparticles were prepared (x = 28, 43, 63, and 98, y = 100-1400) and characterized via transmission electron microscopy (TEM), dynamic light scattering (DLS), and small-angle X-ray scattering (SAXS). It was found that the degree of polymerization of both the PGMA stabilizer and core-forming PTFEMA had a strong influence on the mean particle diameter, which ranged from 20 to 250 nm. Furthermore, SAXS was used to determine radii of gyration of 1.46 to 2.69 nm for the solvated PGMA stabilizer blocks. Thus, the mean effective density of these sterically stabilized particles was calculated and determined to lie between 1.19 g cm-3 for the smaller particles and 1.41 g cm-3 for the larger particles; these values are significantly lower than the solid-state density of PTFEMA (1.47 g cm-3). Since analytical centrifugation requires the density difference between the particles and the aqueous phase, determining the effective particle density is clearly vital for obtaining reliable particle size distributions. Furthermore, selected DCP data were recalculated by taking into account the inherent density distribution superimposed on the particle size distribution. Consequently, the true particle size distributions were found to be somewhat narrower than those calculated using an erroneous single density value, with smaller particles being particularly sensitive to this artifact.

Structural interconnections and the role of heptagonal rings in endohedral trimetallic nitride template fullerenes.

Recent experiments indicate that fullerene isomers outside the classical definition can also encapsulate metallic atoms or clusters to form endohedral metallofullerenes. Our systematic study using DFT calculations, suggests that many heptagon-including nonclassical trimetallic nitride template fullerenes are similar in stability to their classical counterparts, and that conversion between low-energy nonclassical and classical parent cages via Endo-Kroto insertion/extrusion of C2 units and Stone-Wales isomerization may facilitate the formation of endohedral trimetallic nitride fullerenes. Close structural connections are found between favored isomers of trimetallic nitride template fullerenes from C78 to C82 . It appears that the lower symmetry and local deformations associated with introduction of a heptagonal ring favor encapsulation of intrinsically less symmetrical mixed metal nitride clusters. © 2016 Wiley Periodicals, Inc.

Perimeter ring currents in benzenoids from Pauling bond orders.

It is shown that the ring currents in perimeter hexagonal rings of Kekulean benzenoids, as estimated within the Randic conjugated-circuit model, can be calculated directly without tedious pairwise comparison of Kekulé structures or Kekulé counting for cycle-deleted subgraphs. Required are only the Pauling bond orders of perimeter bonds and the number of Kekulé structures of the benzenoid, both readily available from the adjacency matrix of the carbon skeleton. This approach provides easy calculation of complete current maps for benzenoids in which every face has at least one bond on the perimeter (as in the example of cata-condensed benzenoids), and allows qualitative evaluation of the main ring-current contributions to (1)H chemical shifts in general benzenoids. A combined Randic-Pauling model for correlation of ring current and bond length through bond order is derived and shown to be consistent with resilience of current under bond alternation.

A new approach to the method of source-sink potentials for molecular conduction.

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue of the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degeneracy.