A Turán-Type Problem on Distances in Graphs

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices pairwise at distance k has at most (n ? k + 1)²/4 pairs of vertices at distance k.     

meso- and β-Pyrrole-Linked Chlorin-Bacteriochlorin Dyads for Promoting Far-Red FRET and Singlet Oxygen Production

A series of chlorin-bacteriochlorin dyads (derived from naturally occurring chlorophyll-a and bacteriochlorophyll-a), covalently connected either through the meso-aryl or β-pyrrole position (position-3) via an ester linkage have been synthesized and characterized as a new class of far-red emitting fluorescence resonance energy transfer (FRET) imaging, and heavy atom-lacking singlet oxygen-producing agents. From systematic absorption, fluorescence, electrochemical, and computational studies, the role of chlorin as an energy donor and bacteriochlorin as an energy acceptor in these wide-band-capturing dyads was established. Efficiency of FRET evaluated from spectral overlap was found to be 95 and 98 % for the meso-linked and β-pyrrole-linked dyads, respectively. Furthermore, evidence for the occurrence of FRET from singlet-excited chlorin to bacteriochlorin was secured from studies involving femtosecond transient absorption studies in toluene. The measured FRET rate constants, k_FRET, were in the order of 10¹¹ s⁻¹, suggesting the occurrence of ultrafast energy transfer in these dyads. Nanosecond transient absorption studies confirmed relaxation of the energy transfer product, ¹BChl*, to its triplet state, ³Bchl*. The ³Bchl* thus generated was capable of producing singlet oxygen with quantum yields comparable to their monomeric entities. The occurrence of efficient FRET emitting in the far-red region and the ability to produce singlet oxygen make the present series of dyads useful for photonic, imaging and therapy applications.     

Calculation of static and dynamic linear magnetic response in approximate time-dependent density functional theory

We report implementations and results of time-dependent density functional calculations (i) of the frequency-dependent magnetic dipole-magnetic dipole polarizability, (ii) of the (observable) translationally invariant linear magnetic response, and (iii) of a linear intensity differential (LID) which includes the dynamic dipole magnetizability. The density functional calculations utilized density fitting. For achieving gauge-origin independence we have employed time-periodic magnetic-field-dependent basis functions as well as the dipole velocity gauge, and have included explicit density-fit related derivatives of the Coulomb potential. We present the results of calculations of static and dynamic magnetic dipole-magnetic dipole polarizabilities for a set of small molecules, the LID for the SF6 molecule, and dispersion curves for M-hexahelicene of the origin invariant linear magnetic response as well as of three dynamic polarizabilities: magnetic dipole-magnetic dipole, electric dipole-electric dipole, and electric dipole-magnetic dipole. We have also performed comparison of the linear magnetic response and magnetic dipole-magnetic dipole polarizability over a wide range of frequencies for H2O and SF6.     

Ultrafast spectroscopic study of the photochemistry and photophysics of arylhalodiazirines: direct observation of carbene and zwitterion formation

Ultrafast photolysis (lambdaex = 270, 350, or 360 nm) of bromophenyl, chlorophenyl, fluorophenyl, and fluoro-para-trifluoromethylphenyl diazirines produces transient species which absorb broadly in the UV and visible regions. Transient decay can be fit to either mono- or biexponential functions (tau1 approximately 0.3-10 ps, tau2 approximately 10-350 ps; dependent on solvent and halogen). Fluoro- and chlorophenylcarbene are formed within the time resolution of the spectrometer (300 fs, 270 nm excitation). Bromophenyl diazirine decay (270 nm excitation) correlates with the growth of bromophenylcarbene. Solvent and substituent effects on the slower decays of the transient absorptions are consistent with assigning the carriers of transient absorption in the visible region to ring-opened zwitterionic species.     

Calculation of origin-independent optical rotation tensor components in approximate time-dependent density functional theory

We outline an implementation of the origin-independent optical rotation tensor, which includes electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability. The method is based on approximate time-dependent density functional theory. We utilize time-periodic magnetic-field-dependent basis functions as well as a modified velocity-gauge formulation of dynamic polarizability tensors in order to obtain a gauge-origin independence. To ensure gauge-origin independence of the results within a given numerical accuracy, density fit coefficient derivatives are employed. A damping constant has been introduced into the linear response equations to treat both resonance and nonresonance regions of optical activity. We present calculations for trans-2,3-dimethyloxirane and derivatives thereof as well as calculations for androst-4,17-dien-3-one. In the Appendix, we derive the equivalence between the common-gauge origin and gauge-including atomic orbitals formulations for the optical rotation tensor in time-dependent DFT.     

A Note on the Abelian Sandpile in $\pmb{\mathbb{Z}}^{d}$

We analyze the abelian sandpile model on ℤ ^d for the starting configuration of n particles in the origin and 2d−2 particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres (J. Stat. Phys. 198:143–159, 2010) that the radius of the toppled cluster of this configuration is O(n ^1/d ).     

Synthesis of polyhydroxylated bicyclic tetrahydrofurans and tetrahydropyrans via a stereoselective domino cyclization/reduction reaction

A novel reaction cascade involving a Lewis acid-induced migration of an isopropylidene protecting group followed by the formation of a pyranose or furanose ring and subsequent reduction of the hemiacetal is described. Depending on the reaction conditions, as well as, the stereochemistry of the substrate, polyhydroxylated tetrahydrofurans or tetrahydropyrans can be obtained in reasonable yields. The synthons used in this transformation were prepared via a highly stereoselective one-pot tandem reaction, consisting of a 1,4-Michael addition of vinylmagnesium bromide to d-glucose-derived cyclohexenone followed by aldol reaction with 2,3-O-isopropylidene-d-glyceraldehyde.     

Single-Ion Magnetic Behaviour in an Iron(III) Porphyrin Complex: A Dichotomy Between High Spin and 5/2–3/2 Spin Admixture

A mononuclear iron(III) porphyrin compound exhibiting unexpectedly slow magnetic relaxation, which is a characteristic of single-ion magnet behaviour, is reported. This behaviour originates from the close proximity (≈550 cm⁻¹) of the intermediate-spin S=3/2 excited states to the high-spin S=5/2 ground state. More quantitatively, although the ground state is mostly S=5/2, a spin-admixture model evidences a sizable contribution (≈15 %) of S=3/2 to the ground state, which as a consequence experiences large and positive axial anisotropy (D=+19.2 cm⁻¹). Frequency-domain EPR spectroscopy allowed the m_S= |±1/2⟩→|±3/2⟩ transitions to be directly accessed, and thus the very large zero-field splitting in this 3d⁵ system to be unambiguously measured. Other experimental results including magnetisation, Mössbauer, and field-domain EPR studies are consistent with this model, which is also supported by theoretical calculations.