Electric and magnetic dipole shielding constants for the ground state of the relativistic hydrogen‐like atom: Application of the Sturmian expansion of the generalized Dirac‐Coulomb Green function

The Sturmian expansion of the generalized Dirac‐Coulomb Green function (Szmytkowski, J Phys B, 1997, 30, 825; erratum 1997, 30, 2747) is exploited to derive closed‐form expressions for electric $(\sigma_{E})$ and magnetic $(\sigma_{M})$ dipole shielding constants for the ground state of the relativistic hydrogen‐like atom with a point‐like and spinless nucleus of charge Ze. It is found that $\sigma_{E}=Z^{-1}$ (as it should be) and where $\gamma_{1}=\sqrt{1-(Z\alpha)^{2}}$ (α is the fine‐structure constant). This expression for $\sigma_{M}$ agrees with earlier findings of several other authors, obtained with the use of other analytical techniques, and is elementary compared to an alternative one presented recently by Cheng et al. (J Chem Phys 2009, 130, 144102), which involves an infinite series of ratios of the Euler's gamma functions. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Combining Modular Ligation and Supramolecular Self‐Assembly for the Construction of Star‐Shaped Macromolecules

A well‐defined random copolymer of styrene (S) and chloromethylstyrene (CMS) featuring lateral chlorine moieties with an alkyne terminal group is prepared (P(S‐co‐CMS), = 5500 Da, PDI = 1.13). The chloromethyl groups are converted into Hamilton wedge (HW) entities (P(S‐co‐HWS), = 6200 Da, PDI = 1.13). The P(S‐co‐HWS) polymer is subsequently ligated with tetrakis(4‐azidophenyl)methane to give HW‐functional star‐shaped macromolecules (P(S‐co‐HWS))₄, = 25 100 Da, PDI = 1.08). Supramolecular star‐shaped copolymers are then prepared via self‐assembly between the HW‐functionalized four‐arm star‐shaped macromolecules ( P(S‐co‐HW )) ₄ and cyanuric acid (CA) end‐functionalized PS (PS–CA, = 3700 Da, PDI = 1.04), CA end‐functionalized poly(methyl methacrylate) (PMMA–CA, = 8500 Da, PDI = 1.13) and CA end‐functionalized polyethylene glycol (PEG–CA, = 1700 Da, PDI = 1.05). The self‐assembly is monitored by ¹H NMR spectroscopy and light scattering analyses.     

Understanding,Controlling and Programming Cooperativity in Self‐Assembled Polynuclear Complexes in Solution

Deviations from statistical binding, that is cooperativity, in self‐assembled polynuclear complexes partly result from intermetallic interactions ΔE^M,M, whose magnitudes in solution depend on a balance between electrostatic repulsion and solvation energies. These two factors have been reconciled in a simple point‐charge model, which suggests severe and counter‐intuitive deviations from predictions based solely on the Coulomb law when considering the variation of ΔE^M,M with metallic charge and intermetallic separation in linear polynuclear helicates. To demonstrate this intriguing behaviour, the ten microscopic interactions that define the thermodynamic formation constants of some twenty‐nine homometallic and heterometallic polynuclear triple‐stranded helicates obtained from the coordination of the segmental ligands L1 – L11 with Zn²⁺ (a spherical d‐block cation) and Lu³⁺ (a spherical 4f‐block cation), have been extracted by using the site binding model. As predicted, but in contrast with the simplistic coulombic approach, the apparent intramolecular intermetallic interactions in solution are found to be i) more repulsive at long distance ( > ), ii) of larger magnitude when Zn²⁺ replaces Lu³⁺ ( > ) and iii) attractive between two triply charged cations held at some specific distance ( <0). The consequences of these trends are discussed for the design of polynuclear complexes in solution.     

Canonical two‐range addition theorem for slater‐type orbitals

The radial Slater‐type orbitals (STO) ${r^\mu }{e^{ - \alpha r}}$ can be simply obtained by repeated parametric differentiation of the Yukawa Potential $({e^{ - \alpha r}}/r)$ with respect to α. A new compact two‐range addition theorem (AdT) for the STO is herein derived by explicit parametric differentiation of the well‐known Yukawa AdT. The resulting addition formula is combined with the single‐range AdT for solid spherical harmonics $({r^l}Y_l^m(\hat r))$ to present a new AdT for three‐dimensional spherical coordinate STOs. We advance the proposition that this formula is “canonical” in the same sense that the Laplace expansion of the Coulomb potential is considered canonical. We demonstrate how this procedure can be employed for all exponential‐type orbitals. © 2012 Wiley Periodicals, Inc.     

Direct Cyclodextrin‐Mediated Ring Opening Polymerization of ϵ‐Caprolactone in the Presence of Yttrium Trisphenolate Catalyst

Unmodified β‐cyclodextrin has been directly used to initiate ring‐opening polymerization of ϵ‐caprolactone in the presence of yttrium trisphenolate. Well‐defined cyclodextrin (CD)‐centered star‐shaped poly(ϵ‐caprolactone)s have been successfully synthesized containing definite average numbers of arms (N_arm = 4–6) and narrow polydispersity indexes (below 1.10). The number‐average molecular weight ( ) and average molecular weight per arm ( ) are controlled by the feeding molar ratio of monomer to initiator. The prepared star‐PCL with of 2.7 × 10³ is in fully amorphous and that with of 13.3 × 10³ is crystallized. In addition, the obtained poly(e‐caprolactone) (PCL) stars with various molecular weights have different solubilities in methanol and tetrahydrofuran, which can be applied for further modifications.     

Naked and Self‐Clickable Propargylic‐Decorated Single‐Chain Nanoparticle Precursors via Redox‐Initiated RAFT Polymerization

Protection of acetylenic monomers is a common practice to avoid parasitic side reactions during polymerization. Herein, we report that redox‐initiated RAFT polymerization allows the direct, room temperature synthesis of a variety of single‐chain nanoparticle precursors (displaying narrow molecular weight dispersity, / = 1.12 –1.37 up to = 100 kDa) containing well‐defined amounts of naked, unprotected acetylenic functional groups available for rapid and quantitative intrachain cross‐linking via metal‐catalyzed carbon–carbon coupling (i.e., C–C “click” chemistry). To illustrate the useful “self‐clickable” character of the new unprotected acetylenic precursors, single‐chain nanoparticles have been prepared for the first time in a facile and highly efficient manner by copper‐catalyzed alkyne homocoupling (i.e., Glaser–Hay coupling) at room temperature under normal air atmosphere.     

Incorporating glycidyl methacrylate into block copolymers using poly(methacrylate‐ran‐styrene) macroinitiators synthesized by nitroxide‐mediated polymerization

Methyl methacrylate/styrene (MMA/S), ethyl methacrylate/styrene (EMA/S) and butyl methacrylate/styrene (BMA/S) feeds (>90 mol % methacrylate) were copolymerized in 50 wt % p‐xylene at 90 °C with 10 mol % of additional SG1‐free nitroxide mediator relative to unimolecular initiator (BlocBuilder^®) to yield methacrylate rich copolymers with polydispersities _w/ _n = 1.23–1.46. k_pK values (k_p = propagation rate constant, K = equilibrium constant) for MMA/S copolymerizations were comparable with previous literature, whereas EMA/S and BMA/S copolymerizations were characterized by slightly higher k_pK's. Chain extensions with styrene at 110 °C initiated by the methacrylate‐rich macroinitiators (number average molecular weight _n = 12.9–33.5 kg mol^?1) resulted in slightly broader molecular weight distributions with _w/ _n = 1.24–1.86 and were often bimodal. Chain extensions with glycidyl methacrylate/styrene/methacrylate (GMA/S/XMA where XMA = MMA, EMA or BMA) mixtures at 90 °C using the same macroinitiators resulted frequently in bimodal molecular weight distributions with many inactive macroinitiators and higher _w/ _n = 2.01–2.48. P(XMA/S) macroinitiators ( _n = 4.9–8.9 kg mol^?1), polymerized to low conversion and purified to remove “dead” chains, initiated chain extensions with GMA/MMA/S and GMA/EMA/S giving products with _w/ _n ～ 1.5 and much fewer unreacted macroinitiators (<5%), whereas the GMA/BMA/S chain extension was characterized by slightly more unreacted macroinitiators (～20%). © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 2574–2588, 2009     

Towards Clathrates: Frozen States of Hydration of tert‐Butylamine

The dilution of tert‐butylamine (tBA) with water and subsequent cooling leads to a large series of different crystalline hydrates by an in situ IR laser melting‐zone procedure. The crystal structures were determined for tBA?n H₂O, with n=0, , 1, 7 , 7 , 9 , 11, and 17. For the two lower hydrates (n= , 1), one‐ and two‐dimensional hydrogen‐bonded networks are formed, respectively. The higher hydrates (n>1) exhibit a clathrate‐like three‐dimensional water framework with the tBA molecules as part of, or sitting inside, the cages. In all cases, tBA is hydrogen‐bonded to the H₂O framework. In the intermediate range (1相似文献   

Photochemistry of RuII 4,4′‐Bi‐1,2,3‐triazolyl (btz) Complexes: Crystallographic Characterization of the Photoreactive Ligand‐Loss Intermediate trans‐[Ru(bpy)(κ2‐btz)(κ1‐btz)(NCMe)]2+

We report the unprecedented observation and unequivocal crystallographic characterization of the meta‐stable ligand loss intermediate solvento complex trans‐[Ru(bpy)(κ²‐btz)(κ¹‐btz)(NCMe)]²⁺ ( 1 a ) that contains a monodentate chelate ligand. This and analogous complexes can be observed during the photolysis reactions of a family of complexes of the form [Ru($\widehat{NN}$ )(btz)₂]²⁺ ( 1 a – d : btz=1,1′‐dibenzyl‐4,4′‐bi‐1,2,3‐triazolyl; $\widehat{NN}$ =a) 2,2′‐bipyridyl (bpy), b) 4,4′‐dimethyl‐2,2′‐bipyridyl (dmbpy), c) 4,4′‐dimethoxy‐2,2′‐bipyridyl (dmeobpy), d) 1,10‐phenanthroline (phen)). In acetonitrile solutions, 1 a – d eventually convert to the bis‐solvento complexes trans‐[Ru($\widehat{NN}$ )(btz)(NCMe)₂]²⁺ ( 3 a – d ) along with one equivalent of free btz, in a process in which the remaining coordinated bidentate ligands undergo a new rearrangement such that they become coplanar. X‐ray crystal structure of 3 a and 3 d confirmed the co‐planar arrangement of the $\widehat{NN}$ and btz ligands and the trans coordination of two solvent molecules. These conversions proceed via the observed intermediate complexes 2 a – d , which are formed quantitatively from 1 a – d in a matter of minutes and to which they slowly revert back on being left to stand in the dark over several days. The remarkably long lifetime of the intermediate complexes (>12 h at 40 °C) allowed the isolation of 2 a in the solid state, and the complex to be crystallographically characterized. Similarly to the structures adopted by complexes 3 a and d , the bpy and κ²‐btz ligands in 2 a coordinate in a square‐planar fashion with the second monodentate btz ligand coordinated trans to an acetonitrile ligand.     

Theoretical study of synthetic reaction of tetrazole and tetrazolate anion

Tetrazole (H₂CN₄) and tetrazolate anion (HCN$_{4}^{-}$) are high‐energy compounds with a five‐membered ring‐type structures, which can be easily synthesized by HCN and HN₃ and by HCN and N$_{3}^{-}$, respectively, in an irreversible reaction. The ab initio methods including MP2/6‐31G**, B3LYP/6‐31G**, B3LYP/6‐311+G(2d,p), and CBS/QB3 from Gaussian 98 program are employed to study the thermochemistry and reaction mechanism. The transition states of both HCN + HN₃ → H₂CN₄ and HCN + N$_{3}^{-}$ → HCN$_{4}^{-}$ reaction are investigated, and it is found that the latter reaction is more favored than the former one in view of the chemical kinetics and thermodynamics, thus indicating that tetrazole (H₂CN₄) and tetrazolate anion (HCN$_{4}^{-}$) are formed more easily in an alkali environment than in other systems. Pentazole (HN₅) is an unknown high‐energy compound and has not yet been synthesized. For comparison, HN₅ and N$_{5}^{-}$, both which have similar type of synthetic reactions to the above‐mentioned reactions, are studied. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 80: 27–37, 2000     

Unprecedented Electron‐Poor Octahedral Ta6 Clusters in a Solid‐State Compound: Synthesis,Characterisations and Theoretical Investigations of Cs2BaTa6Br15O3

The crystal structure of Cs₂BaTa₆Br₁₅O₃ has been elucidated by using synchrotron X‐ray powder diffraction and absorption experiments. It is built from edge‐bridged octahedral [(Ta₆${{\rm Br}{{{\rm i}\hfill \atop 9\hfill}}}$ ${{\rm O}{{{\rm i}\hfill \atop 3\hfill}}}$ )${{\rm Br}{{{\rm a}\hfill \atop 6\hfill}}}$ ]^4? cluster units with a singular poor metallic electron (ME) count equal to thirteen. This leads to a paramagnetic behaviour related to one unpaired electron. The arrangement of the Ta₆ clusters is similar to that of Cs₂LaTa₆Br₁₅O₃ exhibiting 14‐MEs per [(Ta₆${{\rm Br}{{{\rm i}\hfill \atop 9\hfill}}}$ ${{\rm O}{{{\rm i}\hfill \atop 3\hfill}}}$ )${{\rm Br}{{{\rm a}\hfill \atop 6\hfill}}}$ ]^5? motif. The poorer electron‐count cluster presents longer metal–metal distances as foreseen according to the electronic structure of edge‐bridged hexanuclear cluster. Density functional theory (DFT) calculations on molecular models were used to rationalise the structural properties of 13‐ and 14‐ME clusters. Periodic DFT calculations demonstrate that the electronic structure of these solid‐state compounds is related to those of the discrete octahedral units. Oxygen–barium interactions seem to prevent the geometry of the octahedral cluster to strongly distort, allowing stabilisation of this unprecedented electron‐poor Ta₆ cluster in the solid state.     

Crystal Structures and Emitting Properties of Trifluoromethylaminoquinoline Derivatives: Thermal Single‐Crystal‐to‐Single‐Crystal Transformation of Polymorphic Crystals That Emit Different Colors

2,4‐Trifluoromethylquinoline (TFMAQ) derivatives that have amine ( 1 ), methylamine ( 2 ), phenylamine ( 3 ), and dimethylamine ( 4 ) substituents at the 7‐position of the quinoline ring were prepared and crystallized. Six crystals including the crystal polymorphs of 2 (crystal GB and YG) and 3 (crystal B and G) were obtained and characterized by X‐ray crystallography. In solution, TFMAQ derivatives emitted relatively strong fluorescence (${\lambda {{{\rm f}\hfill \atop {\rm max}\hfill}}}$ =418–469 nm and Φ_f(s)=0.23–0.60) depending on the solvent polarity. From Lippert–Mataga plots, Δμ values in the range of 7.8–14 D were obtained. In the crystalline state, TFMAQ derivatives emitted at longer wavelengths (${\lambda {{{\rm f}\hfill \atop {\rm max}\hfill}}}$ =464–530 nm) with lower intensity (Φ_f(c)=0.01–0.28) than those in n‐hexane solution. The polymorphous crystals of 2 and 3 emitted different colors: 2 , ${\lambda {{{\rm f}\hfill \atop {\rm max}\hfill}}}$ =470 and 530 nm with Φ_f(c)=0.04 and approximately 0.01 for crystal GB and YG, respectively; and 3 , ${\lambda {{{\rm f}\hfill \atop {\rm max}\hfill}}}$ =464 and 506 nm with Φ_f(c)=0.28 and approximately 0.28 for crystal B and G, respectively. In both crystal polymorphs of 2 and 3 , crystals GB and G showed emission color changes by heating/melting/cooling cycles that were representative. By following the color changes in heating at the temperature below the melting point with X‐ray diffraction measurements and X‐ray crystallography, the single‐crystal‐to‐single‐crystal transformations from crystal GB to YG for 2 and from crystal B to G for 3 were revealed.     

Dynamic Mirror‐Symmetry Breaking in Bicontinuous Cubic Phases

Chiral segregation of enantiomers or chiral conformers of achiral molecules during self‐assembly in well‐ordered crystalline superstructures has fascinated chemists since Pasteur. Here we report spontaneous mirror‐symmetry breaking in cubic phases formed by achiral multichain‐terminated diphenyl‐2,2′‐bithiophenes. It was found that stochastic symmetry breaking is a general phenomenon observed in bicontinuous cubic liquid crystal phases of achiral rod‐like compounds. In all compounds studied the ${{\it Im}\bar 3m}$ cubic phase is always chiral, while the ${Ia\bar 3d}$ phase is achiral. These intriguing observations are explained by propagation of homochiral helical twist across the entire networks through helix matching at network junctions. In the ${Ia\bar 3d}$ phase the opposing chiralities of the two networks cancel, but not so in the three‐networks ${{\it Im}\bar 3m}$ phase. The high twist in the ${{\it Im}\bar 3m}$ phase explains its previously unrecognized chirality, as well as the origin of this complex structure and the transitions between the different cubic phases.     

On the 4JHH long‐range coupling in 2‐bromocyclohexanone: conformational insights

2‐Bromocyclohexanone is a model compound in which a ⁴J_{H2, H6} coupling constant is observed, whereas the corresponding ⁴J_{H2, H4} is absent. The observed long‐range coupling is not only a result of the known W‐type coupling, in the axial conformation, but also because of the less usual diaxial spin–spin coupling in the equatorial conformer. The carbonyl group plays a determining role in describing the coupling pathway, as concluded by natural bond orbital (NBO) analysis; although the and interactions in the axial conformer contribute for transmitting the spin information associated with the W‐type coupling, the strong and hyperconjugations in the equatorial conformer define an enhanced coupling pathway for ⁴J_{H2, H6}, despite the inhibition of this coupling because of interaction and the large carbonyl angle. These findings provide the experimental evidence that orbital interactions contribute for the conformational isomerism of 2‐bromocyclohexanone. Copyright © 2008 John Wiley & Sons, Ltd.     

Enhanced Electrogenerated Chemiluminescence of Tris(2,2′‐bipyridyl) Ruthenium(II)/tripropylamine in the Presence of Pyridine and Its Analogues

In the present work, we conducted an investigation on the electrochemical and ECL behavior of Ru(bpy) /TPrA system in the presence of pyridine and its analogues on platinum and gold electrode. Results showed that pyridine and its analogues enhanced Ru(bpy) /TPrA ECL signal and exhibited different enhancement effects on different electrodes. On platinum electrode, the maximum enhancement factor of about 5 was obtained. On gold electrode, a low‐oxidation‐potential (LOP) ECL signal occurred and increased.     

One‐Pot Hydrothermal Synthesis of Heterostructured ZnO/ZnS Nanorod Arrays with High Ethanol‐Sensing Properties

ZnO/ZnS heterostructured nanorod arrays with uniform diameter and length were synthesized from zinc substrates in a one‐pot procedure by using a simple hydrothermal method. Structural characterization by HRTEM indicated that the heterostructured nanorods were composed of parallel segments of wurtzite‐type ZnO and zinc‐blende ZnS, with a distinct interface along the axial direction, which revealed the epitaxial relationship, ZnO (10$\bar 1$ 0) and ZnS ($\bar 1$ 1$\bar 1$ ). The as‐prepared ZnO/ZnS nanorods showed only two green emissions at around 523 nm and 576 nm. We also found that the nanorods exhibited high sensitivity to ethanol at relatively low temperatures, owing to their smaller size and structure.     

(1+1) Resonance‐Enhanced Multiphoton Ionization and Photodissociation Study of CS2 via the 1B2 State

(1+1) resonance‐enhanced multiphoton ionization (REMPI) spectra of CS₂ and molecular dissociation dynamics are investigated using a time‐of‐flight mass spectrometer equipped with velocity imaging detection. The REMPI spectra via a linear‐bent →¹B₂( ) transition are acquired in the wavelength range of 208–217 nm. Each ro‐vibrational band profile of the ¹B₂( ) state is deconvoluted to yield the corresponding predissociative lifetime from 0.3 to 3 ps. Upon excitation at 210.25 and 212.54 nm, the resulting images of S⁺ and CS⁺ fragments are analyzed to give individual translational energy distributions, which are resolved into two components corresponding to the CS+S(³P) and CS+S(¹D) channels. The product branching ratios of S(³P)/S(¹D) are evaluated to be 5.7±1.0 and 9.6±2.5 at 210.25 and 212.54 nm, respectively. Despite the difficulty avoiding the effect of multiphoton absorption, the molecular dissociation channel is verified to prevail over the dissociative ionization channel of CS₂. The anisotropy parameters for the triplet and singlet channels are determined to be ～0.8 and 1.1–1.3, respectively, suggesting that the predissociative state should have a bent configuration with a short lifetime.     

Design of a Solid‐State Electrochemiluminescence Biosensor for Detection of PML/RARα Fusion Gene Using Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$‐AuNPs Aggregations on Gold Electrode

A solid‐state electrochemiluminescence (ECL) biosensor based on special ferrocene‐labeled molecular beacon (Fc‐MB) for highly sensitive detection of promyelocytic leukemia/retinoic acid receptor alpha (PML/RARα) fusion gene was developed successfully using Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$ /2‐(dibutylamino)ethanol (DBAE) as detecting pattern. Such a special sensor involves two main parts, an ECL substrate and an ECL intensity switch. The ECL substrate was made by modifying the complex of Ruthenium (II) tris‐(bipyridine) and Au nanoparticles (Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$ ‐AuNPs) onto the Au electrode (AuE) surface. The molecular beacon probe in which the ferrocene tag could effectively quench the ECL of the Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$ acted as ECL intensity switch. The molecular beacon probe was designed with special base sequence, which could hybridize with its complementary target DNA. In the absence of a target, the hairpin structure of the probe forced the ferrocene (Fc) into close proximity with the ECL substrate, thus reducing ECL intensity. Target binding allowed the Fc away from the ECL substrate and resulted in an obvious increment in ECL intensity due to the decreased Fc quenching effect. The effect of the amount of Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$ and the mixing procedure of Ru(bpy)${{{2+\hfill \atop 3\hfill}}}$ and AuNPs solution on the fabrication of ECL film had been investigated. As a result, the change of ECL intensity had a direct relationship with the logarithm of PML/RARα fusion gene concentration in the range of 0.05–500 pM with a detection limit of 7 fM, and the developed biosensor possessed good molecular recognizability in human serum. Thus, the approach holds promise for the early diagnostics and prognosis monitoring of APL and other diseases.     

Bound‐states of diatomic molecules in the Dirac equation with the q‐deformed Morse potential

We present the solutions of the ro‐vibrational motion of a diatomic molecule with a spatially dependent mass by solving the Dirac equation with position‐dependent mass for repulsive vector $V(r)$ and attractive scalar $S(r)$ q‐deformed Morse potential for any $\kappa$ value, within the framework of Pekeris approximation of the spin‐orbitcoupling term. The relativistic energy spectra are obtained using theNikiforov‐Uvarov method and the two‐component spinor wavefunctions are obtained in terms of the Laguerre polynomials. It is found that there exist only negative energy states for bound states, and the energy values for a fixed value of $n_r$ increase with decrease in $\kappa$ . © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

New Nuclear‐Spin‐Induced Cotton–Mouton Effect in Fluids at High DC Magnetic Field

Based on Buckingham and Pople’s theory of magnetic double refraction, a theoretical expression is derived for a new Cotton–Mouton effect ${\phi _{{\rm{C}} - {\rm{M}}}^{(IB)} }$ in liquid induced by the crossed effect between the high dc magnetic field B₀ and the nuclear magnetic moment ${m_z^{(I)} }$ . It contains temperature‐independent and ‐dependent parts. The latter is proportional to the product between anisotropy of polarizability and the nuclear magnetic shielding tensor. For this new effect ${\phi _{{\rm{C}} - {\rm{M}}}^{(IB)} }$ , its order in magnitude for a molecule with large polarizability anisotropy is estimated to be comparable to the nuclear‐spin‐induced optical Faraday rotation (NSOFR). In the multipass approach, ${\phi _{{\rm{C}} - {\rm{M}}}^{(IB)} }$ can be eliminated by time‐reversal symmetry arguments, but NSOFR is enhanced.