Every N-Dimensional Separable Metric Space Contains a Totally Disconnected (n-1)-Dimensional Subset with no True Quasi-Components

Владимир Тодоров, Петър Стоев - Тази бележка съдържа елементарна конструкция на множество с указаните в заглавието свойства. Да отбележим в допълнение, че така полученото множество остава напълно несвързано дори и след като се допълни с краен брой елементи.

Interacting quantum-dissipative tunnelling systems

Die Untersuchung von dissipativen Quantensystemen erm¨oglicht es, Quantenph¨anomene auch auf makroskopischen L¨angenskalen zu beobachten. Das in dieser Dissertation gew¨ahlte mikroskopische Modell erlaubt es, den bisher nur ph¨anomenologisch zug¨anglichen Effekt der Quantendissipation mathematisch und physikalisch herzuleiten und zu untersuchen. Bei dem betrachteten mikroskopischen Modell handelt es sich um eine 1-dimensionale Kette von harmonischen Freiheitsgraden, die sowohl untereinander als auch an r anharmonische Freiheitsgrade gekoppelt sind. Die F¨alle einer, respektive zwei anharmonischer Bindungen werden in dieser Arbeit explizit betrachtet. Hierf¨ur wird eine analytische Trennung der harmonischen von den anharmonischen Freiheitsgraden auf zwei verschiedenen Wegen durchgef¨uhrt. Das anharmonische Potential wird als symmetrisches Doppelmuldenpotential gew¨ahlt, welches mit Hilfe der Wick Rotation die Berechnung der ¨Uberg¨ange zwischen beiden Minima erlaubt. Das Eliminieren der harmonischen Freiheitsgrade erfolgt mit Hilfe des wohlbekannten Feynman-Vernon Pfadintegral-Formalismus [21]. In dieser Arbeit wird zuerst die Positionsabh¨angigkeit einer anharmonischen Bindung im Tunnelverhalten untersucht. F¨ur den Fall einer fernab von den R¨andern lokalisierten anharmonischen Bindung wird ein Ohmsches dissipatives Tunneln gefunden, was bei der Temperatur T = 0 zu einem Phasen¨ubergang in Abh¨angigkeit einer kritischen Kopplungskonstanten Ccrit f¨uhrt. Dieser Phasen¨ubergang wurde bereits in rein ph¨anomenologisches Modellen mit Ohmscher Dissipation durch das Abbilden des Systems auf das Ising-Modell [26] erkl¨art. Wenn die anharmonische Bindung jedoch an einem der R¨ander der makroskopisch grossen Kette liegt, tritt nach einer vom Abstand der beiden anharmonischen Bindungen abh¨angigen Zeit tD ein ¨Ubergang von Ohmscher zu super- Ohmscher Dissipation auf, welche im Kern KM(τ ) klar sichtbar ist. F¨ur zwei anharmonische Bindungen spielt deren indirekteWechselwirkung eine entscheidende Rolle. Es wird gezeigt, dass der Abstand D beider Bindungen und die Wahl des Anfangs- und Endzustandes die Dissipation bestimmt. Unter der Annahme, dass beide anharmonischen Bindung gleichzeitig tunneln, wird eine Tunnelwahrscheinlichkeit p(t) analog zu [14], jedoch f¨ur zwei anharmonische Bindungen, berechnet. Als Resultat erhalten wir entweder Ohmsche Dissipation f¨ur den Fall, dass beide anharmonischen Bindungen ihre Gesamtl¨ange ¨andern, oder super-Ohmsche Dissipation, wenn beide anharmonischen Bindungen durch das Tunneln ihre Gesamtl¨ange nicht ¨andern.

Low-dimensional hydrogen-bonded structures in the 1:1 and 1:2 proton-transfer compounds of 4,5-dichlorophthalic acid with the aliphatic Lewis bases triethylamine, diethylamine, n-butylamine and piperidine

The structures of proton-transfer compounds of 4,5-dichlorophthalic acid (DCPA) with the aliphatic Lewis bases triethylamine, diethylamine, n-butylamine and piperidine, namely triethylaminium 2-carboxy-4,5-dichlorobenzoate C~6~H~16~N^+^ C~8~H~3~Cl~2~O~4~^-^ (I), diethylaminium 2-carboxy-4,5-dichlorobenzoate C~4~H~12~N^+^ C~8~H~3~Cl~2~O~4~^-^ (II), bis(n-butylaminium) 4,5-dichlorophthalate monohydrate 2(C~4~H~12~N^+^) C~8~H~2~Cl~2~O~4~^2-^ . H~2~O (III) and bis(piperidinium) 4,5-dichlorophthalate monohydrate 2(C~5~H~12~N^+^) C~8~H~2~Cl~2~O~4~^2-^ . H~2~O (IV)have been determined at 200 K. All compounds have hydrogen-bonding associations giving in (I) discrete cation-anion units, linear chains in (II) while (III) and (IV) both have two-dimensional structures. In (I) a discrete cation-anion unit is formed through an asymmetric R2/1(4) N+-H...O,O' hydrogen-bonding association whereas in (II), one-dimensional chains are formed through linear N-H...O associations by both aminium H donors. In compounds (III) and (IV) the primary N-H...O linked cation-anion units are extended into a two-dimensional sheet structure via amide N-H...O(carboxyl) and ...O(carbonyl) interactions. In the 1:1 salts [(I) and (II)], the hydrogen 4,5-dichlorophthalate anions are essentially planar with short intramolecular carboxylic acid O-H...O(carboxyl) hydrogen bonds [O...O, 2.4223(14) and 2.388(2)A respectively]. This work provides a further example of the uncommon zero-dimensional hydrogen-bonded DCPA-Lewis base salt and the one-dimensional chain structure type, while even with the hydrate structures of the 1:2 salts with the primary and secondary amines, the low dimensionality generally associated with 1:1 DCPA salts is also found.

catena-[μ-tris(1,2-bis(tetrazol-1-yl)ethane-N4,N4')iron(II)] bis(tetrafluoroborate): Synthesis, structure, spectroscopic and magnetic characterisation of a chain-type coordination polymer spin-crossover compound.

In analogy to a common synthesis of 1-substituted 5-H tetrazoles (Tetrahedron Lett. 36 (1995)1759; Beloruss. Gos. Univ., Minsk, USSR. Khim. Geterotsikl. Soedin. 11 (1985) 1521; Beloruss. Gos. Univ., Minsk, USSR. Khim. Geterotsikl. Soedin. 1 (1991) 66; BGU, Belarus. Vestsi Akad. Navuk Belarusi, Ser. Khim. Navuk 1 (1992) 73), the new bidentate ligand 1,2-bis(tetrazol-1-yl)ethane [endi] was synthesized and characterized by X-ray diffraction, NMR, IR and UV–Vis spectroscopy. By using iron(II) tetrafluoroborate hexahydrate the complexation with this ligand yields a 1-dimensional linear coordination polymer similar to the recently published chain compound (Inorg. Chem. 39 (2000) 1891) exhibiting a thermally induced spin-crossover phenomenon. Similar to the 1,2-bis(tetrazol-1-yl)propane-bridged compound, our 1,2-bis(tetrazol-1-yl)ethane-bridged compound shows a gradual spin transition, but the spin-crossover temperature T1/2≈140 K is found to be 10 K above the other T1/2. The T1/2 was determined by temperature-dependent 57Fe-Mössbauer, far FT-IR and UV–Vis spectroscopy as well as by temperature-dependent magnetic susceptibility measurements. Single crystals of the complex were grown in situ from a solution of the ligand and iron(II) tetrafluoroborate. The X-ray structure determinations of both the high spin as well as the low spin state of the compound revealed a solid state structure, which is comparable to that of catena-[Fe(1,2-bis(tetrazole-1-yl)propane)3](ClO4)2 (Inorg. Chem. 39 (2000) 1891; 2nd TMR-TOSS Meeting, 4th Spin Crossover Family Meeting, Lufthansa Training Center, Seeheim/Germany, April 30–May 2, 1999). Both the 1,2-bis(tetrazol-1-yl)propane-bridged and our compound do not show a thermal hysteresis effect (J. Am. Chem. Soc. 115 (1993) 9810; Inorg. Chim. Acta 37 (1979) 169; Chem. Phys. Lett. 93 (1982) 567). The synthesis of the complex described in the experimental section yielded a fine powdered product being poorly soluble in most common solvents. The single crystal measurements were done with crystals obtained by various diffusion methods. Most of them yielded either thin needles or small hexagonal prism crystals depending on the specific conditions.

Proton-transfer versus nontransfer in compounds of the diazo-dye precursor 4-(phenyldiazenyl) aniline (aniline yellow) with strong organic acids: the 5-sulfosalicylate and the dichroic benzenesulfonate salts, and the 1:2 adduct with 3,5-dinitrobenzoic

The structures of two 1:1 proton-transfer red-black dye compounds formed by reaction of aniline yellow [4-(phenyldiazenyl)aniline] with 5-sulfosalicylic acid and benzenesulfonic acid, and a 1:2 nontransfer adduct compound with 3,5-dinitrobenzoic acid have been determined at either 130 or 200 K. The compounds are 2-(4-aminophenyl)-1-phenylhydrazin-1-ium 3-carboxy-4-hydroxybenzenesulfonate methanol solvate, C12H12N3+.C7H5O6S-.CH3OH (I), 2-(4-aminophenyl)-1-hydrazin-1-ium 4-(phenydiazinyl)anilinium bis(benzenesulfonate), 2C12H12N3+.2C6H5O3S-, (II) and 4-(phenyldiazenyl)aniline-3,5-dinitrobenzoic acid (1/2) C12H11N3.2C~7~H~4~N~2~O~6~, (III). In compound (I) the diaxenyl rather than the aniline group of aniline yellow is protonated and this group subsequently akes part in a primary hydrogen-bonding interaction with a sulfonate O-atom acceptor, producing overall a three-dimensional framework structure. A feature of the hydrogen bonding in (I) is a peripheral edge-on cation-anion association involving aromatic C--H...O hydrogen bonds, giving a conjoint R1/2(6)R1/2(7)R2/1(4)motif. In the dichroic crystals of (II), one of the two aniline yellow species in the asymmetric unit is diazenyl-group protonated while in the other the aniline group is protonated. Both of these groups form hydrogen bonds with sulfonate O-atom acceptors and thee, together with other associations give a one-dimensional chain structure. In compound (III), rather than proton-transfer, there is a preferential formation of a classic R2/2(8) cyclic head-to-head hydrogen-bonded carboxylic acid homodimer between the two 3,5-dinitrobenzoic acid molecules, which in association with the aniline yellow molecule that is disordered across a crystallographic inversion centre, result in an overall two-dimensional ribbon structure. This work has shown the correlation between structure and observed colour in crystalline aniline yellow compounds, illustrated graphically in the dichroic benzenesulfonate compound.

Molecular cocrystals of 5-(4-bromophenyl)-1,3,4-thiadiazol-2-amine: hydrogen bonding in the structures of the 1:1 adduct with 2-(naphthalen-2-yloxy)acetic acid and the salt with 3,5-dinitrobenzoic acid

The structures of the anhydrous products from the interaction of 2-amino-5-(4-bromophenyl)-1,3,4-thiadiazole with (2-naphthoxy)acetic acid, the 1:1 adduct C8H6BrN3S . C12H10O3 (I) and 3,5-dinitrobenzoic acid, the salt C8H7BrN3S+ C7H3N2O6- (II) have been determined. In the adduct (I), a heterodimer is formed through a cyclic hydrogen-bonding motif [graph set R2/2(8)], involving carboxylic acid O-H...N(hetero)and amine N-H...O(carboxyl) interactions. The heterodimers are essentially planar with a thiadiazole to naphthyl ring dihedral angle of 15.9(2)deg. and the intramolecular thiadiazole to phenyl ring angle of 4.7(2)deg. An amine N-H...N(hetero) hydrogen bond between the heterodimers generates a one-dimensional chain structure extending down [001]. Also present are weak benzene-benzene and naphthalene-naphthalene pi-pi stacking interactions down the b axis [minimum ring centroid separation, 3.936(3) Ang.]. With the salt (II), the cation-anion association is also through a cyclic R2/2(8) motif but involving duplex N-H...O(carboxyl) hydrogen bonds, giving a heterodimer which is close to planar [dihedral angles between the thiadiazole ring and the two benzene rings, 5.00(16)deg. (intra) and 7.23(15)deg. (inter)]. A secondary centrosymmetric cyclic N-H...O(carboxyl) hydrogen-bonding association involving the second amino H-atom generates a heterotetramer. Also present in the crystal are weak pi-pi i-\p interactions between thiadiazolium rings [minimum ring centroid separation, 3.936(3)Ang.], as well as a short Br...O(nitro) interaction [3.314(4)Ang.]. The two structures reported here now provide a total of three crystallographically characterized examples of co-crystalline products from the interaction of 2-amino-5-(4-bromophenyl)-1,3,4-thiadiazole with carboxylic acids, of which only one involves proton-transfer.

Crystal structures of the co-crystalline adduct 5-(4-bromophenyl)-1,3,4-thiadiazol-2-amine--4-nitrobenzoic acid (1/1) and the salt 2-amino-5-(4-bromophenyl)-1,3,4-thiadiazol-3-ium 2-carboxy-4,6-dinitrophenolate

The structures of the 1:1 co-crystalline adduct C8H6BrN3S . C7H5NO4 (I) and the salt C8H7BrN3S+ C7H3N2O7- (II) from the interaction of 5-(4-bromophenyl)-1,3,4-thiadiazol-2-amine with 4-nitrobenzoic acid and 3,5-dinitrosalicylic acid, respectively, have been determined. The primary inter-species association in both (I) and (II) is through duplex R2/2(8) (N-H...O/O-H...O) or (N-H...O/N-H...O) hydrogen bonds, respectively, giving heterodimers. In (II), these are close to planar [dihedral angles between the thiadiazole ring and the two phenyl rings are 2.1(3)deg. (intra) and 9.8(2)deg. (inter)], while in (I) these angles are 22.11(15) and 26.08(18)deg., respectively. In the crystal of (I), the heterodimers are extended into a one-dimensional chain along b through an amine N-...N(thiadiazole) hydrogen bond but in (II), a centrosymmetric cyclic heterotetramer structure is generated through N-H...O hydrogen bonds to phenol and nitro O-atom acceptors and features, together with the primary R2/2(8) interaction, conjoined R4/6(12), R2/1(6) and S(6) ring motifs. Also present in (I) are pi--pi interactions between thiadiazole rings [minimum ring centroid separation, 3.4624(16)deg.] as well as short Br...O(nitro) interactions in both (I) and (II) [3.296(3)A and 3.104(3)A, respectively].

One-dimensional fermions with incommensuration close to dimerization

We study a model of fermions hopping on a chain with a weak incommensuration close to dimerization; both q, the deviation of the wave number from pi, and delta, the strength of the incommensuration, are assumed to be small. For free fermions, we show that there are an infinite number of energy bands which meet at zero energy as q approaches zero. The number of states lying inside the q = 0 gap remains nonzero as q/delta --> 0. Thus the limit q --> 0 differs from q = 0, as can be seen clearly in the low-temperature specific heat. For interacting fermions or the XXZ spin-(1/2) chain, we use bosonization to argue that similar results hold. Finally, our results can be applied to the Azbel-Hofstadter problem of particles hopping on a two-dimensional lattice in the presence of a magnetic field.

Spin state and exchange in the quasi-one-dimensional antiferromagnet KFeS2

We report the optical spectra and single crystal magnetic susceptibility of the one-dimensional antiferromagnet KFeS2. Measurements have been carried out to ascertain the spin state of Fe3+ and the nature of the magnetic interactions in this compound. The optical spectra and magnetic susceptibility could be consistently interpreted using a S = 1/2 spin ground state for the Fe3+ ion. The features in the optical spectra have been assigned to transitions within the d-electron manifold of the Fe3+ ion, and analysed in the strong field limit of the ligand field theory. The high temperature isotropic magnetic susceptibility is typical of a low-dimensional system and exhibits a broad maximum at similar to 565 K. The susceptibility shows a well defined transition to a three dimensionally ordered antiferromagnetic state at T-N = 250 K. The intra and interchain exchange constants, J and J', have been evaluated from the experimental susceptibilities using the relationship between these quantities, and chi(max), T-max, and T-N for a spin 1/2 one-dimensional chain. The values are J = -440.71 K, and J' = 53.94 K. Using these values of J and J', the susceptibility of a spin 1/2 Heisenberg chain was calculated. A non-interacting spin wave model was used below T-N. The susceptibility in the paramagnetic region was calculated from the theoretical curves for an infinite S = 1/2 chain. The calculated susceptibility compares well with the experimental data of KFeS2. Further support for a one-dimensional spin 1/2 model comes from the fact that the calculated perpendicular susceptibility at 0K (2.75 x 10(-4) emu/mol) evaluated considering the zero point reduction in magnetization from spin wave theory is close to the projected value (2.7 x 10(-4) emu/mol) obtained from the experimental data.

Fidelity susceptibility of one-dimensional models with twisted boundary conditions

Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used todetect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility chi(F) can detect a QCP provided that the correlation length exponent satisfies nu < 2. We then show that chi(F) can be used to locate a QCP even if nu >= 2 if we introduce boundary conditions labeled by a twist angle N theta, where N is the system size. If the QCP lies at g = 0, we find that if N is kept constant, chi(F) has a scaling form given by chi(F) similar to theta(-2/nu) f (g/theta(1/nu)) if theta << 2 pi/N. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which nu = q, and in a XY spin-1/2 chain in which nu = 2. Finally we show that when q is very large, the model has two additional QCPs at h = +/- 2 which cannot be detected by studying the energy spectrum but are clearly detected by chi(F). The peak value and width of chi(F) seem to scale as nontrivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy. DOI: 10.1103/PhysRevB.86.245424

An unusual organic-inorganic chain-like hybrid complex [(CuCl)(2)(o-phen)](infinity) (o-phen=o-phenanthroline)

A novel organic-inorganic hybrid complex [(CuCl)(2) (o-phen)](infinity) 1 (o-phen = o-phenanthroline) has been hydrothermally synthesized and structurally characterized by elemental analyses, XPS spectrum, TG analysis, and single-crystal X-ray diffraction. Compound I crystallizes in the monoclinic system, space group P2(1)/n, a = 3.7285(7) Angstrom, b = 19.603(4) Angstrom, c = 16.757(3) Angstrom, beta = 95.83(3)degrees, V = 1218.4(4) Angstrom(3), Z = 4, lambda(MoKalpha) = 0.71073 Angstrom (R(F) = 0.0643 for 2559 reflections). Data were collected on an R-axis RAPID diffractometer at 293 K in the range of 1.60 < θ < 27.48degrees. The title compound exhibits a one-dimensional chain-like scaffolding constructed by the unusual [Cu3Cl3] hexagon motifs by, sharing opposite edges. Only Cu(1) sites of the [Cu3Cl3] hexagon are coordinated with N donors of o-phen groups. Furthermore, the three-dimensional supermolecular architecture is formed by C-H...Cl hydrogen bonds between o-phen groups and CuCl chains.

A novel one-dimensional manganese phosphomolybdate with rectangular cavities: [H3N(CH2)(4)NH3] (H3O)(2){[Mn(phen)(2)](4) [(MnMo12O30)-O-v(HPO4)(6)(H2PO4)2} center dot 4H(2)O

A novel manganese phosphomolybdate, [H3N(CH2)(4)NH3](H3O)(2){[Mn(phen)(2)](4)[(MnMovO30)-O-12(HPO4)(6)(H2PO4)(2)]} . 4H(2)O 1, has been hydrothermally synthesized and structurally characterized by single crystal X-ray diffraction. The crystal data: triclinic, P (1) over bar, a = 14.172(7) Angstrom, b = 16.547(2) Angstrom, c = 16.679(3) Angstrom, alpha = 62.881(12)degrees, beta = 73.83(3)degrees, gamma = 88.81(3)degrees. X-ray crystallography shows that the [Mn(phen)(2)] fragments are covalently bonded to the [Mn(Mo6P4)(2)] dimers leading to a one-dimensional chain with rectangular cavities occupied by tetramethylene-diamine cations and water molecules. (C) 2002 Elsevier Science B.V. All rights reserved.

Interactions of thiamine monophosphate (TMP) with one- and two-dimensional polymeric halogenomercurate anions. Crystal structures of (TMP)(Hg2Br5)center dot 0.5H(2)O and (TMP)(2)(Hg3I8)

An investigation into the interactions between thiamine monophosphate (TMP) and anions has resulted in the preparation and X-ray characterization of the compounds (TMP)(Hg2Br5).0.5H(2)O (1) and (TMP)(2)(Hg3I8) (2). In each compound the TMP molecule exists as a monovalent cation in the usual F conformation. The halogenomercurate anions occur in two-dimensional (2-D) network in 1 or one-dimensional (1-D) chain in 2. In both 1 and 2, the structures consist of alternating cationic sheets of the hydrogen-bonded TMP molecules and anionic sheets of the polymeric halogenomercurate anions. The TMP molecule binds to the polymeric anions through the characteristic 'anion bridge I', C(2)-H..X...pyrimidinium (X = Br in 1 and 1 in 2), and electrostatic interactions between electropositive S(1) and halogen atoms. The 'anion bridge II' of the type N(4'1)-H...X...thiazolium (X = phosphate group) plays a role in stabilizing the molecular conformation. The biological implication of the host-guest-like complexation between TMP and polymeric anions is discussed.

Zinc(II) mediated synthesis of an N-substituted 2-(2-pyridyl)imidazole from a 1,2-diketone and 2-(aminomethyl)pyridine and its ligational behaviour

Reaction of anhydrous ZnCl2 with the 1:2 condensate (L) of benzil and 2-(aminomethyl)pyridine in methanol gives monomeric ZnL'Cl-2 (1) where L' is 2-[(4,5-diphenyl-2-pyridin-2-yl-1H-imidazol-1-yl)-methyl]pyridine. In the X-ray crystal structure, 1 is found to contain tetrahedral zinc with an N2Cl2 coordination sphere and the N-substituent methylpyridine fragment hanging as a free arm. A tentative mechanism is proposed for the zinc mediated conversion of L-->L'. Demetallation of 1 by the action of aqueous NaOH yields L' in the free state. When L' is reacted with Zn(ClO4)(2).6H(2)O in a 1:2 molar proportion, [Zn(L')(2)](n)(ClO4)(2n).(H2O)(n/2).(CH2Cl2)(n/2) (2) is obtained. The zinc atom in 2, as revealed by X-ray crystallography, has a trigonal bipyramidal N-5 coordination sphere. There are two independent ligands in the asymmetric unit of 2. One of them bonds only to one zinc atom in a bidentate mode with the N-substituent methylpyridine hanging free while the other ligand binds to two different zinc atoms in a tridentate fashion, employing the N-substituent methylpyridine nitrogen atom to form the polymeric one-dimensional chain cation.

A novel cation induced polymeric chain in Na-8[{Cu(gly)(2)}(2)]{H-2(H2W12O42)}] center dot 24H(2)O: hydrothermal synthesis, spectroscopic characterization and X-ray structure analysis

A novel bis(glycinato) copper(II) paradodecatungstate Na-8[{Cu(gly)(2)}(2)]-{H-2(H2W12O42)}] center dot 24H(2)O (1) has been synthesized under hydrothermal conditions. The crystal structure of 1 reveals an infinite one-dimensional chain along the [100] direction and is built from paradodecatungstate (H2W12O42)(10-) clusters joined through [Cu(gly)(2)] moieties. Parallel chains are interlinked by NaO6 octahedra to generate a two-dimensional network.