Dihydropyran Ring Conformations. I. Structures of 2-Methoxy- and 2-Hydroxy-2, 4-dimethyl-3, 4-dihydro-2\u3ci\u3eH\u3c/i\u3e, 5\u3ci\u3eH\u3c/i\u3e-Pyrano[3,2-\u3ci\u3ec\u3c/i\u3e][1]benzopyran-5-ones

Conformations of embedded 3,4-dihydro-2H-pyrans (DHP\u27s) are studied in a closely related series of nine molecular structures, including the cis and trans isomers of both racemic and resolved homologs of 2,4-dimethyl-3,4-dihydro-2H,5H-pyrano[3,2-c][l]-benzopyran-5-one. DHP rings in these structures display a range of conformations near the half-chair form but distorted variously toward each of the two inequivalent d,e- and e,f-diplanar forms. Factor analysis reveals the coordination of torsional motions associated with a principal ring-bending mode based on the various ring conformations. Two methyl ketals and an hemiketal structure are reported. (I): (2S,4S)-cis-2-methoxy, C15H16O4, Mr = 260·30, trigonal, P3221, a = 9·579 (5), c = 24·938 (15) Å (hexagonal axes), V = 1981·7 (33) Å, Z = 6, Dx = 1·309 g cm-3, λ(Mo Kᾱ)= 0·71073 Å, μ = 0·884 cm- 1, F(000) = 828, T = 295 K, R = 0·040 for 1470 observations. (II): (2R,4S)-trans-2-methoxy, C15H16O4, Mr = 260·30, monoclinic, P21, a = 9·254 (4), b = 11·970 (5), c = 12·736 (5) Å, β = 105·64 (4)°, V = 1358·5 (22) Å3, Z = 4 (two molecules per asymmetric unit), Dx = 1·274 g cm-3, λ(Mo Kᾱ) = 0·71073 Å, μ = 0·860 cm-1, F(000) = 552, T = 295 K, R = 0·043 for 2923 observations. (III): (2S,4S)-cis-2-hydroxy, C14H14O4, Mr = 246·27, orthorhombic, P212121, a = 6·816 (3), b = 12·826 (7), c = 13·949 (6) Å, V = 1219·4 (17) Å3 , Z = 4, Dx = 1·340 g cm-3, λ(Mo Kᾱ) = 0·71073 Å, μ= 0·919 cm-1, F(000) = 520, T = 293 K, R = 0·046 for 1291 observations

Structure of (Phenyl)bis( 4-hydroxybenzo-2\u3ci\u3eH\u3c/i\u3e-pyran-2-one-3-yl)methane*

A derivative of dicoumarol, C25H16O6, Mr, = 412·41, orthorhombic, P212121, a = 7·959(2), b = 12·865 (3), c = 18·606 (6) Å, V = 1905·3 (22) Å3, Z = 4, Dx = 1.44 g cm-3, λ(Mo Kα) = 0·71073 Å , μ = 0·965 cm-1, F(000) = 856, T = 293 K, final R = 0·042 for 2031 observations. The 4-hydroxycoumarins are intramolecularly hydrogen bonded between hydroxyls and carbonyls, O···O separations are 2·624 (3) and 2·718 (3)Å, a scheme which imparts a dis-symmetry to the otherwise achiral molecule and underlies packing in a polar space group

α-L-Aspartylglycine Monohydrate

C6H10N2O5, H2O, orthorhombic, P212121, a = 4·844 (5), b = 9·916 (3), c = 18·070(4) Å, V = 868·05 Å3, Z = 4, Dc = 1·59, Dm (flotation in chloroform/iodoform) = l·60 (1) Mg m-3; R1 = 0·040, R2 = 0·033 for 1088 observations. The dipeptide crystallizes as a zwitterion with the main-chain carboxyl ionized and the side-chain amino group protonated. The overall dipeptide conformation is highly extended and the molecule is extensively hydrogen bonded

α-L-Glutamylglycine

C7H12N 2O5, orthorhombic, P212121, a = 5·525(5), b = 12·565(4), c = 13·211(6) Å, Z = 4, Dc = l·48, Dm (flotation in chloroform/ methylene chloride) = 1·48(1) Mg m-3, R1 = 0·039, R2 = 0·040 for 1172 observations. The dipeptide crystallizes as a zwitterion with the main-chain carboxyl ionized and the amino terminus protonated. The conformation of the peptide group is trans; the glutamyl side chain is extended, but the carboxy terminus is held by hydrogen bonding in a non-extended conformation with a torsional angle ΦGly = -74.1°

Structures of Five \u3cem\u3etrans\u3c/em\u3e-2-Hydroxy and \u3cem\u3etrans\u3c/em\u3e-2-Methoxy-2-methyl-3,4-dihydro-4-aryl- 2\u3cem\u3eH\u3c/em\u3e,5\u3cem\u3eH\u3c/em\u3e-pyrano[3,2-\u3cem\u3ec\u3c/em\u3e]benzopyran-5-ones

Derivatives of 2-methyl-3,4-dihydro-2H,5H-pyrano[3,2-c][1]benzopyran-5-one. (1) Racemic trans-methoxy-4-phenyl, Mr = 322·4, monoclinic, P21/n, a = 5·858 (1), b = 16· 732 (9), c = 16·383 (9) Å, β = 94·82 (3)°, V = 1600·1 Å3, Z = 4, Dx = 1·338 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·866 cm-1, F(000) = 680, T = 298 K, final R = 0·047 for 1513 intensities. The trans methyl ketal of warfarin (C20H18O4) contains a half-chair dihydropyran ring distorted toward the d,e-diplanar conformation. (2) Resolved (2R,4R)-trans-2-methoxy-4-phenyl, Mr = 322·4, orthorhombic, P212121, a = 11·521(1), b = 14·061 (2), c = 10·055(2) Å, V = 1628·9 Å3 , Z = 4, Dx= 1·314 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·851 cm-1, F(000) = 680, T = 298 K, final R = 0·056 for 1660 intensities. The trans methyl ketal (C20H18O4) from R(+)-warfarin; its dihydropyran ring is a half- chair distorted toward the d,e-diplanar conformation. (3) Resolved trans-2-hydroxy-4-(4-methoxyphenyl), Mr = 338·4, orthorhombic, P212121, a = 10·584 (1), b = 10·621(5), c = 14·778 (2) Å, V = 1661·2 Å3, Z = 4, Dx = 1·353 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·908 cm-1, F(000) = 712, T = 298 K, final R = 0·044 for 1165 unique intensities. Compound C20H18O5 spontaneously resolves on crystallization from acetone and water, and absolute configuration of data specimen was not determined. It has a dihydropyran ring with a half-chair conformation; hydroxyl and lactone carbonyl are intermolecularly H-bonded at O···O = 2·79 Å. (4) Racemic trans-2-hydroxy-4-(2-methyl-6-methoxyphenyl), Mr = 372·4, monoclinic, P21/n, a = 9·637 (2), b = 14·345 (4), c = 13·224 (2) Å, β = 91·09 (1)°, V = 1827·8 Å3, Z = 4, Dx = 1·353 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·961 cm-1, F(000) = 784, T = 298 K, final R= 0·048 for 2339 unique intensities. A synthetic derivative of warfarin, it crystallizes as the hydrate C20H18O6.H2O and has a dihydropyran ring with a half-chair conformation and intermolecular H bonds with O···O distances between 2·67 and 2·77 Å. (5) Racemic trans-2-hydroxy-4-(4-dimethylaminophenyl), Mr = 369·4, triclinic, PĪ, a = 9·066 (3), b = 9·509 (2)°, c = 12·681 (3) Å, α = 98·50 (2), β = 91·25 (3), γ = 116·96 (2)°, V = 958·8 Å3, Z = 2, Dx = 1·279 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·854 cm-1, F(000) = 392, T = 298 K, final R = 0·045 on 2773 unique intensities. A synthetic derivative of warfarin, it also crystallizes as a hydrate, C21H21NO4.H2O, and has a dihydropyran ring with a half-chair conformation and intermolecular H bonds with O···O distances between 2·70 and 3·07 Å

Structures of Four \u3ci\u3etrans\u3c/i\u3e-2-Hydroxy- and Methoxy-2-methyl-3,4-dihydro-4-alkyl-2\u3ci\u3eH\u3c/i\u3e,5\u3ci\u3eH\u3c/i\u3e-pyrano[3,2-c][1]benzopyran-5-ones

Derivatives of 2-methyl-3,4-dihydro-2H,5H-pyrano[ 3,2-c][1]benzopyran-5-one. ( l) Racemic trans- 2-methoxy-4-(2-propyl), Mr= 288·3, monoclinic, P2i/c, a = 13·737 (3), b = 13·228 (6), c = 17·229(4) Å, β = l02·93(2)° , V = 3051·4 Å3, Z = 8 (two molecules/asymmetric unit), Dx = 1·255 g cm- 3, λ(Mo Kα) = O·71073 Å, μ = 0·908 cm-1 , F(OOO) = 1232, T = 298 K, final R = 0·050 for 3988 unique intensities. Dihydropyran rings in C17H2004 are halfchairs, one being distorted toward the d,e-diplanar form. (2) Resolved trans-2-methoxy-4-cyclohexyl, Mr = 328-4, orthorhombic, P212121, a = 10·468 (5), b = 11·245 (5). c = 14·465 (4) Å, V= 1702·7 Å3, Z=4, Dx= 1·281 g cm-3, λ(Mo Kα) = 0·71073 Å, μ = 0·823 cm- 1 , F(OOO) = 704, T = 298 K, final R = 0·051 for 2481 unique intensities. Compound, C20H2404, spontaneously resolves on crystallization from methanol: acetone; data specimen determined to be 2R,4R by circular dichroism spectrum and comparison with structures of known configuration. Dihydropyran ring has a d,e-diplanar conformation. (3) Racemic trans- 2-hydroxy-4-(2-propyl). Compound crystallizes as the hemihydrate C16H1804. ½H20; Mr= 283-3, triclinic, P1, a = 9·015 (4), b = 10·216 (4), c = 16·208 (5) Å, α = 103·08 (3), β = 95·42 (3), γ = 95·28 (3)° , V = 1437·6 Å3, Z = 4 (two molecules/asymmetric unit), Dx = 1·309 g cm-3 , λ(Mo Kα) = O·71073 Å, μ = 0·890 cm-1, F(OOO) = 604, T = 298 K, final R = 0·040 for 4656 unique reflections. One dihydropyran ring is a half-chair, the other has an ef-diplanar conformation. Intermolecular hydrogen bonding occurs between the water and the hydroxyls and lactone carbonyls of each coumarin with O···O distances between 2·82 and 2·90 Å. (4) Racemic trans-2-hydroxyl-4-(2-propyl) derivative also cocrystallizes with 4-hydroxy-2H-benzopyran- 2-one (1:1), Mr = 436·4, triclinic, P1, a= 8·669 (2), b = 10·506 (4), c = 12.559 (2) Å, α = 102·98 (2), β = 107·56 (2), γ = 93·63 (2)°, V = 1052·0 Å3, Z=2, Dx= 1·378g cm-3, γ(Mo Kα) = 0·71073 Å, μ = 0·941 cm-1, F(OOO) = 460, T = 298 K, final R = 0·041 for 3322 unique reflections. Cocrystalline C16H1804C9H603 shows chains of H bonds linking the hydroxyls of the coumarins alternately with the lactone carbonyls, O···O distances 2·68 and 2·75 Å. The dihydropyran ring has a half-chair conformation

A Partition of Unity Method for Divergence-Free or Curl-Free Radial Basis Function Approximation

Divergence-free (div-free) and curl-free vector fields are pervasive in many areas of science and engineering, from fluid dynamics to electromagnetism. A common problem that arises in applications is that of constructing smooth approximants to these vector fields and/or their potentials based only on discrete samples. Additionally, it is often necessary that the vector approximants preserve the div-free or curl-free properties of the field to maintain certain physical constraints. Div/curl-free radial basis functions (RBFs) are a particularly good choice for this application as they are meshfree and analytically satisfy the div-free or curl-free property. However, this method can be computationally expensive due to its global nature. In this paper, we develop a technique for bypassing this issue that combines div/curl-free RBFs in a partition of unity framework, where one solves for local approximants over subsets of the global samples and then blends them together to form a div-free or curl-free global approximant. The method is applicable to div/curl-free vector fields in ℝ2 and tangential fields on two-dimensional surfaces, such as the sphere, and the curl-free method can be generalized to vector fields in ℝd. The method also produces an approximant for the scalar potential of the underlying sampled field. We present error estimates and demonstrate the effectiveness of the method on several test problems

Structural Variations in 3,4-Dihydro-2\u3ci\u3eH\u3c/i\u3e-pyran Ketals: Acyl and Aryl Warfarin Derivatives

The crystal structures of ( ± )-cis-2-methyl-5-oxo-4- phenyl-3,4-dihydro-2H,5H-pyrano[3,2-c][1]benzopyran-2-yl acetate [C21H18O5, Mr = 350·37, monoclinic, P21/n, a = 12·091 (4), b = 8·288 (3), c = 17·840 (5) Å, β = 106·34 (2)°, V = l715(2) Å3, Z = 4, Dx = 1·356 g cm-3, λ(Mo Kᾱ) = 0·7107 Å, μ = 0·904 cm-1, F(000) = 736, T = 295 K, R = 0·050 for 2767 observations with I ≥ 3σ(I)] and (6R,12S)-( — )-6,8-dimethyl-6, 12-methano-6H,l2H,l3H-[1] benzopyran[4,3-d][1,3]benzodioxocin-13-one [C20H16O4, Mr = 320·36, tetragonal, P43, a = 10·788 (4), c = 13·587 (9) Å, V = 1581 (2) Å3 , Z = 4, Dx = 1·345 g cm-3, λ(Mo Kᾱ) = 0·7107 Å, μ = 0·873 cm-1, F(000) = 672, T = 295 K, R = 0·049 for 1425 observations with I ≥ 2·5σ(I)] are described. They are acyl and aryl ketals of warfarin, respectively, and contain an embedded dihydropyran ring. The molecules were studied as part of a series of axial 2-O-substituted-2-methyl-3,4-dihydro-2H-pyran structures which show (hemi)ketal C—O bond-length variations indentified through factor analysis with the systematic geometrical changes associated with a spontaneous elimination (El-like) reaction from the ketal leading to 2-methyl-4H-pyran. As in α-tetrahydropyranyl acetals the C—0 lengths in dihydropyranyl ketals can be expressed as a function of the electron-withdrawing ability of the substituent conjugate base, and the slopes of the relat1onsh1ps for the two systems are similar. Corresponding endocyclic C—0 lengths are about 0·052 Å longer in these model dihydropyranyl ketals