Matrix Least-squares Research Articles

The Crystal Structure of Dimethylsulfoxidepentammineruthenium(II) Hexafluorophosphate

The crystal structure of dimethylsulfoxidepentammineruthenium(II) hexafluorophosphate ( [Ru(NH3)5DMSO]2+ 2PF6−) has been determined from three-dimensional X-ray data. The crystals are monoclinic, space group P21/c, with four formula units in a unit cell of dimensions a = 8.805(2), b = 17.335(5), c = 12.373(2) Å, β = 94.24(2)°. The structure has been refined by full matrix least-squares methods to a final residual R of 0.054 for 1987 reflections greater than 3σ. The DMSO ligand coordinates through the sulfur atom and the S—O bond distance (1.527(7) Å) is considerably longer than that found for other sulfur–donor complexes of the ligand. The Ru—S bond distance is 2.188(3) Å.

Crystal and molecular structure of dimeric bis(2,4-pentanedionato)cyclohexylaminecobalt(II)

The structure of bis(2,4-pentanedionato)cyclohexylaminecobalt(II) has been determined by a single-crystal x-ray diffraction study. The compound crystallizes as monoclinic crystals ( a=8.886(2) Å, b=9.347(2) Å, a=23.020(4) Å, and β=109.92(4)°) of space group P2 1/c with two dimers per unit cell (ϱ obs=1.31(2) g/cm 3 and ϱ calc=1.32 g/cm 3). The structure was solved from Patterson and electron density maps and refined by full matrix least-squares methods to a conventional R value of 0.057 for 1413 independent, non-zero reflections. The centrosymmetric dimer consists of two octahedra sharing an edge, the two chelate rings coordinated to one cobalt are not coplanar, and the second oxygen of each bridging 2,4-pentanedionato group is in the plane of the four-membered cobalt-oxygen ring.

Determination of the generalized spin-hamiltonian parameters of Gd 3+ in lanthanum ethylsulphate at 290°K

The generalized spin Hamiltonian for Gd 3+ (4 f 7; 8S 7 2 ) in lanthanum ethylsulphate with C 3 h site symmetry has been derived using tensor decomposition algebra. The parameters in this generalized spin Hamiltonian were determined from the 290°K EPR spectra measured at ∼35 GHz for the parallel, perpendicular, and intermediate directions to the crystal symmetry axis by solving the resulting secular matrices using a digital computer program of exact matrix diagonalization and least-squares fit. The linear-magnetic field dependent parameters associated with a third- or higher-rank tensor operator are found to be six orders of magnitude smaller than the parameter associated with the first-rank spin tensor operator (the g factor). However, the introduction of these parameters improves the sum of square deviations of the fit by an order of magnitude compared with that obtained using the conventional spin Hamiltonian.

Carbon compounds of the transition metals. Part XXI. The crystal and molecular structure of bis[tricarbonyl-(3-chloro-2-methylnitrosobenzene)iron

The product isolated from the reaction of 3-chloro-2-methylnitrobenzene with pentacarbonyliron is shown to be dimeric and of formula C20H12Cl2N2O8Fe2. Crystals are monoclinic with a= 18·23, b= 7·25, c= 8·71 Å(±0·03%)β= 90·85°, spacegroup P21/n(C2h5, No. 14), and Z= 2. The structure was determined from 815 non-zero and visually estimated independent reflexions. The refinement, by large-block matrix least-squares, yielded a final R of 5·8%. The molecules are centrosymmetric with the two iron and two nitrogen atoms coplar. The oxygen atoms lie on either side of this plane in positions such that each is bonded to one iron atom only. Thus the nitroso-groups bridge the iron atoms unsymmetrically. The Fe ⋯ Fe distance is 3·13 Å, which corresponds to a null bond.

Further examples of unbalanced packing of chiral molecules and the crystal and molecular structure of tris(triphenylphosphine)nitrosyl-iridium

Crystals of CuCIL3, CuBrL3, and Ir(NO)L3, isomorphous with the trigonal polymorph of Pt(CO)L3(L = PPh3), represent further examples of unbalanced packing of chiral molecules. The structure of Ir(NO)(PPh3)3 has been fully determined from 1929 independent reflexions measured by counter methods. This complex crystallizes in space group P3 and has a unit cell of dimensions: a= 19·94(1), and c= 10·780(5)A, with Z= 3. The structure has been refined by full matrix least-squares to a final R of 0·0475. The absolute configuration of the crystal sample has been determined unambiguously and designated R,S,S. The three independent molecules present in the unit cell each possess C3 symmetry and display almost the same conformation. The Ir–N, Ir–P, and N–O bond lengths are 1·67(2), 2·31(1), and 1·24(3)A. The N–Ir–P and P–Ir–P angles are 116·8(5) and 101·3(6)°; there is indirect evidence of linearity in the Ir–N–O interaction.

Crystal and Molecular Structure of N,N ′-Diethyl- N,N ′-Diphenylurea

N,N'-diphenyl-N,N'-diethylurea (C(17)H(20)N(2)O) crystallizes in the space group P2(1)/c. The unit cell constants are: a = 10.42 +/- 0.01 A, b = 16.86 +/- 0.02 A, c = 10.66 +/- 0.001 A, beta = 125 degrees 16' +/- 5'; Z = 4, D(x) = 1.16 g.cm(-3), D(meas) = 1.16 +/- 0.01 g.cm(-3). Data for 1392 reflections were collected at room temperature on a Picker automated diffractometer. The crystal structure was solved by direct methods and refined by bloc-diagonalized matrix least-squares calculations. The molecule is characterized by a pseudo C(2) symmetry; both phenyl groups are trans with respect to the oxygen atom. The hybridization of the two nitrogen atoms is intermediate between trigonal and tetrahedral; the nonplanar distortion of the amide groups is about 30 degrees . The amide C-N bond lengths are 1.37 A.

Platinum complexes with unsaturated amines.XI. the crystal and molecular stucture of trichloro(cis-but-2-enylammonium)platinum(II)

A three-dimensional X-ray structure determination of trichloro(cis-but-2-enylammonium)platinum(II), PtCl3(C4H10N), is reported. The compound crystallizes in the space group C2h5−P21c of the monoclinic system with four formula units in a cell of dimensions a = 7.2254(4), b = 7.521(1), c = 16.596(1) Å, β = 97.77(2)°. The calculated density (2.777 g. cm-3is good agreement with the observed density of 2.79± 0.01 g. cm−3. Data were collected by the equi-inclination Weissenberg technique and intensities were estimated. visually. The structure was refined by block-diagonal matrix least-squares methods to a final conventional R factor of 8.0% for 1381 observed reflactions. The structue consists of discrete squareplanar PtCl3(C4H10N) molecules in which the platinum the three chlorine atoms and the midpoint of the olefinic double bond form a plane, with the platinum, at the centre. The structure is compared with that of the analogous trichloro (trans-but-2-enylammonium)platinum(II).

Small Gaussian Expansions of Atomic Orbitals

Expansions of Clementi STO SCF AO's for some first-row atoms with GTO's have been obtained by the method of least squares. Expansion lengths vary from two to five Gaussians for each AO. In addition, Gaussian sets with constraints that the exponential parameters be shared for 1s and 2s and 2s and 2p orbital expansions are reported. For all cases a full matrix least-squares procedure was employed whereby the expansion coefficients and exponential parameters were simultaneously varied. All orbital expansions were constrained to a normalization condition. These expansions are useful for computing x-ray scattering factors and can be used in quantum-chemical studies.

Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness

Mg[Al(OCH 3) 2(CH 3) 2] 2·C 4H 8O 2. A novel coordination compound of a metal alkoxide and a donor molecule

The crystal structure of Mg[Al(OCH 3) 2(CH 3) 2] 2·C 4H 8O 2 has been determined by single-crystal X-ray diffraction methods. Two molecules crystallize in an orthorhombic unit cell with dimensions a = 13.42±0.02, b = 10.74±0.02, c = 7.30± 0.02 Å. The space groups is Pnnm. Full matrix least-squares refinement with magnesium and aluminum atoms anisotropic, resulted in a fianl R value of R 2=12.1%. The complex is an infinite polymer of Mg[Al(OCH 3) 2(CH 3) 2] 2 units held together by dioxane molecules. About the magnesium atom the point symmetry appears to be 2/ m, with the dioxane rings disordered about the two-fold axis. The metal atoms are bridged together by essentially sp 2 hybridized oxygen atoms, which are coplanar with the methoxy methyl groups and the metal atoms.

The crystal structure determination of π-C 5H 5Fe(CO) 2Mn(CO) 5

The crystal structure of π-C 5H 5Fe(CO) 2Mn(CO) 5, has been determined by single-crystal X-ray diffraction methods. Eight molecules crystal in a monoclinic unit cell with dimensions a = 7.220 ± 0.006, b = 30.387 ± 0.008, c = 12.498 ± 0.002 Å and γ = 90.21 ± 0.10°. The space group symmetry is P2 1/ b. Full matrix least-squares refinement with Fe and Mn atoms anisotropic, resulted in a final R value of 6.4% for 1201 “observed” reflections. The π-C 5H 5Fe(CO) 2Mn(CO) 5 molecules contain a MnFe bond of length 2.843 ± 0.004 Å and no bridging carbonyls. Both metal atoms may be envisioned as having distorted octahedral coordination. The isx sites about the iron atoms are occupied by two carbonyl groups, the manganese atom, and the cyclopentadienyl ring (which essentially occupies three sites); about the manganese atoms there are five carbonyl groups and the iron atom.

Stereochemistry of nucleic acid constituents: I. Refinement of the structure of cytidylic acid b

The crystal structure of cytidylic acid b, ‡ cytidine-3′-phosphate, has been refined by the method of full matrix least-squares to an R factor of 0·045, R = Σ | F o | − Σ | F c |/ Σ | F o | where | F o | and | F c | are respectively the observed and calculated diffraction amplitudes. The nucleotide is found to exist as a zwitter ion. The cytosine ring is slightly nonplanar, with N (3) protonated. The ribose ring is puckered with C (2′) -endo, and displaced (on the same side of C (5′) ) about 0·6 A from the plane of the remaining ring atoms. These latter appear to deviate slightly from a plane. Because the base in nucleic acids need not necessarily be planar and because C (1′) can be significantly displaced from the least-squares plane of the base, we suggest that the definition of the torsion angle Φ CN defined by Donohue & Trueblood (1960) , be modified as follows. The torsion angle Φ CN of the bond C (1′) −N is the angle formed by the projection of C (1′) −O (1′) relative to the projection of N (1′) −C (6) (in pyrimidine ring) and N (9) −C (8) (in purine ring) when viewed along C (1′) −N. This angle is taken as zero when O (1′) is anti-planar to C (2) of the pyrimidine or C (4) of the purine ring, and positive angles are taken as those measured in a clockwise direction when viewing from C (1′) to N. For this structure Φ CN = −42·1° and the conformation anti . All 14 hydrogen atoms in the structure appear in a ΔF synthesis and their parameters have also been refined. Several of the hydrogen isotropic thermal parameters have an anomalously low value. The standard deviation of the atomic co-ordinates from the least-squares refinement are P = 0·0011 A, O = 0·0036 A, N = 0·0042 A, C = 0·005 A and H = 0·06 A.