Metric coordination

A complete and detailed description of molecular structure includes statements concerning the metric coordinates ) of atomic nuclei supplemented by electron density distribution data. Although a large amount of data is involved, such representation is not particularly suitable for the chemically relevant structural features of molecules. 

Figure 4. Schematic of [Cp2Nd(ketim)] 2 in case of one particular phase of unsym-metric coordination...

Stilbene is an unusual arene as a ligand in that it offers the possibility of coordination to an arene ring or to the double bond. In spite of the abundance of rare earth naphthalene complexes in hterature (Bochkarev, 2002 Bochkarev et al., 1997 Fryzuk et al., 2000), ( )-stilbene complexes are rare. Evans et al. reported the synthesis of [(C5Me5)2Sm]2((E)-stilbene) from the direct reaction of (C5Me5)2Sm and ( )-stUbene (Chart 23 Evans et al., 1990). Due to the poor quality of the crystal, the structure of the molecule could not be unambiguously determined but the authors suggested an unsym-metrical coordination mode based on coimectivity. The analogous samarium styrene and butadiene complexes (styrene complex shown in Chart 23) showed two-electron reduction of the C=C bond and concomitant oxidation of Sm(ll) to Sm(lll) (Evans et al., 2001). Related yttrium and lutetium complexes of the readily available tetraphenylethylene dianion have been reported with similar stmctural features (Chart 23 Roitershtein et al., 1998, 2004). 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

The diagonal elements of are the eigenvalues of G and the columns of V are i eigenvectors. The atomic coordinates can be derived from the metric matrix by rewritir... 

The matrix gp, represents the components of a covariant second-order tensor called the metric tensor , because it defines distance measurement with respect to coordinates To illustrate the application of this definition in the... 

Units. The SI system of units and conversion factors (qv) has been formally adopted worldwide, with the exception of Bmnei, Burma, Yemen, and the United States. The participation of the United States in the metrication movement is evident by the passage of the Metric Acts of 1866 and 1975 and the subsequent estabUshment of the American National Metric Council (private) and the U.S. Metric Board (pubHc) to plan, coordinate, monitor, and encourage the conversion process. 

The 1976 CIELAB Color Space. Defiaed at the same time as the CIELUV space, the CIELAB space, propedy designated CIE E i , is a nonlinear transformation of the 1931 CIE X, Y, Z space. It also uses the metric lightness coordinate E, together with ... 

The 1976 CIE Metric Color Spaces. Both the CIELUV and CIELAB spaces can have their Cartesian coordinates converted to cylindtical coordinates, called metric or hue-angle coordinates, with E unchanged. These coordinates are designated CIE LYC h and CIE LYC, b, respectively. 

Dyes and Pigments. Several thousand metric tons of metallated or metal coordinated phthalocyanine dyes (10) are sold annually in the United States. The partially oxidized metallated phthalocyanine dyes are good conductors and are called molecular metals (see Semiconductors Phthalocyanine compounds Colorants forplastics). Azo dyes (qv) are also often metallated. The basic unit for a 2,2 -azobisphenol dye is shown as stmcture (11). Sulfonic acid groups are used to provide solubiHty, and a wide variety of other substituents influence color and stabiHty. Such complexes have also found appHcations as analytical indicators, pigments (qv), and paint additives. 

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

We shall denote the space time coordinates by a (which as a four-vector is denoted by a light face x) with x° — t, x1 = x, af = y, xz = z x — ai0,x. We shall use a metric tensor grMV = gliV with components... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

The coordination of [Me2Sn(IV)f to captopril (cap) [(2S)-l-[(2S)-2-methyl-3-sulfanyl propanoyl]pyrrolidine-2-carboxylic acid] in aqueous solution was studied by means of pH-metric titration, electrospray mass spectrometry, H NMR, and Mossbauer spectroscopies in the 2-11 pH range. The results obtained proved that only monomeric complexes are formed in solution. In the acidic pH... 

If the inverse in Eq. (2.8) does not exist then the metric is singular, in which case the parameterization of the manifold of states is redundant. That is, the parameters are not independent, or splitting of the manifold occurs, as in potential curve crossing in quantum molecular dynamics. In both cases, the causes of the singularity must be studied and revisions made to the coordinate charts on the manifold (i.e. the way the operators are parameterized) in order to proceed with calculations. 

Similar expressions can be derived for second spatial derivatives. The final form of the equations that result after a generalized coordinate transformation depends on the degree of differentiation by using the chain rule, i.e. on the treatment of the metrics x, x, and y. For more details we refer to the... 

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

With the above choice of coordinates, the internal Gqq), Coriolis (Gy ), and rotational (Coxb), parts of the metric are constant, linear, and quadratic functions... 

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