Triangle Coordinates

It is evident that the two fullerenes are strongly electronically communicating in spite of the interposed presence of the Rh6 cluster. A partial contribution to such a strong interaction could, however, arise also from the slight unequivalence of the two C6o environments. In fact, one fullerene is linked to a Rh3 triangle coordinated to a carbon atom (of one isocyanide ligand) and a phosphorus atom (of one diphosphine), respectively, whereas the other fullerene is linked to a Rh3 triangle which coordinates two phosphorus atoms (of the two diphosphines). 

A different orientation of the coordinated alkene is observed in the structure of Ir3(CO)7(/t-PPhCH=CHPPh2) (244). The alkene ligand lies above the metal triangle coordinating to one Ir atom while the two substituent P atoms bond to one and two metals, respectively (Fig. 20). The olefinic C-C distance is 1.44(2) A, and the average C-C-P angle is 118(1)°. The difference in orientation of the alkene presumably results from the interactions of the P substituents with the cluster which are not present in the other alkene-substituted molecules. 

In chemistry, we use often the Gibbs triangle coordinates or tetrahedron coordinates. We emphasize here that the three-dimensional analogue of the triangle coordinates is the tetrahedra coordinates. Triangle coordinates can plot only the mole fractions. However, with the transformation... 

However, we cannot map a three-dimensional event into two dimensions. This comes because within the triangle coordinates we cannot change all the three coordinates independently. In Fig. 1.6, it can be clearly seen that the distances a,, c are mutually dependent on each other. 

There have been several types of graphical illustrations to represent the state of orientation of structural units. Stein has proposed orthogonal equilateral triangle coordinates for the uniaxial symmetric system and Desper has proposed equilateral triangle coordinates for the orthogonally biaxial symmetric system. ... 

Let us consider the second moments of the orientation distribution of the jth vector with respect to the X, axis, i.e. , for which equation (29) may be obtained. Any point within the triangle coordinates, as shown in Figure 4 in terms of cos 0xj>, [Pg.464]

It is now convenient to introduce hyperspherical coordinates (p, 0, and <])), which specify the size and shape of the ABC molecular triangle and the Euler... 

Figure 4. The H3 and H4 loops. Ac the center, the conical intersections are shown schematically an equilateral triangle for H3 and a perfect tetrahedron for Kt, <2p> Jid Q, are the phase-preserving and phase-inverting coordinates, respectively.

The two coordinates defined for H4 apply also for the H3 system, and the conical intersection in both is the most symmetric structure possible by the combination of the three equivalent structures An equilateral triangle for H3 and a perfect tetrahedron for H4. These sbnctures lie on the ground-state potential surface, at the point connecting it with the excited state. This result is generalized in the Section. IV. 

The coordinates of interest to us in the following discussion are Qx and Qy, which describe the distortion of the molecular triangle from Dy, symmetry. In the harmonic-oscillator approximation, the factor in the vibrational wave... 

It can be readily shown that L,J 1,3 satisfy the requirements for shape functions (as stated in Equation 2.3) associated with the triangoilar element. The area of a triangle in tenns of the Cartesian coordinates of its vertices is written as... 

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala) (triangles) and its conformational constraint counterpart, cyclic-CAla) (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)...

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.)...

N occupy an sp lone-pair in the plane of the ring (or the plane of the local PNP triangle) as in Fig. 12.26a. The situation at P is less clear mainly because of uncertainties concerning the d-orbital energies and the radial extent (size) of these orbitals in the bonding situation (as distinct from the free atom). In so far as symmetry is concerned, the sp lone-pair on each N can be involved in coordinate bonding in the jcy plane... 

Dreieck, n. triangle, dreieckig, a. triangular, three-cornered. Dreiecks-koordinaten, f.pl. triangular coordinates. -lehre, /. trigonometry. Dreier-gemisch, n. triple mixture, -gruppe, /. 

Figure 7, Schematic representation of the 1-TS (solid) and 2-TS (dashed) (where TS = transition state) reaction paths in the reaction Ha + HbHc Ha He + Hb- The H3 potential energy surface is represented using the hyperspherical coordinate system of Kuppermann [54], in which the equilateral-triangle geometry of the Cl is in the center (x), and the linear transition states ( ) are on the perimeter of the circle the hyperradius p = 3.9 a.u. The angle is the internal angular coordinate that describes motion around the CL...

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. 

In Figure 4.9 Snyder s solvent selectivity triangle is presented. The solvents of Table 4.2 are marked in the plot with triangular coordinates for the eight groups. 

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