Shape coordinates

Bowl-shaped coordination compounds have been assembled spontaneously from 10 small components, such as six (en)Pd2+ units and four 2,4,6-tri(3-pyridinyl)triazene moieties the overall structure approaches 3oA <00JA2665>. A different construct was derived from the 4-pyridinyl isomer and Pt(bpy)2+ and shown to facilitate self-assembly within the coordination... 

Fig. 15.4B Siereoview of ihe structure of the ](Ph3P)jKhp cation, showing the planar, approximately T-shaped coordination about the rhodium mom. Note the unusual manner in which the phenyl ring ai the lower right is drawn inward ihe rhodium atom. [From Yured, Y. W., Miles. S. L. Buu. R. Reed. C A. J. Am. Clwm. Sac 1977, 99.707f.-7()78 Reproduced with permission ...

A chloride-bridged structure has been proposed for the 1 1 adduct, [HgCl2(btaH)]2 (172). Mercury(II) benzotriazolate, Hg(bta)2, has been obtained from mercury(II) perchlorate, benzotriazole, and triethyl-amine in ethanol (172), and from mercuric oxide and benzotriazole (158). Mercuric acetate and chloride react with benzotriazole to form mixed mercury(II) salts, HgX(Ka) (X = MeC02 or Cl) (139). A dimeric structure (Fig. 13) with T-shaped coordination at mercury has been determined by X-ray diffraction methods (203) for the methyl derivative Hg(Me)(bta) (172). Cyclooctane-1,2-dione dihydrazide reacts with HgO... 

Figure 16 Schematic structure of T-shaped coordination of Cu1 by a tridentate pyrazole-type ligand89...

The shape, rotational, and translational coordinates are used to properly account for the different types of molecular motion. The word molecule refers to any /V-point particle system moving under the influence of the potential V, which is a function of the shape coordinates only. Furthermore, the molecules may rotate and translate freely, but their shape may be subject to P constraints. We call them free molecules, because the constraints in shape can be thought to follow from the potential of the molecule. For example, a bond becomes rigid, if the bond stretching force constant tends to infinite. However, it is simple to generalize the treatment given here to include the rotational and translational constraints, if needed. 

The shape of the molecule can be described by Ay = A — 6 shape coordinates 51,52, , A4V - Typical examples of shape coordinates used in the molecular spectroscopy are the interparticle distance... 

Because of the translational invariance of the shape coordinates and Euler angles [Eqs. (13) and (25)], it follows from Eqs. (4) and (9) that translation is separated from the vibrational and rotational degrees of freedom that is, the matrix [g,y] is partitioned into an internal block gint (depending only on the shape coordinates and Euler angles) of the size (A — 3) x (A — 3) and to a translational block gtransl of the size 3x3 (depending only on the center-of-mass coordinates) as... 

The Lagrangian in standard shape coordinates can be formed in the following way. We obtain the tangents by differentiating the linear combinations... 

It is possible to reduce the amount of work needed in applying the chain rule with a suitable choice of Cartesian coordinates, shape coordinates, and body frames (see Refs. 26-28). For example, in Ref. 26, the chain rule analogous to Eq. (113), namely... 

The reciprocality condition in Eq. (120) can be used to deduce some general properties of the measuring vectors associated with the shape coordinates and Euler angles. For example, because the measuring vector e 1 = u, is the same for any nucleus a, it follows that... 

Shape Coordinates. The gradients of any explicitly defined curvilinear shape coordinate can be found by the methods of geometric algebra. The results given in Table I are useful in this context. We solve the gradients of a bond angle 0paT by direct vectorial differentiation as a simple example ... 

The interested reader should consult Ref. 36 for other explicitly defined curvilinear shape coordinates. There are two ways to read off coordinate gradients Vps, for the standard internal coordinates from the vector derivatives... 

In terms of the shape coordinates, the Euler angles, and the Cartesian coordinates of the center of mass, the kinetic energy operator can be written as the sum... 

R. G. Littlejohn and M. Reinsch, Internal or shape coordinates in the A-body problem. Phys. 

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