Out-of-plane coordinate

Various other ways to incorporate the out-of-plane bending contribution are possible. For e3plane bend involves a cakulation of the angle between a bond from the central atom and the plane defined by I he central atom and the other two atoms (Figure 4.10). A value of 0° corresponds to all four atoms being coplanar. A third approach is to calculate the height of the central atom above a plane defined by the other three atoms (Figure 4.10). With these two definitions the deviation of the out-of-plane coordinate (be it an angle or a distance) can be modelled Lt ing a harmonic potential of the form... 

Some crystal structures of chelate complexes have been reported. An O-acryloyl-lactate-TiCU complex (Fig. 3) [26,27] has rare out-of-plane (Fig. 4) coordination of the acryloyl carbonyl group to the titanium a further study has been conducted [28]. Diethyl phthalate-TiCU [29], l,2-diketone-TiCl4 [25], and achiral [24] or chiral [30] acyloxazolidinone-TiCU complexes have been reported to involve in-plane coordination as shown in Fig. 5. The /S-alkoxyketone-TiCU complex shown in Fig. 6 [31] is characterized by a rare out-of-plane coordination geometry (dihedral bond angle of... 

Ti-0-C3-C4 = 57.6 °). This out-of-plane coordination was proved by NOE experiments to persist in solution. Treatment of the diastereomeric /8-alkoxyketone with TiCU generates the titanium chelate with in-plane coordination geometry (Eq. 1) [31]. NMR study of these out-of-plane and in-plane complexes of the j8-alk-oxyketones revealed that the titanium portion in the former complex acts as a stronger Lewis acid than that of the latter [31,32]-... 

It is more difficult to account for the remarkable anti-Cram selectivity observed in the MAT-mediated nucleophilic additions to a-chiral aldehydes, although out of plane coordination may play an important role (Figure 49). ... 

Figure 10. Pyridazine is easily distorted along the out-of-plane coordinate in the excited S, and T, states [95]. The broken line indicates the nonradiative transition. The rates from the higher vibrational levels are higher than those from the lower levels.

Figure 5 Definitions of out-of-plane coordinate. The molecule shown is the formate anion. The improper torsion definition is nonphysical and is used only because it can be easily adapted to existing torsional models and programs.

This is essentially the group-theoretical background to the description of out-of-plane distortions of the cyclopentane ring by Pitzer and Donath [13]. Most other authors [14] prefer to use torsion angles (Oj about bonds instead of z, s of atoms as the out-of-plane coordinates. The tu s and z s transform in the same way, except that the special forms corresponding to the S and symmetry coordinates are interchanged. In the torsion angle description. 

The three conditions that reduce the five z or m coordinates to only two out-of-plane coordinates are of the same mathematical form ... 

The three-dimensional properties of a laminate given by Eqns (6.11), (6.12), and (6.32) are needed in situations where out-of-plane stresses develop. Besides the obvious case of out-of-plane loading such as the local indentation and the associated solution of contact stresses in an impact problem, out-of-plane stresses typically arise near free edges of laminates, in the immediate vicinity of plydrops and near matrix cracks or delaminations. Typical examples are shown in Figure 6.4. The red lines indicate regions in the vicinity of which out-of-plane stresses [Pg.132]

Similarly the y component could be expressed in terms of the p/s and suitable out-of-plane coordinates. From (30) one immediately obtains... 

In MM3 the out-of-plane coordinate is defined as the angle between bond i- j and a point located in the plane formed by atoms i, k, and /, where, as before, atom j is the sp hybridized atom. Note, however, that this is a distinctly different definition from the Wilson angle described above. There are three of these coordinates for each center, and the value of these coordinates are inserted into the MM3 in-plane bending angle equation that was discussed above. 

The AMBER, CHARMM, and MM4 force fields u,se an improper torsion angle to define the out-of-plane coordinate. In this case the torsion angle defined by atoms i-j-k-l in the trigonal center is employed in the twofold torsion energy function term in equation (8) to calculate the out-of-plane-deformation energy. Only one of the three possible improper torsion angles is selected as the out-of-plane coordinate in these force fields. The other two improper torsions are ignored. 

In practice, these different definitions of the out-of-plane coordinate yield very similar results. 

Here, the K denotes force constants, while bond lengths, bond angles, torsion angles, and out-of-plane coordinates are denoted by b,6,, and x> respectively. The subscript zero denotes reference values. In the last two summations, which represent the nonbonded energy, q is the atomic partial charge and e,y and r, are the van der Waals well depth and radius. The nonbonded intemuclear separation is nj. 

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