Tangent sphere

Draw qualitative shapes of the (1) s, (3) p and (5) d "tangent sphere" atomie orbitals (note that these orbitals represent only the angular portion and do not eontain the radial... 

Figure 4.10 AX4, AX3E, and AX2E2 molecules (a) tangent sphere models or domain models with spherical domains B is a bonding pair and E is a lone pair and (b) conventional bond line structures.

For AX molecules with no lone pairs in the valence shell of A, both the VSEPR model and the LCP model predict the same geometries, namely AX2 linear, AX3 equilateral triangular, AX4 tetrahedral, AX5 trigonal bipyramidal, and AX octahedral. Indeed Bent s tangent sphere model can be used equally as a model of the packing of spherical electron pair domains and as a model of the close packing of spherical ligands around the core of the central atom. 

The shapes of atomic orbitals are routinely confused with graphs of the angular factors in wave functions [60] and shown incorrectly. The graph of a py orbital, for example, gives tangent spheres lying on the y-axis. 

The three-dimensional probability density that represents the shape of the orbital consists of a pair of distorted ellipsoids, and not two tangent spheres. The shape of the complex orbitals p2p l derives from... 

The angular functions for the s and p. orbital are illustrated in Fig. 2.5. For an s orbital, cl> is independent of angle and is of constant value. Hence this graph is circular or, more properly, in three dimensions—spherical. For the p. orbital we obtain two tangent spheres. The px and py orbitals are identical in shape but are oriented along the x and y axes, respectively. We shall defer extensive treatment of the d orbitals (Chapter 11) and / orbitals (Chapter 14) until bond formation in coordination compounds is discussed, simply noting here that the basic angular function for tl orbitals is fout-iobed and that for / orbitals is six-lobed (see Fig. 2.91... 

In Fig. 14.6, which holds in two dimensions, the energies of all faceted surfaces with inclinations between B and C fall on the dashed tangent circle shown. In three dimensions, a comparable construction would show that faceting would occur on three facet planes, such as in Fig. 14.76, and that the counterpart to the tangent circle would be a tangent sphere. 

Fig. 15. Tangent-sphere models of CF4, SiF4, hypothetical CPI , and SiFg-, based on conventional ionic and covalent radii (columns 1 and 2) and the electride ion model (column 3)...

Figure 5.3 The shape domains of relative local convexity of a MIDCO surface G(a) of Figure 5.1, relative to a tangent sphere T of curvature b (radius 1/b) are shown. A geometrical interpretation of the classitication of points r of G(a) into locally concave Dq, locally saddle-type D, and locally convex D2 domains relative to b is given when comparing local neighborhoods of the surface to the tangent sphere T. The classification depends on whether at point r the surface G(a) is curved more in all directions, or more in some and less in some other directions, or less in all directions, than the test sphere T of radius 1/b. In the corresponding three types of domains Do(b). Dm,), and D2(b), or in short Dp, D, and D2, the molecular contour surface G(a) is locally concave, of the saddle-type, and convex, respectively, relative to curvature b.

Within the general scheme of relative convexity, the conventional, ordinary local convexity is obtained as a special, degenerate case of relative local convexity, with a tangent sphere of infinite radius as reference, that is, with a tangent plane of reference curvature b = 0. 

However, much more detailed shape description is obtained if the tangent planes are systematically replaced by some other objects. Typically, a MIDCO is compared to a series of tangent spheres of various radii r, but one may find advantageous in direction-dependent problems to use a series of oriented tangent ellipsoids T, especially if a characterization itself involves some reference directions. In the case of oriented tangent ellipsoids, we assume that they can be translated but not rotated as they are brought into tangential contact with the MIDCO surface G K,a). 

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