Angular dependence of orbitals

Fig. 4-10.—The effect of hybridization of a w orbital. At the left is shown the angular dependence of orbital strength for a pure pic orbital, px (z axis vertical). At the right is shown a 7r orbital with 4.5 percent d character. It is seen that the d character increases overlap of the orbital with a similar orbital to the right (bonding overlap) and decreases overlap with a similar orbital to the left (nonbonding overlap).

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Three-dimensional representations of the angular dependence of some of the atomic orbitals are shown in Fig. 1. The coordinate axes and angles are defined in Fig. 6-5. 

It is well known that the angular dependence of each Cartesian p orbital is simply a unit vector in the corresponding Cartesian direction (px = ux, etc.). We can therefore associate each p, with a Cartesian direction vector dh... 

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

Figures 6.16(b) and (c) show the angular dependence of the Clx and Nix NEXAFS spectra, respectively. The most salient feature of Fig. 6.16(c) is the intensity increase of the n n b3g, Uu)) + cr (7t(big, bin)) peak for increasing 9 values. Flowever, the benzenic-type n a , biv) and n b2g) peaks remain nearly unchanged. For TCNQ the C2 molecular axis forms an angle of about 36 with the c -direction. Since CN bonds form an angle of 60° with this C2 molecular axis within the molecular plane, the intensity associated to o jt b3g, a )) should increase for larger 0. However, the intensity associated with jt n) orbitals, the benzenic-hke orbitals and n n b3g, a )), should exhibit the opposite behaviour because they are perpendicular to the o jt) orbitals. From Fig. 6.16(c) it is clear that the n n b3g, a )) -I- o n b g, 2u)) peak increases with regard to the benzenic-type orbitals for increasing 9e values because of the increasing a ( r) and...

Fig. 2.14 The angular dependence of the s, p2 end d z-, orbitals. The origin of the Cartesian axes passes through the centre of each individual orbital in the directions drawn.

Figure 8.8 shows the resultant angular dependence of this three-atom contribution for the s, p, and sp cases respectively. As expected, the s orbitals... 

Fig. 4-2.—The angular dependence of a tetrahedral orbital with bond direction along the x axis.

Fig. 5-4.—The angular de- Fig. 5-5.—The angular dependence of the dz orbital. pendence of the dx orbital.

The variation of iff with the angles 9 and 0 is exactly the same as illustrated in Fig 3.14 in Section 3.4. The angular dependence of atomic orbitals is often represented in another way, using the relation between spherical polar and cartesian coordinates. For example, the cosd function appropriate to l = 1 and m = 0 can be expressed as... 

Fig. 2.8. Angular dependence of the pseudocontact shift for an axial system, shown as a surface of constant absolute value of S1. In the example S1 is positive along the z axis and negative in the xy plane. The three-dimensional shape of the surface is similar to the representation of a d 2 orbital.

In the case of blue proteins there is a plane with Cu(II), two histidine nitrogens and one cysteine sulfur, with the x and y directions defined as in Fig. 2.16A. The dx2 y2 orbital contains the unpaired electron. The sulfur donor atom has a ir orbital which overlaps the dx2 y2 orbital (Fig. 2.16(B)). Large downfield shifts are observed (see also Chapter 5), and again the angular dependence of the shifts of the CH2 protons is expected to be mainly of sin2 type [61]. 

The second-order effect of spin-orbit coupling on the angular dependence of the zero-field splitting has been investigated by Hall and Hameka.457... 

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