Referring to equations (10-2), lobe I of the hybrid sp3 states combines the spherically symmetric s state with the p state that is oriented along the z axis, and thus sticks out in the +z direction (see Exercises 28 and 33). If Figure 10.12 is a true picture, then in a coordinate system rotated counterclockwise about the y-axis by the tetrahedral angle, lobe II should become lobe I. In the new frame, y-values are unaffected, but what had been values in the zx-plane become values in the z'x'-plane, according to x = x' cos α + z' sin α and z = z' cos α - x' sin α, where α is 109.5˚, or cos-1(-⅓).
(a) Show that lobe II becomes lobe I. Note that since neither the 2s state nor the radial part of the p states is affected by a rotation, only the angular parts given in equations (10-1) need be considered.
(b) Show that if lobe II is instead rotated about the z-axis by simply shifting φ by ± 120˚, it becomes lobes III and IV.
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