Wavefunctions three-dimensional rotations

Both close-coupling approaches (h q)erspherical or with absorbing potentials) and iterative/time-dependent absorbing-potential arrangement-decoupling approaches are readily extended to three-dimensional atom-molecule and molecule-molecule scattering. The wavefunction representation becomes more complicated and includes rotational matrices, but the essence and application of the method remains analogous [58. 65. 75. 76]. 

The shape of the 2p orbital can be deduced from equation (6.29). The presence of the Cartesian coordinate jc in the expression for 2p means that there will be a node in the wavefunction in the yz plane. The surface of constant jf is illustrated in Figure 6.10. The full three-dimensional orbital shape can be obtained by rotating the profile about the x axis. It is seen to consist of two doughnut-shaped lobes of opposite sign. The 2p orbital will be similar, but oriented along the y axis. 

Constant probability density contours of a 2pz AO are shown to the left in Fig. 1.3. The points of maximum density are indicated by filled squares on the positive and negative z-axes. The density is zero in the xy plane. Since the wavefunction is independent of the density has cylinder symmetry around the z-axis. Rotation of the figure around this axis would sweep out two three-dimensional surfaces with constant probability density. 

The wavefunctions V /, may be classified according to their symmetry properties. If we take the symmetry point group of this system to be C3V, there are three symmetry species in this group A (symmetric with respect to all operations of this group), A2 (symmetric with respect to the threefold rotations but antisymmetric with respect to the vertical symmetry planes), and E (a two-dimensional representation). 

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