The angular variation of atomic orbitals

Key points: The boundary surface of an orbital indicates the region of space within which the electron is most likely to be found; orbitals with the quantum number l have l nodal planes. The angular wavefunction expresses the variation of angle around the nucleus and this de scribes the orbital’s angular shape. An s orbital has the same amplitude at a given distance from the nucleus whatever the angular coordinates of the point of interest: that is, an s or bital is spherically symmetrical. The orbital is normally represented by a spherical surface with the nucleus at its centre. The surface is called the boundary surface of the orbital, and defines the region of space within which there is a high (typically 90 per cent) probability of finding the electron. The planes on which the angular wavefunction passes through zero are called angular nodes or nodal planes. An electron will not be found anywhere on a nodal plane. A nodal plane cuts through the nucleus and separates the regions of positive and negative sign of the wavefunction. In general, an orbital with the quantum number l has l nodal planes. An s orbital, with l 0, has no nodal planes and the boundary surface of the orbital is spherical (Fig. 1.13). All orbitals with l>0 have amplitudes that vary with angle. In the most common graphical representation, the boundary surfaces of the three p orbitals of a given shell are identical apart from the fact that their axes lie parallel to each of the three different Cartesian axes centred on the nucleus, and each one possesses a nodal plane passing through the nucleus (Fig. 1.14). This representation is the origin of the labels px, py, and pz, which are

Figure 1.14 The boundary surfaces of p orbitals. Each orbital has one nodal plane running through the nucleus. For example, the nodal plane of the p orbital is the xy-plane. The lightly shaded lobe has a positive amplitude, the more darkly shaded one is negative.

Figure 1.15 One representation of the boundary surfaces of the d orbitals. Four of the orbitals have two perpendicular nodal planes that intersect in a line passing through the nucleus. In the dz2 orbital, the nodal surface forms two cones that meet at the nucleus.

Figure 1.16 One representation of the boundary surfaces of the f orbitals. Other representations (with different shapes) are also sometimes encountered.

alternatives to the use of ml to label the individual orbitals. Each p orbital, with l single nodal plane. The boundary surfaces and labels we use for the d and f orbitals are shown in Figs 1.15 and 1.16, respectively. The dz2 orbital looks different from the remaining d orbitals. There are in fact six possible combinations of double dumb-bell shaped orbitals around three axes: three with lobes between the axes, as in dxy, dyz, and dzx, and three with lobes along the axis. One of these orbitals is dx2 y2. The dz2 orbital can be thought of as the superposition of two contributions, one with lobes along the z- and x-axes and the other with lobes along the z- and y-axes. Note that a d orbital with (l=2) has two nodal planes that inter sect at the nucleus; a typical f orbital (l=3) has three nodal planes.

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