Orbital shape can be spherical {s orbital}, with zero crossing points. Because spheres are radially symmetric, with electron orbits in all directions and so filling space, spherical orbits have no net orientation, so no interaction with spin makes added angular momentum 0. There can only be one kind of spherical orbital, because it must have radial symmetry.
Orbital shape can be double ellipsoidal along straight line {p orbital}, with one crossing point and one rotation axis. p orbital has two elongated lobes along line with one crossing in middle. Double-ellipsoidal orbit can orient in three spatial directions. If axis is along z-axis, aligned with spin, added angular momentum is 0. If axis is along x-axis or y-axis, perpendicular to spin, added angular momentum is -1 or +1. There can only be three kinds of double-ellipsoidal orbital, because one axis can have only three independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be quadruple-ellipsoidal four-leaf clover {d orbital}, with two crossing points and two rotation axes. d orbitals have four elongated lobes, two each along both orthogonal lines, with two crossings in middle. Four-leaf-clover quadruple ellipsoidal orbit can align with x-axis and y-axis; between xy-axis, xz-axis, or yz-axis; or with z-axis, as double ellipsoid and torus. If with x and y or between xy, added angular momentum is -2 or +2, because both axes are perpendicular to z-axis. If between xz or yz, added angular momentum is -1 or +1, because one axis is perpendicular to z-axis. If with z, added angular momentum is 0, because axis aligns with spin axis. There can only be five kinds of quadruple-ellipsoidal orbital, because axes can have only five independent spatial orientations, which fill space. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
Orbital shape can be octuple-ellipsoidal eight-lobed clover {f orbital}, with three crossing points and three rotation axes. f orbitals have six elongated lobes, with two each along three orthogonal lines, with three crossings in middle. Successive and more complex clover-leaf-shaped orbits can have 7, 9, or 11 distinct orientations. For same shell and same orbital angular momentum, all orientations are equally probable and have equal energy. All orientations add to make spherical orbital with zero net angular momentum.
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Date Modified: 2022.0225