Page 23 - Mechanism and Theory in Organic Chemistry

surfaces
are
they
of all ad.jacent poin~~hickaLnct~-zerazera~~eec~~d~o~es;
in three-dimensional space, and most of the important ones for our purposes are
planes, like those shown in Figure 1 .I for the P and d orbitals illustrated. (Nodes
can also be spherical, and of other shapes, but these are of less concern to us.)
The Physical Significance of Atomic Orbital Functions
The fact that an orbital function p is of different algebraic sign in different regions
has no particular physical significance for the behavior of an electron that finds
itself in the state defincd by the orbital. (We shall scc shortly that the significance
of the signs comes from the way in which orbitals can be combined with each
other.) The quantity that has physical meaning is the value at each point of the
function qP, which is positive everywhere, since the square of a negative number
is positive. T_he ssquared functionLP2, gives the probability of findingthe electron
at various points in space. Diagrams like that in Figure 1.2, with shading of
<arying density showing the relative probabilityoffinding the electron in various
regions or, more succinctly, the electron dzrtrzbuLzon or electron denrzty, are ~tually
w2. not of o, itself. The genemlshpeaf~ will be similaa~heshape
pi-res_of
of2. The orbitals and their squares have no edges, even though definite outlines
are usually drawn in diagrams; the values merely approach closer and closcr to
zero as one goes farther and farther from the nucleus.
Extension to Other Atoms
The hydrogen atomic orbitals would not do us a great deal of good if orbitals of
other atoms were radically different, since in that case different pictures would
be required for each atom. But the feature of the hydrogen atom problem that
determines the most important characteristics of the hydrogen atom orbitals is
the spherical symmetry. Since all the atoms are spherically symmetric, the atomic
orbitals of all atoms are similar, the main difference being in their radial depen-
dence, that is, in how rapidly they approach zero as one moves away from the
nucleus. Because the radial dependence is of minimal importance in qualitative
Figure 1.2 Electron density, v2, for 1s and 2p atomic orbitals. The density of shading is
roughly proportional to v2.