be the only factor determining the order of orbital occupation. But in multi-
electron atoms, there is an additional factor. Not only is the absolute energy of
the orbital a factor, but the amount an electron in that orbital interacts with the
other electrons and the nucleusis also a criterion in determining the total en-
ergy of an atom.
To illustrate this point, Figure 12.5 shows the probabilities of the radial
functions in spherically symmetric shells about the nucleus (that is, 4 r^2 R^2
versus r) for the 3dand 4shydrogen-like wavefunctions on the same scale.
Both wavefunctions show a maxima several angstroms from the nucleus.*
However, note that the 4sorbital has three relative maxima before its ab-
solute maximum, and that several of these maxima indicate that an electron
in a 4sorbital has a considerable probability of being closer to the nucleus
than an electron in a 3dorbital. An electron in a 4sorbital is said to pene-
trateinward toward the nucleus. The increased penetration of the negatively
charged 4selectron toward the positively charged nucleus means an addi-
tional energy stabilization of the system as a whole, and as a result the final
electron in K occupies the 4sorbital. This allows the entire K atom to have a
lower total energy. And even though some energy of repulsion erases some
of that energy gain, the last electron of Ca, the next largest atom, also occu-
pies a 4sorbital instead of a 3dorbital, pairing up with the first electron in
the 4sorbital. (However, there are a few exceptions, as a review of electron
configurations will show.) Only with the introduction of another electron, for
an atom of scandium, does the electron occupy a 3dorbital instead of a
4 porbital.
This building up of electron configurations by placing electrons in orbitals
is called the aufbau principle(the name of the principle comes from the
German word aufbauen,meaning “to build up”). Although it might seem at
this point that there is little regularity in building up electron configurations
of larger atoms, there is some level of consistency. For example, the periodic
table’s shape is dictated by the filling of the orbitals by electrons. The number
of valence electrons almost never exceeds 8, due to the eventual regularity in
the filling of orbitals. There are several mnemonic devices used to remember12.5 Other Atoms and the Aufbau Principle 3850
Distance from nucleus (Å)84 (^2) r
R
(^2)
4
(^2) r
R
(^2)
4 s 3 d
(^76543210)
Distance from nucleus (Å)
7654321 8
Figure 12.5 Plots of 4 r^2 R^2 for the 3dand 4swavefunctions. Note that the plots have the
same x-axis, and that the 4selectron has some probability of being rather close to the nucleus.
For multielectron atoms, the penetration of the 4selectron combined with the shielding effect of
the other electrons serves to make the 4sorbital the next one occupied by electrons, rather than
the 3d.
*Actually, in a multielectron atom the maxima would be somewhat farther out because
of a shielding effect on the nucleus by the electrons occupying the inner shells.