Example 12.5
The third electron in Li goes into the 2sorbital. Assuming a (re)normaliza-
tion constant of 1/ 6 , construct a proper antisymmetric wavefunction for Li
in terms of a Slater determinant.Solution
The rows will represent electrons 1, 2, and 3; the columns will represent the
spin orbitals 1s ,1s , and 2s (or 2s ). Following the determinant setup
above, the antisymmetric wavefunction is1 s 1 1 s 1 2s 1 Li
1
6
1 s 2 1 s 2 2 s 2
1 s 3 1 s 3 2 s 3
Because there are two possible wavefunctions for Li (depending on whether
the spin orbital for the last column is 2s or 2s ), we conclude that this
energy level is doubly degenerate.12.5 Other Atoms and the Aufbau Principle
We have presumed, more than proved, that multielectron atoms can be con-
ceptually approximated as combinations of hydrogen-like orbitals (even though
our helium example showed that the predicted energies are not very close).
Further, the Pauli principle restricts orbitals to having only two electrons, each
with different spin. As we consider larger and larger atoms, electrons in these
atoms will occupy orbitals described with larger and larger principal quantum
numbers.
Recall that in the hydrogen atom, the principal quantum number is the only
quantum number that affects the total energy. This is not the case with multi-
electron atoms, because interelectronic interactions affect the energies of the
orbitals, and now the subshells within the shells have different energies. Figure
12.4 illustrates what happens to the electronic energy levels of atoms. In the
case of hydrogen, energies of orbitals are determined by a single quantum
number. In multielectron atoms, the principal quantum number is an impor-
tant factor in the energy of an orbital, but the angular momentum quantum
number is also a factor. (To a much lesser extent, the mand msquantum num-
bers also affect the exact energy of a spin orbital, but their effect on the energy
is more noticeable in molecules. Their effect on the exact energies of electrons
is practically negligible for atoms outside of magnetic fields. See Figure 12.2
for an example.)
When assigning electrons to orbitals in multielectron atoms, it might be as-
sumed that they will occupy the available shell and subshell having the lowest
energy. This is a misstatement. Electrons reside in the next available spin orbital
that yields the lowest total energy for the atom.The placement is not necessarily
determined by the individual energy of the spin orbital. Instead, the total en-
ergy of the atom must be considered. When an atom’s electrons occupy orbitals
that yield the lowest total energy, the atom is said to be in its ground state.Any
other electronic state, which by definition would have a higher total energy, is
considered an excited state.The electrons in an atom can reach excited states by
absorbing energy; this is one of the basic processes in spectroscopy.
Consider an atom of the element beryllium, which has four electrons. Two
of the electrons occupy the orbital labeled 1s. The two remaining electrons
occupy an orbital in the second shell, but which? The n2 shell has 0382 CHAPTER 12 Atoms and Molecules