Page 234 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

8.4 Non-interacting particles 225

In the ground state of helium, according to this model, the two electrons are
in the 1s orbital with opposing spins. The ground-state wave function is

Ø 0 (1, 2) ˆ 2 ÿ1=2 1s(1)1s(2)[á(1)â(2) ÿ á(2)â(1)]

and the ground-state energy is ÿ108:8 eV. The energy of the ground state of
the helium ion He , for which n 1 ˆ 1 and n 2 ˆ1,is ÿ54.4 eV. In Section
‡
9.6, we consider the contribution of the electron±electron repulsion term to the
ground-state energy of helium and obtain more realistic values.
Although the orbital energies for a hydrogen-like atom depend only on the
principal quantum number n, for a multi-electron atom these orbital energies
increase as the azimuthal quantum number l increases. The reason is that the
electron probability density near the nucleus decreases as l increases, as shown
in Figure 6.5. Therefore, on average, an electron with a larger l value is
screened from the attractive force of the nucleus by the inner electrons more
than an electron with a smaller l value, thereby increasing its energy. Thus, the
2s orbital has a lower energy than the 2p orbitals.
Following this argument, in the ®rst- and second-excited states, the electrons
are placed in the 1s and 2s orbitals. The antisymmetric spatial wave function
has the lower energy, so that the ®rst-excited state Ø 1 (1, 2) is a triplet state,
8
á(1)á(2)
<
Ø 1 (1, 2) ˆ 2 ÿ1=2 [1s(1)2s(2) ÿ 1s(2)2s(1)] â(1)â(2)
: ÿ1=2
2 [á(1)â(2) ‡ á(2)â(1)]
and the second-excited state Ø 2 (1, 2) is a singlet state
1
Ø 2 (1, 2) ˆ [1s(1)2s(2) ‡ 1s(2)2s(1)][á(1)â(2) ÿ á(2)â(1)]
2
Similar constructions apply to higher excited states. The triplet states are called
orthohelium, while the singlet states are called parahelium. For a given pair of
atomic orbitals, the orthohelium has the lower energy. In constructing these
excited states, we place one of the electrons in the 1s atomic orbital and the
other in an excited atomic orbital. If both electrons were placed in excited
orbitals (n 1 > 2, n 2 > 2), the resulting energy would be equal to or greater
‡
than ÿ27.2 eV, which is greater than the energy of He , and the atom would
ionize.
This same procedure may be used to explain, in a qualitative way, the
chemical behavior of the elements in the periodic table. The application of the
Pauli exclusion principle to the ground states of multi-electron atoms is
discussed in great detail in most elementary textbooks on the principles of
chemistry and, therefore, is not repeated here.