which is low by 37.8% compared to experiment. Ignoring the repulsion
between the electrons leads to a significant error in the total energy of the
system, so a good model of the He atom should notignore electron-electron
repulsion.The example above shows that assuming that the electrons in helium—and
any other multielectron atom—are simple combinations of hydrogen-like elec-
trons is naive assumption, and predicts quantized energies that are far from the
experimentally measured values. We need other ways to better estimate the
energies of such systems.
12.4 Spin Orbitals and the Pauli Principle
Example 12.3 for the helium atom assumed that both electrons have a princi-
pal quantum number of 1. If the hydrogen-like wavefunction analogy were
taken further, we might say that both electrons are in the ssubshell of the first
shell—that they are in 1sorbitals. Indeed, there is experimental evidence
(mostly spectra) for this assumption. What about the next element, Li? It has
a third electron. Would this third electron also go into an approximate 1s
hydrogen-like orbital? Experimental evidence (spectra) shows that it doesn’t.
Instead, it occupies what is approximately the ssubshell of the second princi-
pal quantum shell: it is considered a 2selectron. Why doesn’t it occupy the 1s
shell?
We begin with the assumption that the electrons in a multielectron atom
can in fact be assigned to approximatehydrogen-like orbitals, and that the
wavefunction of the complete atom is the productof the wavefunctions of each
occupied orbital. These orbitals can be labeled with the nquantum number
labels: 1s,2s,2p,3s,3p, and so on. Each s,p,d,f,...subshell can also be labeled
by an mquantum number, where mranges from to (21 possible
values). But it can also be labeled with a spin quantum number ms, either ^12
or ^12. The spin part of the wavefunction is labeled with either or ,de-
pending on the value ofmsfor each electron. Therefore, there are several sim-
ple possibilities for the approximate wavefunction for, say, the lowest-energy
state (the ground state) of the helium atom:
He(1s 1 )(1s 2 )
He(1s 1 )(1s 2 )
He(1s 1 )(1s 2 )
He(1s 1 )(1s 2 )where the subscript on 1srefers to the individual electron. We will assume that
each individual Heis normalized. Because each Heis a combination of a
spin wavefunction and an orbital wavefunction,He’s are more properly called
spin orbitals.
Because spin is a vector and because vectors can add and subtract from
each other, one can easily determine a total spinfor each possible helium spin
orbital. (It is actually a total z componentof the spin.) For the first spin or-
bital equation above, both spins are , so the total spin is (^12 ) (^12 ) 1.
Similarly, for the last spin orbital, the total spin is (^12 ) (^12 ) 1. For
the middle two spin orbitals, the total (z-component) spin is exactly zero. To
summarize:
12.4 Spin Orbitals and the Pauli Principle 377