The term 1/ 2 multiplies the entire determinant, like it multiplies the entire
wavefunction in equation 12.8. Such determinants used to represent an anti-
symmetric wavefunction are called Slater determinants,after J. C. Slater, who
pointed out such constructions for wavefunctions in 1929.
The use of Slater determinants to express wavefunctions that are automati-
cally antisymmetric stems from the fact that when two rows (or two columns)
of a determinant are exchanged, the determinant of the matrix becomes
negated. In the Slater determinant shown in equation 12.10, the possible spin
orbitals for electron 1 are listed in the first row and the spin orbitals of elec-
tron 2 are listed in the second row. Switching these two rows would be the same
thing as exchanging the two electrons in the helium atom. When this happens,
the determinant changes sign, which is what the Pauli principle requires for ac-
ceptable wavefunctions of fermions. Writing a wavefunction in terms of a
proper Slater determinant guarantees an antisymmetric wavefunction.
The Slater-determinant form of a wavefunction guarantees something else,
which leads to the simplified version of the Pauli principle. Suppose both elec-
trons in helium had the exact same spin orbital. The determinant part of the
wavefunction would have the form1 s 1 1 s 1
1 s 2 1 s 2 which is exactly 0 (12.11)The determinant being exactly zero is a general property of determinants. (If
any two columns or rows of a determinant are the same, the value of the de-
terminant is zero.) Therefore the wavefunction is identically zero and this state
will not exist. The same conclusion can be reached if the spin on both elec-
trons is. Consider, then, the lithium atom. Assuming that all three electrons
were in the 1sshell, the only two possible determinant forms of the wavefunc-
tion would be (depending on the spin function on the third electron):1 s 1 1 s 1 1 s 1 1 s 1 1 s 1 1 s 1 1 s 2 1 s 2 1 s 2 or 1 s 2 1 s 2 1 s 2 (12.12)
1 s 3 1 s 3 1 s 3 1 s 3 1 s 3 1 s 3 Note that in both cases, two columns of the determinant represent the same
spin orbitals for two of the three electrons (1st and 3rd columns for the first
determinant, 2nd and 3rd columns for the second determinant). The mathe-
matics of determinants requires that if any two rows or columns are exactly the
same, the value of the determinant is exactly zero. One cannot have a wave-
function for Li having three electrons in the 1sshell. The third electron, in-
stead,mustbe in a different shell. The next shell and subshell are 2s.
As we have been assigning a set of four quantum numbers to electrons in
hydrogen-like orbitals, we can do so for the spin orbitals of multielectron
atoms where we are approximating hydrogen-like orbitals. In the first row of
equation 12.11, the two spin orbitals can be represented by the set of four
quantum numbers (n,,m,ms) as being (1, 0, 0,^12 ) and (1, 0, 0,^12 ): the same
four quantum numbers. (Can you see how these numbers were determined
from the expression for the spin orbital?) In the first row of equation 12.12,
the three spin orbitals in the first case have the sets (1, 0, 0, 2 ^1 ), (1, 0, 0,^12 ),
and (1, 0, 0,^12 ): the first and third spin orbitals are the same. In the second case,
for the first row, the spin orbitals can be represented by the quantum numbers
(1, 0, 0,^12 ), (1, 0, 0,^12 ), and (1, 0, 0,^12 ), with the second and third spin or-
bitals having the same set of four quantum numbers. In all three cases, other
rows of the Slater determinant can have quantum numbers assigned to them,380 CHAPTER 12 Atoms and Molecules