Essentials of Computational Chemistry

4.5 MANY-ELECTRON WAVE FUNCTIONS 123
are orthonormal and are typically denoted as α and β (not to be confused with the α and
β of Hückel theory!) The spin quantum number is a natural consequence of the application
of relativistic quantum mechanics to the electron (i.e., accounting for Einstein’s theory of
relativity in the equations of quantum mechanics), as first shown by Dirac. Another consequence
of relativistic quantum mechanics is the so-called Pauli exclusion principle, which
is usually stated as the assertion that no two electrons can be characterized by the same set
of quantum numbers. Thus, in a given MO (which defines all electronic quantum numbers
except spin) there are only two possible choices for the remaining quantum number, α or β,
and thus only two electrons may be placed in any MO.
Knowing these aspects of quantum mechanics, if we were to construct a ground-state
Hartree-product wave function for a system having two electrons of the same spin, say α,
we would write
3
HP = ψa(1)α(1)ψb(2)α(2) (4.40)
where the left superscript 3 indicates a triplet electronic state (two electrons spin parallel)
and ψa and ψb are different from one another (since otherwise electrons 1 and 2 would have
all identical quantum numbers) and orthonormal. However, the wave function defined by
Eq. (4.40) is fundamentally flawed. The Pauli exclusion principle is an important mnemonic,
but it actually derives from a feature of relativistic quantum field theory that has more general
consequences, namely that electronic wave functions must change sign whenever the coordinates
of two electrons are interchanged. Such a wave function is said to be ‘antisymmetric’.
For notational purposes, we can define the permutation operator Pij as the operator that
interchanges the coordinates of electrons i and j. Thus, we would write the Pauli principle
for a system of N electrons as
Pij [q1(1),...,qi(i), . . . , qj(j), . . . , qN(N)]
= [q1(1),...,qj(i), . . . , qi(j), . . . , qN(N)]
=−[q1(1),...,qi(i), . . . , qj(j),...,qN(N)] (4.41)
where q now includes not only the three Cartesian coordinates but also the spin function.
If we apply P12 to the Hartree-product wave function of Eq. (4.40),
P12[ψa(1)α(1)ψb(2)α(2)] = ψb(1)α(1)ψa(2)α(2)
= −ψa(1)α(1)ψb(2)α(2) (4.42)
we immediately see that it does not satisfy the Pauli principle. However, a slight modification
to HP can be made that causes it to satisfy the constraints of Eq. (4.41), namely
3 SD = 1
√ 2 [ψa(1)α(1)ψb(2)α(2) − ψa(2)α(2)ψb(1)α(1)] (4.43)