can’t be in the same place at the same time. This is reflected in the Hartree–Fock
formulation of the wavefunction as a determinant (Section 5.2.3.1). Because the
spatial and spin coordinates of two such electrons would then be the same, the
Slater determinant representing the total molecular wavefunction would vanish,
since a determinant is zero if two rows or columns are the same (Section 4.3.3). This
is just a consequence of the antisymmetry of the wavefunction: switching rows or
columns of a determinant changes its sign; if two rows/columns are the same
thenD 1 ¼D 2 (the determinant before and the determinant after switching) and
D 1 ¼#D 2 , soD 1 ¼D 2 ¼0. If the wavefunction were to vanish so would the
electron density, which can be calculated from the wavefunction; this seems
physically unreasonable. This is one way of looking at the Pauli exclusion principle.
The probability of finding an electron in a small region centered on a point defined
by a triplet of spatial coordinates can in principle be calculated from the wavefunc-
tion. Now, since the probability is zero that at any moment two electrons of like spin
are at thesamepoint in space, and since the wavefunction is continuous, the
probability of finding them at a given separation should decrease smoothly with
that separation. This means thateven if electrons were uncharged, with no electro-
static repulsion between them, around each electron there would still be a region
increasingly (the closer we approach the electron) unfriendly to other electrons of
the same spin. This quantum mechanically engendered “Pauli exclusion zone”
around an electron is called aFermi hole, after Enrico Fermi; it applies to fermions
(Section 5.2.2) in general. Besides the quantum mechanical Fermi hole, each
electron is surrounded by a region unfriendly to all other electrons, regardless of
spin, because of the classical electrostatic (Coulomb) repulsionbetween point
particles(¼electrons). For electrons of opposite spin, to which the Fermi hole
effect does not apply, this electrostatic exclusion zone is called a Coulomb hole (of
course, electrons of the same spin also repel one another electrostatically). Since the
HF method does not treat the electrons as discrete point particles it largely ignores
the existence of the Coulomb hole, allowing electrons to get too close on the
average. This is the main source of the overestimation of electron–electron repul-
sion in the HF method. Post-HF calculations attempt to allow electrons, even of
different spin, to avoid one another better than in the HF approximation.
Hartree–Fock calculations give an electronic energy (and thus a total internal
energy, Section 5.5.2.1a) that is too high (the variation theorem,Section 5.2.3.3,
assures us that the Hartree–Fock energy will never be toolow). This is partly
because of the overestimation of electronic repulsion and partly because of the
fact that in any real calculation the basis set is not perfect. For sensibly-developed
basis sets, as the basis set size increases the Hartree–Fock energy gets smaller, i.e.
more negative. The limiting energy that would be given by an infinitely large basis
set is called the Hartree–Fock limit (i.e. the energy in theHartree–Fock limit).
Table5.4and Fig.5.18show the results of some Hartree–Fock and post-Hartree–
Fock calculations on the hydrogen molecule; the limiting energies are close to the
accepted ones [ 78 ]. Errors in energy, or in any other molecular feature, that can
be ascribed to using a finite basis set are said to be caused by basis set truncation.
Basis set truncationdoes not always cause serious errors; for example, the small
256 5 Ab initio Calculations