The idea of a Fermi hole requires some background in the idea of antisymmetrized wavefunctions. The Pauli exclusion principle is the "rule" that no more than two electrons can be in the same orbital. The "rule" traces to a deep algebraic property of nature that has nothing whatsoever to do with the charge on electrons. The essence is that manyelectron wave functions must change sign when the labels on any two electrons are interchanged. This property is called antisymmetry, and its essential consequence is that electrons either stay out of one another's way, forming what is called a Fermi hole, or clump together, forming what is called a Fermi heap. Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of manyelectron atoms.
It is useful to consider an excited state of the helium atom in which electron 1 is in the 1s orbital and electron 2 has been excited to the 2s orbital. It is not possible, in principle, to distinguish electron 1 from electron 2. In other words, electron 2 might be in the 1s orbital with electron 1 in the 2s orbital. While there are 4 possible spin states for this system, only the ones in which the spins of both electrons are aligned (pointing in the same direction) will be considered. (This is the triplet state, there exists a singlet state with the spins paired).
Because electrons are fermions, they must be antisymmetric with respect to exchange. This means that if electrons 1 and 2 are switched, the exact same wavefunction would be obtained, but with a negative sign in front. This antisymmetry can arise either from the spin part (the intrinsic angular momentum of the electron) or the spatial part (the position of the electron as a function of r, theta, and phi) of the wavefunction. If the spatial part of the wavefunction is antisymmetric, the spatial wavefunction will look something like this (for the helium atom described above):
1s(1) 2s(2) – 1s (2) 2s(1)
where we cannot distinguish which electron is in which orbital (so we have separate terms for each case), and if we exchange the electrons, the wavefunction becomes:
1s(2) 2s(1) – 1s(1) 2s(2)
which is equal to:
 [1s(1) 2s(2) – 1s (2) 2s(1)]
Thus this spatial wavefunction is antisymmetric. Because of this result, if electrons 1 and 2 occupy exactly the same point in space, the wavefunction will vanish! Because the wavefunction squared gives the probability density for the electron, this antisymmetry means that the two electrons will never be found directly on top of each other. This gives rise to the phenomenon called the Fermi hole – the region around an electron in which no other electron with parallel spin will come.
Fermi holes give rise to the Pauli exclusion principle and are responsible for the spaceoccupying properties of matter. (This principle does not hold for bosons, which may all occupy a single state as in lasers and BoseEinstein condensates.)
A related phenomenon, called the Fermi heap, occurs when the wavefunction antisymmetry arises from the spin part of the wavefunction, giving the spatial wavefunction (this is the singlet state described above):
1s(1) 2s(2) + 1s (2) 2s(1)
In this case, for paired spins, there is actually a slightly higher probability of finding the electrons together. Fermi heaps play an important role in chemical bonding by allowing both electrons to be localized in the internuclear region and thus shielding the positively charged nuclei from electrostatic repulsion with one another.
Since electrons repel one another electrically, Fermi holes and Fermi heaps have drastic effects on the energy of manyelectron atoms. The most profound result is the periodic properties of the elements.
Animations of Fermi holes and Fermi heaps in the carbon atom are here^{[1]}. Details of the origin and significance of Fermi holes and Fermi heaps in the structure of atoms are discussed here^{[2]}.
