What Is the Exchange-Correlation Hole in DFT?

The exchange-correlation (XC) hole is a way to describe how electrons are distributed around a chosen electron compared with the average electron density. It combines two effects: the exchange hole, caused by the antisymmetry of the electronic wavefunction, and the correlation hole, caused by electrons avoiding one another through their Coulomb interaction.

The hole is useful because it turns the exchange-correlation part of the energy into a picture of a missing or depleted charge distribution around each electron. Its normalization and its role in the energy also explain why approximate density functionals are often described as models for the XC hole.

Definition: a conditional density minus the average density

Let $n(\mathbf r)$ be the electron density, normalized so that its integral is the number of electrons $N$. Let $n_2(\mathbf r,\mathbf r')$ be the spin-summed pair density: it describes the probability of finding one electron at $\mathbf r$ and a different electron at $\mathbf r'$.

When an electron is fixed at $\mathbf r$, the conditional density of the other electrons at $\mathbf r'$ is $n_2(\mathbf r,\mathbf r')/n(\mathbf r)$, wherever $n(\mathbf r)>0$. The XC hole is

$$h_{\mathrm{xc}}(\mathbf r,\mathbf r')=\frac{n_2(\mathbf r,\mathbf r')}{n(\mathbf r)}-n(\mathbf r').$$

In words, it is the density of other electrons conditioned on finding one at $\mathbf r$, minus the density you would expect without that condition. A negative value means a depletion relative to the average density; a positive value means an enhancement.

Why the hole integrates to minus one

For a system with exactly $N$ electrons, fixing one electron leaves $N-1$ other electrons. Thus the conditional density integrates to $N-1$, while the ordinary density integrates to $N$. Their difference integrates to $-1$:

$$\int h_{\mathrm{xc}}(\mathbf r,\mathbf r')\,d\mathbf r'=(N-1)-N=-1.$$

This is the sum rule for the spin-summed XC hole. It expresses the exclusion of the reference electron from the density of its neighbors. The rule is not a statement that a particular hole must be negative everywhere: local positive regions can occur, as long as the total integral is $-1$.

How it relates to exchange-correlation energy

The classical Hartree energy treats the electron density as a continuous charge distribution and includes an electron’s spurious interaction with itself. The physical electron-electron repulsion can instead be written using the conditional density, or equivalently the XC hole:

$$V_{\mathrm{ee}}=\frac12\int n(\mathbf r)\int\frac{n(\mathbf r')+h_{\mathrm{xc}}(\mathbf r,\mathbf r')}{|\mathbf r-\mathbf r'|}\,d\mathbf r'\,d\mathbf r.$$

The factor $1/2$ avoids counting each electron pair twice. Subtracting the Hartree energy leaves the part of the electron-electron interaction energy beyond the classical density-density term.

There is an important distinction: the physical hole at full interaction strength accounts for this potential-energy difference, but the usual DFT exchange-correlation energy also includes the change in kinetic energy relative to the noninteracting Kohn–Sham system. The exact expression for the full $E_{\mathrm{xc}}$ uses a coupling-constant-averaged hole, denoted $\bar h_{\mathrm{xc}}$:

$$E_{\mathrm{xc}}=\frac12\int n(\mathbf r)\int\frac{\bar h_{\mathrm{xc}}(\mathbf r,\mathbf r')}{|\mathbf r-\mathbf r'|}\,d\mathbf r'\,d\mathbf r.$$

The average is taken while the electron-electron interaction is scaled from zero to its physical strength, with the density held fixed. This is the adiabatic-connection formulation of DFT. The averaged hole also integrates to $-1$ around each reference electron.

Exchange hole and correlation hole

A common decomposition is $h_{\mathrm{xc}}=h_{\mathrm{x}}+h_{\mathrm{c}}$. The exchange hole comes from antisymmetry and is already present for a noninteracting Kohn–Sham determinant. It integrates to $-1$. The correlation hole describes the additional change beyond exchange and, under the usual spin-summed definition, integrates to zero. It redistributes density rather than removing another whole electron.

These pieces can have different shapes for same-spin and opposite-spin electron pairs. The exchange hole is associated with same-spin exclusion, while Coulomb correlation also produces avoidance between opposite-spin electrons. The details of the decomposition depend on the reference system and convention, but the total XC hole and its sum rule are the central quantities.

Worked example: a one-electron system

For $N=1$, there are no other electrons. Therefore the conditional density of other electrons is zero:

$$\frac{n_2(\mathbf r,\mathbf r')}{n(\mathbf r)}=0.$$

It follows directly that

$$h_{\mathrm{xc}}(\mathbf r,\mathbf r')=-n(\mathbf r'), \qquad \int h_{\mathrm{xc}}(\mathbf r,\mathbf r')\,d\mathbf r'=-1.$$

In this case the hole exactly cancels the Hartree self-interaction. For the exact functional, $E_{\mathrm{x}}=-E_{\mathrm H}$ and $E_{\mathrm c}=0$ for a one-electron system. This is a simple test for an approximate functional: if it does not cancel this self-interaction, it has a self-interaction error.

Common mistakes

• Calling the hole an extra electron. It is a difference between two densities, so it can be negative and is not itself a probability density.
• Saying it is negative everywhere. Only its integral is fixed at $-1$; the local shape can include positive regions.
• Equating the physical hole with the full $E_{\mathrm{xc}}$. The full energy formula uses a coupling-constant average, which includes the kinetic-correlation contribution.
• Confusing correlation with a second missing electron. In the standard spin-summed decomposition, the correlation hole integrates to zero; exchange supplies the net $-1$ normalization.

Thinking in terms of holes gives a physical interpretation of why exchange-correlation functionals depend on the density and its local environment. It also provides constraints, such as the sum rule and one-electron self-interaction cancellation, that help assess approximate functionals.