Exchange-correlation hole

What the exchange-correlation hole means in common words?

Assuming that the probability is the common word, one could say that the exchange-correlation hole, is a region of space around an electron in which the probability of finding another electron is close to zero due to electron correlation.

How do you explain it to a non-literate person?

Long story short is that mathematically electron correlation arises as a consequence of the instantaneous Coulomb repulsive forces between electrons and also as a consequence of the fundamental principle of antisymmetry of an electronic (fermionic in general) wave function. Note that the first kind of electron correlation which is due to Coulomb repulsions is known as the Coulomb correlation while the second one which is due to antisymmetry of a wave function is knows as the exchange correlation. However, since the root of all ab-initio wave function methods, the HF method, already accounts for exchange correlation, the Coulomb correlation is usually also referred to as just the correlation.

With respect to these two different kinds of electron correlation note that exchange-correlation hole does not refer specifically to just the exchange correlation, rather it is the hole due to both types of the electron correlation mentioned. I think that one would better call it, say, exchange-Coulomb correlation hole to avoid an ambiguity, but I'm afraid we're stuck with the exchange-correlation hole term.

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Long story

Consider the simplest many-electron system, a two-electron one, and let us examine the following two important events

• r⃗ 1${\overrightarrow{r}}_1$ - the event of finding electron-one at a point r⃗ 1${\overrightarrow{r}}_1$;
• r⃗ 2${\overrightarrow{r}}_2$ - the event of finding electron-two at a point r⃗ 2${\overrightarrow{r}}_2$.

Probability theory reminds us that in the general case the so-called joint probability of two events, say, r⃗ 1${\overrightarrow{r}}_1$ and r⃗ 2${\overrightarrow{r}}_2$, i.e. the probability of finding electron-one at point r⃗ 1${\overrightarrow{r}}_1$ and at the same time electron-two at point r⃗ 2${\overrightarrow{r}}_2$, is given by

Pr(r⃗ 1∩r⃗ 2)=Pr(r⃗ 1|r⃗ 2)Pr(r⃗ 2)=Pr(r⃗ 1)Pr(r⃗ 2|r⃗ 1),$$Pr({\overrightarrow{r}}_1\cap {\overrightarrow{r}}_2)=Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)Pr({\overrightarrow{r}}_2)=Pr({\overrightarrow{r}}_1)Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1),$$

where

• Pr(r⃗ 1)$Pr({\overrightarrow{r}}_1)$ is the probability of finding electron-one at point r⃗ 1${\overrightarrow{r}}_1$ irrespective of the position of electron-two;
• Pr(r⃗ 2)$Pr({\overrightarrow{r}}_2)$ is the probability of finding electron-two at point r⃗ 2${\overrightarrow{r}}_2$ irrespective of the position of electron-one;
• Pr(r⃗ 1|r⃗ 2)$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)$ is the probability of finding electron-one at point r⃗ 1${\overrightarrow{r}}_1$, given that electron-two is at r⃗ 2${\overrightarrow{r}}_2$;
• Pr(r⃗ 2|r⃗ 1)$Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1)$ is the probability of finding electron-two at point r⃗ 2${\overrightarrow{r}}_2$, given that electron-one is at r⃗ 1${\overrightarrow{r}}_1$.

The first two of the above probabilities are referred to as unconditional probabilities, while the last two are referred to as conditional probabilities, and in general 

Pr(A|B)≠Pr(A)$Pr(A|B)\ne Pr(A)$, unless events A$A$and B$B$ are independent of each other.

If the above mentioned events r⃗ 1${\overrightarrow{r}}_1$ and r⃗ 2${\overrightarrow{r}}_2$ were independent, then the conditional probabilities would be equal to their unconditional counterparts

Pr(r⃗ 1|r⃗ 2)=Pr(r⃗ 1),Pr(r⃗ 2|r⃗ 1)=Pr(r⃗ 2),$$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)=Pr({\overrightarrow{r}}_1),Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1)=Pr({\overrightarrow{r}}_2),$$

and the joint probability of r⃗ 1${\overrightarrow{r}}_1$ and r⃗ 2${\overrightarrow{r}}_2$ would simply be equal to the product of unconditional probabilities,

Pr(r⃗ 1∩r⃗ 2)=Pr(r⃗ 1)Pr(r⃗ 2).$$Pr({\overrightarrow{r}}_1\cap {\overrightarrow{r}}_2)=Pr({\overrightarrow{r}}_1)Pr({\overrightarrow{r}}_2).$$

Note that this is the picture in which electrons are sort of billiard balls, i.e. neutral macroscopic particles.

In reality, electrons are, of course, not like billiard balls; they are charged microscopic particles, and consequently,

Pr(r⃗ 1|r⃗ 2)≠Pr(r⃗ 1),Pr(r⃗ 2|r⃗ 1)≠Pr(r⃗ 2),$$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)\ne Pr({\overrightarrow{r}}_1),Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1)\ne Pr({\overrightarrow{r}}_2),$$

and

Pr(r⃗ 1∩r⃗ 2)≠Pr(r⃗ 1)Pr(r⃗ 2).$$Pr({\overrightarrow{r}}_1\cap {\overrightarrow{r}}_2)\ne Pr({\overrightarrow{r}}_1)Pr({\overrightarrow{r}}_2).$$

And since the spatial probabilities of finding electrons are inevitably related to their states, the inequality above means that the state of electron-one is not independent of the state of electron-two and vice versa. This interdependency of the states of electrons is referred to as the electron correlation.

The inequalities above, which are the mathematical essence of electron correlation, hold true for two reasons. First, electrons repel each other by Coulomb forces, and, as a consequence, at small distances between the electrons the conditional probability is less than half of the corresponding unconditional one

Pr(r⃗ 1|r⃗ 2)<Pr(r⃗ 1),Pr(r⃗ 2|r⃗ 1)<Pr(r⃗ 2),$$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)<Pr({\overrightarrow{r}}_1),Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1)<Pr({\overrightarrow{r}}_2),$$

while at large distances between the electrons the conditional probability is greater than half of the corresponding unconditional one

Pr(r⃗ 1|r⃗ 2)>Pr(r⃗ 1),Pr(r⃗ 2|r⃗ 1)>Pr(r⃗ 2),$$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_2)>Pr({\overrightarrow{r}}_1),Pr({\overrightarrow{r}}_2|{\overrightarrow{r}}_1)>Pr({\overrightarrow{r}}_2),$$

In the extreme case when r⃗ 1=r⃗ 2${\overrightarrow{r}}_1={\overrightarrow{r}}_2$, the Coulomb repulsion between the electrons becomes infinite, and thus, Pr(r⃗ 1|r⃗ 1)=0$Pr({\overrightarrow{r}}_1|{\overrightarrow{r}}_1)=0$ and consequently

Pr(r⃗ 1∩r⃗ 1)=0$Pr({\overrightarrow{r}}_1\cap {\overrightarrow{r}}_1)=0$, i.e. the probability of finding two electrons at the same point in space is zero. The correlation due to Coulomb repulsion is termed as Coulomb correlation and one sometimes speaks about the presence of the so-called Coulomb holearound each electron - a region of space around it in which the probability of finding another electron is close to zero due to Coulomb repulsion.

Secondly, as a consequence of the Pauli exclusion principle, electrons in the same spin state can not be found at the same location in space, so that for electrons in the same spin state there is an additional contribution into the inequalities between conditional and unconditional probabilities above. This effect is relatively localized as compared to one due to Coulomb repulsion, but bearing in mind the relationship between probabilities and wave functions and taking into account the continuity of the later, it is still noticeable when electrons are close to each other and not just at the same location in space. This type of electron correlation, which is a consequence of the Pauli exclusion principle, is referred to as Fermi correlation, or exchange correlation, and one sometimes speaks about the presence of the so-called Fermi hole around each electron - a region of space around it in which the probability of finding another electron in the same spin state is close to zero.