# Exchange-correlation hole

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## Solution 1

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.

Long story

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

• $\vec{r}_{1}$ - the event of finding electron-one at a point $\vec{r}_{1}$;
• $\vec{r}_{2}$ - the event of finding electron-two at a point $\vec{r}_{2}$.

Probability theory reminds us that in the general case the so-called joint probability of two events, say, $\vec{r}_{1}$ and $\vec{r}_{2}$, i.e. the probability of finding electron-one at point $\vec{r}_{1}$ and at the same time electron-two at point $\vec{r}_{2}$, is given by \begin{equation*} \Pr(\vec{r}_{1} \cap \vec{r}_{2}) = \Pr(\vec{r}_{1}\,|\,\vec{r}_{2}) \Pr(\vec{r}_{2}) = \Pr(\vec{r}_{1}) \Pr(\vec{r}_{2}\,|\,\vec{r}_{1}) \, , \end{equation*} where

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

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

If the above mentioned events $\vec{r}_{1}$ and $\vec{r}_{2}$ were independent, then the conditional probabilities would be equal to their unconditional counterparts \begin{equation*} \Pr(\vec{r}_{1}\,|\,\vec{r}_{2}) = \Pr(\vec{r}_{1}) \, , \quad \Pr(\vec{r}_{2}\,|\,\vec{r}_{1}) = \Pr(\vec{r}_{2}) \, , \end{equation*} and the joint probability of $\vec{r}_{1}$ and $\vec{r}_{2}$ would simply be equal to the product of unconditional probabilities, \begin{equation*} \Pr(\vec{r}_{1} \cap \vec{r}_{2}) = \Pr(\vec{r}_{1}) \Pr(\vec{r}_{2}) \, . \end{equation*} 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, \begin{equation*} \Pr(\vec{r}_{1}\,|\,\vec{r}_{2}) \neq \Pr(\vec{r}_{1}) \, , \quad \Pr(\vec{r}_{2}\,|\,\vec{r}_{1}) \neq \Pr(\vec{r}_{2}) \, , \end{equation*} and $$\Pr(\vec{r}_{1} \cap \vec{r}_{2}) \neq \Pr(\vec{r}_{1}) \Pr(\vec{r}_{2}) \, .$$ 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 $$\Pr(\vec{r}_{1}\,|\,\vec{r}_{2}) < \Pr(\vec{r}_{1}) \, , \quad \Pr(\vec{r}_{2}\,|\,\vec{r}_{1}) < \Pr(\vec{r}_{2}) \, ,$$ while at large distances between the electrons the conditional probability is greater than half of the corresponding unconditional one $$\Pr(\vec{r}_{1}\,|\,\vec{r}_{2}) > \Pr(\vec{r}_{1}) \, , \quad \Pr(\vec{r}_{2}\,|\,\vec{r}_{1}) > \Pr(\vec{r}_{2}) \, ,$$ In the extreme case when $\vec{r}_{1} = \vec{r}_{2}$, the Coulomb repulsion between the electrons becomes infinite, and thus, $\Pr(\vec{r}_{1}\,|\,\vec{r}_{1}) = 0$ and consequently $\Pr(\vec{r}_{1} \cap \vec{r}_{1}) = 0$, 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 hole around 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.

## Solution 2

The exchange-correlation hole refers to the fact the probability of finding at electron at a certain position (r1) given that there is an electron at another position (r2) decreases when the distance between the two electrons decrease.

Put another way, for a two electron system, the least likely position of one electron on top of the other

## Solution 3

Just want to leave a note that the probability of finding two electrons at the same spatial spot is not zero. The argument of infinite repulsion does not fit. Details see here: Can two electron occupy the same spatial spot in a statistical way?

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### Another.Chemist

Updated on February 22, 2020

• Another.Chemist over 3 years

I did a google and chemistry.exchange search and found several technical definitions of the exchange correlation hole.

The questions that I want to proposes are:

• What the exchange-correlation hole means in common words?
• How do you explain it to a non-literate person?
• Philipp over 8 years
Could you add a little more context please: How "common" shall the words be that are used to explain it? No math at all or are some formulas allowed? How "non-literate" is the person supposed to be that it shall be explained to? Can any prior knowledge about quantum mechanics and density functional theory be assumed (especially: is it clear what exchange and correlation actually mean?) or has everything to be explained from scratch and in very simple words?
• Eka over 8 years
is ECH is only for electron or it can be extended to proton also?
• Wildcat over 8 years
Well, protons are also fermions, so there will be Coulomb as well as exchange correlation, and consequently, I think the answer is yes, each proton is also surrounded by ECH, though, of a different kind than that of an electron.
• Eka over 8 years
What about protons inside nucleus? will they have ECH
• Wildcat over 8 years
@Eka, it does not matter where the protons are. Whenever you have a many particle system composed of charged fermions (protons, electrons, etc.) each and every particle will have the corresponding ECH around it.
• Eka over 8 years
In nucleus protons are "tightly packed" so its ECH will be lower comparing to ECH of free moving electron right? ECH is Zero probability of finding fermions around similar fermions.
• Wildcat over 8 years
@Eka, sorry, I'm a chemist, so I have no idea about nuclear structure. :D But I think, that ECH is always the same for each and every proton no matter how tight protons are "packed". ECH is the region around a charged fermion where the probability of finding another charged fermion of the same kind is close to zero. For a charged fermion the probability of finding another charged fermion is exactly zero only at one point, at the location of the particle, but it is also negligibly small in the close proximity to it.