Is there relation between “Plane wave basis” “Correlation-Consistent” basis and “Fourier basis”
This was supposed to be a comment, but there is too much text, so it will be an "answer".


  Is there a relation between "Plane wave basis"
  "Correlation-Consistent" basis and "Fourier basis"?


Well, mathematically, they are entities of the same type that serve for the exact same purpose: 


they all are concrete examples of a well-known idea of linear algebra &amp; functional analysis, the idea of a basis;
and they all are used to expand square-integrable functions.


Does this counts as a relation?


  I was looking for 'a kind' of basis (instead of Fourier) for some work
  and I personally gave the name for my basis as "correlation basis".
  When I typed my thoughts in google to my surprise I got the link
  mentioned in my question and I tried to understand at what context it
  is using the basis. From their I tried to relate the three basis to check whether I can use it for my work.


Are you just afraid of a terminological ambiguity out there? If so, don't worry too much. First, only the phrase "correlation-consistent basis", not just "correlation basis" has a particular well established meaning in quantum chemistry community. This is reflected even in the names of correlation-consistent bases that all starts with "cc" prefix. Secondly, "correlation-consistent basis" has a well established meaning only in quantum chemistry community concerning electron correlation and complete basis set extrapolation, so that unless you doing quantum chemistry, you shouldn't be interested in correlation-consistent bases.



Electron correlation is interdependency of the states of individual electrons in a many-electron system. Consider the simplest many-electron system, a two-electron one, and the following two events:


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


Then mathematically the idea of electron correlation can be simply expressed as the inequality of the joint probability of $1$ and $2$ on the one side and the product of unconditional probabilities of $1$ and $2$ on the other side
$$
    \Pr(1 \cap 2) \neq \Pr(1) \Pr(2) \, .
$$
Similarly, in terms of electron densities, we could indeed write
$$
    \rho_2(\vec{r}_{1}, \vec{r}_{2}) \neq \rho(\vec{r}_{1}) \rho(\vec{r}_{2}) \, ,
$$
where 


$\rho(\vec{r}_{1})$ is the probability density of finding electron-one with arbitrary spin at $\vec{r}_{1}$ while having electron-two with arbitrary spin at arbitrary position. 
Similarly, $\rho(\vec{r}_{2})$ is the probability density of finding electron-two with arbitrary spin at $\vec{r}_{2}$ while having electron-one with arbitrary spin at arbitrary position. 
And $\rho_2$ is the joint probability density of finding a pair of electrons one and two with arbitrary spins simultaneously at $\vec{r}_{1}$ and $\vec{r}_{2}$ respectively.


I think $\vec{r}_{1}$ and $\vec{r}_{2}$ can indeed be thought of as random variables and that essentially electron correlation is about these variables being in a statistical relationship with each other (they are unlikely to be close to each other and likely to be as different as possible). But if you think that a special basis set is needed to account for electron correlation, that is wrong: taking electron correlation in account has very little to do with a basis set choice. There is no need to use a correlation-consistent basis set to treat electron correlation: Pople basis sets can be used to treat it, as well as correlation-consistent one.