# Is there relation between “Plane wave basis” “Correlation-Consistent” basis and “Fourier basis”

This question is related to the following link I found while browsing the internet: http://www.gutenberg.us/articles/Basis_set_(chemistry) The question: Is there a relation between "Plane wave basis" "Correlation-Consistent" basis and "Fourier basis"? Would anyone please add appropriate tags as well? Any comment would be appreciated.

One can think of Fourier basis as constructing the function with different frequencies. I was looking for constructing the function with vectors with same "correlation-consistent" . Is such concept possible? Is "correlation-consistent basis" is something similar?

• To help others get you an answer or answers that are informative, can you give us an idea about your level of understanding/education in regards to quantum chemistry and computational methods? As it stands, the question is a little bit broad or likely to elicit answers that might not be pertinent, so it would help if you can amend your post to narrow the scope a bit. – Todd Minehardt Nov 29 '15 at 0:50
• 1) Do not post the same question on few SE sites. 2) What is the actual meaning of "correlation" in your case. – Wildcat Nov 29 '15 at 10:27
• @Wildcat correlation as in co-variance matrix. Fourier can be expressed by exponential, can we easily derive "correlation-consistent basis" similarly? – Creator Nov 29 '15 at 18:11
• @Creator - In quantum chemistry (and related fields), the term "correlation" describes electronic correlation and has nothing to do with the statistical term you reference - see this Wikipedia page for electronic correlation for some clarification. – Todd Minehardt Nov 29 '15 at 22:01
• @Creator, I still don't understand what you're doing. "Correlation" in "correlation-consistent bases" has nothing to do with correlation of two random variables. – Wildcat Nov 29 '15 at 22:04

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 & 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.