3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – Todd Minehardt Nov 29 '15 at 0:50
  • $\begingroup$ 1) Do not post the same question on few SE sites. 2) What is the actual meaning of "correlation" in your case. $\endgroup$ – Wildcat Nov 29 '15 at 10:27
  • $\begingroup$ @Wildcat correlation as in co-variance matrix. Fourier can be expressed by exponential, can we easily derive "correlation-consistent basis" similarly? $\endgroup$ – Creator Nov 29 '15 at 18:11
  • $\begingroup$ @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. $\endgroup$ – Todd Minehardt Nov 29 '15 at 22:01
  • $\begingroup$ @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. $\endgroup$ – Wildcat Nov 29 '15 at 22:04
4
$\begingroup$

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.

| improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.