Exploring DF vs. RI Approximations

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Discussion Overview

The discussion centers on the differences between density fitting (DF) and resolution of the identity (RI) approximations in quantum chemistry, particularly in the context of computational methods. Participants explore the mathematical and practical implications of these approximations, as well as their usage in software like Molpro.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that density fitting mathematically resembles resolution of the identity under specific conditions, but highlights that RIs involve summation over states and an overlap metric, which are absent in density fitting.
  • Another participant suggests that DF is often used synonymously with RI, based on their interpretation of the literature, but acknowledges that the precise mathematical definition of RI is not typically applied in practice.
  • A participant raises a question regarding the necessity of specifying different basis sets for DF and RI in Molpro, expressing confusion over their equivalence.
  • It is explained that density fitting is used to speed up calculations of two-electron integrals in quantum chemistry, with a focus on the computational scaling advantages it provides.
  • The potential for error in calculations due to the choice of auxiliary basis set in density fitting is also mentioned, emphasizing the need for careful selection to avoid significant inaccuracies.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of DF and RI, with some suggesting they are synonymous while others highlight distinct aspects that warrant separate consideration. The discussion remains unresolved regarding the implications of these differences in practical applications.

Contextual Notes

Participants reference specific computational practices and software requirements, indicating that the discussion is influenced by practical experiences and software implementations, which may not align perfectly with theoretical definitions.

Juanchotutata
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Hi everybody!

I am trying to find the difference between density fitting (DF) and resolution of the Identity (RI) approximations. I have read the following in the article [J. Chem. Phys. 118, 8149 (2003)]:

"Density fitting mathematically resembles a resolution of the identity RI in the specific case where the fitting criterion and target integral type coincide. However, RIs in quantum mechanics usually involve a summation over states and an implied overlap metric, neither of which appear in density fitting. Furthermore RIs do not offer a framework in which to discuss fitting criteria, constraints or robust fitting."

I still do not know what this exactly means. Could anyone give me a hand?

Thank you in advance!
 
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Juanchotutata said:
Hi everybody!

I am trying to find the difference between density fitting (DF) and resolution of the Identity (RI) approximations. I have read the following in the article [J. Chem. Phys. 118, 8149 (2003)]:

"Density fitting mathematically resembles a resolution of the identity RI in the specific case where the fitting criterion and target integral type coincide. However, RIs in quantum mechanics usually involve a summation over states and an implied overlap metric, neither of which appear in density fitting. Furthermore RIs do not offer a framework in which to discuss fitting criteria, constraints or robust fitting."

I still do not know what this exactly means. Could anyone give me a hand?

Thank you in advance!
I agree, as written this is very confusing. In fact, the key comes later in that same paragraph:
Werner et al said:
In this work we therefore use the term DF-MP2 as a synonym for RI-MP2, and hope that other authors will accept this as the standard name.
So DF is a synonym for RI (which also meshes with my experience). I think what the authors meant by the quote above is that, while many researchers use "resolution of the identity" to refer to the density fitting approximation, the precise mathematical "resolution of the identity"
$$1=\sum_n |n\rangle\langle n|$$
is never actually used. Therefore, they're arguing that "density fitting" should be the term of art, as opposed to "resolution of the identity." At least that's my takeaway.
 
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Thank you very much for your reply!

But I am confused, because when I perform my calculations on Molpro (maybe you have used it), I have to specify basis sets for density fitting and basis sets for resolution of the identity. So, if they are the same, why do I have to specify it twice?
 
Juanchotutata said:
Thank you very much for your reply!

But I am confused, because when I perform my calculations on Molpro (maybe you have used it), I have to specify basis sets for density fitting and basis sets for resolution of the identity. So, if they are the same, why do I have to specify it twice?
From the https://www.molpro.net/info/2012.1/doc/manual/node334.html:
" RI-MP2 is an alias for the command DF-MP2. "
The DF procedure invokes two basis sets. The first is the main basis set that is required for all calculations. The second is an auxiliary basis set used for the actual density fitting.

Density fitting is a way to speed up calculations of the two-electron integrals that appear in quantum chemistry calculations:
$$(ab|ij)=\int d\mathbf{r}_1 \int d\mathbf{r}_2 \frac{\phi_{a}(\mathbf{r}_1)\phi_{b}(\mathbf{r}_1)\phi_{i}(\mathbf{r}_2)\phi_{j}(\mathbf{r}_2)}{r_{12}}$$
The main issue is that transforming this integral from the atomic orbital basis to the molecular orbital basis scales as ##O(N^5)##. However, taking, e.g., ##\phi_{a}(\mathbf{r}_1)\phi_{b}(\mathbf{r}_1) = \rho_{ab}## and expanding ##\rho_{ab} = \sum_n (d_{ab})_n \chi_n## using an easily calculable auxiliary basis set ##\chi_n##, we can "cheat" and knock the integral transformation down to ##O(N^4)## (but if you choose your auxiliary basis set wisely, it ends up being closer to ##O(N^3)## in practice). So DF gives a nice speedup when doing things like MP2 calculations. The disadvantage is that you have to choose an auxiliary basis wisely in order to correctly approximate the density, or you might end up with a sizeable error in your calculations.
 
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I think I understand it now. Thank you very much again!
 

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