# Difference between density fitting (DF) and resolution of the Identity (RI) approximations

#### Juanchotutata

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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#### jim mcnamara

Mentor
Thread moved to Chemistry forum.

#### TeethWhitener

Gold Member
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.

#### Juanchotutata

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?

#### TeethWhitener

Gold Member
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 Molpro manual:
" 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.

#### Juanchotutata

I think I understand it now. Thank you very much again!

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