We looked already into MP2 and also discussed its formal computational scaling. In the following we want to discuss
a popular technique to reduce the prefactor of an MP2 implementation. This technique is known as resolution-of-the-identity
approximation (RI) or density fitting (DF). Some authors prefer the name RI other DF, but RI-MP2 and DF-MP2 usually
refers to the same method.
Introduction
The resolution-of-the-identity approximation (RI) [1-3] allows a more efficient handling of the 4 index integrals required
to construct the MP2 doubles amplitudes. This approximation allows to factorize the integral in 3 and 2 index integrals
by introducing a projection onto an auxiliary basis
\begin{equation}
\hat{I} \approx \sum\limits_{PQ} {\left. {\left| P \right.} \right){{\left[ {{V^{ - 1}}} \right]}_{PQ}}\left( {\left. Q \right|} \right.\frac{1}{{\left| {{r_1} - {r_2}} \right|}}}
\label{eqn:Idenity}
\end{equation}
where
\begin{equation}
\left( {\left. P \right|\left. Q \right)} \right. = {V_{PQ}} = \int {\frac{{P({r_1}) \cdot Q({r_2})}}{{\left| {{r_1} - {r_2}} \right|}}d{r_1}d{r_2}} \quad .
\end{equation}
The matrix \({\left[ {{V^{ - 1}}} \right]}_{PQ}\) ensures the ortohonormality of the auxiliary basis and for \(\hat{I}\) being idempotent e.g. \(\hat{I}^{2}=\hat{I}\).
\(\hat{I}\) is only a resolution of the identity of a complete auxiliary basis is used. Thus, we usually have an approximation. But the RI approximation
allows us to reformulate the 4 index integral as
\begin{equation}
\left( {\left. {ia} \right|\left. {jb} \right)} \right. \approx {\left( {\left. {ia} \right|\left. {jb} \right)} \right._{RI}} = {\sum\limits_{PQ} {\left( {\left. {ia} \right|\left. Q \right)} \right.\left[ {{V^{ - 1}}} \right]} _{PQ}}\left( {\left. {jb} \right|\left. P \right)} \right.
\label{eqn:RI}
\end{equation}
It is convenient to introduce the so-called B intermediates
\begin{equation}
{B_{Q,ai}} = \sum\limits_P {{{\left[ {{V^{ - 1/2}}} \right]}_{QP}}} (\left. P \right|ai)
\end{equation}
\begin{equation}
{B_{Q,bj}} = \sum\limits_P {{{\left[ {{V^{ - 1/2}}} \right]}_{QP}}} (\left. P \right|bj)
\end{equation}
so that we have a symmetric expression for constructing the 4 index integrals
\begin{equation}
\begin{split}
{\left( {\left. {ia} \right|\left. {jb} \right)} \right._{RI}} = \sum\limits_{Q} {{B_{Q,ai}}} {B_{Q,bj}} \quad.
\end{split}
\end{equation}
The cost for calculating the integrals is \(\mathcal O(N_O^2 N_v^2 N_x)\) and thus again \(\mathcal O(N^5)\) and
we could not reduce the computational scaling, but the RI approximation is in case of MP2 often an efficiency boost.
Let us compare the operation count for RI-MP2 and MP2.
\begin{equation}
\frac{\text{RI-MP2}}{\text{MP2}} = \frac{N_O^2 N_v^2 N_x}{N_oN^4+N_O^2N^3+N_O^2N_VN^2+N_O^2N_V^2N} \approx \frac{N_o N_x}{\left(N+3N_O\right)N}
\end{equation}
Usually the auxiliary basis is 2 -- 3 times larger than the orbital basis. Thus, especially for calculations with few occupied orbitals
and a large basis set large speed ups can be expected. Besides this theoretical arguments additionally the reduced I/O has to be considered,
which helps in practice a lot to make a calculation feasible.
References
- J. L. Whitten. Coulombic Potential Energy Integrals and Approximations. The Journal of Chemical Physics, 58(10):4496–4501, 1973.
- O. Vahtras, J. Almlöf, and M. W. Feyereisen. Integral Approximations for LCAO-SCF Calculations. Chemical Physics Letters, 213:514–518, 1993.
- Florian Weigend, Marco Häser, Holger Patzelt, and Reinhart Ahlrichs. RI-MP2: Optimized Auxiliary Basis Sets and Demonstration of Efficiency. Chemical Physics Letters, 294(1-3):143–152, 1998.
