# Formulation of an Efficient 𝒪 (M 4)-Scaling Explicitly Correlated MP2-F12 Correction by Combining Numerical Quadrature with Density Fitting and CABS-RI

## TL;DR

This paper introduces a new computational method that reduces the cost of quantum chemistry calculations by achieving O(M⁴) scaling.

## Contribution

A novel hybrid method combining numerical quadrature, density fitting, and CABS-RI to achieve efficient MP2-F12 calculations.

## Key findings

- The new method achieves O(M⁴) scaling for MP2-F12 corrections.
- Mean errors for noncovalent interactions are below 0.01 kcal/mol with modest grid sizes.
- Speedups of one order of magnitude are achieved for expensive computational steps.

## Abstract

We present a novel approach that combines numerical quadrature
with density fitting and CABS-RI for the evaluation of exchange-type
intermediates in RI-MP2-F12 theory, rigorously reducing the formal
and practical scaling of the total correction from
O(M5)
to
O(M4)
. Our new hybrid NQ/DF/CABS-RI ansatz is
based directly on our previously developed NQ/CABS-RI method for the
efficient evaluation of 6c3e integrals [Urban, L.; Laqua, H; Thompson,
T. H.; Ochsenfeld, C. J. Chem. Theory Comput.
2024, 20, 3706–3718] and extends this
approach to the optimized computation of products of 4c2e integrals.
In this framework, the main exchange-type intermediates
V
,
X
, and
B
are reformulated, resulting in more compact
expressions, increased shared computations, and fewer CABS-RI insertions.
We introduce efficient algorithms that cover all exchange-type contributions,
including advantageous batching of integrals. Benchmarks show that
NQ/DF/CABS-RI achieves mean errors below 0.01 kcal/mol for noncovalent
interaction and isomerization energies already with small to modest
grid sizes, while the numerical precision can be adjusted to balance
computational cost. Empirical scaling was determined using linear
glycine chains, demonstrating the expected
O(M4)
behavior for the rate-determining steps,
with the remaining exchange-type expressions scaling nearly linearly.
Compared with an idealized DF/CABS-RI implementation, our approach
achieves speedups of roughly one order of magnitude for the most expensive
steps with virtually no loss of numerical accuracy. Systems with strongly
delocalized electronic structures benefit particularly. For a nanotube
with 168 carbon atoms, the computational time for the most demanding
expressions is reduced from 9.97 to 1.25 days, bringing the cost much
closer to that of conventional DF-MP2. At present, NQ/DF/CABS-RI achieves
efficient
O(M4)
scaling, and further cost reductions are
anticipated through the introduction of integral screening based on
Cholesky orbitals, which will be explored in future work.