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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>5</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:math>. 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. <i>J. Chem. Theory Comput.</i> <b>2024</b>, <i>20</i>, 3706-3718] and extends this approach to the optimized computation of products of 4c2e integrals. In this framework, the main exchange-type intermediates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>V</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>M</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:math> scaling, and further cost reductions are anticipated through the introduction of integral screening based on Cholesky orbitals, which will be explored in future work.