Local Second-Order Møller–Plesset Theory with a Single Threshold Using Orthogonal Virtual Orbitals: Theory, Implementation, and Assessment

Abstract

It has long been clear that electron correlation methods exhibit unphysical compute scalings with molecular size, which has motivated the development of local correlation methods to discard effectively zero contributions in a controlled way to yield an approximate correlation energy. The ideal local correlation method should have a single numerical threshold that controls the dropping of terms with the ability to have that threshold set small enough so that the correlation energy is reproduced to enough significant figures such that the result is chemically identical. This work reports such a method for the second-order Møller–Plesset (MP2) theory. The theory, implementation, and testing of this local MP2 theory are reported. Thresholds ranging from 10–5 to 10–8 and basis sets ranging from split valence plus polarization through to quadruple-ζ are assessed for local MP2 calculations on a range of molecules, including linear chains and molecules with two- and three-dimensional character. The implementation is shared memory parallel via OpenMP and yields roughly 50% parallel efficiency with 16 cores for a large job. Considerable efforts were made to minimize memory demands, which increased as thresholds were tightened. A variety of relative energy calculations are presented as a function of threshold to provide some guidance to users on how to obtain adequate precision at a low compute cost. It is particularly clear that derivative properties require tighter thresholds in order to achieve an adequate precision.

Analytic Gradients for Density Fitting MP2 Using Natural Auxiliary Functions

“…We tested the smoothness of the potential energy surface (PES) on the example of the torsional PES of C 10 H 22 with ε NAF = max, 10 –2 , and 10 –3 a.u. with the cc-pVDZ basis compared to the corresponding DF-MP2 curves . We rotated the two fragments of the molecule around the central C–C bond and plotted the potential energy relative to the all-trans conformer (180° torsional angle), the analytic first derivative of the energy, and the numerical second derivative of the energy calculated from the analytic gradient (with 0.05° step size) in Figure .…”

“…The most important developments along this line include the spin-scaled MP2 approaches, which improve the correlation energy by separate scaling of the Coulomb- and the exchange-like terms, , dispersion-corrected MP2, where the long-range behavior of MP2 is corrected, orbital-optimized MP2, where the MP2 energy is made stationary with respect to the rotation of the MOs, , and explicitly correlated MP2, where the wave function explicitly includes the interelectronic distances in the form of pair functions . These improved MP2 models were also combined with the aforementioned cost-reduction approximations resulting in low- or linear-scaling methods. − …”

Numerical Accuracy Matters: Applications of Machine Learned Potential Energy Surfaces

“…46 Truncations by neglecting matrix elements below a certain magnitude, in order to speed up the calculations, lead to numerical imprecision for gradients and, even more acutely, for second derivatives. 47 Changing from single-to double-precision accuracy for training and inference appreciably increases compute times. Training times for H 2 CO using F64 were several weeks compared with a few days when F32 is used.…”

“…The increased sensitivity of the gradients and higher-order derivatives to the roughness of the PES is reminiscent of approximate correlated ab initio methods such as local MP2 . Truncations by neglecting matrix elements below a certain magnitude, in order to speed up the calculations, lead to numerical imprecision for gradients and, even more acutely, for second derivatives …”