# Laplace-transformed multi-reference second-order perturbation theories in the atomic and active molecular orbital basis

## TL;DR

This paper introduces a Laplace-transformed formulation of NEVPT2 energies in atomic and active molecular orbital bases, enabling linearly-scaling computations and improved efficiency for localized orbitals.

## Contribution

It presents a novel approach to formulate NEVPT2 energies using Laplace transformation in AO and active orbital bases, facilitating scalable and efficient multi-reference perturbation calculations.

## Key findings

- Formulation of NEVPT2 energies in AO basis using Laplace transformation.
- Achieved linearly-scaling NEVPT2 implementation for localized orbitals.
- Controlled numerical errors by adaptive quadrature point selection.

## Abstract

In the present article, we show how to formulate the partially contracted n-electron valence second order perturbation theory (NEVPT2) energies in the atomic and active molecular orbital basis by employing the Laplace transformation of orbital-energy denominators (OED). As atomic-orbital (AO) basis functions are inherently localized and the number of active orbitals is comparatively small, our formulation is particularly suited for a linearly-scaling NEVPT2 implementation. Some of the NEVPT2 energy contributions can be formulated completely in the AO basis as single-reference second-order M{\o}ller-Plesset perturbation theory and benefit from sparse active-pseudo density matrices - particularly if the active molecular orbitals are localized only in parts of a molecule. Furthermore, we show that for multi-reference perturbation theories it is particularly challenging to find optimal parameters of the numerical Laplace transformation as the fit range may vary among the 8 different OEDs by many orders of magnitude. Selecting the number of quadrature points for each OED separately according to an accuracy-based criterion allows us to control the errors in the NEVPT2 energies reliably.