# Occurrence and Impact of Electric-Field-Induced Discontinuities in Correlation Energies from Localized Pair-Natural-Orbital Methods

## TL;DR

This paper investigates how electric fields cause small but significant errors in energy calculations using a specific quantum chemistry method.

## Contribution

The study reveals that electric-field-induced discontinuities in correlation energies can lead to large errors in computed molecular properties.

## Key findings

- Discontinuities in correlation energy under electric fields can cause over 100% discrepancies in hyperpolarizability calculations.
- Weak electric fields can magnify errors due to small field-displacement denominators.
- Diffuse basis sets and fixed PNO dimensions do not resolve the issue and may worsen the errors.

## Abstract

We report an investigation
of discontinuities in the correlation
energy produced by external static electric fields within the local
pair-natural-orbital coupled-cluster singles and doubles (LPNO-CCSD)
method. Such discontinuities arise as a result of variations in both
the dimensions and character of the pairwise virtual-orbital domains
resulting from changes in the strength of the field. Using several
small-molecule test cases – water, fluoroethylene, hypofluorous
acid and cis-1,3-butadiene we observe that, although
the discontinuities in the correlation energy are small (typically
1 μE

h
), they can
yield substantial errors in higher-order electric-field-dependent
properties computed using finite-difference techniques. For the static
hyperpolarizability (third derivative of the energy with respect to
the field) of water, for example, the discrepancies between LPNO-CCSD
and canonical-MO CCSD methods can exceed 100%. Furthermore, weak-field
displacements that should normally decrease errors in numerical differentiation
can yield orders-of-magnitude errors due to magnification of the energy
discontinuities by small field-displacement denominators. For larger
molecules, such fields can produce dramatic errors in the static polarizability
(second derivative of the energy with respect to the field) and hyperpolarizability
even with very tight PNO cutoffs. The use of basis sets containing
diffuse functions, which are essential for reliable predictions of
field-dependent response properties, tend to exacerbate the observed
errors. In addition, the use of fixed virtual PNO dimensions does
no resolve the problem due to mixing of the PNOs relative to zero-field
orbitals as a result of large condition numbers of the pair-correlation
densities.