# Frozen-Core Analytical Gradients within the Adiabatic Connection Random-Phase Approximation from an Extended Lagrangian

## TL;DR

This paper introduces a frozen-core method to reduce computational costs in the random-phase approximation (RPA) while maintaining accuracy for molecular properties.

## Contribution

The novel combination of frozen-core and RPA analytic gradients with an extended Lagrangian framework is introduced.

## Key findings

- Frozen-core reduces matrix size and computational cost for RPA analytic gradients.
- Bond lengths and angles change minimally, showing the method's accuracy.
- Speedups of 35–55% are achieved for various molecular systems.

## Abstract

The implementation
of the frozen-core option in combination with
the analytic gradient of the random-phase approximation (RPA) is reported
based on a density functional theory reference determinant using resolution-of-the-identity
techniques and an extended Lagrangian. The frozen-core option reduces
the dimensionality of the matrices required for the RPA analytic gradient,
thereby yielding a reduction in computational cost. A frozen core
also reduces the size of the numerical frequency grid required for
accurate treatment of the correlation contributions using Curtis–Clenshaw
quadratures, leading to an additional speedup. Optimized geometries
for closed-shell, main-group, and transition metal compounds, as well
as open-shell transition metal complexes, show that the frozen-core
method on average elongates bonds by at most a few picometers and
changes bond angles by a few degrees. Vibrational frequencies and
dipole moments also show modest shifts from the all-electron results,
reinforcing the broad usefulness of the frozen-core method. Timings
for linear alkanes, a novel extended metal atom chain and a palladacyclic
complex show a speedup of 35–55% using a reduced grid size
and the frozen-core option. Overall, our results demonstrate the utility
of combining the frozen-core option with RPA to obtain accurate molecular
properties, thereby further extending the range of application of
the RPA method.