# Extending the Projection-Based Embedding Technique to Open-Shell Systems Using the Huzinaga Equation

## TL;DR

This paper extends a computational chemistry method to handle open-shell systems using a new formulation based on the Huzinaga equation.

## Contribution

The paper introduces a new theory for spin-restricted projection-based embedding in open-shell systems, building on prior work.

## Key findings

- The restricted PbE formulation was developed and implemented for open-shell systems.
- PbE schemes were tested and compared with other multilevel methods like ONIOM and MLC.
- MLC showed the best performance, while PbE and ONIOM variants with GGA low-level methods were competitive.

## Abstract

In this work, we present an approach for the embedding
of wave
function theory (WFT) and density functional theory (DFT) methods
in a lower-level density functional approximation using the projection-based
embedding (PbE) technique for open-shell systems. Our method is based
on the Huzinaga equation, which is implemented in both spin-restricted
and spin-unrestricted forms. While the unrestricted PbE approach has
been previously reported in the literature and follows naturally from
the theory for closed-shell systems, the restricted formulation required
the development of a new theory, building on earlier work by Roothaan
(Rev. Mod. Phys., 1960, 32, 179) as well as Shaik and Filatov (Chem. Phys. Lett., 1999, 304, 429). Our implementation
allows for the use of arbitrary combinations of restricted and unrestricted
wave functions for the high- and low-level methods, which can be advantageous
for the full-system low-level calculations. The various spin-restricted
and unrestricted wave function-based PbE schemes are thoroughly tested,
examining how the error in reaction energies depends on the size of
the subsystem treated at the high level. Additionally, we compared
the performance of PbE to that of other focused multilevel approaches,
such as vacuum embedding, “our-own n-layered integrated molecular
orbital and molecular mechanics” (ONIOM), and multilevel local
correlation (MLC). The results showed that MLC performed the best
among the tested methods, while only those PbE and ONIOM variants
were proved to be competitive whose low-level methods employed at
most a generalized gradient approximation (GGA). It is not straightforward
to determine whether PbE or ONIOM is generally more advantageous:
the latter can sometimes be more accurate and computationally cheaper,
while PbE offers greater robustness and the possibility of systematic
improvement.