# Kohn-Sham Density Functional Theory with Complex, Spin-Restricted Orbitals: Accessing a New Class of Densities without the Symmetry Dilemma

## TL;DR

This paper introduces complex, spin-restricted orbitals in Kohn-Sham DFT, enabling access to new densities and improving triplet-singlet gap predictions, while highlighting limitations of current functionals.

## Contribution

It demonstrates that complex, spin-restricted orbitals can access new density classes and improve certain predictions, offering a novel approach to overcome the symmetry dilemma in DFT.

## Key findings

- cRKS orbitals improve triplet-singlet gap predictions
- RKS and UKS yield qualitatively wrong densities
- Modern functionals show limited improvement with cRKS densities

## Abstract

We show that using complex, spin-restricted orbitals (cR) in Kohn-Sham density functional theory (KS-DFT) allows one to access a new class of densities that is not accessible by either spin-restricted (RKS) or spin-unrestricted (UKS) orbitals. We further show that the real part of a cRKS density matrix can be non-idempotent when the imaginary part of the density matrix is not zero. Using cRKS orbitals shows significant improvements in the triplet-singlet gaps of the TS12 benchmark set for the SPW92, PBE, BLYP, and TPSS functionals. Moreover, it was shown that RKS and UKS yield qualitatively wrong charge densities and spin densities, respectively, leading to worse energetics. We demonstrate that the modern functionals SCAN, MN15-L, and B97M-V show surprisingly no improvement even with a qualitatively more accurate density from cRKS orbitals. To this end, our work not only paves the way to escape the symmetry dilemma whenever there exists a cRKS solution, but also suggests a new route to design better approximate XC functionals.