# The Three-Legged Tree Tensor Networks with SU(2)- and molecular point group symmetry

## TL;DR

This paper enhances the three-legged tree tensor network state (T3NS) by incorporating spin and molecular point group symmetries, enabling more accurate and efficient quantum chemical calculations with better entanglement representation.

## Contribution

The authors extend T3NS to include spin and point group symmetries, improving accuracy and efficiency in quantum chemistry simulations compared to previous methods.

## Key findings

- Achieved more accurate calculations of copper-oxygen isomers.
- Reduced computational effort through symmetry inclusion.
- Demonstrated the method's effectiveness on specific molecular isomers.

## Abstract

We extend the three-legged tree tensor network state (T3NS) [J. Chem. Theory Comput. 2018, 14, 2026-2033] by including spin and the real abelian point group symmetries. T3NS intersperses physical tensors with branching tensors. Physical tensors have one physical index and at most two virtual indices. Branching tensors have up to three virtual indices and no physical index. In this way, T3NS combines the low computational cost of matrix product states and their simplicity for implementing symmetries, with the better entanglement representation of tree tensor networks. By including spin and point group symmetries, more accurate calculations can be obtained with lower computational effort. We illustrate this by presenting calculations on the bis($\mu$-oxo) and $\mu-\eta^2:\eta^2$ peroxo isomers of $[\mathrm{Cu}_2\mathrm{O}_2]^{2+}$. The used implementation is available on github.