  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.
