  We present a new variational tree tensor network state (TTNS) ansatz, the
three-legged tree tensor network state (T3NS). Physical tensors are
interspersed with branching tensors. Physical tensors have one physical index
and at most two virtual indices, as in the matrix product state (MPS) ansatz of
the density matrix renormalization group (DMRG). Branching tensors have no
physical index, but up to three virtual indices. In this way, advantages of
DMRG, in particular a low computational cost and a simple implementation of
symmetries, are combined with advantages of TTNS, namely incorporating more
entanglement. Our code is capable of simulating quantum chemical Hamiltonians,
and we present several proof-of-principle calculations on LiF, N$_2$ and the
bis($\mu$-oxo) and $\mu - \eta^2 : \eta^2$ peroxo isomers of
$[\mathrm{Cu}_2\mathrm{O}_2]^{2+}$.
