Inversion symmetry of singular values and a new orbital ordering method in tensor train approximations for quantum chemistry

The tensor train approximation of electronic wave functions lies at the core of the QC-DMRG (Quantum Chemistry Density Matrix Renormalization Group) method, a recent state-of-the-art method for numerically solving the $N$-electron Schrodinger equation. It is well known that the accuracy of TT approximations is governed by the tail of the associated singular values, which in turn strongly depends on the ordering of the one-body basis. Here we find that the singular values $s_1ge s_2ge ... ge s_d$ of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, $s_1s_d=s_2s_{d-1}=s_3s_{d-2}=...$, thus reducing the tail behaviour to a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wavefunctions, we find that the tail is upper bounded by a suitable superposition of the invariants. Optimizing the invariants or their superposition thus provides a new ordering scheme for QC-DMRG. Numerical tests on simple examples, i.e. linear combinations of a few Slater determinants, show that the new scheme reduces the tail of the singular values by several orders of magnitudes over existing methods, including the widely used Fiedler order.