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

The tensor train (TT) approximation of electronic wave functions lies at the core of the quantum chemistry density matrix renormalization group (QC-DMRG) method, a recent stateof- the-art method for numerically solving the N-electron Schrödinger equation. It is well known that the accuracy of TT approximations is governed by the decay 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 s1 ≥ s2 ≥ ≥ sd of tensors representing ground states of noninteracting Hamiltonians possess a surprising inversion symmetry, s1sd = s2sd 1= s3sd 2 = . . . , thus reducing the tail behavior to the leading singular value and a single hidden invariant, which moreover depends explicitly on the ordering of the basis. For correlated wave functions, 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 magnitude over existing methods, including the widely used Fiedler order.