Generalized tensor methods in quantum chemistry

The computation of the electronic structure is of utmost importance for the task of molecular engineering in modern chemistry and material science. In this context, the accurate computation of the electron correlation is a fundamental and extremely difficult problem. In contrast to the tremendous progress made in calculating weakly correlated systems by Density Functional Theory (DFT) for extended systems or Coupled Cluster Methods for highly accurate calculations, there are two major types of systems for which current quantum chemical methods have deficiencies: (1) Open-shell systems with a large number of unpaired electrons, as they occur in multiple transition metal complexes or in molecular magnets; (2) Extended or periodic systems without a band gap, where the limit of the applicability of the available size consistent methods is reached. The aim of this proposal is to develop a general tensor network state (TNS) based algorithm that can be applied efficiently to these open problems of quantum chemistry. Realization of such an algorithm relies on carrying out a variety of complex tasks. Several new formal methods and methodological concepts of tensor decompositions will have to be designed to comply with the specific, nonlocal nature of the Hamiltonian, and to this end, the applicants will join their rather complementary expertise regarding the powerful DMRG method and similar recent developments from physics, mathematics and information technology. To arrive at an efficient implementation of the quantum chemistry TNS algorithm, our contributions will be implemented and tested based on existing program structures of the QC-DMRG-Budapest and the TTNS-Vienna codes.

The main objective of the project Generalized tensor methods in quantum chemistry was to develop novel computational methods for simulating systems in quantum chemistry where strong quantum correlations are important. This has been the central problem in theoretical quantum chemistry since the advent of quantum mechanics, but insights originating from the field of quantum entanglement and quantum information theory have led to novel ways of describing quantum corrrelations. Technically, this has led to the introduction of a calculus of tensor networks, and those tensor networks can serve as very powerful ansatze for the variational calculation of energies and ground state wavefunctions of problems relevant in quantum chemistry. This proposal was centered around this novel methodology, and has successfully demonstrated the power of those tensor networks.Amongst many topics, we have been able to calculate the dissociation curve of Beryllium ring-shaped clusters, studied the ionic-neutral curve crossing of LiF, developed novel tensor methods for simulating systems in the continuum using a non-commutative version of the Gross-Pitaevskii equation, discovered intriguing connections between the concept of quantum marginals, reduced density matrices and quantum phase transitions, and started using the tensor network tools developed to problems in atomtronics.