The computation of the electronic structure is an utmost important task for molecular engineering in modern chemistry and material science. The accurate computation of the electron correlation is a fundamental and extremely difficult problem in this context. Due to the enormous developments in the past decade -- made from different perspectives for different purposes in distinct communities -- tensor network state methods have already matured to provide a variety of tools to attack highly challenging strongly correlated electronic problems in quantum chemistry. Therefore, the treatment of high-dimensional problems, such as the SchrÃ¶dinger equation, can be efficiently approached by concepts of tensor product approximation. The aim of the 4-day workshop was to bring together theoretical chemists with their expertise in molecular structure theory, condensed matter physicists in many body physics and mathematicians in numerical analysis to intensify the exploration of this continuously growing new field of research, and to stimulate further developments of tensor network methods.

Unfortunately, the workshop organized originally for 2020 had to be shifted by a year due to emergence of Covid pandemic, and finally converted to an online event in 2021. The technical support received from MPI-PKS, however, provided an excellent scientific environment for presenting lectures, contributed talks and posters online, and for discussions as well. Talks have been organized into a morning and an evening session, taking into account different time zones of speakers and participants from Europe, Asia and America. In addition, talks have been recorded, and have been made available for the participants for a one month time period. Altogether there were more than 60 participants, including 16 invited speakers, from three continents (A. Alavi (DE), G. Booth (UK), G. Kin-Lic Chan (US), J. Eisert (DE), E. Fromager (FR), K. Hallberg (AR), S. Knecht (CH), K. Kowalski (US), S. Kvaal (NO), N.J. Mayhall (US), J. Pittner (CZ), Ch. Schilling (UK), S. Sharma (DE), F. Verstraete (BE), S.R. White (US), T. Yanai (JP)).

The spectrum of the workshop was very broad, covering various subtopics on electronic correlation theory. Among the key questions, the most appropriate existing tensor network state structures, or variants to be developed, for treating strongly correlated molecules have been discussed. Furthermore, state of the art for different multireference generalizations of conventional single reference methods like CC or DFT, utilization of concepts of quantum information theory, treatment of relativistic effects, consideration of electron dynamics and time dependent phenomena for molecules via tensor network state methods have also been addressed. In addition, different embedding approaches, basis optimization protocols, identification and measurement of entanglement, various approaches based on first and second quantization, and reformulation of the underlying tensor algebra have also been among the main topics. Rigorous mathematical analysis, novel algorithmic developments and applications to model systems, as well as large scale applications to real materials demonstrated the tremendous progress achieved since density matrix renormalization group (DMRG) has been introduced by Steven R. White. The two poster sessions provided a friendly atmosphere for young colleagues to present their research and results, and the longer breaks between the morning and evening sessions gave the participants a great opportunity for private discussions.

The scientific results of the workshop in the broader sense helped to intensify the communication among the different scientific communities, and expected to give a new impetus toward new developments in the field.