Motivated by the recent increase in the energy resolution of the best (gas-phase, high-resolution) spectroscopy experiments, sophisticated control of molecular quantum states, and synchronisation of spectrometers to atomic clocks, it became necessary to develop ultra-precise quantum chemistry. Solving the existing equations more precisely was not sufficient; it was necessary to go beyond the two fundamental approximations, the Born-Oppenheimer and the non-relativistic approximations, common in molecular computations even today. In the POLYQUANT project, theoretical, algorithmic and methodological developments have been carried out along these lines. The newly-developed methodologies include the computation of 1) the non-adiabatic mass for isolated and coupled electronic states; 2) regularised leading-order relativistic corrections; 3) regularised leading-order QED corrections through the non-relativistic quantum electrodynamics framework; 4) rovibrational and rovibronic computations in diatomics, including non-adiabatic and relativistic and QED corrections and couplings; 5) rovibrational computations for modelling hyperfine and Zeeman effects in polyatomic systems. Furthermore, variational relativistic computations became possible with unprecedented accuracy, and a systematic study was conducted to compare and validate the variational relativistic results with respect to benchmark perturbative relativistic values. All in all, theoretical and methodological progress has been made towards an ultra-precise quantum chemistry framework that can keep up with the progress of ultra-precise spectroscopy and provide possible guidance for related fields, including quantum technology developments operating on a molecular platform or condensed-phase simulations considering non-adiabatic and magnetic effects.
