  The multi-layer multi-configurational time-dependent Hartree (MCTDH) in
optimized second quantization representation (oSQR) approach combines the
tensor contraction scheme of the multi-layer MCTDH approach with the use of an
optimized time-dependent orbital basis. Extending the original work on the
subject [Manthe, Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR
propagation in imaginary time and properties related to particle number
conservation are studied. Difference between the orbital equation of motion in
real and imaginary time are highlighted and a new gauge operator which
facilitates efficient imaginary time propagation is introduced. Studying
Bose-Hubbard models, particle number conservation in MCTDH-oSQR calculations is
investigated in detail. Interesting properties of the single-particle functions
used in the multi-layer MCTDH representation are identified. Based on these
results, a tensor contraction scheme which explicitly utilizes particle number
conservation is suggested.
