The multi-configurational time-dependent Hartree approach in optimized second quantization: imaginary time propagation and particle number conservation

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.