The multilayer multiconfigurational time-dependent Hartree (MCTDH) in optimized second quantization representation (oSQR) approach combines the tensor contraction scheme of the multilayer MCTDH approach with the use of an optimized time-dependent orbital basis. Extending the original work on the subject [U. Manthe and T. Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR propagation in imaginary time and properties related to particle number conservation are studied. Differences 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 multilayer MCTDH representation are identified. Based on these results, a tensor contraction scheme, which explicitly utilizes particle number conservation, is suggested.

中文翻译：

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经过优化的第二量化中的多配置时间相关Hartree方法：虚时传播和粒子数守恒。

优化的第二量化表示（oSQR）方法中的多层多配置时变哈特里（MCTDH）方法结合了多层MCTDH方法的张量收缩方案，并使用了最佳时变轨道基础。扩展关于该主题的原始著作[U. Manthe和T.Weike，化学杂志。物理 146，064117（2017）]，此处研究了MCTDH-oSQR在假想时间内的传播以及与粒子数守恒有关的性质。重点介绍了实时和虚时运动的轨道运动方程之间的差异，并介绍了一种新的轨距算子，该算子运算符有助于有效的虚时时间传播。通过研究Bose-Hubbard模型，详细研究了MCTDH-oSQR计算中的粒子数守恒。确定了多层MCTDH表示法中使用的单粒子函数的有趣属性。基于这些结果，提出了一种张量收缩方案，该方案显式地利用了粒子数守恒。