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

Abstract

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.

Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy

“…We note that SQ formulations have previously been introduced in the context of (ML-)MCTDH for systems of indistinguishable particles. 9,27,28,30,31 Our SQ treats distinguishable degrees of freedom (modes) and, thus, differs in the sense that a set of elementary operators is introduced for each mode.…”

MR-MCTDH[n]: Flexible Configuration Spaces and Nonadiabatic Dynamics within the MCTDH[n] Framework

“…Such systems have been treated successfully with different variants of MCTDH [45][46][47][48][49] . As vibrational modes are distinguishable, permutational symmetry is not an issue in quantum molecular dynamics.…”

Non-adiabatic quantum dynamics without potential energy surfaces based on second-quantized electrons: application within the framework of the MCTDH method

“…( 2). Recently, Weike and Manthe 48 showed that if the initial SPFs are eigenfunctions of the local particle-number operator of the corresponding k-th sub-Fock space n(k) =…”

“…Since, as pointed out by Manthe 48 , the SPFs conserve the local particle number, and the propagation of the expansion coefficients of the A-vector conserves the total particle number, MCTDH-SQR calculations based on SPFs with an integer particle number have entries of the A-vector that stay equal to zero at all times. These coefficients refer to configurations of the whole Fock space perpendicular to the initially populated Hilbert space.…”

“…Here, we focus on the representation and the propagation of the N-SQR wavefunction within the framework of the multiconfiguration time-dependent Hartree MCTDH [39][40][41][42][43][44] method and machinery for second quantized Hamiltonians (MCTDH-SQR) [45][46][47][48] (while including first-quantized nuclear degrees of freedom) and note that other methodological alternatives might be suitable as well. Combining first quantized vibrations and second quantized electrons has been successfully achieved by Wang and Thoss within the context of the MCTDH-SQR approach in applications to model Hamiltonians 45,46 .…”