Time-dependent vibrational coupled cluster theory: Theory and implementation at the two-mode coupling level

Abstract

Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.

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

“…However, the MCTDH[n] methods do by construction not behave correctly with respect to separability as opposed to the TDVCC method introduced in Ref. 26. We, thus, expect the TDVCC method to perform better for larger systems and the current investigation of constraint operators in MCTDH[n] is an important step on the way to introducing time-dependent modals in TDVCC.…”

“…The MCTDH[n] methods together with the full MCTDH method have been implemented in the MidasCpp program, 45 which also features efficient implementations of TDH 4 and TDVCC, 26 an array of methods for time-independent vibrational-structure calculations, 23,25,35 tools for automatic PES generation, [46][47][48] coordinate optimization, 49,50 etc. The MCTDH module has been implemented to interact with the time-independent wave-function modules of MidasCpp such that it can obtain initial wave functions and modal bases from prior vibrational self-consistent field (VSCF), 35,51,52 VCI, and MCTDH calculations.…”

“…Before deriving the MCTDH EOMs, we introduce the following definitions. The notation is a necessary hybrid of what is standard in the MCTDH community 3 with our formulation (m, u m ) used in previous VCC, VCI, and TDVCC 26 works.…”

“…This has recently been extended to the time-dependent context in Ref. 26 where a detailed discussion of the separability properties of TDVCC is given.…”

“…Thus, MCTDH[n] converges toward full MCTDH in the limit where n equals the number of degrees of freedom in the system, M. The equations of motion (EOMs) for MCTDH and MCTDH[n] are derived using a SQ formalism 21 to keep the formulation in line with our previous work on time-independent theory, TDH, 4 and time-dependent vibrational coupled cluster (TDVCC). 26 In addition, once the SQ is in place, SQ provides a short derivation compared to first quantization with natural introduction of key quantities for both MCTDH and MCTDH[n] bringing also some resemblance to SQ formulations in electronic-structure theory. We note that SQ formulations have previously been introduced in the context of (ML-)MCTDH for systems of indistinguishable particles.…”