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The Journal of Chemical Physics

Time-dependent vibrational coupled cluster with variationally optimized time-dependent basis sets

Niels Kristian Madsen, Mads Bøttger Hansen, Ove Christiansen A, Alberto Zoccante

Published Nov 3, 2020• 16 citations
Publisher

Abstract

We develop time-dependent vibrational coupled cluster with time-dependent modals (TDMVCC), where an active set of one-mode basis functions (modals) is evolved in time alongside coupled-cluster wave-function parameters. A biorthogonal second quantization formulation of many-mode dynamics is introduced, allowing separate biorthogonal bases for the bra and ket states, thus ensuring complex analyticity. We employ the time-dependent bivariational principle to derive equations of motion for both the one-mode basis functions and the parameters describing the cluster (T) and linear de-excitation (L) operators. The choice of constraint (or gauge) operators for the modal time evolution is discussed. In the case of untruncated cluster expansion, the result is independent of this choice, but restricting the excitation space removes this invariance; equations for the variational determination of the constraint operators are derived for the latter case. We show that all single-excitation parts of T and L are redundant and can be left out in the case of variationally determined constraint-operator evolution. Based on a pilot implementation, test computations on Henon-Heiles model systems, the water molecule, and a reduced-dimensionality bi-thiophene model are presented, showing highly encouraging results for TDMVCC. It is demonstrated how TDMVCC in the limit of a complete cluster expansion becomes equivalent to multiconfiguration time-dependent Hartree for the same active-space size. Similarly, it is discussed how TDMVCC generally gives better and more stable results than its time-independent-modals counterpart, while equivalent results are obtained for complete expansions and full one-mode basis sets.

DOI
10.1063/5.0024428

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We develop time-dependent vibrational coupled cluster with time-dependent modals (TDMVCC), where an active set of one-mode basis functions (modals) is evolved in time alongside coupled-cluster wave-function parameters. A biorthogonal second quantization formulation of many-mode dynamics is introduced, allowing separate biorthogonal bases for the bra and ket states, thus ensuring complex analyticity. We employ the time-dependent bivariational principle to derive equations of motion for both the one-mode basis functions and the parameters describing the cluster (T) and linear de-excitation (L) operators. The choice of constraint (or gauge) operators for the modal time evolution is discussed. In the case of untruncated cluster expansion, the result is independent of this choice, but restricting the excitation space removes this invariance; equations for the variational determination of the constraint operators are derived for the latter case. We show that all single-excitation parts of T and L are redundant and can be left out in the case of variationally determined constraint-operator evolution. Based on a pilot implementation, test computations on Henon-Heiles model systems, the water molecule, and a reduced-dimensionality bi-thiophene model are presented, showing highly encouraging results for TDMVCC. It is demonstrated how TDMVCC in the limit of a complete cluster expansion becomes equivalent to multiconfiguration time-dependent Hartree for the same active-space size. Similarly, it is discussed how TDMVCC generally gives better and more stable results than its time-independent-modals counterpart, while equivalent results are obtained for complete expansions and full one-mode basis sets.
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.
ABSTRACT In recent decades, coupled cluster theory has proven valuable in accurately describing correlation in many‐body systems, particularly in time‐independent computations of molecular electronic structure and vibrations. This review describes recent advancements in using coupled cluster parameterizations for time‐dependent wave functions for the efficient computation of the quantum dynamics associated with the motion of nuclei. It covers time‐dependent vibrational coupled cluster (TDVCC) and time‐dependent modal vibrational coupled cluster (TDMVCC), which employ static and adaptive basis sets, respectively. We discuss the theoretical foundation, including many‐mode second quantization, bivariational principles, and various parameterizations of time‐dependent bases. Additionally, we highlight key features that make TDMVCC promising for future quantum dynamical simulations. These features include fast configuration‐space convergence, the use of a compact adaptive basis set, and the possibility of efficient implementations with a computational cost that scales only polynomially with system size.
Recently, [JCP 161, 114114 (2024)] successfully derived numerically stable and converging (to the time-dependent complete active space self-consistent-field) time-dependent orthogonal coupled cluster (CC) methods, namely, time-dependent orbital optimized CC (TD-ooCC) methods for fermion-mixtures with arbitrary fermion kinds and numbers, via the time-dependent bivariational principle and setting constraints in the ansatz. In this work, we extend the TD-ooCC methods to eighteen non-bivariational time-dependent orbital-quasi-optimized coupled-cluster (TD-oqoCC-NB) methods. In TD-oqoCC-NB methods, the constraints in the ansatz are also used to avoid overparameterization in the complete active space limit and eliminate the numerical instability in the finite truncation case. The equations of motion are not determined by the time-dependent bivariational principle. Instead, all equations ensuring the convergence to the complete active space limit can be selected flexibly. The applications to electron dynamics and vibrational dynamics are also discussed. In particular, we find that all NB methods violate the Ehrenfest theorem even when they are gauge-invariant, which distinguishes them from TD-ooCC methods. The stationary condition of EOMs of each NB methods also provides novel CC methods for time-independent electronic structure calculations.
A dynamical formulation of coupled cluster theory is derived using a variational principle. By allowing time-dependent single-particle functions, a high degree of adaptivity is introduced, allowing complex systems to be simulated with high accuracy. Equations of motion are derived which are shown to be suitable for computer implementation. The method, called adaptive time-dependent coupled cluster, is a strict generalization of the formulation used in standard coupled cluster response theory, and also represents a systematic hierarchy of size-consistent approximations, including standard time-dependent Hartree--Fock as a trivial case.
We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration interaction wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules.
We apply the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X) to systems of bosons or fermions in lattices described by Hubbard-type Hamiltonians with long-range or short-range interparticle interactions. The wave function is expanded in a variationally optimized time-dependent many-body basis generated by a set of effective creation operators that are related to the original particle creation operators by a time-dependent unitary transform. We use the time-dependent variational principle for the coefficients of this transform and the expansion coefficients of the wave function expressed in the time-dependent many-body basis as variational parameters to derive equations of motion. The convergence of MCTDH-X is shown by comparing its results to the exact diagonalization of one-, two-, and three-dimensional lattices filled with bosons with contact interactions. We use MCTDH-X to study the buildup of correlations in the long-time splitting dynamics of a Bose-Einstein condensate loaded into a large two-dimensional lattice subject to a barrier that is ramped up in the center. We find that the system is split into two parts with emergent time-dependent correlations that depend on the ramping time---for most barrier-raising times the system becomes twofold fragmented, but for some of the very fast ramps, the system shows revivals of coherence.