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Wiley Interdisciplinary Reviews Computational Molecular Science

Time‐Dependent Vibrational Coupled Cluster Theory With Static and Dynamic Basis Functions

Mads Greisen Højlund, Alberto Zoccante, Andreas Buchgraitz Jensen, Ove Christiansen A

Published Jan 1, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1002/wcms.70001

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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.
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.
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.
We advance the Coupled Cluster method’s in the time-dependent realm, by imple- menting a robust solver based on the orbital-adaptive time-dependent Coupled Cluster (OATDCC)[1] method. This involves implementing both a simplified static orbital time-dependent coupled cluster solver with single and double excitations (TDCCSD) and an orbital-adaptive scheme with double excitations (OATDCCD). To supplement the time-dependent methods we implement several ground state solvers based on the Lagrangian Coupled Cluster formulation, with single and double excitations, as well as a non-orthogonal orbital-optimised Coupled Cluster (NOCC) solver[2]. We construct several quantum dot basis sets with different potential functions in one- and two dimensions, including interactions with magnetic fields. What is more, we also implement an interface with popular quantum chemistry software modules PySCF[3] and Psi4[4] for extraction of additional basis sets for atoms and molecules. The quantum systems are allowed to vary with time by addition of a time-dependent term to the Hamiltonian, with which we simulate a laser field in the dipole approxima- tion. As a validation we reproduce results from the scientific literature, both for atoms, molecules and quantum dots. We show that our methods lead to convergence in the ever-increasing basis set size limit, for simple quantum dot systems. For the same quantum dot system, we show how sensitive a system is to changes in the frequency of a driving oscillating field. Frequencies closer to the resonant frequency lead to exctiations and increased energy. We are able to simulate systems that are fairly large - quantum dots in one- and two dimensions with up to twelve electrons. For systems that meander far from the reference state, we show that the orbital-adaptive method has far superior stability, compared with the method with static orbitals. For all quantum dot systems we find strong comformity with the harmonic potential theorem[5], yet we see a slight many-body effect for a two-dimensional double dot system. By subjecting the two-dimensional quantum dots to a homogoenous, static magnetic field in the form of an angular momentum operator, we see two frequencies in the dipole spectrum, instead of one frequency. This is also in accordance with the harmonic potential theorem. The difference between the two frequencies in the new spectrum is the same as the Larmor frequency of the magnetic field, within acceptable tolerance levels.
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.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.