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Ab intio information processing and model extraction. a The ab initio linear vibronic Hamiltonian describes how each vibration mode of frequency ω i (outer circles) couples to the system (S). The coupling matrix for each mode could be different, shown as the colour of connecting lines. b A K-means learning (clustering) algorithm groups modes according to their common coupling type (colour grouping in figure). c A unitary transform (linear combination of vibration modes) of each cluster creates an equivalent nearest-neighbour chain representation of the Hamiltonian, ideal for tensor network simulation
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The simulation of open quantum dynamics is a critical tool for understanding how the non-classical properties of matter might be functionalised in future devices. However, unlocking the enormous potential of molecular quantum processes is highly challenging due to the very strong and non-Markovian coupling of ‘environmental’ molecular vibrations to...
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... makes for an ideal system for studying the role of vibronic quantum dynamics in SF, as, crucially, the molecular symmetry of the ground-state geometry suppresses all electronic couplings between these excited states. Thus, although SF is strongly exergonic in this dimer, it is strictly forbidden unless this symmetry is broken by vibrational motion (see Supplementary Figure 2). However, as in many SF systems, even symmetry breaking cannot induce a significant direct cou- pling between LE + and TT, as the matrix element for this two- electron process is almost always very small 5,6 . ...
Context 2
... λ i ¼ ðλ i 1 ; ¼ ; λ i n Þ, and a y i;0 is termed the reaction coordinate (RC) of chain i, which directly reflects collective motion and time scales of each independent environment. The matrices W i and parameters of our custer model are given in the Supplementary Note 2, and an overview of the processing steps is shown in Fig. ...
Citations
... Spectra are either computed within the well-known cumulant method, 27 the recently introduced Gaussian Non-Condon Theory (GNCT) for describing non-Condon effects in weakly coupled adiabatic states, 25 or tensor network-based approaches for describing strongly coupled diabatic states. 24,26,[28][29][30] Crucially, the MD-based sampling of the system retains a full coupling of the electronic states to environmental degrees of freedom, without the need to invoke frozen solvent environments. Our approaches leverage graphics processing units (GPUs) at all stages, leading to a massive speed-up in computations and enabling high-throughput calculations in complex systems. ...
... In collaboration with others, 24,26 we have recently shown that finite-temperature absorption and emission lineshapes in the presence of a conical intersection can be computed by combining an MD-based sampling approach of spectral densities in complex environments with the thermalized time-evolving density operator with orthogonal polynomials algorithm (T-TEDOPA) [28][29][30]58 for numerically exact quantum dynamics simulations. Here, we briefly summarize the main features of the formalism, focusing specifically on computational tools and algorithms developed by us to apply the approach to a wide range of complex condensed-phase systems. ...
We outline a computational workflow to model linear optical spectra of molecules in complex environments in the presence of nonadiabatic effects. The approach relies on computing excitation energy and transition dipole fluctuations along molecular dynamics (MD) trajectories, treating molecular and environmental degrees of freedom on the same footing. Spectra are then generated following two distinct strategies: In the recently developed Gaussian Non-Condon Theory (GNCT), the linear response functions are computed in terms of independent adiabatic excited states, with non-Condon effects described through spectral densities of transition dipole fluctuations. For strongly coupled excited states, we instead parameterize a linear vibronic coupling (LVC) Hamiltonian directly from spectral densities of energy fluctuations and diabatic couplings computed along the MD trajectory. The optical spectrum is then calculated using powerful numerically exact tensor-network approaches. Both the electronic structure calculations to sample system fluctuations and the quantum dynamics simulations using tensor-network methods are carried out on graphics processing units (GPUs), enabling rapid calculations on complex condensed phase systems. We showcase the strengths of the workflow on a series of two-mode model systems in the presence of a conical intersection (CI), and the pyrazine molecule in different solvent environments.
... To efficiently simulate the quantum dynamics of both, coupled electronic and vibrational, as well as environmental DOFs, the decomposition of the many-body wavefunction into local tensors has proven to be an extremely fruitful approach [30][31][32][33][34][35][36][37][38] . The unitary evolution of excitonic states coupled to O(100) discrete vibrational DOFs is a remarkable milestone facilitating, for instance, the unbi-ased investigation of ultrafast photo-induced dynamics of large molecules [39][40][41] . Recently, wavefunction-based tensor network (TN) methods for open quantum-systems with both Markovian and non-Markovian environments have been developed 42-44 to describe continuous and smooth spectral densities. ...
The ultrafast quantum dynamics of photophysical processes in complex molecules is an extremely challenging computational problem with a wide variety of fascinating applications in quantum chemistry and biology. Inspired by recent developments in open quantum systems, we introduce a pure-state unraveled hybrid-bath method that describes a continuous environment via a set of discrete, effective bosonic degrees of freedom using a Markovian embedding. Our method is capable of describing both, a continuous spectral density and sharp peaks embedded into it. Thereby, we overcome the limitations of previous methods, which either capture long-time memory effects using the unitary dynamics of a set of discrete vibrational modes or use memoryless Markovian environments employing a Lindblad or Redfield master equation. We benchmark our method against two paradigmatic problems from quantum chemistry and biology. We demonstrate that compared to unitary descriptions, a significantly smaller number of bosonic modes suffices to describe the excitonic dynamics accurately, yielding a computational speed-up of nearly an order of magnitude. Furthermore, we take into account explicitly the effect of a $\delta$-peak in the spectral density of a light-harvesting complex, demonstrating the strong impact of the long-time memory of the environment on the dynamics.
... Moreover, methods based on reduced density techniques also become computationally expensive as the memory time and complexity (rank) of the system-environment interactions increase. In response, several numerical methods to simulate open quantum system (OQS) dynamics in this nonperturbative regime have been developed, such as the Time Evolving Matrix Product Operator (TEMPO) 9,10 method (implemented in the OQuPy package 11 ), the Automated Compression of Environments (ACE) al-gorithm [12][13][14] , the Hierarchical Equations Of Motions (HEOM) technique 15,16 (implemented in the QuTip package 17 ), the Multi-Layer-Multi-Configuration Time-Dependent Hartree (ML-MCTDH) method 18 (implemented in the Quantics package 19,20 ), collision models 21 , the Dissipation-Assisted Matrix Product Factorization (DAMPF) method 22 and the T-TEDOPA technique [23][24][25][26][27][28] to name a few. TEMPO, ACE, HEOM, and T-TEDOPA belong to the specific class of numerically exact methods (i.e. ...
... Several widely used model Hamiltonians are already implemented as built-in functions (see methods section of the different MPOs in the online documentation for a complete list), and we encourage users to push their particular models of interest to make this a live repository for timely open system problems. The time-evolution methods conserve the unitarity of the dynamics and are adaptive, and will also be updated to follow new theoretical advances in time-evolution methods, particularly in the case of tree tensor networks, where long-range interactions might be tackled more effectively with entanglement renormalization methods (work in progress) 27 . A straight forward implementation of DMRG-like methods to find interacting ground states will also allow dynamics arising from highly correlated/entangled systemenvironment grounds states to explored in the future, which is a major challenge for reduced density matrix approaches, due to the inherently non-Gaussian nature of the bath in such initial conditions 1 . ...
The MPSDynamics.jl package provides an easy to use interface for performing open quantum systems simulations at zero and finite temperatures. The package has been developed with the aim of studying non-Markovian open system dynamics using the state-of-the-art numerically exact Thermalized-Time Evolving Density operator with Orthonormal Polynomials Algorithm (T-TEDOPA) based on environment chain mapping. The simulations rely on a tensor network representation of the quantum states as matrix product states (MPS) and tree tensor network (TTN) states. Written in the Julia programming language, MPSDynamics.jl is a versatile open-source package providing a choice of several variants of the Time-Dependent Variational Principle (TDVP) method for time evolution (including novel bond-adaptive one-site algorithms). The package also provides strong support for the measurement of single and multi-site observables, as well as the storing and logging of data, which makes it a useful tool for the study of many-body physics. It currently handles long-range interactions, time-dependent Hamiltonians, multiple environments, bosonic and fermionic environments, and joint system-environment observables.
... In extremely complex calculations such as those based on the tensor network approach with huge basis sets, 28,30 the assumption that the FC parameters are the same for different electronic transitions is an inevitable simplification. However, this conjecture is not physically justifiable. ...
... Therefore, for analyzing the vibrational coherence, that mechanism may be viewed as a black box, as was actually done in refs 29,31,32. At this point, it is useful to consider the distinctions between the TIPS-Pent film of refs 12,27 and the dimer DPMes, 28,30 for which the direct comparison with Scheme 2 was introduced. The photophysically active system is here a single molecule. ...
... To predict the behaviour of such concerted, history dependent actions in a extended system under dissipation, we need to go beyond the usual assumptions of Markovian (i.e. memoryless) bath dynamics: the dynamics are manifestly non-Markovian 1 [34,35], implying that they are also non-perturbative and can only be simulated with state-of-the-art numerical techniques [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52]. Despite the challenge, the need to explore such transient out-of-equilibrium phenomena is highlighted by physical examples in which environment (structure)-mediated communication and feedback are thought to play a key role, such as the coordination of multiscale, multielectron processes across photosynthetic proteins, electron transfer in metabolism and multistage catalysis [32,54,55,56,57]. ...
... Other -somewhat arbitrarily chosen -realworld 1D systems where strong environmental signalling effects could be expected include entangled triplet exciton dynamics in pi-conjugated polymers and coupled quantum dot emitters grown in nanowires [33,61]. In general, but particularly in molecular matter, injecting 'system' excitations causes the local structure to relax to a new equilibrium position, and in doing so key system properties such as energy gaps or couplings to other systems can be strongly modified in the new conformation [39,62]. As the establishment of this new global conformation must proceed through the propagation of local reorganization dynamics, dramatic changes at distal locations can be effected at later times [63]. ...
... In another direction, our model could be refined in multiple ways to describe realistic (bio-)chemical systems. For instance, our present theoretical tools could be used to approach structured spectral densities found by ab initio methods [77,78], anharmonic effects [79,80], or Hamiltonian topologies that account for more complex connectivity between systems and/or environments [39]. Another perspective for this model is to provide a new set of processes to analyse the intricate physical effects happening in biological systems made of complexes of organic molecules such as allostery [81,82,56,83] or multi-electron processes [54]. ...
Nanodevices exploiting quantum effects are critically important elements of future quantum technologies (QT), but their real-world performance is strongly limited by decoherence arising from local `environmental' interactions. Compounding this, as devices become more complex, i.e. contain multiple functional units, the `local' environments begin to overlap, creating the possibility of environmentally mediated decoherence phenomena on new time-and-length scales. Such complex and inherently non-Markovian dynamics could present a challenge for scaling up QT, but – on the other hand – the ability of environments to transfer `signals' and energy might also enable sophisticated spatiotemporal coordination of inter-component processes, as is suggested to happen in biological nanomachines, like enzymes and photosynthetic proteins. Exploiting numerically exact many body methods (tensor networks) we study a fully quantum model that allows us to explore how propagating environmental dynamics can instigate and direct the evolution of spatially remote, non-interacting quantum systems. We demonstrate how energy dissipated into the environment can be remotely harvested to create transient excited/reactive states, and also identify how reorganisation triggered by system excitation can qualitatively and reversibly alter the `downstream' kinetics of a `functional' quantum system. With access to complete system-environment wave functions, we elucidate the microscopic processes underlying these phenomena, providing new insight into how they could be exploited for energy efficient quantum devices.
... TDVP-DMRG has first been used for MPSs (and later for standard MCTDH [97,98]). The first application of TDVP-DMRG to TTNSs we are aware of is Ref. [147]. For a mathematical analysis, see Ref. [148]. ...
... See Refs. [110,[147][148][149][150] for more details. 12. ...
The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalisation group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimising unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilisation of ideas between ML-MCTDH and DMRG.
... [9][10][11][12][13] Theoretically, with the development of high-level quantum chemistry theory, precise electronic structure calculations can be performed in the SF materials, and the effective model Hamiltonian can be constructed and analyzed. 74,80,[83][84][85][86][87][88][89][90][91][92][93][94][102][103][104] For instance, the relative energetics of the low-lying states of the pentacene monomer were once investigated by various electronic structure methods. [9][10][11][12][13]74 Particularly, the pentacene dimer receives considerable research attention because it provides a prototype to study the SF mechanism. ...
... 74,80,[83][84][85][86][87][88][89][90][91][92][93][94][102][103][104] For instance, the relative energetics of the low-lying states of the pentacene monomer were once investigated by various electronic structure methods. [9][10][11][12][13]74 Particularly, the pentacene dimer receives considerable research attention because it provides a prototype to study the SF mechanism. Different strategies were proposed to construct the diabatic Hamiltonian of pentacene dimers. ...
... 80 Besides, different dynamics methods, from the exact fullquantum to semi-classical ones, were used to simulate the SF reaction and explore the possibility of various mechanisms proposed. 73,74,79,83,85,86,88,92,93,103,[106][107][108][109][110][111][112][113][114] For example, several groups, including Ratner, 103,108 Teichen and Eaves, 83 Reichman, 85,86,88,107 and Grozema, 115,116 used the quantum master equation method based on the reduced density matrix to explore the intrinsic mechanism behind the SF reaction. Zhao and co-workers once employed the stochastic Schrödinger equation to study the SF dynamics. ...
Singlet fission (SF) is a very significant photophysical phenomenon and possesses potential applications. In this work, we try to give a rather detailed theoretical investigation of the SF process in the stacked polyacene dimer by combining the high-level quantum chemistry calculations and the quantum dynamics simulations based on the tensor network method. Starting with the construction of the linear vibronic coupling model, we explore the pure electronic dynamics and the vibronic dynamics in the SF processes. The role of vibrational modes in nonadiabatic dynamics is addressed. The results show that the super-exchange mechanism mediated by the charge-transfer state is found in both pure electronic dynamics and the nonadiabatic dynamics. Particularly the vibrational modes with the frequencies resonance with the adiabatic energy gap play very import roles in the SF dynamics. This work not only provides a deep and detailed understanding of the SF process but also verifies the efficiency of the tensor network method with the train structure that can serve as the reference dynamics method to explore the dynamics behaviors of complex systems.
... Other directions include Gaussian-based MCTDH variants (G-MCTDH) and other multi-configurational methods, including time-dependent Density Matrix Renormalization Group (TD-DMRG). [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] See also Ref. 40 for a review of multiple spawning methods. TDMVCC seeks to incorporate the benefits of MCTDH and the TDVCC Ansatz. ...
The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper, we report an efficient implementation of TDMVCC, covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: (i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and (ii) the use of a single function as an active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH (time-dependent Hartree) approach that can obtain even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] is illustrated for polyaromatic hydrocarbons with up to 264 modes. Finally, computations on the internal vibrational redistribution of benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.
... 32−39 However, the scientific papers combining ML and SF are intermittent. In 2019, Schroder et al. 40 used ML to explore the quantum dynamics in pentacene dimers. Later in 2021, Ma and co-workers 41 developed general transferable multilevel attention neural network for prediction of properties like the energy of HOMO and tested it with the 262 SF candidates of Troisi et al. 27 The authors reported that the prediction power of their approach is significantly decreased for the SF data set. ...
... [9][10][11][12][13] Theoretically, with the development of high-level quantum chemistry theory, the precise electronic structure calculations can be performed in the SF materials, and the effective model Hamiltonian can be constructed and analyzed. 74,80,[83][84][85][86][87][88][89][90][91][92][93][94][95][96][97] For instance, the relative energetics of the low-lying states for the pentacene monomer were once investigated by various electroinc structure methods. [9][10][11][12][13] Particularly, the pentacene dimer receives considerable research attention, because it provides a prototype to study the SF mechanism. ...
... 93 The ultrafast nonadiabatic dynamics for the SF process were also extensively investigated by using the quantum dynamics methods that treat all involved electronic and nuclear degrees of freedom explicitly. 60,73,74,79,107 Different quantum dynamics approaches were taken, such as the multilayer multiconfigurational time-dependent Hartree method by Burghardt, Tamura and co-workers, 60 Thoss and coworkers 73 and ourselves, 107 and the tensor network by Chin and co-workers, 74 Ma and co-workers. 79 The trajectory-based method is also a practical way to treat the SF dynamics in complex system. ...
... 93 The ultrafast nonadiabatic dynamics for the SF process were also extensively investigated by using the quantum dynamics methods that treat all involved electronic and nuclear degrees of freedom explicitly. 60,73,74,79,107 Different quantum dynamics approaches were taken, such as the multilayer multiconfigurational time-dependent Hartree method by Burghardt, Tamura and co-workers, 60 Thoss and coworkers 73 and ourselves, 107 and the tensor network by Chin and co-workers, 74 Ma and co-workers. 79 The trajectory-based method is also a practical way to treat the SF dynamics in complex system. ...
Singlet fission (SF) is a very significant photophysical phenomenon and possesses potential applications. In this work, we try to give the rather detailed theoretical investigation of the SF process in the stacked polyacene dimer by combining the high-level quantum chemistry calculations, and the quantum dynamics simulations based on the tensor train decomposition method. Starting from the construction of the linear vibronic coupling model, we explore the pure electronic dynamics and the vibronic dynamics in the SF processes. The role of vibrational modes in nonadiabatic dynamics is addressed. The results show that the super-exchange mechanism mediated by the charge-transfer state is found in both pure electronic dynamics and the nonadiabatic dynamics. Particularly, the vibrational modes with the frequency resonance with the adiabatic energy gap play very import roles in the SF dynamics. This work not only provides a deep and detailed understanding of the SF process, but also verifies the efficiency of the tensor train decomposition method that can serve as the reference dynamics method to explore the dynamics behaviors of complex systems.
