Abstract
The simulation of open quantum dynamics is a critical tool for understanding how the nonclassical 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 nonMarkovian coupling of ‘environmental’ molecular vibrations to the electronic ‘system’ degrees of freedom. Here, we present an advanced but general computational strategy that allows tensor network methods to effectively compute the nonperturbative, realtime dynamics of exponentially large vibronic wave functions of real molecules. We demonstrate how ab initio modelling, machine learning and entanglement analysis can enable simulations which provide realtime insight and direct visualisation of dissipative photophysics, and illustrate this with an example based on the ultrafast process known as singlet fission.
Introduction
Ultrafast open quantum dynamics have recently attracted considerable attention in the context of molecular lightharvesting materials, where experimental evidence has begun to highlight the role of coherent and nonequilibrium vibronic dynamics in systems ranging from organic photovoltaics to the pigment–protein complexes of photosynthesis^{1,2}. Exploiting this complex interplay of vibrational and electronic quantum dynamics in tailored materials has been suggested to offer an exciting route towards advanced, nextgeneration lightharvesting devices^{3,4}, and the process known as singlet fission (SF) has recently been shown to be a very promising area in which to explore this. In this article, we present a powerful and general numerical approach designed to capture the full complexity of common molecular open system models, using the example of singlet fission to show how manybody simulation techniques can be combined with ab initio methods to elucidate and visualise the realtime sequence of system–environment processes that drive efficient lightharvesting processes.
In organic semiconductors, such as pentacene, a photogenerated singlet exciton undergoes singlet fission into a pair of entangled triplet excitons on sub100 fs time scales, efficiently generating two electronic excitations from a single incoming photon^{5,6}. Harnessing this carrier multiplication could help to overcome the Shockley–Queisser limit in photovoltaics^{7}, and understanding the links between the dynamics and efficiency of SF has emerged as an active interdisciplinary field of research^{8,9,10,11}. Recently, ultrafast optical spectroscopy experiments have revealed how nonequilibrium and nonperturbative open quantum dynamics contribute to the kinetics of SF, highlighting the role of the molecular vibrational environment in driving ultrafast SF through rapid energy relaxation, conical intersections and vibronic mixing effects^{12,13,14,15,16}. Importantly, these results exemplify the emerging idea that the evolving, highdimensional quantum states of molecular environments can become quantum mechanically entangled with the fate of electronic photoexcitations, and that these correlations may transiently change electronic properties, such as spatial coherence and coupling to other parts of the environment or external probes^{1,3,4}. These nonMarkovian ‘memory’ effects could open novel pathways for controlling or optimising lightharvesting processes, but they arise from manybody physics which can only be explored with numerical techniques that move substantially beyond standard ‘heat bath’ treatments of the environment^{6,17}.
However, obtaining accurate realtime information about the environmental quantum state in open quantum systems is a major theoretical challenge, as it requires (1) the inclusion of a large number of vibrational modes or bath degrees of freedom to capture the effective irreversiblity of dissipative dynamics and (2) the simulation of vibronic wave functions whose sizes diverge exponentially in the number of vibrational modes. A powerful approach to this from chemical physics is the Multilayer MultiConfigurational TimeDependent Hartree (MLMCTDH) algorithm^{11,18,19,20}, but, here, we draw upon the deep insights into manybody wave functions and the theory of lowrank tensor approximations that underlie the highly efficient matrixproduct state (MPS) and tensor network state (TNS) ansatz used for strongly correlated problems in condensed matter^{21,22,23,24,25,26}. Encouragingly, MPS ansätze has already been shown to provide highly accurate results for toy models of open systems with hundreds of quantised and highly excited vibrations^{27,28,29,30,31,32,33}, and have also very recently been applied to the equations of motion of reduced density matrices^{34,35}. Unfortunately, MPS techniques are mostly effective for 1D systems, placing strong constraints on the types of the environment that can be simulated (vide infra). Molecular lightharvesting systems are characterised by much more structured vibronic couplings, and in order to profit from the availability of ab initio parametrizations of molecular vibronic Hamiltonians, a higherdimensional MPSlike method is required. While extensions to higher dimensions have proved challenging in condensed matter settings, we will demonstrate here that many molecular open system models, especially those with defined symmetry properties, can be efficiently simulated with higherdimensional methods.
To acheive this, we introduce here a computational strategy based on tree tensor network states (TTNS), and give a stepbystep demonstration of how we can manipulate ab initio data with machinelearning techniques to formulate the problem in a form suitable for TTNS simulation. To evolve the resulting TTNS representation of the full manybody wave function, we employ a recent variational principle that allows fast and efficient processing^{26,31,36}, and—in a critical step—we show how an analysis of the entanglement ‘topology’ of the tensor network allows for a ‘rewiring’ of the network that can compress the memory requirements by up to six orders of magnitude. We shall then show that the formalism of TTNS enables a completely general ‘on the fly’ method to discover the dominant manybody/multidimensional configurations of the environment that drive open dynamics, and how this allows us to visualise nonequilibrium processes on just a few lowdimensional environmental energy surfaces. With this new capability to identify and distinguish effective ‘reaction coordinates’ according to their evolving relevance, we pinpoint the sequence of ultrafast vibrational motions that drive singlet fission in an ab initio parametreised dimer molecule known to undergo efficient SF, and highlight the potential utility of our technique for unravelling complex nonMarkovian dynamics in a wide range of open quantum systems across physics and chemistry.
Results
Electronic structure and vibronic model
We demonstrate our integrated computational strategy on an ab initio DFTparameterised model of 13,13′bis(mesityl)6,6′dipentacenyl (DPMes, Fig. 1a), a pentacenebased dimer molecule in which quantitative SF has been observed on ultrafast (subps) time scales, despite SF being symmetry forbidden at its groundstate geometry^{37,38}. Ab initio TDDFT calculations have been performed on DPMes to derive a microscopic Hamiltonian model spanned by five electronic diabatic states, henceforth denoted as the ‘system’, which are directly relevant to SF at the ‘cruciform’ geometry of its ground state (see Fig. 1a and Supplementary Note 1). These states are the (anti)symmetrised local exciton states LE^{±}, the (anti)symmetrised chargetransfer states CT^{±} and the final tripletpair TT state. The LE^{±} and CT^{±} states are coherently delocalised over the two monomer units, leading to an optically bright LE^{+} and dark LE^{−} state (Jdimer)^{37,38}. The spontaneous generation of pairs of triplet excitons results from the decay of the optically excited LE^{+} bright state to the lowerenergy TT states. We find excitedstate energies of 2.07 eV (LE^{+}), 2.20 eV (LE^{−}), 2.75 eV (CT^{±}) and 1.83 eV (TT), which define the diagonal system Hamiltonian H_{S}. This makes for an ideal system for studying the role of vibronic quantum dynamics in SF, as, crucially, the molecular symmetry of the groundstate 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 coupling between LE^{+} and TT, as the matrix element for this twoelectron process is almost always very small^{5,6}. Instead, SF is indirectly mediated by the higherlying CT states via superexchange^{5,6}, which is why they must be included in both the ab initio and dynamical simulations, even if they are never populated^{6}. As couplings between states arise exclusively from vibronic effects, superexchange must also be accompanied by the creation and destruction of virtual vibrational quanta in modes that link the LE, TT and CT manifolds, which we term vibronic superexchange (VSEX). Figure 1b shows some of the potential pathways and modes involved in the VSEX for DPMes, showing that different modes may participate in this secondorder process to yield both coherent couplings (same mode mediates VSEX) and creation/emission of real quanta (two distinct modes mediate VSEX).
To explore these effects, we then employ further DFT calculations to construct a linear vibronic Hamiltonian^{39}
based on 252 quantum harmonic normal modes \(\hat b_n^\dagger\) of the dimer’s groundstate potential. This describes the emerging interstate electronic couplings W_{n}, as the system is perturbed from the groundstate geometry (see Supplementary Note 1). Note that the number of modes considered the model, which is slightly smaller than the total number of modes of the dimer structure shown in Fig. 1a (318). This is due to the removal of lowfrequency modes (< 50 cm^{−1}) for which the harmonic approximation is inappropriate, and also the exclusion of the weakly coupled, very highfrequency modes associated with the pendant side group atoms. We also found that our results showed only small quantitative changes if we reduced our model to just the (75) modes that account for 90% of the system–bath coupling in each environment cluster (see below), although this reduction does not lead to any advantages in terms of simulation speeds or memory, as shown and discussed in Supplementary Note 7.
Although for clarity we illustrate our technique with a particular example, our method is designed to deal with linear vibronic models for fairly arbitrary parameter sets. This greatly extends the domain of applications for our method, and the general form of Hamiltonian in Eq. (1) has been applied to systems as diverse as photosynthetic proteins and the strongly coupled electron–phonon–photon systems found in semiconductor heterostructures^{28,40,41,42,43}. The model also remains unsolved and an important testing ground for new theory and computational techniques^{44}, and we also note that its rich phenomenology has recently been the focus of experimental quantum simulations using superconducting and ion trap qubit architectures^{45,46}.
Interface to TTNS
Following the intuition developed in MPS approaches to model systems, we then perform a unitary transformation to cast \(\hat H_{{\mathrm{VIB}}}\) into an equivalent system of independent 1Dchain environments that interact with the system through just one site of the chain (Fig. 1c). This orthogonal polynomial chain transformation^{27,30,31} directly exposes the lowentanglement properties of the environments, i.e., their effective 1D and nearestneighbour coupling structure, which are essential for their efficient simulation with tensor wave functions (vide infra). However, this type of transformation can only be applied to modes that, up to a scaling factor, couple to the system in essentially the same way, so that their couplings can be factorised as \(H_I = W{\sum} \lambda _n(a_n + a_n^\dagger )\). While this is often the case in toy models or solidstate systems (allowing the entire environment to be lumped into a single 1D chain), the molecular ab initio couplings W_{n} generally show significant ndependent variations that grow with the irregularity and size of the molecular structure.
In order to reduce the computational complexity as much as possible, we introduce a clustering operation over the set of W matrices. Defining a ‘coupling pattern’ \(\overline W _n\) such that \(W_n = \lambda _n\overline W _n\), we group modes with similar \(\overline W _n\) by employing unsupervised machine learning in the form of Kmeans clustering^{47}. This determines the minimal required set of subenvironments i and their coupling operators \(\overline W _i\) that optimally approximate \(\overline W _n\) of assigned modes, giving the best decomposition of the systemmode coupling of the form \(H_I = \mathop {\sum}\nolimits_i \overline W _i\mathop {\sum}\nolimits_n \lambda _{in}(a_{in} + a_{in}^\dagger )\), where the index n runs over the total number of modes in cluster i. The clustering algorithm shows that the DPMes couplings require a minimum of four clusters, which is perhaps not so surprising; by group theory, modes should naturally break up into clusters labelled by each 1D irreducible representation (irrep) A_{1}, A_{2}, B_{1} and B_{2} of the approximate point group D_{2d} of the DPMes groundstate geometry (see Fig. 1a). This validates our more general clustering approach, suggesting the particular efficiency of clustering for systems with welldefined symmetries. However, there can also be large variances of the coupling patterns around the centroid \(\overline W _i\) within a given symmetry class. For DPMes, the clustering algorithm determines that seven clusters are needed for a faithful representation of the original ab initio model, which can be seen clearly in the visualisation of the coupling clusters presented in Supplementary Figure 3. To the best of our knowledge, Kmeans clustering has never before been applied to open quantum system problems, and exploring the formal aspects of this type of analysis may generate interesting new insights into efficient representations of system–bath Hamiltonians. When referring to individual cluster environments, we use the symmetry label first, then an additional index to label each subdivision within the symmetry class. For example, the two subclusters of A_{1} symmetry modes are denoted as A_{1,1} and A_{12}.
The transformation of each cluster into a chain \(\hat H_{{\mathrm{c}},i}\) can now be performed, yielding the starlike Hamiltonian (see Fig. 1c)
where \(\lambda _i = (\lambda _{i_1}, \ldots ,\lambda _{i_n})\), and \(a_{i,0}^\dagger\) is termed the reaction coordinate (RC) of chain i, which directly reflects collective motion and time scales of each independent environment. The matrices \(\overline W _i\) and parameters of our custer model are given in the Supplementary Note 2, and an overview of the processing steps is shown in Fig. 2.
Tensor networks for vibronic quantum dynamics
The critical challenge for simulating the complete wave function dynamics of Eq. 2 is its sheer size: each vibrational mode must be described by a Fock state basis of n_{b} > 100 states, leading to a manybody state formally described by \(\approx n_b^{252}\) amplitudes. This is a generic problem, the socalled curse of dimensionality. However, for the shortrange Hamiltonian of Eq. (2), most of this exponentially large Hilbert space is never explored^{48}. This realisation motivates us to apply a tree tensor network state (TTNS) ansatz^{49,50}, which is a powerful lowrank tensor approximation scheme that decomposes the wave function to yield a connected network of small, individually updatable tensors, as sketched in Fig. 3a. We note that very important parallel develoments in the theory of lowrank approximations and tensor decompositions have also been made in the applied mathematics community, with work on Hierarchical Tucker formats and Tensor Trains (formally equivalent to MPS) being of particular relevance for the representation of large quantum states^{25,26,51,52}. Full and formal details of our wave function decomposition are given in Supplementary Note 3, and references within, so here we shall just give a sketch of the concept.
The exponentially large tensor (multidimensional array) C that stores all of the configuration amplitudes is decomposed into a product of smaller tensors, one for each of the vibrational modes and the electronic system. The ordering of the product of tensors normally follows the topology of the underlying Hamiltonian, and is easy to define for 1D nearestneighbour systems. The Hamiltonian we simulate is a collection of chains joined at one site, hence the ordering of our tensors takes a branching treelike form (Fig. 3). For the vibrational modes, the tensors take the form A_{ijk} where the k index runs over the n_{b} basis states (the ‘physical’ index). For a specified physical state (fixed k), the two other indices (dimensions) describe a matrix (as in matrix product states^{21,22,29,30,31,33,53}), and the contraction over these indices when the matrices of neighbouring vibrational modes are multiplied together yields the amplitude of a configuration (Fig. 3a). Thus, for a fixed bond dimension that is sufficiently large to capture the entanglement appearing over the dynamics, this type of ansatz effectively creates a linearscaling problem in system size. However, this becomes a higherorder polynomial when including the overheads for tensor contractions, the sequential updating of each tensor and the extraction of observable quantities when simulating the dynamics^{21,22,23}.
A central parameter is the size of these matrices, quantified by the ‘bond dimension’ D. The degree of correlation that can be described by a tensor network ansatz is set by D^{21,22,23}, with D = 1 corresponding to a simple product state (mean field) ansatz. We find that D values in the range of 10–100 are needed for converged results, indicating the entangled nature of the dynamics of our model. The chain tensors A are relatively ‘cheap’ to simulate (each is of size n_{b}D^{2}), which is the essential motivation for the chain transformation. However, the central tensor which represents the electronic system (S in Fig. 3a), suffers a curse of dimensionality, as it is connected to all of the chains and scales as D^{7} in our fission example. Although clustering and chain transformations reduce its Nneighbours from 252 in \(\hat H_{{\mathrm{LVC}}}\) down to 7 in \(\hat H_{{\mathrm{Star}}}\), the central tensor still requires 57 TB of memory.
Remarkably, we can compress the tensor down to 40 MB by further decomposing it into a loopfree tree network of smaller auxiliary entanglement renormalisation (ER) tensors^{23,54} (see Fig. 3b). Unlike the tensors associated to the physical vibrations and electronic system, these ER tensors are a decomposition of the system tensor purely in terms of the correlations that exist between it and the environments. Consequently, they do not carry physical indices, and the number and connectivity of these objects is chosen to exploit interenvironment entanglement (blue lines in Fig. 3) to minimise the von Neumann entropy S = −Tr(ρ ln ρ) across all bonds. In the example shown in Fig. 3b, the sum of the entropies of the reduced states of chain A_{11} and A_{12} is seen to be greater than the entropy of their joint reduced state, suggesting that they act on the system in a correlated or collective way. This further suggests that we can apply a type of isometry (renormalisation), which is implemented by connecting them together by an auxillary tensor that then transfers a reduced number of effective correlated states into the network where they will eventually act on the system. This general procedure maximises the achievable accuracy at fixed D, reducing the total complexity of any tensor in the network to \({\mathrm{O}}(D^3)\). In our SF example, this decreases the amount of memory required by six orders of magnitude, which is the key enabler for applying TTNS to N > 3 environment models. We note that while entanglement renormalisation methods have been proposed before in condensed matter settings^{23,54}, the open system problem appears to be particularly wellsuited for their application. This is due to the fact that very strong correlations are ‘concentrated’ around the electronic system and can be decomposed with just a few layers of ER tensors after which the systems can be described with simple Atype matrix tensors for the independent (noninteracting) chains. Full details and further discussion are given in Supplementary Note 3. Finally, our TTNS is timeevolved with a recent timedependent variational principle (TDVP) that has been especially adapted to handle the topology and mixed tensor composition (physical, ER, isometries) of our wave function^{31,36,55,56}. TDVP has proven to be faster than competing algorithms^{31,56,57}, and allows simulations to run for longer times without rapid growth of bond dimensions^{58}. The essential feature of this approach is that each tensor is sequentially timeevolved with a local effective Hamiltonian H^{eff}(t) that is constructed at each time step from the full manybody state (see Supplementary Note 3). This is not only a very efficient numerical strategy for evolving the state; we shall also show that the H_{eff}(t) acting on the system tensor also contains valuable information that can be used to visualise multidimensional dynamics.
We now present numerical results for singlet fission in DPMes. All simulations are performed at zero temperature, with no environmental excitations, and start initially from the optically bright LE^{+}.
Fission dynamics in DPMes
Figure 4 presents numerically exact electronic diabatic population dynamics (a) and the environmental occupation of the RCs across different orders of magnitude (b, c). Vibrationally mediated SF happens on a time scale of 200 fs (Fig. 4c), with a final TT yield of ≈90%, comparable with experiments where yields around 94% and time scales of 400–700 fs have been reported^{37}. Considering typical instrument response times (100 fs), the limitations of TDDFT in vacuum and the fact that experiments were performed in solutions, this result appears entirely reasonable. The dynamics are multiscale; at early times < 20 fs they are dominated by ultrafast damped Rabi oscillations (LE^{+}\(\underleftrightarrow {B_1}\) CT^{+}, LE^{+}\(\underleftrightarrow {B_2}\)LE^{−}, see Fig. 4a), with significant population transfer between LE^{+} and LE^{−} accompanied by rapid oscillatory excitation of the RCs of the B_{2} environments that couples these states (see Supplementary Figure 10). By Rabi oscillation, we make the analogy to the Rabi model of a quantum photon mode that drives an atomic transition between two levels, with the quantised molecular vibrations playing the role of photon. The oscillations arise from reversible exchange of quanta between the transition and field, effectively resulting in the emergence of entangled vibronic (dressed) states. From another point of view, what is being resolved here in the diabatic basis could also be seen as the realtime formation and early motion of the adiabatic states from LE^{+} along the couplingmode coordinates of B_{1} and B_{2}. The entanglement generated by the formation of these states can also be seen in rapid (<20 fs) rise of the von Neumann entropy of the reduced electronic state (see Supplementary Figure 9). This is interesting, as the time scales of this evolution just fall within the time resolution of ultrafast spectroscopies, suggesting these non Born–Oppenheimer dynamics might indeed be observable.
The Rabi oscillations of CT^{+} also correlate with the initial increase of TT population dρ_{TT}TT/dt indicating a small coherent mixing of TT character into the photoecxited state via superexchange path I mediated by the A_{2} modes (Fig. 5). However, the more important mixing is that with LE^{−}, which we now show opens the dominant superexchange pathway (2) that drives efficient fission from the initial photoexcitation^{12,59}. After > 50 fs, the population of the TT state quickly begins to rise, mirroring the decay of the hybridised LE^{+} and LE^{−} states, and accompanied by strong excitation of the B_{1} RCs. The B_{1} vibrations only couple LE^{−}, TT and CT^{−}, and thus the mixing of LE^{−} character into the initial state opens the vibronic superexchange channel, labelled path 2 in Fig. 1. As expected from superexchange, the CT^{−} state acts only as a virtual mediator for the effective vibronic coupling between LE^{+} and TT; it is never appreciably populated, as seen in Fig. 4a. The appearance of a strong imaginary coherence Im(ρ_{TT},_{LE−}) in the reduced density matrix of the electronic system indicates coherent transport between these states, which is only possible because the same environment (B_{1}) mediates both couplings that lead to the superexchange interaction (no real vibrational quanta are emitted in the process). Figure 5 shows that the dynamics of this coherence also matches the population current in the TT state dρ_{TT}/dt, providing strong evidence that the B_{1} superexchange drives SF in DPMes.
The spontaneously generated coherence between TT and LE^{−} is the result of a higherorder vibronic interaction (superexchange), and might give a characteristic signature in twodimensional Fourier transform spectroscopies^{12}. We further note that after a rapid rise in entropy, the formation of TT does not purify the reduced electronic state, i.e., the electronic and environmental degrees of freedom are entangled (cannot be factorised). This cannot be explained by a simple relaxation process from a highenergy to lowenergy state with the incoherent emission of vibrational quanta (at T = 0K); we shall indeed demonstrate that the ultrafast coherent mechanism described above can also be seen as a type of adiabatic avoided crossing phenomenon.
Finally in this section, we note the following observation of relevance for future experiments. The residual population (≈10%) in the higherlying diabatic states shows that we reach a bound TT state which is lower in energy than a T+T (or nonrelaxed TT state) at long times, and has a weakly mixed character^{60,61,62}. In particular, the nonvanishing LE^{±} character indicates that the equilibrium ‘TT’ state can exhibit singlet character in its excitedstate absorption^{37}, and the TT population is modulated over the dynamics by the fast A_{1,2} tuning modes, which could be observable in ultrafast vibrational spectroscopy^{12,13}.
Dynamic energy surfaces
The preceding section demonstrated how we could use standard observables of both the system and environment to unravel the superexchange pathway and various modes that contribute to fission. Now, we introduce another powerful way of visualing and analysing the dynamics based on energy surfaces. We do this by calculating dynamical adiabatic potential energy surfaces (PES) and also total energy surfaces (TES), which include the nonadiabatic nuclear kinetic energy operator. A major difficulty of visualising energy surfaces for N > 2 vibrational modes is the multidimensional nature of the dynamics; trying to visualise dynamics—where possible—requires physical intuition, or some other prior knowledge, to identify a sufficiently small number of ‘reaction coordinate’ modes that are relevant for the process under study. Here, we exploit the fact that our pure wave function can always be Schmidtdecomposed into a system–bath bipartition of the form \(\psi (t)\rangle = \mathop {\sum}\nolimits_{i = 1}^5 \lambda _ii_{\mathrm{S}}(t)\rangle \phi _i(t)\rangle\), where the system states i_{S}(t)〉 and manybody environmental states ϕ_{i}(t)〉 are mutually orthogonal in the separate system and environment Hilbert spaces and λ_{i}(t)^{2} is the probability of finding the environment in the ith state at time t. Obtaining this decomposition is particularly easy in a tensor network ansatz, and follows from applying a singular value decomposition (SVD) of the system tensor. The SVD is actually used throughout the algorithm to update and normalise the state, so access to this information is essentially ‘free’. We then calculate the energy surfaces at each time point as the eigenvalues of the effective Hamiltonian H^{eff}(t) and potential V^{eff}(t) of the system states generated by the bath configurations ϕ_{i}(t)〉
where i, k are index bath states and j, l denote the electronic states. In our SF example, the effective Hamiltonian \(H^{{\mathrm{eff}}}(t) \in {\Bbb C}^{25 \times 25}\) leads to 25 surfaces (five electronic states lead to five distinct environment configurations). However, it is important to note that these surfaces are effectively an energy spectrum parameterised just by time and therefore easy to visualise. The exact dynamics provide us with the dominant environment configurations at each point in time, enabling us to simply follow these motions without reference to any paticular molecular coordinate or normal mode basis. With some further manipulation (see Supplementary Note 5), we can now look into the realtime energetics of the dynamics (Fig. 6a, b) by showing the occupations and diabatic character of the excitations on these surfaces. In Figure 6a, b, we represent energy surfaces as lines, the overall population of each surface by the width of a fill around the line and the diabatic composition of these occupations by the colours of the fills.
Returning to singlet fission, we see that the dynamics in both the PES and TES are confined to approximately two lowenergy surfaces. The SF dynamics in the adiabatic PES (Fig. 6a) are evident as the population transfers from the lowest LE^{+} surface to the lowest TT surface. Since the population transfers between noncrossing surfaces, this displays the nonadiabatic nature of the dynamics. Additionally, the adiabatic PES of the populated TT (blue) stays consistently below the LE^{+} PES (red, Fig. 6a), thus SF in DPMes should not be caused by a conical intersection as proposed for other SF systems^{13,15}. However, the actual sequence of processes leading to fission is best resolved in the nonadiabatic TES (Fig. 6b). Here, SF occurs by an avoided crossing appearing at 80 fs (black arrow) which is caused by three key ‘events’: (1) the unoccupied TT TES is pushed above the LE^{+} TES at 20 fs (red arrow) by A_{12} and B_{22} motions; (2) the TT is increasingly coupled to the LE^{+} by superexchange caused by growing B_{1} displacements; (3), the same B_{1} motion causes level repulsion with the highlying CT states (see Fig. 6a) and a slow descent of the TT surface towards that of the occupied LE^{+} state. The combination of (2) and (3) results in an avoided crossing with a splitting of ≈11 meV at 80 fs, in which the population on the LE^{+} surface evolves adiabatically (in the Landau–Zener sense) into a 95% TT population. Indeed, estimates based on the Landau–Zener formula for the diabatic transition probability predict a coupling of 10 meV, which is encouragingly close to our result (see Supplementary Note 10).
This last picture can be further complemented by looking at the dynamical time scales associated with the reaction coordinates mediating these events: in (1), the energy increase of the (unoccupied) TT TES at 20 fs is caused by the response of the collective modes \(\omega _{RC,A_{12}} \approx 1377\,{\mathrm{cm}}^{  1}\sim 24\,{\mathrm{fs}}\) and \(\omega _{RC,B_2} \approx 1100\,{\mathrm{cm}}^{  1}\sim 30\,{\mathrm{fs}}\) to the sudden occupation of the LE^{+} state; in (2), the effective vibronic coupling emerges from displacement of B_{1} with \(\omega _{B_1} \approx 400\,{\mathrm{cm}}^{  1}\sim 83\,{\mathrm{fs}}\), setting the longer time scale for SF (Fig. 4a–c); in (3), the slow reorganisation of B_{1} increases the effective TTLE^{+} coupling from 3 meV at 20 fs to 6 meV at 80 fs, as seen from the avoided crossing gaps. We therefore see that no population is transfered when the TT surface initially rises through the LE^{+} surface, as this is caused by A_{1} mode displacements, which respond much faster than the B_{1} modes. Consequently, the superexchange coupling between surfaces is small at the first crossing (small B_{1} displacment) and transfers little population due to the very fast crossing (nonadiabatic, in the LZ sense). However, the initial motion of the TT surface above the LE^{+} surface is crucial for the subsequent descent to the avoided crossing, so both A_{1} and B_{1} modes play important, albeit different, roles in SF in DPMes.
Remarks on simulation performance
For the fission results presented above, convergence was obtained with 100 bosonic Fock states per vibrational mode. The dynamics of all observables converged to ≈1% for a maximum TTNS bond dimension D between ERnodes of D_{Node,max} = 80, while along the chains D_{chain,max} = 30 (see Supplementary Figure 5). Thus we cover a Hilbert space of 10^{500} states using only 10^{7} parameters. Moreover, the typical simulation times for these parameters were ~96 h on a single Intel Xeon E52670 core for a trajectory of a picosecond, which also included the computation of all observables and the energy surface visualisations that have not been previously reported in other methods. A complete set of convergence data w.r.t. bond dimensions and CPU times can be found in Supplementary Note 4. As a systemspecific model of a particular molecule, we are not able to compare this numerical performance with another method directly, but on the basis of published results on the spinboson model (SBM)^{19,20,28,30,31}, we believe that our algorithm should be broadly comparable with MLMCTDH in terms of resources and speed, although SBM simulations with over 10 K modes have been demonstrated with MLMCTDH. However, we note that the procedure of creating ‘balanced trees’ for efficient MLMCTDH calculations appears, for the moment, to be a matter of trial and error, as discussed at length in the context of singlet fission by Zheng et al.^{11}. Generically, efficient tree constructions are discussed in greater depth in a recent review by Wang^{20}. Our entanglement (von Neumann entropy) analysis of the core tensor and the construction of an optimal threelegged network using ER nodes is based on a clear mathematical framework and might also be applicable to MLMCTDH problems. Compared with MLMCTDH and the present method, the recent MPObased TEMPO algorithm of Strathearn et al. appears to require slightly longer computational time and more computational resources for the Ohmic spinboson model at zero temperature and α;= 0.5^{34}, as do techniques based on hierarchical equations of motion^{35}. However, these latter methods are also efficient in treating the effects of finite temperatures and can reach long times without recurrences, which make them very attractive for looking at experimentally relevant reduced system dynamics.
We finally remark that we have also explored convergence w.r.t. the numbers of modes retained in the clusters (see Supplementary Note 4). We found that a reduction of the number of modes to just 75 still gave quantitatively similar dynamics, although the computational time and resources needed for the simulation did not greatly decrease. We previously mentioned that each 1D environment is relatively ‘cheap’ to extend, but we believe that the ability to shrink the environments dramatically is related to the dominance of corrrelations between the system, the RC modes and a relatively small number of nearby oscillators in each chain on the physics. Indeed, we have also explored the effect of deliberately altering the parameters of the chains to yield RC modes that are damped by a simpler, extended ‘outer’ environment (490 modes). This leads to fairly good qualitative agreement with the full ab initio model, and could provide an interesting way to benchmark a range of theories based on simpler RC master equations^{63,64,65}.
Discussion
The results above demonstrate that manybody tensor network methods and entanglement analysis can provide a powerful bridge between ab initio electronic struture and realtime open quantum dynamics, and provide a truly microscopic elucidation of mechanisms underlying ultrafast lightharvesting processes. Moreover, as a general computational strategy, our technique should also be applicable to a wide range of complex nonMarkovian systems in physics and chemistry, and may allow similar insights into timely problems where the boundary between systems and environments may become a subtle point, such as fewbody quantum thermodynamics and nonequilibirum phenomena in manybody systems. Indeed, from an open system perspective, the coordinated sequences of processes seen in SF—each involving different types of environment and the emergence of different types of interactions between the electronic states—is a good example of nonMarkovianity in action.
For lightharvesting systems, a number of further developments also appear possible at this point, including the possibility of reversing several of the transformations we perform to extract information about the original molecular normal modes that drive the dynamics. With this information, physical reaction coordinates could be identified and targeted in future experiments. An intriguing idea is that lightharvesting processes might then be controlled via these motions, either through quantum control protocols or strong light–matter (polaritonic) interactions^{66,67}. In both cases, access to detailed and accurate dynamics is essential, and we further note that—as the singlecore computational costs of our technique allow easy parallelisation—potentially highthroughput simulations could be implemented for dynamicsbased screening of such SF materials. We therefore hope that the approach we have set out could provide a useful tool in the emerging drive towards dissipative coherent devices constructed from rationally designed, nanostructured organic materials^{1,2,3,4}.
Methods
DFT
All DFT calculations were performed with the NWChem electronic structure code in vacuum^{68}. The groundstate structure and vibrational modes were obtained employing analytical Hessians at the ccPVDZ/B3LYP level of theory. Excitedstate energies and forces (corresponding to the diagonal elements of W) for LE^{±} and CT^{±} were calculated from (linearresponse) TDDFT gradients at the ccPVDZ/LCBLYP level of theory. The longrange corrected functional is required to correctly describe the S_{1} state of pentacene as well as the states with chargetransfer character. An optimised rangeseparation parameter μ = 0.29 was used in the LCBLYP functional. This choice gives a good description of the energy of the first excited singlet of the pentacene molecule^{69}. For TT, the quintet state (total spin 2) was used as a proxy for the purpose of calculating energy and forces. The frontier molecular orbitals at the ccPVDZ/LCBLYP level were also used as inputs for the evaluation of the offdiagonal couplings. Modes below 110 cm^{−1} and above 1500 cm^{−1} were disregarded due to either unrealistic ab initio parameters arising from the neglect of their anharmonicities or irrelevance for present timeresolved experiments, respectively. Limitations of the method are discussed in the Supplementary Note 1.
Tensor network states
To prepare the linear vibronic DPMes Hamiltonian for the TTNS simulations, the vibrational modes were clustered to independent environments by using a weighted Kmeans algorithm^{47}. These environments were then mapped onto chain Hamiltonians with the orthogonal polynomials transformation to facilitate the tensor tree network decomposition and numerical evolution of the complete vibronic wave function^{27}.
The model was simulated with a TTNS, as depicted in Supplementary Figure 4, incorporating entanglement renormalisation (ER) tensors connecting the vibrational chain environments to the excitonic states and using an optimised boson basis (OBB) to allow for a large and expandable Fock basis. The timeevolution is performed in the timedependent variational principle (TDVP)^{36} to simulate exciton–phonon dynamics at zero temperature with no initial environmental excitations at a time step of 0.33 fs. Details about the TDVP algorithm we have developed for the TTNS evolution are given in the Supplementary Note 3, and further background can be found in refs. ^{31,36,49,50,55}. Energy surfaces were calculated from the effective Hamiltonian and effective potential of the system tensor. Further computational details are provided in Supplementary Notes 3, 4 and 5.
Code availability
The custom tree tensor networks state code used for simulations is available by written request to the corresponding author.
Data availability
The data underlying this publication are available in the University of Cambridge data repository at www.repository.cam.ac.uk.
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Acknowledgements
We thank R.H. Friend for making this work possible and A. Rao for useful discussions. F.A.Y.N.S., D.H.P.T., N.D.M.H. and A.W.C. gratefully acknowledge the support of the Winton Programme for the Physics of Sustainability and the Engineering and Physical Sciences Research Council (EPSRC). A. J. M. acknowledges support from EPSRC grant EP/M025330/1.
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F.A.Y.N.S. developed the TTNS method and performed calculations and D.H.P.T. performed DFT calculations. F.A.Y.N.S., D.H.P.T. and A.W.C. interpreted results and wrote the paper. A.J.M. initiated the project and provided experimental insight into DPMes. A.J.M., N.D.M.H. and A.W.C. supervised the project. All authors discussed results and contributed to the paper.
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Schröder, F.A.Y.N., Turban, D.H.P., Musser, A.J. et al. Tensor network simulation of multienvironmental open quantum dynamics via machine learning and entanglement renormalisation. Nat Commun 10, 1062 (2019). https://doi.org/10.1038/s41467019090397
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DOI: https://doi.org/10.1038/s41467019090397
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