Multiconfiguration Time-dependent Hartree Method Research Articles

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems.

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Quantum rates in dissipative systems with spatially varying friction.

We investigate whether making the friction spatially dependent on the reaction coordinate introduces quantum effects into the thermal reaction rates for dissipative reactions. Quantum rates are calculated using the numerically exact multi-configuration time-dependent Hartree method, as well as the approximate ring-polymer molecular dynamics (RPMD), ring-polymer instanton methods, and classical molecular dynamics. By conducting simulations across a wide range of temperatures and friction strengths, we can identify the various regimes that govern the reactive dynamics. At high temperatures, in addition to the spatial-diffusion and energy-diffusion regimes predicted by Kramer's rate theory, a (coherent) tunneling-dominated regime is identified at low friction. At low temperatures, incoherent tunneling dominates most of Kramer's curve, except at very low friction, when coherent tunneling becomes dominant. Unlike in classical mechanics, the bath's influence changes the equilibrium time-independent properties of the system, leading to a complex interplay between spatially dependent friction and nuclear quantum effects even at high temperatures. More specifically, a realistic friction profile can lead to an increase (or decrease) of the quantum (classical) rates with friction within the spatial-diffusion regime, showing that classical and quantum rates display qualitatively different behaviors. Except at very low frictions, we find that RPMD captures most of the quantum effects in the thermal reaction rates.

Quantum rates in dissipative systems with spatially varying friction.

We investigate whether making the friction spatially dependent on the reaction coordinate introduces quantum effects into the thermal reaction rates for dissipative reactions. Quantum rates are calculated using the numerically exact multi-configuration time-dependent Hartree method, as well as the approximate ring-polymer molecular dynamics (RPMD), ring-polymer instanton methods, and classical molecular dynamics. By conducting simulations across a wide range of temperatures and friction strengths, we can identify the various regimes that govern the reactive dynamics. At high temperatures, in addition to the spatial-diffusion and energy-diffusion regimes predicted by Kramer's rate theory, a (coherent) tunneling-dominated regime is identified at low friction. At low temperatures, incoherent tunneling dominates most of Kramer's curve, except at very low friction, when coherent tunneling becomes dominant. Unlike in classical mechanics, the bath's influence changes the equilibrium time-independent properties of the system, leading to a complex interplay between spatially dependent friction and nuclear quantum effects even at high temperatures. More specifically, a realistic friction profile can lead to an increase (or decrease) of the quantum (classical) rates with friction within the spatial-diffusion regime, showing that classical and quantum rates display qualitatively different behaviors. Except at very low frictions, we find that RPMD captures most of the quantum effects in the thermal reaction rates.

Using a pruned basis and a sparse collocation grid with more points than basis functions to do efficient and accurate MCTDH calculations with general potential energy surfaces.

We propose a new collocation multi-configuration time-dependent Hartree (MCTDH) method. It reduces point-set error by using more points than basis functions. Collocation makes it possible to use MCTDH with a general potential energy surface without computing any integrals. The collocation points are associated with a basis larger than the basis used to represent wavefunctions. Both bases are obtained from a direct product basis built from single-particle functions by imposing a pruning condition. The collocation points are those on a sparse grid. Heretofore, collocation MCTDH calculations with more points than basis functions have only been possible if both the collocation grid and the basis set are direct products. In this paper, we exploit a new pseudo-inverse to use both more points than basis functions and a pruned basis and grid. We demonstrate that, for a calculation of the lowest 50 vibrational states (energy levels and wavefunctions) of CH2NH, errors can be reduced by two orders of magnitude by increasing the number of points, without increasing the basis size. This is true also when unrefined time-independent points are used.

Ultrafast Excited-State Nonadiabatic Dynamics in Pt(II) Donor-Bridge-Acceptor Assemblies: A Quantum Approach for Optical Control.

The ultrafast nonadiabatic excited state dynamics of (PTZ-N-benzyl-acetylide) (trans-bis-trimethylphosphine) Pt(II) (acetylide-NDI-bis-methyl) 1, representative of a series of Pt(II) donor-bridge-acceptor assemblies experimentally studied by the Weinstein group, University of Sheffield, is investigated by means of wavepacket propagations based on the multiconfiguration time-dependent Hartree (MCTDH) method. On the basis of electronic structure data obtained at the time-dependent density functional theory (TD-DFT) level, the subpicosecond decay is simulated by solving an 11 electronic states multimode problem, up to 18 vibrational normal modes, including both spin-orbit coupling (SOC) and vibronic coupling. A careful analysis of the results, within the diabatic representation, provides the key features of the spin-vibronic mechanism at work in this complex, distinguishing between the spin-orbit and vibronically activated ultrafast processes within the excited states manifold. The knowledge of the key active normal modes that promote selectively the population of specific electronic excited states opens a route toward optical control by selectively exciting these modes in order to drive the associated nonadiabatic processes. Relevant simulations, over 2 ps, are proposed to assess the impact of these selective vibrational excitations on the branching ratio between the primary photoproducts, namely, bridge-acceptor charge-transfer (CT) and donor-acceptor charge-separated (CS) electronic states. Whereas the excitation of the localized acetylide bridge C≡C bond stretching does not modify drastically the population of the low-lying electronic states within the first two ps, vibrational excitation of the out-of-plane twisting motion of the N-benzyl group linked to the donor entity favors the population of the 1,3CS states at the expense of the lowest 1,3CT states. This quantum study opens the route to IR optical control experiments based on the specific alteration of vibrational normal modes that activate vibronic couplings between key electronic excited states. However, the presence of critical crossings along the PES channels associated with these normal modes and the role of concurrent SOC driven ultrafast transfers of population should not be underestimated.

Effect of zero-point motion on properties of quantum particles adsorbed on a substrate

We qualitatively investigate the effect of zero-point motion (ZPM) on the structure and properties of a film composed of quantum particles adsorbed on a graphite substrate. The amplitude of ZPM is controlled by a change of the particle mass while keeping the interactions fixed. In that sense it is assumed that the interactions can be controlled by future doping methods. The worm-algorithm path integral Monte Carlo (WAPIMC) method is applied to simulate this system in the grand-canonical ensemble, where particles can be exchanged with the external particle reservoir. Another method, namely the multiconfigurational time-dependent Hartree method for bosons is additionally applied to verify some of the WAPIMC results and to provide further information on the entropy and the condensate fraction. Several important findings are reported. It is found that ZPM plays an important role in defining order and disorder in the crystalline structure of the adsorbed film. The total energy of the film drops with a reduction in the amplitude of ZPM, that is, it becomes more negative which is an indication to stronger adsorption. For a few particle numbers, a significant condensate fraction is detected that however drops sharply at critical values of the ZPM amplitude. Most importantly, a connection is established between chaos, in coordinate as well as momentum space, and the Heisenberg uncertainty principle. The importance of the present study lies in the fact that adsorbed two-dimensional films serve as an excellent experimental testbed for demonstrating low-dimensional quantum phenomena in the ground state. The present examination contributes also to a further understanding of the properties of heavy quantum particles adsorbed on substrates.

A tensor network view of multilayer multiconfiguration time-dependent Hartree methods

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.

Compact sum-of-products form of the molecular electronic Hamiltonian based on canonical polyadic decomposition.

We propose an approach to represent the second-quantized electronic Hamiltonian in a compact sum-of-products (SOP) form. The approach is based on the canonical polyadic decomposition of the original Hamiltonian projected onto the sub-Fock spaces formed by groups of spin-orbitals. The algorithm for obtaining the canonical polyadic form starts from an exact sum-of-products, which is then optimally compactified using an alternating least squares procedure. We discuss the relation of this specific SOP with related forms, namely the Tucker format and the matrix product operator often used in conjunction with matrix product states. We benchmark the method on the electronic dynamics of an excited water molecule, trans-polyenes, and the charge migration in glycine upon inner-valence ionization. The quantum dynamics are performed with the multilayer multiconfiguration time-dependent Hartree method in second quantization representation. Other methods based on tree-tensor Ansätze may profit from this general approach.

Coherent Transient Localization Mechanism of Interchain Exciton Transport in Regioregular P3HT: A Quantum-Dynamical Study.

Transient localization has been proposed as a transport mechanism in organic materials, for both charge carriers and excitons. Here, we characterize a quantum coherent transient localization mechanism using full quantum simulations of an H-aggregated model system representative of regioregular poly(3-hexylthiophene) (rrP3HT). A Frenkel-Holstein Hamiltonian parametrized from first principles is considered, including local high-frequency modes and anharmonic, site-correlated interchain modes. Quantum-dynamical calculations are carried out using the Multi-Layer Multi-Configuration Time-Dependent Hartree (ML-MCTDH) method for a 13-site system with 195 vibrational modes, under periodic boundary conditions. It is shown that temporary localization of exciton polarons alternates with resonant transfer driven by interchain modes. While the transport process is mainly determined by exciton-polarons at the low-energy band edge, persistent coupling with the excitonic manifold is observed, giving rise to a nonadiabatic excitonic flux. This elementary transport mechanism remains preserved for limited static disorder and gives way to Anderson localization when the static disorder becomes dominant.

Propagating multi-dimensional density operators using the multi-layer-ρ multi-configurational time-dependent Hartree method

Solving the Liouville–von-Neumann equation using a density operator provides a more complete picture of dynamical quantum phenomena than by using a wavepacket and solving the Schrödinger equation. As density operators are not restricted to the description of pure states, they can treat both thermalized and open systems. In practice, however, they are rarely used to study molecular systems as the computational resources required are even more prohibitive than those needed for wavepacket dynamics. In this paper, we demonstrate the potential utility of a scheme based on the powerful multi-layer multi-configurational time-dependent Hartree algorithm for propagating multi-dimensional density operators. Studies of two systems using this method are presented at a range of temperatures and including up to 13 degrees of freedom. The first case is single proton transfer in salicylaldimine, while the second is double proton transfer in porphycene. A comparison is also made with the approach of using stochastic wavepackets.

Quantum and classical molecular dynamics for H atom scattering from graphene.

This work presents systematic comparisons between classical molecular dynamics (cMD) and quantum dynamics (QD) simulations of 15-dimensional and 75-dimensional models in their description of H atom scattering from graphene. We use an experimentally validated full-dimensional neural network potential energy surface of a hydrogen atom interacting with a large cell of graphene containing 24 carbon atoms. For quantum dynamics simulations, we apply Monte Carlo canonical polyadic decomposition to transform the original potential energy surface (PES) into a sum of products form and use the multi-layer multi-configuration time-dependent Hartree method to simulate the quantum scattering of a hydrogen or deuterium atom with an initial kinetic energy of 1.96 or 0.96 eV and an incident angle of 0°, i.e., perpendicular to the graphene surface. The cMD and QD initial conditions have been carefully chosen in order to be as close as possible. Our results show little differences between cMD and QD simulations when the incident energy of the H atom is equal to 1.96 eV. However, a large difference in sticking probability is observed when the incident energy of the H atom is equal to 0.96 eV, indicating the predominance of quantum effects. To the best of our knowledge, our work provides the first benchmark of quantum against classical simulations for a system of this size with a realistic PES. Additionally, new projectors are implemented in the Heidelberg multi-configuration time-dependent Hartree package for the calculation of the atom scattering energy transfer distribution as a function of outgoing angles.

Accurate Potential Energy Surfaces Using Atom-Centered Potentials and Minimal High-Level Data.

We demonstrate that a Δ-density functional theory (Δ-DFT) approach based on atom-centered potentials (ACPs) represents a computationally inexpensive and accurate method for representing potential energy surfaces (PESs) for the HONO and HFCO molecules and vibrational frequencies derived therefrom. Using as few as 100 CCSD(T)-F12a reference energies, ACPs developed for use with B3LYP/def2-TZVPP are shown to produce PESs for HONO and HFCO with mean absolute errors of 27.7 and 5.8 cm-1, respectively. Application of the multiconfigurational time-dependent Hartree (MCTDH) method with ACP-corrected B3LYP/def2-TZVPP PESs produces vibrational frequencies for cis- and trans-HONO with mean absolute percent errors (MAPEs) of 0.8 and 1.1, compared to 0.8 obtained for the two isomers with CCSD(T)-F12a/cc-pVTZ-F12/MCTDH. For HFCO, the vibrational frequencies obtained using the present (Δ-DFT)/MCTDH approach give a MAPE of 0.1, which is the error obtained with CCSD(T)-F12a/cc-pVTZ-F12/MCTDH. The ACP approach is therefore successful in representing a PES calculated at a high level of theory (CCSD(T)-F12a) and a promising method for the development of a general protocol for the representation of accurate molecular PESs and the calculation of molecular properties from them.

Temperature Dependence of the Electronic Absorption Spectrum of NO2.

The nitrogen dioxide (NO2) radical is composed of the two most abundant elements in the atmosphere, where it can be formed in a variety of ways including combustion, detonation of energetic materials, and lightning. Relevant also to smog and ozone cycles, together these processes span a wide range of temperatures. Remarkably, high-resolution NO2 electronic absorption spectra have only been reported in a narrow range below about 300 K. Previously, we reported [ J. Phys. Chem. A 2021, 125, 5519-5533] the construction of quasi-diabatic potential energy surfaces (PESs) for the lowest four electronic states (X̃, Ã, B̃, and C̃) of NO2. In addition to three-dimensional PESs based on explicitly correlated MRCI(Q)-F12/VTZ-F12 ab initio data, the geometry dependence of each component of the dipoles and transition dipoles was also mapped into fitted surfaces. The multiconfigurational time-dependent Hartree (MCTDH) method was then used to compute the 0 K electronic absorption spectrum (from the ground rovibrational initial state) employing those energy and transition dipole surfaces. Here, in an extension of that work, we report an investigation into the effects of elevated temperature on the spectrum, considering the effects of the population of rotationally and vibrationally excited initial states. The calculations are complemented by new experimental measurements. Spectral contributions from hundreds of rotational states up to N = 20 and from 200 individually-characterized vibrational states were computed. A spectral simulation tool was developed that enables modeling the spectrum at various temperatures─by weighting individual spectral contributions via the partition function, or for pure excited initial states, which can be probed via transient absorption spectroscopy. We validate these results against experimental absorption spectroscopy data at high temperatures, as well as via a new measurement from the (1,0,1) initial vibrational state.

A Structure-Based Gaussian Expansion for Quantum Reaction Dynamics in Molecules: Application to Hydrogen Tunneling in Malonaldehyde.

We developed an approximate method for quantum reaction dynamics simulations, namely, a structure-based Gaussian (SBG) expansion approach, where SBG bases for the expansion of the wave function Ψ, expressed by a product of single-atom Cartesian Gaussians centered at the positions of respective nuclei, are mainly placed around critical structures on reaction pathways such as on the intrinsic reaction coordinate (IRC) through a transition state. In the present approach, the "pseudo-lattice points" at which SBGs are deployed are selected in a perturbative manner so as to make moderate the expansion length. We first applied the SBG idea to a two-dimensional quadruple-well model and obtained accurate tunneling splitting values between the lowest four states. We then applied it to hydrogen tunneling in malonaldehyde and achieved a tunneling splitting of 27.1 cm-1 with only 875 SBGs at the MP2/6-31G(d,p) level of theory, in good agreement with 25 cm-1 by the more elaborate multiconfiguration time-dependent Hartree method. Reasonable results were also obtained for singly and doubly deuterated malonaldehyde. We analyzed the tunneling states by utilizing expansion coefficients of individual SBGs and found that 40-45% of the SBGs in Ψ are nonplanar structures and SBGs away from the IRC contribute a little to hydrogen transfer.

Nonlinear Molecular Electronic Spectroscopy via MCTDH Quantum Dynamics: From Exact to Approximate Expressions.

We present an accurate and efficient approach to computing the linear and nonlinear optical spectroscopy of a closed quantum system subject to impulsive interactions with an incident electromagnetic field. It incorporates the effect of ultrafast nonadiabatic dynamics by means of explicit numerical propagation of the nuclear wave packet. The fundamental expressions for the evaluation of first- and higher-order response functions are recast in a general form that can be used with any quantum dynamics code capable of computing the overlap of nuclear wave packets evolving in different states. Here we present the evaluation of these expressions with the multiconfiguration time-dependent Hartree (MCTDH) method. Application is made to pyrene, excited to its lowest bright excited state S2 which exhibits a sub-100-fs nonadiabatic decay to a dark state S1. The system is described by a linear vibronic coupling Hamiltonian, parametrized with multiconfiguration electronic structure methods. We show that the ultrafast nonadiabatic dynamics can have a remarkable effect on the spectral line shapes that goes beyond simple lifetime broadening. Furthermore, a widely employed approximate expression based on the time scale separation of dephasing and population relaxation is recast in the same theoretical framework. Application to pyrene shows the range of validity of such approximations.

Fragmentation and correlations in a rotating Bose–Einstein condensate undergoing breakup

The theoretical investigation of rotating Bose–Einstein condensates has mainly focused on the emergence of quantum vortex states and the condensed properties of such systems. In the present work, we concentrate on other facets by examining the impact of rotation on the ground state of weakly interacting bosons confined in anharmonic potentials computed both at the mean-field level and particularly at the many-body level of theory. For the many-body computations, we employ the well-established many-body method known as the multiconfigurational time-dependent Hartree method for bosons. We present how various degrees of fragmentation can be generated following the breakup of the ground state densities in anharmonic traps without ramping up a potential barrier for strong rotations. The breakup of the densities is found to be associated with the acquisition of angular momentum in the condensate due to the rotation. In addition to fragmentation, the presence of many-body correlations is examined by computing the variances of the many-particle position and momentum operators. For strong rotations, the many-body variances become smaller than their mean-field counterparts, and one even finds a scenario with opposite anisotropies of the mean-field and many-body variances. Further, it is observed that for higher discrete symmetric systems of order k, namely three-fold and four-fold symmetry, breakup to k sub-clouds and emergence of k-fold fragmentation take place. All in all, we provide a thorough many-body investigation of how and which correlations build up when a trapped Bose–Einstein condensate breaks up under rotation.

Information theoretic measures for interacting bosons in optical lattice.

This work reports the different information theoretic measures, i.e., Shannon information entropy, order, disorder, complexity, and their dynamical measure for the interacting bosons in an optical lattice with both commensurate and incommensurate filling factor. We solve the many-body Schrödinger equationfrom first principles by multiconfigurational time-dependent Hartree method which calculates all the measures with high level of accuracy. We find for both relaxed state as well as quenched state the López-Ruiz-Mancini-Calbet (LMC) measure of complexity is the most efficient depictor of superfluid (SF) to Mott-insulator transition. In the quench dynamics, the distinct structure of LMC complexity can be used as a "figureof merit" to obtain the timescale of SF to Mott state entry, Mott holding time, and the Mott state to SF state entry in the successive cycles. We also find that fluctuations in the dynamics of LMC complexity measure for incommensurate filling clearly establish that superfluid to Mott-insulator transition is incomplete. We overall conclude that distinct structure in the complexity makes it more sensitive than the standard use of Shannon information entropy.

Charged ultralong-range Rydberg trimers

We show that the recently observed class of long-range ion-Rydberg molecules can be divided into two families of states, which are characterised by their unique electronic structures resulting from the ion-induced admixture of quantum defect-split Rydberg $n$P states with different low-field seeking high-$l$ states. We predict that in both cases these diatomic molecular states can bind additional ground state atoms lying within the orbit of the Rydberg electron, thereby forming charged ultralong-range Rydberg molecules (ULRM) with binding energies similar to that of conventional non-polar ULRM. To demonstrate this, we consider a Rydberg atom interacting with a single ground state atom and an ion. The additional atom breaks the system's cylindrical symmetry, which leads to mixing between states that would otherwise be decoupled. The electronic structure is obtained using exact diagonalisation over a finite basis and the vibrational structure is determined using the Multi-Configuration Time-Dependent Hartree method. Due to the lobe-like structure of the electronic density, bound trimers with both linear and nonlinear geometrical configurations of the three nuclei are possible. The predicted trimer binding energies and excitation series are distinct enough from those of the ion-Rydberg dimer to be observed using current experimental techniques.

Optimal Mode Combination in the Multiconfiguration Time-Dependent Hartree Method through Multivariate Statistics: Factor Analysis and Hierarchical Clustering.

The multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extension (ML-MCTDH) are powerful algorithms for the efficient computation of nuclear quantum dynamics in high-dimensional systems. By providing time-dependent variational orbitals and an optimal choice of layered effective degrees of freedom, one is able to reduce the computational cost to an amenable number of configurations. However, choices related to selecting properly the mode grouping and tensor tree are strongly system dependent and, thus far, subjectively based on intuition and/or experience. Therefore, herein we detail a new protocol based on multivariate statistics─more specifically, factor analysis and hierarchical clustering─for a reliable and convenient guiding in the optimal design of such complex "system-of-systems" tensor-network decompositions. The advantages of employing the new algorithm and its applicability are tested on water and two floppy protonated water clusters with large amplitude motions.

Rovibrational states calculations of the H2O-HCN heterodimer with the multiconfiguration time dependent Hartree method.

Water and hydrogen cyanide are two of the most common species in space and the atmosphere with the ability of binding to form dimers such as H2O-HCN. In the literature, while calculations characterizing various properties of the H2O-HCN cluster (equilibrium distance, vibrational frequencies and rotational constants) have been done in the past, extensive calculations of the rovibrational states of this system using a reliable quantum dynamical approach have yet to be reported. In this work, we intend to mend that by performing the first calculation of the rovibrational states of the H2O-HCN van der Waals complex on a recently developed potential energy surface. We use the block improved relaxation procedure implemented in the Heidelberg MultiConfiguration Time-Dependent Hartree (MCTDH) package to compute the states of the H2O-HCN isomer, from which we extract the transition frequencies and rotational constants of the complex. We further adapt an approach first suggested by Wang and Carrington-and supported here by analysis routines of the Heidelberg MCTDH package-to properly characterize the computed rovibrational states. The subsequent assignment of rovibrational states was done by theoretical analysis and visual inspection of the wavefunctions. Our simulations provide a Zero Point Energy (ZPE) and intermolecular vibrational frequencies in good agreement with past ab initio calculations. The transition frequencies and rotational constants obtained from our simulations match well with the available experimental data. This work has the broad aim to propose the MCTDH approach as a reliable option to compute and characterize rovibrational states of van der Waals complexes such as the current one.