A multi-layer multi-configurational time-dependent Hartree approach to lattice models beyond one dimension.

The multi-layer multi-configurational time-dependent Hartree (MCTDH) approach is an efficient method to study quantum dynamics in real and imaginary time. The present work explores its potential to describe quantum fluids. The multi-layer MCTDH approach in second quantization representation is used to study lattice models beyond one dimension at finite temperatures. A scheme to map the lattice sites onto the MCTDH tree representation for multi-dimensional lattice models is proposed. A statistical sampling scheme previously used in MCTDH calculations is adapted to facilitate an efficient description of the thermal ensemble. As example, a two-dimensional hard-core Bose-Hubbard model is studied considering up to 64 × 64 lattice sites. The single particle function basis set size required to obtain converged results is found to not increase with the lattice size. The numerical results properly simulate the finite temperature Berezinskii-Kosterlitz-Thouless phase transition.

A non-hierarchical multi-layer multi-configurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces.

The correlation discrete variable representation (CDVR) enables multi-layer multi-configurational time-dependent Hartree (MCTDH) quantum dynamics simulations on general potential energy surfaces. In a recent study [R. Ellerbrock and U. Manthe, J. Chem. Phys. 156, 134107 (2022)], an improved CDVR that can account for the symmetry properties of a tree-shaped wavefunction representation has been introduced. This non-hierarchical CDVR drastically reduces the number of grid points required in the time-dependent quadrature used to evaluate all potential energy matrix elements. While the first studies on the non-hierarchical CDVR approach have been restricted to single-layer calculations, here the complete theory required for the implementation of the non-hierarchical CDVR approach in the multi-layer MCTDH context will be presented. Detailed equations facilitating the efficient recursive computation of all matrix elements are derived, and a new notation adapted to the symmetry properties of the tree-shaped representation is introduced. Calculations studying the non-adiabatic quantum dynamics of photoexcited pyrazine in 24 dimensions illustrate the properties of the non-hierarchical multi-layer CDVR.

Accurate Quantum Dynamics Calculations for the Cl + CH4/CHD3/CD4 Reaction Rates.

Full-dimensional quantum dynamics simulations of the reaction of Cl with methane and its isotopomers are reported. Thermal rate constants are computed for the Cl + CH4 → HCl + CH3, Cl + CHD3 → HCl + CD3, and Cl + CD4 → DCl + CD3 reactions. Temperatures between 200 and 500 K are considered. In this temperature range, excellent agreement with the experiment is obtained. A detailed analysis of the kinetic isotope effect reveals the crucial importance of the CH3/CD3 umbrella motion. Comparison with approximate ring-polymer molecular dynamics simulations shows significant differences depending on the isotope studied.

QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

Eigenstate calculation in the state-averaged (multi-layer) multi-configurational time-dependent Hartree approach.

A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems.

Straightforward Synthesis of Halopyridine Aldehydes via Diaminomethylation.

A novel two-step method for formylation of fluoropyridines with silylformamidine Me3 SiC(=NMe)NMe2 (1) under catalyst-free conditions was developed. A series of all possible 18 fluoropyridines featuring one to four fluorine atoms were subjected to the reaction with 1 existing in equilibrium with its carbenic form Me2 NC(:)N(Me)SiMe3 (1'). Among them, 12 fluoropyridines were shown to react via C-H insertion. The reaction proceeded either at ß- or γ-positions affording the corresponding aminals. The more fluorine atoms in pyridines, the easier the reaction proceeded. We also hypothesized that the pyridines in which the fluorine was substituted by other halogens would react in a similar manner. To test the hypothesis, a set of 3,5-disubstituted pyridines with various combination of halogen atoms was prepared. 3,5-Difluoropyridine was taken as a compound for comparison. All the pyridines in the series also reacted likewise. In most cases, hydrolysis of the aminals afforded the corresponding aldehydes. As DFT calculations indicate, the reaction mechanism includes deprotonation of pyridine by 1' as a strong base and the following rearrangement of the formed tight ionic pair to the final product. An alternative reaction pathway involving addition of 1' to the pyridine carbon with the following hydrogen transfer via a three-membered transition state structure required much higher activation energy.

A numerically exact correlation discrete variable representation for multi-configurational time-dependent Hartree calculations.

The correlation discrete variable representation (CDVR) enables (multilayer) multi-configurational time-dependent Hartree (MCTDH) calculations with general potentials. The CDVR employs a set of grids corresponding to single-particle functions to efficiently evaluate all potential matrix elements appearing in the MCTDH equations of motion. In standard CDVR approaches, the number of grid points employed is tied to the number of corresponding single-particle functions. This limits the accuracy of the quadrature, which can be achieved for a given single-particle function basis. In this work, an extended CDVR approach that facilitates a numerically exact quadrature of all potential matrix elements is introduced. The number of grid points employed can be increased independent of the number of corresponding single-particle function to achieve any desired quadrature accuracy. The properties of the new scheme are illustrated by numerical calculations studying the photodissociation of NOCl and the vibrational states of CH3. Fast convergence with respect to the number of additional quadrature points is observed: Employing a single additional point in each physical or logical coordinate already ensures negligible quadrature errors.

Latent Carbene in Diaminomethylation of Benzenes: Mechanism and Practical Application.

Silylformamidine 1 exists in equilibrium with its carbenic form 1' due to an easy migration of the silyl group. The reaction of 1 with variously substituted fluorobenzenes proceeds as an insertion of the nucleophilic carbene 1' into the most acidic C-H bond upon mixing the reagents and does not require any catalyst. According to DFT calculations, the classical interpretation of the insertion reaction proceeding via a three-membered transition state structure requires high activation energy. Instead, low activation barriers are predicted for a transfer of the most acidic proton in the aromatic substrate to the carbene carbon. As the next step, a barrierless rearrangement of the formed ion pair toward the product completes the process. The reactivity of substituted benzenes in the reaction with silylformamidine can be roughly assessed by calculated pKa (DMSO) values for the C-H hydrogens. Benzene derivatives having pKa approx. less than 31 can undergo C-H insertion. The reaction provides aminals as the first products, which can easily be transformed into the corresponding aldehydes via acidic hydrolysis. As silylformamidine 1 is tolerant to many functional groups, the reaction can be applied to numerous benzene derivatives, making it a reliable strategy for application in organic synthesis.

A non-hierarchical correlation discrete variable representation.

The correlation discrete variable representation (CDVR) facilitates (multi-layer) multi-configurational time-dependent Hartree (MCTDH) calculations with general potentials. It employs a layered grid representation to efficiently evaluate all potential matrix elements appearing in the MCTDH equations of motion. The original CDVR approach and its multi-layer extension show a hierarchical structure: the size of the grids employed at the different layers increases when moving from an upper layer to a lower one. In this work, a non-hierarchical CDVR approach, which uses identically structured quadratures at all layers of the MCTDH wavefunction representation, is introduced. The non-hierarchical CDVR approach crucially reduces the number of grid points required, compared to the hierarchical CDVR, shows superior scaling properties, and yields identical results for all three representations showing the same topology. Numerical tests studying the photodissociation of NOCl and the vibrational states of CH3 demonstrate the accuracy of the non-hierarchical CDVR approach.

First-Principles Theory for the Reaction of Chlorine with Methane.

A full-dimensional quantum dynamics simulation of the Cl + CH4 → HCl + CH3 reaction based on first-principles theory is reported. Accurate thermal rate constants are calculated, and perfect agreement with experiment is obtained. Despite the heavy atoms present in both reactants, the passage of the reaction barrier is found to occur within only a few tens of femtoseconds. This surprisingly short time scale results from correlated motion of the transferring hydrogen atom and the hydrogen atoms in the methyl fragment which facilitates irreversible barrier passage without relevant participation of heavy atoms. Resonance effects resulting from the heavy-light-heavy characteristics of the reaction system, which were observed in reactive scattering studies, do not affect the thermal rate constant.

Vibrational control of the reaction pathway in the H + CHD3 → H2 + CD3 reaction.

An accurate full-dimensional quantum state-to-state simulation of the six-atom title reaction based on first-principles theory is reported. Counterintuitive effects are found: Increasing the energy in the reactant's CD3 umbrella vibration reduces the energy in the corresponding product vibration. An in-depth analysis reveals the crucial role of the effective dynamical transition state: Its geometry is controlled by the vibrational states of the reactants and subsequently controls the quantum state distribution of the products. This finding enables generalizing the concept of transition state control of chemical reactions to the quantum state-specific level.

Full-dimensional quantum stereodynamics of the non-adiabatic quenching of OH(A2Σ+) by H2.

The Born-Oppenheimer approximation, assuming separable nuclear and electronic motion, is widely adopted for characterizing chemical reactions in a single electronic state. However, the breakdown of the Born-Oppenheimer approximation is omnipresent in chemistry, and a detailed understanding of the non-adiabatic dynamics is still incomplete. Here we investigate the non-adiabatic quenching of electronically excited OH(A2Σ+) molecules by H2 molecules using full-dimensional quantum dynamics calculations for zero total nuclear angular momentum using a high-quality diabatic-potential-energy matrix. Good agreement with experimental observations is found for the OH(X2Π) ro-vibrational distribution, and the non-adiabatic dynamics are shown to be controlled by stereodynamics, namely the relative orientation of the two reactants. The uncovering of a major (in)elastic channel, neglected in a previous analysis but confirmed by a recent experiment, resolves a long-standing experiment-theory disagreement concerning the branching ratio of the two electronic quenching channels.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation.

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

Direct product-type grid representations for angular coordinates in extended space and their application in the MCTDH approach.

Multi-configurational time-dependent Hartree (MCTDH) calculations using time-dependent grid representations can be used to accurately simulate high-dimensional quantum dynamics on general ab initio potential energy surfaces. Employing the correlation discrete variable representation, sets of direct product type grids are employed in the calculation of the required potential energy matrix elements. This direct product structure can be a problem if the coordinate system includes polar and azimuthal angles that result in singularities in the kinetic energy operator. In the present work, a new direct product-type discrete variable representation (DVR) for arbitrary sets of polar and azimuthal angles is introduced. It employs an extended coordinate space where the range of the polar angles is taken to be [-π, π]. The resulting extended space DVR resolves problems caused by the singularities in the kinetic energy operator without generating a very large spectral width. MCTDH calculations studying the F·CH4 complex are used to investigate important properties of the new scheme. The scheme is found to allow for more efficient integration of the equations of motion compared to the previously employed cot-DVR approach [G. Schiffel and U. Manthe, Chem. Phys. 374, 118 (2010)] and decreases the required central processing unit times by about an order of magnitude.

Eight-Dimensional Wave Packet Dynamics Within the Quantum Transition-State Framework: State-to-State Reactive Scattering for H2 + CH3 ⇆ H + CH4.

The present work investigates the calculation of S-matrix elements for six-atom reactions combining reduced-dimensional wave packet dynamics and the quantum transition-state framework. We employ the eight-dimensional (8D) model Hamiltonian developed by Palma and Clary [J. Chem. Phys. 2000, 112, 1859-1867] and reduce basis set sizes as well as the number of wave packets by exploiting space inversion and permutation symmetry. Mode-specific chemistry in the H2 + CH3 ⇆ H + CH4 reaction is studied with full quantum-state resolution. Results for the H + CH4 reaction are compared to full-dimensional benchmark results. Detailed state-to-state results for the H2 + CH3 reaction are presented for the first time. Although the "loss of memory" effect dominates for total energies up to 0.6 eV, more complex patterns emerge at higher energies. The agreement between the present reduced-dimensional and the accurate full-dimensional results is generally good. However, shortcomings in the reduced-dimensional model can also be noted. They are related to the description of the symmetric and asymmetric C-H stretch motion in the CH4 molecule.

Non-adiabatic transitions in the reaction of fluorine with methane.

Reactions of methane with different atoms are benchmark examples of elementary reaction processes intensively studied by theory and experiment. Due to the presence of conical intersections and spin-orbit coupling, non-adiabatic transitions can occur in reactions with F, Cl, or O atoms. Extending detailed quantum theory beyond the Born-Oppenheimer approximation for polyatomic reaction processes, non-adiabatic wave packet dynamics calculations studying the F(2P3/2)/F*(2P1/2) + CHD3 → HF + CD3 reaction on accurate vibronically and spin-orbit coupled diabatic potential energy surfaces are presented. Non-adiabatic transitions are found to increase the reactivity compared to Born-Oppenheimer theory and are more prominent than in triatomic reactions previously studied. Furthermore, the lifetimes of reactive resonances are reduced. The reactivity of F(2P3/2) is found to exceed the one of F*(2P1/2) even at low collision energies.

The multi-configurational time-dependent Hartree approach in optimized second quantization: Imaginary time propagation and particle number conservation.

The multilayer multiconfigurational time-dependent Hartree (MCTDH) in optimized second quantization representation (oSQR) approach combines the tensor contraction scheme of the multilayer MCTDH approach with the use of an optimized time-dependent orbital basis. Extending the original work on the subject [U. Manthe and T. Weike, J. Chem. Phys. 146, 064117 (2017)], here MCTDH-oSQR propagation in imaginary time and properties related to particle number conservation are studied. Differences between the orbital equation of motion in real and imaginary time are highlighted and a new gauge operator, which facilitates efficient imaginary time propagation, is introduced. Studying Bose-Hubbard models, particle number conservation in MCTDH-oSQR calculations is investigated in detail. Interesting properties of the single-particle functions used in the multilayer MCTDH representation are identified. Based on these results, a tensor contraction scheme, which explicitly utilizes particle number conservation, is suggested.

Vibronic coupling in the F·CH4 prereactive complex.

The F + CH4 → HF + CH3 reaction shows a counter-intuitive mode-selective chemistry and prominent resonances. The prereactive F·CH4 complex formed in the entrance channel is assumed to play an important role in the dynamics of the reaction. The present work investigates the effect of nonadiabatic transitions and the geometric phase on the low-lying quasibound states of the F·CH4 complex. Quantum dynamics calculations employing the multiconfigurational time-dependent Hartree approach and accurately accounting for vibronic as well as spin-orbit coupling are performed. Extending previous work [D. Schäpers and U. Manthe, J. Phys. Chem. A 120, 3186 (2016)], which was restricted to the dynamics on a single adiabatic potential energy surface and found the relative rotation of F and CH4 to proceed almost freely, we found chaotic patterns if vibronic coupling is included. While nonadiabatic transitions strongly affect individual resonances, their effect on averaged quantum state densities and the photodetachment spectrum of F⋅CH4 - is found to be minor.

Vibronically and spin-orbit coupled diabatic potentials for X(2P) + CH4 → HX + CH3 reactions: Neural network potentials for X = Cl.

Vibronically and spin-orbit (SO) coupled diabatic potentials for the Cl(2P) + CH4 → HCl + CH3 reaction are constructed based on a recently developed approach [T. Lenzen and U. Manthe, J. Chem. Phys. 150, 064102 (2019)]. Diabatic potentials and couplings describing the entrance channel of the reaction are obtained based on ab initio data using a diabatization by an ansatz scheme. A detailed investigation of the electronic structure in the entrance channel using multireference configuration interaction (MRCI), coupled cluster [CCSD/CCSD(T)], and SO-MRCI calculations is presented. Neural networks using permutationally invariant polynomials as inputs are employed to represent the elements of the diabatic potential energy matrix. The same set of diabatic states is also used in the transition state region and all four exit channels. Here, the lowest adiabatic potential energy surface (PES) derived from the diabatic model is chosen to reproduce an adiabatic PES recently developed by Li and Guo. The accuracy of the resulting PES is evaluated, and the properties of the newly developed coupled diabatic potentials are analyzed in detail.

A Quasi-Classical Evaluation of the J-Shifting Approximation for the Reactive Cross Sections of F + CHD3 and F + CH4.

We evaluated the accuracy of the J-shifting approximation to estimate reactant state-selected cross sections for the F+CH4 → HF+CH3 and F+CHD3 → HF+CD3/DF+CHD2 reactions. In particular, we analyzed how the rotational state of methane influences the quality of the approximation. The systems were considered in full dimensionality. Since full-quantum scattering calculations are still unfeasible for these reactions, we employed quasi-classical trajectories (QCT) to calculate the cross sections. The characteristics of the Born-Oppenheimer potential energy surface of these reactions pose a great challenge to the assumptions of the J-shifting approach. In spite of this, we found that it performs well for both reactions if the methane molecule is in the rotational ground state. However, when methane is rotationally excited, the approach affords good results for the F+CH4 system but clearly fails for F+CHD3. The reasons for this failure will be discussed, and a simple procedure to recover good estimators for the cross sections from J = 0 calculations will be introduced.