https://www.pci.uni-heidelberg.de/tc/usr/mctdh/lit/rev.pdf

The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets

A multilayer (ML) formulation of the multiconfiguration time-dependent Hartree (MCTDH) theory is presented. In this new approach, the single-particle (SP) functions in the original MCTDH method are further expressed employing a time-dependent multiconfigurational expansion. The Dirac–Frenkel variational principle is then applied to optimally determine the equations of motion. Following this strategy, the SP groups are built in several layers, where each top layer SP can contain many more Cartesian degrees of freedom than in the previous formulation of the MCTDH method. As a result, the ML-MCTDH method has the capability of treating substantially more physical degrees of freedom than the original MCTDH method, and thus significantly enhances the ability of carrying out quantum dynamical simulations for complex molecular systems. The efficiency of the new formulation is demonstrated by converged quantum dynamical simulations for systems with a few hundred to a thousand degrees of freedom.

Multilayer formulation of the multiconfiguration time-dependent Hartree theory

Nonadiabatic effects play an important role in many areas of physics and chemistry. The coupling between electrons and nuclei may, for example, lead to the formation of a conical intersection between potential energy surfaces, which provides an efficient pathway for radiationless decay between electronic states. At such intersections the Born-Oppenheimer approximation breaks down, and unexpected dynamical processes result, which can be observed spectroscopically. We review the basic theory required to understand and describe conical, and related, intersections. A simple model is presented, which can be used to classify the different types of intersections known. An example is also given using wavepacket dynamics simulations to demonstrate the prototypical features of how a molecular system passes through a conical intersection.

Beyond Born-Oppenheimer: molecular dynamics through a conical intersection.

We define the hierarchical singular value decomposition (SVD) for tensors of order $d\geq2$. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in $d=2$), and we prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format ($\mathcal{H}$-Tucker) which requires only $\mathcal{O}((d-1)k^3+dnk)$ parameters, where $d$ is the order of the tensor, $n$ the size of the modes, and $k$ the (hierarchical) rank. The $\mathcal{H}$-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank $k$ tensors. Based on this new concept of a hierarchical SVD we present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank $k$ tensors) is in $\mathcal{O}((d-1)k^4+dnk^2)$ and the attainable accuracy is just 2-3 digits less than machine precision.

https://www.mis.mpg.de/preprints/2009/preprint2009_27.pdf

Hierarchical Singular Value Decomposition of Tensors

Knowing the underlying photophysics of thermally activated delayed fluorescence (TADF) allows proper design of high efficiency organic light-emitting diodes. We have proposed a model to describe reverse intersystem crossing (rISC) in donor–acceptor charge transfer molecules, where spin–orbit coupling between singlet and triplet states is mediated by one of the local triplet states of the donor (or acceptor). This second order, vibronically coupled mechanism describes the basic photophysics of TADF. Through a series of measurements, whereby the energy ordering of the charge transfer (CT) excited states and the local triplet are tuned in and out of resonance, we show that TADF reaches a maximum at the resonance point, substantiating our model of rISC. Moreover, using photoinduced absorption, we show how the populations of both singlet and triplet CT states and the local triplet state change in and out of resonance. Our vibronic coupling rISC model is used to predict this behaviour and describes how rISC and TADF are affected by external perturbation.

/pdf/revealing-the-spin-vibronic-coupling-mechanism-of-thermally-20s1drezx3.pdf

Revealing the spin-vibronic coupling mechanism of thermally activated delayed fluorescence.

An efficient and robust integration scheme tailored to the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method is presented. An error estimation allows the automatical adjustment of the step size and hence controls the integration error. The integration scheme decouples the MCTDH equations of motion into several disjoined subsystems, of which one determines the time evolution of the MCTDH-coefficients. While the conventional MCTDH equations are non-linear, the working equation for the MCTDH-coefficients becomes linear in the present integration scheme. To investigate the integrator’s performance it is applied to the photodissociation process of methyl iodide. The results of the novel integration scheme are in perfect agreement to those obtained by solving the MCTDH working equations conventionally. The computation time, however, is reduced by a factor of about ten when the new integration scheme is used to propagate large systems.

An efficient and robust integration scheme for the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method

We present a new approach for determining bound-state spectra of molecules or clusters. In our approach a wave packet propagation is performed, which exploits the efficiency of the multi-configuration time-dependent Hartree scheme, to produce an autocorrelation function. From this, an accurate spectrum is extracted employing the filter-diagonalization procedure. The accuracy of this hybrid method is demonstrated by applying it to the spectrum of carbon dioxide. Compared with the filter-diagonalization scheme based on a numerically exact wave packet propagation and with a matrix diagonalization using the Lanczos algorithm, our approach turns out to be more efficient. The method can easily be generalized to the treatment of resonant states.

Efficiently computing bound-state spectra: A hybrid approach of the multi-configuration time-dependent Hartree and filter-diagonalization methods

We present an extension of the filter-diagonalization (FD) approach which allows the extraction of accurate bound-state spectra from a time-dependent wave function determined by an approximate propagation scheme. To investigate the method, an approximate wave function is generated by a wave packet propagation on a two-dimensional potential energy surface for the LiCN molecule, employing a second-order differencing integrator with large step size. When applied to this inaccurate wave function, our formulation of the FD technique is capable of yielding eigenenergies considerably more precise than those obtained with the original FD method, while the effort is almost the same. The present FD scheme contains some additional refinements that increase the method’s efficiency, accuracy, and practicability. We demonstrate that the accuracy of the FD procedure can be improved by using cosinetype filters. The quality of different variational principles employed to determine the eigenenergies is examined. These variational principles can also be utilized to estimate the error of the energies and intensities inexpensively and reliably.

Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method

A hybrid approach of the multi-configuration time-dependent Hartree (MCTDH) and the filter-diagonalisation (FD) methods for computing bound-state spectra is applied to the study of the HO2 radical. We investigate the efficiency and accuracy of this approach for the case where the potential energy surface is not given as a sum of products of one-dimensional functions. As the MCTDH scheme requires such a product form in order
 to be efficient the potential energy surface was replaced by a potential fit of the required form. The performance of our approach is compared with that of the Lanczos algorithm.

A hybrid approach of the multi-configuration time-dependent Hartree and filter-diagonalisation methods for computing bound-state spectra. Application to HO2

M. H. Beck

Papers

The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets

An efficient and robust integration scheme for the equations of motion of the multiconfiguration time-dependent Hartree (MCTDH) method

Efficiently computing bound-state spectra: A hybrid approach of the multi-configuration time-dependent Hartree and filter-diagonalization methods

Extracting accurate bound-state spectra from approximate wave packet propagation using the filter-diagonalization method

A hybrid approach of the multi-configuration time-dependent Hartree and filter-diagonalisation methods for computing bound-state spectra. Application to HO2