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Important Notes

A new book entitled "Applications of Quantum Dynamics in Chemistry" written by Fabien Gatti, Benjamin Lasorne, Hans-Dieter Meyer and André Nauts has been published (Sep. 2017)  in the series "Lectures Notes in Chemistry" (Springer).

This book explains the usage and application of Molecular Quantum Dynamics, the methodology where both the electrons and the nuclei in a molecule are treated with quantum mechanical calculations. The reader learns how to apply example simulations with the Heidelberg MCTDH program package (Multi Configuration Time Dependent Hartree calculations).

For further details, see the following link:


There will be a "High Dimensional Quantum Dynamics" conference, the HDQD-2024, in Hamburg, July 16-19, organized by Peter Schmelcher.
The registration is open! See

There has been a 7th edition of the "High dimensional quantum dynamics: challenges and opportunities" workshop. It will take place in Groningen (The Netherlands) from the 12th to the 15st of July 2022, organized by Shirin Faraji. See

There has been a "High dimensional quantum dynamics: challenges and opportunities>" workshop took place in Lille, France,  28-31 August 2018. The workshop was organized by Daniel Pelaez Ruiz.

There has been a "High dimensional quantum dynamics: challenges and opportunities" workshop in Rostock, 31. August - 3. September 2016. The workshop was organized by Oliver Kühn.

There has been a "High dimensional quantum dynamics: challenges and opportunities" workshop in Mittelwihr near Strasbourg, September 2 - 6, 2014, organized by Etienne Gindensperger and Roberto Marquardt.

There has been a "High dimensional quantum dynamics: challenges and opportunities" workshop in Birmingham, April 12-14, 2012, organized by Graham Worth.

There has been a "High dimensional quantum dynamics: challenges and opportunities"  workshop in Montpellier, February 25-28, 2008, organized by Fabien Gatti.

There has been a "High dimensional quantum dynamics: challenges and opportunities"  workshop in Leiden,  September 28 - October 1, 2005, organized by  Gert-Jan Kroes and Hans-Dieter Meyer.


April 15th, 2009 there has appeared the book:
Multidimensional Quantum Dynamics:
MCTDH Theory and Applications
H.-D. Meyer, F. Gatti, and G. A. Worth, editors
For a preview (cover page, list of contents, and introduction) see here (PDF).
For an announcement and order form click here (PDF).
You can view the book at .


For further reading go to Literature Downloads. There you will find, among others, the MCTDH review and my lecture notes on MCTDH (intro_MCTDH).


Brief Description of MCTDH

MCTDH stands for Multi Configuration Time Dependent Hartree. MCTDH is a general algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles. MCTDH can thus determine the quantal motion of the nuclei of a molecular system evolving on one or several coupled electronic potential energy surfaces. MCTDH by its very nature is an approximate method. However, it can be made as accurate as any competing method, but its numerical efficiency deteriorates with growing accuracy.

MCTDH is designed for treating multi-dimensional problems, in particular problems that are difficult or even impossible to attack in a conventional way. There is no or only little gain when treating systems with less than three degrees of freedom by MCTDH. However, for convenience − not for numerical speed − be have used the MCTDH package even for one-dimensional problems. MCTDH will in general be best suited for systems with 4 to 12 degrees of freedom. Because of hardware limitations it may in general not be possible to treat much larger systems. For a certain class of problems, however, one can go much further. Already some time ago (1998/1999) we have performed converged MCTDH calculations on the absorption spectrum of pyrazine accounting for the motion of all 24 (!) internal degrees of freedom [30,35,36]. Similar calculations have been done for the ionisation spectra of Allene [56], Butatrien [57] and Benzene [66]. (The numbers is square brackets refer to the List of MCTDH Publications ). The spin-boson model was treated by Haobin Wang including 80 (!!) vibrational modes [49], and we have recently used MCTDH to study the multi-dimensional Henon-Heiles Hamiltonian including up to 32 degrees of freedom [70] and a Morse oscillator coupled to a harmonic bath including up to 61 degrees of freedom [76].
The MCTDH program package has been generalised to enable the propagation of density operators [39,45,46,99], and recently it has been generalized to compute eigenstates ("improved relaxation") [75,128].

MCTDH has the following advantages

  • The MCTDH algorithm can be very fast. In fact it may be several orders of magnitude faster than a conventional treatment. Whether MCTDH is fast or not depends on the system under consideration (see below).
  • The MCTDH algorithm is small, i.e it requires an amazingly small amount of central memory (RAM). The MCTDH representation of the wave function is also very compact, making it possible to store a large number of wave functions for later analysis.

MCTDH has the following disadvantages

  • The MCTDH algorithm requires the Hamiltonian to be expanded as a sum of products of one-particle operators. The kinetic energy is usually already in this MCTDH form. The potential is in MCTDH form only when dealing with model problems. In general one has to transform [19,31] the potential (obtained e.g. by quantum chemistry calculations) to MCTDH form. There exist an efficient algorithm to accomplish this transformation. However, the MCTDH computation time grows linearly with the number of Hamiltonian terms and MCTDH will become slow, if an accurate representation of the potential requires very many terms.
  • The MCTDH algorithm is fast only if the time dependent wavepacket can - at each instant of time - be expanded into a small (but optimised) product basis set. This makes it difficult for MCTDH to propagate a wavepacket when the system is highly chaotic. It is also difficult for MCTDH to propagate a wavepacket for very long times (hundreds of vibrational periods, say). However, we recently [70] have propagated a wavepacket on a 4D Henon-Heiles potential over 300 time units (i.e. about 50 vibrational periods). Thus even a chaotic system may be studied for rather long times. The 14D Henon-Heiles potential was also investigated. Here the propagation was stopped after 25 time units, as this was enough to reproduce the spectrum.

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MCTDH Applications

Problems treated with MCTDH are:

  • Photodissociation: absorption spectra and product distributions. (NOCl, NO2, CH3I, ICN, ArHBr, Ar2HBr, HCl/ice) [2,3,4,9,13,71,104,106].
  • Computation of absorption (or ionisation) spectra of polyatomic molecules. Vibronically coupled systems. Pyrazine [20,30,35,36]; Allene [56]; Butatriene [57]; Benzene [66], Furan [93], Pentatetraene [114], Cyclopropane [135], .Phenide [139],  Difluorobenzene [171].
  • Computation of resonance Raman spectra of polyatomic molecules. (CH3I) [25].
  • Vibrational predissociation. Cl2Ne [12] ; I2...Ne2 cluster [60].
  • Molecule-surface scattering : rotational excitation and diffractive scattering of diatomic molecules colliding with a hard but corrugated surface. (H2/LiF, N2/LiF) [17,23,50]. (See also [42,43]). Dissociation from a surface. (CH4/Ni, CH3I/MgO) [11,16,28]. Dissociative adsorption of H2 on Cu(100) [92], N2 on stepped Ru(0001) [105], and H2 on Pt(111) [115].
  • Inelastic scattering scattering, cross−sections: H2 + H2  [112,148,157,187].
  • Reactive scattering, cross−sections: H + H2 -> H2 + H , H + D2 -> HD + D [15,29,34,54,65]. H2 + OH -> H2O + H [195].
  • Reactive scattering, thermal reaction rates: [24,27,33,37,47,48,62,67,74,124,140,155,193].
  • Computation of eigenenergies by combining MCTDH with the filtering method. CO2 [53], HO2 [55], toluene [88], and HFCO [118].
  • Spin-Boson model [49,58,59,77].
  • Nuclear dynamics during electron-scattering processes: CO2 + e [62,70], H2O + e [86,127,133].
  • Proton transfer reactions [63,80,98,100,102].
  • Isomerisation of HONO [83,84] and IVR of CF3H [85], Toluene [88], HFCO [118] and DFCO [134].
  • High-dimensional Henon-Heiles potentials [70].
  • Morse oscillator coupled to a bath of 60 oscillators [76].
  • Small bosonic systems [129,130,141,142,158,161,162,178,197]
  • Optimal control [126,161].
  • Computation of eigenstates with improved relaxation [128,174].
  • Density operator propagation on model systems (up to 4D) [39,45,46] and on vibrational relaxation of CO on Cu (6D) [99].
  • Dynamics and infrared spectra of the Zundel cation H5O2+ [143,152,153,169,176].

There are more applications. See the List of MCTDH Publications for a comprehensive overview. (The citations [...] refer to this list.)

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The Heidelberg MCTDH Package

The Heidelberg MCTDH package is a set of programs for multi-dimensional quantum dynamics, and can do much more than wavefunction (or density operator) propagation using the MCTDH algorithm. For example, numerically exact propagations are also possible using a short-iterative Lanczos integrator. As a by-product of the improved relaxation method, it is also possible to generate a desired eigenfunction of an operator, and for smaller systems the spectrum of a Hamiltonian may be obtained by Lanczos diagonalisation of a Hamiltonian in a DVR basis. The package consists of about 60 programs, the largest and most important of which is called simply mctdh. Using keyword input read from text files, it is able to set up a system using a range of DVRs or FFT for the primitive basis, form an initial wavefunction (density operator), and propagate this wavefunction (density operator) in time. Perhaps one of the most powerful features of the program is that it uses a text input to generate the operator. In fact, if the operator has a simple analytic form, it is often possible to implement it without having to touch the code. Routines coding for more complicated functions can also be linked to the program. If potential functions are not in MCTDH form, there is the potfit program to make the transformation. It is also possible to use the CDVR method, or even to use the potential as it is, of course with resulting inefficient propagation. Time-dependent Hamiltonians can also be used. Other important programs of the package are a set of analysis tools. These include filter, which performs a filter analysis of the autocorrelation function, and flux, which does the flux analysis. Various programs plot one-- and two--dimensional graphs of the wavefunction and the potential energy surface, and simple movies can be made. Other programs can be used to check the convergence of a calculation, generate a spectrum from the autocorrelation function, etc. All plotting uses the freely available Gnuplot program, often driven using interactive menus. The package consists of more than 250,000 lines of code (mainly FORTRAN 77, some C, and including documentation and examples). The documentation consists of both on-line documentation (in HTML) and the MCTDH User's Guide (in Latex). The installation is performed by convenient installation scripts. We have run MCTDH on DEC−alpha, IBM−RS6000, Cray, Sun, Silicon Graphics and HP computers, and in particular on Linux−PC's.

The MCTDH program package is distributed on request to interested researchers. If you want to work with MCTDH, please send an e-mail to
Hans-Dieter.Meyer at

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Literature Downloads

The first description of the MCTDH algorithm was given in [1] (see the list of MCTDH publications). A reader interested in the MCTDH algorithm should start with reading [2]. The ref. [26] discusses all technical developments up to mid 1997. A comprehensive overview on MCTDH can be found in [40]. This review article, published in Physics Reports, is the most detailed description of MCTDH available. It does not only describe MCTDH as such, but also some of the analyse techniques, as well as some technical aspects (e.g. DVR's). The preprint version (LaTeX/postscript) of this review can be downloaded (see below).
More recently we wrote a feature article on MCTDH [72] which appeared in Theoretical Chemistry Accounts (TCA). This article covers all the new developments (e.g. density operator propagation) of MCTDH.

Some small errors have been found in the review published as Physics Reports 324, 1 (2000). A correction list can be downloaded from below.
(The download version of the review is already corrected).

A brief description of the MCTDH method was prepared for an CCP6 workshop (March 2001). This write-up can be downloaded from below.

In 2008 there appeared two review articles, one on multimode nonadiabatic dynamics (Worth et al) and one one computation of vibrational energies (Bowman et al), which you may find interesting. The preprint form of these articles can be downloaded from below.

In May 2009 there will appear a book on MCTDH:
Multidimensional Quantum Dynamics: MCTDH Theory and Applications The first few pages of the book can be downloaded from below.

The notes of a 2010 lecture "Introduction to MCTDH" can also be found below. These notes have recently been corrected (June 2013).

To get an impression of the MCTDH project see the MCTDH documentation. (Not all links will work! If you are a MCTDH user, please use the HTML-documentation which comes with the package.)

There is also the MCTDH Users Guide. In contrast to the review, which describes the MCTDH algorithm, the guide describes the use of the Heidelberg MCTDH program package.

Some sets of slides from talks at the MCTDH workshop July 2001 are available: The MCTDH Program: Structure and Development and Operators and the MCTDH Program

Finally, a MCTDH bibtex file and the results of some benchmark calculations can be downloaded from this site.


The following files are for download
gzipped Postscript PDF Description
CCP6, 116kB CCP6, 110kB Short description of MCTDH, prepared for an CCP6 workshop (2001)
Review, 460kB Review, 708kB MCTDH Review, preprint version, published in: Physics Reports 324 (2000), 1.
corrections, 44kB corrections, 28kB List of corrections for the review. (The download version is already corrected).
guide86, 875kB guide86, 1.4MB MCTDH Users Guide. Description of the use of the Heidelberg MCTDH Package (8.6 branch).
guide85, 875kB guide85, 1.4MB MCTDH Users Guide. Description of the use of the Heidelberg MCTDH Package (8.5 branch).
guide84, 484kB guide84, 964kB MCTDH Users Guide. Description of the use of the Heidelberg MCTDH Package (8.4 branch).
guide83, 956kB guide83, 480kB MCTDH Users Guide. Description of the use of the Heidelberg MCTDH Package (8.3 branch).
feature, 200kB feature, 252kB MCTDH Feature Article, preprint version, published in: Theor. Chem. Acc. 109 (2003), 251
conical intersection, 1.3 MB conical intersection, 2.4 MB Multidimensional dynamics involving a conical intersection: Wavepacket calculations using the MCTDH method Worth, Meyer, and Cederbaum, in: Conical Intersections, W. Domcke, D. R. Yarkony, and H. Köppel, Eds., World Scientific Co. (2004) ISBN 981-238-672-6
  MCTDH-book Multidimensional Quantum Dynamics: MCTDH Theory and Applications
H.-D. Meyer, G. A. Worth, and F. Gatti, editors
Wiley-VCH, Weinheim, April 2009. First few pages.
  bowman review Variational Quantum Approaches for Computing Vibrational Energies of Polyatomic Molecules
J. M. Bowman, T. Carrington, and H.-D. Meyer
Molecular Physics 106, 2145-2182 (2008).
  IRPC-review Using the MCTDH wavepacket ptopagation method to describe multimode non-adiabatic dynamics
G. A. Worth, H.-D. Meyer, H. Köppel, L.S. Cederbaum and I. Burghardt
International Reviews in Physical Chemistry, Vol. 27, No. 3, (2008) 569-606. (Taylor and Francis).
program, 94kB program, 127kB Slides (MCTDH workshop): The MCTDH Program: Structure and Development.
operators, 128kB operators, 183kB Slides (MCTDH workshop): Operators and the MCTDH Program.
  intro_MCTDH, 1.7MB Introduction to MCTDH, lecture notes (HDM 2010), written by DPR  (2011). Corrected/enlarged (2013,2014,2016,2017,2018).
  Numerical_Methods, 493KB Numerical Methods, DVR, Integrators,   lecture notes (HDM 2011).
  meyer_rev_2011, 1.1MB Studying molecular quantum dynamics with the multiconfiguration time-dependent Hartree (MCTDH) method . H.-D. Meyer, 2012.
Wiley Interdisciplinary Reviews: Computational Molecular Science 2 (2012) 351.
mctdh.bib.gz, 12kB mctdh.bib, 32kB Bibtex file "mctdh.bib" containing references to MCTDH articles.
MCTDH benchmarks On this directory one finds the data of some MCTDH benchmark calculations. (tar-gz)

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The MCTDH program package is distributed on request to interested researchers. If you want to work with MCTDH, please send an e-mail to
Hans-Dieter.Meyer at

We have recently investigated the multi-dimensional Henon-Heiles system with MCTDH (J.Chem.Phys. 117 (2002), 10499). As the results of these benchmark calculations may be of interest to others (e.g. for checking their method against converged fully quantal results), the data may be downloaded from MCTDH benchmarks. There you will find also the results of the Morse oscillator coupled to a bath calculation (J.Chem.Phys. 119 (2003), 24) and of the two pyrazine calculations.

If you are writing an article, the MCTDH bibtex file may be of some help. The bibtex file can be found under Literature Downloads

Prof. Dr. H.-D. Meyer, Universität Heidelberg, Im Neuenheimer Feld 229
e-mail: Hans-Dieter.Meyer at
Phone: +49 − 6221 − 54 52 10

Please note, I am already (formally) retired and cannot take new students.
I can take postdocs only if they come with their own grant.

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Latest Revision: 2024-02-05