Multi-layer Multi-configuration time-dependent Hartree (ML-MCTDH) is a method widely used for simulation of vibrationally resolved electronic spectra that incorporate the coupling between vibrational modes and the electronic structure of the system.

What is the largest system that has been simulated using ML-MCTDH, recorded in literature?


2 Answers 2


When you say MCTDH, I am assuming you mean the vanilla MCTDH without employing the multi-layer structure. In which case 24 modes seem to be the best known result (higher modes can be done, I have tried things myself with more than 24 modes, but they are unpublished results!) However, the true power of MCTDH comes with ML-MCTDH, which is sort of adding another layer of MCTDH on top of MCTDH and so on (think tensor networks).

With ML-MCTDH, for isolated gas phase molecules, photoelectron spectra have been simulated for naphthalene (48D) and anthracene (66D)! [J. Chem. Phys., 138, 014313 (2013)]

  • $\begingroup$ +1 great answer, and good that you contributed your knowledge from personal experience. This is a good place to mention unpublished results now that we have the "cite" button and can have our answers appear on Google Scholar! In the case of 189 modes, how many electronic states were there, how many levels were included in each mode, and did they calculate vibrationally resolved electronic spectra? For napthalene I see that 3N-6 = 48, but how many electronic states and how many vibrational states were included? $\endgroup$ Commented May 13, 2020 at 1:01
  • $\begingroup$ Both the studies mentioned have performed "full-dimensional" calculations, i.e. all the vibrational modes have been included. For the photoelectron spectrum of Napthalene and Anthracene, 6 doublet electronic states were considered in the simulations! $\endgroup$ Commented May 13, 2020 at 1:06
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    $\begingroup$ Since MCTDH solves time-dependent Schrodinger equation (TDSE), it does not make a direct reference to vibrational levels. What one specifies is number of quanta in each mode which defines the so-called "primitive basis". This depends on the problem and the importance of a ptarticular vibrational mode. For the case of Napthalene and Anthracene, most modes have between 7-12 quanta, which would be sufficient to model photoelectron spectra, which is a short-time process ) and thus only Franck-Condon region needs to be captured). $\endgroup$ Commented May 13, 2020 at 1:42
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    $\begingroup$ In that case 80 modes and 2 electronic states were used 20 years ago, and 1000 modes and 2 electronic states were used 17 years ago (see Fig. 1), which seems to be far more than (6 electronic x 66 nuclear). $\endgroup$ Commented May 13, 2020 at 2:13
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    $\begingroup$ Number of electronic states do not play a big part in computational expense as that is a multiplicative factor, as opposed to number of modes which gives rise to exponential increase, causing the famous "curse of dimensionality". $\endgroup$ Commented May 13, 2020 at 2:18

The developers of MCTDH from Heidelberg reported in the year 2000 [1]

The largest system treated with MCTDH to date is the pyrazine molecule, where all 24 (!) vibrational modes were accounted for. The particular representation of the MCTDH wavefunction requires special techniques for generating an initial wavepacket and for analysing the propagated wavefunction

However, this is almost certainly outdated (20 years old). I'm just posting this here to get the ball rolling in case anyone else knows better.


  1. M.H.Beck, A.J├Ąckle, G.A.Worth, H.-D.Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets, Physics Reports, 2000, 324 DOI:10.1016/S0370-1573(99)00047-2
  • $\begingroup$ Did they do vibrationally resolved electronic spectra in that paper though? $\endgroup$ Commented May 12, 2020 at 21:51

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