Programs for generating a cluster expansion

Cluster-expansion of a potential energy surface

For many systems with more then a few degrees of freedom the potential is not given in a form suitable for MCTDH calculations. In this case one can expand the exact potential in terms of n-particle interaction terms called the clusters. The expansion can be developed around several so-called reference points Qref,i. The approximated potential is then given as a (weighted) sum over all reference points.

There are several flavors of cluster-expansions, for instance, expansion around a single reference point or a reference coordinate (with the other coordinates fixed to possibly different values). The implementation of the programs documented here does not strictly distinguish between those two, instead the user is free to mix them. A cluster or n-particle interaction term is identified only by the coordinates or combined modes it spans. The clusters form a hierarchical tree of n-particle interaction terms with several layers. The position, i.e, layer, in the tree is determined by the number of (combined) modes the cluster spans. This is independent from whether or not a combined mode is included completely or just partly. The hierarchy of clusters must have one single root term, i.e., a single cluster which is of the lowest order - usually this is a cluster containing a single point or mode. Here, mode can refer to a single degree of freedom ('coordinate') or a combined mode (' logical coordinate')

Generation of clusters

The clusters are generated in hierarchical order, starting from the lowest order cluster, the root of the hierarchy. This cluster is given as the cut through the potential energy surface (PES), i.e., the potential energy at the reference point or reference mode with all other coordinates fixes to the reference values. In case of a reference mode the cut is coordinate dependent. In a next step, for all clusters being one layer up the hierarchy, also PES cuts are obtained, where the coordinate dependence is now extended to at least one further mode. From those PES cuts now the lowest order cluster is subtracted. This procedure is repeated for all layers in the hierarchy. In general, the algorithm works as follows:  1. Calculate PES cuts for all clusters in a layer.  2. Obtain clusters for this layer by subtracting all lower layer clusters from the PES cut.  3. Repeat in the next higher level.


This package defines a number of tools to find a particular cluster-expansion and relevant terms of the expansion and - when found - to sample those clusters on a DVR grid. To find a suitable cluster expansion the tools for statistical analysis can be used. Here a trial expansion can be tested for convergence. If a suitable expansion is found, the clusters of interest can be sampled on a DVR grid and be potfitted. Finally the complete approximated potential sampled and potfitted within MCTDH can be checked against the exact potential.

All programs are written in Python and rely on the NumPy and SciPy packages which are open-source and freely available for download. Both packages are included in many Linux distributions but usually not installed by default. Note: for Apple computers the Numpy/SciPy developers strongly recommend using the official Python distribution and not the version shipped with Mac OS X.

Statistical analysis - data generation.

With the Python script a Monte Carlo algorithm is used to explore the potential energy surface in the most relevant regions. Alternatively an already given trajectory of coordinate values can be used. The main purpose of this script is the calculation of the contribution of the clusters along the trajectory. This data can then be further analyzed by the script

Statistical analysis - data analysis

The data produced by can be further analyzed with the Python script This allows detailed statistics for single clusters, a particular expansion etc.

Sampling of clusters on a DVR grid

Generating the relevant clusters on a DVR grid is performed with the python script The script can also create (initial) potfits of the clusters.

Checking a PES file

To check an already processed PES by MCTDH, the script may be used. Here again a Monte Carlo sampling is used to test the potential energy surface against the exact potential. Alternatively an already given trajectory of DVR indices can be used.

Technical Remarks

The current implementation has been tested with Python versions 2.4.4 and 2.5.2 on Linux machines and assumes that the path to the Python intepreter is found in /usr/bin/env python. On some architectures this could be different. In this case the path can be changed in the first line of the Python scripts in $MCTDH_DIR/bin/python/clusters/.

All scripts rely on many features of the NumPy/SciPy package which is not installed by default on many platforms. Please make sure that NumPy 1.1 or later and a recent version of SciPy are installed.
Note: for Apple computers the Numpy/SciPy developers strongly recommend using the official Python distribution and not the version shipped with Mac OS X.