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Extension to the multiset formalism, important for non-adiabatic calculations can be made, remembering that the single-particle function sets of the various electronic states are not orthonormal to each other. 2. g. erent integrators, or the comparison of a numerically exact wavefunction with an MCTDH one. erences of MCTDH wavefunctions can be simply evaluated. erence between two wavefunctions, """ ! I "" , (159) can be evaluated using the relationship "" ! 2Re1 " I 2 , (160) and use then made of Eq.
As a set of ab initio points) the product representation scheme can also be used to interpolate multi-dimensional potential energy surfaces (see Ref.  for more details). ort in terms of CPU time and memory. This has been shown in Ref.  in the example of a Coulomb potential. 8. Applications In order to illustrate the product representation scheme of potential energy surfaces, we will give some example calculations. The "rst one treats the S -potential energy surface used for the photo-dissociation process of NOCl .
Iterative optimisation In contrast to the two-dimensional case the expansion (95) is in general not optimal for more than two degrees of freedom, and there may exist another product basis for which the L-error may be smaller for the same set of expansion orders +m ,. In order to arrive at these optimal G potential functions one may iteratively improve the natural potentials by a multidimensional optimisation procedure. H. Beck et al. / Physics Reports 324 (2000) 1}105 where * G denotes the natural potentials obtained by diagonalisation of the density matrix G (see H Eq.
Physics Reports vol.324