The Multi-configuration Time-Dependent Hartree Method

The SPFs t) are d-dimensional functions, i.e d system coordinates have 

The MCTDH ansatz Eq. (4.22) is not unique. Applying linear transformations among the SPFs 

In order to ensure the uniqueness of the MCTDH equations of motion, constraints are introduced on the SPFs 

From this last equation, it follows that if the constraint operator g acting on the Kth particle, are chosen to be Hermitian, then a set of orthonormal SPF will remain orthonormal at all time 

Assuming the simplest choice for the constraints, i.e 3 = 0, the MCTDH equations of motion can be derived by inserting the MCTDH ansatz of Eq. (4.22) into the Dirac-Frenkel time-dependent variational principle of Eq. (4.20). Other choices are possible for the constraints, see Ref. [28] for details. We first introduce the projector on the space spanned by the SPFs of the k particle 

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When considering the femtosecond photoionization dynamics of complex systems, a completely exact evaluation of the time and energy resolved photoelectron spectrum is often not really necessary. Approximative schemes which require significantly lower computational effort are valuable in such cases. Within the nonperturbative formalism, Meier et al. have proposed an efficient computational scheme which incorporates the multi-configuration time-dependent Hartree method.An approximate method which is based on a classical-trajectory description of the nuclear dynamics has been elaborated by Hartmann, Heidenreich, Bonacic-Koutecky and coworkers and applied, among other systems,to the time-resolved photoionization spectroscopy of conical intersections in sodium fluoride clusters. 

M. Sala, F. Gatti and S. Guerin, Coherent destruction of tunneling in a six-dimensional model of NHD2 a computational study using the multi-configuration time-dependent Hartree method , J. Chem. Phys. 141, 164326 (2014)... 

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