A Collocation-Based Multi-Configuration Time-Dependent Hartree Method Using Mode Combination and Improved Relaxation
Although very useful, the original multi-configuration time-dependent Hartree (MCTDH) method has two weaknesses: (1) its cost scales exponentially with the number of atoms in the system; (2) the standard MCTDH implementation requires that the PES be in sum-of-product (SOP) form in order to reduce the cost of computing integrals in the MCTDH basis. One way to deal with (1) is to lump coordinates into groups. This is called mode combination (MC). One way to deal with (2) is to reformulate MCTDH using collocation so that there are no integrals. In this paper we combine MC and collocation to formulate a mode combination collocation multiconfiguration time-dependent Hartree method (MC-C-MCTDH). In practice, its cost does not scale exponentially with the number of atoms in the system and it can be easily used with any general potential energy surfaces (PES); the PES need not be a SOP and need not have a special form. No integrals and hence no quadratures are necessary. We demonstrate the accuracy and eciency of the new method by computing vibrational energy eigenstates of the methyl radical, methane, and acetonitrile. To do this, we use MC-C-MCTDH with a variant of improved relaxation, derived by evaluating a residual at points rather than starting from a variational principle. Because the MC basis functions are multivariate, collocation points in multidimensional spaces are required. We use two types of collocation points: 1) DVR (discrete variable representation)-like points obtained from (approximate) simultaneous diagonalisation of matrices; and 2) Leja points, which are known to be good interpolation points, determined from a generalised recipe suitable for any (not necessarily polynomial-type) basis.