The correlation discrete variable representation revisited

Abstract

The correlation discrete variable representation (CDVR) enables efficient quantum dynamics calculation with the multi-layer multi-configurational time-dependent Hartree (MCTDH) approach on general potential energy surfaces. It employs a time-dependent quadrature to compute potential energy matrix elements, thereby eliminating the need to refit the potential to a sum of products form. The non-hierarchical CDVR conserves the inherent symmetry properties of tree-shaped wavefunction representations and drastically reduces the number of grid points compared to the original hierarchical CDVR. However, it requires projection on the space spanned by the single-hole functions (SHFs) at each node of the tree, which can introduce unphysical couplings for unconverged basis sets. In this work, the non-hierarchical CDVR is revisited and a revised approach that avoids explicit projection on the single-hole space is introduced. The computational costs of the revised approach scale favorably with the number of single-particle functions (SPFs): for a tree with three edges at each node and $n$ SPFs at each edge, a n^4 scaling is achieved. Furthermore, a revised scheme that uses artificial SPFs to systematically increase the accuracy of the CDVR quadrature is presented. Computations studying the photodissociation of NOCl, the vibrational states of methyl, and the non-adiabatic quantum dynamics of photoexcited pyrazine demonstrate the accuracy and efficiency of the revised non-hierarchical CDVR. Notably, for the 24-dimensional pyrazine system the use of the CDVR does not increase the required CPU time compared to calculations utilizing the sum of products form of the vibronic coupling model.

Figures (6)

• Figure 1: The multi-layer MCTDH approach: the grouping of physical coordinates $x_i$ into logical coordinates ${\bf q}^{l;\kappa_1,..,\kappa_{l-1}}$ and the labeling of notes and edges are illustrated.
• Figure 2: Photodissociation of NOCl: real part of the overlap of a MCTDH wavefunction and the numerically exact reference wavefunction. Results of different types of MCTDH calculations are shown using different line styles: solid lines for calculations using exact potential energy matrix elements, dashed lines for calculations using the original non-hierarchical CDVR of Refs.ElM5ElHoM2, and plus signs for calculations using the revised non-hierarchical CDVR. MCTDH calculations with different SPF numbers $n$ are displayed using different colors as indicated in the legend.
• Figure 3: Real part of the overlap $\langle \psi_{B6,SOP}(t) | \psi(t) \rangle$ between the MCTDH wavefunctions $\psi(t)$ obtained by different calculations with reference results obtained with the basis R and a SOP representation of the Hamiltonian: Panels (a) to (d) show results of MCTDH calculations using a SOP representation, the original non-hierarchical CDVR approach ElHoM2, the revised non-hierarchical CDVR approach of the present work, and the revised non-hierarchical CDVR approach with optimized unoccupied SPFs (see text for details), respectively. Results obtained with different SPF basis sets size as indicated in the legend are displayed.
• Figure 4: The absolute values of the autocorrelation function obtained by different MCTDH calculations are displayed as a function of time. Panels (a) to (d) show results of MCTDH calculations using a SOP representation, the original non-hierarchical CDVR approach ElM5ElHoM2, the revised non-hierarchical CDVR approach of the present work, and the revised non-hierarchical CDVR approach with optimized unoccupied SPFs (see text for details), respectively. Results obtained with different SPF basis sets size as indicated in the legend are displayed.
• Figure 5: The population of the second diabatic electronic state computed by different MCTDH calculations with different basis sets (indicated in the legend) is displayed as a function of time. Panels (a) to (d) show results of MCTDH calculations using a SOP representation, the original non-hierarchical CDVR approachElM5ElHoM2, the revised non-hierarchical CDVR approach of the present work, and the revised non-hierarchical CDVR approach with optimized unoccupied SPFs (see text for details), respectively. Results obtained with different SPF basis sets size as indicated in the legend are displayed.