A collocation-based multi-configuration time-dependent Hartree method using mode combination and improved relaxation

Using a pruned basis and a sparse collocation grid with more points than basis functions to do efficient and accurate MCTDH calculations with general potential energy surfaces

We propose a new collocation multi-configuration time-dependent Hartree (MCTDH) method. It reduces point-set error by using more points than basis functions. Collocation makes it possible to use MCTDH with a general potential energy surface without computing any integrals. The collocation points are associated with a basis larger than the basis used to represent wavefunctions. Both bases are obtained from a direct product basis built from single-particle functions by imposing a pruning condition. The collocation points are those on a sparse grid. Heretofore, collocation MCTDH calculations with more points than basis functions have only been possible if both the collocation grid and the basis set are direct products. In this paper, we exploit a new pseudo-inverse to use both more points than basis functions and a pruned basis and grid. We demonstrate that, for a calculation of the lowest 50 vibrational states (energy levels and wavefunctions) of CH2NH, errors can be reduced by two orders of magnitude by increasing the number of points, without increasing the basis size. This is true also when unrefined time-independent points are used.

Eigenstate calculation in the state-averaged (multi-layer) multi-configurational time-dependent Hartree approach

A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems.

Computing vibrational spectra using a new collocation method with a pruned basis and more points than basis functions: Avoiding quadrature

We present a new collocation method for computing the vibrational spectrum of a polyatomic molecule. Some form of quadrature or collocation is necessary when the potential energy surface does not have a simple form that simplifies the calculation of the potential matrix elements required to do a variational calculation. With quadrature, better accuracy is obtained by using more points than basis functions. To achieve the same advantage with collocation, we introduce a collocation method with more points than basis functions. Critically important, the method can be used with a large basis because it is incorporated into an iterative eigensolver. Previous collocation methods with more points than functions were incompatible with iterative eigensolvers. We test the new ideas by computing energy levels of molecules with as many as six atoms. We use pruned bases but expect the new method to be advantageous whenever one uses a basis for which it is not possible to find an accurate quadrature with about as many points as there are basis functions. For our test molecules, accurate energy levels are obtained even using non-optimal, simple, equally spaced points.

Using Collocation to Solve the Schrödinger Equation

We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.

A non-hierarchical correlation discrete variable representation

The correlation discrete variable representation (CDVR) facilitates (multi-layer) multi-configurational time-dependent Hartree (MCTDH) calculations with general potentials. It employs a layered grid representation to efficiently evaluate all potential matrix elements appearing in the MCTDH equations of motion. The original CDVR approach and its multi-layer extension show a hierarchical structure: the size of the grids employed at the different layers increases when moving from an upper layer to a lower one. In this work, a non-hierarchical CDVR approach, which uses identically structured quadratures at all layers of the MCTDH wavefunction representation, is introduced. The non-hierarchical CDVR approach crucially reduces the number of grid points required, compared to the hierarchical CDVR, shows superior scaling properties, and yields identical results for all three representations showing the same topology. Numerical tests studying the photodissociation of NOCl and the vibrational states of CH₃ demonstrate the accuracy of the non-hierarchical CDVR approach.

Symmetries in the multi-configurational time-dependent Hartree wavefunction representation and propagation

In multi-configurational time-dependent Hartree (MCTDH) approaches, different multi-layered wavefunction representations can be used to represent the same physical wavefunction. Transformations between different equivalent representations of a physical wavefunction that alter the tree structure used in the multi-layer MCTDH wavefunction representation interchange the role of single-particle functions (SPFs) and single-hole functions (SHFs) in the MCTDH formalism. While the physical wavefunction is invariant under these transformations, this invariance does not hold for the standard multi-layer MCTDH equations of motion. Introducing transformed SPFs, which obey normalization conditions typically associated with SHFs, revised equations of motion are derived. These equations do not show the singularities resulting from the inverse single-particle density matrix and are invariant under tree transformations. Based on the revised equations of motion, a new integration scheme is introduced. The scheme combines the advantages of the constant mean-field approach of Beck and Meyer [Z. Phys. D 42, 113 (1997)] and the singularity-free integrator suggested by Lubich [Appl. Math. Res. Express 2015, 311]. Numerical calculations studying the spin boson model in high dimensionality confirm the favorable properties of the new integration scheme.

A rectangular collocation multi-configuration time-dependent Hartree (MCTDH) approach with time-independent points for calculations on general potential energy surfaces

We introduce a collocation-based multi-configuration time-dependent Hartree (MCTDH) method that uses more collocation points than basis functions. We call it the rectangular collocation MCTDH (RC-MCTDH) method. It does not require that the potential be a sum of products. RC-MCTDH has the important advantage that it makes it simple to use time-independent collocation points. When using time-independent points, it is necessary to evaluate the potential energy function only once and not repeatedly during an MCTDH calculation. It is inexpensive and straightforward to use RC-MCTDH with combined modes. Using more collocation points than basis functions enables one to reduce errors in energy levels without increasing the size of the single-particle function basis. On the contrary, whenever a discrete variable representation is used, the only way to reduce the quadrature error is to increase the basis size, which then also reduces the basis-set error. We demonstrate that with RC-MCTDH and time-independent points, it is possible to calculate accurate eigenenergies of CH₃ and CH₄.

Direct product-type grid representations for angular coordinates in extended space and their application in the MCTDH approach

Multi-configurational time-dependent Hartree (MCTDH) calculations using time-dependent grid representations can be used to accurately simulate high-dimensional quantum dynamics on general ab initio potential energy surfaces. Employing the correlation discrete variable representation, sets of direct product type grids are employed in the calculation of the required potential energy matrix elements. This direct product structure can be a problem if the coordinate system includes polar and azimuthal angles that result in singularities in the kinetic energy operator. In the present work, a new direct product-type discrete variable representation (DVR) for arbitrary sets of polar and azimuthal angles is introduced. It employs an extended coordinate space where the range of the polar angles is taken to be [-π, π]. The resulting extended space DVR resolves problems caused by the singularities in the kinetic energy operator without generating a very large spectral width. MCTDH calculations studying the F·CH₄ complex are used to investigate important properties of the new scheme. The scheme is found to allow for more efficient integration of the equations of motion compared to the previously employed cot-DVR approach [G. Schiffel and U. Manthe, Chem. Phys. 374, 118 (2010)] and decreases the required central processing unit times by about an order of magnitude.

Using collocation to study the vibrational dynamics of molecules

In this paper, I review collocation methods for solving the time-independent and the time-dependent Schroedinger equation. Unlike traditional variational methods, collocation methods do not require integrals and quadrature. Either collocation or quadrature is necessary if the potential does not have a special form. If the basis is a direct product of univariate bases and the quadrature grid is also a direct product, there exist variational methods that do not require quadrature approximations for potential energy matrix elements. These methods, however, do require storing, in computer memory, vectors with as many components as there are quadrature points. For this reason direct-product variational methods are poor for problems with more than five atoms. There are well established ideas for reducing the size of the basis in a variational calculation. Three such ideas are: 1) prune the direct product basis; 2) use basis functions that are products of multivariate functions; 3) optimise the basis functions (e.g. Multiconfiguration time-dependent Hartree). Reducing the basis size, however, is not enough to the make variational methods tractable because, for all three of these ideas, quadrature rears its ugly head. Collocation is an attractive alternative to variational methods.