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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 potential energy surface (PES) be in the 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 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 MC collocation multi-configuration time-dependent Hartree (MC-C-MCTDH) method. In practice, its cost does not scale exponentially with the number of atoms, and it can be used with any general PES; the PES need not be an SOP and need not have a special form. No integrals and, hence, no quadratures are necessary. We demonstrate the accuracy and efficiency of the new method by computing vibrational energy eigenstates of 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. Because the MC basis functions are multivariate, collocation points in multi-dimensional spaces are required. We use two types of collocation points: (1) discrete variable representation-like points obtained from (approximate) simultaneous diagonalization of matrices and (2) Leja points, which are known to be good interpolation points, determined from a generalized recipe suitable for any basis. [ABSTRACT FROM AUTHOR]

Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH. [ABSTRACT FROM AUTHOR]

We propose a pruned multi-configuration time-dependent Hartree (MCTDH) method with systematically expanding nondirect product bases and use it to solve the time-independent Schrödinger equation. No pre-determined pruning condition is required to select the basis functions. Using about 65 000 basis functions, we calculate the first 69 vibrational eigenpairs of acetonitrile, CH3CN, to an accuracy better than that achieved in a previous pruned MCTDH calculation which required more than 100 000 basis functions. In addition, we compare the new pruned MCTDH method with the established multi-layer MCTDH (ML-MCTDH) scheme and determine that although ML-MCTDH is somewhat more efficient when low or intermediate accuracy is desired, pruned MCTDH is more efficient when high accuracy is required. In our largest calculation, the vast majority of the energies have errors smaller than 0.01 cm-1. [ABSTRACT FROM AUTHOR]

In this paper, we propose a pruned, nondirect product multi-configuration time dependent Hartree (MCTDH) method for solving the Schrödinger equation. MCTDH uses optimized 1D basis functions, called single particle functions, but the size of the standard direct product MCTDH basis scales exponentially with D, the number of coordinates. We compare the pruned approach to standard MCTDH calculations for basis sizes small enough that the latter are possible and demonstrate that pruning the basis reduces the CPU cost of computing vibrational energy levels of acetonitrile (D = 12) by more than two orders of magnitude. Using the pruned method, it is possible to do calculations with larger bases, for which the cost of standard MCTDH calculations is prohibitive. Pruning the basis complicates the evaluation of matrix-vector products. In this paper, they are done term by term for a sum-of-products Hamiltonian. When no attempt is made to exploit the fact that matrices representing some of the factors of a term are identity matrices, one needs only to carefully constrain indices. In this paper, we develop new ideas that make it possible to further reduce the CPU time by exploiting identity matrices. [ABSTRACT FROM AUTHOR]

A rigorous and efficient approach for the calculation of eigenstates in polyatomic molecular systems with potentials displaying multiple wells is introduced. The scheme is based on the multi-configurational time-dependent Hartree (MCTDH) approach and uses multiple MCTDH wavefunctions with different single-particle function bases to describe the quantum dynamics in the different potential wells. More specifically, an iterative block Lanczos-type diagonalization scheme utilizing state-averaged MCTDH wavefunctions localized in different wells is employed to obtain the energy eigenvalues and eigenstates. The approach does not impose any formal restriction on the symmetry of the potential or the number of wells. A seven-dimensional model system of tetrahedral symmetry, which is inspired by A·CH4 type complexes and displays four equivalent potential minima, is used to study the numerical performance of the new approach. It is found that the number of configurations in the MCTDH wavefunctions required to obtain converged results is decreased by roughly one order of magnitude compared to standard MCTDH calculations employing a block-relaxation scheme. [ABSTRACT FROM AUTHOR]

We present a new collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrödinger equation required to compute (ro-)vibrational spectra, photodissociation cross sections, reaction rate constants, etc., that can be used with general potential energy surfaces. Collocation obviates the need for quadrature and facilitates using complicated kinetic energy operators. When the basis is good, the accuracy of collocation solutions to the Schrödinger equation is not sensitive to the choice of the collocation points. We test the collocation MCTDH equations we derive by showing that they can be used to compute accurate vibrational energy levels of CH3. It is possible to choose (imaginary) time-independent collocation points with which collocation-based MCTDH energies are accurate. It is therefore not necessary to calculate potential values many times during the propagation. [ABSTRACT FROM AUTHOR]

A schemeto efficiently calculate ro-vibrational (J> 0)eigenstates within the framework of the multiconfigurational time-dependentHartree (MCTDH) approach is introduced. It employsa basis of MCTDH wave packets which is generated in the calculationof vibrational (J= 0) eigenstates via existing MCTDH-basediterative diagonalization approaches. The subsequent ro-vibrationalcalculations for total angular momenta J> 0 usedirect products of these wave packets and the Wigner rotation matrices.In this ro-vibrational basis, the Hamiltonian matrix can be computedand diagonalized with minor numerical effort for any value of J. Accurate ro-vibrational states are obtained if the numberof iterations in the J= 0 calculations and the basisset sizes in the MCTDH wave function representation are converged.Test calculations studying CH2D show that ro-vibrationaleigenstates for moderately large Jcan be convergedwithin wavenumber accuracy with the same MCTDH basis sets and onlyslightly increased iteration counts compared to purely vibrational(J= 0) calculations. If large J’s are considered or very high accuracies are required, thenumber of iterations required to obtain convergence increases significantly.Comparing the theoretical results with experimental data for the out-of-planebend, symmetric stretch, and antisymmetric stretch fundamentals, theaccuracy of the ab initio potential energy surface employed is investigated. [ABSTRACT FROM AUTHOR]

Motivated by recent photodetachment experiments studyingresonance structures in the transition-state region of the F + CH4→ HF + CH3reaction, the vibrational dynamicsof the precursor complex CH4·F–isinvestigated. Delocalized vibrational eigenstates of CH4·F–are computed in full dimensionality employingthe multiconfigurational time-dependent Hartree (MCTDH) approach anda recently developed iterative diagonalization approach for generalmultiwell systems. Different types of stereographic coordinates areused, and a corresponding general N-body kineticenergy operator is given. The calculated tunneling splittings of theground and the lower vibrational excited states of the CH4·F–complex do not significantly exceed 1cm–1. Comparing the converged MCTDH results forlocalized vibrational excitations with existing results obtained bynormal-mode-based (truncated) vibrational configuration interactioncalculations, significantly lower frequencies are found for excitationsin the intermolecular modes. [ABSTRACT FROM AUTHOR]