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Methodological progress is reported in the challenging direction of a black-box-type variational solution of the (ro)vibrational Schrödinger equation applicable to floppy, polyatomic systems with multiple large-amplitude motions. This progress is achieved through the combination of (i) the numerical kinetic-energy operator (KEO) approach of Mátyus et al. [J. Chem. Phys. 130, 134112 (2009)] and (ii) the Smolyak nonproduct grid method of Avila and Carrington, Jr. [J. Chem. Phys. 131, 174103 (2009)]. The numerical representation of the KEO makes it possible to choose internal coordinates and a body-fixed frame best suited for the molecular system. The Smolyak scheme reduces the size of the direct-product grid representation by orders of magnitude, while retaining some of the useful features of it. As a result, multidimensional (ro)vibrational states are computed with system-adapted coordinates, a compact basis- and grid-representation, and an iterative eigensolver. Details of the methodological developments and the first numerical applications are presented for the CH4·Ar complex treated in full (12D) vibrational dimensionality. [ABSTRACT FROM AUTHOR]

Molecular rotation, vibration, internal rotation, isomerization, tunneling, intermolecular dynamics of weakly and strongly interacting systems, intra-to-inter-molecular energy transfer, hindered rotation and hindered translation over surfaces are important types of molecular motions. Their fundamentally correct and detailed description can be obtained by solving the nuclear Schrödinger equation on a potential energy surface. Many of the chemically interesting processes involve quantum nuclear motions which are 'delocalized' over multiple potential energy wells. These 'large-amplitude' motions in addition to the high dimensionality of the vibrational problem represent challenges to the current (ro)vibrational methodology. A review of the quantum nuclear motion methodology is provided, current bottlenecks of solving the nuclear Schrödinger equation are identified, and solution strategies are reviewed. Technical details, computational results, and analysis of these results in terms of limiting models and spectroscopically relevant concepts are highlighted for selected numerical examples. [ABSTRACT FROM AUTHOR]

In this paper we propose and test a method for computing numerically exact vibrational energy levels of a molecule with six atoms. We use a pruned product basis, a non-product quadrature, the Lanczos algorithm, and the exact normal-coordinate kinetic energy operator (KEO) with the πtμπ term. The Lanczos algorithm is applied to a Hamiltonian with a KEO for which μ is evaluated at equilibrium. Eigenvalues and eigenvectors obtained from this calculation are used as a basis to obtain the final energy levels. The quadrature scheme is designed, so that integrals for the most important terms in the potential will be exact. The procedure is tested on C2H4. All 12 coordinates are treated explicitly. We need only ∼1.52 × 108 quadrature points. A product Gauss grid with which one could calculate the same energy levels has at least 5.67 × 1013 points. [ABSTRACT FROM AUTHOR]

In this paper we propose a new quadrature scheme for computing vibrational spectra and apply it, using a Lanczos algorithm, to CH3CN. All 12 coordinates are treated explicitly. We need only 157'419'523 quadrature points. It would not be possible to use a product Gauss grid because 33 853 318 889 472 product Gauss points would be required. The nonproduct quadrature we use is based on ideas of Smolyak, but they are extended so that they can be applied when one retains basis functions θn1(r1)...θnD(rD) that satisfy the condition α1n1 + ··· + αDnD <= b, where the αk are integers. We demonstrate that it is possible to exploit the structure of the grid to efficiently evaluate the matrix-vector products required to use the Lanczos algorithm. [ABSTRACT FROM AUTHOR]

The size of the quadrature grid required to compute potential matrix elements impedes solution of the vibrational Schrödinger equation if the potential does not have a simple form. This quadrature grid-size problem can make computing (ro)vibrational spectra impossible even if the size of the basis used to construct the Hamiltonian matrix is itself manageable. Potential matrix elements are typically computed with a direct product Gauss quadrature whose grid size scales as ND, where N is the number of points per coordinate and D is the number of dimensions. In this article we demonstrate that this problem can be mitigated by using a pruned basis set and a nonproduct Smolyak grid. The constituent 1D quadratures are designed for the weight functions important for vibrational calculations. For the SF6 stretch problem (D=6) we obtain accurate results with a grid that is more than two orders of magnitude smaller than the direct product Gauss grid. If D>6 we expect an even bigger reduction. [ABSTRACT FROM AUTHOR]

In this paper, we present a new collocation method for solving the Schroedinger equation. Collocation has the advantage that it obviates integrals. All previous collocation methods have, however, the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant, we device a collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions are increased. [ABSTRACT FROM AUTHOR]

In this paper, we present new basis pruning schemes and compatible quadrature grids for solving the vibrational Schrödinger equation. The new basis is designed to include the product basis functions coupled by the largest terms in the potential and important for computing low-lying vibrational levels. To solve the vibrational Schrödinger equation without approximating the potential, one must use quadrature to compute potential matrix elements. For a molecule with more than five atoms, the use of iterative methods is imperative, due to the size of the basis and the quadrature grid. When using iterative methods in conjunction with quadrature, it is important to evaluate matrix-vector products by doing sums sequentially. This is only possible if both the basis and the grid have structure. Although it is designed to include only functions coupled by the largest terms in the potential, the new basis and also the quadrature for doing integrals with the basis have enough structure to make efficient matrix-vector products possible. When results obtained with a multimode approximation to the potential are accurate enough, full-dimensional quadrature is not necessary. Using the quadrature methods of this paper, we evaluate the accuracy of calculations made by making multimode approximations. [ABSTRACT FROM AUTHOR]