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In this paper, we improve the collocation method for computing vibrational spectra that was presented in the work of Avila and Carrington, Jr. [J. Chem. Phys. 143, 214108 (2015)]. Known quadrature and collocation methods using a Smolyak grid require storing intermediate vectors with more elements than points on the Smolyak grid. This is due to the fact that grid labels are constrained among themselves and basis labels are constrained among themselves. We show that by using the so-called hierarchical basis functions, one can significantly reduce the memory required. In this paper, the intermediate vectors have only as many elements as the Smolyak grid. The ideas are tested by computing energy levels of CH2NH. [ABSTRACT FROM AUTHOR]

In this paper, we improve the collocation method for computing vibrational spectra that was presented in Avila and Carrington, Jr. [J. Chem. Phys. 139, 134114 (2013)]. Using an iterative eigensolver, energy levels and wavefunctions are determined from values of the potential on a Smolyak grid. The kinetic energy matrix-vector product is evaluated by transforming a vector labelled with (nondirect product) grid indices to a vector labelled by (nondirect product) basis indices. Both the transformation and application of the kinetic energy operator (KEO) scale favorably. Collocation facilitates dealing with complicated KEOs because it obviates the need to calculate integrals of coordinate dependent coefficients of differential operators. The ideas are tested by computing energy levels of HONO using a KEO in bond coordinates. [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]

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]