Gustavo Avila

Using a pruned basis, a non-product quadrature grid, and the exact Watson normal-coordinate kinetic energy operator to solve the vibrational Schrödinger equation for C2H4.

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 π(t)μπ 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 C(2)H(4). All 12 coordinates are treated explicitly. We need only ~1.52 × 10(8) quadrature points. A product Gauss grid with which one could calculate the same energy levels has at least 5.67 × 10(13) points.

Nonproduct quadrature grids for solving the vibrational Schrödinger equation.

The size of the quadrature grid required to compute potential matrix elements impedes solution of the vibrational Schrodinger 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 N(D), 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 SF(6) 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.

Solving the vibrational Schrödinger equation using bases pruned to include strongly coupled functions and compatible quadratures

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.

Solving the Schroedinger equation using Smolyak interpolants

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.

A multi-dimensional Smolyak collocation method in curvilinear coordinates for computing vibrational spectra

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.

Using multi-dimensional Smolyak interpolation to make a sum-of-products potential.

We propose a new method for obtaining potential energy surfaces in sum-of-products (SOP) form. If the number of terms is small enough, a SOP potential surface significantly reduces the cost of quantum dynamics calculations by obviating the need to do multidimensional integrals by quadrature. The method is based on a Smolyak interpolation technique and uses polynomial-like or spectral basis functions and 1D Lagrange-type functions. When written in terms of the basis functions from which the Lagrange-type functions are built, the Smolyak interpolant has only a modest number of terms. The ideas are tested for HONO (nitrous acid).

Applying a Smolyak collocation method to Cl2CO

ABSTRACT Phosgene (Cl2C=O) is extremely poisonous, underscoring the importance of accurate infrared detection. Here, the computed vibrational energy levels of phosgene are reported for the first time from a six-dimensional potential energy surface (PES) that was constructed from 25,000 single-point energy computations at the CCSD(T)/cc-pVTZ level of theory. The computed points were fit using a neural network method, and the resulting PES was employed in the determination of vibrational energies and wavefunctions. Bond coordinates were utilised in conjunction with a collocation method to minimise problems that arise from the complicated nature of the kinetic energy operator. The collocation method makes possible the computation of energy levels without integral evaluation, and without the need to solve a generalised eigenvalue problem. Moreover, it is built on a nondirect product-pruned basis that is much smaller than the direct product basis that would be required to obtain the same accuracy.

Reducing the cost of using collocation to compute vibrational energy levels: Results for CH2NH

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.

Computing vibrational energy levels of CH4 with a Smolyak collocation method

In this paper, we demonstrate that it is possible to apply collocation to compute vibrational energy levels of a five-atom molecule using an exact kinetic energy operator (with cross terms and coordinate-dependent coefficients). This is made possible by using (1) a pruned basis of products of univariate functions; (2) a Smolyak grid made from nested sequences of grids for each coordinate; (3) a collocation method that obviates the need to solve a generalized eigenvalue problem; (4) an efficient sequential transformation between the (nondirect product) grid and the (nondirect product) basis representations; and (5) hierarchical univariate functions that make it possible to avoid storing large intermediate vectors. The accuracy of the method is confirmed by computing 500 vibrational energy levels of methane.

Comparing Nested Sequences of Leja and PseudoGauss Points to Interpolate in 1D and Solve the Schroedinger Equation in 9D

In this article, we use nested sets of weighted Leja points, which have previously been studied as interpolation points, as collocation points to solve a 9D vibrational Schroedinger equation. Collocation has the advantage that it obviates the need to compute integrals with quadrature. A multi-dimension sparse grid is built from the Leja points and Hermite-type basis functions by restricting sparse grid levels i c using ∑ c g c (i c ) ≤ H, where g c (i c ) is a non-decreasing function and H is a parameter that controls the accuracy. Results obtained with Leja points are compared to those obtained with PseudoGauss points. PseudoGauss points are also nested. They are chosen to improve the accuracy of the Gram matrix. With both Leja and PseudoGauss points it is possible to add one point per level. We also compare Lebesgue constants for weighted Leja and PseudoGauss points.