# Department of Chemistry Faculty Publications

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Item Computing Vibrational Spectra Using a New Collocation Method With a Pruned Basis and More Points Than Basis Functions: Avoiding Quadrature(AIP Publishing, 2023-04-13) Simmons, Jesse; Carrington, Tucker Jr.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.Item Orders of Coupling Representations as a Versatile Framework for Machine Learning from Sparse Data in High-Dimensional Spaces(Elsevier, 2023-07-17) Manzhos, Sergei; Carrington, Tucker Jr.; Ihara, ManabuMachine learning (ML) techniques are already widely and increasingly used in diverse applications in science and technology, including computational chemistry. Specifically in computational chemistry, neural networks (NN) and kernel methods such as Gaussian process regressions (GPR) have been increasingly used for the construction of potential functions and functionals for density functional theory. While ML techniques have a number of advantages vs intuition-based models, notably their generality and black-box nature, they are still challenged when faced with high dimensionality of the feature space or low and uneven data density – in part because of their general nature. We review recent works using methods such as NNs and GPR as building blocks of composite methods in the framework of an expansion over orders of coupling. We introduce models using NN or GPR-based components as part of HDMR (high-dimensional model representations)-based structures. HDMR is a formalization of orders-of-coupling representations that include the many-body and N-mode representations well known in computational chemistry and allows, in particular, building all terms from one dataset of arbitrarily distributed data. The resulting HDMR-NN and HDMR-GPR combinations and NN with HDMR-GPR derived neuron activation functions not requiring non-linear optimization enhance machine learning capabilities in high dimensional spaces and or with sparse data.Item Computing Excited OH Stretch States of Water Dimer in 12D Using Contracted Intermolecular and Intramolecular Basis Functions(AIP Publishing, 2023-01-24) Wang, Xiao-Gang; Carrington, Tucker Jr.Due to the ubiquity and importance of water, water dimer has been intensively studied. Computing the (ro-)vibrational spectrum of water dimer is challenging. The potential has eight wells separated by low barriers, which makes harmonic approximations of limited utility. A variational approach is imperative, but difficult because there are 12 coupled vibrational coordinates. In this paper, we use a product contracted basis whose functions are products of intramolecular and intermolecular functions computed using an iterative eigensolver. An intermediate matrix F facilitates calculating matrix elements. Using F, it is possible to do calculations on a general potential without storing the potential on the full quadrature grid. We find that surprisingly many intermolecular functions are required. This is due to the importance of coupling between inter- and intra-molecular coordinates. The full G16 symmetry of water dimer is exploited. We calculate, for the first time, monomer excited stretch states and compare P(1) transition frequencies with their experimental counterparts. We also compare with experimental vibrational shifts and tunneling splittings. Surprisingly, we find that the largest tunneling splitting, which does not involve the interchange of the two monomers, is smaller in the asymmetric stretch excited state than in the ground state. Differences between levels we compute and those obtained with a [6+6]D adiabatic approximation [Leforestier et al. J. Chem. Phys. 137 014305 (2012)] are ∼0.6 cm−1 for states without monomer excitation, ∼4 cm−1 for monomer excited bend states, and as large as ∼10 cm−1 for monomer excited stretch states.Item Computing Vibrational Energy Levels Using a Canonical Polyadic Tensor Method with a Fixed Rank and a Contraction Tree(AIP Publishing, 2023-06-01) Kallullathil, Sangeeth Das; Carrington, Tucker Jr.In this paper, we use the previously introduced CP-Multiple Shift Block Inverse Iteration (MSBII) eigensolver [J. Chem. Phys. 155, 234105 (2021)] in conjunction with a contraction tree to compute vibrational spectra. The CP-MSBII eigensolver uses the Canonical Polyadic (CP) format. The memory cost scales linearly with the number of coordinates. A tensor in CP format represents a wavefunction constrained to be a sum of products (SOP). An SOP wavefunction can be made more and more accurate by increasing the number of terms, the rank. When the required rank is large, the runtime of a calculation in CP format is long, although the memory cost is small. To make the method more efficient we break the full problem into pieces using a contraction tree. The required rank for each of the sub-problems is small. To demonstrate the effectiveness of the ideas, we computed vibrational energy levels of acetonitrile (12-D) and ethylene oxide (15-D).Item Computing vibrational spectra using a new collocation method with a pruned basis and more points than basis functions: avoiding quadrature(2023-02-14) Simmons, Jesse; Carrington, TuckerWe 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 6 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.