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Manzhos S, Ihara M, Carrington T. Using Collocation to Solve the Schrödinger Equation. J Chem Theory Comput 2023; 19:1641-1656. [PMID: 36974479 DOI: 10.1021/acs.jctc.2c01232] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/10/2023]
Abstract
We review the collocation approach to the solution of the Schrödinger equation and its uses in applications. Interrelations between collocation and other methods are highlighted. We also stress advantages and disadvantages of the rectangular collocation formulation. Using collocation makes it possible to use any, e.g. optimized, coordinates and basis functions, including nonintegrable basis functions, and provides a straightforward way of dealing with singularities in the potential. In addition, we stress that using collocation facilitates tuning the shape of basis functions and the placement of points, both of which can be done with machine-learning methods. Applications to electronic and vibrational problems are reviewed focusing on calculations for molecules on surfaces for which there are few variational calculations. Collocation has advantages when potential energy surfaces are unavailable, in particular, for molecule-surface systems, and for systems for which standard direct product quadrature grids, often used with variational methods, are costly.
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Affiliation(s)
- Sergei Manzhos
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan
| | - Manabu Ihara
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan
| | - Tucker Carrington
- Department of Chemistry, Queen’s University, 90 Bader Lane, Kingston, Ontario K7L 3N6, Canada
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Manzhos S, Ihara M. The loss of the property of locality of the kernel in high-dimensional Gaussian process regression on the example of the fitting of molecular potential energy surfaces. J Chem Phys 2023; 158:044111. [PMID: 36725493 DOI: 10.1063/5.0136156] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023] Open
Abstract
Kernel-based methods, including Gaussian process regression (GPR) and generally kernel ridge regression, have been finding increasing use in computational chemistry, including the fitting of potential energy surfaces and density functionals in high-dimensional feature spaces. Kernels of the Matern family, such as Gaussian-like kernels (basis functions), are often used which allow imparting to them the meaning of covariance functions and formulating GPR as an estimator of the mean of a Gaussian distribution. The notion of locality of the kernel is critical for this interpretation. It is also critical to the formulation of multi-zeta type basis functions widely used in computational chemistry. We show, on the example of fitting of molecular potential energy surfaces of increasing dimensionality, the practical disappearance of the property of locality of a Gaussian-like kernel in high dimensionality. We also formulate a multi-zeta approach to the kernel and show that it significantly improves the quality of regression in low dimensionality but loses any advantage in high dimensionality, which is attributed to the loss of the property of locality.
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Affiliation(s)
- Sergei Manzhos
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan
| | - Manabu Ihara
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan
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Manzhos S, Tsuda S, Ihara M. Machine learning in computational chemistry: interplay between (non)linearity, basis sets, and dimensionality. Phys Chem Chem Phys 2023; 25:1546-1555. [PMID: 36562317 DOI: 10.1039/d2cp04155c] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Machine learning (ML) based methods and tools have now firmly established themselves in physical chemistry and in particular in theoretical and computational chemistry and in materials chemistry. The generality of popular ML techniques such as neural networks or kernel methods (Gaussian process and kernel ridge regression and their flavors) permitted their application to diverse problems from prediction of properties of functional materials (catalysts, solid state ionic conductors, etc.) from descriptors to the building of interatomic potentials (where ML is currently routinely used in applications) and electron density functionals. These ML techniques are assumed to have superior expressive power of nonlinear methods, and are often used "as is", with concepts such as "non-parametric" or "deep learning" used without a clear justification for their need or advantage over simpler and more robust alternatives. In this Perspective, we highlight some interrelations between popular ML techniques and traditional linear regressions and basis expansions and demonstrate that in certain regimes (such as a very high dimensionality) these approximations might collapse. We also discuss ways to recover the expressive power of a nonlinear approach and to help select hyperparameters with the help of high-dimensional model representation and to obtain elements of insight while preserving the generality of the method.
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Affiliation(s)
- Sergei Manzhos
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan.
| | - Shunsaku Tsuda
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan.
| | - Manabu Ihara
- School of Materials and Chemical Technology, Tokyo Institute of Technology, Ookayama 2-12-1, Meguro-ku, Tokyo 152-8552, Japan.
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Manzhos S, Ihara M. Computational vibrational spectroscopy of molecule-surface interactions: what is still difficult and what can be done about it. Phys Chem Chem Phys 2022; 24:15158-15172. [DOI: 10.1039/d2cp01389d] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Interactions of molecules with solid surfaces are responsible for key functionalities for a range of currently actively pursued technologies, including heterogeneous catalysis for synthesis or decomposition of molecules, sensitization, surface...
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Manzhos S, Carrington T. Neural Network Potential Energy Surfaces for Small Molecules and Reactions. Chem Rev 2020; 121:10187-10217. [PMID: 33021368 DOI: 10.1021/acs.chemrev.0c00665] [Citation(s) in RCA: 110] [Impact Index Per Article: 27.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
Abstract
We review progress in neural network (NN)-based methods for the construction of interatomic potentials from discrete samples (such as ab initio energies) for applications in classical and quantum dynamics including reaction dynamics and computational spectroscopy. The main focus is on methods for building molecular potential energy surfaces (PES) in internal coordinates that explicitly include all many-body contributions, even though some of the methods we review limit the degree of coupling, due either to a desire to limit computational cost or to limited data. Explicit and direct treatment of all many-body contributions is only practical for sufficiently small molecules, which are therefore our primary focus. This includes small molecules on surfaces. We consider direct, single NN PES fitting as well as more complex methods that impose structure (such as a multibody representation) on the PES function, either through the architecture of one NN or by using multiple NNs. We show how NNs are effective in building representations with low-dimensional functions including dimensionality reduction. We consider NN-based approaches to build PESs in the sums-of-product form important for quantum dynamics, ways to treat symmetry, and issues related to sampling data distributions and the relation between PES errors and errors in observables. We highlight combinations of NNs with other ideas such as permutationally invariant polynomials or sums of environment-dependent atomic contributions, which have recently emerged as powerful tools for building highly accurate PESs for relatively large molecular and reactive systems.
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Affiliation(s)
- Sergei Manzhos
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, 1650, Boulevard Lionel-Boulet, Varennes, Québec City, Québec J3X 1S2, Canada
| | - Tucker Carrington
- Chemistry Department, Queen's University, Kingston Ontario K7L 3N6, Canada
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Boussaidi MA, Ren O, Voytsekhovsky D, Manzhos S. Random Sampling High Dimensional Model Representation Gaussian Process Regression (RS-HDMR-GPR) for Multivariate Function Representation: Application to Molecular Potential Energy Surfaces. J Phys Chem A 2020; 124:7598-7607. [DOI: 10.1021/acs.jpca.0c05935] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Affiliation(s)
- Mohamed Ali Boussaidi
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, 1650 boulevard Lionel-Boulet, Varennes QC J3X 1S2, Canada
- Ecole Nationale d’Ingénieurs de Tunis, Rue Béchir Salem Belkhiria Campus universitaire, BP 37, 1002, Le Bélvédère, Tunis, Tunisia
| | - Owen Ren
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, 1650 boulevard Lionel-Boulet, Varennes QC J3X 1S2, Canada
- Purefacts Inc., 48 Yonge Street, Suite 400, Toronto, ON M5E 1G6, Canada
| | | | - Sergei Manzhos
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, 1650 boulevard Lionel-Boulet, Varennes QC J3X 1S2, Canada
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Manzhos S. Machine learning for the solution of the Schrödinger equation. MACHINE LEARNING-SCIENCE AND TECHNOLOGY 2020. [DOI: 10.1088/2632-2153/ab7d30] [Citation(s) in RCA: 36] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Ku J, Kamath A, Carrington T, Manzhos S. Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation. J Phys Chem A 2019; 123:10631-10642. [DOI: 10.1021/acs.jpca.9b09732] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Jonas Ku
- Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore
| | - Aditya Kamath
- Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
| | - Sergei Manzhos
- Centre Énergie Matériaux Télécommunications, Institut National de la Recherche Scientifique, 1650, boulevard Lionel-Boulet, Varennes QC J3X 1S2, Canada
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Manzhos S, Carrington T. Using rectangular collocation with finite difference derivatives to solve electronic Schrödinger equation. J Chem Phys 2018; 149:204105. [DOI: 10.1063/1.5052196] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sergei Manzhos
- Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, Singapore 117576, Singapore
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation. MATHEMATICS 2018. [DOI: 10.3390/math6110253] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We explore the use of inverse multiquadratic (IMQ) functions as basis functions when solving the vibrational Schrödinger equation with the rectangular collocation method. The quality of the vibrational spectrum of formaldehyde (in six dimensions) is compared to that obtained using Gaussian basis functions when using different numbers of width-optimized IMQ functions. The effects of the ratio of the number of collocation points to the number of basis functions and of the choice of the IMQ exponent are studied. We show that the IMQ basis can be used with parameters where the IMQ function is not integrable. We find that the quality of the spectrum with IMQ basis functions is somewhat lower that that with a Gaussian basis when the basis size is large, and for a range of IMQ exponents. The IMQ functions are; however, advantageous when a small number of functions is used or with a small number of collocation points (e.g., when using square collocation).
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Manzhos S, Wang X, Carrington T. A multimode-like scheme for selecting the centers of Gaussian basis functions when computing vibrational spectra. Chem Phys 2018. [DOI: 10.1016/j.chemphys.2017.10.006] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Manzhos S, Carrington T. Using an internal coordinate Gaussian basis and a space-fixed Cartesian coordinate kinetic energy operator to compute a vibrational spectrum with rectangular collocation. J Chem Phys 2016; 145:224110. [DOI: 10.1063/1.4971295] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sergei Manzhos
- Department of Mechanical Engineering, National University of Singapore, Block EA #07-08, 9 Engineering Drive 1, 117576 Singapore
| | - Tucker Carrington
- Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada
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Meier P, Oschetzki D, Pfeiffer F, Rauhut G. Towards an automated and efficient calculation of resonating vibrational states based on state-averaged multiconfigurational approaches. J Chem Phys 2015; 143:244111. [DOI: 10.1063/1.4938280] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Patrick Meier
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Dominik Oschetzki
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Florian Pfeiffer
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany
| | - Guntram Rauhut
- Institut für Theoretische Chemie, Universität Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany
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Manzhos S, Carrington T, Laverdure L, Mosey N. Computing the Anharmonic Vibrational Spectrum of UF6 in 15 Dimensions with an Optimized Basis Set and Rectangular Collocation. J Phys Chem A 2015; 119:9557-67. [DOI: 10.1021/acs.jpca.5b07627] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Sergei Manzhos
- Department
of Mechanical Engineering, National University of Singapore, Block EA
#07-08, 9 Engineering Drive 1, Singapore 117576
| | - Tucker Carrington
- Department of Chemistry, Queen’s University, Chernoff Hall, 90 Bader Lane, Kingston, ON, Canada K7L 3N6
| | - Laura Laverdure
- Department of Chemistry, Queen’s University, Chernoff Hall, 90 Bader Lane, Kingston, ON, Canada K7L 3N6
| | - Nicholas Mosey
- Department of Chemistry, Queen’s University, Chernoff Hall, 90 Bader Lane, Kingston, ON, Canada K7L 3N6
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Strobusch D, Scheurer C. Adaptive sparse grid expansions of the vibrational Hamiltonian. J Chem Phys 2014; 140:074111. [DOI: 10.1063/1.4864356] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Chan M, Carrington T, Manzhos S. Anharmonic vibrations of the carboxyl group in acetic acid on TiO2: implications for adsorption mode assignment in dye-sensitized solar cells. Phys Chem Chem Phys 2013; 15:10028-34. [DOI: 10.1039/c3cp00065f] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Chan M, Yamashita K, Carrington T, Manzhos S. Towards Accurate Spectroscopic Identification of Species at Catalytic Surfaces: Anharmonic Vibrations of Formate on AuPt. ACTA ACUST UNITED AC 2012. [DOI: 10.1557/opl.2012.1623] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023]
Abstract
ABSTRACTWe present a calculation of vibrational frequencies of formate on the AuPt(111) surface alloy including full anharmonicity and coupling of all six intramolecular degrees of freedom. This species is a key intermediate in methanol oxidation on this material. We use a modified version of the method of Manzhos and Carrington to compute the spectrum directly from a small number (<10,000) of DFT single-point energies, bypassing the construction of a potential energy surface. This is the first such calculation for a 4-atomic species at a surface. The spectrum is obtained using rectangular collocation and a small basis set of parameterized Hermite functions. The achievable accuracy of the order of several cm-1 corresponds to the typical experimental resolution. Using normal coordinates makes the equations simple and general and easily applicable to other systems. This calculation is doable on a PC. We predict that anharmonicity and coupling lower the fundamental frequencies by dozens of cm-1, which could affect species assignment.
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