Towards Accurate Spectroscopic Identification of Species at Catalytic Surfaces: Anharmonic Vibrations of Formate on AuPt

Chan, Matthew; Yamashita, Koichi; Carrington, Tucker; Manzhos, Sergei

doi:10.1557/opl.2012.1623

Cited by 12 publications

(24 citation statements)

References 24 publications

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“…Rectangular collocation has been used to solve both the vibrational and the electronic Schrödinger equations. [1][2][3][4][5][6][7][8][9][10][11][12][13] We solve the time-independent Schrödinger equation (SE)…”

Section: Introductionmentioning

confidence: 99%

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation

Kamath

Carrington

et al. 2019

J. Phys. Chem. A

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The rectangular collocation approach makes it possible to solve the Schrödinger equation with basis functions that do not have amplitude in all regions in which wavefunctions have significant amplitude. Collocation points can be restricted to a small region of space. As no integrals are computed, there are no problems due to discontinuities in the potential, and there is no need to use integrable basis functions. In this paper, we show, for the Kohn-Sham equation, that machine learning can be used to drastically reduce the size of the collocation point set. This is demonstrated by solving the Kohn-Sham equations for CO and H2O. We solve the Kohn-Sham equation on a given effective potential which is a critical part of all DFT calculations, and monitor orbital energies and orbital shapes. We use a combination of Gaussian process regression and a genetic algorithm to reduce the collocation point set size by more than an order of magnitude (from about 51,000 points to 2,000 points) while retaining mHartree accuracy.

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Section: Introductionmentioning

confidence: 99%

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation

Kamath

Carrington

et al. 2019

J. Phys. Chem. A

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“…Gaussian functions are general and form a sufficiently small basis set (especially if their parameters are fitted) for the calculation to be doable on a personal computer for low-dimensional problems, such as triatomic molecules, and for a small number of states [13,20,27]. For larger molecules and large numbers of states, either other types of basis functions need to be used, such as parameterized harmonic functions, which allow for a small basis set size, as was tested in up to 15 dimensions [11][12][13]19,31], or a much larger Gaussian basis would be needed. For example, a sub-cm −1 accuracy has been achieved for the vibrational spectrum of formaldehyde (a six-dimensional SE), with about 40,000 basis functions and with more than 100,000 collocation points [30,33,34].…”

Section: Problem Statement and The Aim Of This Workmentioning

confidence: 99%

“…Note that M can be smaller if the basis is better. The rectangular collocation approach has been applied to computations of anharmonic spectra of polyatomic molecules, including those on surfaces directly from ab initio samples of V(r) [11,12,19,20,31].…”

Section: Introductionmentioning

confidence: 99%

Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Kamath

Manzhos

2018

Mathematics

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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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“…Note that M can be smaller if the basis is better. The rectangular collocation approach has been applied to computations of anharmonic spectra of polyatomic molecules, including those on surfaces, directly from ab initio samples of ( ) [11,12,19,20,31].…”

Section: Introductionmentioning

confidence: 99%

Inverse Multiquadratic Functions as Basis for Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Kamath¹,

Manzhos²

2018

Preprint

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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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Towards Accurate Spectroscopic Identification of Species at Catalytic Surfaces: Anharmonic Vibrations of Formate on AuPt

Cited by 12 publications

References 24 publications

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation

Machine Learning Optimization of the Collocation Point Set for Solving the Kohn–Sham Equation

Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

Inverse Multiquadratic Functions as Basis for Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

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