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Szalay V, Lengyel K, Kovács L, Timón V, Hernández-Laguna A. Vibrations of H+(D+) in stoichiometric LiNbO3 single crystal. J Chem Phys 2011; 135:124501. [PMID: 21974529 DOI: 10.1063/1.3626839] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A first principles quantum mechanical calculation of the vibrational energy levels and transition frequencies associated with protons in stoichiometric LiNbO(3) single crystal has been carried out. The hydrogen contaminated crystal has been approximated by a model one obtains by translating a supercell, i.e., a cluster of LiNbO(3) unit cells containing a single H(+) and a Li(+) vacancy. Based on the supercell model an approximate Hamiltonian operator describing vibrations of the proton sublattice embedded in the host crystal has been derived. It is further simplified to a sum of uncoupled Hamiltonian operators corresponding to different wave vectors (ks) and each describing vibrations of a quasi-particle (quasi-proton). The three dimensional (3D) Hamiltonian operator of k=0 has been employed to calculate vibrational levels and transition frequencies. The potential energy surface (PES) entering this Hamiltonian operator has been calculated point wise on a large set of grid points by using density functional theory, and an analytical approximation to the PES has been constructed by non-parametric approximation. Then, the nuclear motion Schrödinger equation has been solved by employing the method of discrete variable representation. It has been found that the (quasi-)H(+) vibrates in a strongly anharmonic PES. Its vibrations can be described approximately as a stretching, and two orthogonal bending vibrations. The theoretically calculated transition frequencies agree within 1% with those experimentally determined, and they have allowed the assignment of one of the hitherto unassigned bands as a combination of the stretching and the bending of lower fundamental frequency.
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Affiliation(s)
- Viktor Szalay
- Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary.
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Malshe M, Narulkar R, Raff LM, Hagan M, Bukkapatnam S, Agrawal PM, Komanduri R. Development of generalized potential-energy surfaces using many-body expansions, neural networks, and moiety energy approximations. J Chem Phys 2009; 130:184102. [PMID: 19449903 DOI: 10.1063/1.3124802] [Citation(s) in RCA: 47] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energy surface with two-layer neural networks (NN) for each M-body term in the summations. The total number of NNs required is significantly reduced by employing a moiety energy approximation. An algorithm is presented that efficiently adjusts all the coupled NN parameters to the database for the surface. Application of the method to four different systems of increasing complexity shows that the fitting accuracy of the method is good to excellent. For some cases, it exceeds that available by other methods currently in literature. The method is illustrated by fitting large databases of ab initio energies for Si(n) (n=3,4,...,7) clusters obtained from density functional theory calculations and for vinyl bromide (C(2)H(3)Br) and all products for dissociation into six open reaction channels (12 if the reverse reactions are counted as separate open channels) that include C-H and C-Br bond scissions, three-center HBr dissociation, and three-center H(2) dissociation. The vinyl bromide database comprises the ab initio energies of 71 969 configurations computed at MP4(SDQ) level with a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (4333/433/4) basis set augmented with split outer s and p orbitals (43321/4321/4) and a polarization f orbital with an exponent of 0.5 for the bromine atom. It is found that an expansion truncated after the three-body terms is sufficient to fit the Si(5) system with a mean absolute testing set error of 5.693x10(-4) eV. Expansions truncated after the four-body terms for Si(n) (n=3,4,5) and Si(n) (n=3,4,...,7) provide fits whose mean absolute testing set errors are 0.0056 and 0.0212 eV, respectively. For vinyl bromide, a many-body expansion truncated after the four-body terms provides fitting accuracy with mean absolute testing set errors that range between 0.0782 and 0.0808 eV. These errors correspond to mean percent errors that fall in the range 0.98%-1.01%. Our best result using the present method truncated after the four-body summation with 16 NNs yields a testing set error that is 20.3% higher than that obtained using a 15-dimensional (15-140-1) NN to fit the vinyl bromide database. This appears to be the price of the added simplicity of the many-body expansion procedure.
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Affiliation(s)
- M Malshe
- Nanotechnology Research Group, Oklahoma State University, 218 Engineering, North Stillwater, Oklahoma 74078, USA
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Manzhos S, Carrington T. Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface. J Chem Phys 2009; 129:224104. [PMID: 19071904 DOI: 10.1063/1.3021471] [Citation(s) in RCA: 81] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We demonstrate that it is possible to obtain good potentials using high-dimensional model representations (HDMRs) fitted with neural networks (NNs) from data in 12 dimensions and 15 dimensions. The HDMR represents the potential as a sum of lower-dimensional functions and our NN-based approach makes it possible to obtain all of these functions from one set of fitting points. To reduce the number of terms in the HDMR, we use optimized redundant coordinates. By using exponential neurons, one obtains a potential in sum-of-products form, which greatly facilitates quantum dynamics calculations. A 12-dimensional (reference) potential surface for vinyl bromide is first refitted to show that it can be represented as a sum of two-dimensional functions. To fit 3d functions of the original coordinates, to improve the potential, a huge amount of data would be required. Redundant coordinates avoid this problem. They enable us to bypass the combinatorial explosion of the number of terms which plagues all HDMR and multimode-type methods. We also fit to a set of approximately 70,000 ab initio points for vinyl bromide in 15 dimensions [M. Malshe et al., J. Chem. Phys. 127, 134105 (2007)] and show that it is possible to obtain a surface in sum-of-products form of quality similar to the quality of the full-dimensional fit. Although we obtain a full-dimensional surface, we limit the cost of the fitting by building it from fits of six-dimensional functions, each of which requires only a small NN.
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Affiliation(s)
- Sergei Manzhos
- Département de chimie, Université de Montréal, Case postale 6128, succursale Centre-ville Montréal, (Québec) H3C 3J7 Canada.
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Isotope Effects in Photodissociation: Chemical Reaction Dynamics and Implications for Atmospheres. ADVANCES IN QUANTUM CHEMISTRY 2008. [DOI: 10.1016/s0065-3276(07)00207-9] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Manzhos S, Carrington T. Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions. J Chem Phys 2007; 127:014103. [PMID: 17627333 DOI: 10.1063/1.2746846] [Citation(s) in RCA: 74] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We propose a method for fitting potential energy surfaces with a sum of component functions of lower dimensionality. This form facilitates quantum dynamics calculations. We show that it is possible to reduce the dimensionality of the component functions by introducing new and redundant coordinates obtained with linear transformations. The transformations are obtained from a neural network. Different coordinates are used for different component functions and the new coordinates are determined as the potential is fitted. The quality of the fits and the generality of the method are illustrated by fitting reference potential surfaces of hydrogen peroxide and of the reaction OH+H(2)-->H(2)O+H.
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Affiliation(s)
- Sergei Manzhos
- Département de Chimie, Université de Montréal, CP 6128, succursale Centre-ville, Montréal (Québec) H3C 3J7, Canada.
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Manzhos S, Wang X, Dawes R, Carrington T. A Nested Molecule-Independent Neural Network Approach for High-Quality Potential Fits†. J Phys Chem A 2006; 110:5295-304. [PMID: 16623455 DOI: 10.1021/jp055253z] [Citation(s) in RCA: 131] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
It is shown that neural networks (NNs) are efficient and effective tools for fitting potential energy surfaces. For H2O, a simple NN approach works very well. To fit surfaces for HOOH and H2CO, we develop a nested neural network technique in which we first fit an approximate NN potential and then use another NN to fit the difference of the true potential and the approximate potential. The root-mean-square error (RMSE) of the H2O surface is 1 cm(-1). For the 6-D HOOH and H2CO surfaces, the nested approach does almost as well attaining a RMSE of 2 cm(-1). The quality of the NN surfaces is verified by calculating vibrational spectra. For all three molecules, most of the low-lying levels are within 1 cm(-1) of the exact results. On the basis of these results, we propose that the nested NN approach be considered a method of choice for both simple potentials, for which it is relatively easy to guess a good fitting function, and complicated (e.g., double well) potentials for which it is much harder to deduce an appropriate fitting function. The number of fitting parameters is only moderately larger for the 6-D than for the 3-D potentials, and for all three molecules, decreasing the desired RMSE increases only slightly the number of required fitting parameters (nodes). NN methods, and in particular the nested approach we propose, should be good universal potential fitting tools.
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Affiliation(s)
- Sergei Manzhos
- Département de chimie, Université de Montréal, C.P. 6128, succursale Centre-ville, Montréal (Québec) H3C 3J7, Canada.
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Blomgren P, Gasner S, Palacios A. Stable second-order scheme for integrating the Kuramoto-Sivanshinsky equation in polar coordinates using distributed approximating functionals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036701. [PMID: 16241608 DOI: 10.1103/physreve.72.036701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2004] [Indexed: 05/05/2023]
Abstract
We present an algorithm for the time integration of nonlinear partial differential equations. The algorithm uses distributed approximating functionals, which are based on an analytic approximation method, in order to achieve highly accurate spatial derivatives. The time integration is based on a second-order unconditionally A -stable Crank-Nicolson scheme with a Newton solver. We apply the integration scheme to the Kuramoto-Sivanshinsky equation in polar coordinates, which presents a significant computational challenge due to the stiffness introduced by the estimation of the spatial derivatives at the origin. We present several stationary and nonstationary solutions of the Kuramoto-Sivanshinsky equation and compare with previous numerical results as well as patterns observed in the combustion front of a circular burner. The numerical results of the proposed scheme reproduces several patterns--rotating two-cell, three-cell, hopping three-cell, stationary two-three-four- and five-cell, stationary 5/1,6/1,7/1,8/2 two-ring patterns, etc.--observed in physical experiments. The scheme is extremely robust and can produce long-term simulations consisting of several thousand frames. Although applied to a very specific problem, the approach of combining the framework of distributed approximating functionals with a Crank-Nicolson based time integration is generalizable to a large class of problems.
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Affiliation(s)
- Peter Blomgren
- Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182, USA.
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Blomgren P, Gasner S, Palacios A. Hopping behavior in the Kuramoto-Sivashinsky equation. CHAOS (WOODBURY, N.Y.) 2005; 15:13706. [PMID: 15836274 DOI: 10.1063/1.1848311] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments but not in truly two-dimensional computer simulations. A modal decomposition analysis of the simulated patterns, via the proper orthogonal decomposition, reveals spatio-temporal behavior in which the overall temporal dynamics is similar to that of equivalent experimental states but the spatial dynamics exhibits a few more features that are not seen in the experiments. Similarities in the temporal behavior and subtle differences in the spatial dynamics between numerical hopping states and their experimental counterparts are discussed in more detail.
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Affiliation(s)
- Peter Blomgren
- Nonlinear Dynamical Systems Group, Department of Mathematics & Statistics, San Diego State University, San Diego, CA 92182, USA.
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Witkoskie JB, Doren DJ. Neural Network Models of Potential Energy Surfaces: Prototypical Examples. J Chem Theory Comput 2004; 1:14-23. [DOI: 10.1021/ct049976i] [Citation(s) in RCA: 68] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- James B. Witkoskie
- Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716
| | - Douglas J. Doren
- Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716
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Olsen RA, Busnengo HF, Salin A, Somers MF, Kroes GJ, Baerends EJ. Constructing accurate potential energy surfaces for a diatomic molecule interacting with a solid surface: H2+Pt(111) and H2+Cu(100). J Chem Phys 2002. [DOI: 10.1063/1.1446852] [Citation(s) in RCA: 124] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/31/2023] Open
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Yu HG, Nyman G. Interpolatedab initioquantum scattering for the reaction of OH with HCl. J Chem Phys 2000. [DOI: 10.1063/1.1319999] [Citation(s) in RCA: 38] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wyatt RE, Kouri DJ, Hoffman DK. Quantum wave packet dynamics with trajectories: Implementation with distributed approximating functionals. J Chem Phys 2000. [DOI: 10.1063/1.481717] [Citation(s) in RCA: 50] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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A generalized discrete variable representation approach to interpolating or fitting potential energy surfaces. Chem Phys Lett 2000. [DOI: 10.1016/s0009-2614(00)00359-6] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Hoffman DK, Gunaratne GH, Zhang DS, Kouri DJ. A method to Fourier filter textured images. CHAOS (WOODBURY, N.Y.) 2000; 10:240-247. [PMID: 12779379 DOI: 10.1063/1.166490] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
An algorithm is introduced to extract an underlying image from a class of textures. It is assumed that the image is bandwidth limited and the noise is broad-band. The initial step of the algorithm extends the signal to a larger periodic image using "Distributed Approximating Functionals." The second step introduces a low-pass filter which allows the identification and elimination of the high-frequency components of the noise. The periodicity of the resulting image allows it to be Fourier filtered without aliasing. The feasibility of the algorithm is demonstrated on several noisy patterns generated in experiments and model systems. (c) 2000 American Institute of Physics.
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Affiliation(s)
- David K. Hoffman
- Department of Chemistry and Ames Laboratory, Iowa State University, Ames, Iowa 50011
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Szalay V. Iterative and direct methods employing distributed approximating functionals for the reconstruction of a potential energy surface from its sampled values. J Chem Phys 1999. [DOI: 10.1063/1.480257] [Citation(s) in RCA: 21] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
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Hollebeek T, Ho TS, Rabitz H. A fast algorithm for evaluating multidimensional potential energy surfaces. J Chem Phys 1997. [DOI: 10.1063/1.473683] [Citation(s) in RCA: 64] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Hoffman DK, Frishman A, Kouri DJ. Distributed approximating functional approach to fitting multi-dimensional surfaces. Chem Phys Lett 1996. [DOI: 10.1016/0009-2614(96)01085-8] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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