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Research Papers

Quantifying Parameter Sensitivity and Uncertainty for Interatomic Potential Design: Application to Saturated Hydrocarbons

[+] Author and Article Information
Mark A. Tschopp

Fellow ASME
U.S. Army Research Laboratory,
Aberdeen Proving Ground, MD 21005
e-mail: mark.a.tschopp.civ@mail.mil

B. Chris Rinderspacher

U.S. Army Research Laboratory,
Aberdeen Proving Ground, MD 21005

Sasan Nouranian

Department of Chemical Engineering,
The University of Mississippi,
University, MS 38677

Mike I. Baskes

Department of Aerospace Engineering,
Mississippi State University,
Starkville, MS 39762

Steven R. Gwaltney

Department of Chemistry,
Mississippi State University,
Starkville, MS 39762

Mark F. Horstemeyer

Fellow ASME
Department of Mechanical Engineering,
Mississippi State University,
Starkville, MS 39762

1Corresponding author.

Manuscript received September 1, 2016; final manuscript received January 25, 2017; published online September 7, 2017. Assoc. Editor: Laura Swiler. This work is in part a work of the U.S. Government. ASME disclaims all interest in the U.S. Government's contributions.

ASME J. Risk Uncertainty Part B 4(1), 011004 (Sep 07, 2017) (17 pages) Paper No: RISK-16-1113; doi: 10.1115/1.4037455 History: Received September 01, 2016; Revised January 25, 2017

The research objective herein is to understand the relationships between the interatomic potential parameters and properties used in the training and validation of potentials, specifically using a recently developed modified embedded-atom method (MEAM) potential for saturated hydrocarbons (C–H system). This potential was parameterized to a training set that included bond distances, bond angles, and atomization energies at 0 K of a series of alkane structures from methane to n-octane. In this work, the parameters of the MEAM potential were explored through a fractional factorial design and a Latin hypercube design to better understand how individual MEAM parameters affected several properties of molecules (energy, bond distances, bond angles, and dihedral angles) and also to quantify the relationship/correlation between various molecules in terms of these properties. The generalized methodology presented shows quantitative approaches that can be used in selecting the appropriate parameters for the interatomic potential, selecting the bounds for these parameters (for constrained optimization), selecting the responses for the training set, selecting the weights for various responses in the objective function, and setting up the single/multi-objective optimization process itself. The significance of the approach applied in this study is not only the application to the C–H system but that the broader framework can also be easily applied to any number of systems to understand the significance of parameters, their relationships to properties, and the subsequent steps for designing interatomic potentials under uncertainty.

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Figures

Grahic Jump Location
Fig. 1

Example of some of the molecules used in the parameterization of the C–H MEAM potential and in the property assessment in this study

Grahic Jump Location
Fig. 2

Example of some of the molecule configuration paths used for the energy–distance relationships (as distance is increased in horizontal direction), which was then used in the parameterization of the C–H MEAM potential and in property assessment in this study

Grahic Jump Location
Fig. 3

Heat map showing all the data collected (right) for the 128-run fractional factorial design (left), which includes (from left to right) energies, bond distances (C–C and C–H), bond angles (C–C–C, C–C–H, and H–C–H), dihedral angles (C–C–C–C, C–C–C–H, and H–C–C–H), and the deviation in the energy versus distance plots from first principles results (mean absolute error (MAE) and root-mean-square error (RMSE)). The data have been normalized by subtracting the mean and dividing by the standard deviation.

Grahic Jump Location
Fig. 4

Summary of ANOVA results for the energies of all the molecules examined. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule.

Grahic Jump Location
Fig. 5

Summary of ANOVA results for the (left) C–H and (right) C–C bond distances of all the molecules examined. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean bond distance for all the C–H (C–C) bonds was used for those molecules with multiple C–H (C–C) bonds.

Grahic Jump Location
Fig. 6

Summary of ANOVA results for the C–C–C bond angles of all the molecules examined. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean bond angle for all the C–C–C bond angles was used for those molecules with multiple C–C–C bond angles.

Grahic Jump Location
Fig. 7

Summary of ANOVA results for the C–C–H bond angles of all the molecules examined. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean bond angle for all the C–C–H bond angles was used for those molecules with multiple C–C–H bond angles.

Grahic Jump Location
Fig. 8

Summary of ANOVA results for the H–C–H bond angles of all the molecules examined. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean bond angle for all the H–C–H bond angles was used for those molecules with multiple H–C–H bond angles.

Grahic Jump Location
Fig. 9

Summary of ANOVA results for the C–C–C–C dihedral angles of all the molecules examined. Since there are primarily two dihedral angles in the tested molecules (60 deg and 180 deg), the absolute value of the minimum difference between each dihedral angle and these two reference angles was used. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean dihedral angle for all the C–C–C–C angles was used for those molecules with multiple C–C–C–C dihedral angles.

Grahic Jump Location
Fig. 10

Summary of ANOVA results for the C–C–C–H dihedral angles of all the molecules examined. Since there are primarily two dihedral angles in the tested molecules (60 deg and 180 deg), the absolute value of the minimum difference between each dihedral angle and these two reference angles was used. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean dihedral angle for all the C–C–C–H angles was used for those molecules with multiple C–C–C–H dihedral angles.

Grahic Jump Location
Fig. 11

Summary of ANOVA results for the H–C–C–H dihedral angles of all the molecules examined. Since there are primarily two dihedral angles in the tested molecules (60 deg and 180 deg), the absolute value of the minimum difference between each dihedral angle and these two reference angles was used. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule. The mean dihedral angle for all the H–C–C–H angles was used for those molecules with multiple H–C–C–H dihedral angles.

Grahic Jump Location
Fig. 12

Summary of ANOVA results for the energy versus distance relationships for various molecules and paths. In this heat map, the MAE between the calculated MEAM curve and first principles data at the same distances was used as a response variable. The percent contribution by each of the MEAM parameters was determined by dividing the sum of squares attributed to each parameter by the total sum of squares (e.g., see Table 3), which was repeated for each molecule.

Grahic Jump Location
Fig. 13

Correlation coefficient R for the energies of all the molecules examined for (left) the 1024-run Latin hypercube sampling (LHS) design and (right) the 128-run fractional factorial design. In this plot, each response is plotted against all of the other responses. A high correlation coefficient indicates the strength of the linear dependence between the two responses (i.e., a high correlation, R = 1, for responses plotted against themselves).

Grahic Jump Location
Fig. 14

Correlation coefficient R for the (left) C–H and (right) C–C bond distances of all the molecules examined. The mean bond distance for all the C–H (C–C) bonds was used for those molecules with multiple C–H (C–C) bonds.

Grahic Jump Location
Fig. 15

Correlation coefficient R for the C–C–C bond angles of all the molecules examined. The mean bond angle for all the C–C–C bond angles was used for those molecules with multiple C–C–C bond angles.

Grahic Jump Location
Fig. 16

Correlation coefficient R for the C–C–H bond angles of all the molecules examined. The mean bond angle for all the C–C–H bond angles was used for those molecules with multiple C–C–H bond angles.

Grahic Jump Location
Fig. 17

Correlation coefficient R for the H–C–H bond angles of all the molecules examined. The mean bond angle for all the H–C–H bond angles was used for those molecules with multiple H–C–H bond angles.

Grahic Jump Location
Fig. 18

Contour plot of the p values for the 38 different parameters (vertical axis) and the 221 different responses (horizontal axis) in this study. The parameters are (vertically) ordered by the number of times that a parameter was statistically significant (p ≤ 0.05) and the responses are (horizontally) ordered by the number of times a response garnered statistically significant parameters.

Grahic Jump Location
Fig. 19

Contour plot of the correlation coefficient R for the 221 responses arranged both in the same order as in Fig. 3 (the upper left inset contour plot) and grouped by various clusters according to a Euclidean distance function (large contour plot)

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