Modified Shepard Interpolation Research Articles

Interpolation Methods for Molecular Potential Energy Surface Construction.

The concept of a potential energy surface (PES) is one of the most important concepts in modern chemistry. A PES represents the relationship between the chemical system's energy and its geometry (i.e., atom positions) and can provide useful information about the system's chemical properties and reactivity. Construction of accurate PESs with high-level theoretical methodologies, such as density functional theory, is still challenging due to a steep increase in the computational cost with the increase of the system size. Thus, over the past few decades, many different mathematical approaches have been applied to the problem of the cost-efficient PES construction. This article serves as a short overview of interpolative methods for the PES construction, including global polynomial interpolation, trigonometric interpolation, modified Shepard interpolation, interpolative moving least-squares, and the automated PES construction derived from these.

Surface for methane combustion: O(3P) +CH4 → OH+CH3**Project supported by the National Natural Science Foundation of China (Grant No. 51574016) and completed while the author was in residence at UNSW, Australia supported by the International Cooperation Training Program for Innovative Talents of USTB.

Kinetic investigations including quasi-classical trajectory and canonical unified statistical theory method calculations are carried out on a potential energy surface for the hydrogen-abstraction reaction from methane by atom O(3P). The surface is constructed using a modified Shepard interpolation method. The ab initio calculations are performed at the CCSD(T) level. Taking account of the contribution of inner core electrons to electronic correlation interaction in ab initio electronic structure calculations, modified optimized aug-cc-pCVQZ basis sets are applied to the all-electrons calculations. On this potential energy surface, the triplet oxygen atom attacks methane in a near-collinear H–CH3 direction to form a saddle point with barrier height of 13.55 kcal/mol, which plays a key role in the kinetics of the title reaction. For the temperature range of 298–2500 K, our calculated thermal rate constants for the O(3P) + CH4 → OH + CH3 reaction show good agreement with relevant experimental data. This work provides detailed mechanism of this gas-phase reaction and a theoretical guidance for methane combustion.

Constructing accurate radio environment maps based on Shepard interpolation

The increased development of information industry and relevant radio communication services is making the spectrum management problem more challenging to solve. A suitable spectrum management method enables transmitters to reuse the frequency efficiently and the user equipment (UE) can select the optimum base station. The radio environment map (REM) concept has been proposed as a tool to solve spectrum scarcity and improve spectrum utilization, making different users share spectrum resource efficiently. The REMs are becoming an increasingly popular method for interference management and resource assignment. The reason for this is that REMs can be constructed without the need for surveys or complex calibration processes which are costly and time consuming. This paper gives Shepard interpolation techniques and modifies it in some respects for estimating radio environments with a limited number of measurement data, thus constructing accurate REMs. Additionally, it presents our systematic investigation of the impact of the number and distribution of measurements using the averaged root mean square error (RMSE) as the performance metric. Simulation results show that increasing the number of measurements with the clustering sampling and the modified Shepard interpolation technique, the most accurate REM is obtained in terms of the performance metric RMSE. And the interpolation accuracy can be improved by almost 14 dB with the modified method, which proves that our approach can effectively determine sharing conditions of radio spectrum use both in time and space.

Dynamics of the CH4 + O(3P) → CH3(ν = 0) + OH(ν′ = 0) reaction**Project supported by the National Natural Science Foundation of China (Grant Nos. 51574016 and 51604018).

The dynamics of the ground-state reaction of CH4 + O(3P) → CH3(ν = 0) + OH(ν′ = 0) have attracted a great deal of attention both theoretically and experimentally. This rapid communication represents extensive quasi-classical trajectory calculations of the vibrational distributions on a unique full-dimensional ab initio potential energy surface for the title reaction, at the collision energy of relevance to previous crossed molecular beam experiments. The surface is constructed using the all electrons coupled-cluster singles and doubles approach plus quasi-perturbative triple excitations with optimized basis sets. A modified Shepard interpolation method is also employed for the construction. Good agreement between our calculations and the available experimental results has been achieved, opening the door for accurate dynamics on this surface.

Interpolating DFT Data for 15D Modeling of Methane Dissociation on an fcc Metal

ABSTRACTDetailed simulation of reactions occurring on and with the surfaces of crystalline materials usually require a continuous representation of the potential energy surface that describes the adsorbate–surface interaction. Only a few techniques are available to describe interactions with polyatomic adsorbates that respect all of the symmetries of the interactions. The modified Shepard interpolation has recently been reformulated to ensure symmetries are rigorously imposed. In this work, the modified Shepard interpolation is used to construct a 15D potential energy surface for the reaction of methane with the {100} surface of a face‐centered cubic metal, in the Born‐‐Oppenheimer static surface (BOSS) approximation. The energy of the system is calculated using density functional theory (DFT), and the geometries around which the potential is expanded are selected by quasi‐classical trajectory calculations. The energy of the resulting continuous potential energy surface exactly matched the DFT energy at these points; there is no fitting error. It is demonstrated that the classical reaction probability converges with a reasonable number of interpolation points for this 15D system.

Local-metrics error-based Shepard interpolation as surrogate for highly non-linear material models in high dimensions.

Many problems in computational materials science and chemistry require the evaluation of expensive functions with locally rapid changes, such as the turn-over frequency of first principles kinetic Monte Carlo models for heterogeneous catalysis. Because of the high computational cost, it is often desirable to replace the original with a surrogate model, e.g., for use in coupled multiscale simulations. The construction of surrogates becomes particularly challenging in high-dimensions. Here, we present a novel version of the modified Shepard interpolation method which can overcome the curse of dimensionality for such functions to give faithful reconstructions even from very modest numbers of function evaluations. The introduction of local metrics allows us to take advantage of the fact that, on a local scale, rapid variation often occurs only across a small number of directions. Furthermore, we use local error estimates to weigh different local approximations, which helps avoid artificial oscillations. Finally, we test our approach on a number of challenging analytic functions as well as a realistic kinetic Monte Carlo model. Our method not only outperforms existing isotropic metric Shepard methods but also state-of-the-art Gaussian process regression.

A Theoretical Study on the Dynamics of the Reaction of CH Radicals with Water.

Quasi-classical trajectory (QCT) and RRKM-SSA calculations are carried out to gain insight into the dynamics of the title reaction. The barrier-less initiation step in this system is governed by the capture probability in the entrance channel to form an energized adduct once the centrifugal barrier is surmounted. The dynamics of the title reaction on its lowest doublet electronic state is studied at the DFT level of MPWB1K/6-31++g(2df,2p). An iteratively modified Shepard interpolation technique implemented in the GROW program suite was used to construct a global potential energy surface for the title reaction. The total and individual cross sections for the main products and corresponding reaction probabilities as a function of initial collision energy ranging from 0.4 kJ mol-1 to 52.5 kJ mol-1 are calculated. These data are used to calculate the total rate constant for the title reaction by means of collision theory. Our calculated QCT rate constant is compared with the calculated rate constant from RRKM-SSA method at the CCSD(T)/Aug-cc-pVTZ//MP2/6-31++g(2df,2p) level. The energy partitioning for the main products CH2O + H and HCOH + H and also for the reactants after nonreactive collisions as a function of initial collision energy are discussed.

“Plug-and-Play” potentials: Investigating quantum effects in (H2)2–Li+–benzene

Quantum and anharmonic effects are investigated in (H2)2-Li(+)-benzene, a model for hydrogen adsorption in metal-organic frameworks and carbon-based materials, using rigid-body diffusion Monte Carlo (RBDMC) simulations. The potential-energy surface (PES) is calculated as a modified Shepard interpolation of M05-2X/6-311+G(2df,p) electronic structure data. The RBDMC simulations yield zero-point energies (ZPE) and probability density histograms that describe the ground-state nuclear wavefunction. Binding a second H2 molecule to the H2-Li(+)-benzene complex increases the ZPE of the system by 5.6 kJ mol(-1) to 17.6 kJ mol(-1). This ZPE is 42% of the total electronic binding energy of (H2)2-Li(+)-benzene and cannot be neglected. Our best estimate of the 0 K binding enthalpy of the second H2 to H2-Li(+)-benzene is 7.7 kJ mol(-1), compared to 12.4 kJ mol(-1) for the first H2 molecule. Anharmonicity is found to be even more important when a second (and subsequent) H2 molecule is adsorbed; use of harmonic ZPEs results in significant error in the 0 K binding enthalpy. Probability density histograms reveal that the two H2 molecules are found at larger distance from the Li(+) ion and are more confined in the θ coordinate than in H2-Li(+)-benzene. They also show that both H2 molecules are delocalized in the azimuthal coordinate, ϕ. That is, adding a second H2 molecule is insufficient to localize the wavefunction in ϕ. Two fragment-based (H2)2-Li(+)-benzene PESs are developed. These use a modified Shepard interpolation for the Li(+)-benzene and H2-Li(+)-benzene fragments, and either modified Shepard interpolation or a cubic spline to model the H2-H2 interaction. Because of the neglect of three-body H2, H2, Li(+) terms, both fragment PESs lead to overbinding of the second H2 molecule by 1.5 kJ mol(-1). Probability density histograms, however, indicate that the wavefunctions for the two H2 molecules are effectively identical on the "full" and fragment PESs. This suggests that the 1.5 kJ mol(-1) error is systematic over the regions of configuration space explored by our simulations. Notwithstanding this, modified Shepard interpolation of the weak H2-H2 interaction is problematic and we obtain more accurate results, at considerably lower computational cost, using a cubic spline interpolation. Indeed, the ZPE of the fragment-with-spline PES is identical, within error, to the ZPE of the full PES. This fragmentation scheme therefore provides an accurate and inexpensive method to study higher hydrogen loading in this and similar systems.

Generating accurate dipole moment surfaces using modified Shepard interpolation.

We outline an approach for building molecular dipole moment surfaces using modified Shepard interpolation. Our approach is highly automated, requires minimal parameterization, and is iteratively improvable. Using the water molecule as a test case, we investigate how different aspects of the interpolation scheme affect the rate of convergence of calculated IR spectral line intensities. It is found that the interpolation scheme is sensitive to coordinate singularities present at linear geometries. Due to the generally monotonic nature of the dipole moment surface, the one-part weight function is found to be more effective than the more complicated two-part variant, with first-order interpolation also giving better-than-expected results. Almost all sensible schemes for choosing interpolation reference data points are found to exhibit acceptable convergence behavior.

Using Hessian update formulae to construct modified Shepard interpolated potential energy surfaces: Application to vibrating surface atoms

Modified Shepard interpolation based on second order Taylor series expansions has proven to be a flexible tool for constructing potential energy surfaces in a range of situations. Extending this to gas-surface dynamics where surface atoms are allowed to move represents a substantial increase in the dimensionality of the problem, reflected in a dramatic increase in the computational cost of the required Hessian (matrix of second derivatives) evaluations. This work demonstrates that using approximate Hessians derived from well known Hessian update formulae and a single accurate Hessian can provide an effective way to avoid this expensive accurate Hessian determination.

Quantum effects and anharmonicity in the H2-Li+-benzene complex: A model for hydrogen storage materials

Quantum and anharmonic effects are investigated in H2-Li(+)-benzene, a model for hydrogen adsorption in metal-organic frameworks and carbon-based materials. Three- and 8-dimensional quantum diffusion Monte Carlo (QDMC) and rigid-body diffusion Monte Carlo (RBDMC) simulations are performed on potential energy surfaces interpolated from electronic structure calculations at the M05-2X/6-31+G(d,p) and M05-2X/6-311+G(2df,p) levels of theory using a three-dimensional spline or a modified Shepard interpolation. These calculations investigate the intermolecular interactions in this system, with three- and 8-dimensional 0 K H2 binding enthalpy estimates, ΔH(bind) (0 K), being 16.5 kJ mol(-1) and 12.4 kJ mol(-1), respectively: 0.1 and 0.6 kJ mol(-1) higher than harmonic values. Zero-point energy effects are 35% of the value of ΔH(bind) (0 K) at M05-2X/6-311+G(2df,p) and cannot be neglected; uncorrected electronic binding energies overestimate ΔHbind (0 K) by at least 6 kJ mol(-1). Harmonic intermolecular binding enthalpies can be corrected by treating the H2 "helicopter" and "ferris wheel" rotations as free and hindered rotations, respectively. These simple corrections yield results within 2% of the 8-dimensional anharmonic calculations. Nuclear ground state probability density histograms obtained from the QDMC and RBDMC simulations indicate the H2 molecule is delocalized above the Li(+)-benzene system at 0 K.

SEVEN-DIMENSIONAL QUANTUM DYNAMICS STUDY OF THEH2+NH2→H+NH3REACTION ON AN INTERPOLATED POTENTIAL ENERGY SURFACE

Initial-state-selected time-dependent wave packet dynamics studies have been performed for the H2+ NH2→ H + NH3reaction with a seven-dimensional model on a new interpolated ab initio potential energy surface (PES). The PES is constructed using modified Shepard interpolation Scheme and contains 1967 data points with ab initio calculations carried out on UCCSD(T)/aug-cc-pVTZ level. In the seven-dimensional model, NH2group keeps C2vsymmetry and two NH bonds are fixed at their equilibrium values. The total reaction probabilities are calculated when (1) the two reactants are initially at their ground states; (2) NH2bending mode is excited, and (3) H2is on its first vibrational excited state. The integral cross sections are also reported for these initial states with centrifugal-sudden approximation. Thermal rate constants are calculated for the temperature range of 200–2000 K and compared with the previous calculated values and available experimental data. Good agreements between theory and experiments for the rate constants at intermediate temperature are achieved on this PES.

Accelerating modified Shepard interpolated potential energy calculations using graphics processing units

The potential energy surfaces constructed with the modified Shepard interpolation scheme have been widely used in studies of chemical reaction dynamics. However, computational costs of interpolation increase rapidly with the size of the system and the number of data points needed to achieve a given accuracy. In this work, we present a naive Graphics Processing Unit (GPU)-accelerated algorithm for modified Shepard interpolated potential energy calculations and its implementation with the PGI CUDA Fortran language. The benchmark tests on a NVIDIA Tesla C2050 using four interpolated potential energy surfaces (one for H+H2O↔H2+OH, two for H+NH3↔H2+NH2 and one for H+CH4↔H2+CH3) demonstrated a speedup of 50-fold over the original CPU implementation on an Intel E5620 processor and the speedup increases with the system size and the number of data points. This work presents a promising GPU application in the field of chemical reaction dynamics.

Rate coefficient for the important interstellar radiative association between [formula omitted] and H2 from classical reaction dynamics

The CH3++H2→CH5++hν radiative associative reaction plays a major role in the synthesis of complex hydrocarbons in interstellar dust clouds and was studied using a part-classical, part-ab initio approach. A first-principles potential energy surface (PES) at CCSD/6-311++G(2df, 2pd) was constructed from a set of 828 data points in configurational space using modified Shepard interpolation, and was used in conjuction with long-range perturbation theory at large intermolecular distances. Data points are chosen so as to optimize PES construction according to the GROW software algorithm. Classical reaction dynamics was employed to infer the reaction rate coefficient.

Modified Shepard interpolation of gas-surface potential energy surfaces with strict plane group symmetry and translational periodicity

A new formulation of modified Shepard interpolation of potential energy surface data for gas-surface reactions has been developed. The approach has been formulated for monoatomic or polyatomic adsorbates interacting with crystalline solid surfaces of any plane group symmetry. The interpolation obeys the two dimensional translational periodicity and plane group symmetry of the solid surface by construction. The interpolation remains continuous and smooth everywhere. The interpolation developed here is suitable for constructing potential energy surfaces by sampling classical trajectories using the Grow procedure. A model function has been used to demonstrate the method, showing the convergence of the classical gas-surface reaction probability.

Growing Fragmented Potentials for Gas–Surface Reactions: The Reaction between Hydrogen Atoms and Hydrogen-Terminated Silicon (111)

Systematic molecular fragmentation of nonconducting crystals has been combined with modified Shepard interpolation to develop an automated procedure to study the dynamics of gas–surface reactions. Any level of molecular electronic structure theory can be used to construct the gas–surface potential energy surface. The new methodology has been applied to the interaction of hydrogen atoms with the hydrogen-terminated silicon (111) surface at normal incidence. Direct hydrogen abstraction, hydrogen atom exchange, hydrogen atom sorption, and inelastic scattering processes were all found to be significant.

A scheme to interpolate potential energy surfaces and derivative coupling vectors without performing a global diabatization

Simulation of non-adiabatic molecular dynamics requires the description of multiple electronic state potential energy surfaces and their couplings. Ab initio molecular dynamics approaches provide an attractive avenue to accomplish this, but at great computational expense. Interpolation approaches provide a possible route to achieve flexible descriptions of the potential energy surfaces and their couplings at reduced expense. A previously developed approach based on modified Shepard interpolation required global diabatization, which can be problematic. Here, we extensively revise this previous approach, avoiding the need for global diabatization. The resulting interpolated potentials provide only adiabatic energies, gradients, and derivative couplings. This new interpolation approach has been integrated with the ab initio multiple spawning method and it has been rigorously validated against direct dynamics. It is shown that, at least for small molecules, constructing an interpolated PES can be more efficient than performing direct dynamics as measured by the total number of ab initio calculations that are required for a given accuracy.

Distributed Multipoles and Energies of Flexible Molecules.

In this work we show that energies and distributed multipoles, up to and including rank two, can be accurately determined via a modified Shepard interpolation of ab initio data for small molecules. The molecules considered here are the amino aldehydes, Gly and Ala, which may be typical smaller fragment molecules in certain molecular energy-based fragmentation schemes. The method is general and should be suitable for applications also involving crystal structure prediction, modeling molecular clusters, and Monte Carlo or molecular/reaction dynamics simulations. The configuration space covered by the interpolation includes that sampled by the Gly and Ala peptides in protein crystal structures, i.e., 12 dimensions for Gly: 3 torsion angles (φ, ψ, ω), 5 bond lengths, and 4 bond angles and 15 dimensions for Ala: 4 torsion angles, 6 bond lengths, and 5 bond angles. In this work we also describe a new method of importance, sampling the relevant configuration spaces, and show that it is possible to interpolate "axis free" multipoles.

Ab initio potential energy surface and quantum dynamics for the H + CH4 → H2 + CH3 reaction

A new full-dimensional potential energy surface for the title reaction has been constructed using the modified Shepard interpolation scheme. Energies and derivatives were calculated using the UCCSD(T) method with aug-cc-pVTZ and 6-311++G(3df,2pd) basis sets, respectively. A total number of 30,000 data points were selected from a huge number of molecular configurations sampled by trajectory method. Quantum dynamical calculations showed that the potential energy surface is well converged for the number of data points for collision energy up to 2.5 eV. Total reaction probabilities and integral cross sections were calculated on the present surface, as well as on the ZBB3 and EG-2008 surfaces for the title reaction. Satisfactory agreements were achieved between the present and the ZBB3 potential energy surfaces, indicating we are approaching the final stage to obtain a global potential energy surface of quantitative accuracy for this benchmark polyatomic system. Our calculations also showed that the EG-2008 surface is less accurate than the present and ZBB3 surfaces, particularly in high energy region.

Using Electronic Energy Derivative Information in Automated Potential Energy Surface Construction for Vibrational Calculations.

The availability of an accurate representation of the potential energy surface (PES) is an essential prerequisite in an anharmonic vibrational calculation. At the same time, the high dimensionality of the fully coupled PES and the adverse scaling properties with respect to the molecular size make the construction of an accurate PES a computationally demanding task. In the past few years, our group tested and developed a series of tools and techniques aimed at defining computationally efficient, black-box protocols for the construction of PESs for use in vibrational calculations. This includes the definition of an adaptive density-guided approach (ADGA) for the construction of PESs from an automatically generated set of evaluation points. Another separate aspect has been the exploration of the use of derivative information through modified Shepard (MS) interpolation/extrapolation procedures. With this article, we present an assembled machinery where these methods are embedded in an efficient way to provide both a general machinery as well as concrete computational protocols. In this framework we introduce and discuss the accuracy and computational efficiency of two methods, called ADGA[2gx3M] and ADGA[2hx3M], where the ADGA recipe is used (with MS interpolation) to automatically define modest sized grids for up to two-mode couplings, while MS extrapolation based on, respectively, gradients only and gradients and Hessians from the ADGA determined points provides access to sufficiently accurate three-mode couplings. The performance of the resulting potentials is investigated in vibrational coupled cluster (VCC) calculations. Three molecular systems serve as benchmarks: a trisubstituted methane (CHFClBr), methanimine (CH2NH), and oxazole (C3H3NO). Furthermore, methanimine and oxazole are addressed in accurate calculations aiming to reproduce experimental results.