Interpolating moving least-squares methods for fitting potential energy surfaces: computing high-density potential energy surface data from low-density ab initio data points

A highly accurate and efficient method for molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating-moving-least-squares (IMLS)-based method is developed using low-density ab initio Hessian values to compute high-density PES parameters suitable for accurate and efficient PES representation. The method is automated and flexible so that a PES can be optimally generated for classical trajectories, spectroscopy, or other applications. Two important bottlenecks for fitting PESs are addressed. First, high accuracy is obtained using a minimal density of ab initio points, thus overcoming the bottleneck of ab initio point generation faced in applications of modified-Shepard-based methods. Second, high efficiency is also possible (suitable when a huge number of potential energy and gradient evaluations are required during a trajectory calculation). This overcomes the bottleneck in high-order IMLS-based methods, i.e., the high cost/accuracy ratio for potential energy evaluations. The result is a set of hybrid IMLS methods in which high-order IMLS is used with low-density ab initio Hessian data to compute a dense grid of points at which the energy, Hessian, or even high-order IMLS fitting parameters are stored. A series of hybrid methods is then possible as these data can be used for neural network fitting, modified-Shepard interpolation, or approximate IMLS. Results that are indicative of the accuracy, efficiency, and scalability are presented for one-dimensional model potentials as well as for three-dimensional (HCN) and six-dimensional (HOOH) molecular PESs.     

Interpolating moving least-squares methods for fitting potential energy surfaces: an application to the H2CN unimolecular reaction

Classical trajectories have been used to compute rates for the unimolecular reaction H2CN-->H+HCN on a fitted ab initio potential energy surface (PES). The ab initio energies were obtained from CCSD(T)/aug-cc-pvtz electronic structure calculations. The ab initio energies were fitted by the interpolating moving least-squares (IMLS) method. This work continues the development of the IMLS method for producing ab initio PESs for use in molecular dynamics simulations of many-atom systems. A dual-level scheme was used in which the preliminary selection of data points was done using a low-level theory and the points used for fitting the final PES were obtained at the desired higher level of theory. Classical trajectories were used on various low-level IMLS fits to tune the fit to the unimolecular reaction under study. Procedures for efficiently picking data points, selecting basis functions, and defining cutoff limits to exclude distant points were investigated. The accuracy of the fitted PES was assessed by comparing interpolated values of quantities to the corresponding ab initio values. With as little as 330 ab initio points classical trajectory rate constants were converged to 5%-10% and the rms error over the six-dimensional region sampled by the trajectories was a few tenths of a kcal/mol.     

Interpolating moving least-squares methods for fitting potential energy surfaces: a strategy for efficient automatic data point placement in high dimensions

An accurate and efficient method for automated molecular global potential energy surface (PES) construction and fitting is demonstrated. An interpolating moving least-squares (IMLS) method is developed with the flexibility to fit various ab initio data: (1) energies, (2) energies and gradients, or (3) energies, gradients, and Hessian data. The method is automated and flexible so that a PES can be optimally generated for trajectories, spectroscopy, or other applications. High efficiency is achieved by employing local IMLS in which fitting coefficients are stored at a limited number of expansion points, thus eliminating the need to perform weighted least-squares fits each time the potential is evaluated. An automatic point selection scheme based on the difference in two successive orders of IMLS fits is used to determine where new ab initio data need to be calculated for the most efficient fitting of the PES. A simple scan of the coordinate is shown to work well to identify these maxima in one dimension, but this search strategy scales poorly with dimension. We demonstrate the efficacy of using conjugate gradient minimizations on the difference surface to locate optimal data point placement in high dimensions. Results that are indicative of the accuracy, efficiency, and scalability are presented for a one-dimensional model potential (Morse) as well as for three-dimensional (HCN), six-dimensional (HOOH), and nine-dimensional (CH4) molecular PESs.     

Interpolating moving least-squares methods for fitting potential energy surfaces: Analysis of an application to a six-dimensional system

The basic formal and numerical aspects of different degree interpolated moving least-squares (IMLS) methods are applied to a six-dimensional potential energy surface (PES) of the HOOH molecule, for which an analytic ("exact") potential is available in the literature. The results of systematic investigations of the effects of weight function parameters, the degree and partial degree of IMLS, the number of data points allowed, and the optimal automatic point selection of data points up to full third-degree IMLS fits are reported. With partial reduction of cross terms and automatic point selection the full six-dimensional HOOH PES can be fit over a range of 100 kcal/mol to an accuracy of less than 1 kcal/mol with approximately 1350 ab initio points.     

Gradient incorporation in one-dimensional applications of interpolating moving least-squares methods for fitting potential energy surfaces

We present several approaches to use gradients in higher degree interpolating moving least squares (IMLS) methods for representing a potential energy surface (PES). General procedures are developed to obtain smooth approximations of the PES and its derivatives from quasi-uniform sets of energy and gradient data points. These methods are illustrated and analyzed for the Morse oscillator and a 1-D slice of the ground-state PES for the HCO radical computed using density functional theory. Variations in the IMLS fits with the number and distribution of points and the degree of the polynomial fitting basis set are examined. We determine the effects of gradient inclusion on the accuracy of the IMLS values of the energy, first and second derivatives for two 1-D test cases. Gradient inclusion reduces the number of data points required by up to 40%.     

Interpolating moving least-squares methods for fitting potential energy surfaces: applications to classical dynamics calculations

As a continuation of our efforts to develop efficient and accurate interpolating moving least-squares (IMLS) methods for generating potential energy surfaces, we carry out classical trajectories and compute kinetics properties on higher degree IMLS surfaces. In this study, we have investigated the choice of coordinate system, the range of points (i.e., the cutoff radius) used in fitting, and strategies for selections of data points and basis elements. We illustrate and test the method by applying it to hydrogen peroxide (HOOH). In particular, reaction rates for the O-O bond breaking in HOOH are calculated on fitted surfaces using the classical trajectory approach to test the accuracy of the IMLS method for providing potentials for dynamics calculations.     

Interpolating moving least-squares methods for fitting potential energy surfaces: Improving efficiency via local approximants

The local interpolating moving least-squares (IMLS) method for constructing potential energy surfaces is investigated. The method retains the advantageous features of the IMLS approach in that the ab initio derivatives are not required and high degree polynomials can be used to provide accurate fits, while at the same time it is much more efficient than the standard IMLS approach because the least-squares solutions need to be calculated only once at the data points. Issues related to the implementation of the local IMLS method are investigated and the accuracy is assessed using HOOH as a test case. It is shown that the local IMLS method is at the same level of accuracy as the standard IMLS method. In addition, the scaling of the method is found to be a power law as a function of number of data points N, N(-q). The results suggest that when fitting only to the energy values for a d-dimensional system by using a Qth degree polynomial the power law exponent q approximately Qd when the energy range fitted is large (e.g., E<100 kcalmol for HOOH), and q>Qd when the energy range fitted is smaller (E<30 kcalmol) and the density of data points is higher. This study demonstrates that the local IMLS method provides an efficient and accurate means for constructing potential energy surfaces.     

用神经网络方法拟合的反应体系H＋CH4←→H2＋CH3的一个全域势能面

利用神经网络力法,基于47783个高精度从头算能量点构建了反应体系H＋CH4←→H2＋CH3的一个全域势能面．通过大最的准经典轨线以及量子散射计算测试了势能面的收敛性质．这个势能面对于拟合过程以及从头算点的数目都已经完全收敛,拟合误差很小县比Shepard插值的势能面计算速度更快,代表了此标志性多原子反应体系最好的势能面．     

Application of second-order density functional methods to the calculation of the BeFH potential energy surface

Second-order density functional methods (where the correlation energy depends on the second-order density matrix and on a density functional) are used to introduce the electron correlation in two-configuration direct minimization (TCDM) ab initio electronic energy calculations of three-dimensional potential energy surfaces (PES). We analyzed the behavior of these methods in PES calculations by applying them to the Be + FH reaction. This system was studied also by the usual techniques, allowing a full comparison for the lowest ¹A′ adiabatic state. In particular, we compared the results obtained using the TCDM and multiple reference single and double excitations configuration interaction (MRDCI) methods with the corresponding results obtained using the Colle-Salvetti (CS) and Moscardó-San Fabián (MSF) procedures, within the correlation factor method, using as starting point the TCDM results. We found that the CS and MSF results are in a good overall agreement with the more accurate ab initio results, including the heights of the saddle points and the transition state. © 1997 John Wiley & Sons, Inc.     

A new "spectroscopic" potential energy surface for formaldehyde in its ground electronic state

We report a new "spectroscopic" potential energy surface (PES) of formaldehyde (H(2)(12)C(16)O) in its ground electronic state, obtained by refining an ab initio PES in a least-squares fitting to the experimental spectroscopic data for formaldehyde currently available in the literature. The ab initio PES was computed using the CCSD(T)/aug-cc-pVQZ method at 30 840 geometries that cover the energy range up to 44 000 cm(-1) above equilibrium. Ro-vibrational energies of formaldehyde were determined variationally for this ab initio PES by means of the program TROVE [Theoretical ROtation-Vibration Energies; S. N. Yurchenko, W. Thiel, and P. Jensen, J. Mol. Spectrosc. 245, 126 (2007)]. The parameter values in the analytical representation of the PES were optimized in fittings to 319 ro-vibrational energies with J = 0, 1, 2, and 5. The initial parameter values in the fittings were those of the ab initio PES, the ro-vibrational eigenfunctions obtained from this PES served as a basis set during the fitting process, and constraints were imposed to ensure that the refined PES does not deviate unphysically from the ab initio one in regions of configuration space not sampled by the experimental data. The resulting refined PES, referred to as H(2)CO-2011, reproduces the available experimental J ≤ 5 data with a root-mean-square error of 0.04 cm(-1).     

Ab initio potential energy surface, variational transition state theory, and quasiclassical trajectory studies of the F+CH4-->HF+CH3 reaction

In this work we present a study of the F+CH(4)-->HF+CH(3) reaction (DeltaHdegrees(298 K)=-32.0 kcal mol(-1)) using different methods of the chemical reaction theory. The ground potential energy surface (PES) is characterized using several ab initio methods. Full-dimensional rate constants have been calculated employing the variational transition state theory and using directly ab initio data. A triatomic analytical representation of the ground PES was derived from ab initio points calculated at the second- and fourth-order M?ller-Plesset levels with the 6-311+G(2df,2pd) basis set, assuming the CH(3) fragment to be a 15 a.m.u. pseudoatom in the fitting process. This is suggested from experiments that indicate that the methyl group is uncoupled to the reaction coordinate. A dynamics study by means of the quasiclassical trajectory (QCT) method and employing this analytical surface was also carried out. The experimental data available on the HF internal states distributions are reproduced by the QCT results. Very recent experimental information about the reaction stereodynamics is also borne out by our QCT calculations. Comparisons with the benchmark F+H(2) and analogous Cl+CH(4) reactions are established throughout.     

Efficiency considerations in the construction of interpolated potential energy surfaces for the calculation of quantum observables by diffusion Monte Carlo

A modified Shepard interpolation scheme is used to construct global potential energy surfaces (PES) in order to calculate quantum observables--vibrationally averaged internal coordinates, fully anharmonic zero-point energies and nuclear radial distribution functions--for a prototypical loosely bound molecular system, the water dimer. The efficiency of PES construction is examined with respect to (a) the method used to sample configurational space, (b) the method used to choose which points to add to the PES data set, and (c) the use of either a one- or two-part weight function. The most efficient method for constructing the PES is found to require a quantum sampling regime, a combination of both h-weight and rms methods for choosing data points and use of the two-part weight function in the interpolation. Using this regime, the quantum diffusion Monte Carlo zero-point energy converges to the exact result within addition of 50 data points. The vibrationally averaged O-O distance and O-O radial distribution function, however, converge more slowly and require addition of over 500 data points. The methods presented here are expected to be applicable to both other loosely bound complexes as well as tightly bound molecular species. When combined with high quality ab initio calculations, these methods should be able to accurately characterize the PES of such species.     

Quasiclassical trajectory calculations of the OH+NO2 association reaction on a global potential energy surface

We report a full-dimensional potential energy surface (PES) for the OH+NO(2) reaction based on fitting more than 55,000 energies obtained with density functional theory-B3LYP6-311G(d,p) calculations. The PES is invariant with respect to permutation of like nuclei and describes all isomers of HOONO, HONO(2), and the fragments OH+NO(2) and HO(2)+NO. Detailed comparison of the structures, energies, and harmonic frequencies of various stationary points on the PES are made with previous and present high-level ab initio calculations. Two hydrogen-bond complexes are found on the PES and confirmed by new ab initio CASPT2 calculations. Quasiclassical trajectory calculations of the cross sections for ground rovibrational OH+NO(2) association reactions to form HOONO and HONO(2) are done using this PES. The cross section to form HOONO is larger than the one to form HONO(2) at low collision energies but the reverse is found at higher energies. The enhancement of the HOONO complex at low collision energies is shown to be due, in large part, to the transient formation of a H-bond complex, which decays preferentially to HOONO. The association cross sections are used to obtain rate constants for formation of HOONO and HONO(2) for the ground rovibrational states in the high-pressure limit.     

Vibrational and rotational cooling of NO+ in collisions with He

A quantum mechanical investigation of the vibrational and rotational deactivation of NO(+) in collisions with He atoms in the cold and ultracold regime is presented. Ab initio potential energy calculations are carried out at BCCD(T) level and a new global 3D potential energy surface (PES) is obtained by fitting ab initio points within the reproducing kernel Hilbert space method. As a first test of this PES the bound state energies of the (3)He-NO(+) and (4)He-NO(+) complexes are calculated and compared to previous rigid rotor calculations. The efficiency of the vibrational and the rotational cooling of this molecular ion using a buffer gas of helium is then investigated by performing close coupling scattering calculations for collision energy ranging from 10(-6) to 2000 cm(-1). The calculations are performed for the two isotopes (3)He and (4)He and the results are compared to the available experimental data.     

Ab initio analytical potential energy surface and quasiclassical trajectory study of the O(+)((4)S)+H(2)(X (1)Sigma(g) (+))-->OH(+)(X (3)Sigma(-))+H((2)S) reaction and isotopic variants

An analytical potential energy surface (PES) representation of the O(+)((4)S)+H(2)(X (1)Sigma(g) (+)) system was developed by fitting around 600 CCSD(T)/cc-pVQZ ab initio points. Rate constant calculations for this reaction and its isotopic variants (D(2) and HD) were performed using the quasiclassical trajectory (QCT) method, obtaining a good agreement with experimental data. Calculations conducted to determine the cross section of the title reaction, considering collision energies (E(T)) below 0.3 eV, also led to good accord with experiments. This PES appears to be suitable for kinetics and dynamics studies. Moreover, the QCT results show that, although the hypotheses of a widely used capture model are not satisfied, the resulting expression for the cross section can be applied within a suitable E(T) interval, due to errors cancellation. This could be a general situation regarding the application of this simple model to ion-molecule processes.     

Nitrous oxide dimer: a new potential energy surface and rovibrational spectrum of the nonpolar isomer

The spectrum of nitrous oxide dimer was investigated by constructing new potential energy surfaces using coupled-cluster theory and solving the rovibrational Schro?dinger equation with a Lanczos algorithm. Two four-dimensional (rigid monomer) global ab initio potential energy surfaces (PESs) were made using an interpolating moving least-squares (IMLS) fitting procedure specialized to describe the interaction of two linear fragments. The first exploratory fit was made from 1646 CCSD(T)/3ZaP energies. Isomeric minima and connecting transition structures were located on the fitted surface, and the energies of those geometries were benchmarked using complete basis set (CBS) extrapolations, counterpoise (CP) corrections, and explicitly correlated (F12b) methods. At the geometries tested, the explicitly correlated F12b method produced energies in close agreement with the estimated CBS limit. A second fit to 1757 data at the CCSD(T)-F12b/VTZ-F12 level was constructed with an estimated fitting error of less than 1.5?cm(-1). The second surface has a global nonpolar O-in minimum, two T-shaped N-in minima, and two polar minima. Barriers between these minima are small and some wave functions have amplitudes in several wells. Low-lying rovibrational wave functions and energy levels up to about 150?cm(-1) were computed on the F12b PES using a discrete variable representation/finite basis representation method. Calculated rotational constants and intermolecular frequencies are in very close agreement with experiment.     

Potential energy surface of HDO up to 25,000 cm-1

A new spectroscopically determined potential energy surface (PES) for HD(16)O is presented. This surface is constructed by adjusting the high accuracy ab initio PES of Polyansky et al. [Science 299, 539 (2003)] by fitting to both published experimental data and our still unpublished data. This refinement used experimentally derived term values up to 25,000 cm(-1) and with J< or =8: a data set of 3478 energy levels once some levels with ambiguous assignment is excluded. To improve the extrapolation properties of the empirical PES, the restraint that the resulting PESs remain close to the ab initio surface was imposed. The new HDO_07 PES reproduces the experimental data, including high J levels not included in the fit, with a root mean square error of 0.035 cm(-1). Predictions for rotation-vibration term values up to J=12 are made.     

Global mapping of equilibrium and transition structures on potential energy surfaces by the scaled hypersphere search method: applications to ab initio surfaces of formaldehyde and propyne molecules

Technical details of a new global mapping technique for finding equilibrium (EQ) and transition structures (TS) on potential energy surfaces (PES), the scaled hypersphere search (SHS) method (Ohno, K.; Maeda, S. Chem. Phys. Lett. 2004, 384, 277), are presented. On the basis of a simple principle that reaction pathways are found as anharmonic downward distortions of PES around an EQ point, the reaction pathways can be obtained as energy minima on the scaled hypersphere surface, which would have a constant energy when the potentials are harmonic. Connections of SHS paths between each EQ are very similar to corresponding intrinsic reaction coordinate (IRC) connections. The energy maximum along the SHS path reaches a region in close proximity to the TS of the reaction pathway, and the subsequent geometry optimization from the SHS maximum structure easily converges to the TS. The SHS method, using the one-after-another algorithm connecting EQ and TS, considerably reduces the multidimensional space to be searched to certain limited regions around the pathways connecting each EQ with the neighboring TS. Applications of the SHS method have been made to ab initio surfaces of formaldehyde and propyne molecules to obtain systematically five EQ and nine TS for formaldehyde and seven EQ and 32 TS for propyne.     

Global potential energy surface, vibrational spectrum, and reaction dynamics of the first excited (Ã (2)A(')) state of HO(2)

The authors report extensive high-level ab initio studies of the first excited (A??(2)A(')) state of HO(2). A global potential energy surface (PES) was developed by spline-fitting 17?000 ab initio points at the internal contracted multireference configuration interaction (icMRCI) level with the AVQZ basis set. To ascertain the spectroscopic accuracy of the PES, the near-equilibrium region of the molecule was also investigated using three interpolating moving least-squares-based PESs employing dynamically weighted icMRCI methods in the complete basis set limit. Vibrational energy levels on all four surfaces agree well with each other and a new assignment of some vibrational features is proposed. In addition, the dynamics of both the forward and reverse directions of the H+O(2)(a??(1)Δ(g))?OH+O reaction (J=0) were studied using an exact wave packet method. The reactions are found to be dominated by sharp resonances.