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: 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: further improvement of efficiency via cutoff strategies

In standard applications of interpolating moving least squares (IMLS) for fitting a potential-energy surface (PES), all available ab initio points are used. Because remote ab initio points negligibly influence IMLS accuracy and increase IMLS time-to-solution, we present two methods to locally restrict the number of points included in a particular fit. The fixed radius cutoff (FRC) method includes ab initio points within a hypersphere of fixed radius. The density adaptive cutoff (DAC) method includes points within a hypersphere of variable radius depending on the point density. We test these methods by fitting a six-dimensional analytical PES for hydrogen peroxide. Both methods reduce the IMLS time-to-solution by about an order of magnitude relative to that when no cutoff method is used. The DAC method is more robust and efficient than the FRC method.     

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．     

Vibrational energies of PH3 calculated variationally at the complete basis set limit

The potential energy surface for the electronic ground state of PH(3) was calculated at the CCSD(T) level using aug-cc-pV(Q+d)Z and aug-cc-pVQZ basis sets for P and H, respectively, with scalar relativistic corrections included. A parametrized function was fitted through these ab initio points, and one parameter of this function was empirically adjusted. This analytical PES was employed in variational calculations of vibrational energies with the newly developed program TROVE. The convergence of the calculated vibrational energies with increasing vibrational basis set size was improved by means of an extrapolation scheme analogous to the complete basis set limit schemes used in ab initio electronic structure calculations. The resulting theoretical energy values are in excellent agreement with the available experimentally derived values.     

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).     

Decomposition of protonated formic acid: One transition state—Two product channels

The unimolecular chemistry of protonated formic acid, [HCOOH]H(+), has been investigated by analyzing the fragmentation of metastable ions (MI) during flight in a sector mass spectrometer, and by proton transfer to formic acid in a Fourier-transform ion cyclotron resonance (FT-ICR) mass spectrometer. High level ab initio calculations have been used to model the relevant parts of the potential energy surface (PES). In addition, ab initio direct dynamics calculations have been conducted, tracing out 60 different reaction trajectories. The only stable isomer in the mass spectrometric experiments is HC(OH)(2)(+), which is the precursor to both observed ionic products, HCO(+) and H(3)O(+), via the same saddle point of the potential energy surface. The detailed motion of the dissociating molecule during passage of the post-transition state region of the PES therefore determines which product ion is formed. After passing the TS a transient HC(O)OH(2)(+) molecule is first formed. High total energy increases the probability that the nascent water molecule will have sufficient speed to escape the HCO(+) moiety. Otherwise, typically at low energies, the two units recombine, upon which intra-complex proton transfer is very likely. Eventually, this will give the more stable H(3)O(+).     

Ab initio intermolecular potential energy surface and second pressure virial coefficients of methane

A six-dimensional potential energy hypersurface (PES) for two interacting rigid methane molecules was determined from high-level quantum-mechanical ab initio computations. A total of 272 points for 17 different angular orientations on the PES were calculated utilizing the counterpoise-corrected supermolecular approach at the CCSD(T) level of theory with basis sets of aug-cc-pVTZ and aug-cc-pVQZ qualities. The calculated interaction energies were extrapolated to the complete basis set limit. An analytical site-site potential function with nine sites per methane molecule was fitted to the interaction energies. In addition, a semiempirical correction to the analytical potential function was introduced to take into account the effects of zero-point vibrations. This correction includes adjustments of the dispersion coefficients and of a single-parameter within the fit to the measured values of the second virial coefficient B(T) at room temperature. Quantitative agreement was then obtained with the measured B values over the whole temperature range of the measurements. The calculated B values should definitely be more reliable at very low temperatures (T<150 K) than values extrapolated using the currently recommended equation of state.     

Ab initio calculation and quasi‐classical dynamics study of the two lowest potential energy surfaces of the O(1D)+HBr system*

The lowest singlet 1¹A′ and 1¹A″ potential energy surfaces (PES) of the O(¹D)+HBr system have been ab initio computed. The complete active space self‐consistent field (CASSCF) method was used in most of the calculations, considering all the valence orbitals as active. The calculations were complemented with both analytical gradient calculations to characterize the stationary points and multireference configuration interaction (MRCI) calculations at selected nuclear geometries to improve the determination of the barrier heights and of the energetics. Electronic energy values for both PESs were then independently fitted by polynomial expansions in bond order coordinates. On the fitted surfaces quasi‐classical trajectories were separately run. Single‐surface calculations behave qualitatively different for the ground and the excited PES at low collision energies. A satisfactory agreement with existing experimental data was obtained by using the ground PES while calculations performed on the excited 1¹A″ PES worsened the agreement. However, when collision energy is increased, detailed experimental distributions are less well reproduced by calculations on the ground PES. This may imply the participation via nonadiabatic transitions of the 2¹A′ PES at higher energies while the adiabatic ground singlet PES well describes reactive scattering at low collision energy. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001     

Ab initio global potential-energy surface for H5(+) --> H3(+) + H2

An accurate global potential-energy surface (PES) is reported for H5(+) based on more than 100,000 CCSD(T)/aug-cc-pVTZ ab initio energies. This PES has full permutational symmetry with respect to interchange of H atoms and dissociates to H3(+) and H2. Ten known stationary points of H5(+) are characterized and compared to previous ab initio calculations. Quantum diffusion Monte Carlo calculations are performed on the PES to obtain the zero-point energy of H5(+) and the anharmonic dissociation energy (D0) of H5(+) --> H3(+) + H2. The rigorous zero-point state of H4D+ is also calculated and discussed within the context of a strictly classical approach to obtain the branching ratio of the reaction H4D+ --> H3(+) + HD and H2D+ + H2. Such an approach is taken using the PES and critiqued based on the properties of the quantum zero-point state. Finally, a simple procedure for adding the long range-interaction energy is described.     

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.     

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.     

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.     

Ab initio intermolecular potential energy surface and thermophysical properties of hydrogen sulfide

A six-dimensional potential energy hypersurface (PES) for two interacting rigid hydrogen sulfide molecules was determined from high-level quantum-mechanical ab initio computations. A total of 4016 points for 405 different angular orientations of two molecules were calculated utilizing the counterpoise-corrected supermolecular approach at the CCSD(T) level of theory and extrapolating the calculated interaction energies to the complete basis set limit. An analytical site-site potential function with eleven sites per hydrogen sulfide molecule was fitted to the interaction energies. The PES has been validated by computing the second pressure virial coefficient, shear viscosity, thermal conductivity and comparing with the available experimental data. The calculated values of volume viscosity were not used to validate the potential as the low accuracy of the available data precluded such an approach. The second pressure virial coefficient was evaluated by means of the Takahashi and Imada approach, while the transport properties, in the dilute limit, were evaluated by utilizing the classical trajectory method. In general, the agreement with the primary experimental data is within the experimental error for temperatures higher than 300 K. For lower temperatures the lack of reliable data indicates that the values of the second pressure virial coefficient and of the transport properties calculated in this work are currently the most accurate estimates for the thermophysical properties of hydrogen sulfide.     

辛准经典轨迹法研究Cl+H2反应

用辛准经典轨迹法模拟了Cl+H2反应在mBW2势能面上的动力学行为。研究了各种初始条件下的反应碰撞截面,产物的能量分配,角度分布和态分布。另外,我们还比较了反应物的三种能量形式(平动能,转动能和振动能)对反庆的有效性。     

Ab initio, VTST and QCT study of the 3A potential energy surface implied in the N(D)+O2(XΣg)→O(P)+NO(XΠ) reaction

A detailed ab initio study based on the CASSCF and CASPT2 methods has been carried out on the 3²A^′ potential energy surface (PES) involved in the reaction between N(²D) and O₂ to produce O(³P) and NO. All the stationary points have been characterised and a grid of more than 800 points has been fitted to an analytical function. This PES has been used to obtain the rate constant of this reaction at different levels of theory (statistical and quasiclassical trajectory methods, taking into account recent data of our own on other PESs involved in the reaction) and has been compared with the experimental results obtaining a good agreement.     

Ab initio molecular dynamics of hydrogen dissociation on metal surfaces using neural networks and novelty sampling

We outline a hybrid multiscale approach for the construction of ab initio potential energy surfaces (PESs) useful for performing six-dimensional (6D) classical or quantum mechanical molecular dynamics (MD) simulations of diatomic molecules reacting at single crystal surfaces. The algorithm implements concepts from the corrugation reduction procedure, which reduces energetic variation in the PES, and uses neural networks for interpolation of smoothed ab initio data. A novelty sampling scheme is implemented and used to identify configurations that are most likely to be predicted inaccurately by the neural network. This hybrid multiscale approach, which couples PES construction at the electronic structure level to MD simulations at the atomistic scale, reduces the number of density functional theory (DFT) calculations needed to specify an accurate PES. Due to the iterative nature of the novelty sampling algorithm, it is possible to obtain a quantitative measure of the convergence of the PES with respect to the number of ab initio calculations used to train the neural network. We demonstrate the algorithm by first applying it to two analytic potentials, which model the H2/Pt(111) and H2/Cu(111) systems. These potentials are of the corrugated London-Eyring-Polanyi-Sato form, which are based on DFT calculations, but are not globally accurate. After demonstrating the convergence of the PES using these simple potentials, we use DFT calculations directly and obtain converged semiclassical trajectories for the H2/Pt(111) system at the PW91/generalized gradient approximation level. We obtain a converged PES for a 6D hydrogen-surface dissociation reaction using novelty sampling coupled directly to DFT. These results, in excellent agreement with experiments and previous theoretical work, are compared to previous simulations in order to explore the sensitivity of the PES (and therefore MD) to the choice of exchange and correlation functional. Despite having a lower energetic corrugation in our PES, we obtain a broader reaction probability curve than previous simulations, which is attributed to increased geometric corrugation in the PES and the effect of nonparallel dissociation pathways.     

Quantum and classical studies of vibrational motion of CH5+ on a global potential energy surface obtained from a novel ab initio direct dynamics approach

We report a full dimensional, ab initio based potential energy surface for CH(5) (+). The ab initio electronic energies and gradients are obtained in direct-dynamics calculations using second-order M?ller-Plesset perturbation theory with the correlation consistent polarized valence triple zeta basis. The potential energy and the dipole moment surfaces are fit using novel procedures that ensure the full permutational symmetry of the system. The fitted potential energy surface is tested by comparing it against additional electronic energy calculations and by comparing normal mode frequencies at the three lowest-lying stationary points obtained from the fit against ab initio ones. Well-converged diffusion Monte Carlo zero-point energies, rotational constants, and projections along the CH and HH bond lengths and the tunneling coordinates are presented and compared with the corresponding harmonic oscillator and standard classical molecular dynamics ones. The delocalization of the wave function is analyzed through comparison of the CH(5) (+) distributions with those obtained when all of the hydrogen atoms are replaced by (2)H and (3)H. The classical dipole correlation function is examined as a function of the total energy. This provides a further probe of the delocalization of CH(5) (+).     

Quasiclassical trajectory calculations of the reaction C+C2H2-->l-C3H, c-C3H+H, C3+H2 using full-dimensional triplet and singlet potential energy surfaces

Full-dimensional, density functional theory (B3LYP/6-311g(d,p))-based potential energy surfaces (PESs) are reported and used in quasi-classical calculations of the reaction of C with C(2)H(2). For the triplet case, the PES spans the region of the reactants, the complex region (with numerous minima and saddle points) and the products, linear(l)-C(3)H+H, cyclic(c)-C(3)H+H and c-(3)C(3)+H(2). For the singlet case, the PES describes the complex region and products l-C(3)H+H, c-C(3)H+H and l-(1)C(3)+H(2). The PESs are invariant under permutation of like nuclei and are fit to tens of thousands of electronic energies. Energies and harmonic frequencies of the PESs agree well the DFT ones for all stationary points and for the reactant and the products. Dynamics calculations on the triplet PES find both l-C(3)H and c-C(3)H products, with l-C(3)H being dominant at the energies considered. Limited unimolecular reaction dynamics on the singlet PES find both products in comparable amounts as well as the C(3)+H(2) product.