Fitting Potential Energy Surfaces Research Articles

A new potential energy surface and microwave and infrared spectra of the He-OCS complex.

A new high quality potential energy surface for the He-OCS van der Waals complex was calculated using the CCSD(T) method and avqz+33221 basis set. It is found that the global minimum energy is -51.33 cm(-1) at R(e) = 6.30a0 and θ(e) = 110.0°, the shallower minimum is located at R = 8.50a0 and θ = 0° with well depth -32.26 cm(-1). Using the fitted potential energy surface, we have calculated bound energy levels of the He-OCS, He-O(13)CS, He-OC(34)S, and (3)He-OCS complexes. The theoretical results are all in better agreement compared to previous theoretical work.

A compact and accurate semi-global potential energy surface for malonaldehyde from constrained least squares regression.

We present a new approach to semi-global potential energy surface fitting that uses the least absolute shrinkage and selection operator (LASSO) constrained least squares procedure to exploit an extremely flexible form for the potential function, while at the same time controlling the risk of overfitting and avoiding the introduction of unphysical features such as divergences or high-frequency oscillations. Drawing from a massively redundant set of overlapping distributed multi-dimensional Gaussian functions of inter-atomic separations we build a compact full-dimensional surface for malonaldehyde, fit to explicitly correlated coupled cluster CCSD(T)(F12*) energies with a root mean square deviations accuracy of 0.3%-0.5% up to 25,000 cm(-1) above equilibrium. Importance-sampled diffusion Monte Carlo calculations predict zero point energies for malonaldehyde and its deuterated isotopologue of 14 715.4(2) and 13 997.9(2) cm(-1) and hydrogen transfer tunnelling splittings of 21.0(4) and 3.2(4) cm(-1), respectively, which are in excellent agreement with the experimental values of 21.583 and 2.915(4) cm(-1).

Permutation invariant polynomial neural network approach to fitting potential energy surfaces. III. Molecule-surface interactions.

The permutation invariant polynomial-neural network (PIP-NN) method for constructing highly accurate potential energy surfaces (PESs) for gas phase molecules is extended to molecule-surface interaction PESs. The symmetry adaptation in the NN fitting of a PES is achieved by employing as the input symmetry functions that fulfill both the translational symmetry of the surface and permutation symmetry of the molecule. These symmetry functions are low-order PIPs of the primitive symmetry functions containing the surface periodic symmetry. It is stressed that permutationally invariant cross terms are needed to avoid oversymmetrization. The accuracy and efficiency are demonstrated in fitting both a model PES for the H2 + Cu(111) system and density functional theory points for the H2 + Ag(111) system.

An overview of nonadiabatic dynamics simulations methods, with focus on the direct approach versus the fitting of potential energy surfaces

We review state-of-the-art nonadiabatic molecular dynamics methods, with focus on the comparison of two general strategies: the “direct” one, in which the potential energy surfaces (PES) and the couplings between electronic states are computed during the integration of the dynamics equations; and the “PES-fitting” one, whereby the PES and couplings are preliminarily computed and represented as functions of the nuclear coordinates. Both quantum wavepacket dynamics (QWD) and classical trajectory approaches are considered, but we concentrate on methods for which the direct strategy is viable: among the QWD ones, we focus on those based on traveling basis functions. We present several topics in which recent progress has been made: quantum decoherence corrections in trajectory methods, the use of quasi-diabatic representations, the sampling of initial conditions and the inclusion of field–molecule interactions and of spin–orbit couplings in the dynamics. Concerning the electronic structure calculations, we discuss the use of ab initio, density functional and semiempirical methods, and their combination with molecular mechanics (QM/MM approaches). Within the semiempirical framework, we provide a concise but updated description of our own method, based on configuration interaction with floating occupation molecular orbitals. We discuss the ability of different approaches to provide observables directly comparable with experimental results and to simulate a variety of photochemical and photophysical processes. In the concluding remarks, we stress how the border between direct and PES-fitting methods is not so sharp, and we briefly discuss recent trends that go beyond this traditional distinction.

Communication: separable potential energy surfaces from multiplicative artificial neural networks.

We present a potential energy surface fitting scheme based on multiplicative artificial neural networks. It has the sum of products form required for efficient computation of the dynamics of multidimensional quantum systems with the multi configuration time dependent Hartree method. Moreover, it results in analytic potential energy matrix elements when combined with quantum dynamics methods using Gaussian basis functions, eliminating the need for a local harmonic approximation. Scaling behavior with respect to the complexity of the potential as well as the requested accuracy is discussed.

Ab initio intermolecular potential energy surfaces for the Ar–NCCN van der Waals complexes

The intermolecular potential energy surface of complex pairing argon with cyanogen molecule (NCCN) was calculated using the coupled cluster with single and double and perturbative triple excitations (CCSD(T)) with aug-cc-pvdz basis set extended with a set of mid-bond (3s3p2d1f1g) functions. The interaction energies were calculated by the supermolecular approach with the full counterpoise correction for the basis set superposition error. The calculated potential energies were fitted to an analytical expression. The calculated Ar–NCCN potential energy surface shows a global minimum at 3.35 Å, the distance between argon and centre of mass of cyanogen, for the T-shaped geometry and two local minimum at distance of 5.54 Å for the linear geometry on one side of cyanogen. Finally, the interaction second virial coefficients were calculated using the fitted potential energy surface and were compared with those obtained by the parameters of the Beattie–Bridgeman equation of states of pure argon and cyanogens fluids, approximately.

Anchor Points Reactive Potential for Bond-Breaking Reactions

We present a new method for fitting potential energy surfaces in molecular-mechanics-like internal coordinates based on data from electronic structure calculations. The method should be applicable to chemical reactions involving either bond dissociation or isomerization and is illustrated here for bond dissociation, in particular the breaking of an O-H bond in methanol and the breaking of an N-H bond in dimethylamine. As compared to previously available systematic methods for fitting global potential energy surfaces, it extends the maximum size of the system than can be treated by at least an order of magnitude.

Potential energy surface fitting by a statistically localized, permutationally invariant, local interpolating moving least squares method for the many-body potential: Method and application to N4

Fitting potential energy surfaces to analytic forms is an important first step for efficient molecular dynamics simulations. Here, we present an improved version of the local interpolating moving least squares method (L-IMLS) for such fitting. Our method has three key improvements. First, pairwise interactions are modeled separately from many-body interactions. Second, permutational invariance is incorporated in the basis functions, using permutationally invariant polynomials in Morse variables, and in the weight functions. Third, computational cost is reduced by statistical localization, in which we statistically correlate the cutoff radius with data point density. We motivate our discussion in this paper with a review of global and local least-squares-based fitting methods in one dimension. Then, we develop our method in six dimensions, and we note that it allows the analytic evaluation of gradients, a feature that is important for molecular dynamics. The approach, which we call statistically localized, permutationally invariant, local interpolating moving least squares fitting of the many-body potential (SL-PI-L-IMLS-MP, or, more simply, L-IMLS-G2), is used to fit a potential energy surface to an electronic structure dataset for N4. We discuss its performance on the dataset and give directions for further research, including applications to trajectory calculations.

Permutation invariant polynomial neural network approach to fitting potential energy surfaces. II. Four-atom systems

A rigorous, general, and simple method to fit global and permutation invariant potential energy surfaces (PESs) using neural networks (NNs) is discussed. This so-called permutation invariant polynomial neural network (PIP-NN) method imposes permutation symmetry by using in its input a set of symmetry functions based on PIPs. For systems with more than three atoms, it is shown that the number of symmetry functions in the input vector needs to be larger than the number of internal coordinates in order to include both the primary and secondary invariant polynomials. This PIP-NN method is successfully demonstrated in three atom-triatomic reactive systems, resulting in full-dimensional global PESs with average errors on the order of meV. These PESs are used in full-dimensional quantum dynamical calculations.

Permutation invariant polynomial neural network approach to fitting potential energy surfaces

A simple, general, and rigorous scheme for adapting permutation symmetry in molecular systems is proposed and tested for fitting global potential energy surfaces using neural networks (NNs). The symmetry adaptation is realized by using low-order permutation invariant polynomials (PIPs) as inputs for the NNs. This so-called PIP-NN approach is applied to the H + H2 and Cl + H2 systems and the analytical potential energy surfaces for these two systems were accurately reproduced by PIP-NN. The accuracy of the NN potential energy surfaces was confirmed by quantum scattering calculations.

On representing chemical environments

We review some recently published methods to represent atomic neighbourhood environments, and analyse their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms, and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighbourhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbours increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether new approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps these difficulties by directly defining the similarity between any two neighbourhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.

Ab Initio Potential Energy and Dipole Moment Surfaces for CS2: Determination of Molecular Vibrational Energies

The ground state potential energy and dipole moment surfaces for CS2 have been determined at the CASPT2/C:cc-pVTZ,S:aug-cc-pV(T+d)Z level of theory. The potential energy surface has been fit to a sum-of-products form using the neural network method with exponential neurons. A generic interface between neural network potential energy surface fitting and the Heidelberg MCTDH software package is demonstrated. The potential energy surface has also been fit using the potfit procedure in MCTDH. For fits to the low-energy regions of the potential, the neural network method requires fewer parameters than potfit to achieve high accuracy; global fits are comparable between the two methods. Using these potential energy surfaces, the vibrational energies have been computed for the four most abundant CS2 isotopomers. These results are compared to experimental and previous theoretical data. The current potential energy surfaces are shown to accurately reproduce the low-lying vibrational energies within a few wavenumbers. Hence, the potential energy and dipole moments surfaces will be useful for future study on the control of quantum dynamics in CS2.

Modified Feed-Forward Neural Network Structures and Combined-Function-Derivative Approximations Incorporating Exchange Symmetry for Potential Energy Surface Fitting

The classical interchange (permutation) of atoms of similar identity does not have an effect on the overall potential energy. In this study, we present feed-forward neural network structures that provide permutation symmetry to the potential energy surfaces of molecules. The new feed-forward neural network structures are employed to fit the potential energy surfaces for two illustrative molecules, which are H(2)O and ClOOCl. Modifications are made to describe the symmetric interchange (permutation) of atoms of similar identity (or mathematically, the permutation of symmetric input parameters). The combined-function-derivative approximation algorithm (J. Chem. Phys. 2009, 130, 134101) is also implemented to fit the neural-network potential energy surfaces accurately. The combination of our symmetric neural networks and the function-derivative fitting effectively produces PES fits using fewer numbers of training data points. For H(2)O, only 282 configurations are employed as the training set; the testing root-mean-squared and mean-absolute energy errors are respectively reported as 0.0103 eV (0.236 kcal/mol) and 0.0078 eV (0.179 kcal/mol). In the ClOOCl case, 1693 configurations are required to construct the training set; the root-mean-squared and mean-absolute energy errors for the ClOOCl testing set are 0.0409 eV (0.943 kcal/mol) and 0.0269 eV (0.620 kcal/mol), respectively. Overall, we find good agreements between ab initio and NN prediction in term of energy and gradient errors, and conclude that the new feed-forward neural-network models advantageously describe the molecules with excellent accuracy.

Optical-optical double resonance spectroscopy of the quasi-linear S2 state of CHF and CDF. I. Spectroscopic analysis

In this work, we report on our full results of the spectroscopic analysis of the quasi-linear S(2) state of the prototypical halocarbene, CHF, and its deuterated isotopomer CDF using optical-optical double resonance spectroscopy through the S(1) state. A total of 51 S(2) state vibrational levels with angular momenta in the range [script-l] = 0-3 were observed for CHF, and 76 levels for CDF. Progressions involving all three fundamental vibrations were observed, and rotational constants were determined for each of these levels by measuring spectra through different intermediate J levels of the S(1) state. Our experimental results are in excellent agreement with the predictions of vibrational calculations using the discrete variable representation method. The variational vibrational calculations were performed with an analytic potential energy surface fit to ab initio data by the method of interpolating moving least squares. The ab initio data are Davidson-corrected multi-reference configuration interaction calculations based on a state-averaged multiconfigurational self-consistent field reference incorporating a generalized dynamic weighting scheme.

The solvent effect in enzyme-substrate interactions: Models of carboxypeptidase

The structure of water in the active site of carboxypeptidase A is being studied with ab initio quantum mechanical calculations and with the Monte Carlo statistical method. The zinc-containing portion of the active site was modeled by [Zn(NH3)2(OH)]+ in a geometry taken from the crystal structure of the enzyme. The potential energy surface was probed by calculating the interaction energy of the model with a water molecule situated at 211 representative positions. Energy points were fit to an analytical expression that was used to determine cross-sectional contour maps of one water molecule moving on the fitted potential energy surface. Expansion of the active site model to include other catalytically important functional groups showed the specific contribution of each to the interaction potential. The analytical potentials were also used in a Monte Carlo simulation of the solvent structure in the active site of carboxypeptidase A. Preliminary results from the last iteration of a Monte Carlo calculation on a portion of the active site indicate a distribution of water molecules that is consistent with experimental data on the native enzyme, and with the inferences drawn from these model studies with single water molecules.

Non-Hermitian Multiconfiguration Molecular Mechanics.

We present a new version of the multiconfiguration molecular mechanics (MCMM) algorithm for fitting potential energy surfaces of complex reactive systems. The main improvement consists in allowing the valence bond configuration interaction matrix to be non-Hermitian, which broadens the range of geometries over which the potential energy surface can be fit accurately. A second improvement is that the new algorithm has simpler gradients and Hessians and executes faster. The performance of the new algorithm is evaluated using the example of two model reactions.

Nonadiabatic dynamics at metal surfaces: Independent-electron surface hopping

Recent experiments have shown convincing evidence for nonadiabatic energy transfer from adsorbate degrees of freedom to surface electrons during the interaction of molecules with metal surfaces. In this paper, we propose an independent-electron surface hopping algorithm for the simulation of nonadiabatic gas-surface dynamics. The transfer of energy to electron-hole pair excitations of the metal is successfully captured by hops between electronic adiabats. The algorithm is able to account for the creation of multiple electron-hole pairs in the metal due to nonadiabatic transitions. Detailed simulations of the vibrational relaxation of nitric oxide on a gold surface, employing a multistate potential energy surface fit to density functional theory calculations, confirm that our algorithm can capture the underlying physics of the inelastic scattering process.

Interpolating moving least-squares methods for fitting potential energy surfaces: Using classical trajectories to explore configuration space

We develop two approaches for growing a fitted potential energy surface (PES) by the interpolating moving least-squares (IMLS) technique using classical trajectories. We illustrate both approaches by calculating nitrous acid (HONO) cis-->trans isomerization trajectories under the control of ab initio forces from low-level HF/cc-pVDZ electronic structure calculations. In this illustrative example, as few as 300 ab initio energy/gradient calculations are required to converge the isomerization rate constant at a fixed energy to approximately 10%. Neither approach requires any preliminary electronic structure calculations or initial approximate representation of the PES (beyond information required for trajectory initial conditions). Hessians are not required. Both approaches rely on the fitting error estimation properties of IMLS fits. The first approach, called IMLS-accelerated direct dynamics, propagates individual trajectories directly with no preliminary exploratory trajectories. The PES is grown "on the fly" with the computation of new ab initio data only when a fitting error estimate exceeds a prescribed tight tolerance. The second approach, called dynamics-driven IMLS fitting, uses relatively inexpensive exploratory trajectories to both determine and fit the dynamically accessible configuration space. Once exploratory trajectories no longer find configurations with fitting error estimates higher than the designated accuracy, the IMLS fit is considered to be complete and usable in classical trajectory calculations or other applications.

Spectroscopy, dissociation dynamics, and potential energy surfaces for CN(A)−Ar

The A (2)Pi-X (2)Sigma(+) band system of CN-Ar has been examined using fluorescence depletion and action spectroscopy techniques. Eight vibronic bands of the complex were observed in association with the monomer 3-0 transition. Pump-probe measurements were used to characterize CN(A (2)Pi(32),nu=3) fragments from direct photodissociation of CN(A (2)Pi,nu=3)-Ar and CN(X (2)Sigma(+),nu=7) fragments from CN(A (2)Pi,nu=3)-Ar predissociation. The latter showed a marked preference for population of positive parity diatomic rotational levels. Bound state calculations were used to assign the A-X bands and to obtain fitted potential energy surfaces for the A state. The average potential obtained from fitting had a well depth of D(e)=137.8 cm(-1). High-level ab initio calculations were used to obtain equilibrium Jacobi coordinates of theta(e)=94 degrees and R(e)=7.25 bohr. The near-symmetric character of the fitted potential energy surface was consistent with the symmetry preference observed in the predissociation dynamics.

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.