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The positions of grid points for representing a multidimensional potential energy surface (PES) have a non-negligible impact on its accuracy and the associated computational effort for its generation. Six different positioning schemes were studied for PESs represented by n-mode expansions as needed for the accurate calculation of anharmonic vibrational frequencies by means of vibrational configuration interaction theory. A static approach, which has successfully been used in many applications, and five adaptive schemes based on Gaussian process regression have been investigated with respect to the number of necessary grid points and the accuracy of the fundamental modes for a small set of test molecules. A comparison with a related, more sophisticated, and consistent approach by Christiansen et al. is provided. The impact of the positions of the ab initio grid points is discussed for multilevel PESs, for which the computational effort of the individual electronic structure calculations decreases for increasing orders of the n-mode expansion. As a result of that, the ultimate goal is not the maximal reduction of grid points but rather the computational cost, which is not directly related. [ABSTRACT FROM AUTHOR]

The accuracy of the training data limits the accuracy of bulk properties from machine-learned potentials. For example, hybrid functionals or wave-function-based quantum chemical methods are readily available for cluster data but effectively out of scope for periodic structures. We show that local, atom-centered descriptors for machine-learned potentials enable the prediction of bulk properties from cluster model training data, agreeing reasonably well with predictions from bulk training data. We demonstrate such transferability by studying structural and dynamical properties of bulk liquid water with density functional theory and have found an excellent agreement with experimental and theoretical counterparts. [ABSTRACT FROM AUTHOR]

The vibrational spectra of simple C H 3 + —Rg (Rg = He, Ne, Ar, Kr) complexes have been studied by vibrational configuration interaction theory relying on multidimensional potential energy surfaces (PESs) obtained from explicitly correlated coupled cluster calculations, CCSD(T)-F12a. In agreement with experimental results, the series of rare gas atoms leads to rather unsystematic results and indicates huge zero point vibrational energy effects for the helium complex. In order to study these sensitive complexes more consistently, we also introduce configuration averaged vibrational self-consistent field theory, which is a generalization of standard vibrational self-consistent field theory to several configurations. The vibrational spectra of the complexes are compared to that of the methyl cation, for which corrections due to scalar-relativistic effects, high-order coupled-cluster terms, e.g., quadruple excitations, and core-valence correlation have explicitly been accounted for. The occurrence of tunneling splittings for the vibrational ground-state of C H 3 + —He has been investigated on the basis of semiclassical instanton theory. These calculations and a direct comparison of the energy profiles along the intrinsic reaction coordinates with that of the hydronium cation, H3O+, suggest that tunneling effects for vibrationally excited states should be very small. [ABSTRACT FROM AUTHOR]

Microcanonical instanton theory offers the promise of providing rate constants for chemical reactions including quantum tunneling of atoms over the whole temperature range. We discuss different rate expressions, which require the calculation of stability parameters of the instantons. The traditional way of obtaining these stability parameters is shown to be numerically unstable in practical applications. We provide three alternative algorithms to obtain such stability parameters for non-separable systems, i.e., systems in which the vibrational modes perpendicular to the instanton path couple to movement along the path. We show the applicability of our algorithms on two molecular systems: H2 + OH → H2O + H using a fitted potential energy surface and HNCO + H → NH2CO using a potential obtained on-the-fly from density functional calculations. [ABSTRACT FROM AUTHOR]

Transition paths characterize chemical reaction mechanisms. In this paper, we present a new method to find mean reaction paths based on the free energy. A nudged elastic band (NEB) is optimized using gradients and Hessians of the free energy, which are obtained from umbrella integration. The transition state can be refined by a Newton-Raphson search starting from the highest point of the NEB path. All optimizations are done using Cartesian coordinates. Independent molecular dynamics (MD) runs are performed at each image used to discretize the path. This makes the method intrinsically parallel. In contrast to other free energy methods, the algorithm does not become more expensive when including more degrees of freedom in the active space. The method is applied to the alanine-dipeptide as a test case and compared to pathways that have been derived from metadynamics and forward flux sampling. [ABSTRACT FROM AUTHOR]

Umbrella integration is a method to analyze umbrella sampling simulations. It calculates free-energy changes from distributions obtained from molecular dynamics. While it can be formulated on the full sampled distributions, they are generally approximated by normal distributions. This is equivalent to the truncation of a power series of the free energy with respect to the reaction coordinate after the quadratic term or by a truncation of a cumulant expansion. Here, expressions for additional terms in the power series are derived. They can be calculated from the central moments of the distributions. This extension allows to test the approximations in applications. [ABSTRACT FROM AUTHOR]

The instanton method allows to accurately calculate tunneling rates down to very low temperature. However, with lowering the temperature, the computational effort steeply increases as many more discretization points are required. This is caused in practical applications by the majority of the discretization points accumulating at a very small region in configuration space. Here, we describe a method to flexibly discretize the instanton path adapted to the temperature. Chosen appropriately, the discretization leads to a much more uniform distribution of the images (control points) along the path which reduces the number of required images by about a factor of two. Combined with a modified Newton-Raphson optimizer and successive updates of the Hessians, the proposed method provides converged reaction rates at computational costs reduced by more than an order of magnitude. We show the success of the method on analytic test potentials and on molecules with energies directly obtained from density functional theory calculations. [ABSTRACT FROM AUTHOR]

Umbrella integration is a method to analyze umbrella sampling simulations by calculating and integrating the mean force. Here, the method is extended to multidimensional reaction coordinates. Approximation of the probability distribution obtained from sampling by a multivariate normal distribution allows to calculate the mean force from the average and the covariance matrix of the reaction coordinate. Integration schemes of the free-energy gradient field are discussed. Integration on a real-space grid is compared to expansion of the gradient in a series of analytic functions (such as a Fourier analysis), which can be integrated, and the expansion of the gradient only at the window means in a series of analytic functions. The Fourier analysis was found particularly useful for periodic reaction coordinates, such as torsion angles. An expression is provided to calculate the Hessian of the free energy with respect to the reaction coordinates from sampling data. The utility of the method is demonstrated at the example of the free-energy surface of the alanine dipeptide in vacuum calculated with respect to the backbone torsion angles [uppercase_phi_synonym] and Ψ. Relevance of the Jacobian term for non-Cartesian reaction coordinates is discussed. [ABSTRACT FROM AUTHOR]

Algorithmic improvements of the dimer method [G. Henkelman and H. Jónsson, J. Chem. Phys. 111, 7010 (1999)] are described in this paper. Using the limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimizer for the dimer translation greatly improves the convergence compared to the previously used conjugate gradient algorithm. It also saves one energy and gradient calculation per dimer iteration. Extrapolation of the gradient during repeated dimer rotations reduces the computational cost to one gradient calculation per dimer rotation. The L-BFGS algorithm also improves convergence of the rotation. Thus, three to four energy and gradient evaluations are needed per iteration at the beginning of a transition state search, while only two are required close to convergence. Moreover, we apply the dimer method in internal coordinates to reduce coupling between the degrees of freedom. Weighting the coordinates can be used to apply chemical knowledge about the system and restrict the transition state search to only part of the system while minimizing the remainder. These improvements led to an efficient method for the location of transition states without the need to calculate the Hessian. Thus, it is especially useful in large systems with expensive gradient evaluations. [ABSTRACT FROM AUTHOR]

Umbrella sampling simulations, or biased molecular dynamics, can be used to calculate the free-energy change of a chemical reaction. We investigate the sources of different sampling errors and derive approximate expressions for the statistical errors when using harmonic restraints and umbrella integration analysis. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an analytical model potential are presented for validation. While the derivations are based on umbrella integration analysis, the final error estimate is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram analysis method. [ABSTRACT FROM AUTHOR]

We present a method to analyze biased molecular-dynamics and Monte Carlo simulations, also known as umbrella sampling. In the limiting case of a strong bias, this method is equivalent to thermodynamic integration. It employs only quantities with easily controllable equilibration and greatly reduces the statistical errors compared to the standard weighted histogram analysis method. We show the success of our approach for two examples, one analytic function, and one biological system. [ABSTRACT FROM AUTHOR]

The protonation of N2 bound to the active center of nitrogenase has been investigated using state-of-the-art density-functional theory calculations. Dinitrogen in the bridging mode is activated by forming two bonds to Fe sites, which results in a reduction of the energy for the first hydrogen transfer by 123 kJ/mol. The axial binding mode with open sulfur bridge is less reactive by 30 kJ/mol and the energetic ordering of the axial and bridged binding modes is reversed in favor of the bridging dinitrogen during the first protonation. Protonation of the central ligand is thermodynamically favorable but kinetically hindered. If the central ligand is protonated, the proton is transferred to dinitrogen following the second protonation. Protonation of dinitrogen at the Mo site does not lead to low-energy intermediates. [ABSTRACT FROM AUTHOR]

We implemented a geometry optimizer based on Gaussian process regression (GPR) to find minimum structures on potential energy surfaces. We tested both a two times differentiable form of the Matérn kernel and the squared exponential kernel. The Matérn kernel performs much better. We give a detailed description of the optimization procedures. These include overshooting the step resulting from GPR in order to obtain a higher degree of interpolation vs. extrapolation. In a benchmark against the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimizer of the DL-FIND library on 26 test systems, we found the new optimizer to generally reduce the number of required optimization steps. [ABSTRACT FROM AUTHOR]

Artificial neural networks are used to fit a potential energy surface (PES). We demonstrate the benefits of using not only energies but also their first and second derivatives as training data for the neural network. This ensures smooth and accurate Hessian surfaces, which are required for rate constant calculations using instanton theory. Our aim was a local, accurate fit rather than a global PES because instanton theory requires information on the potential only in the close vicinity of the main tunneling path. Elongations along vibrational normal modes at the transition state are used as coordinates for the neural network. The method is applied to the hydrogen abstraction reaction from methanol, calculated on a coupled-cluster level of theory. The reaction is essential in astrochemistry to explain the deuteration of methanol in the interstellar medium. [ABSTRACT FROM AUTHOR]

The calculation of free-energy barriers by umbrella sampling and many other methods is hampered by the necessity for an a priori choice of the reaction coordinate along which to sample. We avoid this problem by providing a method to search for saddle points on the free-energy surface in many coordinates. The necessary gradients and Hessians of the free energy are obtained by multidimensional umbrella integration. We construct the minimum free-energy path by following the gradient down to minima on the free-energy surface. The change of free energy along the path is obtained by integrating out all coordinates orthogonal to the path. While we expect the method to be applicable to large systems, we test it on the alanine dipeptide in vacuum. The minima, transition states, and free-energy barriers agree well with those obtained previously with other methods. [ABSTRACT FROM AUTHOR]

The vibrational spectra of simple CH 3 + -Rg (Rg = He, Ne, Ar, Kr) complexes have been studied by vibrational configuration interaction theory relying on multidimensional potential energy surfaces (PESs) obtained from explicitly correlated coupled cluster calculations, CCSD(T)-F12a. In agreement with experimental results, the series of rare gas atoms leads to rather unsystematic results and indicates huge zero point vibrational energy effects for the helium complex. In order to study these sensitive complexes more consistently, we also introduce configuration averaged vibrational self-consistent field theory, which is a generalization of standard vibrational self-consistent field theory to several configurations. The vibrational spectra of the complexes are compared to that of the methyl cation, for which corrections due to scalar-relativistic effects, high-order coupled-cluster terms, e.g., quadruple excitations, and core-valence correlation have explicitly been accounted for. The occurrence of tunneling splittings for the vibrational ground-state of CH 3 + -He has been investigated on the basis of semiclassical instanton theory. These calculations and a direct comparison of the energy profiles along the intrinsic reaction coordinates with that of the hydronium cation, H 3 O + , suggest that tunneling effects for vibrationally excited states should be very small.

We calculated reaction rate constants including atom tunneling of the reaction of dihydrogen with the hydroxy radical down to a temperature of 50 K. Instanton theory and canonical variational theory with microcanonical optimized multidimensional tunneling were applied using a fitted potential energy surface [J. Chen et al., J. Chem. Phys. 138, 154301 (2013)]. All possible protium/deuterium isotopologues were considered. Atom tunneling increases at about 250 K (200 K for deuterium transfer). Even at 50 K the rate constants of all isotopologues remain in the interval 4 ⋅ 10(-20) to 4 ⋅ 10(-17) cm(3) s(-1), demonstrating that even deuterated versions of the title reaction are possibly relevant to astrochemical processes in molecular clouds. The transferred hydrogen atom dominates the kinetic isotope effect at all temperatures.