Chemical Modelling: Applications and Theory comprises critical literature reviews of molecular modelling, both theoretical and applied. Molecular modelling in this context refers to modelling the structure, properties and reactions of atoms, molecules and materials. Each chapter is compiled by experts in their fields and provides a selective review of recent literature, incorporating sufficient historical perspectives for the non-specialist to gain an understanding.
About the Author
Prof. Dr. Michael Springborg heads up of the three groups in Physical Chemistry at the University of Saarland where the main activities concentrate on teaching and research. The major part of Prof. Dr. Michael Springborg's research concentrates on the development and application of theoretical methods, including accompanying computer programs, for the determination of materials properties. Quantum theory forms the theoretical foundation for most of our work. The materials of the group's interest range from atoms, via clusters and polymers, to solids. They study their structural, electronic, energetic, and opitcal properties.
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Chemical Modelling Applications and Theory Volume 7
A Review of the Literature Published Between June 2008 and December 2009
By M. Springborg
The Royal Society of ChemistryCopyright © 2010 The Royal Society of Chemistry
All rights reserved.
Neural network potential-energy surfaces for atomistic simulations
Studying chemical reactions in computer simulations requires a reliable description of the atomic interactions. While for systems of moderate size precise electronic structure calculations can be carried out to determine the energy and the forces, for large systems it is necessary to employ more efficient potentials. In past decades a huge number of such potentials has been developed for a variety of systems. Still, for the investigation of many chemical problems the accuracy of the available potentials is not yet satisfactory. In particular, chemical reactions at surfaces, strongly varying bonding patterns in materials science, and the complex reactivity of metal centers in coordination chemistry are prominent examples where most existing potentials are not sufficiently accurate. In recent years, a new class of interatomic potentials based on artificial neural networks has emerged. These potentials have a very flexible functional form and can therefore accurately adapt to a reference set of electronic structure energies. To date, neural network potentials have been constructed for a number of systems. They are promising candidates for future applications in large-scale molecular dynamics simulations, because they can be evaluated several orders of magnitude faster than the underlying electronic structure energies. However, further methodical developments are needed to reach this goal. In this review the current status of neural network potentials is summarized. Open problems and limitations of the hitherto proposed methods are discussed, and some possible solutions are presented.
Molecular dynamics (MD) and Monte Carlo simulations have significantly contributed to the detailed understanding of a variety of chemical processes at the atomic level. However, the outcome of the simulations critically depends on the accuracy of the energies and atomic forces, i.e., on the quality of the underlying potential-energy surface (PES). The PES is defined as a high-dimensional function providing the potential-energy as a function of the atomic positions. Individual points on the PES can be calculated using a variety of quantum chemical methods like e.g. Hartree Fock theory, Møller Plesset perturbation theory or coupled cluster theory. However, these methods are computationally too demanding to be applicable in molecular simulations on a routine basis. The only first-principles electronic structure method, which is sufficiently fast to perform MD simulations "on-the-fly" for systems of moderate size is density-functional theory (DFT). In spite of some limitations of currently available approximate exchange-correlation functionals, the resulting "ab initio molecular dynamics" is without doubt the most accurate method to follow chemical reactions in complex systems dynamically without relying on the construction of intermediate PESs. Nevertheless, in many cases even the most efficient implementations of ab initio MD are computationally simply too expensive to be carried out on the currently available supercomputers, and this situtation is unlikely to change in the next decade. Additionally, it is a frustrating fact that in ab initio MD simulations a lot of time is spent on recalculating similar structures again and again, even if closely related atomic configurations have been visited before. Therefore, it would be desirable to collect and reuse the information about the PES gained in these simulations.
To extend the time and length scales of molecular simulations, a huge number of more efficient approximate potentials for various applications has been developed in the past decades. For very simple systems like diatomic molecules or weakly interacting noble gas atoms very accurate analytic forms can be constructed based on chemical knowledge and intuition. These potentials, e.g. the Lennard Jones potential or the Morse potential, depend only on a few parameters that can be determined from experiment or ab initio calculations. However, these simple pair potentials already fail for three-atomic systems, because usually the interactions between atoms are not pairwise additive.
The most basic approach to carry out MD simulations for larger systems is to use classical force fields. A variety of different force fields for molecular mechanics (MM) simulations has been developed, which are mainly intended to describe the non-reactive dynamics of large systems. In particular in the field of biochemistry force fields play an essential role to study the complex properties of large biomolecules. However, classical force fields require the specification of the connectivity of the atoms. Therefore, they are not able to describe chemical reactions, i.e., the making and breaking of bonds. To describe reactions, they can be combined with quantum mechanical (QM) methods in so-called QM/MM simulations. In recent years also "reactive force fields", e.g. ReaxFF, have been introduced, which overcome this limitation. However, these reactive force fields are typically highly adapted to specific systems by analytic terms customized to describe e.g. certain bonding situations, and only a few applications have been reported so far.
Mainly in the field of materials science various types of potentials have been developed based on the concept of the bond order. Like for reactive force fields also for the application of these potentials a specification of the atomic positions is sufficient. Although many of these potentials like the Tersoff potential, the Stillinger-Weber potential, the Brenner potential and many others have been introduced already one or two decades ago, they are still frequently used in materials simulations, in particular for semiconductors. For metallic systems the embedded atom method (EAM) and the modified embedded atom method (MEAM) introduced by Baskes and coworkers are widely distributed.
In parallel to these methods also a variety of simplified electronic structure methods like tight binding and semiempirical methods have been developed. Since these methods still contain essential parts of the underlying quantum mechanics, they usually provide a good transferability at the expense of larger computational costs.
In general, the construction of accurate potentials is a tedious task and can result in several months of "laborious iterative fitting". Once an acceptable potential has been found, an extension to describe further bonding situations can be very difficult because of the complex interdependence of all parameters. Often a complete restart of the work is necessary. In particular in the case of force fields and simple empirical potentials an extension to new systems often requires the introduction of new energy terms on a trial and error basis.
For this reason, also alternative approaches have been suggested, which do not build on physically motivated functional forms, but apply purely mathematical fitting techniques to reproduce a set of reference data as closely as possible. This data set is typically obtained by the most accurate and still affordable electronic structure calculations. While for low-dimensional systems simple fitting schemes like splines can be used, for complex PESs involving many atoms, only a few methods provide the required accuracy. Probably the most common approach currently used is based on Taylor expansions of the energy. Here the potential-energy of an atomic configuration is expressed as a weighted average of Taylor expansions about close reference points in a precalculated data set. The PES can be iteratively improved further by computing additional trajectories and adding new reference configurations to the fitting set.
In comparison with the empirical potentials and the electronic structure methods mentioned above, such purely mathematical fitting procedures are still much less commonly used. Nevertheless, a lot of methodical work is going on to improve the accuracy and to extend the applicability of potential-energy surfaces without a physically derived (and constrained) functional form. The advantage of this type of potentials is that no approximations have to be introduced which could limit the accuracy. On the other hand, a lot of effort has to be made to ensure that all physical features of a PES are correctly included.
A promising new "mathematical" approach to construct PESs is based on artificial neural networks (NN), which can "learn" the topology of a potential-energy surface from a set of reference points. The first artificial neural networks have been developed in 1943 by McCulloch and Pitts to investigate the neural signal processing in the brain. The early models, put forward e.g. by Rosenblatt through the introduction of the perceptron already contained many important features of modern artificial NNs. Still the early NNs were limited to algorithms being essentially equivalent to linear regression. This limitation was overcome in the early eighties by the introduction of NNs operating in a nonlinear way, which extended the applicability to a wide range of problems including the fitting of arbitrary functions.
Nowadays, NNs are common tools in computer science and mathematics. They are mainly used in classification problems. This is an obvious application, because, depending on the stimulation, biological neurons either send or do not send a signal to neighboring neurons. Similarly, also the original artificial NNs produced a binary response by applying a step funtion to some accumulated incoming signal. This constraint to a binary output has been lifted in later applications by replacing the step function by sigmoidally shaped functions, which provide a continuous output.
In all applications, the general purpose of NNs is to construct some input-output relations and to use these relations to analyze and classify data sets. NNs are now frequently used in many fields as diverse as financial market analysis, optimization tasks like solving the travelling salesman problem, finger print identification, speech recognition, text recognition and weather forecast, just to mention a few examples.
Also in chemistry artificial neural networks have found wide use. They have been used to fit spectroscopic data, to investigate quantitative structure-activity relationships (QSAR), to predict deposition rates in chemical vapor deposition, to predict binding sites of biomolecules, to derive pair potentials from diffraction data on liquids, to solve the Schrödinger equation for simple model potentials like the harmonic oscillator, to estimate the fitness function in genetic algorithm optimizations, in experimental data analysis, to predict the secondary structure of proteins, to predict atomic energy levels, and to solve classification problems from clinical chemistry, in particular the differentiation between diseases on the basis of characteristic laboratory data.
Apart from this variety of applications, NNs have another important property. It has been shown that artificial neural networks are universal approximators, i.e., NNs can be used to approximate unknown functions of many variables very accurately. Specifically, it has been proven that any continuous, real-valued function of n dimensions can be fitted to arbitrary accuracy by feed-forward neural networks with only one hidden layer. For this purpose neural networks can be regarded as a nested function of rather simple functional elements, which can adapt very accurately to a set of known reference points. No knowledge about the form of the underlying function is required. This function approximation is achieved by optimizing the values of a comparably large number of fitting parameters called weights.
This fitting capability makes NNs an attractive tool for the construction of potential-energy surfaces. This is done using a number of reference points, which are typically obtained in electronic structure calculations. In this review, various methods to construct NN potential-energy surfaces for atomistic simulations are discussed. All methods have in common that they attempt to set up a direct functional relation between the atomic positions and the potential-energy of a system. This is still a relatively new and not widely distributed approach, but we will see that there is already a number of very successful applications. Still, as with any new method, several methodical problems still have to be solved, to make NN potentials a general purpose tool for all types of systems.
It should be noted that apart from the direct mapping of the energies onto structures, NNs have been used to evaluate many physical quantities and properties, which are just indirectly related to the potential-energy surface. Examples are the construction of the relationship between experimental vibrational spectra and a multidimensional PES of macromolecules, the prediction of the outcome of a reaction without computing the individual MD trajectories, the prediction of probabilities and rates of chemical reactions, the prediction of force constants and vibrational frequencies in large organic molecules, and the prediction of the outcome of trajectories in atomic and molecular scattering processes. Since in these applications the PES does not explicitly appear, these applications are not covered in this review. Further, NN-based "empirical" improvements of energetics using molecular descriptors similar to the ones employed in QSAR techniques, are not discussed here. These studies comprise e.g. the refinement of heats of formation obtained in DFT calculations, the improvement of calculated absorption energies, the estimation of bond enthalpies and Gibbs free energies, the improvement of exchange-correlation functionals, the estimation of correlation energies of diatomic molecules and heavy atoms, the improvement of ionization energies and electron affinities, the prediction of DFT energies and the extrapolation of results of electronic structure calculations to higher level methods with converged basis sets. The relation between the descriptors used in some of these methods, e.g. the total number of atoms, the number of specific bond types, energy eigenvalues, electrostatic moments, and many others, and the target quantity is often not transparent. Consequently, these methods are very different from the construction of a PES as a function of the atomic configuration, for which the existence of a functional relation to the energy is obvious, although it may be very complicated.
There is nothing like "the neural network". Many types of neural networks have been proposed and used for various purposes, like Hopfield networks, the adaptive bidirectional associative memory, the Kohonen network, and radial basis function NNs. For the construction of multidimensional PESs for chemical reactions the class of multilayer-feed forward neural networks is by far most frequently used. Therefore we will focus on the discussion of the applicability and limitations of this NN type. Our goals are to illustrate how NNs can be used to construct a direct mathematical relation between the system, i.e., the atomic coordinates and the potential-energy, to summarize successful applications and to point out open challenges.
In the following section, the general structure and the mathematical form of feed-forward neural networks will be introduced. In the next two sections, low-dimensional NN PESs for molecules and molecule-surface interactions are discussed, respectively, and some important technical concepts to deal with the symmetry are presented. In Section 5 applications of NN potentials to high-dimensional PESs are summarized, which so far are very rare but crucial to establish a general-purpose method. The current status, the scope and the limitations of NN potentials are discussed in Section 6.
Excerpted from Chemical Modelling Applications and Theory Volume 7 by M. Springborg. Copyright © 2010 The Royal Society of Chemistry. Excerpted by permission of The Royal Society of Chemistry.
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Table of Contents
Preface; Modelling Photochemical Pharmaceutics and Photodegrading; Proton transport; Polarizabilities and hyperpolarizabilities; Numerical Methods in Chemistry; Elongation method; Quantum Monte Carlo Methods; Neural Networks; Protein Folding; Mechanically Induced Chemistry: First-Principles Simulation; Nanoelectronics; Orbital Dependent Exact Exchange Methods in Denisty Functional Theory; Computer-Aided Drug Design