Interatomic potential

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies, adsorption, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions. V_\mathrm{} = \sum_{i = 1}^N V_1(\vec r_i) + \sum_{i,j = 1}^N V_2(\vec r_i,\vec r_j) + \sum_{i,j,k = 1}^N V_3(\vec r_i,\vec r_j,\vec r_k) + \cdots Here \textstyle V_1 is the one-body term, \textstyle V_2 the two-body term, \textstyle V_3 the three body term, \textstyle N the number of atoms in the system, \vec r_i the position of atom i , etc. i , j and k are indices that loop over atom positions. Note that in case the pair potential is given per atom pair, in the two-body term the potential should be multiplied by 1/2 as otherwise each bond is counted twice, and similarly the three-body term by 1/6. Alternatively, the summation of the pair term can be restricted to cases \textstyle i and similarly for the three-body term \textstyle i , if the potential form is such that it is symmetric with respect to exchange of the j and k indices (this may not be the case for potentials for multielemental systems). The one-body term is only meaningful if the atoms are in an external field (e.g. an electric field). In the absence of external fields, the potential V should not depend on the absolute position of atoms, but only on the relative positions. This means that the functional form can be rewritten as a function of interatomic distances \textstyle r_{ij} = |\vec r_i-\vec r_j| and angles between the bonds (vectors to neighbours) \textstyle \theta_{ijk} . Then, in the absence of external forces, the general form becomes V_\mathrm{TOT} = \sum_{i,j}^N V_2(r_{ij}) + \sum_{i,j,k}^N V_3(r_{ij},r_{ik},\theta_{ijk}) + \cdots In the three-body term \textstyle V_3 the interatomic distance \textstyle r_{jk} is not needed since the three terms \textstyle r_{ij},r_{ik},\theta_{ijk} are sufficient to give the relative positions of three atoms i, j, k in three-dimensional space. Any terms of order higher than 2 are also called many-body potentials. In some interatomic potentials the many-body interactions are embedded into the terms of a pair potential (see discussion on EAM-like and bond order potentials below). In principle the sums in the expressions run over all N atoms. However, if the range of the interatomic potential is finite, i.e. the potentials \textstyle V(r) \equiv 0 above some cutoff distance \textstyle r_\mathrm{cut} , the summing can be restricted to atoms within the cutoff distance of each other. By also using a cellular method for finding the neighbours, the MD algorithm can be an O(N) algorithm. Potentials with an infinite range can be summed up efficiently by Ewald summation and its further developments. \vec{F}_i = -\nabla_{\vec {r}_i} V_\mathrm{TOT} For two-body potentials this gradient reduces, thanks to the symmetry with respect to ij in the potential form, to straightforward differentiation with respect to the interatomic distances \textstyle r_{ij} . However, for many-body potentials (three-body, four-body, etc.) the differentiation becomes considerably more complex since the potential may not be any longer symmetric with respect to ij exchange. In other words, also the energy of atoms k that are not direct neighbours of i can depend on the position \textstyle \vec{r}_{i} because of angular and other many-body terms, and hence contribute to the gradient \textstyle \nabla_{\vec{r}_{k}} . ===Parametric potentials=== V_\mathrm{LJ}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right] where \textstyle \varepsilon is the depth of the potential well and \textstyle \sigma is the distance at which the potential crosses zero. The attractive term proportional to \textstyle 1/r^6 in the potential comes from the scaling of van der Waals forces, while the \textstyle 1/r^{12} repulsive term is much more approximate (conveniently the square of the attractive term). On its own, this potential is quantitatively accurate only for noble gases and has been extensively studied in the past decades, but is also widely used for qualitative studies and in systems where dipole interactions are significant, particularly in chemistry force fields to describe intermolecular interactions - especially in fluids. Another simple and widely used pair potential is the Morse potential, which consists simply of a sum of two exponentials. V_\mathrm{M}(r) = D_e ( e^{-2a(r-r_e)}-2e^{-a(r-r_e)} ) Here \textstyle D_e is the equilibrium bond energy and \textstyle r_e the bond distance. The Morse potential has been applied to studies of molecular vibrations and solids, and also inspired the functional form of more accurate potentials such as the bond-order potentials. Ionic materials are often described by a sum of a short-range repulsive term, such as the Buckingham pair potential, and a long-range Coulomb potential giving the ionic interactions between the ions forming the material. The short-range term for ionic materials can also be of many-body character. Pair potentials have some inherent limitations, such as the inability to describe all 3 elastic constants of cubic metals or correctly describe both cohesive energy and vacancy formation energy. Therefore, quantitative molecular dynamics simulations are carried out with various of many-body potentials. V(r_{ij}) = { 1 \over 4 \pi \varepsilon_0} {Z_1Z_2 e^2 \over r_{ij}} \varphi(r/a) Here, \varphi(r) \to 1 when r \to 0 . Z_1 and Z_2 are the charges of the interacting nuclei, and a is the so-called screening parameter. A widely used popular screening function is the "Universal ZBL" one. and more accurate ones can be obtained from all-electron quantum chemistry calculations In a comparative study of several quantum chemistry methods, it was shown that pair-specific "NLH" repulsive potentials with a simple three-exponential screening function are accurate to within ~2% above 30 eV, while the universal ZBL potential differs by ~5%–10% from the quantum chemical calculations above 100 eV. In binary collision approximation simulations this kind of potential can be used to describe the nuclear stopping power. V_\mathrm{TOT} = \sum_{i,j}^N V_2(r_{ij}) + \sum_{i,j,k}^N V_3(r_{ij},r_{ik},\theta_{ijk}) where the three-body term describes how the potential energy changes with bond bending. It was originally developed for pure Si, but has been extended to many other elements and compounds and also formed the basis for other Si potentials. Metals are very commonly described with what can be called "EAM-like" potentials, i.e. potentials that share the same functional form as the embedded atom model. In these potentials, the total potential energy is written V_\mathrm{TOT} = \sum_i^N F_i \left(\sum_{j} \rho (r_{ij}) \right) + \frac{1}{2} \sum_{i, j}^N V_2(r_{ij}) where \textstyle F_i is a so-called embedding function (not to be confused with the force \textstyle \vec F_i ) that is a function of the sum of the so-called electron density \textstyle \rho (r_{ij}) . \textstyle V_2 is a pair potential that usually is purely repulsive. In the original formulation the electron density function \textstyle \rho (r_{ij}) was obtained from true atomic electron densities, and the embedding function was motivated from density-functional theory as the energy needed to 'embed' an atom into the electron density. However, many other potentials used for metals share the same functional form but motivate the terms differently, e.g. based on tight-binding theory or other motivations. EAM-like potentials are usually implemented as numerical tables. A collection of tables is available at the interatomic potential repository at NIST (see here). Covalently bonded materials are often described by bond order potentials, sometimes also called Tersoff-like or Brenner-like potentials. These have in general a form that resembles a pair potential: V_{ij}(r_{ij}) = V_\mathrm{repulsive}(r_{ij}) + b_{ijk} V_\mathrm{attractive}(r_{ij}) where the repulsive and attractive part are simple exponential functions similar to those in the Morse potential. However, the strength is modified by the environment of the atom i via the b_{ijk} term. If implemented without an explicit angular dependence, these potentials can be shown to be mathematically equivalent to some varieties of EAM-like potentials. Thanks to this equivalence, the bond-order potential formalism has been implemented also for many metal-covalent mixed materials. EAM potentialshave also been extended to describe covalent bonding by adding angular-dependent terms to the electron density function \rho , in what is called the modified embedded atom method (MEAM). Current research in interatomic potentials involves using systematically improvable, non-parametric mathematical forms and increasingly complex machine learning methods. The total energy is then written V_\mathrm{TOT} = \sum_i^N E(\mathbf{q}_i) where \mathbf{q}_i is a mathematical representation of the atomic environment surrounding the atom i , known as the descriptor. E is a machine-learning model that provides a prediction for the energy of atom i based on the descriptor output. An accurate machine-learning potential requires both a robust descriptor and a suitable machine learning framework. The simplest descriptor is the set of interatomic distances from atom i to its neighbours, yielding a machine-learned pair potential. However, more complex many-body descriptors are needed to produce highly accurate potentials. It is also possible to use a linear combination of multiple descriptors with associated machine-learning models. Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including neural networks, Gaussian process regression, and linear regression. A non-parametric potential is most often trained to total energies, forces, and/or stresses obtained from quantum-level calculations, such as density functional theory, as with most modern potentials. However, the accuracy of a machine-learning potential can be converged to be comparable with the underlying quantum calculations, unlike analytical models. Hence, they are in general more accurate than traditional analytical potentials, but they are correspondingly less able to extrapolate. Further, owing to the complexity of the machine-learning model and the descriptors, they are computationally far more expensive than their analytical counterparts. Non-parametric, machine learned potentials may also be combined with parametric, analytical potentials, for example to include known physics such as the screened Coulomb repulsion, or to impose physical constraints on the predictions. ==Machine-learned interatomic potentials== Since the 1990s, machine learning programs have been employed to construct interatomic potentials, mapping atomic structures to their potential energies. These are generally referred to as 'machine learning potentials' (MLPs) or as 'machine-learned interatomic potentials' (MLIPs). Such machine learning potentials help fill the gap between highly accurate but computationally intensive simulations like density functional theory and computationally lighter, but much less precise, empirical potentials. Early neural networks showed promise, but their inability to systematically account for interatomic energy interactions limited their applications to smaller, low-dimensional systems, keeping them largely within the confines of academia. However, with continuous advancements in artificial intelligence technology, machine learning methods have become significantly more accurate, increasing the use of machine learning in the field. Modern neural networks have revolutionized the construction of highly accurate and computationally light potentials by integrating theoretical understanding of materials science into their architectures and preprocessing. Almost all are local, accounting for all interactions between an atom and its neighbor up to some cutoff radius. These neural networks usually intake atomic coordinates and output potential energies. Atomic coordinates are sometimes transformed with atom-centered symmetry functions or pair symmetry functions before being fed into neural networks. Encoding symmetry has been pivotal in enhancing machine learning potentials by drastically constraining the neural networks' search space. Conversely, message-passing neural networks (MPNNs), a form of graph neural networks, learn their own descriptors and symmetry encodings. They treat molecules as three-dimensional graphs and iteratively update each atom's feature vectors as information about neighboring atoms is processed through message functions and convolutions. These feature vectors are then used to directly predict the final potentials. In 2017, the first-ever MPNN model, a deep tensor neural network, was used to calculate the properties of small organic molecules. Another class of machine-learned interatomic potential is the Gaussian approximation potential (GAP), which combines compact descriptors of local atomic environments with Gaussian process regression to machine learn the potential energy surface of a given system. To date, the GAP framework has been used to successfully develop a number of MLIPs for various systems, including for elemental systems such as Carbon Silicon, and Tungsten, as well as for multicomponent systems such as Ge 2 Sb 2 Te 5 and austenitic stainless steel, Fe 7 Cr 2 Ni.

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