In the context of chemistry, molecular physics, physical chemistry, and molecular modelling, a force field is a computational model that is used to describe the forces between atoms (or collections of atoms) within molecules or between molecules as well as in crystals. Force fields are a variety of interatomic potentials. More precisely, the force field refers to the functional form and parameter sets used to calculate the potential energy of a system on the atomistic level. Force fields are usually used in molecular dynamics or Monte Carlo simulations. The parameters for a chosen energy function may be derived from classical laboratory experiment data, calculations in quantum mechanics, or both. Force fields utilize the same concept as force fields in classical physics, with the main difference being that the force field parameters in chemistry describe the energy landscape on the atomistic level. From a force field, the acting forces on every particle are derived as a gradient of the potential energy with respect to the particle coordinates.
A large number of different force field types exist today (e.g. for organic molecules, ions, polymers, minerals, and metals). Depending on the material, different functional forms are usually chosen for the force fields since different types of atomistic interactions dominate the material behavior.
There are various criteria that can be used for categorizing force field parametrization strategies. An important differentiation is 'component-specific' and 'transferable'. For a component-specific parametrization, the considered force field is developed solely for describing a single given substance (e.g. water). For a transferable force field, all or some parameters are designed as building blocks and become transferable/ applicable for different substances (e.g. methyl groups in alkane transferable force fields). A different important differentiation addresses the physical structure of the models: All-atom force fields provide parameters for every type of atom in a system, including hydrogen, while united-atom interatomic potentials treat the hydrogen and carbon atoms in methyl groups and methylene bridges as one interaction center. Coarse-grained potentials, which are often used in long-time simulations of macromolecules such as proteins, nucleic acids, and multi-component complexes, sacrifice chemical details for higher computing efficiency.
Force fields for molecular systems
The basic functional form of potential energy for modeling molecular systems includes intramolecular interaction terms for interactions of atoms that are linked by covalent bonds, and intermolecular (i.e. nonbonded also termed noncovalent) terms that describe the long-range electrostatic and van der Waals forces. The specific decomposition of the terms depends on the force field, but a general form for the total energy in an additive force field can be written as
where the components of the covalent and noncovalent contributions are given by the following summations:
The bond and angle terms are usually modeled by quadratic energy functions that do not allow bond breaking. A more realistic description of a covalent bond at higher stretching is provided by the more expensive Morse potential. The functional form for dihedral energy is variable from one force field to another. Additionally, "improper torsional" terms may be added to enforce the planarity of aromatic rings and other conjugated systems, and "cross-terms" that describe the coupling of different internal variables, such as angles and bond lengths. Some force fields also include explicit terms for hydrogen bonds.
The nonbonded terms are computationally most intensive. A popular choice is to limit interactions to pairwise energies. The van der Waals term is usually computed with a Lennard-Jones potential or the Mie potential and the electrostatic term with Coulomb's law. However, both can be buffered or scaled by a constant factor to account for electronic polarizability. A large number of force fields based on this or similar energy expressions have been proposed in the past decades for modeling different types of materials such as molecular substances, metals, glasses etc. - see below for a comprehensive list of force fields.
Bonded terms
Bond potential As it is rare for bonds to deviate significantly from their equilibrium values, the most simplistic approaches utilize a Hooke's law formula:
where is the force constant, is the bond length, and is the value for the bond length between atoms and when all other terms in the force field are set to 0. The term is at times differently defined or taken at different thermodynamic conditions.
The bond stretching constant can be determined from the experimental infrared spectrum, Raman spectrum, or high-level quantum-mechanical calculations. The constant determines vibrational frequencies in molecular dynamics simulations. The stronger the bond is between atoms, the higher is the value of the force constant, and the higher the wavenumber (energy) in the IR/Raman spectrum.
Though the formula of Hooke's law provides a reasonable level of accuracy at bond lengths near the equilibrium distance, it is less accurate as one moves away. In order to model the Morse curve better, one could employ cubic and higher powers. However, for most practical applications these differences are negligible, and inaccuracies in predictions of bond lengths are on the order of the thousandth of an angstrom, which is also the limit of reliability for common force fields. A Morse potential can be employed instead to enable bond breaking and higher accuracy, even though it is less efficient to compute. For reactive force fields, bond breaking and bond orders are additionally considered.
Angle potential Angular vibrations are usually described through a spring equation as well:
where is the force constant, the angle between three atoms and the equilibrium angle.
Dihedral potential A dihedral potential, also called a torsion potential, is often described by the following function:
where is the force constant, the dihedral angle, the phase and defines the number of minima or maxima between and .
Dihedral terms admit a range of mathematical representations, as illustrated by the multiple dihedral functional forms provided in OpenMM's standard forces.
Non-bonded terms
Electrostatic interactions Electrostatic interactions are represented by a Coulomb energy, which utilizes atomic charges to represent chemical bonding ranging from covalent to polar covalent and ionic bonding. The typical formula is the Coulomb law:
where is the distance between two atoms and . The total Coulomb energy is a sum over all pairwise combinations of atoms and usually excludes .
Atomic charges can make dominant contributions to the potential energy, especially for polar molecules and ionic compounds, and are critical to simulate the geometry, interaction energy, and the reactivity. The assignment of charges usually uses some heuristic approach, with different possible solutions.
Historically the electrostatic potential (ESP) has been used to determine partial charges with the Merz-Kollman method being one of the first. Other methods uses population based analysis schemes of wavefunctions like the Mulliken or Löwdin population analysis.
Another method to derive partial charges are charge equilibration methods like EEM, SQE or ACKS2. The idea is to assign geometry-dependent charges by letting electronegativity differences drive charge flow between atoms until a self-consistent minimum of the electrostatic energy is reached.
Force fields for crystal systems Atomistic interactions in crystal systems significantly deviate from those in molecular systems, e.g. of organic molecules. For crystal systems, particularly multi-body interactions, these interactions are important and cannot be neglected if a high accuracy of the force field is the aim. For crystal systems with covalent bonding, bond order potentials are usually used, e.g. Tersoff potentials. For metal systems, usually embedded atom potentials are used. Additionally, Drude model potentials have been developed, which describe a form of attachment of electrons to nuclei.
Parameterization In addition to the functional form of the potentials, a force fields consists of the parameters of these functions. Together, they specify the interactions on the atomistic level. The parametrization, i.e. determining of the parameter values, is crucial for the accuracy and reliability of the force field. Different parametrization procedures have been developed for the parametrization of different substances, e.g. metals, ions, and molecules. For different material types, usually different parametrization strategies are used. In general, two main types can be distinguished for the parametrization, either using data/ information from the atomistic level, e.g. from quantum mechanical calculations or spectroscopic data, or using data from macroscopic properties, e.g. the hardness or compressibility of a given material. Often a combination of these routes is used. Hence, one way or the other, the force field parameters are always determined in an empirical way. Nevertheless, the term 'empirical' is often used in the context of force field parameters when macroscopic material property data was used for the fitting. Experimental data (microscopic and macroscopic) included for the fit, for example, the enthalpy of vaporization, enthalpy of sublimation, dipole moments, and various spectroscopic properties such as vibrational frequencies. Often, for molecular systems, quantum mechanical calculations in the gas phase are used for parametrizing intramolecular interactions and parametrizing intermolecular dispersive interactions by using macroscopic properties such as liquid densities. The assignment of atomic charges often follows quantum mechanical protocols with some heuristics, which can lead to significant deviation in representing specific properties.
A large number of workflows and parametrization procedures have been employed in the past decades using different data and optimization strategies for determining the force field parameters. They differ significantly, which is also due to different focuses of different developments. The parameters for molecular simulations of biological macromolecules such as proteins, DNA, and RNA were often derived/transferred from observations for small organic molecules, which are more accessible for experimental studies and quantum calculations.
Atom types are defined for different elements as well as for the same elements in sufficiently different chemical environments. For example, oxygen atoms in water and an oxygen atoms in a carbonyl functional group are classified as different force field types. Typical molecular force field parameter sets include values for atomic mass, atomic charge, Lennard-Jones parameters for every atom type, as well as equilibrium values of bond lengths, bond angles, and dihedral angles. The bonded terms refer to pairs, triplets, and quadruplets of bonded atoms, and include values for the effective spring constant for each potential.
Heuristic force field parametrization procedures have been very successful for many years, but have recently been criticized since they are usually not fully automated and therefore subject to some subjectivity of the developers, which also brings problems regarding the reproducibility of the parametrization procedure.
Efforts to provide open source codes and methods include openMM and openMD. The use of semi-automation or full automation, without input from chemical knowledge, is likely to increase inconsistencies at the level of atomic charges, for the assignment of remaining parameters, and likely to dilute the interpretability and performance of parameters.