Editing Molecular dynamics (section)

== Potentials in MD simulations ==
{{Main|Interatomic potential|Force field (chemistry)|l2=Force field|Comparison of force field implementations}}
A molecular dynamics simulation requires the definition of a [[potential function]], or a description of the terms by which the particles in the simulation will interact. In chemistry and biology this is usually referred to as a [[force field (chemistry)|force field]] and in materials physics as an [[interatomic potential]]. Potentials may be defined at many levels of physical accuracy; those most commonly used in chemistry are based on [[molecular mechanics]] and embody a [[classical mechanics]] treatment of particle-particle interactions that can reproduce structural and [[conformational change]]s but usually cannot reproduce [[chemical reaction]]s.

The reduction from a fully quantum description to a classical potential entails two main approximations. The first one is the [[Born–Oppenheimer approximation]], which states that the dynamics of electrons are so fast that they can be considered to react instantaneously to the motion of their nuclei. As a consequence, they may be treated separately. The second one treats the nuclei, which are much heavier than electrons, as point particles that follow classical Newtonian dynamics. In classical molecular dynamics, the effect of the electrons is approximated as one potential energy surface, usually representing the ground state.

When finer levels of detail are needed, potentials based on [[quantum mechanics]] are used; some methods attempt to create hybrid [[QM/MM|classical/quantum]] potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, usually undergoing a chemical transformation.

=== Empirical potentials ===

Empirical potentials used in chemistry are frequently called [[Force field (chemistry)|force fields]], while those used in materials physics are called [[interatomic potential]]s.

Most [[Force field (chemistry)|force fields]] in chemistry are empirical and consist of a summation of bonded forces associated with [[chemical bond]]s, bond angles, and bond [[dihedral angle|dihedrals]], and non-bonded forces associated with [[van der Waals force]]s and [[electrostatic charge]].<ref>{{cite journal | vauthors = Rizzuti B | title = Molecular simulations of proteins: From simplified physical interactions to complex biological phenomena | journal =   Biochimica et Biophysica Acta (BBA) - Proteins and Proteomics| year = 2022 | volume = 1870 | issue = 3 | pages = 140757 | pmid = 35051666 | doi = 10.1016/j.bbapap.2022.140757 | s2cid = 263455009 }}</ref> Empirical potentials represent quantum-mechanical effects in a limited way through ad hoc functional approximations. These potentials contain free parameters such as [[electrostatic charge|atomic charge]], van der Waals parameters reflecting estimates of [[atomic radius]], and equilibrium [[bond length]], angle, and dihedral; these are obtained by fitting against detailed electronic calculations (quantum chemical simulations) or experimental physical properties such as [[Young's modulus|elastic constants]], lattice parameters and [[spectroscopy|spectroscopic]] measurements.

Because of the non-local nature of non-bonded interactions, they involve at least weak interactions between all particles in the system. Its calculation is normally the bottleneck in the speed of MD simulations. To lower the computational cost, [[Force field (chemistry)|force fields]] employ numerical approximations such as shifted cutoff radii, [[reaction field method|reaction field]] algorithms, particle mesh [[Ewald summation]], or the newer particle–particle-particle–mesh ([[P3M]]).

Chemistry force fields commonly employ preset bonding arrangements (an exception being ''[[ab initio quantum chemistry methods|ab initio]]'' dynamics), and thus are unable to model the process of chemical bond breaking and reactions explicitly. On the other hand, many of the potentials used in physics, such as those based on the [[bond order potential|bond order formalism]] can describe several different coordinations of a system and bond breaking.<ref>{{cite journal | vauthors = Sinnott SB, Brenner DW |author-link1=Susan Sinnott|year= 2012 |title= Three decades of many-body potentials in materials research |journal= MRS Bulletin |volume= 37 |issue= 5 |pages= 469–473 |doi=10.1557/mrs.2012.88|doi-access= free |bibcode=2012MRSBu..37..469S }}</ref><ref>{{cite journal | vauthors = Albe K, Nordlund K, Averback RS |year= 2002 |title= Modeling metal-semiconductor interaction: Analytical bond-order potential for platinum-carbon |journal= Phys. Rev. B |volume= 65 |issue= 19 |page= 195124 |doi=10.1103/physrevb.65.195124|bibcode= 2002PhRvB..65s5124A}}</ref> Examples of such potentials include the [[Brenner potential]]<ref>{{cite journal | vauthors = Brenner DW | title = Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films | journal = Physical Review B | volume = 42 | issue = 15 | pages = 9458–9471 | date = November 1990 | pmid = 9995183 | doi = 10.1103/physrevb.42.9458 | url = https://apps.dtic.mil/sti/pdfs/ADA230023.pdf | url-status = live | bibcode = 1990PhRvB..42.9458B | archive-url = https://web.archive.org/web/20170922092328/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA230023 | archive-date = September 22, 2017 }}</ref> for hydrocarbons and its
further developments for the C-Si-H<ref>{{cite journal|doi=10.1080/01418619608240734|title= Empirical potentials for C-Si-H systems with application to C<sub>60</sub> interactions with Si crystal surfaces|journal= Philosophical Magazine A|volume= 74|issue= 6|pages= 1439–1466|year= 1996| vauthors = Beardmore K, Smith R |bibcode= 1996PMagA..74.1439B}}</ref> and C-O-H<ref>{{cite journal|title=A reactive empirical bond order (rebo) potential for hydrocarbon oxygen interactions|journal=Journal of Physics: Condensed Matter|volume=16|issue=41|pages=7261–7275|doi=10.1088/0953-8984/16/41/008|year=2004| vauthors = Ni B, Lee KH, Sinnott SB |bibcode= 2004JPCM...16.7261N|s2cid=250760409 }}</ref> systems. The [[ReaxFF]] potential<ref>{{cite journal | vauthors = Van Duin AC, Dasgupta S, Lorant F, Goddard WA |title=ReaxFF: A Reactive Force Field for Hydrocarbons |journal=The Journal of Physical Chemistry A |date=October 2001 |volume=105 |issue=41 |pages=9396–9409 |doi=10.1021/jp004368u |bibcode= 2001JPCA..105.9396V |citeseerx=10.1.1.507.6992}}</ref> can be considered a fully reactive hybrid between bond order potentials and chemistry force fields.

=== Pair potentials versus many-body potentials ===

The potential functions representing the non-bonded energy are formulated as a sum over interactions between the particles of the system. The simplest choice, employed in many popular [[force field (chemistry)|force fields]], is the "pair potential", in which the total potential energy can be calculated from the sum of energy contributions between pairs of atoms. Therefore, these force fields are also called "additive force fields". An example of such a pair potential is the non-bonded [[Lennard-Jones potential]] (also termed the 6–12 potential), used for calculating van der Waals forces.

:<math>
U(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]
</math>

Another example is the Born (ionic) model of the ionic lattice. The first term in the next equation is [[Coulomb's law]] for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is also included.<ref>{{cite journal | vauthors = Cruz FJ, Lopes JN, Calado JC, Minas da Piedade ME | title = A molecular dynamics study of the thermodynamic properties of calcium apatites. 1. Hexagonal phases | journal = The Journal of Physical Chemistry B | volume = 109 | issue = 51 | pages = 24473–24479 | date = December 2005 | pmid = 16375450 | doi = 10.1021/jp054304p }}</ref><ref>{{cite journal | vauthors = Cruz FJ, Lopes JN, Calado JC |title=Molecular dynamics simulations of molten calcium hydroxyapatite |journal=Fluid Phase Equilibria |date=March 2006 |volume=241 |issue=1–2 |pages=51–58 |doi=10.1016/j.fluid.2005.12.021 |bibcode=2006FlPEq.241...51C }}</ref> When ''n<sub>l</sub>'' = 6, this potential is also called the [[Buckingham potential|Coulomb–Buckingham potential]].

:<math>U_{ij}(r_{ij}) = \frac {z_i z_j}{4 \pi \epsilon_0} \frac {1}{r_{ij}} + A_l \exp \frac {-r_{ij}}{p_l} + C_l r_{ij}^{-n_l} + \cdots
</math>

In [[Many-body problem|many-body potentials]], the potential energy includes the effects of three or more particles interacting with each other.<ref name="ReferenceA">{{cite journal| vauthors = Justo JF, Bazant MZ, Kaxiras E, Bulatov VV, Yip S |title=Interatomic potential for silicon defects and disordered phases|journal=Phys. Rev. B|date=1998|volume=58|issue=5|pages=2539–2550|doi=10.1103/PhysRevB.58.2539|arxiv= cond-mat/9712058 |bibcode= 1998PhRvB..58.2539J|s2cid=14585375}}</ref> In simulations with pairwise potentials, global interactions in the system also exist, but they occur only through pairwise terms. In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms, as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the dependency between the variables cannot in general be expressed using only pairwise products of the degrees of freedom. For example, the [[Tersoff potential]],<ref>{{cite journal | vauthors = Tersoff J | title = Modeling solid-state chemistry: Interatomic potentials for multicomponent systems | journal = Physical Review B| volume = 39 | issue = 8 | pages = 5566–5568 | date = March 1989 | pmid = 9948964 | doi = 10.1103/physrevb.39.5566 | bibcode = 1989PhRvB..39.5566T }}</ref> which was originally used to simulate [[carbon]], [[silicon]], and [[germanium]], and has since been used for a wide range of other materials, involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential. Other examples are the [[embedded atom model|embedded-atom method]] (EAM),<ref>{{cite journal | vauthors = Daw MS, Foiles SM, Baskes MI |author1-link=Murray S. Daw|author2-link=Stephen M. Foiles|author3-link=Michael Baskes |title=The embedded-atom method: a review of theory and applications |journal=Materials Science Reports |date=March 1993 |volume=9 |issue=7–8 |pages=251–310 |doi=10.1016/0920-2307(93)90001-U |url=https://zenodo.org/record/1258631 |doi-access=free }}</ref> the EDIP,<ref name="ReferenceA"/> and the Tight-Binding Second Moment Approximation (TBSMA) potentials,<ref>{{cite journal | vauthors = Cleri F, Rosato V | title = Tight-binding potentials for transition metals and alloys | journal = Physical Review B | volume = 48 | issue = 1 | pages = 22–33 | date = July 1993 | pmid = 10006745 | doi = 10.1103/physrevb.48.22 | bibcode = 1993PhRvB..48...22C }}</ref> where the electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms, and the potential energy contribution is then a function of this sum.

=== Semi-empirical potentials ===

[[Semi-empirical quantum chemistry methods|Semi-empirical]] potentials make use of the matrix representation from quantum mechanics. However, the values of the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical formulae are used once again to determine the energy contributions of the orbitals.

There are a wide variety of semi-empirical potentials, termed [[Tight binding (physics)|tight-binding]] potentials, which vary according to the atoms being modeled.

=== Polarizable potentials ===

{{Main|Force field (chemistry)|l1=Force field}}
Most classical force fields implicitly include the effect of [[polarizability]], e.g., by scaling up the partial charges obtained from quantum chemical calculations. These partial charges are stationary with respect to the mass of the atom. But molecular dynamics simulations can explicitly model polarizability with the introduction of induced dipoles through different methods, such as [[Drude particle]]s or fluctuating charges. This allows for a dynamic redistribution of charge between atoms which responds to the local chemical environment.

For many years, polarizable MD simulations have been touted as the next generation. For homogenous liquids such as water, increased accuracy has been achieved through the inclusion of polarizability.<ref name="Lamoureux3">{{cite journal | vauthors = Lamoureux G, Harder E, Vorobyov IV, Roux B, MacKerell AD |year=2006 |title=A polarizable model of water for molecular dynamics simulations of biomolecules |journal=Chem Phys Lett |volume=418 |issue=1 |pages=245–249 |doi=10.1016/j.cplett.2005.10.135|bibcode= 2006CPL...418..245L }}</ref><ref name="Sokhan2015">{{cite journal | vauthors = Sokhan VP, Jones AP, Cipcigan FS, Crain J, Martyna GJ | title = Signature properties of water: Their molecular electronic origins | journal = Proceedings of the National Academy of Sciences of the United States of America | volume = 112 | issue = 20 | pages = 6341–6346 | date = May 2015 | pmid = 25941394 | pmc = 4443379 | doi = 10.1073/pnas.1418982112 | doi-access = free | bibcode = 2015PNAS..112.6341S }}</ref><ref name="Cipcigan">{{cite journal | vauthors = Cipcigan FS, Sokhan VP, Jones AP, Crain J, Martyna GJ | title = Hydrogen bonding and molecular orientation at the liquid-vapour interface of water | journal = Physical Chemistry Chemical Physics | volume = 17 | issue = 14 | pages = 8660–8669 | date = April 2015 | pmid = 25715668 | doi = 10.1039/C4CP05506C | doi-access = free | bibcode = 2015PCCP...17.8660C | hdl = 20.500.11820/0bd0cd1a-94f1-4053-809c-9fb68bbec1c9 | hdl-access = free }}</ref> Some promising results have also been achieved for proteins.<ref>{{cite journal | vauthors = Mahmoudi M, Lynch I, Ejtehadi MR, Monopoli MP, Bombelli FB, Laurent S | title = Protein-nanoparticle interactions: opportunities and challenges | journal = Chemical Reviews | volume = 111 | issue = 9 | pages = 5610–5637 | date = September 2011 | pmid = 21688848 | doi = 10.1021/cr100440g }}</ref><ref name=Patel2004b>{{cite journal | vauthors = Patel S, Mackerell AD, Brooks CL | title = CHARMM fluctuating charge force field for proteins: II protein/solvent properties from molecular dynamics simulations using a nonadditive electrostatic model | journal = Journal of Computational Chemistry | volume = 25 | issue = 12 | pages = 1504–1514 | date = September 2004 | pmid = 15224394 | doi = 10.1002/jcc.20077 | s2cid = 16741310 | doi-access = free }}</ref> However, it is still uncertain how to best approximate polarizability in a simulation.{{Citation needed|date=April 2009}} The point becomes more important when a particle experiences different environments during its simulation trajectory, e.g. translocation of a drug through a cell membrane.<ref>{{cite journal | vauthors = Najla Hosseini A, Lund M, Ejtehadi MR | title = Electronic polarization effects on membrane translocation of anti-cancer drugs | journal = Physical Chemistry Chemical Physics | volume = 24 | issue = 20 | pages = 12281–12292 | date = May 2022 | pmid = 35543365 | doi = 10.1039/D2CP00056C | bibcode = 2022PCCP...2412281N | s2cid = 248696332 }}</ref>

=== Potentials in ''ab initio'' methods ===
{{Main|Quantum chemistry|List of quantum chemistry and solid state physics software}}

In classical molecular dynamics, one potential energy surface (usually the ground state) is represented in the force field. This is a consequence of the [[Born–Oppenheimer approximation]]. In excited states, chemical reactions or when a more accurate representation is needed, electronic behavior can be obtained from first principles using a quantum mechanical method, such as [[density functional theory]]. This is named ''Ab Initio Molecular Dynamics'' (AIMD). Due to the cost of treating the electronic degrees of freedom, the computational burden of these simulations is far higher than classical molecular dynamics. For this reason, AIMD is typically limited to smaller systems and shorter times.

''[[Ab initio]]'' [[quantum mechanical]] and [[Quantum chemistry|chemical]] methods may be used to calculate the [[potential energy surface|potential energy]] of a system on the fly, as needed for conformations in a trajectory. This calculation is usually made in the close neighborhood of the [[reaction coordinate]]. Although various approximations may be used, these are based on theoretical considerations, not on empirical fitting. ''Ab initio'' calculations produce a vast amount of information that is not available from empirical methods, such as density of electronic states or other electronic properties. A significant advantage of using ''ab initio'' methods is the ability to study reactions that involve breaking or formation of covalent bonds, which correspond to multiple electronic states. Moreover, ''ab initio'' methods also allow recovering effects beyond the Born–Oppenheimer approximation using approaches like [[mixed quantum-classical dynamics]].

=== Hybrid QM/MM ===
{{Main|QM/MM}}
QM (quantum-mechanical) methods are very powerful. However, they are computationally expensive, while the MM (classical or molecular mechanics) methods are fast but suffer from several limits (require extensive parameterization; energy estimates obtained are not very accurate; cannot be used to simulate reactions where covalent bonds are broken/formed; and are limited in their abilities for providing accurate details regarding the chemical environment). A new class of method has emerged that combines the good points of QM (accuracy) and MM (speed) calculations. These methods are termed mixed or hybrid quantum-mechanical and molecular mechanics methods (hybrid QM/MM).<ref>The methodology for such methods was introduced by Warshel and coworkers. In the recent years have been pioneered by several groups including: [[Arieh Warshel]] ([[University of Southern California]]), Weitao Yang ([[Duke University]]), Sharon Hammes-Schiffer ([[The Pennsylvania State University]]), Donald Truhlar and Jiali Gao ([[University of Minnesota]]) and Kenneth Merz ([[University of Florida]]).</ref>

The most important advantage of hybrid QM/MM method is the speed. The cost of doing classical molecular dynamics (MM) in the most straightforward case scales O(n<sup>2</sup>), where n is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with every other particle). However, use of cutoff radius, periodic pair-list updates and more recently the variations of the particle-mesh Ewald's (PME) method has reduced this to between O(n) to O(n<sup>2</sup>). In other words, if a system with twice as many atoms is simulated then it would take between two and four times as much computing power. On the other hand, the simplest ''ab initio'' calculations typically scale O(n<sup>3</sup>) or worse (restricted [[Hartree–Fock]] calculations have been suggested to scale ~O(n<sup>2.7</sup>)). To overcome the limit, a small part of the system is treated quantum-mechanically (typically active-site of an enzyme) and the remaining system is treated classically.

In more sophisticated implementations, QM/MM methods exist to treat both light nuclei susceptible to quantum effects (such as hydrogens) and electronic states. This allows generating hydrogen wave-functions (similar to electronic wave-functions). This methodology has been useful in investigating phenomena such as hydrogen tunneling. One example where QM/MM methods have provided new discoveries is the calculation of hydride transfer in the enzyme liver [[alcohol dehydrogenase]]. In this case, [[quantum tunneling]] is important for the hydrogen, as it determines the reaction rate.<ref>{{cite journal | vauthors = Billeter SR, Webb SP, Agarwal PK, Iordanov T, Hammes-Schiffer S | title = Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and role of enzyme motion | journal = Journal of the American Chemical Society | volume = 123 | issue = 45 | pages = 11262–11272 | date = November 2001 | pmid = 11697969 | doi = 10.1021/ja011384b }}</ref>

=== Coarse-graining and reduced representations ===

At the other end of the detail scale are [[Coarse-grained modeling|coarse-grained]] and lattice models. Instead of explicitly representing every atom of the system, one uses "pseudo-atoms" to represent groups of atoms. MD simulations on very large systems may require such large computer resources that they cannot easily be studied by traditional all-atom methods. Similarly, simulations of processes on long timescales (beyond about 1 microsecond) are prohibitively expensive, because they require so many time steps. In these cases, one can sometimes tackle the problem by using reduced representations, which are also called [[Coarse-grained modeling|coarse-grained models]].<ref name=":0">{{cite journal | vauthors = Kmiecik S, Gront D, Kolinski M, Wieteska L, Dawid AE, Kolinski A | title = Coarse-Grained Protein Models and Their Applications | journal = Chemical Reviews | volume = 116 | issue = 14 | pages = 7898–7936 | date = July 2016 | pmid = 27333362 | doi = 10.1021/acs.chemrev.6b00163 | doi-access = free }}</ref>

Examples for coarse graining (CG) methods are discontinuous molecular dynamics (CG-DMD)<ref>{{cite journal | vauthors = Voegler Smith A, Hall CK | title = alpha-helix formation: discontinuous molecular dynamics on an intermediate-resolution protein model | journal = Proteins | volume = 44 | issue = 3 | pages = 344–360 | date = August 2001 | pmid = 11455608 | doi = 10.1002/prot.1100 | s2cid = 21774752 }}</ref><ref>{{cite journal | vauthors = Ding F, Borreguero JM, Buldyrey SV, Stanley HE, Dokholyan NV | title = Mechanism for the alpha-helix to beta-hairpin transition | journal = Proteins | volume = 53 | issue = 2 | pages = 220–228 | date = November 2003 | pmid = 14517973 | doi = 10.1002/prot.10468 | s2cid = 17254380 }}</ref> and Go-models.<ref>{{cite journal | vauthors = Paci E, Vendruscolo M, Karplus M | title = Validity of Gō models: comparison with a solvent-shielded empirical energy decomposition | journal = Biophysical Journal | volume = 83 | issue = 6 | pages = 3032–3038 | date = December 2002 | pmid = 12496075 | pmc = 1302383 | doi = 10.1016/S0006-3495(02)75308-3 | bibcode = 2002BpJ....83.3032P }}</ref> Coarse-graining is done sometimes taking larger pseudo-atoms. Such united atom approximations have been used in MD simulations of biological membranes. Implementation of such approach on systems where electrical properties are of interest can be challenging owing to the difficulty of using a proper charge distribution on the pseudo-atoms.<ref>{{cite journal | vauthors = Chakrabarty A, Cagin T |title=Coarse grain modeling of polyimide copolymers |journal=Polymer |date=May 2010 |volume=51 |issue=12 |pages=2786–2794 |doi=10.1016/j.polymer.2010.03.060 }}</ref> The aliphatic tails of lipids are represented by a few pseudo-atoms by gathering 2 to 4 methylene groups into each pseudo-atom.

The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data or all-atom simulations. Ideally, these parameters should account for both [[enthalpy|enthalpic]] and [[entropy|entropic]] contributions to free energy in an implicit way.<ref>{{cite journal | vauthors = Foley TT, Shell MS, Noid WG | title = The impact of resolution upon entropy and information in coarse-grained models | journal = The Journal of Chemical Physics | volume = 143 | issue = 24 | pages = 243104 | date = December 2015 | pmid = 26723589 | doi = 10.1063/1.4929836 | bibcode = 2015JChPh.143x3104F }}</ref> When coarse-graining is done at higher levels, the accuracy of the dynamic description may be less reliable. But very coarse-grained models have been used successfully to examine a wide range of questions in structural biology, liquid crystal organization, and polymer glasses.

Examples of applications of coarse-graining:
* [[protein folding]] and [[protein structure prediction]] studies are often carried out using one, or a few, pseudo-atoms per amino acid;<ref name=":0" />
* [[liquid crystal]] phase transitions have been examined in confined geometries and/or during flow using the [[Gay-Berne potential]], which describes anisotropic species;
* [[Polymer]] glasses during deformation have been studied using simple harmonic or [[FENE]] springs to connect spheres described by the [[Lennard-Jones potential]];
* [[Supercoiling|DNA supercoiling]] has been investigated using 1–3 pseudo-atoms per basepair, and at even lower resolution;
* Packaging of [[DNA|double-helical DNA]] into [[bacteriophage]] has been investigated with models where one pseudo-atom represents one turn (about 10 basepairs) of the double helix;
* RNA structure in the [[ribosome]] and other large systems has been modeled with one pseudo-atom per nucleotide.

The simplest form of coarse-graining is the ''united atom'' (sometimes called ''extended atom'') and was used in most early MD simulations of proteins, lipids, and nucleic acids. For example, instead of treating all four atoms of a CH<sub>3</sub> methyl group explicitly (or all three atoms of CH<sub>2</sub> methylene group), one represents the whole group with one pseudo-atom. It must, of course, be properly parameterized so that its van der Waals interactions with other groups have the proper distance-dependence. Similar considerations apply to the bonds, angles, and torsions in which the pseudo-atom participates. In this kind of united atom representation, one typically eliminates all explicit hydrogen atoms except those that have the capability to participate in hydrogen bonds (''polar hydrogens''). An example of this is the [[CHARMM]] 19 force-field.

The polar hydrogens are usually retained in the model, because proper treatment of hydrogen bonds requires a reasonably accurate description of the directionality and the electrostatic interactions between the donor and acceptor groups. A hydroxyl group, for example, can be both a hydrogen bond donor, and a hydrogen bond acceptor, and it would be impossible to treat this with one OH pseudo-atom. About half the atoms in a protein or nucleic acid are non-polar hydrogens, so the use of united atoms can provide a substantial savings in computer time.

=== Machine Learning Force Fields ===
Machine Learning Force Fields] (MLFFs) represent one approach to modeling interatomic interactions in molecular dynamics simulations.<ref name="Unke_2021">{{cite journal | vauthors = Unke OT, Chmiela S, Sauceda HE, Gastegger M, Poltavsky I, Schütt KT, Tkatchenko A, Müller KR | title = Machine Learning Force Fields | journal = Chemical Reviews | volume = 121 | issue = 16 | pages = 10142–10186 | date = August 2021 | pmid = 33705118 | pmc = 8391964 | doi = 10.1021/acs.chemrev.0c01111 }}</ref> MLFFs can achieve accuracy close to that of [[Ab initio quantum chemistry methods|ab initio methods]]. Once trained, MLFFs are much faster than direct quantum mechanical calculations. MLFFs address the limitations of traditional force fields by learning complex potential energy surfaces directly from high-level quantum mechanical data. Several software packages now support MLFFs, including [[Vienna Ab initio Simulation Package|VASP]]<ref name="Hafner_2008">{{cite journal | vauthors = Hafner J | title = Ab-initio simulations of materials using VASP: Density-functional theory and beyond | journal = Journal of Computational Chemistry | volume = 29 | issue = 13 | pages = 2044–78 | date = October 2008 | pmid = 18623101 | doi = 10.1002/jcc.21057 }}</ref> and open-source libraries like DeePMD-kit<ref name = "Wang_2018">{{cite journal | vauthors = Wang H, Zhang L, Han J, Weinan E |title=DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics |journal=Computer Physics Communications |date=July 2018 |volume=228 |pages=178–184 |doi=10.1016/j.cpc.2018.03.016|arxiv=1712.03641 }}</ref><ref name="Zeng_2023">{{cite journal | vauthors = Zeng J, Zhang D, Lu D, Mo P, Li Z, Chen Y, Rynik M, Huang L, Li Z, Shi S, Wang Y, Ye H, Tuo P, Yang J, Ding Y, Li Y, Tisi D, Zeng Q, Bao H, Xia Y, Huang J, Muraoka K, Wang Y, Chang J, Yuan F, Bore SL, Cai C, Lin Y, Wang B, Xu J, Zhu JX, Luo C, Zhang Y, Goodall RE, Liang W, Singh AK, Yao S, Zhang J, Wentzcovitch R, Han J, Liu J, Jia W, York DM, E W, Car R, Zhang L, Wang H | title = DeePMD-kit v2: A software package for deep potential models | journal = The Journal of Chemical Physics | volume = 159 | issue = 5 | pages = | date = August 2023 | pmid = 37526163 | pmc = 10445636 | doi = 10.1063/5.0155600 }}</ref> and [https://schnetpack.readthedocs.io/en/latest/ SchNetPack].<ref name="Schütt_2019">{{cite journal | vauthors = Schütt KT, Kessel P, Gastegger M, Nicoli KA, Tkatchenko A, Müller KR | title = SchNetPack: A Deep Learning Toolbox For Atomistic Systems | journal = Journal of Chemical Theory and Computation | volume = 15 | issue = 1 | pages = 448–455 | date = January 2019 | pmid = 30481453 | doi = 10.1021/acs.jctc.8b00908 | arxiv = 1809.01072 }}</ref><ref name="Schütt_2023">{{cite journal | vauthors = Schütt KT, Hessmann SS, Gebauer NW, Lederer J, Gastegger M | title = SchNetPack 2.0: A neural network toolbox for atomistic machine learning | journal = The Journal of Chemical Physics | volume = 158 | issue = 14 | pages = 144801 | date = April 2023 | pmid = | doi = 10.1063/5.0138367  | arxiv = 2212.05517 }}</ref>