Force field (computer physics)

A force field (English: forcefield ) is computational physics and related disciplines, such as theoretical chemistry , a parameterization of the potential energy and is used particularly in the description of molecules used. When reference is made to a specific force field, both the functional form of the force field and a special (defined) set of parameters are meant.

Force fields often contain terms for contributions to the potential energy that are mediated by chemical bonds and terms for interactions that are not mediated by chemical bonds:

${\displaystyle E=E_{\text{bonded}}+E_{\text{nonbonded}}}$.

The article often contains a Lennard-Jones potential term and a Coulomb potential term . The article often contains terms that describe the torsion of bonds, bond angles, and bond lengths. ${\displaystyle E_{\text{nonbonded}}}$ ${\displaystyle E_{\text{bonded}}}$

The term describing the bond length between types of atoms of types A and B can e.g. B. take the form where the spring constant and the equilibrium distance are parameters. Since, for example, carbon atoms have different equilibrium distances and spring constants, depending on whether there is a double or single bond, not only element symbols, but atom types are used to characterize the parameters to be applied. With the (sole) choice of the above functional form to describe the bond length, the breaking of bonds would not be possible. However, there are reactive force fields (such as ReaxFF ) that can describe the breaking of bonds. ${\displaystyle E_{AB}=k_{AB}(r_{AB}^{0}-r)^{2}}$ ${\displaystyle k_{AB}}$ ${\displaystyle r_{AB}^{0}}$

The parameters of a force field are selected in such a way that it can reproduce certain aspects as precisely as possible in computer simulations.

Force fields offer the advantage that relatively large systems can be modeled. Furthermore, the splitting of the energy into its individual contributions can contribute to the understanding (e.g. the molecular structure). The accuracy of the description, however, depends crucially on how well the selected parameter set fits the system to be examined.

Potentials

First generation force fields (or "class I" force fields ) are divided into bonding potentials and non-bonding potentials :

${\displaystyle \ E_{\text{total}}=E_{\text{bonded}}+E_{\text{nonbonded}}}$

Bonding potentials

Binding potentials are usually divided into elongation, bending and torsion potentials

${\displaystyle \ E_{\text{bonded}}=E_{\text{bond}}+E_{\text{angle}}+E_{\text{dihedral}}}$

Stretching of covalent bonds

The binding energy of two atoms i and j is calculated as follows:

${\displaystyle E_{\text{bond}}^{ij}=k_{ij}\,(|r_{ij}|-r_{eq})^{2}}$

With

• ${\displaystyle E_{\text{bond}}^{ij}}$... the binding energy between i and j
• ${\displaystyle k_{ij}}$... the binding stiffness
• ${\displaystyle r_{ij}}$... the current distance between the atoms i and j
• ${\displaystyle r_{eq}}$... the equilibrium distance.

Bending of two covalent bonds

The bending energy of three atoms i, j and k is calculated as follows:

${\displaystyle E_{\text{angle}}^{ijk}=k_{ijk}\,(\theta _{ijk}-\theta _{eq})^{2}}$

With

• ${\displaystyle E_{\text{angle}}^{ijk}}$... the bending energy between i, j and k
• ${\displaystyle k_{ijk}}$... the bending stiffness
• ${\displaystyle \theta _{ijk}}$... the current angle between the atoms i, j and k
• ${\displaystyle \theta _{eq}}$... the equilibrium angle.

Twisting of three covalent bonds

The torsional energy of four atoms i, j, k and l is calculated as follows:

${\displaystyle E_{\text{dihedral}}^{ijkl}=k_{ijkl}\left[1+\cos(n\,\phi )\,\cos(\psi )+\sin(n\,\phi )\,\sin(\psi )\right]}$

With

• ${\displaystyle E_{\text{dihedral}}^{ijkl}}$... the torsional energy between i, j, k and l
• ${\displaystyle k_{ijkl}}$... the torsional stiffness
• ${\displaystyle n}$... the periodicity (usually n∈ {1,2,3})
• ${\displaystyle \phi }$... the current angle between the connections i with j and k with l, when projecting into a plane orthongoal to the jk connection.
• ${\displaystyle \psi }$... the equilibrium angle.

Non-binding potential

Binding potentials are usually divided into electrostatic and van der Waals potentials

${\displaystyle \ E_{\text{nonbonded}}=E_{\text{electrostatic}}+E_{\text{van der Waals}}}$

Van der Waals

The van der Waals potentials and hydrogen bonds are often approximated with a Lennard-Jones potential :

${\displaystyle E_{\text{van der Waals}}^{ij}={\frac {a_{i,j}}{\|r_{ij}\|^{12}}}-{\frac {b_{i,j}}{\|r_{ij}\|^{6}}}}$

With

• ${\displaystyle E_{\text{van der Waals}}^{ij}}$... van der Waals potential (approximated by a Lennard-Jones potential )
• ${\displaystyle a_{i,j}}$, ... Lennhard-Jones parameters ${\displaystyle b_{i,j}}$
• ${\displaystyle r_{ij}}$... the current distance between i and j.

Electrostatics

The Coulomb potential takes into account the electrostatic interactions and is calculated as follows:

${\displaystyle E_{\text{electrostatic}}^{ij}={\frac {1}{\|r_{ij}\|}}\,k_{C}\,Q_{i}\,Q_{j}}$

With

• ${\displaystyle E_{\text{electrostatic}}^{ij}}$... Coulomb potential
• ${\displaystyle k_{C}}$... Coulomb constant (usually implemented in modified loads for performance reasons)
• ${\displaystyle Q_{i}}$, ... charges of the atoms i and j ${\displaystyle Q_{j}}$
• ${\displaystyle r_{ij}}$... the current distance between i and j.

Classification

First and second generation

Force fields (English or classes (. In different generations class )) are divided. First generation force fields (or "class I" force fields ) are calculated according to:

${\displaystyle \ E_{\text{total}}=E_{\text{bonded}}+E_{\text{nonbonded}}}$

${\displaystyle \ E_{\text{bonded}}=E_{\text{bond}}+E_{\text{angle}}+E_{\text{dihedral}}}$

${\displaystyle \ E_{\text{nonbonded}}=E_{\text{electrostatic}}+E_{\text{van der Waals}}}$

Second generation force fields (or "class II" force fields ) continue to use correlation terms to calculate the potentials, which take into account, among other things, simultaneous bond elongation / compression and a change in the bond angle.

First generation (or "class I" force fields):

• AMBER
• OPLS
• CHARMM
• ...

Second generation (or "class II" force fields):

• MM2, MM3, MM4
• COMPASS
• UFF (universal force field)
• CFF (consistent force field)
• MMFF (Merck molecular force field)
• ...

Other classification

Classic force fields

• AMBER (Assisted Model Building and Energy Refinement) - Commonly used in protein and DNA calculations. Partly also suitable for polarized force fields.
• CHARMM (Chemistry at Harvard Molecular Mechanics) - Widely used. Partly also suitable for polarized force fields.
• OPLS (Optimized Potential for Liquid Simulations) - variants e.g. B. OPLS-AA, OPLS-UA, OPLS-2001, OPLS-2005
• UFF (Universal Force Field) - force field parameters for the entire periodic table.
• CFF (Consistent Force Field) - Commonly used for organic compounds, also for polymers and metals.
• COMPASS (Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies)
• MMFF (Merck Molecular Force Field) - Widely used.
• MM2, MM3, MM4 - Developed by Norman Allinger , widely used.

Polarizable force fields

• CFF / ind and ENZYMIX - application in biological systems.
• DRF90
• PIPF (Polarizable Intermolecular Potential for Fluids) - For organic liquids and biopolymers.
• PFF (Polarizable Force Field)
• CPE (basis Chemical Potential Equalization)

Reactive force fields

• ReaxFF (Reactive Force Field) - Fast, suitable for very large and long simulations (>> 1,000,000 atoms) of chemical reactions.
• EVB (Empirical Valence Bond) - wide application.

Coarsened force fields

• VAMM (Virtual Atom Molecular Mechanics) - Coarsened force field for conformational analysis.

Water force fields

• TIP3P, TIP4P
• Flexible SPC (Flexible Simple Point Charge water model)

Individual evidence

1. ^ Norman L. Allinger: Molecular Structure . July 19, 2010, doi : 10.1002 / 9780470608852 ( wiley.com [accessed December 16, 2018]).
2. ↑ Introduction to Computational Chemistry, 3rd Edition. Retrieved December 16, 2018 (American English). 
3. ↑ ^{a b c d e f g h} M.A. González: Force fields and molecular dynamics simulations . In: École thématique de la Société Française de la Neutronique . tape 12 , 2011, ISSN  2107-7223 , p. 169–200 , doi : 10.1051 / sfn / 201112009 ( neutron-sciences.org [accessed July 14, 2018]).
4. ↑ ^{a b c d e f g h i} Johannes Kalliauer, Gerhard Kahl, Stefan Scheiner, Christian Hellmich: A new approach to the mechanics of DNA: Atoms-to-beam homogenization . In: Journal of the Mechanics and Physics of Solids . tape 143 , June 4, 2020, p. 104040 , doi : 10.1016 / j.jmps.2020.104040 .
5. ^ Norman Allinger : Conformational analysis. 130. MM2. A hydrocarbon force field utilizing V1 and V2 torsional terms . In: Journal of the American Chemical Society . 99, No. 25, 1977, pp. 8127-8134. doi : 10.1021 / ja00467a001 .
6. ↑ Wendy D. Cornell, Piotr Cieplak, Christopher I. Bayly, Ian R. Gould, Kenneth M. Merz: A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules . In: Journal of the American Chemical Society . tape 117 , no. May 19 , 1995, ISSN  0002-7863 , pp. 5179-5197 , doi : 10.1021 / ja00124a002 .
7. ↑ Patel, S., Brooks, CL, III (2004) "CHARMM fluctuating charge force field for proteins: I parameterization and application to bulk organic liquid simulations." J. Comput. Chem. 25 1-16.
8. ↑ Patel, S., MacKerell, AD, Jr., Brooks, CL, III (2004) "CHARMM fluctuating charge force field for proteins: II Protein / solvent properties from molecular dynamics simulations using a nonadditive electrostatic model." J. Comput. Chem. 25 1504-1514.
9. ↑ William L. Jorgensen, Julian Tirado-Rives: The OPLS [optimized potentials for liquid simulations] potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin . In: Journal of the American Chemical Society . tape 110 , no. 6 , March 1988, pp. 1657-1666 , doi : 10.1021 / ja00214a001 .
10. ↑ AK Rappe, CJ Casewit, KS Colwell, WA Goddard, WM Skiff: UFF, a full periodic table forcefield for molecular mechanics and molecular dynamics simulations . In: Journal of the American Chemical Society . 114, No. 25, December 1992, pp. 10024-10035. doi : 10.1021 / ja00051a040 .
11. ^ H. Sun: COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase ApplicationsOverview with Details on Alkane and Benzene Compounds . In: Journal of Physical Chemistry B . 102, No. 38, 1998, pp. 7338-7364. doi : 10.1021 / jp980939v .
12. ↑ Thomas A. Halgren: Merck molecular force field. I. Basis, form, scope, parameterization, and performance of MMFF94 . In: Journal of Computational Chemistry . tape 17 , no. 5-6 , 1996, ISSN  1096-987X , pp. 490-519 , doi : 10.1002 / (SICI) 1096-987X (199604) 17: 5 / 63.0.CO; 2-P .
13. ↑ Norman L. Allinger, Young H. Yuh, Jenn Huei Lii: Molecular mechanics. The MM3 force field for hydrocarbons. 1 . In: Journal of the American Chemical Society . tape 111 , no. 23 , November 1, 1989, ISSN  0002-7863 , pp. 8551-8566 , doi : 10.1021 / ja00205a001 .
14. ^ Jenn Huei Lii, Norman L. Allinger: Molecular mechanics. The MM3 force field for hydrocarbons. 2. Vibrational frequencies and thermodynamics . In: Journal of the American Chemical Society . tape 111 , no. 23 , November 1, 1989, ISSN  0002-7863 , pp. 8566-8575 , doi : 10.1021 / ja00205a002 .
15. ^ Jenn Huei Lii, Norman L. Allinger: Molecular mechanics. The MM3 force field for hydrocarbons. 3. The van der Waals' potentials and crystal data for aliphatic and aromatic hydrocarbons . In: Journal of the American Chemical Society . tape 111 , no. 23 , November 1, 1989, ISSN  0002-7863 , pp. 8576-8582 , doi : 10.1021 / ja00205a003 .
16. ^ Warshel A. and Levitt M. (1976) "Theoretical Studies of Enzymatic Reactions: Dielectric Electrostatic and Steric Stabilization of the Carbonium Ion in the Reaction of Lysozyme". J. Mol. Biol. 103 227-249.
17. ↑ Gao, J., Habibollahzadeh, D., and Shao, L. (1995) "A Polarizable Intermolecular Potential Functions for Simulations of Liquid Alcohols". J. Phys. Chem. 99 16460-16467.
18. ↑ Xie, W., Pu, J., MacKerell, AD, Jr., and Gao, J. (2007) "Development of a Polarizable Intermolecular Potential Function (PIPF) for Liquid Amides and Alkanes". J. Chem. Theory Comput. 3 1878-1889.
19. ↑ Chelli, R., Procacci, P. (2002) "A Transferable Polarizable Electrostatic Force Field for Molecular Mechanics based on the Chemical Potential Equalization Principle." J. Chem. Phys. 17 9175-9189
20. ^ Adri CT van Duin, Siddharth Dasgupta, Francois Lorant, William A. Goddard: ReaxFF: A Reactive Force Field for Hydrocarbons . In: Journal of Physical Chemistry A . tape 105 , no. 41 , October 1, 2001, ISSN  1089-5639 , p. 9396-9409 , doi : 10.1021 / jp004368u .
21. ^ Arieh Warshel, Robert M. Weiss: An empirical valence bond approach for comparing reactions in solutions and in enzymes . In: Journal of the American Chemical Society . tape 102 , no. 20 , September 1, 1980, ISSN  0002-7863 , pp. 6218–6226 , doi : 10.1021 / ja00540a008 .
22. ^ A. Korkut, WA Hendrickson (2009). "A force field for virtual atom molecular mechanics of proteins." Proc. Natl. Acad. Sci. USA 106 15667-15672.