# 5.3. Fourier Dihedral¶

## 5.3.1. Functional Form¶

The Fourier dihedral potential has the functional forms:

$$E={{K}_{1,ijkl}}\left[ 1+\cos \left( {{N}_{1}}{{\phi }_{ijkl}}-{{D}_{1,ijkl}} \right) \right]+{{K}_{2,ijkl}}\left[ 1+\cos \left( {{N}_{2}}{{\phi }_{ijkl}}-{{D}_{2,ijkl}} \right) \right]$$ $$\qquad +{{K}_{3,ijkl}}\left[ 1+\cos \left( {{N}_{3}}{{\phi }_{ijkl}}-{{D}_{3,ijkl}} \right) \right]+{{K}_{4,ijkl}}\left[ 1+\cos \left( {{N}_{4}}{{\phi }_{ijkl}}-{{D}_{4,ijkl}} \right) \right]$$ $$\qquad +{{K}_{5,ijkl}}\left[ 1+\cos \left( {{N}_{5}}{{\phi }_{ijkl}}-{{D}_{5,ijkl}} \right) \right]$$

$$E={{K}_{1,ijkl}}\left[ 1-\cos \left( {{N}_{1}}{{\phi }_{ijkl}}-{{D}_{1,ijkl}} \right) \right]+{{K}_{2,ijkl}}\left[ 1-\cos \left( {{N}_{2}}{{\phi }_{ijkl}}-{{D}_{2,ijkl}} \right) \right]$$ $$\qquad +{{K}_{3,ijkl}}\left[ 1-\cos \left( {{N}_{3}}{{\phi }_{ijkl}}-{{D}_{3,ijkl}} \right) \right]+{{K}_{4,ijkl}}\left[ 1-\cos \left( {{N}_{4}}{{\phi }_{ijkl}}-{{D}_{4,ijkl}} \right) \right]$$ $$\qquad +{{K}_{5,ijkl}}\left[ 1-\cos \left( {{N}_{5}}{{\phi }_{ijkl}}-{{D}_{5,ijkl}} \right) \right]$$

The force-field parameters for this potential and units are given by:

 Equation Symbol Parameter Definition Units $$K_{1,ijkl}$$ Dihedral coefficient for atoms [i,j,k,l] energy $$K_{2,ijkl}$$ Dihedral coefficient for atoms [i,j,k,l] energy $$K_{3,ijkl}$$ Dihedral coefficient for atoms [i,j,k,l] energy $$K_{4,ijkl}$$ Dihedral coefficient for atoms [i,j,k,l] energy $$K_{5,ijkl}$$ Dihedral coefficient for atoms [i,j,k,l] energy $$N_{1}$$ Nonnegative integer coefficient N/A $$N_{2}$$ Nonnegative integer coefficient N/A $$N_{3}$$ Nonnegative integer coefficient N/A $$N_{4}$$ Nonnegative integer coefficient N/A $$N_{5}$$ Nonnegative integer coefficient N/A $$D_{1,ijkl}$$ Equilibrium Dihedral for atoms [i,j,k,l] degrees $$D_{2,ijkl}$$ Equilibrium Dihedral for atoms [i,j,k,l] degrees $$D_{3,ijkl}$$ Equilibrium Dihedral for atoms [i,j,k,l] degrees $$D_{4,ijkl}$$ Equilibrium Dihedral for atoms [i,j,k,l] degrees $$D_{5,ijkl}$$ Equilibrium Dihedral for atoms [i,j,k,l] degrees

## 5.3.2. XML Schema¶

The XML schema for the Fourier dihedral potential has the following representation (design mode representation using Liquid XML Studio):

The relationship between the equation symbols and XML schema notations are given by:

 Parameter Definition Equation Symbol Schema Notation Atom type of atom [i] $$i$$ AT-1 Atom type of atom [j] $$j$$ AT-2 Atom type of atom [k] $$k$$ AT-3 Atom type of atom [l] $$l$$ AT-4 Dihedral coefficient for atoms [i,j,k,l] $$K_{1,ijkl}$$ K1 Dihedral coefficient for atoms [i,j,k,l] $$K_{2,ijkl}$$ K2 Dihedral coefficient for atoms [i,j,k,l] $$K_{3,ijkl}$$ K3 Dihedral coefficient for atoms [i,j,k,l] $$K_{4,ijkl}$$ K4 Dihedral coefficient for atoms [i,j,k,l] $$K_{5,ijkl}$$ K5 Nonnegative integer coefficient $$N_{1}$$ N1 Nonnegative integer coefficient $$N_{2}$$ N2 Nonnegative integer coefficient $$N_{3}$$ N3 Nonnegative integer coefficient $$N_{4}$$ N4 Nonnegative integer coefficient $$N_{5}$$ N5 Equilibrium dihedral angle for atoms [i,j,k,l] $$D_{1,ijkl}$$ D1 Equilibrium dihedral angle for atoms [i,j,k,l] $$D_{2,ijkl}$$ D2 Equilibrium dihedral angle for atoms [i,j,k,l] $$D_{3,ijkl}$$ D3 Equilibrium dihedral angle for atoms [i,j,k,l] $$D_{4,ijkl}$$ D4 Equilibrium dihedral angle for atoms [i,j,k,l] $$D_{5,ijkl}$$ D5

The general attributes (describing the entire data set) are given by:

 General Attributes Cardinality Value style Fixed Fourier formula Fixed Enumerations specified in schema convention Optional Enumerations specified in schema Kn-units Required Enumerations specified in schema Dn-units Required Enumerations specified in schema

The specific attributes (attached to each set of parameters) are given by:

 Specific Attributes Cardinality Definition comment Optional Comment attached to parameter set version Optional Version number of parameter set reference Optional Reference attached to parameter set

Note that an XML document will be rejected from being entered into the WebFF database if a required attribute is left unspecified.