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.