# 6.5. Fourier Improper¶

## 6.5.1. Functional Form¶

The Fourier improper potential has the functional form:

$$E={{K}_{i,ijkl}}\left[ {{C}_{0,ijkl}}+{{C}_{1,ijkl}}\cos \left( {{\omega }_{ijkl}} \right)+{{C}_{2,ijkl}}\cos \left( 2{{\omega }_{ijkl}} \right) \right]$$

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

 Equation Symbol Parameter Definition Units $$K_{i,ijkl}$$ Improper coefficient for atoms [i,j,k,l] energy $$C_{0,ijkl}$$ Real coefficient for cosine term #0 [i,j,k,l] N/A $$C_{1,ijkl}$$ Real coefficient for cosine term #1 [i,j,k,l] N/A $$C_{2,ijkl}$$ Real coefficient for cosine term #2 [i,j,k,l] N/A

## 6.5.2. XML Schema¶

The XML schema for the Fourier improper 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 Improper coefficient for atoms [i,j,k,l] $$K_{i,ijkl}$$ Ki Real coefficient for cosine term #0 [i,j,k,l] $$C_{0,ijkl}$$ C0 Real coefficient for cosine term #1 [i,j,k,l] $$C_{1,ijkl}$$ C1 Real coefficient for cosine term #2 [i,j,k,l] $$C_{2,ijkl}$$ C2

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

 General Attributes Cardinality Value style Fixed Fourier formula Fixed Ki*[C0+C1*cos(w)+C2*cos(2*w)] convention Optional Enumerations specified in schema Ki-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.