Distance is defined by points and :
The first derivatives of with respect to Cartesian coordinates are:
Angle is defined by points , , and , and spanned by vectors and :
The first derivatives of α with respect to Cartesian coordinates are:
These equations for the derivatives have a numerical instability when the angle goes to 0 or to 180°. Presently, the problem is ‘solved’ by testing for the size of the angle; if it is too small, the derivatives are set to 0 in the hope that other restraints will eventually pull the angle towards well behaved regions. Thus, angle restraints of 0 or 180° should not be used in the conjugate gradients or molecular dynamics optimizations.
Dihedral angle is defined by points , , , and ():
The first derivatives of χ with respect to Cartesian coordinates are:
These equations for the derivatives have a numerical instability when the angle goes to 0. Thus, the following set of equations is used instead [van Schaik et al., 1993]:
The only possible instability in these equations is when the length of the central bond of the dihedral, , goes to 0. In such a case, which should not happen, the derivatives are set to 0. The expressions for an improper dihedral angle, as opposed to a dihedral or dihedral angle, are the same, except that indices are permuted to . In both cases, covalent bonds , , and are defining the angle.
This is the accessibility value calculated by the PSA algorithm (see Model.write_data()). This is usually set by the last call to Restraints.make() or Restraints.make_distance(). First derivatives are not calculated, and are always returned as 0.
Atomic density for a given atom is simply calculated as the number of atoms within a distance EnergyData.contact_shell of that atom. First derivatives are not calculated, and are always returned as 0.
The absolute atomic coordinates , and are available for every point , primarily for use in anchoring points to planes, lines or points. Their first derivatives with respect to Cartesian coordinates are of course simply 0 or 1.