Example: The TRIPOS Force Field

Equations to calculate the energy as function of the molecular geometry:

E = E_str + E_bend + E_oop + E_tors + E_vdw ( + E_ele + E_{dist_c} + E_{ang_c} + E_{tors_c} + E_{range_c} + E_multi + E_{field_fit} )

Main terms:

E_str : energy of a bond stretched or compressed from its natural bond length.
E_str = _{all bonds} 0.5 * k_b,i * ( d_i - d_i⁰ ) ²

where:

• d_i : the length of the i-th bond (Angstrom)
  
• d_i⁰ : the equilibrium length of the i-th bond(Angstrom)
  
• k_b,i: the bond stretching force constant (kcal/(mole)(Angstrom)²)
  

E_bend : energy of bending bond angles from their natural values.
E_bend = _{all angles} 0.5 * k_theta,i * ( theta_i - theta_i⁰ ) ²

where:

• theta_i : the angle between two adjacent bonds (degrees)
  
• theta_i⁰ : the equilibrium value for the i-th angle
  
• k_theta,i: the angle bending force constant (kcal/(mole)(degrees)²)
  

E_oop : energy of bending planar atoms out of the plane.
E_oop = _{all trigonal atoms} 0.5 * k_oop,i * d_i²

where:

• d_i : the distance between the center atom and the plane of its substituents (Angstrom)
• k_oop,i : the out of plane bending constant (kcal/(mole)(degrees)²)

E_tors : torsional energy due to twisting about bonds.
E_tors = _{all torsions} 0.5 * V_omega,i * [ 1 + S_i * cos ( |n_i| * omega_i ]

where:

• V_omega,i : the torsional barrier (kcal/mole)
• S_i : +1 for staggered minimum energy and -1 for eclipsed minimum energy
• |n_i| : the periodicity
• omega_i : the torsion angle

E_vdw : energy due to van der Waals non-bonded interactions.
E_vdw = _{all non-bonded atom pairs} E_ij * { [ 1 / a_ij¹² ] - [ 2 / a_ij⁶ ] }

where:

• E_ij : the van der Waals constant (kcal/mole) = (E_i * E_j)^0.5
• a_ij : equals r_ij / ( R_i + R_i )
• r_ij : the distance between atoms i and j (Angstrom)
• R_i : the van der Waals radius of the i-th atom (Angstrom)

Optional terms:

E_ele : energy due to electrostatic interactions.
E_ele = 332.17 * _{all non-bonded atom pairs} Q_i*Q_j / ( D_ij*r_ij )

where:

• D_ij : the value of the dielectric function for atoms i and j
• Q_i : the net atomic charge at the i-th atom
• r_ij : the distance between atoms i and j (Angstrom)
• 332.17 : a unit conversion factor

E_{dist_c} : energy associated with distance constraints.
E_{dist_c} = 0.5 * k_d * ( d - d⁰)²

where:

• d : the distance between two specified atoms (Angstrom)
• d⁰ : the requested distance between the two atoms (Angstrom)
• k_d : the force constant (kcal/(mole)(Angstrom)²)

E_{ang_c} : energy associated with angle constrains.
E_{ang_c} = 0.5 * k_theta * ( theta - theta⁰)²

where:

• theta : the angle between three specified atoms (degrees)
• theta⁰ : the requested angle between the three atoms (degrees)
• k_theta : the force constant (kcal/(mole)(degrees)²)

E_{tors_c} : energy associated with torsion angle constraints.
E_{tors_c} = 0.5 * k_omega * ( omega - omega⁰)²

where:

• omega : the angle between four specified atoms (degrees)
• omega⁰ : the requested angle between the foru atoms (degrees)
• k_theta : the force constant (kcal/(mole)(degrees)²)

E_{range_c} : energy associated with range constraints.
E_{range_c} = 0 for d^low < d < d^high

E_{range_c} = 0.5 * k_r * ( d - d^low)² for d^low < d < d^high

E_{range_c} = 0.5 * k_r * ( d^high - d)² for d^low < d < d^high

where:

• d^low : the minimum distance between two specified atoms (Angstrom)
• d^high : the maximum distance between two specified atoms (Angstrom)
• k_r : the force constant (kcal/(mole)(Angstrom)²)

E_multi : energy associated with multifitting.
E_multi = _{all reference pairs} k_s,i * d_i²

where:

• k_s,i : the spring constant (kcal/(mole)(Angstrom)²
• d_i : the distance between the atom and the reference point (Angstrom)

E_{field_fit} : energy associated with fitting fields.
The magnitude of any field fit energy penalty is calculated as the sum of the squared differences in field values over all intersections of a three-dimensional laatice embedded in Cartesian space. Actually two field values, steric and electrostatic, are calculated at each lattice intersection, using the TRIPOS force field, as the sum of interactions between an artificial probe atom at that intersection and each of the atoms in the target or template molecule.