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Transferable Potentials for Phase Equilibria

Standard Conventions for 1-4 Bonded Interactions

TraPPE uses two standard conventions for dihedral angles:

The first convention (similar to IUPAC) uses the symbol \phi_{ijkl} to represent the dihedral angle as illustrated in the figure above. Synperiplanar (cis) conformations correspond to \phi_{ijkl} = 0^{\circ}, while antiperiplanar (trans) conformations correspond to \phi_{ijkl} = 180^{\circ}. Clockwise rotations about the dihedral axis are considered positive, while counterclockwise rotations are negative. Note that in this convention bond vectors are defined sequentially from end to end through the dihedral. When the vector normal to the plane formed by \vec{\mathbf{v}}_{ij} and \vec{\mathbf{v}}_{jk} points in the opposite direction from the vector normal to the plane formed by \vec{\mathbf{v}}_{jk} and \vec{\mathbf{v}}_{kl}, \phi_{ijkl} = 180^{\circ} and the conformation is trans.

\cos \phi_{ijkl} = { \left( \vec{\mathbf{v}}_{ij} \times \vec{\mathbf{v}}_{jk} \right)\cdot \left( \vec{\mathbf{v}}_{jk} \times \vec{\mathbf{v}}_{kl} \right) \over\left| \vec{\mathbf{v}}_{ij} \times \vec{\mathbf{v}}_{jk} \right|\left| \vec{\mathbf{v}}_{jk} \times \vec{\mathbf{v}}_{kl} \right| }

The second convention uses the symbol \psi_{ijkl} and defines each plane with bond vectors that point towards the center of the dihedral (note the red vectors in the figure above). Using this convention, synperiplanar (cis) conformations correspond to \psi_{ijkl} = 180^{\circ}, while antiperiplanar (trans) conformations correspond to \psi_{ijkl} = 0^{\circ}. Clockwise rotations about the dihedral axis are again considered positive, and counterclockwise rotations are negative.When the vector normal to the plane formed by \vec{\mathbf{v}}_{ij} and \vec{\mathbf{v}}_{jk} points in the opposite direction from the vector normal to the plane formed by \vec{\mathbf{v}}_{lk} and \vec{\mathbf{v}}_{kj}, \psi_{ijkl} = 180^{\circ} and the conformation is cis.

\cos \psi_{ijkl} = { \left( \vec{\mathbf{v}}_{ij} \times \vec{\mathbf{v}}_{jk} \right)\cdot \left( \vec{\mathbf{v}}_{lk} \times \vec{\mathbf{v}}_{kj} \right) \over\left| \vec{\mathbf{v}}_{ij} \times \vec{\mathbf{v}}_{jk} \right|\left| \vec{\mathbf{v}}_{lk} \times \vec{\mathbf{v}}_{kj} \right| }

Throughout the TraPPE website, these conventions are noted clearly for each model by the distinguishing symbol of the dihedral angle (\phi_{ijkl} or \psi_{ijkl}). When a particular torsion type includes a phase shift, this is also explicitly noted in the section of the website describing 1-4 bonded interactions for that specific model.