Flexible molecules

Modelling of internal degrees of freedom, usual techniques:

Bond distances

Atoms linked by a chemical bond (stretching) using the harmonic spring approximation:

$\Phi _{str}(r_{12})={\frac {1}{2}}K_{str}(r_{12}-b_{0})^{2}$

However, this internal coordinates are very often kept constrained (fixed bond distances)

Bond Angles

Bond sequence: 1-2-3:

Bond Angle: $\left.\theta \right.$

$\cos \theta ={\frac {{\vec {r}}_{21}\cdot {\vec {r}}_{23}}{|{\vec {r}}_{21}||{\vec {r}}_{23}|}}$

Two typical forms are used to model the bending potential:

$\Phi _{bend}(\theta )={\frac {1}{2}}k_{\theta }\left(\theta -\theta _{0}\right)^{2}$

$\Phi _{bend}(\cos \theta )={\frac {1}{2}}k_{c}\left(\cos \theta -c_{0}\right)^{2}$

Dihedral angles. Internal Rotation

Bond sequence: 1-2-3-4 Dihedral angle ($\left.\phi \right.$) definition:

Consider the following vectors:

• ${\vec {a}}\equiv {\frac {{\vec {r}}_{3}-{\vec {r}}_{2}}{|{\vec {r}}_{3}-{\vec {r}}_{2}|}}$; Unit vector in the direction of the 2-3 bond

• ${\vec {b}}\equiv {\frac {{\vec {r}}_{21}-({\vec {r}}_{21}\cdot {\vec {a}}){\vec {a}}}{|{\vec {r}}_{21}-({\vec {r}}_{21}\cdot {\vec {a}}){\vec {a}}|}}$; normalized component of ${\vec {r}}_{21}$ ortogonal to ${\vec {a}}$

• ${\vec {e}}_{34}\equiv {\frac {{\vec {r}}_{34}-({\vec {r}}_{34}\cdot {\vec {a}}){\vec {a}}}{|{\vec {r}}_{34}-({\vec {r}}_{34}\cdot {\vec {a}}){\vec {a}}|}}$; normalized component of ${\vec {r}}_{34}$ ortogonal to ${\vec {a}}$

• ${\vec {c}}={\vec {a}}\times {\vec {b}}$

• $e_{34}=(\cos \phi ){\vec {a}}+(\sin \phi ){\vec {c}}$

For molecules with internal rotation degrees of freedom (e.g. n-alkanes), a torsional potential is usually modelled as:

• $\Phi _{tors}\left(\phi \right)=\sum _{i=0}^{n}a_{i}\left(\cos \phi \right)^{i}$

or

• $\Phi _{tors}\left(\phi \right)=\sum _{i=0}^{n}b_{i}\cos \left(i\phi \right)$

Van der Waals intramolecular interactions

For pairs of atoms (or sites) which are separated by a certain number of chemical bonds:

Pair interactions similar to the typical intermolecular potentials are frequently used (e.g. Lennard-Jones potentials)