# Flexible molecules

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Modelling of internal degrees of freedom, usual techniques:

## Bond distances

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

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

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

## Bond Angles

Bond sequence: 1-2-3:

Bond Angle: ${\displaystyle \left.\theta \right.}$

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

Two typical forms are used to model the bending potential:

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

## Dihedral angles. Internal Rotation

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

Consider the following vectors:

• ${\displaystyle {\vec {a}}\equiv {\frac {{\vec {r}}_{3}-{\vec {r}}_{2}}{|{\vec {r}}_{3}-{\vec {r}}_{2}|}}}$; Unit vector in the direction of the 2-3 bond
• ${\displaystyle {\vec {b}}\equiv {\frac {{\vec {r}}_{21}-({\vec {r}}_{21}\cdot {\vec {a}}){\vec {a}}}{|{\vec {r}}_{21}-({\vec {r}}_{21}\cdot {\vec {a}}){\vec {a}}|}}}$; normalized component of ${\displaystyle {\vec {r}}_{21}}$ ortogonal to ${\displaystyle {\vec {a}}}$
• ${\displaystyle {\vec {e}}_{34}\equiv {\frac {{\vec {r}}_{34}-({\vec {r}}_{34}\cdot {\vec {a}}){\vec {a}}}{|{\vec {r}}_{34}-({\vec {r}}_{34}\cdot {\vec {a}}){\vec {a}}|}}}$; normalized component of ${\displaystyle {\vec {r}}_{34}}$ ortogonal to ${\displaystyle {\vec {a}}}$
• ${\displaystyle {\vec {c}}={\vec {a}}\times {\vec {b}}}$
• ${\displaystyle e_{34}=(\cos \phi ){\vec {a}}+(\sin \phi ){\vec {c}}}$

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

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

or

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

## 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)