Flexible molecules

Modelling of internal degrees of freedom, usual techniques:

Bond distances

Atoms linked by a chemical bond (stretching):

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

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

Bond Angles

Bond sequence: 1-2-3:

Bond Angle: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:

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

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

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:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{tors} \left(\phi\right) = \sum_{i=0}^n a_i \left( \cos \phi \right)^i }

or

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{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)