Flexible molecules

Modelling of internal degrees of freedom, usual techniques:

Bond distances

Atoms linked by a chemical bond (stretching):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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: $\left.\theta \right.$

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

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 (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\phi \right.} ) definition:

Consider the following vectors:

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

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

• 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 \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 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 \vec{r}_{34} } ortogonal to 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 \vec{a} }

• 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 \vec{c} = \vec{a} \times \vec{b} }

• 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 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 \Phi_{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 \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)