Flexible molecules

Modelling of internal degrees of freedom, usual techniques:

Bond distances

• Atoms linked by a chemical bond (stretching):

$V_{str}(r_{12})=\frac{1}{2}{K}_{str}(r_{12}-b_0{)}^2$

Bond Angles

Bond sequence: 1-2-3:

Bond Angle: $\theta$

$\cos \theta =\frac{{\overrightarrow{r}}_{21}\cdot {\overrightarrow{r}}_{23}}{\left|{\overrightarrow{r}}_{21}\right|\left|{\overrightarrow{r}}_{23}\right|}$

Two typical forms are used to model the bending potential:

$V_{bend}(\theta )=\frac{1}{2}{k}_{\theta }{(\theta -{\theta }_0)}^2$

$V_{bend}(\cos \theta )=\frac{1}{2}{k}_c{(\cos \theta -c_0)}^2$

Dihedral angles. Internal Rotation

Bond sequence: 1-2-3-4 Dihedral angle ($\varphi$) definition:

Consider the following vectors:

• $\overrightarrow{a}\equiv \frac{{\overrightarrow{r}}_3-{\overrightarrow{r}}_2}{|{\overrightarrow{r}}_3-{\overrightarrow{r}}_2|}$; Unit vector in the direction of the 2-3 bond

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

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

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

• $e_{34}=(\cos \varphi )\overrightarrow{a}+(sin\varphi )\overrightarrow{c}$

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

• $V_{tors}(\varphi )=\sum _{i=0}^n{a}_i{(\cos \varphi )}^i$

or

• $V_{tors}(\varphi )=\sum _{i=0}^n{b}_i\cos (i\varphi )$