Flexible molecules

Modelling of internal degrees of freedom, usual techniques:

[edit] 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 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:

$\Phi_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2$

$\Phi_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2$

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 $\vec{a}$