Flexible molecules: Difference between revisions

Revision as of 13:06, 21 June 2007

Modelling of internal degrees of freedom, usual techniques:

Atoms linked by a chemical bond (stretching):

$Φ_{s t r} (r_{12}) = \frac{1}{2} K_{s t r} (r_{12} - b_{0})^{2}$

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

Bond sequence: 1-2-3:

Bond Angle: $θ$

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

Two typical forms are used to model the bending potential:

$Φ_{b e n d} (θ) = \frac{1}{2} k_{θ} {(θ - θ_{0})}^{2}$

$Φ_{b e n d} (\cos θ) = \frac{1}{2} k_{c} {(\cos θ - c_{0})}^{2}$

Bond sequence: 1-2-3-4 Dihedral angle ( $ϕ$ ) 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 $\vec{a}$

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

or

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)

@@ Line 4: / Line 4: @@
 Atoms linked by a chemical bond (stretching):
-<math> V_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} - b_0 )^2 </math>
+<math> \Phi_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} - b_0 )^2 </math>
 However, this internal coordinates are very often kept constrained (fixed bond distances)
@@ Line 20: / Line 20: @@
 <math>
-V_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2
+\Phi_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2
 </math>
 <math>
-V_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2
+\Phi_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2
 </math>
@@ Line 50: / Line 50: @@
 *<math>
-V_{tors} \left(\phi\right) = \sum_{i=0}^n a_i \left( \cos \phi \right)^i
+\Phi_{tors} \left(\phi\right) = \sum_{i=0}^n a_i \left( \cos \phi \right)^i
 </math>
@@ Line 56: / Line 56: @@
 * <math>
-V_{tors} \left(\phi\right) = \sum_{i=0}^n b_i  \cos \left( i \phi \right)
+\Phi_{tors} \left(\phi\right) = \sum_{i=0}^n b_i  \cos \left( i \phi \right)
 </math>