Flexible molecules: Difference between revisions

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== Bond distances ==  
== Bond distances ==  
* Atoms linked by a chemical bond (stretching):
Atoms linked by a chemical bond (stretching) using the [[harmonic spring approximation]]:


<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)


== Bond Angles  ==
== Bond Angles  ==
Line 11: Line 12:
Bond sequence:  1-2-3:
Bond sequence:  1-2-3:


Bond Angle: <math> \theta </math>
Bond Angle: <math> \left. \theta \right. </math>


<math> \cos \theta = \frac{ \vec{r}_{21} \cdot \vec{r}_{23} } {|\vec{r}_{21}| |\vec{r}_{23}|}  
:<math> \cos \theta = \frac{ \vec{r}_{21} \cdot \vec{r}_{23} } {|\vec{r}_{21}| |\vec{r}_{23}|}  
</math>
</math>


Two typical forms are used to model the ''bending'' potential:
Two typical forms are used to model the ''bending'' potential:


<math>
:<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>


<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>
</math>


== Internal Rotation ==
== Dihedral angles. Internal Rotation ==
 
Bond sequence: 1-2-3-4
Dihedral angle (<math> \left. \phi \right. </math>) definition:
 
Consider the following vectors:
 
* <math> \vec{a}  \equiv \frac{\vec{r}_3 -\vec{r}_2}{|\vec{r}_3 -\vec{r}_2|} </math>; Unit vector in the direction of the 2-3 bond
 
* <math> \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} | } </math>;  normalized component of <math> \vec{r}_{21} </math> ortogonal to <math> \vec{a} </math> 
 
* <math> \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} | } </math>; normalized component of <math> \vec{r}_{34} </math> ortogonal to <math> \vec{a} </math>
 
*<math> \vec{c} = \vec{a} \times \vec{b} </math>
 
*<math> e_{34} = (\cos \phi) \vec{a} + (\sin \phi) \vec{c} </math>
 
For molecules with internal rotation degrees of freedom (e.g. ''n''-alkanes), a ''torsional'' potential is
usually modelled as:
 
*<math>
\Phi_{tors} \left(\phi\right) = \sum_{i=0}^n a_i \left( \cos \phi \right)^i
</math>
 
or
 
* <math>
\Phi_{tors} \left(\phi\right) = \sum_{i=0}^n b_i  \cos \left( i \phi \right)
</math>
 
== 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 model|Lennard-Jones]] potentials)
[[category: force fields]]
[[category: models]]

Latest revision as of 15:32, 30 July 2007

Modelling of internal degrees of freedom, usual techniques:

Bond distances[edit]

Atoms linked by a chemical bond (stretching) using the harmonic spring approximation:

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

Bond Angles[edit]

Bond sequence: 1-2-3:

Bond Angle:

Two typical forms are used to model the bending potential:

Dihedral angles. Internal Rotation[edit]

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

Consider the following vectors:

  • ; Unit vector in the direction of the 2-3 bond
  • ; normalized component of ortogonal to
  • ; normalized component of ortogonal to

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

or

Van der Waals intramolecular interactions[edit]

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)