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| == Bond distances == | | == Bond distances == |
| Atoms linked by a chemical bond (stretching) using the [[harmonic spring approximation]]: | | * Atoms linked by a chemical bond (stretching): |
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| :<math> \Phi_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} - b_0 )^2 </math>
| | <math> V_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} - b_0 )^2 </math> |
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| However, this internal coordinates are very often kept constrained (fixed bond distances)
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| == Bond Angles == | | == Bond Angles == |
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| Bond sequence: 1-2-3: | | Bond sequence: 1-2-3: |
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| Bond Angle: <math> \left. \theta \right. </math> | | Bond Angle: <math> \theta </math> |
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| :<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> |
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| Two typical forms are used to model the ''bending'' potential: | | Two typical forms are used to model the ''bending'' potential: |
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| :<math>
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| \Phi_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2
| | V_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2 |
| </math> | | </math> |
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| :<math>
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| \Phi_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2
| | V_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2 |
| </math> | | </math> |
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| == Dihedral angles. Internal Rotation == | | == Internal Rotation == |
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| Bond sequence: 1-2-3-4
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| Dihedral angle (<math> \left. \phi \right. </math>) definition:
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| Consider the following vectors:
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| * <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
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| * <math> \vec{b} \equiv \frac{ \vec{r}_{21} - (\vec{r}_{21}\cdot \vec{a} ) \vec{a} }
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| { |\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>
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| * <math> \vec{e}_{34} \equiv \frac{ \vec{r}_{34} - (\vec{r}_{34}\cdot \vec{a} ) \vec{a} }
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| { |\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>
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| *<math> \vec{c} = \vec{a} \times \vec{b} </math>
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| *<math> e_{34} = (\cos \phi) \vec{a} + (\sin \phi) \vec{c} </math>
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| For molecules with internal rotation degrees of freedom (e.g. ''n''-alkanes), a ''torsional'' potential is | | For molecules with internal rotation degrees of freedom (e.g. ''n''-alkanes), a ''torsional'' potential is |
| usually modelled as: | | usually modelled as: |
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| *<math>
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| \Phi_{tors} \left(\phi\right) = \sum_{i=0}^n a_i \left( \cos \phi \right)^i
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| </math>
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| or
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| * <math>
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| \Phi_{tors} \left(\phi\right) = \sum_{i=0}^n b_i \cos \left( i \phi \right)
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| </math>
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| == Van der Waals intramolecular interactions ==
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| For pairs of atoms (or sites) which are separated by a certain number of chemical bonds:
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| Pair interactions similar to the typical intermolecular potentials are frequently
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| used (e.g. [[Lennard-Jones model|Lennard-Jones]] potentials)
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| [[category: force fields]]
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| [[category: models]]
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