Cluster diagrams

2007-07-04T22:50:08Z

222.66.97.75:

'''Diagrams''' are sometimes known as ''graphs''.
==Chain clusters==
A ''chain'' is a cluster with at least one node.
All routes from one base point to the other pass through at least one nodal
field point. Cutting at this point separates the diagram into two parts,
each of which contains a base point. A ''simple chain'' is one in which
every field point is a node. A ''netted chain'' is one that can
be formed from a simple chain by adding not more than one field point
across each link of the simple chain.

==Bundles==

A ''bundle'' is a parallel collection of links.
It has no nodes since there is always more than one independent path
from one base point to the other.

==Elementary Clusters==

An ''elementary'' cluster is that which is neither a chain or a bundle.

:<math>\left.h(r)\right. = C(r) B(r) E(r)</math>

where ''C(r)'' is the set of ''chain'' clusters, ''B(r)'' is the set of ''bundles'', and
''E(r)'' is the set of ''elementary'' clusters (Eq. 5.2 Ref. 1).
Similarly,

:<math>\left.c(r)\right. = B(r) E(r)</math>

==Lemmas==
There are five lemmas (see Ref.s 2,3 and 4).
==References==
#[http://dx.doi.org/10.1088/0034-4885/28/1/306 J. S. Rowlinson "The equation of state of dense systems", Reports on Progress in Physics ''28''' pp. 169-199 (1965)]
#[http://dx.doi.org/10.1143/PTP.25.537 Tohru Morita and Kazuo Hiroike "A New Approach to the Theory of Classical Fluids. III General Treatment of Classical Systems", Progress of Theoretical Physics '''25''' pp. 537-578 (1961)]
#[http://dx.doi.org/10.1063/1.1724313 Cyrano De Dominicis "Variational Formulations of Equilibrium Statistical Mechanics", Journal of Mathematical Physics '''3''' pp. 983-1002 (1962)]
#[http://dx.doi.org/10.1063/1.1703949 Cyrano De Dominicis "Variational Statistical Mechanics in Terms of "Observables" for Normal and Superfluid Systems", Journal of Mathematical Physics '''4''' pp. 255-265 (1963)]
[[category: integral equations]]
[[category: statistical mechanics]]

Flexible molecules

2007-07-04T20:19:54Z

222.66.97.75:

Modelling of internal degrees of freedom, usual techniques:

== Bond distances ==
Atoms linked by a chemical bond (stretching):

<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 sequence: 1-2-3:

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>

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

<math>
\Phi_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2
</math>

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

== 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]]

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Cluster diagrams

Flexible molecules