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Interaction Energy

Methods For Configuration Integral Calculation

We need to calculate configuration integral

$\begin{displaymath} Z_N = \int\exp\bigl(-U_N(\tilde q)/kT)\,d\tilde q\end{displaymath}$

1.

Analytic methods:

(a): Exactly solvable problems (ideal gas, ideal crystal, 2D-lattice and others)
(b): Expansions around solvable models: virial expansion, high- and low-temperature expansions, imperfect solids...
(c): Mean-field theories: neglecting fluctuations
(d): Symmetry tricks--renormalization group etc.

2.

Computer simulation (Monte Carlo)

In all cases we need U.

Assumption:: The total energy is the sum of pair interactions:
$\begin{displaymath} U_N(q_1,q_2,\dots,q_N)= \sum_{i=1}^N\sum_{j\gt i} u(q_i,q_j) \end{displaymath}$
A good assumption for the majority of cases.
Symmetry:: For unstructured isotropic particles,
$\begin{displaymath} u(q_1,q_2)=u(r_{12}),\quad r_{12}=\left\lvert \mathbf{q}_1-\mathbf{q}_2\right\rvert \end{displaymath}$

Let us discuss non-polar neutral monoatomic molecules. Let r be the distance between them.

1.: Molecules cannot come too close:
$\begin{displaymath} \lim_{r\to0} u(r)= \infty\end{displaymath}$
2.: At large distances molecules do not interact
$\begin{displaymath} \lim_{r\to\infty} u(r)=0 \end{displaymath}$
3.: Molecules repeal each other at small distances and attract at large distances.
4.: Exact result: at $r\to\infty$ the energy is $u\sim-1/r^6$ (dispersive forces)

``Real Potential'':

$\begin{figure} \InputIfFileExists{u.pstex_t}{}{} \end{figure}$

Hard Core Potential:

The simplest one

$\begin{displaymath} u(r) = \begin{cases} \infty,& r<\sigma\ 0,& r\gt\sigma \end{cases} \end{displaymath}$

$\begin{figure} \InputIfFileExists{hardcore.pstex_t}{}{} \end{figure}$

Hard Core Plus Attraction:

Slightly more complex

$\begin{displaymath} u(r) = \begin{cases} \infty,& r<\sigma\ -\epsilon(\sigma/r)^6,& r\gt\sigma \end{cases} \end{displaymath}$

(2)

$\begin{figure} \InputIfFileExists{soft.pstex_t}{}{} \end{figure}$

Lennard-Jones Potential:

The favorite among simulators

$\begin{displaymath} u(r) = \epsilon\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right] \end{displaymath}$

At $r\gg\sigma$ potential behaves as -1/r⁶--correct. At $r\ll\sigma$ potential diverges $u\to\infty$ --correct.

Why 12? Because computers calculate x² very fast!

If you want to be realistic, you could add

Complex interaction--Coulomb potential (very difficult!), dipole-dipole interactions, multipole terms...
Bonds: chemical, hydrogen,...
``Realistic'' potentials--QM Computations (Computer only!)

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