** Next:** Virial Expansion
**Up:** Imperfect Gases and Liquids.
** Previous:** Quiz Answers

- Methods For Configuration Integral Calculation
- Pair Interactions
- Intermolecular Potential: Simple Case
- Intermolecular Potential: Other Cases

We need to calculate *configuration integral*

- 1.
- Analytic methods:
- (a)
- Exactly solvable problems (ideal gas, ideal crystal,
2
*D*-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: A good assumption for the majority of cases.
**Symmetry:**- For unstructured isotropic particles,

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

- 1.
- Molecules cannot come too close:
- 2.
- At large distances molecules do not interact
- 3.
- Molecules repeal each other at small distances and attract at large distances.
- 4.
- Exact result: at the energy is
*(dispersive forces)*

**``Real Potential'':**-

**Hard Core Potential:**- The simplest one

**Hard Core Plus Attraction:**- Slightly more complex
(2)

**Lennard-Jones Potential:**- The favorite among simulators
At potential behaves as -1/
*r*--correct. At potential diverges --correct.^{6}Why 12? Because computers calculate

*x*very fast!^{2}

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!)

© 1997

Thu Sep 18 22:50:29 EDT 1997