Computer simulation is a bridge between theory and experiment:
Computer simulation 
``exact" results for a particular model
Compare to experimental results test model
Compare to theoretical predictions for this model test theories
Theories and models will help understand and interpret experimental results
Computer simulation is a bridge between microscopic and
macroscopic properties:
Computer simulation
microscopic details (i.e. atomic and molecular motion)
Microscopic details macroscopic properties
Computer simulation can provide information not available from real experiments:
-measurements under extreme conditions that are experimentally prohibitive or difficult
-details of molecular motion and structure impossible or difficult to study experimentally
-molecular events that are too fast or too slow
to study experimentally
The complete mathematical description of a system
Nonrelativistic Schrödinger equation:
: coordinates of nuclei
: coordinates of electrons
: wavefunction for the system
: total energy of the system
: Hamiltonian for the system
and 
:
kinetic energies of the nuclei and electrons, respectively
:
potential of interaction between all nuclei and electrons
This equation is too complicated for general use, so approximations are invoked.
Born-Oppenheimer approximation:
Since electrons are much lighter than nuclei, they move much faster,
so we can decouple the motion of the electrons
from the nuclei to get 2 separate equations.
The first describes the motion of the electrons (for fixed nuclei)
where
:
electronic wave function (depends only parametrically on the positions of the nuclei)
: electronic energy (function of only the nuclear coordinates)
can be viewed as the potential energy surface on which the nuclei move.
The second equation describes the motion of the nuclei on the potential
energy surface :
where
In Eq. 6
the electronic coordinates are averaged over the electronic wavefunction.
The nuclei are moving in the average field of the electrons.
This is valid if the electrons move much faster than the nuclei.
E is an approximation of
from Eq. 1 and includes electronic, vibrational, rotational,
and translational energy.
The corresponding approximation to the wavefunction is
In principle, Eq. 3 could be solved for ,
and then Eq. 6 could be solved for the nuclear motion.
But solution of Eq. 3 requires a large amount of computation
(using ab initio quantum chemistry codes such as Gaussian or semiempirical
codes such as MNDO, MOPAC, etc.). Thus, usually an empirical fit to
(a forcefield) is used.
Solution of Eq. 6
is called quantum dynamics and also requires a large amount of computation. Since the nuclei are relatively heavy, the quantum mechanical effects are often insignificant, so Eq. 6
can be replaced with the Newton's equations of motion
(i.e. 
).
Solving this equation for the nuclei moving classically on a
single potential surface
is called classical molecular dynamics.
If we are not interested in the time evolution of the system
then we can use
to calculate static properties such as equilibrium structures, transition states, relative energies, etc. This is called molecular mechanics.
Typical assumptions of classical molecular dynamics
1. Born-Oppenheimer approximation: nuclei move in the average field of the electrons
2. nuclei move on a single potential surface (i.e. a single electronic state)
3. potential surface can be approximated by an empirical fit
4. nuclear motion can be described by classical mechanics
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