Energy minimisation used to optimise geometry

How does the minimisation routine find its way to the most stable structure?

You should read Chapter 3 of Goodman's 'Chemical Applications of Molecular
Modelling' about this

Suppose we are minimising a diatomic molecule

This is a onedimensional minimisation: energy depends only on one
bond length r

Calculate slope at the starting point, dE/dr

If positive, need to decrease r. If negative, increase r

A polyatomic molecule has 3N6 internal coordinates, where N
is number of atoms

Therefore have to minimise in 3N6 dimensional space

First derivative of energy with respect to distance, dE/dr,
is a force

Instead of one force, we will have a whole array of them:
dE/dr_{1},
dE/dr_{2}
...

This defines a vector which tells us relatively how much and in which sense
to alter each dimension in the next step, to go in the direction of steepest
descent

Actual size of step taken is reduced as forces get smaller, near bottom
of valley
How does the program know when it has arrived?

Should be negligible forces to go anywhere else

The next step to be made would be negligibly small
Both of these tests have to be satisfied in the program Gaussian
What can go wrong?

For a large molecule with several rotatable groups where there is little
hindrance to rotation, the energy minimum can be very flat with respect
to these rotations and accompanying breathing movements of the molecule

The forces for going anywhere can be far smaller than the required maximum,
but they still produce displacements which are too large

Energies are probably good enough: they are not changing much by
then

You cannot publish the results of a nonconverged modelling experiment

In some modelling programs, you need to be careful that you have not reached
some builtin limit on the number of cycles, without actually reaching
a minimum: be careful to look at the text output

You may be in a saddle point instead of a minimum

Imagine walking down a hillside in thick fog

You reach some level ground

You do not know whether you are at the bottom of a hole, or whether you
are in a pass, with a further dropoff to come

If you are in a true minimum, all the force constants for vibration should
be positive: if you distort the molecule, it should fall back towards
the minimum

If you are in a saddle, then one vibration will go and not come back 
it has a negative force constant, and the vibration is imaginary

Good journals often require you to calculate vibration frequencies to make
sure that none are imaginary, before they will publish a calculated geometry

You may be in a small minimum of little chemical importance, even though
it is a true minimum

The structure may look very reasonable but there may be one or more much
more stable ones you have not thought of

For this you need at least a human protocol of Conformation
searching : it is not permissible to do nothing but hope you
have landed on the required model

The bigger modelling packages may provide automated conformation searching,
so that this can be done objectively