However, how can we be sure that the
optimized geometry given by the calculation really is an energy minimum?
It could be a different type of stationary point (a saddlepoint).
Remember a (first-order) saddlepoint is
a maximum in one direction, and a minimum in all other directions.
For a simple function in one dimension,
at a minimum the second derivative (f'') is positive whereas at a maximum it is
negative.
For a multidimensional function, we
need to calculate the eigenvalues of
the Hessian matrix.
The eigenvalues, l, are given by:
(H – lI)x = 0
x is an eigenvector, which we
will deal with later.
For a non-trivial solution, the
determinant must be zero, i.e.
For large matrices, this is
done by diagonalizing the matrix.
This involves finding the
matrix U so that
U-1HU
=
E
where E is the matrix of eigenvalues.
Standard computational methods have
been developed for diagonalizing matrices, which we do not need to go
into.
There are 3N eigenvalues (N being
the number of atoms) if Cartesian coordinates are used.