### 1B: Normal coordinates

If the energy can be calculated as a function of the cartesian coordinates, then
the first derivative represents the forces acting upon each atom in each
direction.

In mimima dE/dx (the *gradient*, a vector of length 3N) is zero for every
x.
The second derivatives, dE^{2}/dxdy, are the force constants for every
combination of x and y; they fill a 3N x 3N matrix, the 'Hessian' matrix.

The (3N)^{2} terms of the Hessian matrix are more than the required 3N-6.
The number can be reduced by diagonalization of the matrix. By subsequent
multiplication and adding/subtraction of rows and columns we get a
diagonalized matrix, with the force constants (which are proportional to the
frequencies squared) of the normal vibrations as eigenvalues along the diagonal,
and combinations of cartesian coordinates as eigenvectors.
These combinations are the normal coordinates for a given structure.

In minima the second derivatives are all positive: the energy increases in
whichever direction the atoms move.

In saddle points, transition states, the first derivatives are zero, and all
but one second derivatives (eigenvalues) are positive.
The one negative eigenvalue represents the reaction coordinate.
Only following this normal coordinate the energy will fall, in all other
directions it increases.
This is the way a transition state is characterized: the normal vibrations are
calculated (keyword FORCE in MOPAC or Gamess) and one (and only one)
'negative' (actually imaginary) frequency
should be found. See paragraph 2A for an example.

Example of
FORCE output.

See paragraph 2A for another example.

Using eigenvectors
in calculations.

This chapter is continued in:

1C: Internal coordinates

This is the first chapter: Types of coordinates, z-matrices, input files

Next chapter How to locate a Transition State.

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