The second derivatives, dE2/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.
See paragraph 2A for another example.
Using eigenvectors in calculations.
This chapter is continued in:
1C: Internal coordinates