The definition of critical points

The gradient vector of the density in the Cartesian coordinate system i, j, and k is defined as 

At a critical point, Vp(r) equals zero because each of the three contributions to Eq. (6.18) are zero. The classification of critical points is based on the second derivatives, which as noted above are all negative for a density maximum, but have different signs for saddle points and minima of the distribution. 

The Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution  

In an isolated molecule, or cluster of atoms, the Poincare-Hopf relationship 

In an assembly of molecules such as a molecular crystal, there are minima, that is, pits, in the voids between the molecules, and the Poincare-Hopf relation is replaced by the Morse equation (Johnson 1992) 

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