Numerical Hessian

The total number of energy calculations for a N-Cartesian-coordinate system is 2N^2+1.

If N=3n assuming each particle has 3 Cartesian coordinates, then the number of energy calculations = 2*(3n)^2+1 = 18*n^2+1


The single point energy of the geometry is evaluated first.


Then the coordinates are disturbed in the following ways:


For each diagonal element, disturb only 1 coordinate (x) at a time. Evaluate E(x+dx) and E(x-dx). The 2nd derivative can be derived in the following ways:


E'(x+dx/2)=[E(x+dx)-E(x)]/dx


E'(x-dx/2)=[E(x)-E(x-dx)]/dx


E" = [E'(x+dx/2)-E'(x-dx/2)]/dx


E" = [E(x+dx) + E(x-dx) - 2E(x)]/dx^2


For each off-diagonal elements, disturb 2 coordinates (x and y) at the same time. Evaluate E(x-dx,y-dy), E(x-dx,y+dy), E(x+dx,y-dy), and E(x+dx,y+dy).


The 2nd derivative can be derived in the following way:


E'(x+dx,y) with respect to y = [E(x+dx,y+dy)-E(x+dx,E(y-dy))]/2dy


E'(x-dx,y) with respect to y = [E(x-dx,y+dy)-E(x-dx,E(y-dy))]/2dy


E"(x,y) = {[E(x+dx,y+dy)-E(x+dx,E(y-dy))]/2dy - [E(x-dx,y+dy)-E(x-dx,E(y-dy))]/2dy}/2dx


E"(x,y) = {[E(x+dx,y+dy)-E(x+dx,E(y-dy))] - [E(x-dx,y+dy)-E(x-dx,E(y-dy))]}/4dxdy


E" = [E(x+dx,y+dy) + E(x-dx,y-dy) - E(x-dx, y+dy) - E(x+dx, y-dy)]/4dxdy


Note that E(x,y) is not used in the above off-diagonal element calculation.


In summary, 2N+1 energy calculations need to be done to obtain the N diagonal elements of the energy second derivative matrix (Hessian); 4[N(N-1)/2] energy calculations need to be done to obtain the N(N-1)/2 off-diagonal elements of the Hessian matrix.


2N+1+4[N(N-1)/2]=2N^2+1


For example, a 4-particle system has 12 Cartesian coordinates. 2(12^2)+1=289 energy calculations need to be carried out to obtain the Hessian matrix. A 5-particle system has 15 Cartesian coordinates. 2(15^2)+1=451 energy calculations need to be carried out to obtain the Hessian matrix.


The use of the internal coordinates reduces the number of coordinates by 5 or 6. A linear system has 3n-5 internal coordinates while a non-linear system has 3n-6 internal coordinates, where n is the number of particles.


The number of coordinates can be further reduced if the studied system has a symmetry element.