It is obvious that depending on the energy expression the derivative will be an affordable task or not. There are available methods to calculate the first, the second and sometimes higher order derivatives of the energy.
In general there are two ways to calculate the energy derivatives, numerically and analytically.
The gradient component of a molecular system with a geometry can be approximated as:
An improvement to the equation 1.67 is displacing the coordinate forward and backward. 2.4
The same argument can be used for second derivatives. In this case we have diagonal and off-diagonal
elements of the Hessian matrix:
The quality of the derivative depends on the magnitude of and the accuracy of potential energy . (See section 3.1.1.2 for the numerical quantities of these magnitudes and some practical recommendations to decrease the high computational cost of numerical derivatives)