 Geometry optimization is then a mathematical optimization problem, in which it is desired to find the value of for which is at a local minimum, that is, the derivative of the energy with respect to the position of the atoms, , is the zero vector and the second derivative matrix of the system, formula_1, also known as the Hessian matrix, which describes the curvature of the PES at , has all positive eigenvalues (is positive definite).A special case of a geometry optimization is a search for the geometry of a transition state; this is discussed below.The computational model that provides an approximate could be based on quantum mechanics (using either density functional theory or semi-empirical methods), force fields, or a combination of those in case of QM/MM.