second derivative matrix of the system, , 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. Using this computational model and an initial guess (or ansatz) of the correct geometry, an iterative optimization procedure is followed, for example: calculate the force on each atom (that is, ) if the force is less than some threshold, finish otherwise, move the atoms by some computed step that is predicted to reduce the force repeat from the start Practical aspects of optimization As described above, some method such as quantum mechanics can be used to calculate the energy, , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, and the second derivative matrix of the system, , also known as the Hessian matrix, which describes the curvature of the PES at . An optimization algorithm can use some or all of , and to try to minimize the forces and this could in theory be any method such as gradient descent, conjugate gradient or Newton's method, but in practice, algorithms which use knowledge of the PES curvature, that is the Hessian matrix, are found to be superior. For most systems of practical interest, however, it may be prohibitively expensive to compute the second derivative matrix, and it is estimated from successive values of the gradient, as is typical in a Quasi-Newton optimization. The choice of the coordinate system can be crucial for performing a successful optimization. Cartesian coordinates, for example, are redundant since a non-linear molecule with atoms has vibrational degrees of freedom whereas the set of Cartesian coordinates has dimensions. Additionally, Cartesian coordinates are highly correlated, that is, the Hessian matrix has many non-diagonal terms that are not close to zero. This can lead to numerical problems in the optimization, because, for example, it is difficult to obtain a good approximation to the Hessian matrix and calculating it precisely is too computationally expensive. However, in case that energy is expressed with standard force fields, computationally