efficient methods have been developed able to derive analytically the Hessian matrix in Cartesian coordinates while preserving a computational complexity of the same order to that of gradient computations. Internal coordinates tend to be less correlated but are more difficult to set-up and it can be difficult to describe some systems, such as ones with symmetry or large condensed phases. Many modern computational chemistry software packages contain automatic procedures for the automatic generation of reasonable coordinate systems for optimization. Degree of freedom restriction Some degrees of freedom can be eliminated from an optimization, for example, positions of atoms or bond lengths and angles can be given fixed values. Sometimes these are referred to as being frozen degrees of freedom. Figure 1 depicts a geometry optimization of the atoms in a carbon nanotube in the presence of an external electrostatic field. In this optimization, the atoms on the left have their positions frozen. Their interaction with the other atoms in the system are still calculated, but alteration the atoms' position during the optimization is prevented. Transition state optimization Transition state structures can be determined by searching for saddle points on the PES of the chemical species of interest. A first-order saddle point is a position on the PES corresponding to a minimum in all directions except one; a second-order saddle point is a minimum in all directions except two, and so on. Defined mathematically, an nth order saddle point is characterized by the following: and the Hessian matrix, , has exactly n negative eigenvalues. Algorithms to locate transition state geometries fall into two main categories: local methods and semi-global methods. Local methods are suitable when the starting point for the optimization is very close to the true transition state (very close will be defined shortly) and semi-global methods find application when it is sought to locate the transition state with very little a priori knowledge of its geometry. Some methods, such as the Dimer method (see below), fall into both categories. Local searches A so-called local optimization requires an initial guess of the transition state that is very close to the true transition state. Very close typically means that the initial guess must have a corresponding Hessian matrix with one negative eigenvalue, or, the negative eigenvalue corresponding to the reaction coordinate must be greater in magnitude than the other negative eigenvalues. Further, the eigenvector with the most negative eigenvalue must correspond