It is often possible to gain new insight into the properties of a molecular system by expressing them in terms of stationary points of the PES, i.e. points where the gradient of the potential vanishes [37,9]. Such a coarse-grained picture may be appropriate if the system spends most of its time in the vicinity of these points and the properties of interest can be expressed in terms of the properties of these points only. In realistic applications it may also be the only way forward, as the corresponding PES's are usually complex.

The most important stationary points are minima and the transition states that connect them. Here we define a minimum as a stationary point where the Hessian, the second derivative matrix, has no negative eigenvalues, while a transition state is a stationary point with precisely one such eigenvalue [38].

The number of stationary points on the PES generally scales exponentially with system size [43,41,40,42,39], which necessitates an appropriate sampling strategy of some sort for larger systems. In particular, to analyse kinetic properties a representative sample is usually obtained, which generally involves extensive use of single-ended and double-ended transition state searching techniques [9,7,30].

Locating transition states on a PES also provides an important tool in the study of dynamics using statistical rate theories [47,44,48,46,45]. Unfortunately, it is significantly harder to locate transition states than local minima, since the system must effectively `balance on a knife-edge' in one degree of freedom. Many algorithms have been suggested for this purpose, and the most efficient method may depend upon the nature of the system. For example, different considerations probably apply if second derivatives can be calculated relatively quickly, as for many empirical potentials [9]. Transformation to an alternative coordinate system may also be beneficial for systems bound by strongly directional forces [51,52,50,56,55,53,54,49,57].

Single-ended transition state searches [67,70,12,66,23,20,54,85,86,17,63,75,78,69,77,79,19,73,81,76,15,62,71,82,18,68,14,74,72,59,21,64,60,65,83,13,61,16,22,84,58,80] only require an initial starting geometry. The result of a single-ended search may be a transition state that is not connected to the starting point by a steepest-descent path, and such methods can be useful for building up databases of stationary points to provide a non-local picture of the potential energy surface, including thermodynamic and dynamic properties [9,86]. However, double-ended searches [107,93,89,91,101,102,87,105,53,88,100,97,94,28,104,106,29,27,98,96,95,90,92,99,103] require two endpoint geometries, a mechanism to generate a set of configurations between them, and a suitable functional (or gradient) to be evaluated and minimised. The most successful single- and double-ended methods currently appear to be based upon hybrid eigenvector-following [17,18,12,14,24,13,19,16,22,21,23,20,15] and the nudged elastic band approach [29,7,108,109,110,27,30], respectively. The two search types are often used together, since double-ended transition state searches do not produce a tightly converged transition state and further refinement may be needed [9,7].