Locating Saddlepoints
With the advent of modern computational techniques, it has become possible to exhaustively search the potential energy surface of individual molecules containing fewer than about 12 rotatable bonds (i.e., degrees of freedom for the dihedral angles that define molecular geometry, along with bond lengths and bond angles, in internal coordinate space) when classical (molecular mechanics-based) potential energy functions are employed. The previous section has broadly described the issues and various algori thmic techniques for finding local and global minima on such complex multidimensional energy surfaces. This section focuses on another aspect of conformational searches: the identification of saddlepoints and their connection to chemical reactions.
In addition to a description of the conformational properties of individual molecules, the potential energy surface can be employed to describe the energetics of chemical reactions. Therefore, searches on the potential energy hypersurface of a molecule can extend to molecular reactions as well (Eksterowicz and Houk, 1993). Reactants and products correspond to energy minima, whereas transition states linking products to reactants usually correspond to first-order saddlepoints on the energy surface (alth ough unusual symmetries can produce higher-order transition states, including those of the "monkey-saddle" type). Thus, the location of stationary points (particularly minima and saddlepoints) on potential energy surfaces represents an important and chal lenging problem in computational chemistry.
In chemical applications, special conformational-space search methods have been devised for locating minima on molecular mechanics-based potential energy surfaces. These methods include stochastic (Saunders, 1987; Chang et al., 1989; Ferguson and Raber , 1989) and deterministic, grid-based (Motoc et al., 1986; Lipton and Still, 1988; Dammkoehler et al., 1989) approaches. Yet, with rare exception (Kolossvary and Guida, 1993), conformational searches have not been performed in such a way that saddlepoint s are located. Nonetheless, the utility and indeed necessity of determining the conformational transition states that link these minima have recently been emphasized (Anet, 1990). Whereas in the past, conformational searches have been synonymous with loc ation of energy minima, it is clear that in order to adequately study the conformational properties of molecules it is essential to locate first-order saddlepoints as well.
Significant effort has addressed the problem of locating transition states on potential energy surfaces derived from quantum mechanics calculations. A number of algorithms have been developed such as those that rely on eigenvector-following techniques (Cerjan and Miller, 1981; Simons et al., 1983, 1984; Bell and Crighton, 1984; Simons, 1985; Baker, 1986). In these methods one begins a saddlepoint search at or near a local minimum that is found by standard minimization techniques. A spectral decompos ition is performed to find all the normal modes of the system (mass scaled eigenvalues and associated eigenvectors of the Hessian matrix (i.e., second-derivative matrix of the potential energy); then one of the normal modes is selected and followed in an "uphill" direction (i.e., a direction that leads to an increase of potential energy) until a saddlepoint is located. Evaluation of the energy gradient and Hessian matrix at each step of the search is performed until a point on the surface is located at which the gradient is zero and the Hessian possesses only one negative eigenvalue. In another approach, the linear synchronous transit method (Halgren and Lipscomb, 1977) has been employed to aid in the location of saddlepoints. I t locates a maximum along a path connecting two structures and thus can be used to provide an initial guess for the transition state structure that connects them. Methods that find the location of saddlepoints by beginning the search at points on the pot ential energy surface that are of higher energy than the saddlepoint one wishes to locate have also been described (Berry et al., 1988). Recent developments (Jorgensen et al., 1988; Culot et al., 1992) have led to improved efficiency in locating transiti on states in calculations based on quantum mechanics-derived potential energy surfaces. Nonetheless, the aforementioned saddlepoint searches sometimes fail to converge, or they converge to critical points that are minima. Clearly, more robust algorithms are still needed, and this is an area that mathematical optimizers may find very interesting.
It is conceivable that algorithms for locating transition states on potential energy surfaces derived from calculations based on quantum mechanics could be employed for the location of conformational transition states on molecular mechanics-derived pote ntial energy surfaces once the minima have been located. However, these algorithms have generally been used to study mechanisms of chemical reactions and have not been adequately tested for locating such conformational transition structures. In a typical conformational search procedure, the potential energy surface is scanned randomly or systematically and a large number of trial structures are generated for energy optimization. These structures can be severely "distorted" geometrically in the sense that bond lengths and angles lie out of the ranges observed experimentally, and van der Waals radii of atoms may overlap. These structures must then be optimized by the standard, "greedy" descent methods of local minimization toward a local minimum or toward a saddlepoint. However, for the quantum chemical calculation of a reaction mechanism, the reactant and product are usually known, and uphill movement toward the interconnecting saddlepoint is sought.
Conformational search procedures that locate first-order saddlepoints and minima with equal efficiency would be of enormous utility. Even though advances in this area have been slow, some progress has been achieved. For example, the so-called self-pen alty walk method (Czerminski and Elber, 1990) provides an example of an algorithm for the calculation of reaction paths in complex molecular systems when molecular mechanics-derived potential energy functions are employed. However, it is likely that addi tional work will be required to develop methods for the efficient conformational searching of saddlepoints. New algorithms for conformational searches in which first-order saddlepoints are efficiently located are clearly urgently needed.
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