# Saddle optimization

Saddle Optimization Method [c.329]

Both the electronic and the geometry optimization problem, particularly the latter, may have more than one solution. For small, rigid molecules, tire approximate molecular geometry is chemically obvious, and the presence of multiple minima is not a serious concern. For large, flexible molecules, however, finding the absolute minimum, or a complete set of low-lying equilibrium structures, is only a partially solved problem. This topic will be discussed in the last section of this chapter. The rest of the article deals with local optimization, i.e., finding a minimum from a reasonably close starting point. We will also discuss the detennination of other stationary points—most importantly saddle points-constrained optimization, and reaction paths. Several reviews have been published on geometry optimization [1, 2]. The optimization of SCF-type wavefiinctions is often highly nonlinear, particularly for the multiconfigurational case, and this has received most attention [3, 4]. [c.2332]

Muller K and Brown L D 1979 Location of saddle points and minimum energy paths by a constrained simplex optimization procedure Theor. Chim. Acta 53 75 [c.2358]

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques [c.253]

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. [c.270]

The subject of force-coupled hamionic oscillation, leading inevitably to normal modes of motion and to the Hessian matr ix, has been developed in far greater detail than other topics in molecular mechanics because the mathematical formalism is basic to almost everything we do in molecular computational chemistr y, extending well beyond the classical mechanics of atomic vibrations. Virtually every mathematical technique described in this book uses some kind of minimization minimization of the error in statistics, minimization of classical mechanical energy in molecular mechanics, or minimization of the electr onic energy in molecular orbital calculations. Less frequently, in the study of reactive intermediates or excited species, a saddle point is sought. The term optimization can be used to include techniques that seek saddle points as well as minima. The term stationary point is used to denote the result of a generalized optimization procedure. [c.143]

Once a PES has been computed, it can be analyzed to determine quite a bit of information about the chemical system. The PES is the most complete description of all the conformers, isomers, and energetically accessible motions of a system. Minima on this surface correspond to optimized geometries. The lowest-energy minimum is called the global minimum. There can be many local minima, such as higher-energy conformers or isomers. The transition structure between the reactants and products of a reaction is a saddle point on this surface. A PES can be used to find both saddle points and reaction coordinates. Figure 20.1 illustrates these topological features. One of the most common reasons for doing a PES computation is to subsequently study reaction dynamics as described in Chapter 19. The vibrational properties of the molecule can also be obtained from the PES. [c.173]

To find a first order saddle point i.e., a transition structure), a maximum must be found in one (and only one) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) being varied. So, a transition structure is characterized by the point where all the first derivatives of energy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. [c.65]

Theory can be used to examine any arrangement of atoms in a molecule. A reaction potential surface can be explored to any desired degree, using the same computational methods as for geometry optimization of reactants and products. In theory, it is possible to determine whether or not a given structure corresponds to a local minimum (stable intermediate) or a saddle point (transition structure). Favored reaction pathways can then be obtained as those involving progression from reactants to products over the lowest-energy transition structures. [c.307]

At both minima and saddle points, the first derivative of the energy, known as the gradient, is zero. Since the gradient is the negative of the forces, the forces are also zero at such a point. A point on the potential eneigy surface where the forces are zero is called a stationary point All successful optimizations locate a stationary point, although not always the one that was intended. [c.40]

Optimization goal (minimum or saddle point] [c.43]

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. [c.70]

One su ested path between the two forms is via a perpendicular transition structure having symmetry. A plausible way to begin an investigation of this reaction is to attempt to locate a saddle point on the potential energy surface corresponding to this hypothesized transition structure. A Hartree-Fock/6-311++G(d,p) calculation succeeds in finding such a transition structure. However, higher level computations using MP2 and QCISD with the same basis set fail to locate a similar stationary point. Instead, these optimizations proceed to a Cj minimum (not a saddle point and thus not a transition structure) in which the hydrogen of the central carbon has migrated to the terminal carbon. This new minimum lies approximately 10 kcal-moT above the equilibrium structure on the potential energy surface. [c.169]

An IRC calculation examines the reaction path leading down from a transition structure on a potential energy surface. Such a calculation starts at the saddle point and follows the path in both directions from the transition state, optimizing the geometry of the molecular system at each point along the path. In this way, an IRC calculation definitively connects two minima on the potential energy surface by a path which passes through the transition state between them. [c.173]

First, we perform an optimization of the transition structure for the reaction, yielding the planar structure at the left. A frequency calculation on the optimized structure confirms that it is a first-order saddle point and hence a transition structure, having a zero-point corrected energy of -113.67941 hartrees. The frequency calculation also prepares for the IRC computation to follow. [c.179]

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. [c.249]

Many problems in computational chemistry can be formulated as an optimization of a multidimensional function/ Optimization is a general term for finding stationary points of a function, i.e. points where tlie first derivative is zero. In the majority of cases the desired stationary point is a minimum, i.e. all the second derivatives should be positive. In some cases the desired point is a first-order saddle point, i.e. the second derivative is negative in one, and positive in all other, directions. Some examples [c.316]

The reaction path formed by a sequence of points generated by constrained optimizations may be discontinuous. For methods where two points are gradually moved from the reactant and product sides (e.g. saddle and LTP), tliis means that tlie distance between end-points does not converge towards zero. [c.332]

The ability of augmented Hessian methods for generating a search toward a first-order saddle point, even when started in a region where the Hessian has all positive eigenvalues, suggests that it may be possible to start directly from a minimum and walk to the TS by following a selected Hessian eigenvector uphill. Such mode followings, however, are only possible if the eigenvector being followed is only weakly coupled to the other eigenvectors (i.e. third and higher derivatives are small). All NR based methods assume that one of the Hessian eigenvectors points in the general direction of the TS, but this is only strictly true when the higher-order derivatives are small. If this is not the case, NR based methods may fail to converge even when started from a good geometry, where the Hessian has one negative eigenvalue. Note also that the magnitude of the higher derivatives depends on the choice of coordinates, i.e. a good choice of coordinates may transform a divergent optimization into a convergent one. [c.335]

A similar method, direct inversion in the iterative subspace (DIIS) [22, 23] tries to minimize the nonn of the error vector (in most cases the gradient) by interpolating in the subspace spanned by the previous vectors. Unlike the CG method, DIIS is able to converge to saddle points. DIIS is now the standard method for die SCF optimization problem. It is also usefiil for geometry optimization [24]. It does not have the conjugate property and therefore requires the storage of previous coordinate and gradient vectors (in practice, usually restricted to about 20 or fewer). However, not using the CG property, which is valid for quadratic surfaces only, probably adds to the stability of the method. [c.2337]

It is also possible to maximize along modes other than the lowest and, in this way perhaps, locate transition states for alternative rearrangements/dissociations from the same initial starting point. For maximization along the Ml mode (instead of the lowest), would be replaced by b, and the sunmiation would now exclude the /rth mode but include the lowest. Since what was originally the Mi mode is the mode along which the negative eigenvalue is required, then this mode will eventually become the lowest mode at some stage of the optimization. To ensure that the original mode is being followed smoothly from one cycle to the next, the mode that is actually followed is the one with the greatest overlap with the mode followed on the previous cycle. This procedure is known as mode following. For more details and some examples, see [ ]. Mode following can work well for small systems, but for larger, flexible molecules there are usually a number of soft modes which lead to transition states for confonnational rearrangements and not to the more interesting reaction saddle points. Moreover, each eigenvector can be followed in two opposite directions and frequently only one leads to a reaction. [c.2352]

To fin d a first order saddle poiri t (i.e., a trail sition structure), a m ax-imiim must be found in on e (and on/y on e) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) bein g varied. So, a tran sition structu re is ch aracterized by th e poin t wh ere all th e first derivatives of en ergy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. [c.65]

The last column in File 5-3 shows that no imaginary frequencies are found in this example. In general, imaginary frequencies are found when optimization settles on a saddle point for a transition from one confomier to another. Because the force constant on a saddle point is negative, it has an imaginary root, leading to an imaginary frequency. Searching the potential energy surface of propene sometimes reveals a saddle point with the double bond eclipsed by one hydrogen of the methyl group as it rotates from one staggered confomiation to another. [c.160]

Variational transition state theory (VTST) is formulated around a variational theorem, which allows the optimization of a hypersurface (points on the potential energy surface) that is the elfective point of no return for reactions. This hypersurface is not necessarily through the saddle point. Assuming that molecules react without a reverse reaction once they have passed this surface [c.166]

An alternative approach to the construction of a smooth reaction pathway with a well-refined transition state is the conjugate peak refinement (CPR) method of Fischer and Karplus [55]. As in the global methods, the path is optimized as a whole and self-consis-tently. However, all points along the path are not treated equally. The computational effort is always directed at bringing the highest energy segment of the path closer to the valley of the energy surface. Starting from some initial guess at the path, a simple set of rules known as a heuristic is applied in each cycle of CPR, when one path point is either added, improved, or removed. This is repeated until the only remaining high energy path points are the acmal saddle points of the transition pathway. [c.217]

Stable=Opt Test the stability of the SCF solution and reoptimize the wavefunction to the lower energy solution if any instability is found. When we speak of optimizing the wavefunction, we are not referring to a geometry optimization, which locates the lowest energy conformation near a specified starting molecular structure. Predicting an SCF energy involves finding the lowest energy solution to the SCF equations. Stability calculations ensure that this optimized electronic wavefunction is a minimum in wavefunction space—and not a saddle point— which is an entirely separate process from locating minima or saddle points on a nuclear potential energy surface. See Appendix A for more details on the internals of SCF calculations. [c.34]

A transition state > 1 imaginary frequency The structure is a higher-order saddle point, but is not a transition structure that connects two minima. QST2 may again be of use. Otherwise, examine the normal modes corresponding to the imaginary frequencies. One of them will (hopefully) point toward the reactants and products. Modify the geometry based on the displacements in the other mode(s), and rerun the optimization. [c.72]

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. [c.147]

As the real function contains tenns beyond second-order, the NR fonmila may be used iteratively for stepping towards a stationary point. Near a minimum, all the Hessian eigenvalues are positive (by definition), and the step direction is opposite to the gradient direction, as it should be. If, however, one of the Hessian eigenvalues is negative, the step in this direction will be along the gradient component, and thus increase the function value. In this case the optimization may end up at a stationary point with one negative Hessian eigenvalue, a first-order saddle point. In general, the NR method will attempt to converge on the nearest stationary point. Another problem is the use of the inverse Hessian for determining the step size. If one of the Hessian eigenvalues becomes close to zero, the step size goes towards infinity (except if the corresponding gradient component / is exactly zero). The NR step is thus without bound, it may take the variables far outside the region where the second-order Taylor expansion is valid. The latter region is often described by a Trust Radius . In some cases the NR step is taken as a search direction along which the function is minimized, analogously to the steepest descent and conjugate gradient methods. The augmented Hessian methods described below are nonnally more efficient. [c.319]

In the Saddle"" algorithm the lowest of the two energy minima, reaetant and product, is first identified. A trial strueture is generated by displaeing the geometry of the low energy species a fraction (for example 0.05) towards the high energy minimum. The trial structure is then optimized, subjeet to the constraint that the distance to the high energy minimum is eonstant. The lowest energy structure on the hypersphere becomes the new interpolation end-point, and the proeedure is repeated. The two geometries will (hopefully) gradually converge on a low energy structure intermediate between the original two minima, as illustrated in Figure 14.7. [c.329]

The Sphere optimization technique is related to the saddle method described in Section 14.5.3, and involves a sequence of constrained optimizations on hyperspheres with increasingly larger radii, using the reactant (or product) geometry as a constant expansion point. The lowest energy point on each successive hypersphere thus traces out a low energy path on the energy surface as illustrated in Figure 14.10. [c.331]

The optimization methods described in Seetions 14.1-14.5 eoncentrate on loeating stationary points on an energy surface. The important points for discussing chemical reactions are minima, corresponding to reactant(s) and product(s), and saddle points, corresponding to transition structures. Once a TS has been located, it should be verified that it indeed connects the desired minima. At the TS the vibrational nonnal coordinate associated with the imaginary frequency is the reaction coordinate (Section 13.1), and an inspection of the corresponding atomic motions may be a strong indication that it is the correct TS. A rigorous proof, however, requires a detemaination of the Minimum Energy Path (MEP) from the TS to tire connecting minima. If the MEP is located in mass-weighted coordinates, it is called the Intrinsic Reaction Coordinate (IRC). The IRC path is of special importance in connection with studies of reaction dynamics, since the nuclei usually will stay close to the IRC, and a model for the reaction surface may be constructed by expanding the energy to second order for example, around points on the IRC (Section 16.2.4). [c.344]

See pages that mention the term

**Saddle optimization**:

**[c.307] [c.395] [c.463] [c.320]**

Introduction to computational chemistry (2001) -- [ c.329 ]