2 Force Field Methods
2.9 Transition Structure Modelling
Structural changes can be divided into two general types: those of a conformational nature and those involving bond breaking/forming. There are intermediate cases, such as bonds involving metal coordination, but since metal coordination is difficult to model anyway, we will neglect such systems at present. The bottleneck for structural changes is the highest energy point along the reaction path, called the Transition State or Transition Structure(TS) (Chapter 13). Conformational TS’s have the same atom types and bonding for both the reactant and product, and can be located on the force field energy surface by standard optimization algorithms. Since conformational changes are often localized to rotation around a single bond, simply locating the maximum energy structure for rotation (so-called “torsional angle driving”, see Section 12.4.1) around this bond often represents a good approximation of the real TS.
Modelling TS’s for reactions involving bond breaking/forming within a force field methodology is much more difficult.53 In this case, the reactant and product are not described by the same set of atom types and/or bonding. There may even be a differ- ent number of atoms at each end of the reaction (for example lone pairs disappear- ing).This means that there are two differentforce field energy functions for the reactant and product, i.e. the energy as a function of the reactant coordinate is not continuous.
Nevertheless, methods have been developed for modelling differences in activation energies between similar reactions by means of force field techniques, and three approaches are described below.
2.9.1 Modelling the TS as a minimum energy structure
One of the early applications of TS modelling was the work on steric effects in SN2 reactions by DeTar and coworkers, and it has more recently been advocated by Houk and coworkers.54The approach consists of first locating the TS for a typical example of the reaction with electronic structure methods, often at the Hartree–Fock or density functional theory level. The force field function is then modified such that an energy minimum is created with a geometry that matches the TS geometry found by the elec- tronic structure method. The modification defines new parameters for all the energy terms involving the partly formed/broken bonds. The stretch energy terms have natural bond lengths taken from the electronic structure calculation, and force constants that are typically half the strength of normal bonds. Bond angle terms are similarly
modified with respect to equilibrium values and force constants, the former taken from the electronic structure data and the latter usually estimated. These modifications often necessitate the definition of new “transition state” atom types. Once the force field parameters have been defined, the structure is minimized as usual. Sometimes a few cycles of parameter adjustments and re-optimizations are necessary for obtaining a set of parameters capable of reproducing the desired TS geometry. P.-O. Norrby has described a partly automated method for simultaneously optimizing all the parame- ters to reproduce the reference structure.55When the modified force field is capable of reproducing the reference TS geometry, it can be used for predicting TS geometries and relative energies of reactions related to the model system. As long as the differ- ences between the systems are purely “steric”, it can be hoped that relative energy dif- ferences (between the reactant and the TS model) will correlate with relative activation energies. Purely electronic effects, such as Hammett-type effects due to para-substitu- tion in aromatic systems, can of course not be modelled by force field techniques.
2.9.2 Modelling the TS as a minimum energy structure on the reactant/product energy seam
There are two principal problems with the above modelling technique. First, the TS is modelled as a minimum on the energy surface, while it should be a first-order saddle point. This has the consequence that changes in the TS position along the reaction coordinate due to differences in the reaction energy will be in the wrong direction (Section 15.6). In many cases, this is probably not important. For reactions having a reasonable barrier, the TS geometry appears to be relatively constant, which may be rationalized in terms of the Marcus equation (Section 15.5). Comparing reactions that differ in terms of the steric hindrance at the TS, however, may be problematic, as the TS changes along the reaction coordinate will be in the wrong direction. The second problem is the more or less ad hocassignment of parameters. Even for quite simple reactions, many new parameters must be added. Inventing perhaps 40 new parameters for reproducing maybe five relative activation energies raises the nagging question as to whether TS modelling is just a fancy way of describing five data points by 40 variables.56
Both of these problems are eliminated in the intersecting potential energy surface modelling technique called SEAM.57The force field TS is here modelled as the lowest point on the seam of the reactant and product energy functions, as shown in Figure 2.21. Locating the minimum energy structure on the seam is an example of a con- strained optimization; the energy should be minimized subject to the constraint that the reactant and product energies are identical. Although this is computationally some- what more complicated than the simple minimization required in the Houk approach, it can be handled in a quite efficient manner.
In the SEAM approach only the force field parameters for describing the reactant and products are necessary, alleviating the problem of assigning parameters specific for the TS. Furthermore, differences in reactivity due to differences in reaction energy or steric hindrance at the TS are automatically included. The question is how accu- rately the lowest energy point on the seam resembles the actual TS. This is difficult to evaluate rigorously as it is intimately connected with the accuracy of the force field used for describing the reactant and product structures. It is clear that the TS will have
bond distances and angles significantly different from equilibrium structures. This method of TS modelling therefore requires a force field that is accurate over a much wider range of geometries than normal. Especially important is the stretch energy, which must be able to describe bond breaking. A polynomial expansion is therefore not suitable, and for example a Morse function is necessary. Similarly, the repulsive part of the van der Waals energy must be fairly accurate, which means that Lennard- Jones potentials are not suitable and should be replaced by, for example, Buckingham- type potentials. Furthermore, many of the commonly employed cross terms (Section 2.2.8) become unstable at long bonds lengths and must be modified. When such mod- ifications are incorporated, however, the intersecting energy surface model appears to give surprisingly good results.
There are of course also disadvantages in this approach: these are essentially the same as the advantages! The SEAM method automatically includes the effect of dif- ferent reaction energies, since a more exothermic reaction will move the TS toward the reactant and lower the activation energy (Section 15.5). This, however, requires that the force field be able to calculate relative energies of the reactant and product, i.e. the ability to convert steric energies to heat of formation. As mentioned in Section 2.2.10, there are only a few force fields that have been parameterized for this. In prac- tice, this is not a major problem since the reaction energy for a prototypical example of the reaction of interest can be obtained from experimental data or estimated. Using the normal force field assumption of transferability of heat of formation parameters, the difference in reaction energy is thus equal to the difference in steric energy. Only the reaction energy for a single reaction of the given type therefore needs to be esti- mated and relative activation energies are not sensitive to the exact value used.
If the minimum energy seam structure does not accurately represent the actual TS (compared for example with that obtained from an electronic structure calculation) the lack of specific TS parameters becomes a disadvantage. In the Houk approach, it is fairly easy to adjust the relevant TS parameters to reproduce the desired TS geom- etry. In the intersecting energy surface method, the TS geometry is a complicated result Figure 2.21 Modelling a transition structure as a minimum on the intersection of two potential energy surfaces
of the force field parameters for the reactant and product, and the force field energy functions. Modifying the force field parameters, or the functional form of some of the energy terms, in order to achieve the desired TS geometry without destroying the description of the reactant/product, is far from trivial. A final disadvantage, which is inherent to the SEAM method, is the implicit assumption that all the geometrical changes between the reactant and product occurs in a “synchronous” fashion, albeit weighted by the energy costs for each type of distortion. “Asynchronous” or “two- stage” reactions (as opposed to two-step reactions that involve an intermediate), where some geometrical changes occur mainly before the TS, and others mainly after the TS, are difficult to model by this method.
Since the TS is given in terms of the diabatic energy surfaces for the reactant and product, it is also clear that activation energies will be too high. For evaluating rela- tive activation energies of similar reactions this is not a major problem since the impor- tant aspect is the relative energies. The overestimation of the activation energy can be improved by adding a “resonance” term to the force field, as discussed in the next section.
2.9.3 Modelling the reactive energy surface by interacting force field functions or by geometry-dependent parameters
Within a valence bond approach (Chapter 7), the reaction energy surface can be con- sidered as arising from the interaction of two diabaticsurfaces. The adiabaticsurface can be generated by solving a 2 ×2 secular equation involving the reactant and product energy surfaces,Erand Ep.
(2.42)
A. Warshel has pioneered the Extended Valence Bond (EVB) method,58 where the reactant and product surfaces are described by force field energy functions, and Truhlar has more recently generalized the approach by the Multi-Configurations Molecular Mechanics(MCMM) method.59In either case, the introduction of the interaction term Vgenerates a continous energy surface for transforming the reactant into the product configuration, and the TS can be located analogously to energy surfaces generated by electronic structure methods. The main drawback of this method is the somewhat arbi- trary interaction element, and the fact that the TS must be located as a first-order saddle point, which is significantly more difficult than locating minima or minima on seams. It can be noted that the SEAM method corresponds to the limiting case where V→0 in the EVB method.
Another way of creating a continous surface connecting the reactant and product energy functions is to make the force field parameters dependent on the geometry, which is an approach used in the ReaxFFmethod.60The force constant for stretching a bond, for example, should decrease and approach zero as the bond length increases towards infinity. The energy function in this case depends directly on the atomic coordinates via the energy term in eq. (2.3), but also indirectly via the geometry
E E V
V E E
E E Ep E E V
r
p
r r p
−
− =
=
[
( + )− ( + ) +]
0
1 4
2
2 2
dependence of the parameters.Achieving a smooth and realistic variation of the energy with geometry requires quite elaborate interpolation functions, which makes the para- meterization non-trivial.