MM models

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

The monomer-monomer (MM) model, for the reaction A -H B —> AB, assumes the following Langmuir-Hinshelwood reaction schema ... 

The phase diagram of the MM model is quite simple for I"a< — 1/2 (7a > 7]a) the catalyst becomes poisoned by A (B) species, respectively. Thus one has a first-order IPT where 7ia = 1/2 is a trivial critical point given by the stoichiometry of the reaction. In contrast to the ZGB model. 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

An epidemic analysis of the MM model reveals that the number of empty sites (and the rate of AB production) decays according to Eq. (7) with... 

The MM model with one species desorption (say B) has also been studied [90]. Due to desorption, the B-poisoned state is no longer observed and the system undergoes a second-order IPT between a reactive regime and an A-poisoned state. The behavior of the MM model with one species desorption is similar to another variant of the MM model which incorporates the Eley-Rideal mechanism [57]. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

The properties of the interface between an A-rich and a B-rich patch in the MM model have also been studied [93], The geometry used is the same as... 

Another multiple-reaction irreversible surface reaction process is the dimer-monomer-monomer (DMM) model as proposed in Ref. 99. This model is suitable for investigating, on the one hand, the influence caused by dimer traces in the MM model, and on the other hand the effect of monomer traces in the ZGB model. In fact, the DMM model assumes the following reaction steps ... 

FIG. 20 Plot of the critical points Yq versus Fa for the DMM model. Second order IPTs (v) and first-order IPTs ( and ). ( ) shows the location of the IPT of the MM model. 

Iti Chapter 1, we dealt at length with molecular mechanics. MM is a classical model where atoms are treated as composite but interacting particles. In the MM model, we assume a simple mutual potential energy for the particles making up a molecular system, and then look for stationary points on the potential energy surface. Minima correspond to equilibrium structures. 

However, the active site is only a conceptual tool and the assignment of the active-site atoms is more or less arbitrary. It is not possible to know beforehand which residues and protein interactions that will turn out to be important for the studied reaction. Hybrid QM/MM methods have been used to extend the active site only models by incorporating larger parts of the protein matrix in studies of enzymatic reactions [19-22], The problem to select active-site residues appears both for active-site and QM/MM models, but in the latter, explicit effects of the surrounding protein (i.e. atoms outside the active-site selection) can at least be approximately evaluated. As this and several other contributions in this volume show, this is in many cases highly desirable. 

In the ONIOM(QM MM) scheme as described in Section 2.2, the protein is divided into two subsystems. The QM region (or model system ) contains the active-site selection and is treated by quantum mechanics (here most commonly the density functional B3LYP [31-34]). The MM region (referred to as the real system ) is treated with an empirical force field (here most commonly Amber 96 [35]). The real system contains the surrounding protein (or selected parts of it) and some solvent molecules. To analyze the effects of the protein on the catalytic reactions, we have in general compared the results from ONIOM QM MM models with active-site QM-only calculations. Such comparisons make it possible to isolate catalytic effects originating from e.g. the metal center itself from effects of the surrounding protein matrix. 

Figure 2-14. Illustration of the different hydrogen bonding patterns for an iron-bound peroxide in IPNS using an active-site model (left) and an ONIOM QM MM model (right)...

Our studies on the three enzymes have involved the use of semi-empirical methods, using published and also SRP parameter sets. For both LADH and MADH (Figures 5-3a and b) hybrid QM/MM models were employed [8, 9, 88-90], In LADH the PES surface was calculated at the AMI level [20] but scaled by data from the HF/3-21G surface [91]. The results of the CVT calculation with the SCT correction show quite modest yet contributory degrees of tunnelling, an RTE... 

This section describes the main methodological advances that will be used in subsequent selected applications, including (1) Development of fast semiempirical methods for multiscale quantum simulations, (2) Directions for development of next-generation QM/MM models, and (3) Linear-scaling electrostatic and generalized solvent boundary methods. 

Optimizing a MM model (MOMEC95, force field396) with regard to axial Ni - -11 and Ni C interactions in a set of structurally characterized bis(diamine)nickel(II) complexes has allowed the deduction of a van der Waals radius for low-spin Ni11 of 1.35 A.3... 

The combined QM/MM model can be used along with Statistical Perturbation Theory to carry out a Monte Carlo simulation of a chemical reaction in solution, with the advantage of allowing solute electronic structure relaxation in solution. Particularly, the combined AM1/TIP3P force field has recently been applied to simulate several chemical processes in solution. We will refer here briefly to the Claisen rearrangement and to the Menshutkin reaction. 

Figure 3. Structural representation of the computational models A-F. Models B, D, and E are combined QM/MM models where the regions enclosed in the dotted polygons represent the QM region. The regions outside the polygons are treated by a molecular mechanics force field. For the electronic structure calculation of the QM region, the covalent bonds that traverse the QM/MM boundary (the dotted polygon), have been capped with hydrogen atoms. In model A the atoms labelled 1 through 4 are the atoms that have been fixed in the calculations of models A through E.

Next let us examine a QM/MM model system of complex 3 where the steric influence of the phenyl substituents and of the ferrocene is accounted for but the electronic effects have been largely eliminated (model B). In other words, the peripheral groups have been delegated to the MM region, while keeping the molecular system used for electronic structure calculation identical to that in model A. 

Finally, we have constructed a QM/MM model of complex 3 whereby the phenyl phosphine groups are contained in the QM region, while the ferrocenyl and phenyl substituents on the pyrazole is accounted for on a steric basis only (model E). The agreement between the X-ray structure of 3 and model E is remarkable. For example, both the Pd-P distance and the twist of the coordination plane of the Pd center, 0, are virtually identical to those of the X-ray structure. In fact, the selected parameters displayed in Table 2 are generally better than those of the full QM calculation. The good agreement between the calculated and experimental structures is important for the detailed mechanistic study of the hydrosilylation that is presented in later sections of this chapter. 

In the X-ray structures of both 3 and 4, the tetrahedral distortion is greater than that found in model A. Since there is an electronic preference for a square planar coordination, the pendant phenyl substituents in 3 and 4 likely result in further steric crowding and therefore in more distorted structures compared to model A. In the QM/MM model B an optimized 0 angle of 30° is found, close to the 34° angle of the X-ray structure. Since the phenyl and trimethyl phenyl groups are accounted for on a steric basis only in model B, the result supports the notion that the severely distorted coordination of the Pd center in 3 is due to a steric effect. 

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