2 Force Field Methods
2.10 Hybrid Force Field Electronic Structure Methods
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.
additional terms corresponding to stretching and bending interactions. The mechani- cal embedding model is rarely a useful level of approximation, as the wave function of the QM region does not respond to changes in the MM region.
The next level of improvement is called electronic embedding, where the atoms in the MM regions are allowed to polarize the QM region. Partial charges on the MM atoms can be incorporated into the QM Hamiltonian analogously to nuclear charges (i.e. adding Vne-like terms to the one-electron matrix elements in eq. (3.56)), and the QM atoms thus feel the electric potential due to all the MM atoms.
(2.45) The non-bonded mechanical term in eq. (2.44) is still needed in order to prevent the MM atoms from drifting into the QM region. The electronic embedding allows the geometry of MM atoms to influence the QM region, i.e. the wave function in the QM region becomes coupled to the MM geometry. An interesting computational issue arises when the number of MM atoms is large and the QM region is small, since the calculation of the one-electron integrals associated with VQM/MMmay become a domi- nating factor, rather than the two-electron integrals associated with the QM region itself, but in most cases the inclusion of the VQM/MMterm only marginally increases the computational effort over a mechanical embedding.
A further refinement, often called polarizable embedding, can be made by allowing the QM atoms also to polarize the MM region, i.e. the electric field generated by the QM region influences the MM electric moments (atomic charges and dipoles). This of course requires that a polarizable force field is employed (Section 2.2.7), and necessi- tates a double iterative procedure for allowing the electric fields in both the QM and MM regions to be determined in a self-consistent fashion. This substantially increases the computational cost, and since polarizable force fields are not yet commonly used anyway, most QM/MM methods employ the electronic embedding approximation. An exception is the effective fragment method, often used for modelling solvation, where both quadrupoles and polarizabilities are included for the MM atoms.62
In many cases, the QM and MM regions belong to the same molecule, and the divi- sion between the two parts must be done by cutting one or more covalent bonds. This leaves one or more unpaired electrons in the QM part, which must be properly ter- minated. In most cases, the dangling bonds are terminated by adding “link” atoms, typ- ically a hydrogen. For semi-empirical methods, it can also be a pseudo-halogen atom with parameters adjusted to provide a special link atom.63Alternatively, the termina- tion can be in the form of a localized molecular or generalized hybrid orbital.64At present, there does not seem to be a clear consensus on whether one or the other approach provides the best results, but the link atom method is somewhat simpler to implement. When the link atom procedure is used, the link atom(s) is only present in the QM calculation, and is not seen by the MM framework. A number of choices must also be made for which and how many of the MM bend and torsional terms that involve one or more QM atoms are included. Bending terms involving two MM and one QM atoms are usually included, but those involving one MM and two QM atoms may be neglected. Similarly, the torsional terms involving only one QM atom are usually included, but those involving two or three QM atoms may or may not be neglected.
VQM MM R r
MM Atoms
= −
∑
− Qa a i aN
The concept of mixing methods of different accuracy has been generalized in the ONIOM(our own n-layered integrated molecular orbital molecular mechanics) method to include several (usually two or three) layers, for example using relatively high-level theory in the central part, a lower level electronic structure theory in an intermediate layer and force field to treat the outer layer.65 The original ONIOM method only employed mechanical embedding for the QM/MM interface, but more recent exten- sions have also included electronic embedding.66 The ONIOM method employs an extrapolation scheme based on assumed additivity, in analogy to the CBS, Gnand Wn methods discussed in Section 5.7. For a two-layer scheme, the small (model) system is calculated at both the low and high levels of theory, while the large (real) system is calculated at the low level of theory. The result for the real system at the high theo- retical level is estimated by adding the change between the high and low levels of theory for the model system to the low level results for the real system, as illustrated in Figure 2.22 and eq. (2.46).
System size
levellaciteroehT
Ehigh(real)
Elow(real) Elow(model)
Ehigh(model)
Figure 2.22 Illustration of the ONIOM extrapolation method
(2.46) A similar extrapolation can be done for multi-level ONIOM models, although it requires several intermediate calculations. It should be noted that derivatives of the ONIOM model can be constructed straightforwardly from the corresponding deriva- tive of the underlying methods, and it is thus possible to perform geometry optimiza- tions and vibrational analysis using the ONIOM energy function.
QM/MM methods are often used for modelling solvent effects, with the solvent treated by MM methods, but in some cases the first solvation shell is included in the QM region. If such methods are used in connection with dynamical sampling of the configurational space, it is possible that MM solvent molecules can enter the QM regions, or QM solvent molecules can drift into the MM region. In order to handle such situations, there must be a procedure for allowing solvent molecules to switch between a QM and MM description. In order to ensure a smooth transition, a transi- tion region can be defined between the two parts, where a switching function is employed to make a continuous transition between the two descriptions.67
The main problem with QM/MM methods is that there is no unique way of decid- ing which part should be treated by force field and which by quantum mechanics, and QM/MM methods are therefore not “black box” methods. The “stitching” together of
E E
E E
ONIOM high level
low level low level
real system, high level model system model system real system
( )= ( )−
( )+ ( )
the two regions is certainly not unique, and the many possible combinations of force field and QM methods make QM/MM methods still somewhat experimental. Fur- thermore, the inability to perform calibration studies of large systems by pure QM methods makes it difficult to evaluate the severity of the approximations included in QM/MM methods.
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