The Link Atom Problem

3 The Link Atom Problem. - As mentioned earlier, the question of how to describe the boundary between the quantum and classical regions is hotly debated. Following the example given by Bakowies and Thiel,142 consider a bimolecular system X+ + Y The question of how to partition this system is trivial. X+ may be treated quantum mechanically and Y may be treated classically, or vice versa. However the partitioning of a covalently bonded, unimolecular system, X-Y, is more difficult as none of the obvious fragments, X+ + Y, X + Y or X- + Y+ accurately describe the electron distribution of either X or Y as part of the whole system X-Y. 

In addition to the hybridised orbital approach, Zhang et al 52 have proposed a pseudobond approach. The boundary atom of the MM region is replaced by a single free valence carbon atom, Cps. Each C, has 7 electrons, a nuclear charge of 7 and an effective core potential parameterised to mimic an sp2 hybridised orbital between it and the QM boundary atom, thereby forming a sort of 

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Slavicek P, Martinez TJ (2006) Multicentered valence electron effective potentials solution to the link atom problem for ground and excited electronic states. J Chem Phys 124 084107... 

To circumvent problems associated with the link atoms different approaches have been developed in which localized orbitals are added to model the bond between the QM and MM regions. Warshel and Levitt [17] were the first to suggest the use of localized orbitals in QM/MM studies. In the local self-consistent field (LSCF) method the QM/MM frontier bond is described with a strictly localized orbital, also called a frozen orbital [43]. These frozen orbitals are parameterized by use of small model molecules and are kept constant in the SCF calculation. The frozen orbitals, and the localized orbital methods in general, must be parameterized for each quantum mechanical model (i.e. energy-calculation method and basis set) to achieve reliable treatment of the boundary [34]. This restriction is partly circumvented in the generalized hybrid orbital (GHO) method [44], In this method, which is an extension of the LSCF method, the boundary MM atom is described by four hybrid orbitals. The three hybrid orbitals that would be attached to other MM atoms are fixed. The remaining hybrid orbital, which represents the bond to a QM atom, participates in the SCF calculation of the QM part. In contrast with LSCF approach the added flexibility of the optimized hybrid orbital means that no specific parameterization of this orbital is needed for each new system. 

The phase problem and the problem of arbitration. Fibrous structures are usually made up of linear polymers with helical conformations. Direct or experimental solution of the X-ray phase problem is not usually possible. However, the extensive symmetry of helical molecules means that the molecular asymmetric unit is commonly a relatively small chemical unit such as one nucleotide. It is therefore not difficult to fabricate a preliminary model (which incidently provides an approximate solution to the phase problem) and then to refine this model to provide a "best" solution. This process, however, provides no assurance that the solution is unique. Other stereochemically plausible models may have to be considered. Fortunately, the linked-atom least-squares approach provides a very good framework for objective arbitration independent refinements of competing models can provide the best models of each kind the final values of n or its components (eqn. xxiv) provide measures of the acceptability of various models these measures of relative acceptability can be compared using standard statistical tests (4) and the decision made whether or not a particular model is significantly superior to any other. 

Such an approach also seems to be quite artificial. An example of employing it is given in [250] (cited in [245]) where it is shown that strong deviations of the link atom equilibrium position from the line connecting atoms forming a covalent bond are possible and lead to serious problems. Also, the vibrational spectra calculated with the optimization of the link atom position are much worse than those derived... 

The dummy junction atom or link atom approach introduces so-called link atoms to satisfy the valence of the atoms on the QM side of the QM/MM interface. Usually this atom is a hydrogen, but other atom types have also been used, e.g. halogens such as fluorine or chlorine. The link atom method can be used with both the Warshel-type QM/MM methods and ONIOM methods. The link atom method has been criticised because it introduces extra unphysical atoms to the system, which come with associated extra degrees of freedom. Another problem is that a C-H bond is clearly not chemically exactly equivalent to a C-C bond. Despite these problems, the simplicity of the link atom method means that it is used widely in the QM/MM modelling of proteins and other biological molecules. ... 

Unfortunately QM/MM potentials are not devoid of problems. The most severe ones are probably the division of covalent bonds across the QM and MM regions and the lack of explicit polarisation of the MM approach. The first of these two difficulties has been looked at by several groups who have proposed different schemes to deal with the problem Warshel and Levitt [299] have used a single hybrid orbital on the MM atom in the QM/MM region a similar approach has been proposed subsequently by Rivail and co-workers [312, 355, 373] with their frozen orbital (or excluded orbital) in which the continuity between the two critical regions is assured by a strictly localised bond orbital (SLBO) obtained from model compounds. Another popular approach introduces link atoms [300, 310, 315] between QM and MM covalently bonded atoms to cap the valency of the QM atoms the link atoms, usually hydrogen, do not interact with the MM atoms. These are not, by any means, the only ways of dealing with this problem. However, so far it does not seem to have an obvious solution. 

Another problem lies in the region between QM and MM. Considerable debate has arisen when the partitioning between the two regions cuts across covalent bonds [58,62,63]. The most common method in modeling this middle space is to use the link atom method. It consists of adding QM hydrogen atoms in order to fill the free valencies of the QM atoms that are connected to the atoms described by MM. These dummy atoms are explicitly treated during the QM calculations but do not interact with the MM atoms. Whether or not these link atoms should interact via Coulombic interactions is still open to debate. 

At the same time such an approach seems quite artificial. Moreover, in Ref. [136] (cited by Ref. [121]) it is stated that the strong deviation of the link atom equilibrium position from the line connecting atoms forming covalent bond leads to serious problems. Moreover, the vibrational spectra calculated by the optimization of the link atom position method are worse than even the MM-force field derived. Also the QM/MM calculated proton affinity for small gas phase aluminosilicate clusters is very sensitive to the length of the bond between boundary QM atom and the hydrogen atom introduced [137]. The problems with positioning of link atoms are... 

When solving the problem for the QM region, the link atoms substitute the ones that stay next to the BA in the MM region. Since included in the bonding (in the zero-order energy Eqm° of Eq. (5)), these substituted atoms should be excluded from the perturbation energy Eqm. Thus with the link atoms included in the zero-order approximation, we get instead of Eqs. (20) and (21) ... 

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