Hybrid frozen orbital

Figure 1 The hybrid frozen orbital and the three hybrid orbitals included in the quantum computation.

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. 

Separation of covalently bonded atoms into QM and MM regions introduces an unsatisfied valence in the QM region this can be satisfied by several different methods. In the frozen-orbital approach a strictly localized hybrid sp2 bond orbital containing a single electron is used at the QM/MM junction [29]. Fro-... 

A subtle but key difference in the methodologies is that the orbital containing the two electrons in the C-X bond is frozen in the LSCF method, optimized in the context of an X-H bond in the link atom method, and optimized subject only to the constraint that atom C s contribution be a particular sp hybrid in the GHO method. In the link atom and LSCF methods, the MM partial charge on atom C interacts with some or all of the quantum system in the GHO method, it is only used to set the population in the frozen orbitals. 

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. 

There are several versions of the HO method. Warshel and Levitt used a hybrid orbital approximation in their landmark hybrid potential study [15]. They do not, however, seem to have used it subsequently. More recently, Rivail and coworkers have developed their local self consistent field (LSCF) method for use with semiempirical and ab initio HF/MM hybrid potentials [29, 30, 31, 32]. In the LSCF scheme, the atomic orbitals on the boundary atom are replaced by four sp hybrid orbitals. One of these, the strictly localized bond orbital (SLBO), is constructed so that it points along the broken QM/MM bond towards the MM atom. This orbital is frozen out of the QM calculation and so has a constant form but the other three orbitals are optimized in the HF calculation. Rivail et al have obtained parameters for the frozen orbital by performing calculations on model systems. An illustration of the LSCF method is shown in figure 5. 

The local self-consistent field (LSCF) or fragment SCF method has been developed for treating large systems [105,134-139], in which the bonds at the QM/MM junction ( frontier bonds ) are described by strictly localized bond orbitals. These frozen localized bond orbitals are taken from calculations on small models, and remain unchanged in the QM/MM calculation. The LSCF method has been applied at the semiempirical level [134-137], and some developments for ab initio calculations have been made [139]. Gao et al. have developed a similar Generalized Hybrid Orbital method for semiempirical QM/MM calculations, in which the semiempirical parameters of atoms at the junction are modified to enhance the transferability of the localized bond orbitals [140]. Recent developments for ab initio QM/MM calculations include the method of Phillip and Friesner [141], who use Boys-localized orbitals in ab initio Hartree-Fock QM/MM calculations. These orbitals are again taken from calculations on small model systems, and kept frozen in QM/MM calculations. 

Finally, the third class of approaches encompasses all methods dealing with frozen electronic density (see Fig. 1.1c). Generally, the electronic density is obtained from orbitals (hybrid orbitals or localized molecular orbitals) determined on small molecules which contain the bond of interest [9,19,20], It is then possible to cut bonds of any polarity (P-0 in DNA for example), or multiphcity. It is even possible to cnt peptide bond, which represent a serious advantage for the study of proteins. The universality of these methods is however accompanied by an inherent coding complexity. Among these methods, the Local Self-Consistent Field approach (LSCF) developed in our group since more than fifteen years is detailed in the next section. 

Fomili A, Loos P-F, Sironi M, Assfeld X (2006) Frozen core orbitals as an alternative to specific frontier bond potential in hybrid quantum mechanics/molecular mechanics methods. Chem Phys Lett 427 236V240... 

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