LSCF method

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 first reported approach along these lines was the localized self-consistent-field (LSCF) method of Ferenczy et al. (1992), originally described for the NDDO level of theory. In this case, the auxiliary region consists of a single frozen orbital on each QM boundary atom. 

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. 

For the majority of enzyme-catalysed reactions, covalently bonded parts of the system must be separated into QM and MM regions. There has been considerable research into methods for QM/MM partitioning of covalently bonded systems. Important methods include the local self-consistent field (LSCF) method,114115 and the generalized hybrid orbital (GHO) technique.116 Alternatively a QM atom (or QM pseudo-atom) can be added to allow a bond at the QM/MM frontier for example, the link atom method or the connection atom method. 

Ferre N, X Assfeld, JL Rivail (2002) Specific force field parameters determination for the hybrid ab initio QM/MM LSCF method. J. Comput. Chem. 23 (6) 610-624... 

Ramos et al. [238] have also used the QM/MM LSCF method [312] to understand the reasons why the pancreatic trypsin inhibitor, PTI, behaves as an inhibitor of trypsin rather than as a substrate. In fact, PTI places a peptidic bond between a lysine and an alanine in the catalytic triad of trypsin with the side chain of the lysine in the binding pocket of the enzyme, exactly as a cleavable peptide would do, and this provokes no reaction between PTI and trypsin. The QM/MM calculations performed show that the geometry adopted by the active site of the enzyme in the complex is such that it prevents the nucleophilic attack of the hydroxyl oxygen on the peptide bond of PTI. 

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. 

The local self-consistent field (LSCF) method108 provides a clear and consistent framework for treating the boundary between covalently bonded QM and MM atoms. In the LSCF method, a strictly localized bond orbital, also often described as a frozen orbital, describes the electrons of the frontier bond. This frozen orbital is used at the QM/MM boundary, i.e. for the QM atom at the frontier between QM and MM regions. The electron density of the orbital is... 

Another strategy to separate the QM part from the MM one is to freeze the pair of electrons in the broken bond (assumed to be a single bond). This has been suggested first by Warshel and Levitt [22] and the method has been developed recently at the semiempirical [23,241 and ab initio levels [26-281 as the local self-consistent field (LSCF) method. 

One can thus consider that the LSCF method is universal in the sense that it can be applied to any MM and QM methods, to bonds of any polarity and multiplicity. 

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. 

Figure 5 A schematic diagram of the LSCF method of Rivail and co-workers [29,30,31,32]. Solid line lobes correspond to active orbitals and the dashed lobe is a frozen orbital.

The LSCF method developed here uses strictly localized molecular orbitals which are obtained by using any standard localization procedure [55-59] followed by the removal of the tails of orthogonalization and delocalization, and renormalizing. A study of the results obtained on small reference molecules by using these localization methods has been done [60], and the comparison with the use of extremely localized molecular orbitals (ELMO) [61-63] allows us to assess the reliability of the methodology. The study shows that the results obtained are rather... 

Various possible improvements have been considered. For instance mixing bonding and antibonding bond orbitals [36] led to an Optimized LSCF method [64] in which each SLBO is combined with its corresponding Strictly Localized Anti Bonding Orbital. 

The occupied molecular orbitals allow us to define the Pq density matrix as in any SCF computational scheme. The total density matrix P (equation 28) is then defined and the Fockian (equation 29) can be computed for another cycle of the local self-consistent field (LSCF) method. One notices that this scheme exhibits little difference with the usual SCF computations. In particular, the B matrix plays the same role as the orthogonalization matrix in the usual methods. This computational scheme is therefore rather easy to implement. 

Stewart s LSCF method is quite different in principle from the D C approach, but it is successful for essentially the same reason, namely, that chemical bonding usually persists only over fairly short distances. In the LSCF approach, the system is defined by a set of localized MOs (LMOs), i =, 2,..., N., where N is the total number of atomic orbital basis functions. At the outset, each LMO is a linear combination of atomic orbitals that span only one or two atoms. Occupied LMOs correspond to a bonds, n bonds, and lone pairs that are present in the Lewis dot structure. Each diatomic bonding LMO has an associated orthogonal antibonding LMO that spans the same two atoms. 

---