Strictly local bond orbitals

Models of this type are present in the literature. The simplest ones are based on the use of local orbitals. It is the local self-consistent field (LSCF) approach [216,231, 265,266]. In it the chemical bonds between QM and MM regions are represented by strictly local bond orbitals (SLBOs). The BOs can be obtained by the a posteriori localization procedures known in the literature. The localized orbitals thus obtained have some degree of delocalization, i.e. they have non-zero contributions of the AOs centered on the atoms not incident to a given bond (or a lone pair) ascribed to this particular BO. These contributions are the so-called tails of the localized orbitals and neglecting them yields the strictly local BOs (SLBOs) which are used in the LSCF scheme. The QM part of the system is described by a set of delocalized MOs while the boundary is modeled by the frozen SLBOs. 

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... 

If one assumes that the electron pair of the broken bond is described by a localized orbital, the electronic structure of the quantum part can be expanded in a set of molecular orbitals that have to be orthogonal to this localized orbital. The assumption that the properties of the bonds are transferable from one equilibrium structure to another can be applied to this bond, which is assumed to remain unchanged even if the structure of the quantum part is modified. This means that this localized orbital may be approximated by a combination of atomic orbitals centered on the pair of atoms that defines this bond only [i.e., we use a strictly localized bond orbital (SLBO), which can be extracted from a molecular orbital study of a model molecule in which the bond of interest is present]. 

The input requires the usual data of a quantum chemical treatment the starting geometry of the system and the basis set to be used, plus the L strictly localized bond orbitals expanded in the basis set chosen for the computation. 

Fig. 1.1 Schematic representations of the three types of QM-MM junctions for the frontier bond X-Y where the X atom is in the QM part and the Y atom is in the MM part. The quantum part is depicted in Balls and Sticks representation and the MM one in Sticks only, a Link Atom (LA) approach, b Connection Atom (CA) method, c Frozen density approach, here a Strictly Localized Bond Orbital is depicted in blue...

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. 

One of the basic assumptions of the NDDO semi-empirical methods is the orthogonality of the atomic orbitals. Therefore, if one describes the bond between the quanffim and the classical part by a strictly localized bond orbital (SLBO), i.e. a localized orbital free of localization tails [35] hosting 2 electrons, the molecular orbitals of the quanffim part are orthogonal to the localized one and are able to describe correctly the chemical group of interest. This property has been put forward by Warshel and Levitt in a basic paper [36]. 

We assume that the quantum subsystem is linked to the classical one by L single bonds to which L strictly localized bond orbitals (SLBO) developed in the basis set used for the quantum computation are attached. Without any loss of generality we can assume that these functions are orthogonal... 

The last method we shall quote has been developed by Th6ry et al. It is based on somewhat different principles. The bond connecting QM and MM portions is described in terms of a strictly localized bond orbital, as given in equation (13). The hybrid orbital /ia is taken from previous calculations with a model molecule, or more simply treated as a pre-defined parameter the same holds for coefficient ca-... 

BOVB Breathing orbital valence bond. A VB computational method. The BOVB wave function is a linear combination of VB structures that simultaneously optimizes the structural coefficients and the orbitals of the structures and allows different orbitals for different structures. The BOVB method must be used with strictly localized active orbitals (see HAOs). When all the orbitals are localized, the method is referred to as L-BOVB. There are other BOVB levels, which use delocalized MO-type inactive orbitals, if the latter have different symmetry than the active orbitals. (See Chapters 9 and 10.)... 

The QM and MM regions are bound with a strictly localized molecular orbital (SLMO) (see Fig. 1). This is realized by defining an MM host (MMH) atom (which is also known as frontier atom in the literature) at the border of the QM and MM subsystems so that it is connected to an QM host (QMH) atom with a bond orbital whose basis functions are located exclusively on the two atoms. The QMH atom contributes to the SLMO with one electron, and its other electrons are part of the optimized wave function. The MMH atom of the SLMO also contributes with a single electron to the SLMO, while its other valence electrons are not treated explicitly. 

The localized bond orbitals ,) are assumed to be strictly localized orbitals (SLO) i.e., to be expanded on atomic orbitals of atoms X, and Y, only. 

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-... 

Although the theory behind BLW is more general, a typical application of the method is the energy calculation of a specific resonance structure in the context of resonance theory. As a resonance structure is, by definition, composed of local bonds plus core and lone pairs, a bond between atoms A and B will be represented as a bonding MO strictly localized on the A and B centers, a lone pair will be an AO localized on a single center, and so on. With these restrictions on orbital extension, the SCF solution can be... 

In the previous section we used quaternions to construct a convenient parameterization of the hybridization manifold, using the fact that it can be supplied by the 50(4) group structure. However, the strictly local HOs allow for the quaternion representation for themselves. Indeed, the quaternion was previously characterized as an entity comprising a scalar and a 3-vector part h = (h0, h) = (s, v). This notation reflects the symmetry properties of the quaternion under spatial rotation its first component ho = s does not change under spatial rotation i.e. is a scalar, whereas the vector part h — v — (hx,hy,hz) expectedly transforms as a 3-vector. These are precisely the features which can be easily found by the strictly local HOs the coefficient of the s-orbital in the HO s expansion over AOs does not change under the spatial rotation of the molecule, whereas the coefficients at the p-functions transform as if they were the components of a 3-dimensional vector. Thus each of the HOs located at a heavy atom and assigned to the m-th bond can be presented as a quaternion ... 

The BOVB method is aimed at combining the qualities of interpretability and compactness of valence bond wave functions with a quantitative accuracy of the energetics. The fundamental feature of the method is the freedom of the orbitals to be different for each VB structure during the optimization process. In this manner, the orbitals respond to the instantaneous field of the individual VB structure rather than to an average field of all the structures. As such, the BOVB method accounts for the differential dynamic correlation that is associated with elementary processes like bond forming/breaking, while leaving the wave function compact. The use of strictly localized orbitals ensures a maximum correspondence between the wave function and the concept of Lewis structure, and makes the method suitable for calculation of diabatic states. 

The above best calculation [11] corresponds to the simplest level of the BOVB method, referred to as L-BOVB. All orbitals, active and inactive, are strictly local, and the ionic structures are of closed-shell type, as represented in 10 and 11. However the theory can be further improved, and the corresponding levels are displayed in Table 2. It appears that the L-BOVB/6-31+G level, yields a fair bonding energy, but an equilibrium distance that is rather too long compared to sophisticated estimations. This is the sign of an incomplete description of the bond. Indeed this simpler level does not fully account for the... 

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