Strictly localized orbitals

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. 

Now we pass to the formal derivations of a hybrid method. We assume that the orbitals forming the basis for the entire molecular system may be ascribed either to the chemically active part of the molecular system (reactive or R-states) or to the chemically inactive rest of the system (medium or M-states). In the present context, the orbitals are not necessarily the basis AO, but any set of their orthonormal linear combinations thought to be distributed between the subsystems. The numbers of electrons in the R-system (chemically active subsystem) Nr and in the M-system (chemically inactive subsystem) NM = Ne — Nr, respectively, are good quantum numbers at least in the low energy range. We also assume that the orbital basis in both the systems is formed by the strictly local orbitals proposed in [59]. The strictly local orbitals are orthonormalized linear combinations of the AOs centered on a single atom. In that sense they are the classical hybrid orbitals (HO) ... 

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. 

Loos PF, Assfeld X (2007) Self-consistent strictly localized orbitals. J Chem Theory Comput 3 1047-1053... 

Fomili A, Moreau Y, Sironi M, Assfeld X (2006) On the suitability of strictly localized orbitals for hybrid QM/MM calculations. J Comput Chem 27 515-523... 

Our QM/MM code uses frozen strictly localized orbitals to separate the covalently bound QM and MM subsystems. Since this method has been described in detail in Ref. [13], here only a summary of the most important features of the method are recapitulated. 

Automatic generation of frozen strictly localized orbitals... 

Loos, P.-R, Assfeld, X. (2007). Self-consistent strictly localized orbitals. Journal of Chemical Theory and Computation, 3,1047. 

CQFF = classical quantum force field CTC = charge transfer complex LSCF = local self-consistent field SLO = strictly localized orbitals. 

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. 

Apart from choosing the force field for the classical subsystem and the level for quantum computations, the only latitude left to the user is the choice of the frontier bond characteristics. These may be extracted from a computation on a model molecule containing the bond of interest. In such a case, only one orbital needs to be localized so that one can use a so-called external localization criterion which is based upon a property of the bond of interest. The Weinstein-Pauncz criterion, which maximizes the overlap population between the two atoms defining the bond, appears to be quite convenient for this purpose. This localized orbital usually contains minor contributions of atoms other than the X and Y atoms. In order to obtain a strictly localized orbital, these contributions are discarded and the orbital renormalized. If the reference frame is different from the frame used for the molecule on which the LSCF computation is being performed, a simple matrix transformation generates the SLO expression in the LSCF basis set and the contribution of the orbital to the Pl density matrix is straightforward. 

Enol ethers (Figure 2-58a) have two electron pairs on the oxygen atom in two different orbitals, one delocalized across the two carbon atoms, the other strictly localized on the oxygen atom (Figure 2-58b). Ionization ftom either of these two orbitals is associated with two quite different ionization potentials, a situation that cannot be handled by the present connection tables. 

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-... 

The reason for this becomes apparent when one compares the shapes of the localized it orbitals with that of the ethylene 7r orbital. All of the former have a positive lobe which extends over at least three atoms. In contrast, the ethylene orbital is strictly limited to two atoms, i.e., the ethylene 7r orbital is considerably more localized than even the maximally localized orbitals occurring in the aromatic systems. This, then, is the origin of the theoretical resonance energy the additional stabilization that is found in aromatic conjugated systems arises from the fact that even the maximally localized it orbitals are still more delocalized than the ethylene orbital. The localized description permits us therefore to be more precise and suggests that resonance stabilization in aromatic molecules be ascribed to a "local delocalization of each localized orbital. One infers that it electrons are more delocalized than a electrons because only half as many orbitals cover the same available space. It is also noteworthy that localized it orbitals situated on joint atoms (n 2, it23, ir l4, n22 ) contribute more stabilization than those located on non-joint atoms, i.e. the joint provides more paths for local delocalization. 

OEOs), thereby performing GVB or SCVB calculations. The orbitals can also be defined by pairs that are allowed to delocalize on only two centers (BDOs), or they can be defined as strictly localized on a single center or fragment (see below). The VBSCF method is implemented in the TURTLE module (now being a part of GAMESS—UK) and in the XMVB package. 

The BOVB method has several levels of accuracy. At the most basic level, referred to as L-BOVB, all orbitals are strictly localized on their respective fragments. One way of improving the energetics is to increase the number of degrees of freedom by permitting the inactive orbitals to be delocalized. This option, which does not alter the interpretability of the wave function, accounts better for the nonbonding interactions between the fragments and is referred to... 

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... 

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.)... 

HAO Hybrid atomic orbitals that are strictly localized on a single atomic center. The HAOs have no delocalization tails. 

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. 

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 ... 

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