Fixed hybrid 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. 

The basis set used is Dunning s cc-pVDZ [16]. The CASSCF wave function was obtained with CAS(4,4). The geometries of the equilibrium and TS structures were determined with this basis set and active space. The orbitals were then localized in the active orbital space. The orbitals were transformed so as to have maximum overlap with two carbon sp2 orbitals and hydrogen Is orbitals. The sp1 orbitals were used with the fixed hybridization ratio of 2s to 2p orbitals (1 2) and with a fixed angle of 120° relative to the CO axis... 

There is still freedom in the choice of atomic orbitals used in Eq. (50). For instance, one can use fixed atomic orbitals, which eliminates the (sometimes costly) orbital optimisation. One can also use fully optimised, potentially delocalised orbitals in the spin-coupled / Coulson-Fischer sense. Finally, one can use real atomic orbitals by limiting each orbital to its own atom. This often gives a clearer physical picture of chemical bonding. It generates for instance optimal hybrids [9,10]. 

Atoms that are linked by electron-pair bonds are positioned so that orbital overlap is maximised. The orbitals used are also sensitive to bond overlap and hybridisation, so that atomic orbitals frequently mix to give hybrid orbitals with greater overlapping power. The shapes of atomic orbitals and hybrid orbitals are quite definite and point in fixed directions. This leads to the fact that covalent bonding is directional. From a geometrical point of view, the array of covalent bonds in a solid resembles a net. 

The carbon-carbon bonds constitute the backbone of a polyolefin chain. In principle, each carbon atom contains four bonds aligning along the tetrahedron sp hybrid orbits with the bond angles fixed at 109°287 In other words, the connection of... 

It should be clear that the idea of fixed, localized hybrid orbitals is simply another approximation used to attempt to rationalize experimental observation. The simplistic qualitative picture presented assumes, for example, that the combination of AOs cannot include fractional contributions to the molecular wave function. As a consequence, as will become clear, angular adjustments must be accounted for qualitatively. It should also be clear that, for the moment, all but the most gross symmetry arguments are ignored. 

Hybrid orbital methods differ from the link atom models in that no extra atoms are introduced into the system. Instead, an atom (usually a tetravalent sp carbon) at the covalent boundary is designed to have both a QM and a MM character. This is typically done by defining a set of hybrid sp orbitals on the atom, some of which are fixed and not included in the QM calculation and the remainder of which are allowed to participate. 

The picture of benzene as a planar framework of ct bonds with six electrons m a delo cahzed rr orbital is a useful but superficial one Six elecfrons cannof simulfaneously occupy any one orbifal be if an afomic orbifal or a molecular orbifal We can fix fhis wifh the more accurate molecular orbital picture shown m Figure 114 We learned m Section 2 4 that when atomic orbitals (AOs) combine to give molecular orbitals (MOs) the final number of MOs musf equal fhe original number of AOs Thus fhe six 2p AOs of SIX sp hybridized carbons combine fo give six tt MOs of benzene... 

The orbitals of the d states in clusters of the 3d, 4d, and 5d transition elements (or in the bulk metals) are fairly localized on the atoms as compared with the sp valence states of comparable energy. Consequently, the d states are not much perturbed by the cluster potential, and the d orbitals of one atom do not strongly overlap with the d orbitals of other atoms. Intraatomic d-d correlations tend to give a fixed integral number of d electrons in each atomic d-shell. However, the small interatomic d-d overlap terms and s-d hybridization induce intraatomic charge fluctuations in each d shell. In fact, a d orbital contribution to the conductivity of the metals and to the low temperature electronic specific heat is obtained only by starting with an extended description of the d electrons.7... 

Electron occupation in the frontier bands of metal crystals varies with different metals as shown in Fig. 2-7. For metallic iron the frontier bands consist of hybridized 4s-3d-4p orbitals, in which 4s and 3d are partially occupied by electrons but 4p is vacant for electrons. Figure 2-8 shows the electron state density curve of metallic iron, where the 3d and 4s bands are partially filled with electrons. Electrons in metals occupy the energy states in a frontier band successively fix>m the lower band edge level to the Fermi level, leaving the higher levels vacant. 

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