Other hybridization schemes

For molecular species with other than linear, trigonal planar or tetrahedral-based structures, it is usual to involve d orbitals within valence bond theory. We shall see later that this is not necessarily the case within molecular orbital theory. We shall also see in Chapters 14 and 15 that the bonding in so-called hypervalent compounds such as PF5 and SFg, can be described without invoking the use of J-orbitals. One should therefore be cautious about using sp d hybridization schemes in compounds of / -block elements with apparently expanded octets around the central atom. Real molecules do not have to conform to simple theories of valence, nor must they conform to the sp d schemes that we consider in this book. Nevertheless, it is convenient to visualize the bonding in molecules in terms of a range of simple hybridization schemes. 

Hybridization of v, p,., Py, p, d i and d,.i yi atomic orbitals gives six sp d hybrid orbitals corresponding to an octahedral arrangement. The bonding in M0F5 can be described in terms of sp d hybridization of the central atom. If we remove the z-components from this set (i.e. p and d i) and hybridize only the s, Px, Py and d i yi atomic orbitals, the resultant set of four sp d hybrid orbitals corresponds to a square planar arrangement, e.g. [PtCLi].  

Each set of hybrid orbitals is associated with a particular shape, although this may not coincide with the molecular shape if lone pairs also have to be accommodated  

The mixing of s, Px, py, Pz and d 2 atomic orbitals gives a set of five sp d hybrid orbitals, the mutual orientations of which correspond to a trigonal bipyramidal arrangement (Fig. 5.7a). The five sp d hybrid orbitals are not eqmvalent and divide into sets of two axial and three equatorial orbitals the axial orbital lobes lie along 

The (T-bonding framework in a square-pyramidal species may also be described in terms of an sp d hybridization scheme. The change in spatial disposition of the five hybrid orbitals from trigonal bipyramidal to square-based pyramidal is a craisequence of the participation of a different d orbital. Hybridization of s, Px, Py, Pz and dx2 yp. atomic orbitals generates a set of five sp d hybrid orbitals (Fig. 5.7b). 

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Other hybridization schemes could be used as long as they result in one radially directed and two tangential orbitals... 

The Ni(CO)4 sp hybridization scheme is based on the formation of the four bonding sp orbitals directed towards the comers of a tetrahedron. The prototype example for such bonding is CH4. The two other hybridization schemes of importance are d sp , giving 6 directed orbitals for octahedral coordination, and dsp, given four directed orbitals for a square planar coordination. The latter is characteristic for complexes which contain 16 valence electrons, e.g. PtCl4. In PtCl4 two a electrons are counted for each Cl ion. 

Up to this point, we have discussed only the sp hybridization scheme and used it to describe the bonding in the CH molecule. Many other hybridization schemes have been devised as a way to explain bonding in molecules of practically any shape. Before using the concept of hybridization or introducing other hybridization schemes to describe bonding in other molecules, it will be helpful to emphasize the following points. 

Ethylene is planar with bond angles close to 120° (Figure 2 15) therefore some hybridization state other than sp is required The hybridization scheme is determined by the number of atoms to which carbon is directly attached In sp hybridization four atoms are attached to carbon by ct bonds and so four equivalent sp hybrid orbitals are required In ethylene three atoms are attached to each carbon so three equivalent hybrid orbitals... 

We use different hybridization schemes to describe other arrangements of electron pairs (Fig. 3.16). For example, to explain a trigonal planar electron arrangement, like that in BF, and each carbon atom in ethene, we mix one s-orbital with two /7-orbitals and so produce three sp2 hybrid orbitals ... 

So far, we have not considered whether terminal atoms, such as the Cl atoms in PC15, are hybridized. Because they are bonded to only one other atom, we cannot use bond angles to predict a hybridization scheme. However, spectroscopic data and calculation suggest that both s- and p-orbitals of terminal atoms take part in bond formation, and so it is reasonable to suppose that their orbitals are hybridized. The simplest model is to suppose that the three lone pairs and the bonding pair are arranged tetrahedrally and therefore that the chlorine atoms bond to the phosphorus atom by using sp hybrid orbitals. 

Now consider the alkynes, hydrocarbons with carbon-carbon triple bonds. The Lewis structure of the linear molecule ethyne (acetylene) is H—O C- H. To describe the bonding in a linear molecule, we need a hybridization scheme that produces two equivalent orbitals at 180° from each other this is sp hybridization. Each C atom has one electron in each of its two sp hybrid orbitals and one electron in each of its two perpendicular unhybridized 2p-orbitals (43). The electrons in the sp hybrid orbitals on the two carbon atoms pair and form a carbon—carbon tr-bond. The electrons in the remaining sp hybrid orbitals pair with hydrogen Ls-elec-trons to form two carbon—hydrogen o-bonds. The electrons in the two perpendicular sets of 2/z-orbitals pair with a side-by-side overlap, forming two ir-honds at 90° to each other. As in the N2 molecule, the electron density in the o-bonds forms a cylinder about the C—C bond axis. The resulting bonding pattern is shown in Fig. 3.23. 

White phosphorus, P4, is so reactive that it bursts into flame in air. The four atoms in P4 form a tetrahedron in which each P atom is connected to three other P atoms, (a) Assign a hybridization scheme to the P4 molecule, (b) Is the P4 molecule polar or nonpolar ... 

The reaction between SbF and CsF produces, among other products, the anion [Sb2F7]. This anion has no F—F bonds and no Sb—Sb bonds, (a) Propose a Lewis structure for the ion. (b) Assign a hybridization scheme to the Sb atoms. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Although the discussions of the preceding molecules have been couched in valence bond terms (Lewis structures, hybridization, etc.), recall that the criterion for molecular shape (rule 2 above) was that the cr bonds of the central atom should be allowed to gel as far from each other as possible 2 at 180°. 3 at 120°, 4 at 109.5°, etc. This is (he heart of the VSEPR method of predicting molecular structures, and is, indeed, independent of valence bond hybridization schemes, although it is most readily applied in a VB context. 

FIGURE 3.19 Two other common hybridization schemes, (a) An s- and two p-orbitals can blend together to give three sp2 hybrid orbitals that point to the corners of an equilateral triangle, (b) An s-orbital and a p-orbital hybridize into two sp hybrid orbitals that point in opposite directions. In each case, the hybrid orbital is shown alongside the arrow indicating its location and direction. 

This result implies that, among the four required atomic orbitals (on the central atom), one must have A i symmetry and the other three must form a 7 2 set. From Areas III and IV of the Td character table, we know that the. v orbital has A symmetry, while the three p orbitals, or the d, dv-, and dxz orbitals, collectively form a T2 set. In other words, the hybridization scheme can be either the well-known sp3 or the less familiar sd3, or a combination of these two schemes. 

As for hybrid modeling, the problem of the foundations of MM is seen from a somewhat different perspective. A priori there is no limitation for employing that or any other MM scheme as a classical component of a hybrid model. In practice, however, different MM schemes behave differently when tailored to a QM treated part. Indeed, it is not clear how to handle the bond-dipole based electrostatic energy employed in the MM2 and MM3 schemes, if some bond must be broken, as their ends are expected to be treated by different methods. It applies even more to the schemes with charge equilibration. We shall try to describe the problems created by these inconsistencies as related to the current hybrid methods in the next section, with the analysis of the current state of the art, from the point of view of the general theory of electron variables separation. 

After showing in the previous sections that the real place of the hybrid schemes in quantum chemistry is much more important than one may think, as almost every quantum chemistry method developed so far is hybrid explicitly or implicitly, we turn to a description of the existing hybrid methods understood in the narrow sense namely, those where a part of a system is described by a quantum chemistry method and other parts by MM methods described in previous sections. 

From the point of view of general theory described in Section 1.7.2 the relevant classification of QM/MM methods should be based on an assessment of the level to which the key elements of this theory are treated in that or any other specific hybrid scheme. The authors of [234] made a step in this direction and proposed a classification of hybrid schemes based on the interaction between the quantum and classical fragments. Such a classification of the QM/MM schemes is much more informative. It is built around the hypothetical representation of the total energy of the complex system comprising the quantitatively and classically treated subsystems in the form ... 

This process could be followed for determining the structures of molecules such as CH4, PF5, SF4, SF6, and many others. Figure 2.14 summarizes the common hybridization schemes and geometrical arrangements for many types of molecules. Also shown are the symmetry types (point groups), and these will be discussed later. Figure 2.14 should be studied thoroughly so that these structures become very familiar. 

The computational advantages of such multigrid methods arise from two key factors. First, microscopic simulations are carried out over microscopic length scales instead of the entire domain. For example, if the size of fine grid is 1% of the coarse grid in each dimension, the computational cost of the hybrid scheme is reduced by 10 2rf, compared with a microscopic simulation over the entire domain, where d is the dimensionality of the problem. Second, since relaxation of the microscopic model is very fast, QSS can be applied at the microscopic grid while the entire system evolves over macroscopic time scales. In other words, one needs to perform a microscopic simulation at each macroscopic node for a much shorter time than the macroscopic time increment, as was the case for the onion-type hybrid models as well. 

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