Hybrid schemes

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... 

One more hybridization scheme is important m organic chemistry It is called sp hybridization and applies when carbon is directly bonded to two atoms as m acetylene The structure of acetylene is shown m Figure 2 18 along with its bond distances and bond angles Its most prominent feature is its linear geometry... 

It should also be remembered that the discretization scheme influences the accuracy of the results. In most CFD codes, different discretization schemes can be chosen for the convective terms. Usually, one can choose between first-order schemes (e.g., the first-order upwind scheme or the hybrid scheme) or second-order schemes (e.g., a second-order upwind scheme or some modified QUICK scheme). Second-order schemes are, as the name implies, more accurate than first-order schemes. However, it should also be remembered that the second-order schemes are numerically more unstable than the first-order schemes. Usually, it is a good idea to start the computations using a first-order scheme. Then, when a converged solution has been obtained, the user can continue the calculations with a second-order scheme. 

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 ... 

Some of the elements in Period 3 and later periods can accommodate five or more electron pairs, as in PCI,. We can devise a hybridization scheme to describe... 

FIGURE 3.16 Three common hybridization schemes shown as outlines of the amplitude of the wavefunction and in terms of the orientations of the hybrid orbitals, (a) An s-orbital and a p-orbital hybridize into two sp hybrid orbitals that >oint in opposite direc tions, forming a linear molecular shape, (b) An s-orbital and two p-orbitals can blend together to give three ip hybrid orbitals that point to the corners of an equilateral triangle, (c) An s-orbital and three p-orbitals can blend together to give four sp hybrid orbitals that point to the corners of a tetrahedron. 

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. 

EXAMPLE 3.5 Sample exercise Assigning a hybridization scheme What is the hybridization of sulfur in PF5 ... 

A hybridization scheme is adopted to match the electron arrangement of the molecule. Valence-shell expansion requires the use of d-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. 

STRATEGY Use the VSEPR model to identify the shape of the molecule and then assign the hybridization consistent with that shape. All single bonds are cr-bonds and multiple i bonds are composed of a cr-bond and one or more TT-bonds. Because the C atom is attached to three atoms, we anticipate that its hybridization scheme is sp1 and that one unhybridized p-orbital remains. Finally, we form cr- and Tr-bonds by allowing the 1 orbitals to overlap. 

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. 

Complexes of d- and /-block metals can be described in terms of hybridization schemes, each associated with a particular shape. Bearing in mind that the number of atomic orbitals hybridized must be the same as the number of hybrid orbitals produced, match the hybrid orbitals sp1d, sp fd , and sp d3f to the following shapes (a) pentagonal bipyramidal ... 

Self-Test 19.2B What are the hybridization schemes of the C and N atoms in formamide, HCONH2 ... 

Mingos DMP, Zhenyang L (1990) Hybridization Schemes for Coordination and Organometallic Compounds. 72 73-112... 

The hybridization schemes match the shapes that we describe in Chapter 9. Notice that the steric numbers of inner atoms uniquely determine both the electron group geometry and the hybridization. This makes sense, because the steric number describes how many electron groups must be accommodated around an inner atom. 

As stated in Sect. 2.3, hybrid schemes suffer from the dilemma that using semiempirical methods gives a wrong description of the primary system whereas ab initio methods are so expensive that the affordable number of time steps does not allow proper statistical-mechanical sampling. For selected systems, however, there is a possibility of using very accurate methods and still executing many time steps. The requirements on the system are that the internal conformational space of the primary system is finite and that the primary system can be described by only a few active degrees of freedom. 

The best results are obtained with the hybrid schemes, the standard B3LYP functional and the new, one-parameter PBE1PBE protocol. Of course, one must keep in mind that independent of the particular functional chosen, large and flexible basis sets must be used. 

Hybrid MPC-MD schemes may be constructed where the mesoscopic dynamics of the bath is coupled to the molecular dynamics of solute species without introducing explicit solute-bath intermolecular forces. In such a hybrid scheme, between multiparticle collision events at times x, solute particles propagate by Newton s equations of motion in the absence of solvent forces. In order to couple solute and bath particles, the solute particles are included in the multiparticle collision step [40]. The above equations describe the dynamics provided the interaction potential is replaced by Vj(rJVs) and interactions between solute and bath particles are neglected. This type of hybrid MD-MPC dynamics also satisfies the conservation laws and preserves phase space volumes. Since bath particles can penetrate solute particles, specific structural solute-bath effects cannot be treated by this rule. However, simulations may be more efficient since the solute-solvent forces do not have to be computed. 

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. 

Abstract ONIOM is a flexible hybrid scheme that can combine the most suitable computational... 

The data obtained has been derived from different experimental designs (namely, the number of collected fractions and the hybridization scheme), different microarray platforms (spotted versus Affymetrix arrays), and different data analysis schemes. In this chapter, we discuss various experimental designs for microarray analyses, provide detailed protocols for various experimental steps when using the spotted microarray platform, and discuss aspects of data analysis. 

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