Hybrid orbital defined

There exists no uniformity as regards the relations between localized orbitals and molecular symmetry. Consider for example an atomic system consisting of two electrons in an (s) orbital and two electrons in a (2px) orbital, both of which are self-consistent-field orbitals. Since they belong to irreducible representations of the atomic symmetry group, they are in fact the canonical orbitals of this system. Let these two self-consistent-field orbitals be denoted by Cs) and (2p), and let (ft+) and (ft ) denote the two digonal hybrid orbitals defined by... 

Fig. 12.12 JcH for hydrocarbons and s-character of C hybrid orbitals defined by the maximum overlap criterion (see also ref. 156).

The two nonbonding electron pairs may be described in two ways. We may choose to accommodate two electrons in a third hybrid orbital h3 with maximum electron density in the plane of the molecule corresponding to the /i3 hybrid orbital defined by equation (13.5) and the last two electrons in an uhybridized p orbital perpendicular to the molecular plane. The other possibility is to combine hs and p to form two new equivalent hybrids, and hs pointing in approximately tetrahedral directions ... 

Let us return to the hybrid orbitals defined by Equation 2.2, in a quadru-polar field exactly strong enough to make s and px degenerate (Fig. 2.5). It is... 

Suppose you use Po,p-y and p+, along with s orbitals to construct hybrid orbitals. Will they be the same hybrid orbitals defined by thep ,pj and p orbitals Justify your answer. 

It is sometimes convenient to combine aos to form hybrid orbitals that have well defined directional character and to then form mos by combining these hybrid orbitals. This recombination of aos to form hybrids is never necessary and never provides any information that could be achieved in its absence. However, forming hybrids often allows one to focus on those interactions among directed orbitals on neighboring atoms that are most important. 

First, the VB part of the description of benzene. Each C atom is sp2 hybridized, with one electron in each hybrid orbital. Each C atom has a p.-orbital perpendicular to the plane defined by the hybrid orbitals, and it contains one electron. Two sp2 hybrid orbitals on each C atom overlap and form cr-bonds with similar orbitals on the two neighboring C atoms, forming the 120° internal angle of the benzene hexagon. The third, outward-pointing sp2 hybrid orbital on each C atom forms a hydrogen atom. The resulting cr-framework is the same as that illustrated in Fig. 3.20. 

All the atoms of butadiene lie in a plane defined by the s p hybrid orbitals. Each carbon atom has one remaining p orbital that points perpendicular to the plane, in perfect position for side-by-side overlap. Figure 10-42 shows that all four p orbitals interact to form four delocalized molecular orbitals two are bonding MOs and two are antibonding. The four remaining valence electrons fill the orbitals, leaving the two p orbitals empty. 

The significance of term A was defined in section 1.17.1. Factor f is analogous (ratio A/ defines the polarity of the bond) and parameters e and s describe the hybridization state of the orbitals. Because cos e = 0.093 (Duncan and Pople, 1953), molecular hybrid orbitals ) and are composed essentially of the wave functions of atomic orbitals 2p of oxygen and D of hydrogen. The value 0.578 obtained for cos s also indicates that orbitals (0 and are essentially of type sp. Figure 8.1C shows the formation of hybrid MOs... 

A nodal plane or surface is the locus of all points at which a wave-function has zero amplitude as a result of its changing sign on passing from one side of the surface to the other the probability of finding an electron on such a surface is zero. o-Bonds and o-orbitaJs are defined as those having no nodal surface which contains the bond axis such bonds and orbitals will be symmetric about the bond axis. In this section we consider which AOs of a central atom A (which is bonded to a set of other atoms) can be combined to form a hybrid orbital which is symmetric about the bond axis and therefore capable of cr-bonding. 

In symmetries lower than cubic the (/-orbitals mix with the donor atom s—p hybrid orbitals to varying extents in molecular orbitals of appropriate symmetry. However, the mixing is believed to be small and the ligand field treatment of the problem proceeds upon the basis that the effective d-orbitals still follow the symmetry requirements as (/-orbitals should. There will be separations between the MOs which can be reproduced using the formal parameters appropriate to free-ion d-orbitals. That is, the separations may be parameterized using the crystal field scheme. Of course, the values that appear for the parameters may be quite different to those expected for a free ion (/-orbital set. Nevertheless, the formalism of the CFT approach can be used. For example, for axially distorted octahedral or tetrahedral complexes we expect to be able to parameterize the energies of the MOs which house the (/-orbitals using the parameter set Dq, Ds and Dt as set out in Section 6.2.1.4 or perhaps one of the schemes defined in equations (11) and (12). 

Hybrid Orbitals. Orbitals, as one-electron energy levels, and corresponding wavefunctions are mathematical concepts only states are physically observable. Nevertheless, the simple picture of orbitals as the rungs of an energy ladder is very helpful, and is in many cases sufficient to account for the photophysical and photochemical properties of molecules. In more accurate pictures of orbitals it is necessary to consider their interactions, as they are not really totally independent. In this respect the concept of hydrid orbitals is important such hybrid orbitals are formed from a combination of elementary orbitals defined by their quantum numbers n, /, and m. The best... 

To confirm that hi, hi, and hs are equivalent to each other, we can calculate their hybridization indices and see that they are identical. The hybridization index n of a hybrid orbital is defined as... 

FIGURE 4. Basis orbitals for (planar) cycloalkanes with ring size n. Basis orbitals comprise the radially oriented spi and spo hybrid orbitals at C, the tangentially oriented Pip (in plane) and Pop (out of plane) orbitals at C as well as in-phase and out-ofphase combinations of the two 1 s(H) orbitals, which combine with spo t and pop orbitals, respectively. Note that ethene is included as a two-membered ring where the ring is symbolized by two bent bonds and the CH bonds are shown to define the orientation of the ring plane (parallel to the drawing plane). The nature of each orbital set (c or n,... 

For more complex molecules, the v.b. wavefimctions are defined using functions describing each formal covalent bond which are similar to Eq. (8.10) (eventually mixed with ionic contributions). In this process, it is often necessary to consider, not the pure s and p (or s, p, and d) atomic orbitals, but linear combinations of them hybrid atomic orbitals. We have already briefly encountered hybrid orbitals in the study of excited H atoms (for n = 2, for example) and in the m.o. study of BeH2. For instance, in the case of CH4, the 2s atomic orbital and the three 2p a.o.s are replaced by four linear combinations four sp hybrid orbitals. They are equivalent and have symmetry axes that conform to the actual tetrahedral geometry of the molecule, as is illustrated in Fig. 8.5. 

---