Atomic orbital hybrid

We said in Section 1.5 that chemists use two models for describing covalent bonds valence bond theory and molecular orbital theory. Having now seen the valence bond approach, which uses hybrid atomic orbitals to account for geometry and assumes the overlap of atomic orbitals to account for electron sharing, let s look briefly at the molecular orbital approach to bonding. We ll return to the topic in Chapters 14 and 15 for a more in-depth discussion. 

In the molecule Li2 the bond involves a hybrid atomic orbital as+bp formed from the 2s orbital and one of the much less stable 2p orbitals. It is shown below that the amount of p character of this bond orbital (equal to b2, with a2 + b2 = 1) is small, being about 8%. On the other hand, if each of the atoms in metallic lithium requires a bond orbital and a metallic orbital and the two are equivalent they will be 2- -p) and 2 t(s —p), with 50 % p character. The analysis of energy quantities supports this conclusion. 

To truly understand the geometry of bonds, we need to understand the geometry of these three different hybridization states. The hybridization state of an atom describes the type of hybridized atomic orbitals (ip, sp, or sp) that contain the valence electrons. Each hybridized orbital can be used either to form a bond with another atom or to hold a lone pair. 

CS INDO [10] (as well as the parent C INDO [9]) shares the same basic idea as the PCILO scheme [29,30] to exploit the conceptual and computational advantages of using a basis set of hybrid atomic orbitals (AOs) directed along, or nearly, the chemical bonds. 

A carbon atom combining with four other atoms clearly does not use the one 2s and the three 2p atomic orbitals that would now be available, for this would lead to the formation of three directed bonds, mutually at right angles (with the three 2p orbitals), and one different, non-directed bond (with the spherical 2s orbital). Whereas in fact, the four C—H bonds in, for example, methane are known to be identical and symmetrically (tetrahedrally) disposed at an angle of 109° 28 to each other. This may be accounted for on the basis of redeploying the 2s and the three 2p atomic orbitals so as to yield four new (identical) orbitals, which are capable of forming stronger bonds (cf. p. 5). These new orbitals are known as sp3 hybrid atomic orbitals, and the process by which they are obtained as hybridisation ... 

Similar, but different, redeployment is envisaged when a carbon atom combines with three other atoms, e.g. in ethene (ethylene) (p. 8) three sp2 hybrid atomic orbitals disposed at 120° to each other in the same plane (plane trigonal hybridisation) are then employed. Finally, when carbon combines with two other atoms, e.g. in ethyne (acetylene) (p. 9) two sp1 hybrid atomic orbitals disposed at 180° to each other (idigonal hybridisation) are employed. In each case the s orbital is always involved as it is the one of lowest energy level. 

Orbital hybridization A mathematical approach that involves the combining of individual wave functions for s and p orbitals to obtain wave functions for new orbitals => hybrid atomic orbitals... 

The use of MO theory to find deep minima in the So surface, or geometries of stable molecules, is well known. A simplified rule would be to choose the geometry so as to allow efficient overlap of valence orbitals of the constituent atoms in a way giving bonding orbitals for all available electrons from pairs or larger sets of suitably hybridized atomic orbitals. No atomic orbitals occupied by one electron should be left over dangling free and unable to interact with others, since that would give radicals, biradicals, etc. Chemical intuition allows one to proceed almost automatically in cases of molecules of familiar types. 

A set of hybridized atomic orbitals holds the same maximum number of electrons as the set of atomic orbitals from which the hybridized atomic orbitals were formed. A hybridized atomic orbital can hold a maximum of 2 electrons having opposite spin. 

Sigma (O) bond A bond formed by the end to end overlap of pure or hybridized atomic orbitals. 

The use of the Lbwdin orthogonalisation technique (or any other method of or-thogonalisation) means inevitably that the final basis of orthogonalised hybrid atomic orbitals (OHAOs) does contain many-centre orbitals in the sense that each OHAO is mainly its HAO parent but necessarily contains (minimal) contributions from overlapping HAOs. 

The use of hybrid atomic orbitals in qualitative valence theory has, in the past, rested on two points (i) an empirical justification of their use involving the concept of the valence state of an atom and (ii) a simple linear transformation technique for the construction of the explicit forms of the orbitals. In this section we show that both of these points can be replaced. The justification can be replaced by a derivation and this derivation can be used to suggest variational forms which render the linear transformation technique redundant. 

Intermediate values of R have eigenfunctions which are Unear combinations of the usual (complex) hydrogenic orbitals with the same values of n and m hybrid atomic orbitals. The actual values of the linear combination coefficients determining the expUcit form of the hybrids depend on R in the case of n = 2 m = 0 we get an inequivalent pair of hybrids pointing in opposite directions of the general form... 

The trigonal bond orbitals in the ten valence electron system as well as the two sets of trigonal lone pair orbitals in the 14 valence electron system are superpositions of it orbitals and o orbitals. The formation of such trigonally symmetric molecular orbitals from a-type and w-type molecular orbitals is entirely analogous in character to the formation of the three (sp2) hybrid atomic orbitals from one (s) and two ip) atomic orbitals which was discussed in the preceding section. This can be visualized by looking at the diatomic molecule... 

Fig. 6. The radially orientated sp hybrid atomic orbital (AO) and tangentially orientated p AO s that a BH unit can supply for skeletal bonding.

We start with Salem s treatment of the Walden inversion Frontier orbital approximation is assumed the major interaction is supposed to be that between the nucleophile s HOMO and the substrate s LUMO. Now, according to ab initio calculations, the latter is essentially an out-of-phase combination of a carbon hybrid atomic orbital 0c with a leaving group hybrid atomic orbital 0x- In the first approximation, the LUMO wave function may be written as ... 

See Fig. 10-7. The C s use sp hybrid atomic orbitals to form tr bonds with each other and with the H s. The remaining p orbitals at right angles to the plane of the C s overlap laterally to form a tt electron cloud. 

See Fig. 20-1. The four C s and the heteroatom Z use ip -hybridized atomic orbitals to form the a bonds. When Z is O or S, one of the unshared pairs of e s is in an sp HO. Each C has a p orbital with one electron and the heteroatom Z has a p orbital with two electrons. These five p orbitals are parallel to each other and overlap side-by-side to give a cyclic rr system with six p electrons. These compounds are aromatic because six electrons fit Hiickel s 4n + 2 rule, which is extended to include heteroatoms. 

Figure 1.7. The symmetry characteristics of (a) s, (b) p, and (c) spn (hybrid) atomic orbitals. The shapes of the electron distributions are similar if one ignores the phases.

Reed and Allen, using their bond polarity index, have assigned values of 0.000, 0.027, and 0.050, respectively (compared to H —0.032 and F 0.189) [108]. Without attempting to be too quantitative, convenient values of the core energies of hybrid atomic orbitals, in units, recognizing that changes in coordination number also occur, are approximately... 

Of the various methods of approximating the correct molecular orbitals, we shall discuss only one- the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by oneor the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

---