Atomic orbitals, combining hybridization

An answer was provided in 1931 by Linus Pauling, who showed how an s orbital and three p orbitals on an atom can combine mathematically, or hybridize, to form four equivalent atomic orbitals with tetrahedral orientation. Shown in Figure 1.10, these tetrahedrally oriented orbitals are called sp3 hybrids. Note that the superscript 3 in the name sp3 tells how many of each type of atomic orbital combine to form the hybrid, not how many electrons occupy it. 

The number of hybrid orbitals obtained equals the number of atomic orbitals combined. 

An answer waa provided in 1931 by Linus Pauling, who showed mathematically how an a orbital and three p orbitala on an atom can combine, or hybridiva, to form four e<]uival [ic atomic orbitals with tetrahedral ori-ematlon. Shown In Figure l.ll. these Tcimhcdrally oncmed orbitals are called sp hybrids. (The superscript 3 in the nmne indicates that threep atomic orbitals combine to form the hybrid, not that 3 electrons occupy it>... 

Hybrid orbital (Section 1.8B) A new orbital that results from the mathematical combination of two or more atomic orbitals. The hybrid orbital is intermediate in energy compared to the atomic orbitals that were combined to form it. 

When we discussed sp hybrid orbitals in Section 1.7, we said that all four of carbon s valence-shell atomic orbitals combine to form four equivalent sp hybrids. Imagine instead that the 2s orbital combines with only two of the three 2p orbitals. Three sp hybrid orbitals result, and one 2p orbital remains unchanged. The three sp orbitals lie in a plane at angles of 120° to one another, with the remaining p orbital perpendicular to the sp plane, as shown in Figure 1.14. 

In order to account for the molecular structures, retaining two-electron bonds and at the same time utilizing the excessive numbers of orbitals available it has been supposed that three or more atomic orbitals combine to form only one bonding orbital. The structures of the simpler boranes may be formulated with 3-centre bonds of three kinds. (Fig. 24.19). In the central (closed) bond, (a), the three B atoms use hybrid orbitals and are situated at the corners of an equilateral triangle. 

Instead of atomic orbitals or hybrids, which have the disadvantage of not being orthogonal to one another, use is often made of linear combinations there of, made orthogonal to one another by the procedure of Landshofi generalized by Lowdin which can be symbolically written as ... 

The number of hybrid orbitals obtained equals the number of atomic orbitals combined. Lone pairs of electrons take up more space than bonding pairs of electrons, thus they compress the bond angles. 

Molecular shapes provide a motivation to consider an extension to the concept of atomic orbitals combining to form molecules. For a molecule such as water, if only p orbitals are available, the bond angle would have to be 90°. But that is not the observed angle, so we must have additional flexibility, and hybrid orbitals provide that. 

The combination of one 2 5 atomic orbital and two 2p atomic orbitals forms three equivalent sp hybrid orbitals (Figure 1.18). Because they are derived from three atomic orbitals, sp hybrid orbitals always occur in sets of three. The third 2p atomic orbital (remember 2p 2py, and 2pP) is not involved in hybridization and consists of two lobes lying perpendicular to the plane of the hybrid orbitals [Figure 1.18(c)]. 

The combination of one 2s atomic orbital and one 2p atomic orbital forms two equivalent sp hybrid orbitals. Because they are derived from two atomic orbitals, sp hybrid orbitals always occur in sets of two (Figure 1.21). 

Hybrid orbitals are formed by combinations of atomic orbitals, a process called hybridization. Mathematically, this is accomplished by combining the wave functions of the 2s > 2) arid three 2p >p2p/ ip > ip) orbital wave functions. The number of hybrid orbitals formed is equal to the number of atomic orbitals combined. Elements of the second period form three types of hybrid orbitals, designated sp, sp, and sp, each of which can contain up to two electrons. 

The mathematical combination of one 2s atomic orbital wave function and two 2p atomic orbital wave functions forms three equivalent sp hybrid orbital wave functions. Because they are derived from three atomic orbitals, sp hybrid orbitals always occur in sets of three. As with sp orbitals, each sp hybrid orbital (three-dimensional plot of (f/) consists of two lobes, one larger than the other. The axes of the three sp hybrid orbitals lie in a plane and are directed toward the corners of an equilateral triangle the angle between sp hybrid orbitals is 120°. The third 2p atomic orbital (remember 2p, 2p 2p) is not involved in hybridization (its wave function is not mathematically combined with the other three) and remains as two lobes lying perpendicular to the plane of the sp hybrid orbitals. Figure 1.14 shows three equivalent sp orbitals along with the remaining unhybridized 2p atomic orbital. Each sp orbital has 33% s-character and 67% p-character (one 2s orbital, two 2p orbitals). 

As in our earlier discussion of s hybridization (p. 55), we will create three new hybrid orbitals that will do a better job of attaching the carbon atom to the three hydrogens than do the pure atomic orbitals. Three hybrid orbitals are needed, and our mathematical operations will therefore involve combining three wave functions (atomic orbitals) to produce the three new hybrid orbitals. Recall that in such quantum mechanical calculations the number of orbitals created always equals the number of orbitals combined in the calculation, here three. So, let s combine the carbon 2s, 2p and 2py orbitals to produce three new, s-f" hybrids. The 2p orbital unused in our calculation remains, unhybridized and waiting to be incorporated in our picture (Fig. 3.3). Remember that the choice of 2p orbitals to be combined is arbitrary. Any pair of orbitals can be used, leaving the third left over. 

Figure 9.4 (a) An s atomic orbital and two p atomic orbitals combine to form three sp hybrid orbitals, (b) The three sp hybrid orbitals on B are arranged in a trigonal plane. (Empty atomic orbitals are shown in white.) (c) Hybrid orbitals on B overlap with 2p orbitals on F. 

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