Description of the Double Bond

Let us examine the structure of the block-diagonal matrix describing orbital interaction within the Hiickel scheme  

3Remember that orthogonal does not mean not interacting (Magnasco, 2004a). 

Since finn fiirin the bond energy of the double bond is lower than the bond energy corresponding to two separate single a bonds. This is the first result obtained from MO theory. 

Since the a and n MOs are mathematical functions obtained in principle as solutions of a differential equation, describing the motion of a single electron in the field provided by nuclei and all remaining electrons, we 

From a geometrical point of view, ft 1 and B2 are obtained by a rotation of 180° around the z axis, but, although the individual form of the MOs is changed, the physical situation is unchanged as we can easily see by evaluating the molecular energy in the new basis. Since the two bond orbitals are equivalent (under reflection across the yz plane) their asso- 

Since the bond energy of the double bond is lower than the 

From a geometrical point of view, B and are obtained by a rotation 

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Our treatment of O2 shows that the extra complexity of the molecular orbital approach explains features that a simpler description of bonding cannot explain. The Lewis structure of O2 does not reveal its two unpaired electrons, but an MO approach does. The simple (t-tt description of the double bond in O2 does not predict that the bond in 2 is stronger than that in O2, but an MO approach does. As we show in the following sections, the molecular orbital model has even greater advantages in explaining bonding when Lewis structures show the presence of resonance. 

When the quantum mechanical equations are examined it is found that the two descriptions of the double bond are identical in the molecular-orbital treatment based on s-p hybrids.76 They are not identical... 

We said in Section L6 that chemists use two models for deseribtng covalent bondfi valence bond theory and molecular orbital Uieoiy- Having now seen a valence bond description of the double bond in ethylene, let s also took at a molecular orbital descTipcion. 

Bi2B22 Bent (or Banana) Description of the Double Bond... 

Figure 2.7 Diagram of the MO orbital energies (S — 0) for the carbon-carbon interaction in the cPn2 description of the double bond (left) and in the equivalent B12B22 bent bond description (right)...

Figure 2.8 Equivalent bent (or banana) bond description of the double bond...

This is true only for the MO approximation. Ab initio VB calculations by Palke (1986) on ethylene show the equivalent banana bond description of the double bond in terms of equivalent nonorthogonal hybrids to be more stable than the a-n description. 

In the cr-71 description of the double bond we must solve the secular equation (2.5) in the basis (tAtB xaXb), assuming S = 0 as before, getting after diagonalization the molecular orbitals ... 

Hence, hybridization increases the angle between the bent bonds. This is the third result given by the quantum mechanical description of the double bond, namely, hybridization increases the angle between the bent bonds, increasing overlap and therefore increasing the bond strength. 

For C2H4 a GVB calculation gave a description of the double bond as composed of one cr and one tt bond, in contrast to the energy-localized MOs (Section 15.10), which are two equivalent bent banana bonds. 

There may be chemists who would contend that one innovation of great significance has been made—the introduction of the ff,n description of the double bond and the triple bond and of conjugated systems, in place of the bent bond description. I contend that the a,n description is less satisfactory than the bent bond description, that this innovation is only ephemeral, and that the use of the a,n description will die out before long.. 

This conclusion was reinforced by a number of investigations. The title of one publication was Double Bonds Are Bent Equivalent Hybrid (Banana) Bonds. Another study concluded that the GVB description of the double bond in (C2F4) is not the traditional picture of o and n bonds but rather a representation in terms of two bent bonds. Still another paper comparing... 

One solution to the use of oversimplified models in organic chemistry is to hybridize complementary conceptual models, just as we hybridize the two Kekule structures for benzene in our minds in order to imderstand and describe aromaticity. The (x,n and bent bond descriptions represent a peiir of models that serve as useful begiiming points or approaches to the description of the double bond. Visualizing a hybrid of these two mental pictures may be more nearly correct than is thinking in terms of either model alone. 

It is interesting that these values for C=C and C=C are within 2 pm of the values that correspond to bent bonds with the normal single-bond length 154 pm and at tetrahedral angles, as indicated in Figure 6-11. This agreement supports the bent-bond description of the double bond and the triple bond. 

The TT-bond description of the double bond agrees well with experiment. The geometric, or cis-trans, isomers of the compound 1,2-dichloroethene (described in the chapter opening) illustrate this. Isomers are compounds of the same molecular formula but with different arrangements of the atoms. (The numbers in the name 1,2-dichloroethene refer to the positions of the chlorine atoms one chlo-... 

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