Scheme 1.2. Molecular Polarizability a
1.2. Molecular Orbital Theory and Methods
1.2.1. The Hückel MO Method
The Hückel MO (HMO) method was very important in introducing the concepts of MO theory into organic chemistry. The range of molecules that the method can treat is quite limited and the approximations are severe, but it does provide insight into a number of issues concerning structure and reactivity. Furthermore, the mathe- matical formulation is simple enough that it can be used to illustrate the nature of the calculations. The HMO method is restricted to planar conjugated systems such as polyenes and aromatic compounds. The primary simplification is that only the 2pz orbitals are included in the construction of the HMOs. The justification is that many of the properties of conjugated molecules are governed by theorbitals that arise from thepz atomic orbitals. A further approximation of the HMO calculations is that only adjacentpzorbitals interact. This allows construction of mathematical formulations for theMOs for such systems as linear and branched-chain polyenes, cyclic polyenes, and fused-ring polyenes. For conjugated linear polyenes such as 1,3,5-hexatriene, the energy levels are given by the equation
E=+mj (1.15)
wheremj=2 cosj/n+1forj=123 n, withnbeing the number of carbon atoms in the conjugated polyene.
The quantity is called the Coulomb integral; it represents the binding of an electron in a 2pzorbital and is considered to be constant for allsp2carbon atoms. The
28
CHAPTER 1 Chemical Bonding and Molecular Structure
quantity is called the resonance integral and represents the energy of an electron distributed over two or more overlapping 2pzorbitals. For linear polyenes, this equation generates a set of HMOs distributed symmetrically relative to the energyassociated with an isolated 2pz orbital. The contribution of each atomic orbital to each HMO is described by a coefficient:
Crj= 2
n+1 1/2
sin rj
n+1
(1.16) Figure 1.7 gives the resulting HMOs for n=2 to n=7. Table 1.7 gives the coefficients for the HMOs of 1,3,5-hexatriene. From these coefficients, the overall shape of the orbitals can be deduced and, in particular, the location of nodes is determined. Nodes represent an antibonding contribution to the total energy of a particular orbital. Orbitals with more nodes than bonding interactions areantibonding and are above the referenceenergy level. The spacing between orbitals decreases with the length of the polyene chain, and as a result, the gap between the HOMO and LUMO decreases as the conjugated chain lengthens.
These coefficients give rise to the pictorial representation of the 1,3,5-hexatriene molecular orbitals shown in Figure 1.8. Note in particular the increase in the energy of the orbital as the number of nodes goes from 0 to 5. The magnitude of each atomic coefficient indicates the relative contribution at that atom to the MO. In1, for example, the central atoms C(3) and C(4) have larger coefficients than the terminal atoms C(1) and C(6), whereas for3the terminal carbons have the largest coefficients.
The equation for the HMOs of completely conjugated monocyclic polyenes is
E=+mj (1.17)
wheremj=2 cos2j/n for j=0±1±2 n−1/2 for n=odd and (n/2) for n=even. This gives rise to the HMO diagrams shown in Figure 1.9 for cyclic polyenes withn=3 ton=7. Table 1.8 gives the atomic coefficients for benzene,n=6, and Figure 1.10 gives pictorial representations of the MOs.
There is an easy way to remember the pattern of MOs for monocyclic systems.
Figure 1.11 shows Frost’s circle.41 A polygon corresponding to the ring is inscribed
2 3 4 5 6 7
. . .
α
Fig. 1.7. HMO orbital diagram for polyenesn=2 ton=7.
41 A. A. Frost and B. Musulin,J. Chem. Phys.,21, 572 (1953).
29
SECTION 1.2 Molecular Orbital Theory and Methods
Table 1.7. Energy Levels and Atomic Coefficients for HMOs of 1,3,5-Hexatriene
orbitalmj c1 c2 c3 c4 c5 c6
1 1802 02319 04179 05211 05211 04179 02319
2 1247 04179 05211 02319 −02319 −05211 −04179
3 0445 05211 02319 −04179 −04179 02319 05211
4 −0445 05211 −02319 −04179 04179 02319 −05211
5 −1247 04179 −05211 02319 02319 −05211 04179
6 −1802 02319 −04179 05211 −05211 04179 −02319
in a circle with one point of the polygon at the bottom. The MO pattern corresponds to each point of contact of the polygon and circle. If the circle is given a radius of 2, the point of contact gives the coefficient ofin the expression for the energy of the MO. Compilations of HMO energy levels and atomic coefficients are available for a number of conjugated systems.42
What do we learn about molecules such as 1,3,5-hexatriene and benzene from the HMO description of theorbitals?
1. The frontier MOs are identified and described. The frontier orbitals are the highest occupiedMO (HOMO) and thelowest unoccupiedMO (LUMO). These orbitals are intimately involved in chemical reactivity, because they are the most available to electrophiles and nucleophiles, respectively. From the atomic coefficients, which can be represented graphically, we see the symmetry and relative atomic contribution
α–1.802β
α–1.247β
α–0.445β
α+0.445β α+1.247β
α+1.802β
Fig. 1.8. Molecular orbitals for 1,3,5-hexatriene.
42 E. Heilbronner and P. A. Straub, Hückel Molecular Orbitals, Springer Verlag, New York, 1966;
C. A. Coulson, A. Streitwieser, Jr., and J. I. Brauman, Dictionary of -Electron Calculations, W H. Freeman, San Francisco, 1965.
30
CHAPTER 1 Chemical Bonding and Molecular Structure
Fig. 1.9. HMO energy levels for cyclic polyenen=3 ton=7.
of the orbitals. Reaction is facilitated by large overlap of interacting orbitals, so we expect reactions to involve atoms with large orbital coefficients.
2.The HOMO-LUMO gap is approximated.Remember that for atoms, radicals, and ions, hardness and softness are defined in relation to the electron affinity (EA) and ionization potential (IP) (see Section 1.1.6). The energies of the HOMO and LUMO are indicators of the IP and EA, respectively, of the molecules. The HOMO-LUMO gap is an indicator of the reactivity of the molecules in terms of hardness or softness.
The smaller the gap, the softer the molecule.
3. The overall stabilization of the molecule as the result of conjugation is estimated. Remember from the resonance concept in VB theory that conjugation is generally associated with additional stabilization (see Section 1.1.7). In HMO theory this stabilization is expressed as the difference between the energy of the conjugated system and the same number of isolated double bonds. The energy of an isolated double bond in the HMO method is equal to 2+2, so for 1,3,5-hexatriene, a stabilization of 0988is computed. For benzene, the computed stabilization is 2:
Three isolated double bonds=32+2=6+6
Hexatriene=2+1802+2+1247+2+0445=6+6988 Stabilization=6+6988−6+6=0988
Benzene=2+2+2++2+=6+8 Stabilization=6+8−6+6=20
Let us consider the significance of this stabilization, which is sometimes called the delocalization energy (DE). The stabilization results from the removal of the restriction that theelectrons be localized between two particular atoms. Comparison of the DE of 1,3,5-hexatriene and benzene would suggest that the triene is stabi- lized by almost half the extent of benzene, but thermodynamic comparisons do
Table 1.8. Energy Levels and Coefficients for HMOs of Benzene
-orbital mj c1 c2 c3 c4 c5 c6
1 2000 04083 04083 04083 04083 04083 04083
2 1000 00000 05000 05000 00000 −05000 −05000
3 1000 05774 02887 −02887 −05774 −02887 02887
4 −1000 00000 −05000 −05000 00000 05000 −05000
5 −1000 05774 −02887 02887 −05774 −02887 02887
6 −2000 −04083 −04083 −04083 −04083 −04083 −04083
31
SECTION 1.2 Molecular Orbital Theory and Methods α–2.000β
α–1.000β
α+1.000β
α+2.000β
Fig. 1.10.Molecular orbitals for benzene.
not support this result (see Section 3.1.1). Relative to three ethene double bonds, 1,3,5-hexatriene is stabilized by about 8 kcal,43whereas for benzene the stabilization is around 30 kcal/mol. Furthermore, the HMO DE for polycyclic aromatic hydrocarbons such as anthracene and phenanthrene continues to increase with molecular size. This is contrary to chemical reactivity and thermodynamic data, which suggest that on a per atom basis, benzene represents the optimum in stabilization. Thus, the absolute value of the DE does not seem to be a reliable indicator of stabilization.
On the other hand, the difference in stabilization between acyclic and cyclic polyenes turns out to be a very useful indicator of the extra stabilization associated with cyclic systems. This extra stabilization oraromaticityis well represented by the difference in the DE of the cyclic compound and the polyene having the same number of conjugated double bonds.44 For 1,3,5-hexatriene and benzene, this difference is 1012. For comparison of molecules of different sizes, the total stabilization energy is divided by the number of electrons.45 We will see in Chapter 9 that this value gives a very useful estimate of the stability of cyclic conjugated systems.
For monocyclic conjugated polyenes, high stabilization is found for systems with 4n+2 electrons but not for systems with (4n) electrons. The relationship is formulated as Hückel’s rule, which states that completely conjugated planar hydro- carbons are strongly stabilized (aromatic) when they have 4n+2 electrons.
Benzene (6electrons) is aromatic but cyclobutadiene (4electrons) and cyclooc- tatetraene (8electrons) are not.
Fig. 1.11. Frost’s circle mnenomic for HMOs of cyclic polyenes.
43 W. Fang and D. W. Rogers,J. Org. Chem.,57, 2294 (1992).
44 M. J. S. Dewar and C. de Llano,J. Am. Chem. Soc.,91, 789 (1969).
45 B. A. Hess, Jr., and L. J. Schaad, Jr.,J. Am. Chem. Soc.,93, 305, 2413 (1971).
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CHAPTER 1 Chemical Bonding and Molecular Structure
Hückel’s rule also pertains to charged cyclic conjugated systems. The cyclo- propenyl (2electrons), cyclopentadienyl anion (6electrons), and cycloheptatrienyl (tropylium) cation (6electrons) are examples of stabilized systems. We say much more about the relationship between MO configuration and aromaticity in Chapter 9.
–
+
+