orbital is the LUMO, an antibonding orbital. The four Hückel molecular orbitals for 1,3-butadiene are* = 0.3720 +0.602d2 +0.60203 + 0.3720 2 = 0.6026 + 0.372, – 0.372d3 - 0.60264 3 = 0.602d - 0.37262 - 0.37203 + 0.60264 a = 0.3726 - 0.60262 + 0.60203 – 0.37264 (11.75) These four molecular orbitals are indicated in Fig. 11.21. Notice that the r orbitals extend the entire length of the molecule. (For the calculation of these coefficients, see Computer Problem 11.C.) The Hückel secular determinant for benzene is a - E B B B E 0. B = 0 (11.76) a – E B a - E B B a - E B The six roots are E = a + 2B E2 = E3 = a +B E4 = Es = a -B E6 = a - 2B (11.77) Since benzene has six 7 electrons, pairs of electrons go in the three lowest energy orbitals (those with plus signs in E;). Thus the n electronic energy in benzene is E, = 2(a + 2B) + 4(a + B) = 6a + 8B (11.78) The equations for the six Hückel molecular orbitals for benzene are not given here, but the corresponding electron densities are shown in Fig. 11.22. Note that the electronic energy in C,H6 is more negative than three times the value in C,H4, indicating that C,H does not contain three double bonds. The difference is called the delocalization energy. The Hückel theory is an example of a semiempirical molecular orbital method. We have used a simple Hamiltonian (neglecting many terms) to find the orbitals and their energies. We now can use experimental quantities to fit a and B for ethylene. We then use the values to make predictions for butadiene, benzene, and so on. This method does not give quantitative results, but it does provide us with qualitative insights about larger systems for which the more com- putationally intensive methods are too costly or time-consuming, and it gives us insight into the excited electronic states of conjugated 7 electron molecules.
orbital is the LUMO, an antibonding orbital. The four Hückel molecular orbitals for 1,3-butadiene are* = 0.3720 +0.602d2 +0.60203 + 0.3720 2 = 0.6026 + 0.372, – 0.372d3 - 0.60264 3 = 0.602d - 0.37262 - 0.37203 + 0.60264 a = 0.3726 - 0.60262 + 0.60203 – 0.37264 (11.75) These four molecular orbitals are indicated in Fig. 11.21. Notice that the r orbitals extend the entire length of the molecule. (For the calculation of these coefficients, see Computer Problem 11.C.) The Hückel secular determinant for benzene is a - E B B B E 0. B = 0 (11.76) a – E B a - E B B a - E B The six roots are E = a + 2B E2 = E3 = a +B E4 = Es = a -B E6 = a - 2B (11.77) Since benzene has six 7 electrons, pairs of electrons go in the three lowest energy orbitals (those with plus signs in E;). Thus the n electronic energy in benzene is E, = 2(a + 2B) + 4(a + B) = 6a + 8B (11.78) The equations for the six Hückel molecular orbitals for benzene are not given here, but the corresponding electron densities are shown in Fig. 11.22. Note that the electronic energy in C,H6 is more negative than three times the value in C,H4, indicating that C,H does not contain three double bonds. The difference is called the delocalization energy. The Hückel theory is an example of a semiempirical molecular orbital method. We have used a simple Hamiltonian (neglecting many terms) to find the orbitals and their energies. We now can use experimental quantities to fit a and B for ethylene. We then use the values to make predictions for butadiene, benzene, and so on. This method does not give quantitative results, but it does provide us with qualitative insights about larger systems for which the more com- putationally intensive methods are too costly or time-consuming, and it gives us insight into the excited electronic states of conjugated 7 electron molecules.
Chemistry: Principles and Practice
3rd Edition
ISBN:9780534420123
Author:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward Mercer
Publisher:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward Mercer
Chapter10: Molecular Structure And Bonding Theories
Section: Chapter Questions
Problem 10.101QE: The molecular orbital diagram of NO shown in Figure 10.47 also applies to OF. Draw the complete...
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What would be the lowest possible excitation energy for benzene in this model? Give your answer in units of β.
Transcribed Image Text:The 27 orbital of butadiene is the HOMO, a bonding orbital, and the 37
orbital is the LUMO, an antibonding orbital.
The four Hückel molecular orbitals for 1,3-butadiene are*
= 0.3720 + 0.602d2 +0.602d3 + 0.3720
2 = 0.60261 + 0.37262-0.37203 - 0.60264
s = 0.602d, - 0.372d2 - 0.372¢3 + 0.602d4
la = 0.3720 - 0.602dz + 0.602ds – 0.37264
(11.75)
These four molecular orbitals are indicated in Fig. 11.21. Notice that the r orbitals
extend the entire length of the molecule. (For the calculation of these coefficients,
see Computer Problem 11.C.)
The Hückel secular determinant for benzene is
a - E
B
B
a - E
B
E
a
= 0
(11.76)
a - E
B
a - E
B
B.
a - E
The six roots are
Ε-α+ 2β
Ez = E3 = a + B
E, = Es = a-B
E α-2β
(11.77)
Since benzene has six 7 electrons, pairs of electrons go in the three lowest energy
orbitals (those with plus signs in E). Thus the r electronic energy in benzene is
E, = 2(a + 2B) + 4(a + B)
= 6a + 8B
(11.78)
The equations for the six Hückel molecular orbitals for benzene are not given
here, but the corresponding electron densities are shown in Fig. 11.22. Note that
the 7 electronic energy in C,H, is more negative than three times the value in
CH,, indicating that C,H, does not contain three double bonds. The difference
is called the delocalization energy.
The Hückel theory is an example of a semiempirical molecular orbital
method. We have used a simple Hamiltonian (neglecting many terms) to find
the orbitals and their energies. We now can use experimental quantities to fit a
and B for ethylene. We then use the values to make predictions for butadiene,
benzene, and so on. This method does not give quantitative results, but it does
provide us with qualitative insights about larger systems for which the more com-
putationally intensive methods are too costly or time-consuming, and it gives us
insight into the excited electronic states of conjugated 7 electron molecules.
*The derivation of these wavefunctions is given in L. N. Levine, Quantum Chemistry, 5th cd., Upper
Saddle River, NJ: Prentice-Hall, 1999.
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