Question

a. Calculate the HMO energy levels and atomic orbital coefficients for $1,3-$ butadiene. b. Estimate the delocalization energy, in units of $\beta$, of the cyclobutadienyl dication $\mathrm{C}_{4} \mathrm{H}_{4}{ }^{2+}$ from HMO theory. c. Estimate, in units of $\beta$, the energy associated with the longest-wavelength UVVIS absorption of $1,3,5,7$-octatetraene. Does it appear at a longer or shorter wavelength than the corresponding absorption in $1,3,5$-hexatriene? 

   a. Calculate the HMO energy levels and atomic orbital coefficients for $1,3-$ butadiene.
b. Estimate the delocalization energy, in units of $\beta$, of the cyclobutadienyl dication $\mathrm{C}_{4} \mathrm{H}_{4}{ }^{2+}$ from HMO theory.
c. Estimate, in units of $\beta$, the energy associated with the longest-wavelength UVVIS absorption of $1,3,5,7$-octatetraene. Does it appear at a longer or shorter wavelength than the corresponding absorption in $1,3,5$-hexatriene?
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Advanced Organic Chemistry. Part A. Structure and Mechanisms
Advanced Organic Chemistry. Part A. Structure and Mechanisms
Francis A. Carey,… 5th Edition
Chapter 1, Problem 17 ↓
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a. Calculate the HMO energy levels and atomic orbital coefficients for $1,3-$ butadiene. b. Estimate the delocalization energy, in units of $\beta$, of the cyclobutadienyl dication $\mathrm{C}_{4} \mathrm{H}_{4}{ }^{2+}$ from HMO theory. c. Estimate, in units of $\beta$, the energy associated with the longest-wavelength UVVIS absorption of $1,3,5,7$-octatetraene. Does it appear at a longer or shorter wavelength than the corresponding absorption in $1,3,5$-hexatriene?
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Key Concepts

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HĂĽckel Molecular Orbital (HMO) Theory
HMO theory is a simplified approach used to describe the ?-electron systems in conjugated hydrocarbons. It approximates the electronic structure by using a secular determinant based on a limited set of atomic orbitals, typically ignoring electron-electron interactions beyond an averaged field. This theory yields molecular orbital energy levels and coefficients that describe the distribution of ?-electron density across the molecule.
Energy Levels in Conjugated Systems
This concept involves determining the quantized energy states of electrons in systems with alternating single and double bonds. These energy levels are obtained by solving the eigenvalue problem associated with the HĂĽckel matrix, where the energy differences correspond to transitions observable in spectroscopic experiments. The arrangement and relative energies of these levels are crucial for understanding reactivity and optical properties.
Atomic Orbital Coefficients
Atomic orbital coefficients are the weights that describe the contribution of each atomic orbital to a molecular orbital obtained by solving the HĂĽckel secular equation. They provide insight into the electron distribution across a conjugated system and determine how electrons are delocalized among the atoms, an important factor in predicting chemical behavior.
Delocalization Energy
Delocalization energy refers to the stabilization that results from the spreading of electrons over multiple atoms or bonds within a molecule, rather than being confined between two atoms. In the context of HMO theory, it is estimated by comparing the total ?-electron energy of the system with that of an analogous hypothetical system where electrons are fully localized, highlighting the energetic benefits of electron delocalization.
UV-VIS Spectroscopy in Conjugated Systems
In conjugated systems, the longest-wavelength (lowest energy) UV-VIS absorption corresponds to electronic transitions between the highest occupied and the lowest unoccupied molecular orbitals. The position of this absorption band is influenced by the extent of conjugation, as longer conjugated chains generally result in lower energy gaps and thus absorptions at longer wavelengths. Comparisons between different polyenes provide insights into how molecular structure affects optical properties.
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Transcript

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00:01 In this exercise, we have the w -ionized lithium atom, and in question a, we have to find the energy levels e -n of this atom.
00:11 So remember that for a hydrogen -like atom, e -n, the energy of the energy level, is given by minus z squared, divided by n squared, times 13 .6 electron volts.
00:29 In this case, the lithium, the w -ionized lithium is a hydrogen, like atom because there's only one electron orbiting the nucleus and z the atomic number of the lithium is equal to three uh so en for the lithium is is minus nine which is three squared times 13 .6 divided by n squared so this is equal to minus 122 .4 electron volts divided by n squared in question b, we have to calculate explicitly the energy of the fourth energy level, so that's e4, which is minus 122 .4, divided by 4 squared.
01:20 So this is minus 7 .65 electron volts.
01:26 In question c, we have to calculate the energy of the second energy level, so that's minus 122 .4, divided by 2 squared.
01:38 Which is 4.
01:40 So this is minus 30 .6 electron volts.
01:46 In question d, we have to calculate the energy of a photon that's emitted after the transition of the atom from the fourth, of the electron in the atom from the fourth energy level to the second one.
02:00 So the electrons transitions from n equals 4 to n equals 2.
02:06 And from conservation of energy, we have that the energy of the photon, e gamma, is equal to the initial energy minus the final energy.
02:17 So this is e4 minus e2.
02:20 This is minus 7 .65 plus 30 .6 electron volts, which is equal to 22 .96, 95 electron volts.
02:34 This is the answer in electron volts.
02:37 And the answer, in order to obtain the answer in joules, we need to multiply this by 1 .6 times 10 to the minus 19 joules per electron volt.
02:52 So we get that the energy of the photon is 3 .672 times 10 to the minus 18 joules...
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