Question

Particle-in-a-box energies take the form (Fig 3):

En = εn2 = (h2/8meL2)n2,

where h = 6.6×10-34 J·s, me = electron mass, and n = quantum number starting at n = 1 for the ground state of an isolated electron. Using this formula, estimate the HOMO→LUMO transition energy ∆Ein kcal/mol for 1,3,5-hexatriene.

Fig 3. Particle-in-box energies.

You can do this in many different unit systems. Atomic units are particularly convenient; in these units, h = 2π and me = 1. The atomic unit of length is the “Bohr radius” = 0.53 Å. The resulting atomic unit of energy is the “Hartree” = 627 kcal/mol.

Answer 1

For 1,3,5 hexatriene (CH₂=CH-CH=CH-CH=CH₂) there is total 6 energy level and there is 6 pi electron. If we fill up the energy level the 3rd energy level is found to be HOMO as we can keep total no of 2 electrons in each level. and 4th energy level is found to be LUMO.

HOMO to LUMO transition energy will be the energy gap between 3rd energy level and 4th energy level of  1,3,5 hexatriene.

from the formula given above, we get E₃ = (h2/8meL2)3² as n=3 for 3rd energy level.

and E₄ = (h2/8meL2)4² as n=4 for 4th energy level

E₄ - E₃ = (h2/8meL2)4² -  (h2/8meL2)3²

= (h2/8meL2)(4² - 3²)

= 7(h2/8meL2)

= 7 (2π/8*(0.53 Å)²) Hartree using atomic unit Atomic units

= 19.56 Hartree we took π = 3.14

= 19.56 * 627 kcal/mol (Hartree” = 627 kcal/mol)

= 12264.12 kcal/mol (ans)