Question

The benzene molecule, C6H6, contains 6 carbon atoms in a ring. Each carbon atom contributes one electron that is free to move round the ring. By treating the electrons as particles moving on a ring of radius r the delocalised electron energies can be estimated by E = h2n 2 / (8π2mer 2 ) (1) where n = 0, ±1, ±2, ±3, … .

(a) Use equation (1) to calculate the n=1 to n=2 electronic transition energy in benzene. Take the radius of the ring to be 0.14 nm. [3]

(b) If this transition was induced by the absorption of a photon, what would the wavelength of the photon be? [2]

(c) The experimentally measured absorption in the ultraviolet spectrum of benzene occurs at 185 nm. Discuss two possible reasons for the discrepancy between this observation and your answer to part . Your answer should make clear the assumptions on which the particle on a ring model is based and discuss their validity in the case of a real molecule such as benzene. [6]

(d) If calculations were attempted using the Self-Consistent Field (SCF) ab initio method, adjustments to the wavefunction would be made so that the energy of the molecule always decreased. Explain briefly why this is done. [3]

(e) Electron correlation is an important feature of electronic motion that is not taken into account by the SCF method. Briefly explain the phrase electron correlation and how including its effects would change the calculated energy of a molecule. [3] (f) Calculate the number of vibrational modes in benzene. [1]

(g) Explain briefly why the force constant for CC stretching in benzene (662 N m–1 ) is significantly larger than that for CH stretching (551 N m–1 ). [2]

Answer 1

a) E = h²n² / (8π²m_er²)

h= 6.626 x 10^-34 m² kg / s

m_e= 9.11 x 10^-31 kilograms

r = 0.14 nm = 0.14 x 10^-9 m

for n=1

E = h²n² / (8π²m_er²)= (6.626 x 10^-34)² x 1 / 8 x (3.14)² x 9.11 x 10^-31 x (0.14 x 10^-9)²

= 3.1141439 x 10^-19 J

for n=2

   E = h²n² / (8π²m_er²)= (6.626 x 10^-34)² x 4 / 8 x (3.14)² x 9.11 x 10^-31 x (0.14 x 10^-9)²

= 1.2456576 x 10^-18 J

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b) E that is transition energy = [(Energy of n=2) -  (Energy of n=1)] = h = hc/

=> (1.2456576 x 10^-18 -  3.1141439 x 10^-19 ) = hc/

   = /

= 6.626 x 10^-34 x 3 x 10⁸ / (9.342 x 10^-19)

= 2.1278 x 10^-7 m

= 212.78 nm

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c) -> Particle in a ring system considers a particle moving in a ring freely. But a real system can contain other particles which can interact themselves. Here , in benzene there are 6 freely movable electrons and each of them can interact with each other.This interaction between the electrons can cause increase in energy of the system that can decrease the wave length of the corresponding radiation.

-> Another reason for the reduction of the wave length may be the presence of positively charged carbon nuclei. The presence of a counter particle is lacked by a simple particle in a ring system and the positively charged nuclei can reduce the energy of the ground state which can increase the energy gap between the states n=1 and n=2 and hence the corresponding wave length can be reduced.

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d) SCF method consists a continuous loop of calculations which converges energy in each of the loops. The corresponding wave function is calibrated using the energy found in each of the loops of the SCF calculations, till we obtain the wave function that is apt for the system that we consider. If the energy is not always decreased, the value of energy won't be converged to a minimum value, and we always find the minimum value of energy to get the apt wave function since the lower the energy the better will be the corresponding wave function. So if the value of energy is not decreased in any of the loops, we can's get a converged minimum value of energy so that we can't find an apt wave function.

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e) Electron correlation can be defined as the interaction between electrons in the electronic structure of a quantum system.

In Haretree-Fock SCF calculations electron correlation is considered as the interaction between a single electron and a mean field that created by all other electrons, instead of talking the interaction between each of the electron pairs. This is called mean field approximation.Because of this approximation taken, the calculation of energy will always show some error, because sum of interactions of each electrons won't be equal to the electron correlation calculated using mean field approximation and the electron correlation hugely affect the total energy of the system.

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f) Vibrational modes for a non-linear molecule = 3n-6 where n is the number of atoms in the molecule   

here n=6

number of vibrational modes = 3 x 6 - 6 = 12

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g) Though the electron negativity difference in C-C and C-H are different the main reason for the rise in the force constant of stretching of C-C bonds is the double bond character of the C-C bonds due to the resonance that benzene possesses. Since the C-C bond contains double bond character, the bond energy will be higher ad hence the force constant of the vibrations will be higher since it will be harder to vibrate.