After the soccer-ball structure for C 60 was first suggested in it became important to obtain as much independent supporting evidence as possible.

## Symmetry in quantum mechanics

This came in the areas of nmr, IR, and Raman spectroscopy. The nmr spectrum showed a single resonance indicating 60 equivalent carbon atoms in the structure.

The IR spectrum was found to have four lines, while Raman spectroscopy yielded ten lines. The second half of this paper will use group theory to demonstrate that a C 60 molecule with a soccer-ball structure must have four IR active and ten Raman active vibrational modes. As is well-known C 60 is a carbon cage consisting of 20 hexagons and 12 pentagons and resembles a soccerball. Removing the leather, but keeping the seams, leaves 60 vertices for the carbon atoms and 90 covalent bonds between them.

## Group theory and its application to the quantum mechanics of atomic spectra - CERN Document Server

Actually C 60 has spheroidal geometry and belongs to the truncated icosahedral symmetry group, l h. Curly and Smalley, codiscoverers of buckminsterfullerene, have described it as the "roundest molecule that can possibly exist" 4 so the model presented here assumes, initially, that Coo is a perfect sphere. Each carbon is sigma bonded to three other carbons using three of its four valence electrons to form these bonds. The remaining electron is considered to be delocalized on the surface of the sphere created by the 60 atom carbon cage.

### 1st Edition

The quantum mechanical behavior of an individual electron restricted to the surface of a sphere is well-known. The energy levels associated with the spherical harmonic states are a function of the radius of the sphere and the angular momentum quantum number. Just as the quantum mechanical solution for the one-electron hydrogen atom can be adapted for qualitative treatments of the electronic structure of multi-electron atoms, so the energy level diagram for the electron moving on a sphere can be used to describe the electronic structure of C The energy level diagram shown in Figure 1 provides a qualitative description of the electronic structure obtained when one applies the aufbau principle, the Pauli exclusion principle, and Hund' s rule to the addition of 60 electrons to the available spherical harmonic energy levels.

Figure 1. Sixty electrons placed in the lowest available energy levels according to the aufbau principle, the Pauli exclusion principle, and Hund's rule. The energies for the individual levels shown on the right are in eV and were calculated using equation 1. As it stands this is not a satisfactory picture because pristine C 60 is an insulator and has no un-paired electrons.

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This difficulty is resolved by recalling that C 60 is not a perfect sphere, but has the lower symmetry of the icosahedral group. The behavior of the spherical harmonics under the rotations of the icosahedral group is given by. It is easy to show that this reducible representation is a linear combination of the fivefold degenerate H u , an the two three-fold degenergate, T 1u and T 2u irreducible representations of the icosahedral group.

Group theory doesn't predict the order of the levels, but Figure 2 shows that if the five-fold degenerate level is placed lowest, an energy level diagram that captures the essentials of the known electronic structure of C 60 is obtained. This information helps us design a better experience for all users. To learn more about cookies, please see our cookie policy. To learn more about how we use and protect your data, please see our privacy policy. Subjects Science Chemistry Nonfiction.

tiougeiskimin.tk Science Chemistry Nonfiction. Group Theory and Quantum Mechanics Embed. Learn how to read digital books for free. Media Group Theory and Quantum Mechanics. OverDrive uses cookies and similar technologies to improve your experience, monitor our performance, and understand overall usage trends for OverDrive services including OverDrive websites and apps. Required Cookies These cookies allow you to explore OverDrive services and use our core features. Request Permissions Exam copy. Overview Author s. Summary This book is intended for theoretical physicists with a desire to understand the value of modern group-theoretical methods in quantum theory.

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Petrashen E.