How Is the Vibrational Frequency of a Carbon Dimer Calculated?

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In summary, the conversation discusses a proposed problem to find the vibrational frequency of a carbon dimer C2 and the steps needed to solve it. It is part of a final project for a computational/numerical methods course, and the student is seeking help as they are not experienced in physics. The equations and parameters necessary for calculating the total energy of the molecule are also mentioned, and the student is unsure of how to construct the matrix and find the hopping parameters. They are seeking resources to aid in their understanding of the problem.
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Homework Statement


The proposed problem is to find the vibrational frequency of a carbon dimer C2. Then I have to write and run an MD simulation to find the period of oscillations and compare the two.

This is part of a final project for a computational/numerical methods course I am taking. Up until now, we have done nothing but numerical computations, but the professor decided it would be good for us to know what it is like to solve a real physics problem. The only prerequisite for this course is experience in programming, so I am not very far along in my physics (through Modern I: intro to quantum and classical dynamics). I am a bit overwhelmed, so any help is greatly appreciated.

Homework Equations


[tex]E_{tot}=E_{bs}+E_{rep}[/tex]

where [tex]E_{bs}[/tex] is the sum of electronic eigenvalues over all occupied states, and
[tex]E_{rep}[/tex] is a short-ranged repulsive energy.

[tex]E_{rep}=\sum_{i}f\left(\sum{j}\phi\left(r_{ij}\right)\right)[/tex]

where [tex]\phi\left(r_{ij}\right)[/tex] is a pairwise potential between atoms i and j, and f is a functional expressed as a 4th-order polynomial with argument [tex]\sum{j}\phi\left(r_{ij}\right)[/tex].

[tex]s\left(r\right)=\left(r_{0}/r\right)^{n}exp\left(n\left[-\left(r/r_{c}\right)^{n_{c}}+\left(r_{0}/r_{c}\right)^{n_{c}}\right] \right)[/tex]

[tex]\phi\left(r\right)=\phi_{0}\left(d_{0}/r\right)^{m}exp\left(m\left[-\left(r/d_{c}\right)^{m_{c}}+\left(d_{0}/d_{c}\right)^{m_{c}}\right] \right)[/tex]

where [tex]r_{0}[/tex] denotes the nearest-neighbor atomic separations, and [tex]n, n_{c}, r_{c}, \phi_{0}, m, d_{c}, and m_{c}[/tex] are parameters that need to be determined.

These equations are from A transferable tight-binding potential for carbon by C H Xu et al. They describe the process of finding the total energy for diamond.


The Attempt at a Solution


In order to calculate the vibrational frequency, I need to calculate the total energy of the molecule. Once I have the total energy, I can plot the energy as a function of the inter-atomic distance and take the second derivative to find the spring constant and calculate the vibrational frequency.

For Ebs, the electronic eigenvalues can be obtained by solving an empirical tight-binding Hamiltonian [tex]H_{TB}[/tex]. The off-diagonal are described by a set of orthogonal sp3 two-center hopping parameters, [tex]V_{ss\sigma}[/tex],
[tex]V_{sp\sigma}[/tex], [tex]V_{pp\sigma}[/tex], and [tex]V_{pp\pi}[/tex], scaled with interatomic separation r as a function of s(r); and the on-site elements are the atomic orbital energies of the corresponding atom.

This is all I know. I know what is necessary to get Ebs, but I don't know how to construct the matrix. I don't know what dimensions it should be, but I suspect 8x8 because carbon is tetravalent, and I don't really understand what the elements should be. Are the diagonal elements found from the Hamiltonian? What are the hopping parameters for the off-diagonal elements?

If anyone has a good article or webpage that might help me, it would also be appreciated.
 
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sorry, I am still having issues with this, so i am bumping the thread to make it more visible
 

Related to How Is the Vibrational Frequency of a Carbon Dimer Calculated?

1. What is the vibrational frequency of C2?

The vibrational frequency of C2, also known as the carbon dimer, is approximately 1280 cm-1.

2. How is the vibrational frequency of C2 determined?

The vibrational frequency of C2 is determined through spectroscopic techniques, such as infrared or Raman spectroscopy, which measure the absorption or scattering of light by the molecule.

3. What factors affect the vibrational frequency of C2?

The vibrational frequency of C2 is affected by factors such as the bond strength between the two carbon atoms, the bond length, and the mass of the atoms.

4. How does the vibrational frequency of C2 change with temperature?

The vibrational frequency of C2 generally increases with higher temperatures due to the increase in molecular motion and bond stretching.

5. Can the vibrational frequency of C2 be used to identify the molecule?

Yes, the vibrational frequency of C2 is a characteristic property of the molecule and can be used in spectroscopic techniques to identify the presence of C2 in a sample.

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