How Is the Vibrational Frequency of a Carbon Dimer Calculated?

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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

$$E_{tot}=E_{bs}+E_{rep}$$

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

$$E_{rep}=\sum_{i}f\left(\sum{j}\phi\left(r_{ij}\right)\right)$$

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

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

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

where $$r_{0}$$ denotes the nearest-neighbor atomic separations, and $$n, n_{c}, r_{c}, \phi_{0}, m, d_{c}, and m_{c}$$ 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 E_bs, the electronic eigenvalues can be obtained by solving an empirical tight-binding Hamiltonian $$H_{TB}$$. The off-diagonal are described by a set of orthogonal sp³ two-center hopping parameters, $$V_{ss\sigma}$$,
$$V_{sp\sigma}$$, $$V_{pp\sigma}$$, and $$V_{pp\pi}$$, 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 E_bs, 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