How Is the Vibrational Frequency of a Carbon Dimer Calculated?

  • Thread starter Thread starter singular
  • Start date Start date
  • Tags Tags
    Frequency
singular
Messages
41
Reaction score
0

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 Ebs, 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 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.
 
Physics news on Phys.org
sorry, I am still having issues with this, so i am bumping the thread to make it more visible
 
Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
Back
Top