DFT Knowledge Check for Posed Problem

Question

Currently I am trying to apply to a graduate program overseas. They require me to include a research proposal in my application. My topic of interest is computational materials science.

So, one application of ab initio methods for modelling matter is density functional theory. In it, a central component is a formulation of the many-body Schrodinger equation known as the Kohm-Sham equation. The equation is

$$ \left( \frac{-\hbar^2\nabla^2}{2m} + \mathcal{V}_N(\textbf{r}) + \mathcal{V}_H(\textbf{r}) + \mathcal{V}_{xc}[\rho(\textbf{r})] \right) \phi_i(\textbf{r}) = \textit{E}_i \phi_i (\textbf{r}) $$

To solve this equation, one can apply the self consistency principle and iteratively solve the equation until convergence:

1. Pick initial $\rho(\textbf{r})$ and decide the exchange-correlation functional
2. Calculate potentials
3. Solve the eigenvalue problem
4. Get new $\rho(\textbf{r})$ and check for convergence
5. Go to step 2 if necessary

The eigenproblem step can get computationally expensive for large systems. The proposed research would speed up this step for certain systems by applying parallelization and relevant numerical techniques like spectral approximation, polynomial filtering, and spectral slicing. We also propose looking into modifying known iterative algorithms for solving the eigenvalue problem.

EDIT: I feel like I'm throwing a dart while blindfolded with writing this research proposal for a graduate school application and would like some input.

Answer 1

I agree with Camps that your best bet is to look at a potential advisor and see what they study. Your given example might be a bit of a stretch for some computational materials science advisors, due to its very fundamental nature of solving the problem rather than applying the problem. This could be exactly what one advisor works on though so you should look around the department.

You should look to see who does applied versus methods work for example, you would want to work with a methods orientated person for that project. Machine learning is also a vague methods based project topic as well.