Projects and Solutions

Disclaimer:

All problems below can have significant difficulty and are for motivated, adventure-seeking students. The problems are not self-contained on purpose and you need to search the literature even to know what the problem is talking about, or what are the related techniques needed to solve the problem! However, it can be a rewarding experience if you can conquer at least one problem in the list during the two weeks.

Theory:

1. Use time independent perturbation theory and demonstrate that the Van der Waals interaction between two charge neutral atoms is O(R^{-6}) where R is the distance between the two atoms,

2. Demonstrate that the spin-orbit coupling effect can be obtained by a perturbation analysis of the Dirac equation when c is large, where c is the speed of light.

Programming:

1. Create a one dimensional model for the H2 molecule, where the Coulomb interaction is replaced by a Gaussian function. Solve this model at the restricted Hartree-Fock level using a numerical discretization of your choice, and study the behavior of the total energy as the H2 molecule dissociates. Perform a configuration interaction calculation by using the lowest two states from the Hartree-Fock calculation.

Notes on configuration interaction: first 11 pages of http://vergil.chemistry.gatech.edu/notes/ci.pdf

2. Use the KSSOLV package and compute the ground state of a SiH4 molecule. Store the Kohn-Sham orbitals in the real space grid, and write a code to localize the Kohn-Sham orbitals using Boys localization method.

3. Create a one dimensional model for the H2 molecule, and obtain the ground state energy using a diffusion Monte Carlo (DMC) method. Study the behavior of the total energy as the H2 molecule dissociates. (DMC Ref:. I. Kosztin, B. Faber, K. Schulten, Introduction to the diffusion Monte Carlo method, Am. J. Phys. 64 (1996) )

Papers: Read one of the following papers, and write a report typed into latex.

E. Cancès (2001), SCF algorithms for Kohn-Sham models with fractional occupation numbers, J. Chem. Phys. 114, 10616-16622.

M. P. Teter, M. C. Payne, and D. C. Allan. (1989) Solution of Schrodinger’s equation for large systems. Phys. Rev. B, 40:12255–12263

A. Niklasson (2011) Density Matrix Methods in Linear Scaling Electronic Structure Theory, in Linear-Scaling Techniques in Computational Chemistry and Physics, Chap 16, pp. 439--473

Y. Zhou, Y. Saad, M. L. Tiago, J. R. Chelikowsky (2006) Self-consistent-field calculations using Chebyshev-filtered subspace iteration, Journal of Computational Physics 219, 172–184

R. Car and M. Parrinello, (1985) Unified Approach for Molecular Dynamics and Density-Functional Theory, Phys. Rev. Lett. 55, 2471. A slightly longer (and more modern) version: J. Hutter (2012), Car-Parrinello molecular dynamics, Wiley Interdiscip. Rev. Comput. Mol. Sci. 2 604–612.

E. Lieb (1983) Density functionals for coulomb systems, Int. J. Quantum Chem. 24, Pages 243–277

G. Friesecke (1997) Pair Correlations and Exchange Phenomena in the Free Electron Gas, Comm. Math. Phys., 184, pp 143-171

A.J. Cohen, P. Mori-Sánchez, W. Yang (2012), Challenges for density functional theory., Chem. Rev. 112 289–320. Chap. 4.

W. Yang (1986) Gradient correction in Thomas-Fermi theory, Phys. Rev. A 34, 4575

G. Nenciu (1983) Existence of the exponentially localised Wannier functions, Comm. Math. Phys. Volume 91, 81-85.

The solution to Programming Project 1 is provided by Jason Kaye.

HF.m, computes the first two eigenfunctions using the HF formulation.

CI.m, uses these eigenfunctions to compute the minimal CI energy, and also outputs the HF energy.

HFvsCIdriver.m is a driver which varies the distances between the atoms, plotting the HF and CI energies for each distance.

The solution to Programming Project 2 is provided by Felix Henneke

Diffution_Monte_Carlo.ipynb: Jupyter notebook written in julia.