Computational Materials Physics

Computational Materials Physics is a graduate-level course on density functional theory, taught openly online by Prof. Stefaan Cottenier (Ghent University, compmatphys.org). I’m working through it independently: lectures, hands-on calculations, and the real research project that closes it out. This page is a snapshot of where I’ve gotten and what I’ve written up along the way.

The premise DFT rests on is a severe one. A solid is nothing but nuclei and electrons interacting electromagnetically under quantum mechanics — atoms aren’t even a necessary construct. Everything a material does, its lattice constant, its bulk modulus, its band gap, is in principle already contained in the Schrödinger equation. Being able to predict material behavior these from ab-initio [“from first principles”] calculations is an incredibly powerful tool, and I’m having a wonderful time learning about that process.

Where I’ve gotten

• Foundations. Born–Oppenheimer approximation; Hartree–Fock and post-HF methods; the Hohenberg–Kohn theorems; Kohn–Sham equations; exchange–correlation functionals and correlation; numerical solution methods and basis sets.
• Calculations in practice. Quantum ESPRESSO driven from Python via the ASE API, run on HPC; pseudopotentials; plane-wave cutoff and k-mesh convergence testing.
• Writeups. A valueable but understated aspect of this course, especially for an undergrad, has been the oppurtunity to practice making professional LaTeX writeups; my reports below double as a showcase for that process.
• Crystallography. CIF files and structure databases; space groups and Wyckoff positions; symmetry constraints on structural degrees of freedom.
• Geometry optimization. E(V) curves and Birch–Murnaghan equation-of-state fits; bulk modulus and pressure off the E–V curve; cell shape; classical and quantum forces via Hellmann–Feynman; ionic relaxation under damped Newtonian dynamics; full variable-cell relaxation through the stress tensor; formation energies and 0 K phase diagrams. The FeAl project was part of this unit.
• Electronic structure. Reciprocal space, reciprocal lattice, and the first Brillouin zone; plane-wave expansion of the electron density; crystal quantum numbers (n, k, s); band structures and DOS.
• Chemical bonding. Bonding types; cohesive energy; charge density difference plots. Current chapter — the rocksalt activity below is from it.

Still ahead: elastic constants, supercells and surfaces, precision versus accuracy, magnetism, phonons and finite temperature, band gaps.

DFT structure optimization of FeAl

The larger of the two write-ups, done mid-course to sharpen my workflow before the research project. A succinct lesson in energy coupling between the various degrees of freedom that are optimized in structure optimization, the difficulty in doing them one at a time, and the power of stress-tensor formalism automation.

The structure: an as-given ideal HCP FeAl crystal in an orthorhombic cell, space group 59 (Pmmn), Fe on Wyckoff position 2a and Al on 2b. Optimized by hand, one degree of freedom at a time, then against a single full relaxation for comparison.

By hand

• Volume. Isotropic scaling across five volumes, E(V) fit to Birch–Murnaghan: V₀ = 48.940 ų, 12.7% below the as-given cell.
• Shape. b/a swept at fixed V₀ with c/a pinned at its ideal 1.7320; quadratic fit put the minimum at b/a = 1.6428, all of 0.6 meV below the original ratio.
• Positions. Ionic relaxation in that frozen cell. Site symmetry pins x and y; the free z parameters moved −0.063 Å (Fe) and −0.101 Å (Al), worth 82.9 meV.

Each step lowered the energy, so each looked like progress.

By stress tensor

I ran one vc-relax from the hand-optimized structure, relaxing all of it at once. From this, a further 264.5 meV worth of strain energy was relaxed, three times what the shape and position steps had turned up between them. Where it came from:

• c/a — the one ratio held fixed by hand—went 1.7320 → 1.5279, −11.8%: a +6.2%, c −6.3%, b unmoved, and volume down only 0.45%. The volume optimization had been sound; the cell shape was nowhere near equilibrium.
• b/a landed at 1.5473, ~6% off the 1.6428 the quadratic fit had predicted. The original fit was minimizing along the wrong c/a.
• The z parameters relaxed back to 0.91984 (Fe) and 0.41550 (Al), within 0.003 of the ideal values they started from. The 0.06–0.10 Å shifts found by hand weren’t a real deviation from ideal HCP at all, just an artifact of relaxing atoms inside a cell that wasn’t itself at equilibrium.

The manual pass had also left out three degrees of freedom it never touched—the cell angles—which the relaxation confirmed stay at 90° (forced by symmetry). However, to be sure of the correct answer, dong the whole thing by hand means optimizing over six cell parameters plus the atomic ones, at 100s+ of runtime per data point. Using stress tensors to automate this optimization makes the whole thing much more convinent.

Read the full report (PDF) →

Room heating from the cohesive energy of rocksalt

An analytic problem from the chemical bonding chapter, and a useful reality check on a quantity that otherwise stays abstract. The puzzle is to connect a box of free Na atoms to a box of free Cl atoms, let them react into one teaspoon of rocksalt, and ask how much the room warms. Also a sample for developing my LaTeX skills.

The solutions hold steady with materials intuition. We all learn that nuclei within ionic salts are held tightly in their crystals, requiring large amounts of energy to break their electrostatic bonds—by thermodynamics, you would expect to see that same large amount of energy released when dissociated particles react to form the same salt.

Read the write-up (PDF) →

What’s next

The course closes with a research project on a real material of current research relevance (last year, it was hydrogen storage in Ti-Fe-Hf alloys), using the full toolkit gained throughout the semester. The work above is the groundwork for it, and I look forward to presenting it here soon.