{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Computational materials\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/computational-materials). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Computational materials science connects atomic structure to predicted energies,\n", "forces, geometries, electronic bands, vibrations, and response properties. Each\n", "result belongs to a particular physical approximation and finite numerical\n", "representation. A band gap or formation energy without that context is not a\n", "reproducible material claim.\n", "\n", "Calculations range from empirical interatomic potentials to quantum\n", "electronic-structure methods. The right method depends on the observable,\n", "chemical environment, system size, temperature, and accuracy needed.\n", "\n", "## Separate nuclei and electrons carefully\n", "\n", "Many electronic-structure calculations use the Born–Oppenheimer approximation:\n", "nuclei are treated as fixed while the electronic state is solved, then nuclear\n", "positions are updated using the resulting forces. This is useful because nuclei\n", "are much heavier than electrons, but it can fail when electronic and nuclear\n", "motion are strongly coupled.\n", "\n", "For fixed nuclear positions\n", "$\\{\\mathbf R_I\\}$,\n", "the electronic solution defines a potential-energy surface\n", "$E(\\{\\mathbf R_I\\})$.\n", "Forces are\n", "\n", "\n", "$$\n", "\\mathbf F_I=-\\frac{\\partial E}{\\partial\\mathbf R_I}\n", "$$\n", "\n", "\n", "at a fully self-consistent electronic state, with additional implementation\n", "details such as Pulay corrections when the basis depends on geometry.\n", "\n", "## Density-functional theory\n", "\n", "The many-electron wavefunction depends on three coordinates per electron and\n", "becomes extremely expensive as electron count grows. Density-functional theory\n", "uses the electron density\n", "$n(\\mathbf r)$\n", "as its central variable. In Kohn–Sham DFT, auxiliary one-electron orbitals obey\n", "\n", "\n", "$$\n", "\\left[-\\frac{\\hbar^2}{2m_e}\\nabla^2+V_{\\mathrm{eff}}[n](\\mathbf r)\\right]\\phi_i(\\mathbf r)=\\epsilon_i\\phi_i(\\mathbf r)\n", "$$\n", "\n", "\n", "and the density is constructed from occupied orbitals:\n", "\n", "\n", "$$\n", "n(\\mathbf r)=\\sum_i f_i|\\phi_i(\\mathbf r)|^2\n", "$$\n", "\n", "\n", "The effective potential includes nuclear attraction, classical electron\n", "repulsion, and an exchange-correlation approximation. The exact\n", "exchange-correlation functional is unknown, so choices such as local,\n", "generalized-gradient, hybrid, or more specialized approximations create model\n", "error even after numerical convergence.\n", "\n", "Kohn–Sham eigenvalues are useful computational quantities, but they are not all\n", "automatically exact excitation energies. Standard approximations often\n", "underestimate semiconductor band gaps.\n", "\n", "## The self-consistent field loop\n", "\n", "Because the potential depends on density and the density depends on the\n", "orbitals, the equations are solved iteratively:" ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "# Two-site mean-field toy model illustrating self-consistency, not a DFT calculation.\n", "density = np.array([0.7, 0.3])\n", "for iteration in range(1000):\n", " H = np.array([[density[0], -1.0], [-1.0, density[1]]])\n", " energies, orbitals = np.linalg.eigh(H)\n", " output_density = abs(orbitals[:, 0]) ** 2\n", " residual = np.linalg.norm(output_density - density)\n", " density = 0.7 * density + 0.3 * output_density\n", " if residual < 1e-10:\n", " break\n", "else:\n", " raise RuntimeError(\"Self-consistency failed to converge.\")\n", "print(\"Converged site populations:\", density, \"residual:\", residual)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Simple replacement can oscillate or diverge, especially in metals. Mixing and\n", "preconditioning improve convergence. Reaching the iteration threshold means the\n", "chosen discrete equations are self-consistent; it does not establish that the\n", "functional, structure, or temperature is physically adequate.\n", "\n", "## Basis sets and pseudopotentials\n", "\n", "Electronic orbitals need a finite basis. Plane waves are systematic for periodic\n", "materials and are controlled by an energy cutoff:\n", "\n", "\n", "$$\n", "\\frac{\\hbar^2|\\mathbf G+\\mathbf k|^2}{2m_e}\\leq E_{\\mathrm{cut}}\n", "$$\n", "\n", "\n", "Increasing $E_{\\mathrm{cut}}$\n", "adds basis functions and usually increases cost. Localized atomic orbitals can\n", "be more compact but require their own size and completeness tests.\n", "\n", "Core electrons oscillate rapidly near nuclei and are often replaced by\n", "pseudopotentials or projector-augmented-wave data. The chosen data set must\n", "match the exchange-correlation approximation and include relevant semicore,\n", "relativistic, and spin–orbit effects. Two calculations with the same cutoff but\n", "different pseudopotentials are not the same model.\n", "\n", "## Reciprocal-space sampling\n", "\n", "Periodic crystals require integration over the Brillouin zone. A finite\n", "$\\mathbf k$-point mesh\n", "approximates that integral:\n", "\n", "\n", "$$\n", "\\frac{1}{\\Omega_{\\mathrm{BZ}}}\\int_{\\mathrm{BZ}}Q(\\mathbf k)\\,d\\mathbf k\\approx\\sum_jw_jQ(\\mathbf k_j)\n", "$$\n", "\n", "\n", "Metals often need denser sampling than wide-gap insulators because occupation\n", "changes sharply near the Fermi surface. Smearing can aid integration, but its\n", "width affects energies and forces and must be converged or extrapolated\n", "appropriately." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "# Brillouin-zone average for a fixed illustrative tight-binding dispersion.\n", "previous = None\n", "for points in [16, 32, 64, 128]:\n", " k = 2 * np.pi * (np.arange(points) + 0.5) / points - np.pi\n", " observable = np.mean(np.sqrt(1 + 0.5 * np.cos(k)))\n", " change = None if previous is None else abs(observable - previous)\n", " print(points, observable, change)\n", " previous = observable\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Converge the actual target observable. Total energy per atom may appear stable\n", "before a small band gap, stress, phonon frequency, or magnetic energy difference\n", "does.\n", "\n", "## Periodic cells and finite-size effects\n", "\n", "Periodic boundaries represent an infinite repeating structure. A primitive cell\n", "is efficient for a perfect crystal. Defects, surfaces, disorder, and isolated\n", "molecules need larger supercells plus checks that repeated images do not\n", "interact significantly.\n", "\n", "A charged defect is especially delicate because periodic copies and compensating\n", "background charge produce long-range artifacts. Formation energies require\n", "finite-size and potential-alignment corrections whose definitions must be\n", "reported.\n", "\n", "For a slab, vacuum thickness, slab thickness, surface termination, dipole\n", "corrections, and in-plane cell size all need convergence. “Twenty angstroms of\n", "vacuum” is an input, not proof of isolation.\n", "\n", "## Relaxing a structure\n", "\n", "A geometry optimization seeks a local stationary structure. Typical conditions\n", "include small maximum force and small stress:\n", "\n", "\n", "$$\n", "\\max_I|\\mathbf F_I| \"\n", " f\"{output_value:.6g} scaled energy units\"\n", " )\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows bond length; the vertical axis shows harmonic bond energy. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_grid = np.linspace(0.7, 1.5, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Bond length (scaled length units)\")\n", "ax.set_ylabel(\"Harmonic bond energy (scaled energy units)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "harmonic bond energy change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1.2 scaled length units is\n", "0.8 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = 4 q - 4\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.204\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"scaled energy units\")\n", "print(\"Full expression:\", full_prediction, \"scaled energy units\")\n", "print(\n", " \"Approximation difference:\",\n", " full_prediction - linear_prediction,\n", " \"scaled energy units\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 1.204 scaled length units. The tangent estimate is\n", "0.0832 scaled energy units, while the full expression gives\n", "0.083232 scaled energy units. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "input_span = 0.8\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "bond length inside the stated interval, predict whether harmonic bond energy will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: This toy potential illustrates optimisation; it is not a density-functional calculation or a prediction for a named material. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of scaled energy units:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1.5) - evaluate(0.7))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.7, 1.5, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-022", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 3*, Chapter 9, “Condensed Matter Physics”](https://openstax.org/books/university-physics-volume-3/pages/9-introduction).\n", "- [OpenStax, *University Physics Volume 3*, §9.5, “Band Theory of Solids”](https://openstax.org/books/university-physics-volume-3/pages/9-5-band-theory-of-solids).\n", "- [Tipler and Llewellyn, *Modern Physics*, molecules and solids](https://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf).\n", "- [Benjamin Crowell, *Light and Matter*, quantum and solid-state foundations](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [Physics LibreTexts, Condensed Matter Physics bookshelf](https://phys.libretexts.org/Bookshelves/Condensed_Matter_Physics).\n", "- [Materials Project documentation, materials methodology](https://docs.materialsproject.org/methodology/materials-methodology)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "computational-materials.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/computational-materials", "source_sha256": "256e100ae019d36e4a6ea360f0b2dba5869e32ed304d8bcf7dc19bf05f9b4bbe" } }, "nbformat": 4, "nbformat_minor": 5 }