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DFT Jacob's Ladder

A visualization of Perdew's Jacob's ladder of density functional theory (DFT), representing a hierarchy of increasingly sophisticated approximations to the exchange-correlation energy (E_xc). The ladder spans from simple local density approximations at the bottom to accurate but expensive double-hybrid functionals near the top, with each rung adding new physical ingredients to improve accuracy at the cost of computational complexity.


DFT Jacob's Ladder

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Code

dft-jacobs-ladder.typ (90 lines)

#import "@preview/cetz:0.5.2": canvas, draw
#import draw: circle, content, line, rect

#set page(width: auto, height: auto, margin: 8pt, fill: none)
#set text(size: 14pt)

#canvas({
  let rung-sep = 2.2 // Vertical separation between rungs
  let width = 12.4 // Width of each rung
  let height = 1.5 // Height of each rung
  let circle-r = 0.8 // Radius of symbol circles
  let circle-offset = -width / 2 + 1.15 // Keep the symbols near the left edge

  let draw-rung(y, color, text-content, symbol) = {
    rect(
      (-width / 2, y),
      (width / 2, y + height),
      fill: color,
      stroke: rgb("888") + 0.4pt,
    )

    // Add text content, left-aligned relative to circle and vertically centered
    content(
      (circle-offset + circle-r + 0.2, y + height / 2),
      text(size: 14pt)[#align(horizon)[#text-content]],
      anchor: "west", // Left align text
    )

    if symbol != none {
      circle(
        (circle-offset, y + height / 2),
        radius: circle-r,
        fill: color,
        stroke: rgb("888") + 0.4pt,
      )
      content((circle-offset, y + height / 2), align(center, symbol))
    }
  }

  draw-rung(0, rgb("b4a7d6"), [0: *Hartree World*], none)

  draw-rung(
    rung-sep,
    rgb("a4c2f4"),
    [1: *Local Density Approx. (LDA)*\ VWN, GPW92],
    $rho(bold(r))$,
  )

  draw-rung(
    2 * rung-sep,
    rgb("f9cb9c"),
    [2: *Generalized Gradient Approx. (GGA)*\ PBE, BLYP],
    $nabla rho(bold(r))$,
  )

  draw-rung(
    3 * rung-sep,
    rgb("ffe599"),
    [3: *Meta-GGA*\ (r/r$""^2$)SCAN],
    [$\ nabla^(2) rho(bold(r))$ \ $tau(bold(r))$],
  )

  draw-rung(
    4 * rung-sep,
    rgb("b6d7a8"),
    [4: *Occupied Hybrid Methods* \ B3LYP, CAM-B3LYP, M06-2X],
    ${phi_i^"occ"}$,
  )

  draw-rung(
    5 * rung-sep,
    rgb("ea9999"),
    [5: *Virtual Double-Hybrids* \ B2PLYP],
    ${phi_i^"virt"}$,
  )

  draw-rung(6 * rung-sep, rgb("ffb7c5"), [*Chemical accuracy*], none)

  line(
    (-width / 2, 0),
    (-width / 2, 6 * rung-sep + height),
    stroke: 0.7pt,
  )
  line(
    (width / 2, 0),
    (width / 2, 6 * rung-sep + height),
    stroke: 0.7pt,
  )
})