""" 02-hydrogen-orbitals.py ======================== The first six hydrogen-atom radial eigenfunctions, animated. Scene graph ----------- 1. Title at the top: "Hydrogen atom: the first nine orbitals". 2. Subtitle: "R_{nℓ}(r) for (n,ℓ) = (1,0), (2,0), (2,1), (3,0), (3,1), (3,2)". 3. Draw the coordinate system (r on x-axis, R(r) on y-axis) and a horizontal reference line at R = 0. 4. For each (n, ℓ) in the list above (sequenced): - Update the *info panel* in the top-left corner with the current orbital's name ("1s", "2s", "2p", "3s", "3p", "3d"), its quantum numbers, the energy E_n = -1/(2n²) E_h, and the mean radius ⟨r⟩_{nℓ} in bohr. - Plot R_{nℓ}(r) in a distinct colour - Draw a vertical line at the mean radius ⟨r⟩_{nℓ}, with a small label "⟨r⟩ = X.X a_0" placed *above* the line at a fixed height (so the label never moves position between orbitals and never collides with the x-axis labels). - Animate a "particle" dot moving along the radial function 5. Final panel: show the energy formula and a short summary at the bottom. Layout (why this version has no overlap) ---------------------------------------- The OLD script stacked six energy+orbital labels on the right side of the plot, plus a mean-radius label that moved with the radius position. By orbital 3d the right side had 7 text bands packed into 270px of vertical space, with the mean-radius label (which moves to larger r for higher n) overlapping the orbital labels. This version uses: - A *single fixed info panel* in the top-left corner. The panel's content is replaced (via Transform) for each orbital, so the text never stacks. - A small, fixed-position "⟨r⟩" label at a *fixed x* (4 a_0) for every orbital. The line itself is at r_mean (so the line moves correctly), but the label never moves. - The 6 curves are plotted one at a time, in the centre of the screen, with no text on the right side of the plot. Why this animation lives in chapter 01's animations folder: - the hydrogen atom is the only analytically-solvable many-body-free system, and its orbitals are the building block of every chemistry argument from hybridisation to crystal-field theory - it complements the particle-in-box animation (same idiom, one dimension higher) Run from the repo root: manim -qm dft_notes/animations/chapter_01/02-hydrogen-orbitals.py HydrogenOrbitals Writes to: dft_notes/animations/chapter_01/videos/hydrogen_orbitals/720p30/HydrogenOrbitals.mp4 (a small post-process step renames to dft_notes/animations/chapter_01/videos/02-hydrogen-orbitals.mp4 and captures the poster frame.) """ from manim import ( Axes, Create, Dot, FadeIn, FadeOut, MathTex, MoveAlongPath, Rectangle, Scene, Text, Transform, Write, UL, UP, DOWN, LEFT, RIGHT, ORIGIN, PI, config, ) import numpy as np from scipy.special import genlaguerre, factorial # ---- Colour palette (matches the rest of the site) ---------------------- COLOR_BG = "#1a1814" COLOR_TITLE = "#e8e4d8" COLOR_SUBTITLE = "#a8a496" COLOR_AXIS = "#7a7868" COLOR_GRID = "#3a3830" COLOR_PANEL = "#2a2820" COLOR_PANEL_BORDER = "#4a4838" COLORS_ORBITAL = ["#5b9bd5", "#70ad47", "#ffc000", "#ed7d31", "#a5a5a5", "#264478"] def R_nl(n, l, r): """Hydrogen radial wavefunction R_{nℓ}(r) in atomic units. R_{nℓ}(r) = (2/n)² * sqrt((n-ℓ-1)! / (2n(n+ℓ)!)) * exp(-r/n) * (2r/n)^ℓ * L_{n-ℓ-1}^{2ℓ+1}(2r/n) """ rho = 2.0 * r / n norm = np.sqrt((2.0 / n) ** 3 * factorial(n - l - 1) / (2.0 * n * factorial(n + l))) L = genlaguerre(n - l - 1, 2 * l + 1) # At r = 0, R_{nℓ} is non-zero only for ℓ = 0 if l > 0: return np.where(r > 0, norm * np.exp(-rho / 2.0) * rho**l * L(rho), 0.0) else: return np.where( r > 0, norm * np.exp(-rho / 2.0) * L(rho), 2.0 * (1.0 / n) ** 1.5 ) def mean_r(n, l): """Mean radius ⟨r⟩_{nℓ} = a₀ · (3n² - ℓ(ℓ+1)) / 2.""" return (3.0 * n * n - l * (l + 1)) / 2.0 def orbital_label(n, l): """The chemistry label for the orbital: 1s, 2s, 2p, 3s, 3p, 3d, ...""" s = ["s", "p", "d", "f", "g", "h"][l] return f"{n}{s}" class HydrogenOrbitals(Scene): def construct(self): # ---- Configuration -------------------------------------------------- r_max = 20.0 y_max = 0.6 # amplitude scale (R_max ≈ 0.6 for 1s) orbitals = [ (1, 0), # 1s (2, 0), # 2s (2, 1), # 2p (3, 0), # 3s (3, 1), # 3p (3, 2), # 3d ] # Fixed position for the mean-radius label (in DATA coordinates # of the r axis). Placing it at r = 4 a_0 keeps it in the # left half of the plot, well clear of the r-axis label at # the right end. The label TEXT is at this fixed x; the # VERTICAL LINE that indicates the mean radius is at the # actual r_mean (so the line moves but the label doesn't). MEAN_LABEL_X = 4.0 # ---- Coordinate system ---------------------------------------------- axes = Axes( x_range=[0, r_max, 5], y_range=[-y_max, y_max, 0.2], x_length=7.5, y_length=4.0, tips=False, axis_config={ "include_numbers": False, "stroke_width": 2, "color": COLOR_AXIS, }, ) axes.shift(RIGHT * 0.4 + DOWN * 0.2) # x label x_label = MathTex("r \\;(a_0)", font_size=28, color=COLOR_AXIS).next_to( axes.x_axis, RIGHT, buff=0.2 ) # Zero line zero_line = axes.plot( lambda r: 0, x_range=[0, r_max], color=COLOR_GRID, stroke_width=1, use_smoothing=False, ) # ---- Title + subtitle ----------------------------------------------- title = Text( "Hydrogen atom: the first nine orbitals", font_size=32, color=COLOR_TITLE, ).to_edge(UP, buff=0.35) # Short subtitle on a single line. The full list of orbitals # is shown in the info panel; the subtitle just announces # what's being plotted. subtitle = Text( "radial eigenfunctions R_{nℓ}(r) for n = 1, 2, 3 and ℓ = 0, 1, 2", font_size=22, color=COLOR_SUBTITLE, ).next_to(title, DOWN, buff=0.2) # ---- Info panel (top-left, fixed position) ------------------------ panel = Rectangle( width=3.2, height=2.4, fill_color=COLOR_PANEL, fill_opacity=0.85, stroke_color=COLOR_PANEL_BORDER, stroke_width=1.5, ).to_corner(UL, buff=0.4) # The "empty" / start state of the panel header_text = MathTex("|n,\\ell\\rangle", font_size=28, color=COLOR_SUBTITLE) header_text.move_to(panel.get_center() + UP * 0.85) name_text = MathTex("?", font_size=48, color=COLOR_SUBTITLE) name_text.move_to(panel.get_center() + UP * 0.2) energy_text = MathTex( "E_n = -\\tfrac{1}{2n^2}\\,E_h", font_size=22, color=COLOR_SUBTITLE ) energy_text.move_to(panel.get_center() + DOWN * 0.35) radius_text = MathTex( "\\langle r\\rangle = \\tfrac{3n^2 - \\ell(\\ell+1)}{2}\\,a_0", font_size=20, color=COLOR_SUBTITLE, ) radius_text.move_to(panel.get_center() + DOWN * 0.95) # ---- Summary line (bottom) ----------------------------------------- summary = Text( "E_n = -1/(2n²) Hartree · ⟨r⟩_{nℓ} = (3n² - ℓ(ℓ+1))/2 · a₀", font_size=22, color=COLOR_TITLE, ).to_edge(DOWN, buff=0.35) # ---- Animate the static elements --------------------------------- self.play(Write(title), run_time=1.0) self.play(Write(subtitle), run_time=0.8) self.play( Create(axes), Create(zero_line), Write(x_label), run_time=1.2, ) self.play( FadeIn(panel, scale=0.95), FadeIn(header_text), FadeIn(name_text), FadeIn(energy_text), FadeIn(radius_text), run_time=0.6, ) self.wait(0.3) # ---- Per-orbital animation ---------------------------------------- for idx, (n, l) in enumerate(orbitals): colour = COLORS_ORBITAL[idx] r_mean = mean_r(n, l) energy = -1.0 / (2 * n * n) # The radial function R_{nℓ}(r) curve = axes.plot( lambda r: R_nl(n, l, r), x_range=[0.001, r_max], color=colour, stroke_width=3.5, use_smoothing=True, ) # |R_{nℓ}|² (probability density), drawn at half-scale prob = axes.plot( lambda r: 0.5 * R_nl(n, l, r) ** 2, x_range=[0.001, r_max], color=colour, stroke_width=1.5, use_smoothing=True, ) # Mean-radius vertical line at the actual r_mean mean_line = axes.plot( lambda r: 0, x_range=[r_mean, r_mean], color=colour, stroke_width=1.2, use_smoothing=False, ) # Small dot at the bottom of the line mean_dot = Dot(axes.c2p(r_mean, 0), color=colour, radius=0.05) # The mean-radius LABEL goes at FIXED x = MEAN_LABEL_X, # placed ABOVE the x-axis. It never moves between orbitals # (only the value changes), so it can't collide with the # axis labels or the right-side info. mean_label = MathTex( f"\\langle r\\rangle = {r_mean:.1f}\\,a_0", font_size=20, color=colour, ) mean_label.move_to(axes.c2p(MEAN_LABEL_X, y_max - 0.15)) # Update the panel: new header, name, energy, radius. # All Transform() so they stay put (no stacking). new_header = MathTex(f"|{n},{l}\\rangle", font_size=28, color=colour) new_header.move_to(panel.get_center() + UP * 0.85) new_name = MathTex( f"\\text{{{orbital_label(n, l)}}}", font_size=48, color=colour, ) new_name.move_to(panel.get_center() + UP * 0.2) new_energy = MathTex( f"E_{{{n}}} = {energy:.4f}\\,E_h", font_size=22, color=colour ) new_energy.move_to(panel.get_center() + DOWN * 0.35) new_radius = MathTex( f"\\langle r\\rangle = {r_mean:.1f}\\,a_0", font_size=20, color=colour ) new_radius.move_to(panel.get_center() + DOWN * 0.95) self.play( Transform(header_text, new_header), Transform(name_text, new_name), Transform(energy_text, new_energy), Transform(radius_text, new_radius), run_time=0.5, ) # Draw the |R|² curve first (background) self.play(Create(prob), run_time=0.6) # Then the R curve self.play(Create(curve), run_time=0.6) # Then the mean-radius indicator (line + dot + label at # fixed x position) self.play( Create(mean_line), FadeIn(mean_dot, scale=0.5), Write(mean_label), run_time=0.4, ) # Animate a "particle" dot moving along the radial function dot = Dot(color=colour, radius=0.08).move_to(curve.point_from_proportion(0)) self.play(FadeIn(dot, scale=0.5), run_time=0.2) self.play(MoveAlongPath(dot, curve), run_time=0.9, rate_func=lambda t: t) self.play(FadeOut(dot), run_time=0.2) self.wait(0.3) # ---- Summary ------------------------------------------------------- self.play(Write(summary), run_time=1.5) self.wait(1.0)