""" 02-kohn-sham-mapping.py ======================= The Hohenberg-Kohn mapping: V_ext(r) <-> rho(r). The bijection that makes DFT possible, drawn as a forward/inverse arrow diagram. Companion to chapter 04 (HK theorems section). Scene graph ----------- 1. Title card. 2. Motivating question: can the density alone reconstruct the potential? 3. Two boxes side by side: v_ext(r) with a well sketch, rho(r) with a density bump sketch. 4. Forward arrow (trivial direction): Schroedinger equation -> unique rho. 5. The theorem: reverse arrow in yellow, rho -> v_ext (up to a constant); the well visibly lifts by a constant. 6. One-to-one correspondence highlighted around both boxes. 7. Consequence chain: v_ext -> H -> Psi -> all observables. 8. End card: all ground-state properties are functionals of rho. Run from the repo root: manim -qm dft_notes/animations/chapter_04/02-kohn-sham-mapping.py KohnShamMapping Writes to: dft_notes/animations/chapter_04/videos/... (see render pipeline in README) """ import numpy as np from manim import ( Scene, MathTex, Tex, VGroup, Write, Create, FadeIn, FadeOut, GrowArrow, Arrow, Line, ReplacementTransform, SurroundingRectangle, RoundedRectangle, ParametricFunction, LaggedStart, UP, DOWN, LEFT, RIGHT, ORIGIN, YELLOW, ) # ---------------------------------------------------------------- palette CORAL = "#cc785c" # site accent 1 -> the potential side TEALC = "#5db8a6" # site accent 2 -> the density side INK = "#e8e4de" # warm off-white text DIM = "#9a958c" # secondary text BOX_W = 4.0 BOX_H = 2.7 BOX_Y = 0.35 LX = -4.15 # left box centre x RX = 4.15 # right box centre x def box_at(cx, stroke_color): b = RoundedRectangle( corner_radius=0.12, width=BOX_W, height=BOX_H, stroke_color=stroke_color, stroke_width=2.5, ) b.set_fill(stroke_color, opacity=0.05) b.move_to(np.array([cx, BOX_Y, 0.0])) return b def r_axis(cx, y): """Tiny r-axis under a sketch.""" ln = Line(np.array([cx - 1.45, y, 0.0]), np.array([cx + 1.45, y, 0.0]), color=DIM, stroke_width=2) lab = Tex("r", color=DIM, font_size=22).next_to(ln, RIGHT, buff=0.10) return VGroup(ln, lab) class KohnShamMapping(Scene): def construct(self): # ------------------------------------------------ 1. title card title = Tex("The Hohenberg--Kohn mapping", color=INK, font_size=40).to_edge(UP, buff=0.45) subtitle = Tex("why DFT is possible", color=DIM, font_size=26).next_to(title, DOWN, buff=0.18) self.play(Write(title), run_time=2.0) self.play(FadeIn(subtitle), run_time=0.8) self.wait(1.4) self.play(FadeOut(subtitle), title.animate.scale(0.72).to_edge(UP, buff=0.25), run_time=0.9) # ------------------------------------------------ 2. motivating question q = Tex("can the density alone reconstruct the potential?", color=DIM, font_size=27).move_to(np.array([0.0, 0.4, 0.0])) self.play(FadeIn(q), run_time=0.9) self.wait(1.8) self.play(FadeOut(q), run_time=0.7) # ------------------------------------------------ 3. the two boxes # left: external potential, drawn as a well box_l = box_at(LX, CORAL) lbl_l = MathTex(r"v_{\mathrm{ext}}(\mathbf{r})", color=CORAL, font_size=40).move_to(np.array([LX, 1.22, 0.0])) desc_l = Tex("external potential", color=DIM, font_size=20).move_to(np.array([LX, 0.90, 0.0])) base_l = r_axis(LX, 0.35) dip = ParametricFunction( lambda t: np.array([ LX + t, 0.35 - 0.85 * np.exp(-(t / 0.62) ** 2), 0.0, ]), t_range=np.array([-1.45, 1.45, 0.04]), color=CORAL, stroke_width=3.5, ) self.play(Create(box_l), run_time=1.5) self.play(Write(lbl_l), FadeIn(desc_l), run_time=1.0) self.play(Create(base_l), run_time=0.6) self.play(Create(dip), run_time=2.0) self.wait(1.6) # right: ground-state density, drawn as a bump box_r = box_at(RX, TEALC) lbl_r = MathTex(r"\rho(\mathbf{r})", color=TEALC, font_size=40).move_to(np.array([RX, 1.22, 0.0])) desc_r = Tex("ground-state density", color=DIM, font_size=20).move_to(np.array([RX, 0.90, 0.0])) base_r = r_axis(RX, -0.55) bump = ParametricFunction( lambda t: np.array([ RX + t, -0.58 + 1.02 * np.exp(-(t / 0.55) ** 2), 0.0, ]), t_range=np.array([-1.40, 1.40, 0.04]), color=TEALC, stroke_width=3.5, ) bump.set_fill(TEALC, opacity=0.22) self.play(Create(box_r), run_time=1.5) self.play(Write(lbl_r), FadeIn(desc_r), run_time=1.0) self.play(Create(base_r), run_time=0.6) self.play(Create(bump), run_time=2.0) self.wait(1.6) # ------------------------------------------------ 4. forward arrow (trivial) fwd = Arrow(np.array([-1.95, 1.02, 0.0]), np.array([1.95, 1.02, 0.0]), color=CORAL, stroke_width=6, buff=0.0) self.play(GrowArrow(fwd), run_time=1.4) mini = Arrow(ORIGIN, RIGHT * 0.45, color=CORAL, stroke_width=5, buff=0.0, tip_length=0.16) cap_fwd = Tex("Schr\u00f6dinger equation $\\Rightarrow$ unique $\\rho$", color=INK, font_size=27) cap_tag = Tex("(trivial direction)", color=DIM, font_size=22) cap_row = VGroup(mini, cap_fwd, cap_tag).arrange(RIGHT, buff=0.20) cap_row.move_to(np.array([0.0, -1.52, 0.0])) self.play(FadeIn(cap_row, lag_ratio=0.25), run_time=1.0) self.wait(2.2) # ------------------------------------------------ 5. the theorem (yellow) rev_ghost = Arrow(np.array([1.95, -0.33, 0.0]), np.array([-1.95, -0.33, 0.0]), color=DIM, stroke_width=5, buff=0.0) self.play(GrowArrow(rev_ghost), run_time=1.5) self.wait(0.8) rev = Arrow(np.array([1.95, -0.33, 0.0]), np.array([-1.95, -0.33, 0.0]), color=YELLOW, stroke_width=9, buff=0.0) self.play(ReplacementTransform(rev_ghost, rev), run_time=1.0) self.wait(0.9) cap_rev = MathTex( r"\rho \;\longrightarrow\; v_{\mathrm{ext}}" r"\;\; \text{(up to a constant)}", color=YELLOW, font_size=30, ).move_to(np.array([0.0, -2.18, 0.0])) self.play(Write(cap_rev), run_time=1.3) self.wait(1.1) cap_hk = Tex("HK theorem: the density determines the potential", color=DIM, font_size=25).move_to(np.array([0.0, -2.76, 0.0])) self.play(FadeIn(cap_hk), run_time=0.9) self.wait(2.0) # "up to a constant", made visible: lift the well, put it back const_tag = MathTex(r"+\;\text{const}", color=DIM, font_size=24).move_to(np.array([LX + 1.02, -0.10, 0.0])) self.play(FadeIn(const_tag), run_time=0.6) self.play(dip.animate.shift(UP * 0.22), run_time=0.8) self.play(dip.animate.shift(DOWN * 0.22), run_time=0.8) self.play(FadeOut(const_tag), run_time=0.5) self.wait(0.9) # ------------------------------------------------ 6. one-to-one highlight bij_rect = SurroundingRectangle( VGroup(box_l, box_r), color=TEALC, buff=0.18, stroke_width=2, corner_radius=0.15, ) bij_lbl = Tex("one-to-one correspondence", color=TEALC, font_size=26).next_to(bij_rect, UP, buff=0.14) self.play(Create(bij_rect), run_time=1.1) self.play(FadeIn(bij_lbl), run_time=0.9) self.wait(2.2) # ------------------------------------------------ 7. consequence chain self.play(FadeOut(bij_rect), FadeOut(bij_lbl), run_time=0.6) chain = VGroup( MathTex(r"v_{\mathrm{ext}}", color=CORAL, font_size=33), MathTex(r"\longrightarrow", color=DIM, font_size=30), MathTex(r"\hat{H}", color=INK, font_size=33), MathTex(r"\longrightarrow", color=DIM, font_size=30), MathTex(r"\Psi", color=INK, font_size=33), MathTex(r"\longrightarrow", color=DIM, font_size=30), MathTex(r"\text{all observables}", color=YELLOW, font_size=33), ).arrange(RIGHT, buff=0.34).move_to(np.array([0.0, -3.42, 0.0])) self.play(LaggedStart(*[Write(c) for c in chain], lag_ratio=0.32), run_time=4.0) self.wait(2.4) # ------------------------------------------------ 8. end card self.wait(1.0) everything = VGroup( title, box_l, lbl_l, desc_l, base_l, dip, box_r, lbl_r, desc_r, base_r, bump, fwd, cap_row, rev, cap_rev, cap_hk, chain, ) self.play(FadeOut(everything), run_time=1.3) end = Tex("all ground-state properties are ", "functionals of $\\rho$", font_size=38).move_to(ORIGIN) end[0].set_color(INK) end[1].set_color(YELLOW) end_box = SurroundingRectangle(end, color=YELLOW, buff=0.22, stroke_width=2.5, corner_radius=0.12) src = Tex("Hohenberg--Kohn theorem \u00b7 chapter 04", color=DIM, font_size=22).next_to(end_box, DOWN, buff=0.45) self.play(Write(end), run_time=2.0) self.play(Create(end_box), run_time=0.9) self.play(FadeIn(src), run_time=0.8) self.wait(3.2)