""" 01-dft-u.py =========== LDA+U: the penalty functional. A two-orbital atom. With U = 0 the (spin-)levels are degenerate and the energy is indifferent to how the electron is spread -- the LDA failure. The penalty term E_U = (U_eff/2) sum_i n_i (1 - n_i) costs energy only for *fractional* occupations, so its minimum sits at integer n, and on the level diagram the same U_eff splits the degenerate pair into lower and upper Hubbard bands: the Mott gap. Scene graph ----------- 1. Title card. 2. The penalty formula, written large (persistent header). 3. LEFT : f(n) = n(1-n) on small axes -- inverted parabola, zero at n = 0, 1; max 1/4 at n = 1/2. 4. The two cases as dots on the parabola: LDA delocalised (1/2, 1/2) -> E_U = U_eff/4 (coral, apex) integer (1, 0) -> E_U = 0 (teal, base) 5. RIGHT: two-level diagram; U = 0 degenerate (metal) -> U ramps up, levels split by U_eff (LHB / UHB). 6. End card. Run from the repo root: manim -qm dft_notes/animations/chapter_13/01-dft-u.py DFTPlusU Writes to: dft_notes/animations/chapter_13/videos/... """ import numpy as np from manim import ( Scene, MathTex, Tex, Text, VGroup, Write, Create, FadeIn, FadeOut, Dot, DashedLine, Axes, DoubleArrow, Arrow, GrowArrow, Line, ValueTracker, Indicate, config, UP, DOWN, LEFT, RIGHT, UL, UR, DR, ORIGIN, YELLOW, smooth, ) # ---------------------------------------------------------------- palette CORAL = "#cc785c" # fractional / LDA accent TEALC = "#5db8a6" # integer / insulator accent INK = "#e8e4de" # warm off-white text DIM = "#9a958c" # secondary text # ------------------------------------------------------- layout constants LEFT_X = -3.55 # left-panel centre RIGHT_X = 3.95 # right-panel centre EPS_Y = -0.55 # U = 0 level energy (right panel) SPAN = 1.05 # half-split at U = 1 (upper +SPAN/2, lower -SPAN/2) XA0, XA1 = 2.95, 3.80 # upper-orbital line span XB0, XB1 = 4.10, 4.95 # lower-orbital line span class DFTPlusU(Scene): def construct(self): # ------------------------------------------------ 1. title card title = Tex("LDA+U: the penalty functional", color=INK, font_size=40).to_edge(UP, buff=0.45) subtitle = Tex("why fractional occupations cost energy", color=DIM, font_size=26).next_to(title, DOWN, buff=0.18) self.play(Write(title), run_time=1.6) self.play(FadeIn(subtitle), run_time=0.7) self.wait(1.6) self.play(FadeOut(subtitle), title.animate.scale(0.75).to_edge(UP, buff=0.25), run_time=0.8) # ------------------------------------------- 2. the big formula formula = MathTex( r"E_U \;=\;", r"\frac{U_{\mathrm{eff}}}{2}", r"\sum_i\;", r"n_i\,(1-n_i)", font_size=52, ).move_to(np.array([0.0, 2.25, 0.0])) formula[0].set_color(INK) formula[1].set_color(TEALC) formula[2].set_color(DIM) formula[3].set_color(CORAL) self.play(Write(formula), run_time=2.2) self.wait(2.6) # --------------------------------------------- 3. LEFT: parabola left_lbl = Tex(r"penalty per orbital: $f(n) = n(1-n)$", color=DIM, font_size=27).move_to(np.array([LEFT_X, 1.15, 0.0])) self.play(FadeIn(left_lbl), run_time=0.7) axes = Axes( x_range=[0, 1.2, 1], y_range=[0, 0.32, 0.1], x_length=4.4, y_length=2.7, tips=True, axis_config={"stroke_width": 2, "include_ticks": False, "tip_width": 0.14, "tip_height": 0.14}, ).move_to(np.array([LEFT_X, -0.75, 0.0])) curve = axes.plot(lambda t: t * (1.0 - t), x_range=[0, 1], color=CORAL, stroke_width=6) self.play(Create(axes), run_time=1.1) self.play(Create(curve), run_time=1.8) self.wait(1.0) xticks = VGroup(*[ MathTex(s, color=DIM, font_size=26).next_to(axes.c2p(v, 0.0), DOWN, buff=0.18) for v, s in [(0.0, "0"), (0.5, r"\tfrac12"), (1.0, "1")] ]) ytick = MathTex(r"\tfrac14", color=DIM, font_size=26).next_to(axes.c2p(0.0, 0.25), LEFT, buff=0.15) xlbl = MathTex("n", color=DIM, font_size=28).next_to( axes.c2p(1.2, 0.0), DR, buff=0.12) flbl = MathTex("f(n)", color=DIM, font_size=28).next_to( axes.c2p(0.0, 0.32), UP, buff=0.12) dash_v = DashedLine(axes.c2p(0.5, 0.0), axes.c2p(0.5, 0.25), color=DIM, stroke_width=2) dash_h = DashedLine(axes.c2p(0.0, 0.25), axes.c2p(0.5, 0.25), color=DIM, stroke_width=2) self.play(FadeIn(xticks), FadeIn(ytick), FadeIn(xlbl), FadeIn(flbl), Create(dash_v), Create(dash_h), run_time=1.0) self.wait(1.0) # ------------------------------------- 4. the two cases as dots dot_lda = Dot(axes.c2p(0.5, 0.25), radius=0.09, color=CORAL) lda_lbl = MathTex(r"n=\tfrac12", color=CORAL, font_size=26)\ .move_to(axes.c2p(0.76, 0.115)) self.play(FadeIn(dot_lda, scale=2.2), run_time=0.9) self.play(FadeIn(lda_lbl), run_time=0.8) self.wait(0.9) dot_int1 = Dot(axes.c2p(0.0, 0.0), radius=0.09, color=TEALC) dot_int2 = Dot(axes.c2p(1.0, 0.0), radius=0.09, color=TEALC) int1_lbl = MathTex(r"n=0", color=TEALC, font_size=24)\ .next_to(dot_int1, UL, buff=0.13) int2_lbl = MathTex(r"n=1", color=TEALC, font_size=24)\ .next_to(dot_int2, UR, buff=0.13) self.play(FadeIn(dot_int1, scale=2.2), FadeIn(dot_int2, scale=2.2), run_time=1.0) self.play(FadeIn(int1_lbl), FadeIn(int2_lbl), run_time=0.8) self.wait(1.4) row_lda = MathTex( r"\text{LDA } (\tfrac12,\tfrac12):\;\;", r"E_U = \frac{U_{\mathrm{eff}}}{2}\Bigl(\tfrac14+\tfrac14\Bigr) = \frac{U_{\mathrm{eff}}}{4}", color=CORAL, font_size=29, ).move_to(np.array([LEFT_X, -2.85, 0.0])) self.play(Write(row_lda), run_time=1.4) self.wait(2.2) row_int = MathTex( r"\text{integer } (1,0):\;\;", r"E_U", r"\;=\;", r"\frac{U_{\mathrm{eff}}}{2}\,(0+0)", r"\;=\;", r"0", color=TEALC, font_size=29, ).move_to(np.array([LEFT_X, -3.50, 0.0])) row_int[5].set_color(YELLOW) self.play(Write(row_int), run_time=1.3) self.wait(2.0) # ----------------------------------- 5. RIGHT: level diagram right_lbl = Tex(r"the same atom, energy levels:", color=DIM, font_size=27).move_to(np.array([RIGHT_X, 1.15, 0.0])) utracker = ValueTracker(0.0) upper = Line(np.array([XA0, EPS_Y, 0]), np.array([XA1, EPS_Y, 0]), color=INK, stroke_width=6) lower = Line(np.array([XB0, EPS_Y, 0]), np.array([XB1, EPS_Y, 0]), color=INK, stroke_width=6) def upd_upper(m): y = EPS_Y + 0.5 * SPAN * utracker.get_value() m.put_start_and_end_on(np.array([XA0, y, 0]), np.array([XA1, y, 0])) def upd_lower(m): y = EPS_Y - 0.5 * SPAN * utracker.get_value() m.put_start_and_end_on(np.array([XB0, y, 0]), np.array([XB1, y, 0])) upper.add_updater(upd_upper) lower.add_updater(upd_lower) cap_metal = Tex("$U = 0$: degenerate levels $\\Rightarrow$ metal", color=DIM, font_size=26)\ .move_to(np.array([RIGHT_X, -2.35, 0.0])) self.play(FadeIn(right_lbl), run_time=0.7) self.play(FadeIn(upper), FadeIn(lower), run_time=1.1) self.play(FadeIn(cap_metal), run_time=0.9) self.wait(2.2) # --- ramp U up: one .animate only, everything else via updaters self.play(utracker.animate.set_value(1.0), run_time=4.6, rate_func=smooth) self.wait(1.2) # gap bracket, built at the final split (arrows cannot be resized) gap_arrow = DoubleArrow( np.array([5.35, EPS_Y - 0.5 * SPAN, 0]), np.array([5.35, EPS_Y + 0.5 * SPAN, 0]), buff=0, stroke_width=3, tip_length=0.16, color=YELLOW) gap_lbl = MathTex(r"U_{\mathrm{eff}}", color=YELLOW, font_size=32)\ .next_to(gap_arrow, RIGHT, buff=0.18) uhb = MathTex(r"\text{UHB}", color=CORAL, font_size=30)\ .move_to(np.array([2.55, EPS_Y + SPAN * 0.5, 0])) lhb = MathTex(r"\text{LHB}", color=TEALC, font_size=30)\ .move_to(np.array([2.55, EPS_Y - SPAN * 0.5, 0])) self.play(FadeIn(uhb), FadeIn(lhb), run_time=1.0) self.wait(0.6) self.play(FadeIn(gap_arrow), FadeIn(gap_lbl), run_time=1.0) self.play(Indicate(gap_lbl, color=YELLOW), run_time=0.9) cap_split = Tex("with $U_{\\mathrm{eff}}>0$ the manifold splits", color=INK, font_size=26)\ .move_to(np.array([RIGHT_X, -2.35, 0.0])) self.play(FadeOut(cap_metal), run_time=0.5) self.play(Write(cap_split), run_time=1.2) self.wait(1.4) cap_gap = Tex("gap $\\approx U_{\\mathrm{eff}}$ $\\Rightarrow$ Mott insulator", color=YELLOW, font_size=28)\ .move_to(np.array([RIGHT_X, -3.00, 0.0])) self.play(Write(cap_gap), run_time=1.4) self.wait(3.0) self.wait(0.8) # ------------------------------------------------ 6. end card everything = VGroup( title, formula, left_lbl, axes, curve, xticks, ytick, xlbl, flbl, dash_v, dash_h, dot_lda, lda_lbl, dot_int1, dot_int2, int1_lbl, int2_lbl, row_lda, row_int, right_lbl, upper, lower, gap_arrow, gap_lbl, uhb, lhb, cap_split, cap_gap, ) self.play(FadeOut(everything), run_time=1.1) end1 = Tex("the penalty forces integer occupations", color=INK, font_size=36).move_to(np.array([0.0, 1.15, 0.0])) end_arrow = Arrow(np.array([0.0, 0.55, 0]), np.array([0.0, -0.30, 0]), buff=0, stroke_width=4, color=DIM) end2 = Tex("the Mott gap opens", color=YELLOW, font_size=52).move_to(np.array([0.0, -1.00, 0.0])) src = Tex("chapter 13 · LDA+U · minimum of $n(1-n)$ is at integer filling", color=DIM, font_size=24).next_to(end2, DOWN, buff=0.55) self.play(Write(end1), run_time=1.5) self.play(GrowArrow(end_arrow), run_time=0.7) self.play(Write(end2), run_time=1.2) self.play(FadeIn(src), run_time=0.8) self.wait(4.0)