""" 01-hellmann-feynman.py ====================== The Hellmann-Feynman theorem: where DFT forces come from. Every algebraic step is shown explicitly -- no skipped algebra. Companion to chapter 04, section 4.7 (signs and formulas verified against dft_notes/chapter_04/00-kohn-sham.md, lines 700-830). Scene graph ----------- 1. Title card. 2. Setup: H(lambda), E(lambda) = . 3. Product rule: the total derivative splits into three terms. 4. Terms (1) and (3) combine to E d/dlambda = 0. 5. The theorem, boxed: dE/dlambda = . 6. Force definition F_I = -dE/dR_I; only V_en and V_nn feel R_I. 7. Their derivatives are classical electrostatics; final force. Run from the repo root: manim -qm dft_notes/animations/chapter_04/01-hellmann-feynman.py HellmannFeynman 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, SurroundingRectangle, Line, UP, DOWN, ORIGIN, YELLOW, ) # ---------------------------------------------------------------- palette CORAL = "#cc785c" # site accent 1 TEALC = "#5db8a6" # site accent 2 INK = "#e8e4de" # warm off-white text DIM = "#9a958c" # secondary text WORK_X = -3.4 # working column centre PANEL_X = 5.05 # facts panel centre LABEL_Y = 2.48 # every section label lives here (title clears it) def panel_row(tex_src, y): """A right-panel results row.""" return MathTex(tex_src, color=INK, font_size=34).move_to(np.array([PANEL_X, y, 0.0])) def work_tex(src, y, color=INK, size=40): return Tex(src, color=color, font_size=size).move_to(np.array([WORK_X, y, 0.0])) def work_math(src, y, color=INK, size=38): return MathTex(src, color=color, font_size=size).move_to(np.array([WORK_X, y, 0.0])) class HellmannFeynman(Scene): def construct(self): # ------------------------------------------------ 1. title card title = Tex("The Hellmann--Feynman theorem", color=INK, font_size=40).to_edge(UP, buff=0.45) subtitle = Tex("where DFT forces come from", 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(1.0) self.play(FadeOut(subtitle), title.animate.scale(0.72).to_edge(UP, buff=0.25), run_time=0.8) # ------------------------------------------------ 2. setup panel_title = Tex("key facts", color=DIM, font_size=28).move_to(np.array([PANEL_X, 2.55, 0.0])) panel_rule = Line(np.array([3.15, 2.32, 0.0]), np.array([6.95, 2.32, 0.0]), color=DIM, stroke_width=1) self.play(FadeIn(panel_title), Create(panel_rule), run_time=0.7) lbl0 = work_tex("setup:", LABEL_Y, color=TEALC, size=30) s0a = work_math( r"E(\lambda) = \Bigl\langle \Psi(\lambda)\Bigm|\hat H(\lambda)\Bigm|\Psi(\lambda)\Bigr\rangle", 1.70, color=INK, size=36) s0b = work_tex(r"$\hat H(\lambda)$ depends on a parameter $\lambda$;", 0.92, color=DIM, size=26) s0c = work_tex(r"$\lambda$ could be a nuclear coordinate $R_{I\alpha}$", 0.44, color=DIM, size=26) self.play(Write(lbl0), run_time=0.8) self.play(Write(s0a), run_time=1.5) self.wait(0.9) self.play(FadeIn(s0b), run_time=0.7) self.play(FadeIn(s0c), run_time=0.7) self.wait(0.8) fact_norm_l = Tex("exact normalised ground state:", color=DIM, font_size=23).move_to(np.array([PANEL_X, 2.02, 0.0])) fact_norm = panel_row(r"\langle\Psi(\lambda)|\Psi(\lambda)\rangle = 1", 1.66) self.play(FadeIn(fact_norm_l), FadeIn(fact_norm), run_time=0.6) self.wait(0.5) # ------------------------------------- 3. total derivative: 3 terms lbl1 = work_tex(r"differentiate $E(\lambda)$ -- product rule:", LABEL_Y, color=CORAL, size=29) tag1 = Tex("(1)", color=CORAL, font_size=26).move_to(np.array([-5.62, 1.62, 0.0])) tag2 = Tex("(2)", color=TEALC, font_size=26).move_to(np.array([-5.62, 0.64, 0.0])) tag3 = Tex("(3)", color=CORAL, font_size=26).move_to(np.array([-5.62, -0.34, 0.0])) d1 = MathTex( r"\frac{dE}{d\lambda} \;=\;" r"\Bigl\langle \frac{\partial\Psi}{\partial\lambda}\Bigm|\hat H\Bigm|\Psi\Bigr\rangle", color=INK, font_size=36, ).move_to(np.array([-3.05, 1.62, 0.0])) d2 = MathTex( r"+\;\Bigl\langle \Psi\Bigm|\frac{\partial\hat H}{\partial\lambda}\Bigm|\Psi\Bigr\rangle", color=INK, font_size=36, ).move_to(np.array([WORK_X, 0.64, 0.0])) d3 = MathTex( r"+\;\Bigl\langle \Psi\Bigm|\hat H\Bigm|\frac{\partial\Psi}{\partial\lambda}\Bigr\rangle", color=INK, font_size=36, ).move_to(np.array([WORK_X, -0.34, 0.0])) self.play(FadeOut(VGroup(lbl0, s0a, s0b, s0c)), run_time=0.5) self.play(Write(lbl1), run_time=0.8) self.play(Write(d1), FadeIn(tag1), run_time=1.4) self.wait(0.8) self.play(Write(d2), FadeIn(tag2), run_time=1.3) self.wait(0.8) self.play(Write(d3), FadeIn(tag3), run_time=1.3) self.wait(1.0) fact_three = Tex(r"product rule $\Rightarrow$ three terms", color=DIM, font_size=23).move_to(np.array([PANEL_X, 1.06, 0.0])) self.play(FadeIn(fact_three), run_time=0.6) self.wait(0.6) # --------------------------------------- 4. terms (1)+(3) combine lbl2 = work_tex("terms (1) and (3) combine:", LABEL_Y, color=TEALC, size=29) c1 = MathTex( r"\Bigl\langle \frac{\partial\Psi}{\partial\lambda}\Bigm|\hat H\Bigm|\Psi\Bigr\rangle" r" + \Bigl\langle \Psi\Bigm|\hat H\Bigm|\frac{\partial\Psi}{\partial\lambda}\Bigr\rangle", color=INK, font_size=33, ).move_to(np.array([WORK_X, 1.55, 0.0])) c2 = MathTex( r"=\; E(\lambda)\,\frac{\partial}{\partial\lambda}\langle\Psi|\Psi\rangle", color=INK, font_size=36, ).move_to(np.array([WORK_X, 0.62, 0.0])) c3 = MathTex(r"=\; 0", color=YELLOW, font_size=44, ).move_to(np.array([WORK_X, -0.31, 0.0])) why = work_tex(r"$\Psi$ is normalised at every $\lambda$", -1.05, color=DIM, size=26) self.play(FadeOut(VGroup(lbl1, tag1, tag2, tag3, d1, d2, d3)), run_time=0.5) self.play(Write(lbl2), run_time=0.8) self.play(Write(c1), run_time=1.5) self.wait(0.8) self.play(Write(c2), run_time=1.5) self.wait(0.8) self.play(Write(c3), run_time=1.0) self.wait(0.6) self.play(FadeIn(why), run_time=0.7) self.wait(0.9) fact_zero = Tex(r"(1)$+$(3) $\to E\,\partial\langle\Psi|\Psi\rangle/\partial\lambda = 0$", color=DIM, font_size=23).move_to(np.array([PANEL_X, 0.56, 0.0])) self.play(FadeIn(fact_zero), run_time=0.6) # ------------------------------------------- 5. the theorem, boxed res_hf = MathTex( r"\frac{dE}{d\lambda} = " r"\Bigl\langle \Psi(\lambda)\Bigm|\frac{\partial\hat H}{\partial\lambda}\Bigm|\Psi(\lambda)\Bigr\rangle", color=YELLOW, font_size=40, ).move_to(np.array([WORK_X, 0.45, 0.0])) self.play(FadeOut(VGroup(lbl2, c1, c2, c3, why)), run_time=0.5) self.play(Write(res_hf), run_time=1.8) self.wait(0.9) box_hf = SurroundingRectangle(res_hf, color=YELLOW, buff=0.16) self.play(Create(box_hf), run_time=0.7) self.wait(0.7) hf_group = VGroup(res_hf, box_hf) fact_hf_l = Tex("Hellmann--Feynman:", color=DIM, font_size=23).move_to(np.array([PANEL_X, 0.10, 0.0])) self.play(hf_group.animate.scale(0.62).move_to(np.array([PANEL_X, -0.52, 0.0])), FadeIn(fact_hf_l), run_time=1.0) self.wait(0.6) # ------------------------------------------------ 6. force on a nucleus lblF = work_tex("force on nucleus $I$:", LABEL_Y, color=CORAL, size=30) f_def = MathTex( r"\mathbf{F}_I = -\frac{\partial E}{\partial \mathbf{R}_I}", color=INK, font_size=42, ).move_to(np.array([WORK_X, 1.70, 0.0])) f_n1 = work_tex(r"Born--Oppenheimer surface $E(\mathbf R)$;", 0.95, color=DIM, size=26) f_n2 = work_tex(r"apply the theorem with $\lambda = R_{I\alpha}$.", 0.47, color=DIM, size=26) self.play(Write(lblF), run_time=0.8) self.play(Write(f_def), run_time=1.3) self.wait(0.9) self.play(FadeIn(f_n1), run_time=0.7) self.play(FadeIn(f_n2), run_time=0.7) self.wait(0.8) fact_force_l = Tex("force definition:", color=DIM, font_size=23).move_to(np.array([PANEL_X, -1.28, 0.0])) fact_force = panel_row(r"\mathbf{F}_I = -\partial E/\partial \mathbf{R}_I", -1.63) self.play(FadeIn(fact_force_l), FadeIn(fact_force), run_time=0.6) self.wait(0.6) # ------------------------------------ 7. which pieces feel R_I? lblV = work_tex(r"only two pieces of $\hat H$ contain $\mathbf R_I$:", LABEL_Y, color=TEALC, size=29) v1a = MathTex( r"\hat V_{en} = -\sum_{i,I} \frac{Z_I}{|\,\mathbf r_i - \mathbf R_I\,|}", color=INK, font_size=34, ).move_to(np.array([WORK_X, 1.62, 0.0])) v1b = MathTex( r"\hat V_{nn} = \frac{1}{2}\sum_{I \neq J} \frac{Z_I Z_J}{|\,\mathbf R_I - \mathbf R_J\,|}", color=INK, font_size=34, ).move_to(np.array([WORK_X, 0.70, 0.0])) v2 = work_tex(r"$\hat T_e$, $\hat V_{ee}$ never mention nuclei.", 0.00, color=DIM, size=26) self.play(FadeOut(VGroup(lblF, f_def, f_n1, f_n2)), run_time=0.5) self.play(Write(lblV), run_time=0.8) self.play(Write(v1a), run_time=1.4) self.wait(0.7) self.play(Write(v1b), run_time=1.4) self.wait(0.7) self.play(FadeIn(v2), run_time=0.7) self.wait(0.9) # ------------------------------- 8. their R_I-derivatives dv_lbl = work_tex("differentiate those two:", LABEL_Y, color=CORAL, size=29) dv_en = MathTex( r"\frac{\partial \hat V_{en}}{\partial \mathbf R_I}" r"= -\sum_i Z_I\,\frac{\mathbf r_i - \mathbf R_I}{|\,\mathbf r_i - \mathbf R_I\,|^3}", color=CORAL, font_size=30, ).move_to(np.array([WORK_X, 1.58, 0.0])) dv_nn = MathTex( r"\frac{\partial \hat V_{nn}}{\partial \mathbf R_I}" r"= -\sum_{J \neq I} Z_I Z_J\," r"\frac{\mathbf R_I - \mathbf R_J}{|\,\mathbf R_I - \mathbf R_J\,|^3}", color=TEALC, font_size=30, ).move_to(np.array([WORK_X, 0.62, 0.0])) self.play(FadeOut(VGroup(lblV, v1a, v1b, v2)), run_time=0.5) self.play(Write(dv_lbl), run_time=0.8) self.play(Write(dv_en), run_time=1.6) self.wait(0.9) self.play(Write(dv_nn), run_time=1.6) self.wait(0.9) sign_note = work_tex( r"the minus in $\mathbf F_I = -\partial E / \partial\mathbf R_I$ flips both signs:", -0.18, color=DIM, size=26) self.play(FadeIn(sign_note), run_time=0.7) self.wait(0.9) # ------------------------------------------- 9. the force, assembled lblA = work_tex("assemble the force:", LABEL_Y, color=TEALC, size=29) g1 = MathTex( r"\mathbf F_I = +Z_I \int \rho(\mathbf r)\," r"\frac{\mathbf r - \mathbf R_I}{|\,\mathbf r - \mathbf R_I\,|^3}\, d\mathbf r", color=INK, font_size=32, ).move_to(np.array([WORK_X, 1.50, 0.0])) g2 = MathTex( r"+\;\sum_{J \neq I} Z_I Z_J\," r"\frac{\mathbf R_I - \mathbf R_J}{|\,\mathbf R_I - \mathbf R_J\,|^3}", color=INK, font_size=32, ).move_to(np.array([WORK_X + 0.6, 0.40, 0.0])) self.play(FadeOut(VGroup(dv_lbl, dv_en, dv_nn, sign_note)), run_time=0.5) self.play(Write(lblA), run_time=0.8) self.play(Write(g1), run_time=1.8) self.wait(0.9) self.play(Write(g2), run_time=1.5) self.wait(0.9) box_force = SurroundingRectangle(VGroup(g1, g2), color=YELLOW, buff=0.18) self.play(Create(box_force), run_time=0.8) interp = work_tex(r"classical Coulomb force on a charge $Z_I$ at $\mathbf R_I$:", -0.78, color=DIM, size=25) self.play(FadeIn(interp), run_time=0.8) self.wait(1.0) fact_classical = Tex("electrons attract; other nuclei repel.", color=DIM, font_size=23).move_to(np.array([PANEL_X, -2.18, 0.0])) self.play(FadeIn(fact_classical), run_time=0.6) self.wait(1.0) # ------------------------------------------------ 10. end card everything = VGroup(title, panel_title, panel_rule, fact_norm_l, fact_norm, fact_three, fact_zero, fact_hf_l, hf_group, fact_force_l, fact_force, fact_classical, lblA, g1, g2, box_force, interp) self.play(FadeOut(everything), run_time=0.9) end = Tex("The Hellmann--Feynman theorem:\\\\", "nuclei move in the classical electrostatic field\\\\", "of the electron density and the other nuclei.", color=INK, font_size=32).move_to(ORIGIN) src = Tex("chapter 04, section 4.7 ยท complete basis set (no Pulay term yet)", color=DIM, font_size=24).next_to(end, DOWN, buff=0.5) self.play(Write(end), run_time=1.5) self.play(FadeIn(src), run_time=0.8) self.wait(2.2)