""" 01-h2-two-configuration.py ========================== Full CI on H2: the two-configuration problem, solved live. Every arithmetic step is shown explicitly -- no skipped algebra. Companion to chapter 02, section 2.5 (all numbers verified against PySCF and dft_notes/python_codes/chapter_02/03-h2-full-ci-toy.py). Scene graph ----------- 1. Title card. 2. The two closed-shell determinants Phi1 = |gg> and Phi4 = |uu>. 3. The MO integrals (h_gg, h_uu, J_gg, J_uu, K_gu) as a reference panel. 4. Matrix elements, one substitution per line: H11 = 2 h_gg + J_gg = -1.8310 H44 = 2 h_uu + J_uu = -0.2537 H14 = K_gu = +0.1813 5. The 2x2 Hamiltonian block assembles itself from those numbers. 6. The eigenvalue formula; each numeric substitution shown: mean -> half-difference -> squares -> sum -> root -> E0(el) -> E0(tot) 7. Ground-state composition and correlation energy. Run from the repo root: manim -qm dft_notes/animations/chapter_02/01-h2-two-configuration.py H2TwoConfiguration Writes to: dft_notes/animations/chapter_02/videos/... (see render pipeline in README) """ import numpy as np from manim import ( Scene, MathTex, Tex, Text, VGroup, Write, Create, FadeIn, FadeOut, TransformMatchingTex, Transform, ReplacementTransform, Circle, Indicate, SurroundingRectangle, Rectangle, Line, Arrow, config, UP, DOWN, LEFT, RIGHT, UL, DR, ORIGIN, PI, RED, GREEN, BLUE, YELLOW, GREY, WHITE, TEAL, MAROON, PURPLE, ) # ---------------------------------------------------------------- palette CORAL = "#cc785c" # site accent 1 TEALC = "#5db8a6" # site accent 2 INK = "#e8e4de" # warm off-white text DIM = "#9a958c" # secondary text TITLE_Y = 3.35 # title baseline (frame half-height is 4) def result_line(tex_src, y): """A right-panel results row.""" return MathTex(tex_src, color=INK, font_size=38).move_to(np.array([5.05, y, 0.0])) class H2TwoConfiguration(Scene): def construct(self): # ------------------------------------------------ 1. title card title = Tex("Full CI on H$_2$: the two-configuration problem", color=INK, font_size=40).to_edge(UP, buff=0.45) subtitle = Tex("every algebraic step shown", 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. the two states phi1 = MathTex(r"\Phi_1 = |\sigma_g \bar\sigma_g\rangle", color=TEALC, font_size=44).move_to(np.array([-3.4, 1.7, 0.0])) phi4 = MathTex(r"\Phi_4 = |\sigma_u \bar\sigma_u\rangle", color=CORAL, font_size=44).move_to(np.array([-3.4, 0.85, 0.0])) note = Tex("both electrons in the bonding / antibonding MO", color=DIM, font_size=26).next_to(phi4, DOWN, buff=0.35) self.play(Write(phi1), run_time=1.2) self.play(Write(phi4), run_time=1.2) self.play(FadeIn(note), run_time=0.6) self.wait(1.2) # --------------------------------------- 3. integrals side panel panel_title = Tex("MO integrals ($E_h$)", color=DIM, font_size=28).move_to(np.array([5.05, 2.55, 0.0])) panel_rule = Line([3.15, 2.32, 0], [6.95, 2.32, 0], color=DIM, stroke_width=1) self.play(FadeIn(panel_title), Create(panel_rule), run_time=0.7) rows = VGroup( result_line(r"h_{gg} = -1.2528", 2.05), result_line(r"h_{uu} = -0.4756", 1.50), result_line(r"J_{gg} = 0.6746", 0.95), result_line(r"J_{uu} = 0.6975", 0.40), result_line(r"K_{gu} = 0.1813", -0.15), result_line(r"V_{NN} = 0.7143", -0.70), ) self.play(FadeIn(rows, lag_ratio=0.12), run_time=1.6) self.wait(1.4) # ------------------------------------------------ 4. H11, step by step work_y = 1.7 # top of the working column dy = -0.62 # line spacing lbl1 = Tex("matrix element $H_{11}$:", color=TEALC, font_size=30).move_to([-3.4, work_y, 0]) s1a = MathTex(r"H_{11} = 2\,h_{gg} + J_{gg}", color=INK, font_size=40).move_to([-3.4, work_y + dy, 0]) s1b = MathTex(r"= 2\,(-1.2528) + 0.6746", color=INK, font_size=40).move_to([-3.4, work_y + 2 * dy, 0]) s1c = MathTex(r"= -2.5056\; + \;0.6746", color=INK, font_size=40).move_to([-3.4, work_y + 3 * dy, 0]) s1d = MathTex(r"= \;-1.8310", color=YELLOW, font_size=44).move_to([-3.4, work_y + 4 * dy, 0]) self.play(FadeOut(note), FadeOut(phi1), FadeOut(phi4), run_time=0.5) self.play(Write(lbl1), run_time=0.8) self.play(Write(s1a), run_time=1.2) self.wait(0.6) self.play(Write(s1b), run_time=1.2) self.wait(0.6) self.play(Write(s1c), run_time=1.0) self.wait(0.6) self.play(Write(s1d), run_time=0.9) self.wait(1.0) res_H11 = result_line(r"H_{11} = -1.8310", -1.25).set_color(YELLOW) box_H11 = SurroundingRectangle(res_H11, color=TEALC, buff=0.10) self.play(FadeOut(VGroup(lbl1, s1a, s1b, s1c)), s1d.animate.move_to(res_H11), run_time=0.8) self.play(FadeIn(box_H11), run_time=0.5) self.add(res_H11) self.remove(s1d) res_H11_row = VGroup(res_H11, box_H11) self.wait(0.6) # ------------------------------------------------ 5. H44 work_y2 = 1.7 lbl2 = Tex("matrix element $H_{44}$:", color=CORAL, font_size=30).move_to([-3.4, work_y2, 0]) s2a = MathTex(r"H_{44} = 2\,h_{uu} + J_{uu}", color=INK, font_size=40).move_to([-3.4, work_y2 + dy, 0]) s2b = MathTex(r"= 2\,(-0.4756) + 0.6975", color=INK, font_size=40).move_to([-3.4, work_y2 + 2 * dy, 0]) s2c = MathTex(r"= -0.9512\; + \;0.6975", color=INK, font_size=40).move_to([-3.4, work_y2 + 3 * dy, 0]) s2d = MathTex(r"= \;-0.2537", color=YELLOW, font_size=44).move_to([-3.4, work_y2 + 4 * dy, 0]) self.play(FadeOut(res_H11_row), run_time=0.4) self.play(Write(lbl2), run_time=0.8) self.play(Write(s2a), run_time=1.2) self.wait(0.6) self.play(Write(s2b), run_time=1.2) self.wait(0.6) self.play(Write(s2c), run_time=1.0) self.wait(0.6) self.play(Write(s2d), run_time=0.9) self.wait(1.0) # park H44 in the results panel next to H11 res_H44_final = result_line(r"H_{44} = -0.2537", -1.80).set_color(YELLOW) self.play(FadeOut(VGroup(lbl2, s2a, s2b, s2c)), s2d.animate.move_to(res_H44_final), run_time=0.8) self.remove(s2d); self.add(res_H44_final) self.wait(0.5) # ------------------------------------------------ 6. H14 lbl3 = Tex("coupling:", color=PURPLE, font_size=30).move_to([-3.4, work_y2, 0]) s3a = MathTex(r"H_{14} = K_{gu} = (\,gu \mid gu\,)", color=INK, font_size=40).move_to([-3.4, work_y2 + dy, 0]) s3b = Tex("direct term dies on spin orthogonality;", color=DIM, font_size=27).move_to([-3.4, work_y2 + 2 * dy, 0]) s3c = Tex("exchange survives.", color=DIM, font_size=27).move_to([-3.4, work_y2 + 3 * dy, 0]) s3d = MathTex(r"= \;+0.1813", color=YELLOW, font_size=44).move_to([-3.4, work_y2 + 4 * dy, 0]) self.play(Write(lbl3), run_time=0.7) self.play(Write(s3a), run_time=1.2) self.wait(0.7) self.play(FadeIn(s3b), run_time=0.7) self.play(FadeIn(s3c), run_time=0.7) self.wait(0.7) self.play(Write(s3d), run_time=0.9) self.wait(1.0) # both parked values now on the right, stacked res_H14 = result_line(r"H_{14} = +0.1813", -2.35).set_color(YELLOW) self.play(FadeOut(VGroup(lbl3, s3a, s3b, s3c)), s3d.animate.move_to(res_H14), FadeOut(res_H44_final), run_time=0.8) self.add(res_H14); self.remove(s3d) self.wait(0.6) # ------------------------------------------------ 7. assemble matrix mat_label = Tex("the $2 \\times 2$ block:", color=INK, font_size=30).move_to(np.array([-3.4, 1.7, 0.0])) mat = MathTex( r"\mathbf{H} \;=\;", r"\begin{pmatrix} -1.8310 & +0.1813 \\ +0.1813 & -0.2537 \end{pmatrix}", color=INK, font_size=44, ).move_to(np.array([-3.4, 0.35, 0.0])) mat[1][1:8].set_color(TEALC) # -1.8310 mat[1][16:23].set_color(CORAL) # -0.2537 self.play(Write(mat_label), run_time=0.7) self.play(Write(mat), run_time=1.8) self.wait(1.4) # ------------------------------------------------ 8. eigen formula eig_lbl = Tex("lower eigenvalue:", color=INK, font_size=30).move_to(np.array([-3.4, -1.15, 0.0])) eig = MathTex( r"E_- = \frac{H_{11}+H_{44}}{2}" r" - \sqrt{\Bigl(\frac{H_{11}-H_{44}}{2}\Bigr)^{\!2} + H_{14}^2}", color=INK, font_size=36, ).move_to(np.array([-3.4, -1.95, 0.0])) self.play(FadeOut(mat_label), mat.animate.scale(0.78).to_edge(UP, buff=1.15), run_time=0.7) self.play(Write(eig_lbl), run_time=0.7) self.play(Write(eig), run_time=2.0) self.wait(1.4) # ------------------------------------------------ 9. substitutions steps = [ (r"\frac{H_{11}+H_{44}}{2} = \frac{-1.8310 + (-0.2537)}{2} = -1.0424",), (r"\frac{H_{11}-H_{44}}{2} = \frac{-1.8310 - (-0.2537)}{2} = -0.7886",), (r"\left(-0.7886\right)^2 = 0.6220, \qquad (0.1813)^2 = 0.0329",), (r"\sqrt{0.6220 + 0.0329} = \sqrt{0.6548} = 0.8092",), ] cur = eig for i, (src,) in enumerate(steps): nxt = MathTex(src, color=INK, font_size=34).move_to(eig) self.play(FadeOut(cur), Write(nxt), run_time=1.1) self.wait(1.0) cur = nxt # final energy chain e_el = MathTex(r"E_-(\mathrm{el}) = -1.0424 - 0.8092 = -1.8516", color=INK, font_size=34).move_to(eig) self.play(FadeOut(cur), Write(e_el), run_time=1.1) self.wait(0.8) e_tot = MathTex( r"E_0 = -1.8516 + 0.7143 = \mathbf{-1.1373}\; E_h", color=YELLOW, font_size=42, ).move_to(eig) self.play(FadeOut(e_el), Write(e_tot), run_time=1.1) box_E = SurroundingRectangle(e_tot, color=YELLOW, buff=0.14) self.play(Create(box_E), run_time=0.7) self.wait(1.6) # ------------------------------------------------ 10. ground state gs_lbl = Tex("ground state:", color=INK, font_size=30).move_to(np.array([-3.4, -0.55, 0.0])) gs = MathTex( r"\Psi_0 = 0.9936\,\Phi_1 \;-\; 0.1128\,\Phi_4", color=INK, font_size=40, ).move_to(np.array([-3.4, -1.30, 0.0])) wts = Tex("weights: 98.7% HF-like + 1.3% doubly excited", color=DIM, font_size=27).move_to(np.array([-3.4, -2.00, 0.0])) corr = MathTex( r"E_{\mathrm{corr}} = -1.1373 - (-1.1167) = -0.0206\;E_h", color=TEALC, font_size=36, ).move_to(np.array([-3.4, -2.75, 0.0])) self.play(FadeOut(VGroup(eig_lbl, box_E)), run_time=0.5) self.play(e_tot.animate.move_to(np.array([-3.4, -0.55, 0.0])).scale(0.8), run_time=0.7) self.remove(gs_lbl) self.play(Write(gs), run_time=1.4) self.play(FadeIn(wts), run_time=0.7) self.wait(1.0) self.play(Write(corr), run_time=1.4) self.wait(1.6) # ------------------------------------------------ 11. end card everything = VGroup(title, panel_title, panel_rule, rows, mat, e_tot, gs, wts, corr) self.play(FadeOut(everything), run_time=0.9) end = Tex("Left--right correlation:\\\\", "the exact wavefunction mixes in the doubly excited configuration.", color=INK, font_size=34).move_to(ORIGIN) src = Tex("chapter 02, section 2.5 ยท verified against PySCF", color=DIM, font_size=24).next_to(end, DOWN, buff=0.5) self.play(Write(end), run_time=1.4) self.play(FadeIn(src), run_time=0.8) self.wait(2.2)