H2 Bond Breaking
A visual explanation of static correlation in H2 covalent bond dissociation. A three-stage orbital storyboard follows H2 from a bonded molecule to separated neutral atoms. Calculated STO-3G restricted Hartree-Fock, unrestricted Hartree-Fock, and full configuration interaction curves show the energy error of a restricted determinant, while interacting natural occupations evolve toward one electron in each orbital. Separated-atom pair-probability maps expose the restricted reference's 50% ionic weight and show how a coherent combination of bonding and antibonding configurations cancels it. The diagram distinguishes the interacting state from the auxiliary Kohn-Sham determinant, which can remain doubly occupied when the exact exchange-correlation functional supplies the missing correlation.
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Code
h2-bond-breaking.typ (716 lines)
#import "@preview/cetz:0.5.2": canvas, draw
#import draw: circle, content, line, rect
// H2 in STO-3G: fixed nuclei, nonrelativistic Hamiltonian, no fitted curve shapes.
// Tabulated data below were calculated from normalized contracted s-Gaussian integrals.
// H exponents: 3.42525091, 0.62391373, 0.16885540 bohr^-2.
// Contraction coefficients: 0.15432897, 0.53532814, 0.44463454.
// RHF: doubly occupied gerade orbital. UHF: minimize over opposite g/u rotations.
// FCI: diagonalize the 2x2 gerade singlet block; checked against the full four-state
// alpha/beta product space at R = 0.45, 0.74, 1.2, 2, 6, 30 A.
// At 0.74 A: RHF = -1.1167593074 Eh; FCI = -1.1372838345 Eh.
// Isolated H = -0.4665818496 Eh in this basis. The chart subtracts twice this value.
// Natural occupations are twice the squared g^2/u^2 configuration amplitudes.
// The atomic probability cartoons and spatial wavefunction formula use R -> infinity;
// A and B are normalized, nonoverlapping 1s orbitals. The singlet spin factor is implicit.
// Sources: Fuchs et al., JCP 122, 094116 (2005), doi:10.1063/1.1858371;
// Burke, The ABC of DFT, chapters 4, 7, 11, 13; Hehre et al., JCP 51, 2657 (1969).
#set page(width: auto, height: auto, margin: 0pt, fill: none)
#set text(font: "Avenir Next", size: 20pt, fill: rgb("#20334C"))
#set par(leading: 0.5em)
#set math.equation(numbering: none)
#let ink = rgb("#19304E")
#let muted = rgb("#344257")
#let blue = rgb("#2764C3")
#let orange = rgb("#BC502D")
#let purple = rgb("#7948AD")
#let teal = rgb("#087F7C")
#let pale_blue = rgb("#c7d3e1")
#let pale_orange = rgb("#e1d2c7")
#let pale_teal = rgb("#c7d8d1")
#let label(left, top, width, body, size: 20pt, color: ink, weight: "regular", centered: false) = {
content(
(left, -top),
block(width: width * 1pt)[
#text(size: size, fill: color, weight: weight, if centered { align(center, body) } else {
body
})
],
anchor: "north-west",
padding: 0pt,
)
}
#let symbol(center_x, center_y, body, size: 24pt, color: ink) = {
content(
(center_x, -center_y),
text(size: size, fill: color, top-edge: "bounds", bottom-edge: "bounds", body),
anchor: "center",
padding: 0pt,
)
}
#let panel(left, top, width, height, fill) = {
rect((left, -top), (left + width, -top - height), radius: 15, fill: fill, stroke: none)
}
#let connect(start, end, color: muted) = {
line(start, end, stroke: 1.7pt + color, mark: (end: "stealth", scale: 0.7, fill: color))
}
#let nucleus(center_x, center_y) = {
circle((center_x, -center_y), radius: 9, fill: ink, stroke: none)
line((center_x - 4, -center_y), (center_x + 4, -center_y), stroke: 1.2pt + white)
line((center_x, -center_y - 4), (center_x, -center_y + 4), stroke: 1.2pt + white)
}
#let electron(center_x, center_y, spin) = {
let direction = if spin == "up" { 1 } else { -1 }
circle((center_x, -center_y), radius: 9, fill: purple, stroke: none)
line(
(center_x, -center_y - direction * 4.5),
(center_x, -center_y + direction * 4.5),
stroke: 1.2pt + white,
mark: (end: "stealth", scale: 0.42, fill: white),
)
}
#let orbital_cloud(center_x, center_y, radius_x, radius_y, color: blue) = {
for idx in range(24) {
let scale = 1 - idx / 30
circle(
(center_x, -center_y),
radius: (radius_x * scale, radius_y * scale),
fill: color.lighten(95% - idx / 23 * 24%),
stroke: none,
)
}
}
// Columns: R [A], E_RHF - 2E_H, E_UHF - 2E_H, E_FCI - 2E_H [Eh], n_g, n_u.
#let curve_data = csv(
bytes(
"
0.3,0.339335940535,0.339335940535,0.331359988305,1.99509254849,0.00490745151211
0.325,0.236279239964,0.236279239964,0.227900681554,1.99464277172,0.00535722828441
0.35,0.152708797009,0.152708797009,0.143894306676,1.9941379224,0.00586207760284
0.375,0.0845649185931,0.0845649185931,0.0752807036227,1.99357359604,0.00642640396028
0.4,0.0288023049251,0.0288023049251,0.0190139944613,1.99294499151,0.00705500849237
0.45,-0.0543494386122,-0.0543494386122,-0.0652518969214,1.99147367515,0.00852632484561
0.475,-0.0849513847429,-0.0849513847429,-0.0964655496514,1.99061928175,0.00938071825293
0.5,-0.10983257544,-0.10983257544,-0.121996095371,1.98967725972,0.010322740281
0.525,-0.129898443818,-0.129898443818,-0.142749956087,1.98864076226,0.0113592377447
0.55,-0.145887037073,-0.145887037073,-0.159466207641,1.98750256653,0.012497433472
0.575,-0.158406797671,-0.158406797671,-0.172754354148,1.98625508455,0.0137449154462
0.6,-0.167964543159,-0.167964543159,-0.183122307762,1.98489036469,0.0151096353108
0.625,-0.174986291082,-0.174986291082,-0.190997220521,1.98340007963,0.0165999203655
0.65,-0.179832846558,-0.179832846558,-0.196741085212,1.98177549814,0.0182245018582
0.675,-0.182811564713,-0.182811564713,-0.200662516032,1.98000743939,0.0199925606118
0.7,-0.184185335876,-0.184185335876,-0.203025754953,1.97808621099,0.0219137890089
0.725,-0.184179570007,-0.184179570007,-0.204057677958,1.97600153366,0.0239984663356
0.74,-0.183595608281,-0.183595608281,-0.204120135374,1.97466774704,0.0253322529562
0.75,-0.182987749822,-0.182987749822,-0.203953368231,1.9737424573,0.0262575427009
0.775,-0.180775962626,-0.180775962626,-0.202880664112,1.97129727461,0.0287027253931
0.8,-0.177686698358,-0.177686698358,-0.20098396756,1.96865343924,0.0313465607553
0.825,-0.173842108796,-0.173842108796,-0.19838750679,1.96579749551,0.0342025044904
0.85,-0.169346854805,-0.169346854805,-0.19519817934,1.96271502629,0.0372849737073
0.875,-0.16429062058,-0.16429062058,-0.191508054447,1.95939062513,0.0406093748655
0.9,-0.158750341899,-0.158750341899,-0.18739658218,1.95580789707,0.0441921029255
0.925,-0.152792175846,-0.152792175846,-0.182932537672,1.95194949172,0.0480505082802
0.95,-0.146473229088,-0.146473229088,-0.178175718616,1.94779717069,0.052202829313
0.975,-0.139843057563,-0.139843057563,-0.173178410166,1.94333191042,0.0566680895787
1,-0.132944950194,-0.132944950194,-0.167986631111,1.93853404041,0.0614659595939
1.025,-0.125817011009,-0.125817011009,-0.162641176892,1.93338341596,0.0666165840427
1.05,-0.118493056522,-0.118493056522,-0.157178477391,1.92785962413,0.0721403758735
1.075,-0.111003347519,-0.111003347519,-0.1516312895,1.92194222067,0.0780577793349
1.1,-0.103375175904,-0.103375175904,-0.146029245847,1.91561099548,0.084389004518
1.125,-0.0956333278956,-0.0956333278956,-0.140399281481,1.9088462635,0.0911537364983
1.15,-0.0878004444537,-0.0878004444537,-0.13476595979,1.90162917726,0.0983708227355
1.175,-0.0798972985993,-0.0801784712832,-0.12915171756,1.89394205698,0.106057943018
1.2,-0.0719430074416,-0.0732088127404,-0.123577047182,1.88576873306,0.114231266943
1.225,-0.063955194439,-0.066842059785,-0.118060631635,1.87709489533,0.122905104672
1.25,-0.0559501149635,-0.0610264547671,-0.112619445427,1.86790844248,0.132091557523
1.275,-0.0479427557704,-0.0557147849465,-0.107268832186,1.85819982431,0.141800175691
1.3,-0.0399469166497,-0.0508639152697,-0.102022567311,1.84796236887,0.152037631126
1.325,-0.0319752804576,-0.0464343894629,-0.0968929120081,1.83719258584,0.162807414157
1.35,-0.0240394759464,-0.0423900866883,-0.0918906633457,1.82589043724,0.174109562765
1.375,-0.0161501363548,-0.0386979231224,-0.087025203486,1.81405956649,0.185940433506
1.4,-0.00831695557921,-0.0353275897783,-0.0823045501653,1.80170747724,0.198292522764
1.45,0.00714652339066,-0.0294436787105,-0.0733232327065,1.77548961591,0.224510384094
1.51594202899,0.0270430118527,-0.0231599631324,-0.0624664898593,1.73804644183,0.261953558171
1.58188405797,0.0462752222731,-0.0182241422939,-0.0527695325009,1.69779892855,0.30220107145
1.64782608696,0.0647631982805,-0.0143473042433,-0.0442282101762,1.65539027766,0.344609722344
1.71376811594,0.082450979081,-0.0113018953429,-0.0368012976597,1.61157319453,0.388426805472
1.77971014493,0.0993022955028,-0.0089090437726,-0.0304197503905,1.56714783794,0.432852162062
1.84565217391,0.115297126157,-0.00702830555667,-0.0249956143654,1.52289773679,0.477102263214
1.9115942029,0.130428938698,-0.00554945978974,-0.020430129755,1.47953421836,0.520465781644
1.97753623188,0.144702412877,-0.00438600966875,-0.0166206483383,1.43765671428,0.562343285721
2,0.14937104485,-0.00404913398153,-0.0154774130585,1.42381726988,0.576182730119
2.04347826087,0.158131470755,-0.00347007224105,-0.0134661253541,1.39773179038,0.602268209622
2.10942028986,0.170737496078,-0.00274837583053,-0.0108711154627,1.36008957314,0.639910426861
2.17536231884,0.182547685276,-0.00217912757237,-0.0087483592458,1.32493353028,0.675066469721
2.24130434783,0.193593521384,-0.00172955830986,-0.00702016021337,1.29235857188,0.707641428116
2.30724637681,0.203909392189,-0.00137399328148,-0.00561881103006,1.26237280241,0.737627197591
2.3731884058,0.213531384651,-0.00109233182523,-0.00448633692002,1.23491935861,0.765080641392
2.43913043478,0.222496283021,-0.000868847014815,-0.00357379849258,1.20989606537,0.790103934632
2.50507246377,0.230840783892,-0.000691237268374,-0.00284035177735,1.18717178323,0.812828216772
2.57101449275,0.238600923828,-0.000549877680824,-0.00225221337773,1.16659916029,0.833400839709
2.63695652174,0.245811699019,-0.000437230394414,-0.00178163209634,1.14802402639,0.851975973609
2.70289855072,0.252506844722,-0.000347381866822,-0.00140592966136,1.13129193583,0.868708064167
2.76884057971,0.258718736328,-0.000275681311676,-0.00110664368613,1.11625245338,0.883747546618
2.8347826087,0.264478373541,-0.000218459522425,-0.00086878523768,1.10276175623,0.897238243766
2.90072463768,0.269815413209,-0.000172811195852,-0.000680209963068,1.09068404591,0.909315954087
2.96666666667,0.274758223075,-0.000136427043853,-0.000531093937819,1.07989216426,0.920107835739
3.03260869565,0.279333936708,-0.000107464611371,-0.000413501649085,1.07026770883,0.929732291173
3.09855072464,0.283568497575,-8.4448922455e-05,-0.000321032449378,1.0617008563,0.938299143702
3.16449275362,0.287486686895,-6.61959270387e-05,-0.000248532358111,1.05409003275,0.945909967255
3.23043478261,0.291112134944,-5.17532639933e-05,-0.000191859480446,1.04734151682,0.952658483185
3.29637681159,0.294467318884,-4.03541242848e-05,-0.000147693038275,1.04136902466,0.958630975342
3.36231884058,0.297573551936,-3.13810205745e-05,-0.000113377745033,1.03609330071,0.96390669929
3.42826086957,0.30045096929,-2.43370756893e-05,-8.67968291719e-05,1.03144172334,0.968558276655
3.49420289855,0.303118515747,-1.88230626545e-05,-6.62683449506e-05,1.02734792616,0.972652073844
3.56014492754,0.305593939151,-1.45188949721e-05,-5.04604915261e-05,1.02375143203,0.976248567966
3.62608695652,0.307893792475,-1.11686080436e-05,-3.83225146e-05,1.02059729635,0.979402703649
3.69202898551,0.310033446176,-8.56811926853e-06,-2.90284271338e-05,1.01783575625,0.982164243746
3.75797101449,0.312027111321,-6.55522950277e-06,-2.19312975731e-05,1.01542188442,0.984578115583
3.82391304348,0.313887873077,-5.00145235627e-06,-1.65262526624e-05,1.01331524699,0.986684753009
3.88985507246,0.315627733443,-3.80534561506e-06,-1.2420657165e-05,1.0114795667,0.988520433303
3.95579710145,0.317257661726,-2.88708248353e-06,-9.3101873222e-06,1.00988239286,0.990117607143
4,0.318293725075,-2.39529333279e-06,-7.66272909436e-06,1.00893072605,0.991069273952
4.02173913043,0.318787650964,-2.18404742425e-06,-6.95972471076e-06,1.00849478062,0.991505219382
4.08768115942,0.320226778517,-1.64727770258e-06,-5.18817324646e-06,1.00729098174,0.992709018261
4.15362318841,0.321583269054,-1.23860079593e-06,-3.85645158874e-06,1.0062481492,0.993751850795
4.21956521739,0.322864558394,-9.28341905793e-07,-2.8580409186e-06,1.00534605757,0.994653942432
4.28550724638,0.324077356831,-6.93496123749e-07,-2.1115771186e-06,1.00456684052,0.995433159485
4.35144927536,0.325227710846,-5.16277261386e-07,-1.55506922028e-06,1.00389474665,0.996105253351
4.41739130435,0.326321062341,-3.82970312574e-07,-1.14140439633e-06,1.003315914,0.996684086004
4.48333333333,0.327362304774,-2.83027378201e-07,-8.34865520494e-07,1.00281816331,0.997181836687
4.54927536232,0.328355835772,-2.08357815668e-07,-6.08441864225e-07,1.00239080987,0.997609190126
4.6152173913,0.329305605982,-1.52772607565e-07,-4.41758368219e-07,1.00202449309,0.997975506913
4.68115942029,0.33021516407,-1.11550679693e-07,-3.19485472411e-07,1.00171102307,0.998288976929
4.74710144928,0.331087697895,-8.11012828006e-08,-2.30120996481e-07,1.00144324314,0.998556756864
4.81304347826,0.331926071968,-5.87017984222e-08,-1.65059163559e-07,1.001214907,0.998785093004
4.87898550725,0.33273286138,-4.2294592606e-08,-1.17880669337e-07,1.00102056951,0.99897943049
4.94492753623,0.333510382398,-3.03299869664e-08,-8.38124754043e-08,1.00085548971,0.999144510286
5.01086956522,0.334260720001,-2.16451532253e-08,-5.93176670005e-08,1.00071554497,0.999284455035
5.0768115942,0.334985752577,-1.53709391881e-08,-4.17847538792e-08,1.00059715504,0.999402844962
5.14275362319,0.33568717405,-1.08603811499e-08,-2.92928868939e-08,1.00049721512,0.999502784884
5.20869565217,0.336366513681,-7.6339909949e-09,-2.0434893111e-08,1.00041303669,0.99958696331
5.27463768116,0.337025153761,-5.33802435587e-09,-1.418427098e-08,1.00034229546,0.999657704541
5.34057971014,0.33766434542,-3.71274955224e-09,-9.79552916558e-09,1.00028298543,0.999717014571
5.40652173913,0.33828522274,-2.56840892998e-09,-6.72976774307e-09,1.00023337849,0.999766621513
5.47246376812,0.338888815358,-1.76708137012e-09,-4.5992989417e-09,1.00019198881,0.999808011194
5.5384057971,0.339476059712,-1.20905563339e-09,-3.12660974888e-09,1.00015754151,0.999842458486
5.60434782609,0.340047809076,-8.22640067177e-10,-2.11407413797e-09,1.00012894513,0.99987105487
5.67028985507,0.340604842509,-5.56579116129e-10,-1.42170086814e-09,1.00010526733,0.999894732673
5.73623188406,0.341147872836,-3.74436814887e-10,-9.50860168381e-10,1.00008571364,0.999914286363
5.80217391304,0.341677553758,-2.50467202534e-10,-6.32452090699e-10,1.00006960876,0.999930391239
5.86811594203,0.342194486184,-1.66581748395e-10,-4.18335255326e-10,1.00005638018,0.999943619819
5.93405797101,0.342699223865,-1.1015377499e-10,-2.75165779051e-10,1.00004554382,0.999954456181
6,0.343192278403,-7.24198478963e-11,-1.79980030879e-10,1.00003669151,0.999963308492
",
),
).map(row => row.map(float))
#let energy_point(distance, energy) = (
88 + (distance - 0.25) / 5.75 * 510,
-625 - (0.4 - energy) / 0.64 * 180,
)
#let occupation_point(distance, occupation) = (
764 + (distance - 0.25) / 5.75 * 270,
-625 - (2 - occupation) / 2 * 180,
)
#let probability_map(left, top, correlated: false) = {
let cell_size = 55
for (col_idx, atom) in ([A], [B]).enumerate() {
symbol(left + col_idx * cell_size + cell_size / 2, top - 17, atom, size: 23pt)
symbol(left - 17, top + col_idx * cell_size + cell_size / 2, atom, size: 23pt)
}
for row_idx in range(2) {
for col_idx in range(2) {
let same_atom = row_idx == col_idx
let cell_fill = if same_atom {
if correlated { rgb("#cdd1d6") } else { orange.lighten(72%) }
} else { blue.lighten(if correlated { 53% } else { 78% }) }
rect(
(left + col_idx * cell_size, -top - row_idx * cell_size),
(left + (col_idx + 1) * cell_size - 3, -top - (row_idx + 1) * cell_size + 3),
radius: 5,
fill: cell_fill,
stroke: none,
)
symbol(
left + col_idx * cell_size + (cell_size - 3) / 2,
top + row_idx * cell_size + (cell_size - 3) / 2,
if not correlated { [$1/4$] } else if same_atom { [$0$] } else { [$1/2$] },
size: 23pt,
color: if same_atom { orange } else { ink },
)
}
}
}
#let centered_label = label.with(centered: true)
#canvas(length: 1pt, {
rect((0, -45), (1100, -2270), fill: none, stroke: none)
rect((0, -45), (1100, -53), fill: teal, stroke: none)
label(
32,
65,
1036,
[Bond breaking is central to chemical reactions; H₂ is the simplest test of whether DFT gets the electron rearrangement right.],
size: 23pt,
color: muted,
)
draw.translate((0, -24))
label(
30,
98,
1040,
[1 Stretch the bond: a second configuration becomes essential],
size: 24pt,
weight: "bold",
)
// === Bond stretching storyboard ===
for (left, title, caption, fill) in (
(28, [BONDED], [Mostly two electrons in the bonding orbital], pale_blue),
(386, [STRETCHED], [Antibonding-pair configuration gains weight], pale_orange),
(744, [SEPARATED], [One electron on each neutral H atom], pale_teal),
) {
panel(left, 126, 328, 178, fill)
label(left + 13, 138, 302, title, size: 23pt, weight: "bold", centered: true)
label(left + 13, 245, 302, caption, size: 20pt, centered: true)
}
orbital_cloud(192, 198, 88, 35)
nucleus(168, 198)
nucleus(216, 198)
orbital_cloud(510, 198, 53, 35)
orbital_cloud(590, 198, 53, 35)
nucleus(510, 198)
nucleus(590, 198)
orbital_cloud(830, 198, 35, 35)
orbital_cloud(986, 198, 35, 35)
nucleus(830, 198)
nucleus(986, 198)
connect((361, -198), (381, -198), color: teal)
connect((719, -198), (739, -198), color: teal)
label(785, 232, 90, [H (A)], size: 17pt, centered: true)
label(941, 232, 90, [H (B)], size: 17pt, centered: true)
draw.translate((0, -38))
// === Decode the orbital names through phase and symmetry ===
panel(28, 280, 686, 310, rgb("#cdd3da"))
panel(744, 280, 328, 310, rgb("#cdd3da"))
label(42, 289, 300, [$sigma_g$: bonding], size: 23pt, color: blue, weight: "bold")
label(
400,
289,
300,
[$sigma_u$: antibonding (also $sigma_u^*$)],
size: 23pt,
color: orange,
weight: "bold",
)
label(758, 289, 300, [READ THE SYMBOLS], size: 23pt, weight: "bold")
orbital_cloud(106, 340, 61, 22)
symbol(78, 340, [$+$], size: 21pt, color: blue)
symbol(134, 340, [$+$], size: 21pt, color: blue)
orbital_cloud(446, 340, 27, 22)
orbital_cloud(504, 340, 27, 22, color: orange)
symbol(446, 340, [$+$], size: 21pt, color: blue)
symbol(504, 340, [$-$], size: 21pt, color: orange)
line((475, -315), (475, -364), stroke: (paint: muted, thickness: 0.8pt, dash: "dashed"))
label(409, 375, 131, [node:\ zero amplitude], size: 17pt, color: muted, centered: true)
label(
182,
320,
162,
[*g = gerade (even)*\ Invert through midpoint:\ orbital keeps its sign.],
size: 20pt,
)
label(
540,
320,
162,
[*u = ungerade (odd)*\ Invert through midpoint:\ orbital flips its sign.],
size: 20pt,
)
label(42, 459, 300, [1s waves add → density builds between nuclei.], size: 20pt, color: blue)
label(400, 459, 300, [1s waves subtract → a node between nuclei.], size: 20pt, color: orange)
label(
42,
530,
658,
[+/− show wavefunction sign (phase), not electric charge. Inversion maps each point through the bond center.],
size: 20pt,
color: muted,
)
label(
758,
317,
298,
[$sigma$: unchanged by rotation around the H–H axis.\ $sigma_g^2$: two opposite-spin electrons in $sigma_g$.\ Configuration = an orbital-occupation pattern.\ g/u label symmetry, not bonding in general; an asterisk marks antibonding.],
size: 20pt,
)
draw.translate((0, -160))
// === Calculated dissociation and natural occupations ===
label(
30,
458,
1040,
[2 Watch the energy and the real-electron occupations],
size: 24pt,
weight: "bold",
)
label(65, 490, 583, [Energy relative to two separated H atoms], size: 23pt, weight: "bold")
for (start_x, color, name, description, dashed) in (
(81, orange, [RHF], [Restricted\ Hartree-Fock], false),
(257, purple, [UHF], [Unrestricted\ Hartree-Fock], true),
(433, teal, [FCI], [Full configuration\ interaction], false),
) {
line((start_x, -535), (start_x + 25, -535), stroke: (
paint: color,
thickness: 2.4pt,
dash: if dashed { "dashed" } else { "solid" },
))
// Anchor actual glyph bounds to the same centerline as the line sample.
content(
(start_x + 34, -535),
text(
size: 20pt,
fill: color,
weight: "bold",
top-edge: "bounds",
bottom-edge: "bounds",
name,
),
anchor: "west",
padding: 0pt,
)
label(start_x + 34, 548, 142, description, size: 20pt, color: color)
}
for energy in (-0.2, 0, 0.2, 0.4) {
let point = energy_point(0.25, energy)
line(point, energy_point(6, energy), stroke: (
paint: if energy == 0 { muted } else { rgb("#E7EBF1") },
thickness: if energy == 0 { 1pt } else { 0.65pt },
dash: "dashed",
))
label(43, -point.at(1) - 8, 37, [#energy], size: 17pt, color: muted)
}
line(
energy_point(0.25, 0.4),
energy_point(0.25, -0.24),
energy_point(6, -0.24),
stroke: 0.8pt + muted,
)
for distance in (1, 2, 3, 4, 5, 6) {
let point = energy_point(distance, -0.24)
line(point, (point.at(0), point.at(1) - 4), stroke: 0.7pt + muted)
label(
point.at(0) - 12,
-point.at(1) + 8,
24,
[#distance],
size: 17pt,
color: muted,
centered: true,
)
}
// Draw FCI last so the correct asymptote remains visible where UHF merges with it.
for (column, color, dashed) in ((1, orange, false), (2, purple, true), (3, teal, false)) {
line(..curve_data.map(row => energy_point(row.at(0), row.at(column))), stroke: (
paint: color,
thickness: 2.6pt,
dash: if dashed { "dashed" } else { "solid" },
))
}
label(35, 601, 40, [$E_h$], size: 17pt, color: muted)
label(603, 727.5, 62, [2 H], size: 17pt, color: muted)
label(328, 612, 270, [RHF: wrong dissociation limit], size: 17pt, color: orange)
label(
238,
746,
360,
[Static correlation: several configurations are needed],
size: 20pt,
color: teal,
)
label(140, 849, 420, [H–H distance $R$ (angstrom)], size: 20pt, color: muted, centered: true)
centered_label(
726,
490,
329,
[Natural occupations],
size: 23pt,
weight: "bold",
)
centered_label(
720,
516,
340,
[Electrons per orbital; total = 2.],
size: 20pt,
color: muted,
)
for occupation in (0, 1, 2) {
let point = occupation_point(0.25, occupation)
line(point, occupation_point(6, occupation), stroke: (
paint: rgb("#DCE3EC"),
thickness: 0.7pt,
dash: "dashed",
))
label(738, -point.at(1) - 8, 20, [#occupation], size: 17pt, color: muted)
}
line(
occupation_point(0.25, 2),
occupation_point(0.25, 0),
occupation_point(6, 0),
stroke: 0.8pt + muted,
)
for (column, color) in ((4, blue), (5, orange)) {
line(
..curve_data.map(row => occupation_point(row.at(0), row.at(column))),
stroke: 2.6pt + color,
)
}
label(894, 656, 133, [bonding $sigma_g$], size: 20pt, color: blue)
label(889, 743, 155, [antibonding $sigma_u$], size: 20pt, color: orange)
for distance in (1, 3, 6) {
let point = occupation_point(distance, 0)
label(point.at(0) - 13, 813, 26, [#distance], size: 17pt, color: muted, centered: true)
}
label(838, 849, 146, [$R$ (angstrom)], size: 20pt, color: muted, centered: true)
centered_label(
727,
575,
329,
[Bonding/antibonding: $(2, 0)$ → $(1, 1)$.],
size: 20pt,
color: ink,
)
centered_label(
88,
880,
968,
[Basis = allowed orbital building blocks; STO-3G uses one 1s (lowest atomic orbital) per H. FCI mixes all allowed configurations and is *exact in this basis*.],
size: 20pt,
color: muted,
)
draw.translate((0, -125))
// === Pair probabilities expose the error in the separated-atom limit ===
label(
30,
831,
1040,
[3 Separated atoms: density can hide wrong pair probabilities],
size: 24pt,
weight: "bold",
)
centered_label(
46,
864,
266,
[Restricted $sigma_g^2$ reference],
size: 23pt,
color: orange,
weight: "bold",
)
centered_label(
371,
864,
266,
[Correlated singlet],
size: 23pt,
color: teal,
weight: "bold",
)
label(96, 1010, 180, [spin-up electron], size: 17pt, color: muted, centered: true)
label(420, 1010, 180, [spin-up electron], size: 17pt, color: muted, centered: true)
probability_map(125, 1060)
probability_map(450, 1060, correlated: true)
label(10, 1095, 80, [spin-down\ electron], size: 17pt, color: muted, centered: true)
label(335, 1095, 80, [spin-down\ electron], size: 17pt, color: muted, centered: true)
connect((278, -1115), (342, -1115), color: teal)
label(272, 1072, 80, [correlate], size: 17pt, color: teal, centered: true)
centered_label(
62,
1182,
255,
[50% ionic weight],
size: 23pt,
color: orange,
weight: "bold",
)
label(387, 1182, 255, [0% ionic weight], size: 23pt, color: teal, weight: "bold", centered: true)
centered_label(
63,
1211,
577,
[Ionic weight = chance both electrons are on the same atom.\ Orange diagonal: charged $"H"^- + "H"^+$; blue off-diagonal: neutral $"H" + "H"$.],
size: 20pt,
color: muted,
)
panel(690, 862, 382, 450, rgb("#cad4e0"))
centered_label(
708,
912,
348,
[A/B: 1s orbitals on atoms A/B; (1)/(2): electron labels.],
size: 20pt,
color: muted,
)
centered_label(
708,
1150,
348,
[$Psi_S$: the pair’s spatial wave; $abs(Psi_S)^2$: probability density.],
size: 20pt,
color: muted,
)
centered_label(
705,
877,
352,
[Cancel the ionic amplitudes],
size: 24pt,
color: blue,
weight: "bold",
)
centered_label(
708,
975,
348,
[$sigma_g = (A + B) / sqrt(2) quad sigma_u = (A - B) / sqrt(2)$],
size: 24pt,
)
centered_label(
700,
1037,
362,
[$Psi_S = frac(1, sqrt(2)) (sigma_g (1) sigma_g (2) - sigma_u (1) sigma_u (2))$],
size: 18pt,
color: blue,
)
centered_label(
708,
1105,
348,
[$= frac(1, sqrt(2)) (A(1) B(2) + B(1) A(2))$],
size: 20pt,
color: teal,
)
centered_label(
709,
1205,
347,
[Superposition: add waves first, then square to get probabilities.\ Same-atom terms cancel; only one-on-each-atom terms survive.],
size: 20pt,
color: muted,
)
centered_label(
46,
903,
266,
[Restricted: both spins share one spatial orbital.],
size: 20pt,
color: muted,
)
centered_label(
371,
903,
266,
[Singlet: spins combine to zero. Measured along any shared axis, they give opposite results.],
size: 20pt,
color: teal,
)
draw.translate((0, -180))
// === The KS distinction is the central DFT takeaway ===
label(
30,
1158,
1040,
[4 What this means for density-functional theory (DFT)],
size: 24pt,
weight: "bold",
)
for (left, fill) in ((28, pale_blue), (386, pale_orange), (744, pale_teal)) {
panel(left, 1188, 328, 270, fill)
}
for (left, color, title, title_size, caption, illustration, body) in (
(
43,
blue,
[KOHN–SHAM ORBITAL],
23pt,
[$n$: electron density · $phi_"KS"$: model orbital],
symbol(192, 1244, [$phi_"KS" = sqrt(n / 2)$], size: 24pt, color: blue),
[Its occupation stays 2. Exact XC supplies correlation; this model electron pair is not the real state.],
),
(
401,
orange,
[UHF: MIXED SPIN STATE],
23pt,
none,
{
circle((510, -1244), radius: 23, fill: rgb("#cdd3da"), stroke: 0.8pt + orange.lighten(65%))
circle((590, -1244), radius: 23, fill: rgb("#cdd3da"), stroke: 0.8pt + orange.lighten(65%))
electron(510, 1244, "up")
electron(590, 1244, "down")
},
[UHF lets up/down spins use different orbitals.\ It reaches two neutral H atoms, but the spins no longer form a pure singlet.],
),
(
759,
teal,
[THE XC FUNCTIONAL],
23pt,
[Exchange–correlation (XC) energy functional],
symbol(908, 1244, [$E_"xc" [n]$], size: 27pt, color: teal),
[A functional maps density to energy. Common approximations can miss static correlation; exact DFT need not.],
),
) {
centered_label(left, 1201, 298, title, size: title_size, color: color, weight: "bold")
if caption != none {
centered_label(left + 1, 1264, 296, caption, size: 20pt, color: color)
}
illustration
centered_label(left + 1, 1320, 296, body, size: 20pt)
}
centered_label(
30,
1500,
1040,
[H₂ exposes a central DFT challenge: the density can look right while the energy is wrong unless XC captures how electrons avoid each other.],
size: 20pt,
color: teal,
)
centered_label(
30,
1560,
1040,
[Fixed nuclei; curves include nuclear repulsion. $E_h$: hartree, the atomic energy unit. Natural occupations: eigenvalues of the one-electron density matrix. Maps/formula: nonoverlapping 1s orbitals; singlet spin factor suppressed. Clouds are schematic.],
size: 20pt,
color: muted,
)
centered_label(
30,
1670,
1040,
[#link("https://dft.uci.edu/pubs/FNGB05.pdf")[Fuchs et al., JCP 122, 094116 (2005)] · #link("https://dft.uci.edu/teaching/lausanne/ABCDFT.pdf")[Burke: The ABC of DFT, §§4, 7, 11, 13] · #link("https://doi.org/10.1063/1.1672392")[Hehre et al., STO-3G (1969)] · #link("https://ocw.mit.edu/courses/5-61-physical-chemistry-fall-2017/resources/mit5_61f17_lec25/")[MIT: orbital symmetry] · #link("https://web.mit.edu/2.111/www/notes09/spring.pdf")[MIT: singlet spins]],
size: 20pt,
color: muted,
)
})