========================================================================== WHY THE PERIODIC TABLE IS THAT PECULIAR SHAPE Science Journaling Club - analysis/periodic-table-shape.py ========================================================================== PART A1. BLOCK WIDTHS FROM THE ALLOWED QUANTUM NUMBERS -------------------------------------------------------------------------- For a given l, m_l runs from -l to +l, which is 2l+1 values. Each of those takes two electrons, one per spin. Seats = 2(2l+1). l subshell m_l values seats = 2(2l+1) block width 0 s 1 2 2 1 p 3 6 6 2 d 5 10 10 3 f 7 14 14 Those four numbers, 2 / 6 / 10 / 14, are the widths of the s, p, d and f blocks. Nothing else in the table is free to choose them. PART A2. THE MADELUNG FILLING ORDER (increasing n+l, then increasing n) -------------------------------------------------------------------------- order orbital n l n+l seats Z range 1 1s 1 0 1 2 1 - 2 2 2s 2 0 2 2 3 - 4 3 2p 2 1 3 6 5 - 10 4 3s 3 0 3 2 11 - 12 5 3p 3 1 4 6 13 - 18 6 4s 4 0 4 2 19 - 20 7 3d 3 2 5 10 21 - 30 8 4p 4 1 5 6 31 - 36 9 5s 5 0 5 2 37 - 38 10 4d 4 2 6 10 39 - 48 11 5p 5 1 6 6 49 - 54 12 6s 6 0 6 2 55 - 56 13 4f 4 3 7 14 57 - 70 14 5d 5 2 7 10 71 - 80 15 6p 6 1 7 6 81 - 86 16 7s 7 0 7 2 87 - 88 17 5f 5 3 8 14 89 - 102 18 6d 6 2 8 10 103 - 112 19 7p 7 1 8 6 113 - 118 20 8s 8 0 8 2 119 - 120 PART A3. PERIOD LENGTHS, READ STRAIGHT OFF THE FILLING ORDER -------------------------------------------------------------------------- period orbitals filled length ends at Z 1 1s 2 2 2 2s 2p 8 10 3 3s 3p 8 18 4 4s 3d 4p 18 36 5 5s 4d 5p 18 54 6 6s 4f 5d 6p 32 86 7 7s 5f 6d 7p 32 118 8 8s 2 120 predicted period lengths : [2, 8, 8, 18, 18, 32, 32] real periods 1-7 : [2, 8, 8, 18, 18, 32, 32] agreement : EXACT VALIDATION 1. NOBLE-GAS ATOMIC NUMBERS, PREDICTED vs OBSERVED -------------------------------------------------------------------------- period predicted Z at period end observed noble gas diff 1 2 He 2 +0 2 10 Ne 10 +0 3 18 Ar 18 +0 4 36 Kr 36 +0 5 54 Xe 54 +0 6 86 Rn 86 +0 7 118 Og 118 +0 The construction used no chemistry. It used l < n, 2(2l+1) seats, and the ordering n+l. It lands on all seven noble gases exactly, and the seven periods sum to 118, the number of known elements. PART A4. THE SHAPE ITSELF, GENERATED FROM THE RULES ALONE -------------------------------------------------------------------------- s-block : 14 elements, 2 columns, 7 rows p-block : 36 elements, 6 columns, 6 rows d-block : 40 elements, 10 columns, 4 rows f-block : 28 elements, 14 columns, 2 rows total : 118 Laid out honestly, with the f-block in line, the table is 2 + 14 + 10 + 6 = 32 columns wide. The familiar 18-column table is that same table with 14 columns cut out and parked underneath. PART A5. WHERE THE RULE BREAKS (Z = 1..103, against NIST v5.12) -------------------------------------------------------------------------- Z sym Madelung predicts NIST measures block 24 Cr 3d4 4s2 3d5 4s1 d 29 Cu 3d9 4s2 3d10 4s1 d 41 Nb 4d3 5s2 4d4 5s1 d 42 Mo 4d4 5s2 4d5 5s1 d 44 Ru 4d6 5s2 4d7 5s1 d 45 Rh 4d7 5s2 4d8 5s1 d 46 Pd 4d8 5s2 4d10 d 47 Ag 4d9 5s2 4d10 5s1 d 57 La 4f1 6s2 5d1 6s2 f 58 Ce 4f2 6s2 4f1 5d1 6s2 f 64 Gd 4f8 6s2 4f7 5d1 6s2 f 78 Pt 5d8 6s2 5d9 6s1 d 79 Au 5d9 6s2 5d10 6s1 d 89 Ac 5f1 7s2 6d1 7s2 f 90 Th 5f2 7s2 6d2 7s2 f 91 Pa 5f3 7s2 5f2 6d1 7s2 f 92 U 5f4 7s2 5f3 6d1 7s2 f 93 Np 5f5 7s2 5f4 6d1 7s2 f 96 Cm 5f8 7s2 5f7 6d1 7s2 f 103 Lr 6d1 7s2 7s2 7p1 d violations found : 20 out of 103 elements (19.4%) obedient elements : 83 out of 103 (80.6%) violations by block : s: 0, p: 0, d: 11, f: 9 block census, Z=1..103: s: 14, p: 30, d: 31, f: 28 Every one sits in the d-block or the f-block. The s-block and the p-block, 44 elements between them, never break the rule once. of the 20, how many move exactly one electron out of an ns shell: 9 Pd (Z=46) moves two, emptying 5s completely: [Kr] 4d10. PART B1. EFFECTIVE NUCLEAR CHARGE FROM SLATER'S RULES -------------------------------------------------------------------------- Zeff = Z - S. For an ns or np electron, S counts 0.35 from each other electron in the same shell, 0.85 from each in the shell below, and 1.00 from everything deeper. Z sym config (valence) S Zeff Zeff(SCF) Slater - SCF 3 Li 1s2 2s1 1.70 1.300 1.279 +0.021 4 Be 1s2 2s2 2.05 1.950 1.912 +0.038 5 B 2s2 2p1 2.40 2.600 2.421 +0.179 6 C 2s2 2p2 2.75 3.250 3.136 +0.114 7 N 2s2 2p3 3.10 3.900 3.834 +0.066 8 O 2s2 2p4 3.45 4.550 4.453 +0.097 9 F 2s2 2p5 3.80 5.200 5.100 +0.100 10 Ne 2s2 2p6 4.15 5.850 5.758 +0.092 .................................................................... 11 Na 2p6 3s1 8.80 2.200 2.507 -0.307 12 Mg 2p6 3s2 9.15 2.850 3.308 -0.458 13 Al 3s2 3p1 9.50 3.500 4.066 -0.566 14 Si 3s2 3p2 9.85 4.150 4.285 -0.135 15 P 3s2 3p3 10.20 4.800 4.886 -0.086 16 S 3s2 3p4 10.55 5.450 5.482 -0.032 17 Cl 3s2 3p5 10.90 6.100 6.116 -0.016 18 Ar 3s2 3p6 11.25 6.750 6.764 -0.014 period 2: Zeff runs 1.30 -> 5.85, a rise of 4.55 over 7 steps Slater step = 0.650 per element, exactly and always SCF step = 0.640 per element on average period 3: Zeff runs 2.20 -> 6.75, same 0.65 step SCF step = 0.608 per element on average mean |Slater - SCF|, period 2: 0.088 period 3: 0.202 Slater runs high through period 2 and low through period 3, and the period-3 error shrinks from -0.307 at Na to -0.014 at Ar. VALIDATION 2. ONE-ELECTRON LIMIT, WHERE THE ANSWER IS ANALYTIC -------------------------------------------------------------------------- quantity model analytic / NIST diff H first ionisation energy 13.6057 eV 13.5984 eV +0.0073 eV (analytic hydrogenic value 13.6057 eV = 1 Ry) He+ ionisation energy 54.4228 eV 54.4178 eV +0.0050 eV (analytic hydrogenic value 54.4228 eV = 4 Ry) With one electron there is nothing to screen, so Slater's rules collapse to the exact hydrogenic formula. Agreement to 0.05% and 0.009% is the arithmetic behaving, not the chemistry. PART B2. PREDICTED vs MEASURED FIRST IONISATION ENERGY -------------------------------------------------------------------------- IE = E(atom) - E(cation), both summed as sum_i Ry*(Zeff_i/n*_i)^2. Z sym predicted measured diff error 1 H 13.606 13.598 +0.007 +0.1% 2 He 24.218 24.587 -0.369 -1.5% ............................................................ 3 Li 5.748 5.392 +0.357 +6.6% 4 Be 7.874 9.323 -1.448 -15.5% 5 B 9.779 8.298 +1.481 +17.8% 6 C 11.463 11.260 +0.203 +1.8% 7 N 12.925 14.534 -1.609 -11.1% 8 O 14.167 13.618 +0.549 +4.0% 9 F 15.187 17.423 -2.235 -12.8% 10 Ne 15.987 21.565 -5.578 -25.9% ............................................................ 11 Na 7.317 5.139 +2.178 +42.4% 12 Mg 9.078 7.646 +1.432 +18.7% 13 Al 10.741 5.986 +4.755 +79.4% 14 Si 12.306 8.152 +4.154 +51.0% 15 P 13.772 10.487 +3.285 +31.3% 16 S 15.140 10.360 +4.780 +46.1% 17 Cl 16.410 12.968 +3.442 +26.5% 18 Ar 17.582 15.760 +1.822 +11.6% ............................................................ 19 K 4.810 4.341 +0.470 +10.8% 20 Ca 5.968 6.113 -0.145 -2.4% mean absolute error, Z = 1..20 : 20.9% mean absolute error, period 2 : 11.9% mean absolute error, period 3 : 38.4% period 3 is where the model falls apart: every prediction is too high, because Slater's n* = 3.0 for the third shell is too small and the 0.85 screening from the n = 2 shell is far too generous. worst single element : Al at +79.4% Direction of travel across period 2: steps where the model rises : 7 of 7 steps where measurement rises : 5 of 7 the two measured reversals : Be -> B (9.323 -> 8.298 eV) N -> O (14.534 -> 13.618 eV) the model predicts both as rises, so it misses both. Slater's rules know nothing about which orbital an electron sits in, and both reversals are orbital effects: the 2p electron of boron is higher in energy than beryllium's 2s, and oxygen's fourth 2p electron has to pair up in an orbital that already holds one. PART B3. PREDICTED vs MEASURED ATOMIC RADIUS -------------------------------------------------------------------------- Hydrogenic scaling: r proportional to n*^2 / Zeff, with a single scale constant k fitted once by least squares over Li..F and Na..Cl against Slater's (1964) empirical radii. One free parameter total. fitted k = 49.79 pm (Bohr radius a0 = 52.92 pm, ratio 0.941) Z sym n*^2/Zeff predicted Slater'64 Cordero'08 resid vs S64 3 Li 3.0769 153.2 145 128 +8.2 4 Be 2.0513 102.1 105 96 -2.9 5 B 1.5385 76.6 85 84 -8.4 6 C 1.2308 61.3 70 76 -8.7 7 N 1.0256 51.1 65 71 -13.9 8 O 0.8791 43.8 60 66 -16.2 9 F 0.7692 38.3 50 57 -11.7 .................................................................. 11 Na 4.0909 203.7 180 166 +23.7 12 Mg 3.1579 157.2 150 141 +7.2 13 Al 2.5714 128.0 125 121 +3.0 14 Si 2.1687 108.0 110 111 -2.0 15 P 1.8750 93.3 100 107 -6.7 16 S 1.6514 82.2 100 105 -17.8 17 Cl 1.4754 73.5 100 102 -26.5 mean absolute residual vs Slater 1964 : 11.2 pm Contraction across a period, measured as r(first) / r(last): period 2, model Li/F = 4.00 period 2, Slater'64 Li/F = 2.90 period 2, Cordero'08 Li/F = 2.25 period 3, model Na/Cl = 2.77 period 3, Slater'64 Na/Cl = 1.80 period 3, Cordero'08 Na/Cl = 1.63 The model gets the direction right everywhere and the amount wrong in period 3, where it shrinks the atoms about 1.7 times harder than the covalent radii do. Covalent radii are contact distances between bonded atoms, and near the right-hand edge they stop shrinking almost entirely. A hydrogenic 1/Zeff law has no way to know that. Down a group, where the model does better: Li -> Na model 153.2 -> 203.7 pm Slater'64 145 -> 180 pm Be -> Mg model 102.1 -> 157.2 pm Slater'64 105 -> 150 pm B -> Al model 76.6 -> 128.0 pm Slater'64 85 -> 125 pm C -> Si model 61.3 -> 108.0 pm Slater'64 70 -> 110 pm F -> Cl model 38.3 -> 73.5 pm Slater'64 50 -> 100 pm PART C. THE CRUDE RELATIVISTIC ESTIMATE -------------------------------------------------------------------------- A Bohr-model 1s electron orbits at v/c = Z*alpha. Its mass goes up by gamma, and the orbit radius goes down by the same factor. Z sym v/c gamma 1s contraction 1 H 0.0073 1.0000 0.0% 26 Fe 0.1897 1.0185 1.8% 47 Ag 0.3430 1.0646 6.1% 54 Xe 0.3941 1.0880 8.1% 79 Au 0.5765 1.2238 18.3% 80 Hg 0.5838 1.2317 18.8% 82 Pb 0.5984 1.2481 19.9% 92 U 0.6714 1.3493 25.9% 118 Og 0.8611 1.9667 49.2% silver -> gold, one row apart in the same group: contraction goes from 6.1% to 18.3%, a factor of 3.0 This is the back-of-an-envelope version. Proper four-component Dirac-Fock calculations give a 6s contraction for gold of the same order, and they are what the chemistry arguments actually rest on (Desclaux 1973; Pyykko & Desclaux 1979). Photon energies, for the colour argument: 2.4 eV -> 517 nm gold, 5d -> 6s absorption edge 3.9 eV -> 318 nm silver, 4d -> 5s absorption edge visible band runs about 380-750 nm, so gold's edge falls inside it and silver's falls in the ultraviolet, outside it. HEADLINE NUMBERS USED IN THE ARTICLE AND THE INTERACTIVE MODEL ========================================================================== block widths : 2 / 6 / 10 / 14 period lengths : 2, 8, 8, 18, 18, 32, 32 period ends : 2, 10, 18, 36, 54, 86, 118 full-width table : 32 columns columns cut out and parked below : 14 Madelung violations, Z = 1..103 : 20 (19.4%) violations in the s and p blocks : 0 Slater Zeff step per element : 0.650 exactly SCF Zeff step, period 2 : 0.640 IE mean abs error, period 2 : 11.9% IE mean abs error, period 3 : 38.4% IE reversals in period 2, measured : 2 (Be->B, N->O) IE reversals the model predicts : 0 radius law, one fitted constant k : 49.79 pm radius mean abs residual : 11.2 pm 1s contraction, Ag / Au / Og : 6.1% / 18.3% / 49.2% gold absorption edge : 2.4 eV = 517 nm silver absorption edge : 3.9 eV = 318 nm ========================================================================== END. Every number printed above comes from this file. ==========================================================================