TWO-BREATHER BOND ENERGY ON DISCRETE LATTICE
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Started: 2026-03-21 13:30:37
gamma = 0.0650898425

STEP 1: Single breather energy (n=1)
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  n=1: E_single = 0.105959 +/- 0.001869, omega = 0.997967 (pred 0.997882)
  n=2: E_single = 0.214184 +/- 0.003753, omega = 0.991336 (pred 0.991539)
  n=3: E_single = 0.327065 +/- 0.005663, omega = 0.979249 (pred 0.980995)
  n=4: E_single = 0.447271 +/- 0.007579, omega = 0.960856 (pred 0.966298)

  Reference: E_single(n=1) = 0.105959 +/- 0.001869

STEP 2: Two-breather interaction V(R)
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Opposite phase (+,-) = bonding
Same phase (+,+) = antibonding

   R       E_bond       E_anti       V_bond       V_anti      V_split    E_std_b
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   3     0.000597     0.423262    -0.211322    +0.211343     0.422666   0.000009
   5     0.002372     0.421490    -0.209546    +0.209571     0.419118   0.000037
   7     0.005286     0.418582    -0.206633    +0.206663     0.413296   0.000082
   9     0.009270     0.414604    -0.202648    +0.202685     0.405334   0.000145
  11     0.014238     0.409644    -0.197681    +0.197725     0.395406   0.000224
  13     0.020082     0.403806    -0.191837    +0.191887     0.383724   0.000319
  15     0.026685     0.397207    -0.185234    +0.185289     0.370523   0.000427
  17     0.033919     0.389975    -0.178000    +0.178056     0.356056   0.000548
  19     0.041656     0.382237    -0.170263    +0.170318     0.340581   0.000678
  21     0.049767     0.374122    -0.162151    +0.162203     0.324355   0.000817
  23     0.058131     0.365753    -0.153787    +0.153835     0.307622   0.000963
  25     0.066633     0.357246    -0.145286    +0.145327     0.290613   0.001112
  27     0.075169     0.348704    -0.136749    +0.136785     0.273534   0.001264
  29     0.083648     0.340220    -0.128271    +0.128301     0.256572   0.001417
  31     0.091989     0.331873    -0.119930    +0.119955     0.239885   0.001567
  33     0.100126     0.323732    -0.111793    +0.111813     0.223606   0.001716
  35     0.108004     0.315850    -0.103915    +0.103931     0.207846   0.001859
  37     0.115581     0.308270    -0.096338    +0.096351     0.192689   0.001998
  39     0.122823     0.301024    -0.089095    +0.089106     0.178201   0.002131
  41     0.129709     0.294136    -0.082209    +0.082217     0.164427   0.002257
  50     0.158510     0.265329    -0.053408    +0.053411     0.106819   0.002783
  60     0.178512     0.245326    -0.033407    +0.033408     0.066814   0.003145
  80     0.199795     0.224043    -0.012124    +0.012124     0.024248   0.003524

STEP 3: Bond energy analysis
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Bonding potential minimum:
  R_eq = 3 lattice sites
  D_e = 0.211322 (lattice units)
  V_bond(R_eq) = -0.211322

At R_eq = 3:
  V_bond = -0.211322
  V_anti = +0.211343
  Splitting = 0.422666

Asymptotic (R=80): V_bond = -0.012124, V_anti = +0.012124

Antibonding/bonding ratio: 1.000
  GWT predicts f_anti = 2d/(2d-1) = 1.200

Single breather energy: 0.105959 (lattice units)
Bond depth / single energy: 1.994370

Pöschl-Teller s = 0.172787
s^2 = 0.029855
D_e / (2*s^2) = 3.539099 (should relate to E_single)

GWT analytical prediction:
  D_e(H2) = pi/d^2 * E_H = 4.749 eV
  R_eq(H2) = 1.401 Bohr

If this IS H2: conversion = 22.472 eV per lattice unit
  Single breather = 2.4 eV = 0.0000 MeV
  (electron mass = 0.511 MeV)

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KEY QUESTION: Does the potential have a WELL?
If V_bond goes negative and comes back to 0, a bond forms.
The well depth IS the bond energy. No formula needed.
======================================================================

POTENTIAL CURVE:
   R       V_bond       V_anti        split
   3    -0.211322    +0.211343     0.422666 <-- min
   5    -0.209546    +0.209571     0.419118
   7    -0.206633    +0.206663     0.413296
   9    -0.202648    +0.202685     0.405334
  11    -0.197681    +0.197725     0.395406
  13    -0.191837    +0.191887     0.383724
  15    -0.185234    +0.185289     0.370523
  17    -0.178000    +0.178056     0.356056
  19    -0.170263    +0.170318     0.340581
  21    -0.162151    +0.162203     0.324355
  23    -0.153787    +0.153835     0.307622
  25    -0.145286    +0.145327     0.290613
  27    -0.136749    +0.136785     0.273534
  29    -0.128271    +0.128301     0.256572
  31    -0.119930    +0.119955     0.239885
  33    -0.111793    +0.111813     0.223606
  35    -0.103915    +0.103931     0.207846
  37    -0.096338    +0.096351     0.192689
  39    -0.089095    +0.089106     0.178201
  41    -0.082209    +0.082217     0.164427
  50    -0.053408    +0.053411     0.106819
  60    -0.033407    +0.033408     0.066814
  80    -0.012124    +0.012124     0.024248

Completed: 2026-03-21 13:30:42
