3D BOND SURFACE FROM REAL TORUS SLICES
======================================================================
Started: 2026-03-25 19:26:58
d = 3, gamma = 0.0650898425

3D torus: N=64, R_major=8, kink_width=3
Torus peak: 1.4410

PART 1: 1D SLICE PROFILES FROM THE 3D TORUS
-------------------------------------------------------
        axial (θ=0°): peak=0.0018, FWHM=7.1, significant=56 pts
       equat (θ=90°): peak=1.4128, FWHM=7.5, significant=64 pts
      diag45 (θ=45°): peak=0.0203, FWHM=11.3, significant=96 pts
      body (θ=54.7°): peak=0.0591, FWHM=10.8, significant=88 pts
               θ=15°: peak=0.0027, FWHM=10.8, significant=70 pts
               θ=30°: peak=0.0057, FWHM=13.2, significant=86 pts
               θ=60°: peak=0.1073, FWHM=9.4, significant=82 pts
               θ=75°: peak=0.5989, FWHM=8.5, significant=70 pts

SLICE PROFILES (phi vs r):
     r        axial (θ=0°)      diag45 (θ=45°)       equat (θ=90°)
   -15              0.0000              0.0001              0.0054
   -14              0.0000              0.0002              0.0134
   -13              0.0000              0.0005              0.0352
   -12              0.0000              0.0011              0.0917
   -11              0.0000              0.0026              0.2883
   -10              0.0000              0.0044              0.6557
    -9              0.0000              0.0081              1.1694
    -8              0.0001              0.0125              1.4128
    -7              0.0001              0.0171              1.1694
    -6              0.0002              0.0202              0.6557
    -5              0.0004              0.0203              0.2883
    -4              0.0007              0.0158              0.1186
    -3              0.0011              0.0106              0.0352
    -2              0.0014              0.0072              0.0134
    -1              0.0017              0.0039              0.0054
     0              0.0018              0.0021              0.0022
     1              0.0017              0.0039              0.0054
     2              0.0014              0.0072              0.0134
     3              0.0011              0.0106              0.0352
     4              0.0007              0.0154              0.0917
     5              0.0004              0.0203              0.2883
     6              0.0002              0.0202              0.6557
     7              0.0001              0.0171              1.1694
     8              0.0001              0.0138              1.4128
     9              0.0000              0.0081              1.1694
    10              0.0000              0.0044              0.6557
    11              0.0000              0.0026              0.2883
    12              0.0000              0.0013              0.1186
    13              0.0000              0.0005              0.0352
    14              0.0000              0.0002              0.0134
    15              0.0000              0.0001              0.0054

PART 2: BOND CURVES FROM REAL SLICE PROFILES
-------------------------------------------------------
        axial (θ=0°): E0=1.000000, bound=1, R_eq=4, D_e=0.00000016
               θ=15°: E0=1.000000, bound=1, R_eq=4, D_e=0.00000059
               θ=30°: E0=0.999999, bound=1, R_eq=4, D_e=0.00000365
      diag45 (θ=45°): E0=0.999974, bound=1, R_eq=4, D_e=0.00014945
      body (θ=54.7°): E0=0.999074, bound=1, R_eq=4, D_e=0.00530627
               θ=60°: E0=0.992427, bound=1, R_eq=4, D_e=0.02835281
               θ=75°: E0=0.141577, bound=2, R_eq=4, D_e=0.46068752
       equat (θ=90°): E0=-0.394268, bound=3, R_eq=6, D_e=0.12008238

PART 3: DETAILED BOND CURVES
-------------------------------------------------------
   R      axial (θ=0°)             θ=30°    diag45 (θ=45°)             θ=60°     equat (θ=90°)
----------------------------------------------------------------------------------------------
   4       -0.00000016       -0.00000365       -0.00014945       -0.02835281       +0.13855618
   6       -0.00000014       -0.00000297       -0.00010190       -0.01465229       -0.12008238
   8       -0.00000012       -0.00000279       -0.00008927       -0.00970974       -0.00905693
  10       -0.00000011       -0.00000244       -0.00008802       -0.00933669       -0.00032961
  12       -0.00000010       -0.00000222       -0.00008874       -0.01158189       -0.00002906
  14       -0.00000010       -0.00000194       -0.00008074       -0.01113837       -0.00000447
  16       -0.00000010       -0.00000175       -0.00006469       -0.00996231       -0.00000053
  18       -0.00000010       -0.00000165       -0.00005577       -0.00639560       -0.00000008
  20       -0.00000010       -0.00000161       -0.00005111       -0.00497945       -0.00000001
  22       -0.00000010       -0.00000160       -0.00004893       -0.00426136       -0.00000000
  24       -0.00000010       -0.00000159       -0.00004827       -0.00408125       -0.00152248
  26       -0.00000010       -0.00000159       -0.00004808       -0.00403332       -0.00024587
  28       -0.00000010       -0.00000159       -0.00004803       -0.00402355       -0.00002894
  30       -0.00000010       -0.00000159       -0.00004802       -0.00402156       -0.00000454
  32       -0.00000010       -0.00000159       -0.00004802       -0.00402121       -0.00000089
  34       -0.00000010       -0.00000159       -0.00004802       -0.00402116       -0.00000061
  36       -0.00000010       -0.00000159       -0.00004802       -0.00402115       -0.00000057
  38       -0.00000010       -0.00000159       -0.00004802       -0.00402114       -0.00000057
  40       -0.00000010       -0.00000159       -0.00004802       -0.00402114       -0.00000056

PART 4: BONDING-ANTIBONDING SPLITTING VS DIRECTION
-------------------------------------------------------
The splitting measures tunnel coupling strength.
At large R it saturates at 2×(binding energy of single well).

   R      axial (θ=0°)             θ=30°    diag45 (θ=45°)             θ=60°     equat (θ=90°)
----------------------------------------------------------------------------------------------
   4        0.00015037        0.00014638        0.00024484        0.03596496        0.11187786
   6        0.00015040        0.00014685        0.00020377        0.02226501        0.16050206
   8        0.00015041        0.00014698        0.00019350        0.01732292        0.01084824
  10        0.00015043        0.00014723        0.00019253        0.01694956        0.00299028
  12        0.00015043        0.00014740        0.00019315        0.01844961        0.23420270
  14        0.00015043        0.00014762        0.00018685        0.01719974        0.47502168
  16        0.00015043        0.00014776        0.00017471        0.01546926        0.51157125
  18        0.00015043        0.00014784        0.00016836        0.01242403        0.47501769
  20        0.00015043        0.00014787        0.00016518        0.01135499        0.23417683
  22        0.00015043        0.00014788        0.00016373        0.01096373        0.00280547
  24        0.00015043        0.00014789        0.00016329        0.01087829        0.00152248
  26        0.00015043        0.00014789        0.00016317        0.01086106        0.00024587
  28        0.00015043        0.00014789        0.00016314        0.01085811        0.00002895
  30        0.00015043        0.00014789        0.00016313        0.01085772        0.00000461
  32        0.00015043        0.00014789        0.00016313        0.01085767        0.00000125
  34        0.00015043        0.00014789        0.00016313        0.01085767        0.00000113
  36        0.00015043        0.00014789        0.00016313        0.01085767        0.00000113
  38        0.00015043        0.00014789        0.00016313        0.01085767        0.00000113
  40        0.00015043        0.00014789        0.00016313        0.01085767        0.00000113

PART 5: V(R, θ) — THE ANGULAR BOND MAP
-------------------------------------------------------
Bond potential as a function of distance and approach angle.

         direction     θ   R_eq          D_e    split_max
----------------------------------------------------------
        axial (θ=0°)   0.0      4   0.00000016     0.000150
               θ=15°  15.0      4   0.00000059     0.000150
               θ=30°  30.0      4   0.00000365     0.000148
      diag45 (θ=45°)  45.0      4   0.00014945     0.000245
      body (θ=54.7°)  54.7      4   0.00530627     0.006274
               θ=60°  60.0      4   0.02835281     0.035965
               θ=75°  75.0      4   0.46068752     0.374029
       equat (θ=90°)  90.0      6   0.12008238     0.511571

PART 6: Oh-WEIGHTED ISOTROPIC AVERAGE
-------------------------------------------------------
V_iso(R) from sin(θ)-weighted angular average:

   R        V_iso      V_axial      V_equat   anisotropy
-------------------------------------------------------
   4  -0.06526470  -0.00000016  +0.13855618      -0.0000
   6  -0.04130514  -0.00000014  -0.12008238      +0.0000
   8  -0.00689902  -0.00000012  -0.00905693      +0.0000
  10  -0.01463807  -0.00000011  -0.00032961      +0.0003
  12  -0.04556179  -0.00000010  -0.00002906      +0.0035
  14  -0.05621266  -0.00000010  -0.00000447      +0.0221
  16  -0.05231505  -0.00000010  -0.00000053      +0.1856
  18  -0.05664705  -0.00000010  -0.00000008      +1.1966
  20  -0.02689341  -0.00000010  -0.00000001     +10.1107
  22  -0.00624439  -0.00000010  -0.00000000     +53.7119
  24  -0.00200986  -0.00000010  -0.00152248      +0.0001
  26  -0.00105802  -0.00000010  -0.00024587      +0.0004
  28  -0.00091332  -0.00000010  -0.00002894      +0.0034
  30  -0.00089436  -0.00000010  -0.00000454      +0.0216
  32  -0.00089207  -0.00000010  -0.00000089      +0.1102
  34  -0.00089186  -0.00000010  -0.00000061      +0.1612
  36  -0.00089183  -0.00000010  -0.00000057      +0.1718
  38  -0.00089183  -0.00000010  -0.00000057      +0.1731
  40  -0.00089183  -0.00000010  -0.00000056      +0.1732

SUMMARY
======================================================================

3D bond surface from real torus slice profiles:

  θ=  0.0°: D_e = 0.00000016
  θ= 15.0°: D_e = 0.00000059
  θ= 30.0°: D_e = 0.00000365
  θ= 45.0°: D_e = 0.00014945
  θ= 54.7°: D_e = 0.00530627
  θ= 60.0°: D_e = 0.02835281
  θ= 75.0°: D_e = 0.46068752
  θ= 90.0°: D_e = 0.12008238

KEY: The torus is NOT spherically symmetric.
Different approach angles see different kink profiles,
leading to direction-dependent bond strengths.
This IS the origin of sigma/pi/delta bond types in chemistry.

Completed: 2026-03-25 19:27:07
