3D BOND SURFACE FROM 1D HESSIAN SLICES
======================================================================
Started: 2026-03-25 19:22:44
d = 3, gamma = 0.0650898425

THREE DIRECTION CLASSES (Oh symmetry):
-------------------------------------------------------

  FACE   (6 dirs): along cube axis
    Approach through torus hole or along ring
    Lattice spacing: a = 1
    Multiplicity: 6

  EDGE   (12 dirs): along face diagonal
    Approach at 45° to torus plane
    Lattice spacing: a√2 = 1.414
    Multiplicity: 12

  VERTEX (8 dirs): along body diagonal
    Approach at 54.7° to torus plane
    Lattice spacing: a√3 = 1.732
    Multiplicity: 8

PART 1: FACE DIRECTION (standard, along cube axis)
-------------------------------------------------------
Single well E0 = -0.37237471
   R         V(R)        split
------------------------------
   4  +0.13161056   0.11218150
   6  +0.04161751   0.27019428
   8  +0.00320981   0.46136080
  10  +0.00026931   0.49276269
  12  +0.00002202   0.49702520
  14  +0.00000149   0.49764951
  16  +0.00000004   0.49774404
  18  -0.00000001   0.49775860 <-- min
  20  -0.00000000   0.49776087
  22  -0.00000000   0.49776123
  24  -0.00000000   0.49776128
  26  -0.00000000   0.49776129
  28  -0.00000000   0.49776129
  30  -0.00000000   0.49776129
  32  -0.00000000   0.49776129
  34  -0.00000000   0.49776129
  36  -0.00000000   0.49776129
  38  +0.00000000   0.49776129
  40  -0.00000000   0.49776129
  42  +0.00000000   0.49776129
  44  -0.00000000   0.49776129
  46  +0.00000000   0.49776129
  48  -0.00000000   0.49776129
  50  +0.00000000   0.49776129

FACE: R_eq = 18, D_e = 0.00000001

PART 2: EDGE DIRECTION (face diagonal)
-------------------------------------------------------
Along the face diagonal, each 1D step = √2 in 3D.
The kink appears WIDER (projected onto the longer axis).

Effective kink width (edge): 4.243
 R_3d   R_1d         V(R)        split
----------------------------------------
  5.7      4  +0.10618720   0.00186640
  8.5      6  -0.30773568   0.36866193
 11.3      8  +0.01800947   0.06055133
 14.1     10  +0.00188739   0.11474568
 17.0     12  +0.00020443   0.12448955
 19.8     14  +0.00002292   0.12577860
 22.6     16  +0.00000255   0.12595135
 25.5     18  +0.00000028   0.12597470
 28.3     20  +0.00000003   0.12597787
 31.1     22  +0.00000000   0.12597830
 33.9     24  +0.00000000   0.12597836
 36.8     26  +0.00000000   0.12597837
 39.6     28  -0.00000000   0.12597837
 42.4     30  -0.00000000   0.12597837
 45.3     32  -0.00000000   0.12597837
 48.1     34  -0.00000000   0.12597837
 50.9     36  -0.00000000   0.12597837
 53.7     38  +0.00000000   0.12597837
 56.6     40  +0.00000000   0.12597837
 59.4     42  +0.00000000   0.12597837
 62.2     44  +0.00000000   0.12597837
 65.1     46  +0.00000000   0.12597837
 67.9     48  +0.00000000   0.12597837
 70.7     50  +0.00000000   0.12597837

EDGE: R_eq(1d) = 6, D_e = 0.30773568
       R_eq(3d) = 8.5

PART 3: VERTEX DIRECTION (body diagonal)
-------------------------------------------------------
Effective kink width (vertex): 5.196
 R_3d   R_1d         V(R)        split
----------------------------------------
  6.9      4  +0.04914571   0.01262244
 10.4      6  +0.02959991   0.00052931
 13.9      8  +0.02068026   0.00399755
 17.3     10  +0.00537370   0.02551805
 20.8     12  +0.00069885   0.04347690
 24.2     14  +0.00009735   0.04740331
 27.7     16  +0.00001396   0.04797966
 31.2     18  +0.00000201   0.04806148
 34.6     20  +0.00000029   0.04807307
 38.1     22  +0.00000004   0.04807471
 41.6     24  +0.00000001   0.04807495
 45.0     26  +0.00000000   0.04807498
 48.5     28  +0.00000000   0.04807498
 52.0     30  +0.00000000   0.04807498
 55.4     32  +0.00000000   0.04807498
 58.9     34  +0.00000000   0.04807498
 62.4     36  +0.00000000   0.04807498
 65.8     38  +0.00000000   0.04807498
 69.3     40  +0.00000000   0.04807498
 72.7     42  +0.00000000   0.04807498
 76.2     44  +0.00000000   0.04807498
 79.7     46  +0.00000000   0.04807498
 83.1     48  -0.00000000   0.04807498
 86.6     50  +0.00000000   0.04807498

VERTEX: R_eq(1d) = 48, D_e = 0.00000000
         R_eq(3d) = 83.1

PART 4: ANGULAR ANISOTROPY
-------------------------------------------------------
At the same 3D distance R, how does V depend on direction?

 R_3d       V_face       V_edge     V_vertex   anisotropy
----------------------------------------------------------
    6  +0.04161751  -0.03330671  +0.00000000       0.0000
    8  +0.00320981  -0.33073536  +0.04028522       0.0000
   10  +0.00026931  -0.13020525  +0.03076065       0.0000
   12  +0.00002202  +0.04612602  +0.02583009       0.0000
   14  +0.00000149  +0.00551242  +0.02013134       0.0000
   16  +0.00000004  -0.00695032  +0.01087090       0.0000
   20  -0.00000000  -0.00026335  +0.00099064       0.0000
   24  -0.00000000  +0.00019610  +0.00011734       0.0000
   30  -0.00000000  +0.00001155  +0.00000626       0.0000
   40  -0.00000000  +0.00000005  -0.00000005       0.0000

PART 5: Oh-WEIGHTED ISOTROPIC AVERAGE
-------------------------------------------------------
V_iso(R) = (6·V_face + 12·V_edge + 8·V_vertex) / 26
This is the spherically-averaged potential (A1g component).

 R_3d     V_face     V_edge   V_vertex      V_iso V_face/V_iso
------------------------------------------------------------
    6  +0.041618  -0.033307  +0.000000  -0.005768      -7.2149
    8  +0.003210  -0.330735  +0.040285  -0.139511      -0.0230
   10  +0.000269  -0.130205  +0.030761  -0.050568      -0.0053
   12  +0.000022  +0.046126  +0.025830  +0.029242       0.0008
   14  +0.000001  +0.005512  +0.020131  +0.008739       0.0002
   16  +0.000000  -0.006950  +0.010871  +0.000137       0.0003
   20  -0.000000  -0.000263  +0.000991  +0.000183      -0.0000
   24  -0.000000  +0.000196  +0.000117  +0.000127      -0.0000
   30  -0.000000  +0.000012  +0.000006  +0.000007      -0.0000
   40  -0.000000  +0.000000  -0.000000  +0.000000      -0.0000

PART 6: MORSE WELL PARAMETERS BY DIRECTION
-------------------------------------------------------
  FACE    : R_eq =  18.0, D_e = 0.00000001, decay = 0.0000
  EDGE    : R_eq =   8.5, D_e = 0.30773568, decay = 0.0000
  VERTEX  : R_eq =  83.1, D_e = 0.00000000, decay = 0.0000

PART 7: 3D BOND SURFACE CROSS-SECTIONS
-------------------------------------------------------
V(R, θ) where θ = angle from torus axis:
  θ = 0°:    FACE direction (through the hole)
  θ = 45°:   EDGE direction (face diagonal)
  θ = 54.7°: VERTEX direction (body diagonal)
  θ = 90°:   FACE direction (in the torus plane)

(θ = 0° and θ = 90° both map to FACE — torus has axial symmetry)

   R       θ=0°    θ=22.5°      θ=45°    θ=54.7°
------------------------------------------------
   6  +0.041618  +0.004155  -0.033307  +0.000000
   8  +0.003210  -0.163763  -0.330735  +0.040285
  10  +0.000269  -0.064968  -0.130205  +0.030761
  12  +0.000022  +0.023074  +0.046126  +0.025830
  14  +0.000001  +0.002757  +0.005512  +0.020131
  16  +0.000000  -0.003475  -0.006950  +0.010871
  20  -0.000000  -0.000132  -0.000263  +0.000991

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

3D bond surface reconstructed from 3 unique 1D Hessian calculations.

Well depths:
  FACE   (6 dirs):  D_e = 0.00000001
  EDGE   (12 dirs): D_e = 0.30773568
  VERTEX (8 dirs):  D_e = 0.00000000

Anisotropy (max/min depth): 4927709412622661.00
The interaction is ANISOTROPIC

Strongest bonding direction: EDGE
  FACE → through torus hole / along ring
  EDGE → at 45° to torus plane
  VERTEX → along body diagonal

Completed: 2026-03-25 19:22:45
