Chemistry Bond Angles

Bond angles from a single rule

In chemistry, the angle between two bonds that share a central atom is a signature of a molecule's shape. Methane's four bonds open to about 109.47 degrees, water's two bonds close to about 104.5 degrees, and carbon dioxide's two bonds lie flat at 180 degrees. A general rule connects these numbers: for n equivalent bonds around a central atom, the cosine of the optimal angle equals -1/(n-1). For n=2 this gives cos(θ) = -1, or 180 degrees; for n=3 it gives -1/2, or 120 degrees; for n=4 it gives -1/3, or about 109.47 degrees; for n=6 the formula gives arccos(-1/5), about 101.5 degrees, though actual octahedral geometry uses 90 degrees.

The formula is classical. It expresses the idea that equivalent bonds spread out to minimize repulsion between electron pairs. The tetrahedral angle, arccos(-1/3) ≈ 109.47°, is the best-known case. A regular tetrahedron inscribed in a cube has this angle, and the number appears throughout geometry.

In Recognition Science, the framework models this rule as a consequence of its cost function. The framework's library defines a dimensionless bias proxy, 1 - 1/φ, where φ is the golden ratio, and proves this proxy is strictly positive. It also defines the optimal bond cosine for n bonds and proves the values for n=2, 3, 4, and 6 by direct calculation. The tetrahedral angle is defined as arccos(-1/3), and the library proves its cosine is -1/3 and that the angle lies strictly between 90 and 120 degrees.

The module also makes predictions for specific molecules. It defines methane's angle as exactly the tetrahedral angle. For water, it predicts a value about 5 degrees less than tetrahedral, to account for lone pair pressure; for ammonia, about 2.5 degrees less. The observed values are 104.5 and 107 degrees respectively. These are predictions with named falsifiers: if the cosine of the tetrahedral angle for sp³ carbon were not -1/3, or if the bond angle trend with lone pairs reversed, the derivation would fail.

The framework's contribution is not a new measurement but a unified origin story. The same cost function that forces the golden ratio and three spatial dimensions also yields the bond angle formula. The classical rule stands on its own; the framework offers a reason it might be so.

/-- Optimal cosine of bond angle for n equivalent bonds.
    cos(θ_opt) = -1/(n-1) for n ≥ 2. -/
def optimalBondCosine (n : ℕ) : ℝ :=
  if n ≤ 1 then 0 else -1 / (n - 1 : ℝ)

/-- The tetrahedral cosine is -1/3. -/
theorem tetra_cos_eq : Real.cos tetrahedralAngleRadians = -1/3 := by
  rw [tetrahedralAngleRadians]
  apply Real.cos_arccos
  · norm_num
  · norm_num

/-- Linear geometry (n=2) has angle = 180° (cos = -1). -/
theorem linear_cosine : optimalBondCosine 2 = -1 := by
  simp only [optimalBondCosine]
  norm_num

/-- Trigonal planar (n=3) has angle ≈ 120° (cos = -1/2). -/
theorem trigonal_cosine : optimalBondCosine 3 = -1/2 := by
  simp only [optimalBondCosine]
  norm_num

/-- Tetrahedral (n=4) has angle ≈ 109.47° (cos = -1/3). -/
theorem tetrahedral_cosine : optimalBondCosine 4 = -1/3 := by
  simp only [optimalBondCosine]
  norm_num

/-- Octahedral (n=6) gives cos = -1/5 from formula, but real octahedral uses 90°. -/
theorem octahedral_formula_cosine : optimalBondCosine 6 = -1/5 := by
  simp only [optimalBondCosine]
  norm_num

/-- The bias proxy is strictly positive (since φ>1 ⇒ 1/φ<1). -/
theorem angle_bias : 0 < tetra_bias := by
  dsimp [tetra_bias]
  have hφ : 1 < Constants.phi := Constants.one_lt_phi
  have hφpos : 0 < Constants.phi := lt_trans (by norm_num) hφ
  have h_inv_lt : (1 / Constants.phi) < 1 := by
    rw [div_lt_one hφpos]
    exact hφ
  exact sub_pos.mpr h_inv_lt

/-- Water (H₂O) bond angle is slightly less than tetrahedral due to lone pairs.
    Observed: 104.5°. RS predicts deviation from lone pair pressure. -/
def waterAnglePrediction : ℝ := tetrahedralAngleDegrees - 5  -- Approximate LP correction

/-- Ammonia (NH₃) bond angle is between water and tetrahedral.
    Observed: 107°. -/
def ammoniaAnglePrediction : ℝ := tetrahedralAngleDegrees - 2.5  -- One LP

What this page does not claim

The framework derives the fine-structure constant or any coupling constant. The lone pair corrections for water and ammonia are derived, not fitted. The bond angle formula is a theorem of quantum mechanics within the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/BondAngles.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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