Where Bond Angles Come From

Methane's four bonds sit at 109.47° — the tetrahedral angle, \(\arccos(-1/3)\). Graphene's three bonds sit at 120°. Acetylene's two at 180°. Sulfur hexafluoride's six at 90°. Standard chemistry obtains these from energy minimisation: VSEPR repulsion arguments or variational mixing of orbitals, calculated case by case, each geometry its own calculation.

In IDWT the electron is a mode of \(\Psi_\infty\) in the \(d=6\) sector, following a genuine orbit in six spatial dimensions; the s and p orbital shapes are 3D projections of that orbit's angular momentum eigenstates (Part 8 of the notes derives this structure, including the Bohr spectrum and all selection rules, from the \(\mathbb{CP}^3\) sector geometry). The companion article describes how the electron settles into the angular-momentum configuration of its 6D orbit that fits its bonding environment. This article does the arithmetic: given that picture, the angles are forced — not minimised into place, but forced by orthogonality, in a few lines of linear algebra.

A bonding electron at a carbon center occupies a hybrid orbit state: a combination of the L=0 state \(|s\rangle\) and the L=1 states \(|p_x\rangle, |p_y\rangle, |p_z\rangle\), pointed along its bond axis n:

\[ |h\rangle = a\,|s\rangle + b\,(\mathbf{n}\cdot|\mathbf{p}\rangle), \qquad a^2 + b^2 = 1, \]

where \(a^2\) is the state's s-share. Every one of these is a full 6D orbit state — the s and p labels name its observable harmonics, not a 3D object. Three premises, stated plainly because they carry the physics:

1. Orbit states. Each σ-bonding electron occupies a state of this form, directed along its internuclear axis. The state space is established in Part 8; that bonding selects states of this form is motivated, not yet derived — the molecular equations of motion are an open part of the framework.
2. Orthogonality. The n bonding electrons at the center are co-present, distinct modes of the single wave \(\Psi_\infty\) — that is what the Pauli principle is in IDWT, a consequence of the spinor anticommutation rather than an external rule — and their orbit states are mutually orthogonal.
3. Equivalence. The n bonds are equivalent, so the n states are symmetry-related and carry equal s-shares.

There is one \(|s\rangle\) state, and the n orthonormal hybrids exhaust it, so completeness fixes every s-share: \(a^2 = 1/n\). Orthogonality of any pair then reads

\[ \langle h_i|h_j\rangle \;=\; \frac{1}{n} + \Bigl(1-\frac{1}{n}\Bigr)(\mathbf{n}_i\cdot\mathbf{n}_j) \;=\; 0 \quad\Longrightarrow\quad \mathbf{n}_i\cdot\mathbf{n}_j = -\frac{1}{n-1}. \]

That is the whole derivation. The angles follow:

• \(n = 2\): \(\cos\theta = -1\) → 180° (acetylene)
• \(n = 3\): \(\cos\theta = -1/2\) → 120° (graphene, ethylene)
• \(n = 4\): \(\cos\theta = -1/3\) → 109.471° (methane)

The inputs are the premises above — nothing else. No repulsion model, no energy functional, no angle put in by hand. The tetrahedral angle is what orthogonality looks like when four equivalent states share one \(|s\rangle\).

The same algebra caps the count. Four mutually orthogonal states exhaust the s+p state set, so a center bonding through these states supports at most four σ bonds. And n equivalent directions in the three observable coordinates with equal pairwise cosine have a Gram matrix whose rank exceeds 3 for n ≥ 5: five equivalent equiangular bonds are geometrically impossible. Five-coordinate molecules comply — PF₅ does not have five equivalent bonds; it splits them into three equatorial and two axial, with different lengths.

Six-coordinate centers like SF₆ bond through L ≤ 2 states — the d-shell joins. For hybrids that are axially symmetric about their own bond axes, the overlap generalises by the spherical-harmonic addition theorem to

\[ \langle h_i|h_j\rangle \;=\; \sum_L c_L^2\, P_L(\mathbf{n}_i\cdot\mathbf{n}_j), \]

with \(P_L\) the Legendre polynomials and \(c_L^2\) the angular-momentum shares. Six equivalent hybrids exhaust \(|s\rangle\), the three \(|p\rangle\) states, and two zonal d states, so completeness forces the shares: \(c_0^2 = 1/6\), \(c_1^2 = 1/2\), \(c_2^2 = 1/3\). Orthogonality of any pair becomes

\[ \tfrac{1}{6} + \tfrac{x}{2} + \tfrac{1}{3}P_2(x) = 0 \;\Longleftrightarrow\; 3x^2 + 3x = 0 \;\Longleftrightarrow\; x \in \{0, -1\}. \]

Every pair of bond directions must be perpendicular or antipodal — and six unit vectors with pairwise cosines in {0, −1} are three orthogonal axes. The octahedron is not selected from candidates; it is the only solution. The 90° of SF₆ comes out of the same orthogonality requirement as the 109.47° of methane, with the shares dictated by counting states.

Water's H–O–H angle is 104.5°, not 109.47°, because oxygen's four hybrids are not equivalent: two reach hydrogen nuclei and two hold lone pairs. Give the bond states s-share \(1/4-\delta\) and the lone pairs correspondingly more (completeness keeps the total at 1), and bond-state orthogonality gives

\[ \cos\theta_{\mathrm{HOH}} = -\frac{1/4-\delta}{3/4+\delta}. \]

The measured 104.5° corresponds to δ = 0.050; ammonia's 107.8° corresponds to δ = 0.016; methane is δ = 0 exactly. The δ values are fitted to the measured angles, not predicted — honesty requires saying so. What the family does deliver is the trend: δ grows monotonically with the number of lone pairs, CH₄ < NH₃ < H₂O, because lone-pair states, which reach no external nucleus, take a larger s-share and push the bond states together. Deriving δ itself from the nuclear charge is an open calculation in Part 11.

The angle identities are exact mathematics given the premises, and every numerical claim here is verified by explicit construction in the project's proof script (idwt.py, STEPs 42–45): the hybrid states are built, their Gram matrices checked orthonormal to machine precision, the angles computed. The premises themselves are largely derived in Part 11 §4. Orthogonality turns out not to be an assumption at all — the antisymmetric many-electron state depends only on the occupied subspace, so the directed states can always be taken orthonormal. The directed form of each bonding state is derived from the axial symmetry of the companion nucleus's potential: within the degenerate shell, its leading multipole couples only the s state to the p state along the bond axis, and the bound combination is the directed hybrid. Equal s-shares follow when the ligands are identical and symmetrically arranged. What remains open is the energetics — that identical ligands adopt the symmetric arrangement, and the lone-pair δ of water and ammonia, which is fitted rather than predicted. The full development is Part 11 of the notes.

The full development, with premises labelled and the six-coordinate and lone-pair cases worked in detail, is Part 11 of the notes: doi:10.5281/zenodo.19767493.