# Geometric analysis

Not every molecular property comes from the electron density. This last properties chapter covers `qc.prop`'s
purely **geometric** descriptors — quantities computable from the nuclear positions (and, for a couple, the
van der Waals radii) alone, without touching the wavefunction at all. They connect qc-rs to rotational
spectroscopy, packing/solvation-accessible-surface reasoning, and radial structural analysis.

## Theory: rotational constants from the moment of inertia

A rigid molecule's rotational energy levels — what a microwave/rotational spectrum measures — are set by its
**moment-of-inertia tensor**, built purely from nuclear masses and positions relative to the centre of mass:

$$
I_{\alpha\beta} = \sum_A m_A\big(|\mathbf r_A|^2\delta_{\alpha\beta} - r_{A,\alpha}r_{A,\beta}\big).
$$

Diagonalizing $\mathbf I$ gives the three **principal moments of inertia** $I_a\le I_b\le I_c$ (in
$\text{amu}\cdot\text{Å}^2$), and each converts to a **rotational constant**

$$
X = \frac{h}{8\pi^2 c\,I_X}, \qquad X \in \{A,B,C\},
$$

conventionally reported in GHz or cm⁻¹. A **linear** molecule has $I_a=0$ (no moment of inertia about its own
axis) and only two independent rotational constants; an **asymmetric top** (the general case, e.g. water) has
all three distinct; a **symmetric top** has two equal. These constants are exactly what a rotational or
microwave spectrum measures directly — computing them from a qc-rs geometry is the bridge from "a computed
structure" to "a predicted rotational spectrum," and (with the analogous mass-weighted machinery from
[Hessians & thermochemistry](../hessian-frequencies-thermo.md)) the rotational partition function that feeds
thermochemistry.

## Usage

```python
import qc
water = "O 0 0 0.117; H 0 0.757 -0.469; H 0 -0.757 -0.469"
m = qc.chk.new(atom=water, ao="sto-3g", unit="angstrom").scf(ref="r").run()

rc = qc.prop.geom.rotational_constants(m)
rc["constants_ghz"]           # [822.15, 437.53, 285.56]  A, B, C (GHz)
rc["moments_amu_angstrom2"]   # [0.615, 1.155, 1.770]      I_a, I_b, I_c
```

Water's three distinct rotational constants confirm it as an **asymmetric top** — consistent with its bent,
low-symmetry shape. This needs only the converged geometry, not the wavefunction, so it works identically on
an optimized or an as-typed structure (though only the former corresponds to a real equilibrium spectrum).

## Other geometric descriptors

Three further leaves round out `qc.prop.geom`, all built from nuclear positions plus van der Waals radii
rather than the wavefunction:

- **`surface_area(mychk)`** — the molecular (solvent-accessible-style) surface area, from the union of
  scaled van der Waals spheres, with a per-atom breakdown. Useful alongside
  [dispersion](../solvation-dispersion.md)/solvation reasoning, where buried vs. exposed surface matters.
- **`free_volume(mychk)`** — partitions a bounding box into *occupied* (inside any atom's vdW sphere) and
  *free* volume, reporting the free fraction — a packing-density descriptor relevant to crystals and cavities.
- **`rdf(mychk)`** — the radial distribution of atoms around a chosen centre, binned by distance: a structural
  fingerprint useful for clusters and disordered/extended systems where "distance to the nearest neighbour of
  each type" is the natural first question.

```python
sa = qc.prop.geom.surface_area(m)
sa["area_angstrom2"], sa["per_atom_angstrom2"]     # total + per-atom surface area

fv = qc.prop.geom.free_volume(m)
fv["free_fraction"]                                 # fraction of the bounding box that is empty
```

## Worked example

```python
import qc
water = "O 0 0 0.117; H 0 0.757 -0.469; H 0 -0.757 -0.469"
m = qc.chk.new(atom=water, ao="sto-3g", unit="angstrom").scf(ref="r").run()

rc = qc.prop.geom.rotational_constants(m)
print("A, B, C (GHz):", [round(x, 2) for x in rc["constants_ghz"]])
print("I_a, I_b, I_c (amu·Å²):", [round(x, 4) for x in rc["moments_amu_angstrom2"]])
# A, B, C (GHz): [822.15, 437.53, 285.56]
# I_a, I_b, I_c (amu·Å²): [0.6147, 1.1551, 1.7698]
```

:::{exercise}
:label: ex-geom

1. Linear CO₂ and bent SO₂ are both triatomic. Without computing anything, predict how many *distinct*
   rotational constants each will have, and why.
2. `rotational_constants` needs only the geometry — no SCF, no basis set choice beyond what you already
   used to optimize the structure. Why does it not need the wavefunction at all?
3. You want to know how "crowded" the interior of a molecular crystal's unit cell is. Which `qc.prop.geom`
   leaf gives you a single descriptive number for that?
:::

:::{solution} ex-geom
:class: dropdown

1. **Linear CO₂**: $I_a=0$ about its own axis, so it has only **2** distinct rotational constants ($B=C$
   from the two equal perpendicular moments). **Bent SO₂**: a genuine asymmetric top with all three moments
   distinct, so it has **3** distinct constants — exactly the water pattern in this chapter.
2. The moment-of-inertia tensor is built purely from **nuclear masses and positions** — no electron density
   or orbitals enter the formula at all. The *geometry* you feed it may have come from a full SCF
   optimization, but the rotational-constant calculation itself is classical mechanics on point masses.
3. **`qc.prop.geom.free_volume(mychk)`** — its `free_fraction` is exactly a single number summarizing how
   much of a bounding region is empty vs. occupied by atomic volume.
:::

That completes the molecular-properties suite tour — from charges and bonds, through topology and weak
interactions, to reactivity, orbital localization, electrostatics, and now geometry. Together with the SCF,
correlation, gradients, and environment chapters earlier in Part III, you have the full day-to-day qc-rs
toolkit.