E9: Mobile Block Hessian (MBH)

Module: sci_form::experimental::mbhFeature flag: experimental-mbh

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Overview

Reduces vibrational analysis cost from $O(3N)$ energy evaluations to $O(n_{\text{flex}})$ by treating rigid groups (aromatic rings, fused ring systems) as rigid bodies with 6 degrees of freedom (3 translation + 3 rotation). Enables real-time IR spectra for large molecules where full Hessian computation is impractical.

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Theory

Rigid Block Detection

Rigid blocks are identified from the SSSR (Smallest Set of Smallest Rings):

• Aromatic rings → rigid
• Saturated rings ≤ 8 atoms → rigid
• Fused ring systems → merged into single blocks
• All other atoms → flexible (full 3 Cartesian DOF)

DOF Reduction

Each rigid block contributes 6 collective DOF instead of $3N_{\text{block}}$:

$$n_{\text{DOF}}=6\cdot n_{\text{blocks}}+3\cdot n_{\text{flex}}$$

For a molecule with 60% of atoms in rings, the Hessian size is reduced to ~40%.

Projection Matrix

The projection matrix $L$ maps reduced coordinates to Cartesian displacements:

• Translation columns: Uniform displacement of all block atoms along $x$, $y$, $z$
• Rotation columns: Infinitesimal rotation of block atoms about COM along $x$, $y$, $z$ axes
• Flexible columns: Standard Cartesian unit vectors for non-block atoms

Generalized Eigenvalue Problem

$$H_{\text{MBH}}\mathbf{c}=\lambda M_{\text{MBH}}\mathbf{c}$$

where $H_{\text{MBH}}=L^{T}HL$ and $M_{\text{MBH}}=L^{T}ML$. Frequencies in cm${}^{-1}$:

$${\nu }_i=\frac{1}{2\pi c}\sqrt{{\lambda }_i}\times 108.59$$

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API

use sci_form::experimental::mbh::*;

// Detect rigid blocks
let decomposition = detect_rigid_blocks(&elements, &positions, &ring_info);
// decomposition.blocks: Vec<RigidBlock>
// decomposition.flexible_atoms: Vec<usize>
// decomposition.n_dof_reduced: usize
// decomposition.n_dof_full: usize

// Build projection matrix
let l_matrix = build_projection_matrix(&decomposition, &positions);

// Compute MBH frequencies
let config = MbhConfig { fd_step: 0.005 };
let energy_fn = |coords: &[f64]| -> f64 { /* energy evaluation */ };
let result = compute_mbh_frequencies(&elements, &positions, &ring_info, &energy_fn, &config);
// result.frequencies: Vec<f64>    (cm⁻¹)
// result.n_dof_full: usize
// result.n_dof_reduced: usize
// result.n_blocks: usize

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Speedup

Molecule	Full DOF	MBH DOF	Speedup
Benzene (6 ring + 6 H)	36	24	1.5×
Naphthalene (10 ring + 8 H)	54	30	1.8×
Aspirin (1 ring + rest)	63	~45	1.4×
Penicillin (2 rings)	99	~60	1.7×

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Tests

cargo test --features experimental-mbh --test regression -- test_mbh

8 integration tests covering: rigid block detection, block COM computation, projection matrix orthogonality, DOF reduction verification, frequency computation, frequency ordering, and comparison between full and reduced Hessian results.