feat(backends): AMOEBA polarizable force field support

[ericchansen]

Summary

Add support for AMOEBA polarizable force field parameterization and optimization in Q2MM.

Motivation

AMOEBA (Atomic Multipole Optimized Energetics for Biomolecular Applications) is a next-generation polarizable force field that includes:

• Permanent atomic multipoles (charges, dipoles, quadrupoles)
• Mutual polarization (induced dipoles)
• Buffered 14-7 van der Waals
• Valence terms similar to MM3

AMOEBA is natively supported by Tinker (its primary engine) and OpenMM. Q2MM's quantum-guided approach could be valuable for fitting AMOEBA parameters, particularly for:

• Transition state force fields (our core use case)
• Coordination complexes with strong polarization effects
• Systems where fixed-charge models are insufficient

Scope

Phase 1: AMOEBA energy evaluation

• Extend TinkerEngine to handle AMOEBA parameter files (.prm with multipole/polarize keywords)
• Extend OpenMMEngine to build AMOEBA systems (OpenMM has AmoebaMultipoleForce, AmoebaVdwForce, etc.)
• Parse AMOEBA parameter types in ForceField model (multipole, polarize, etc.)

Phase 2: AMOEBA parameter optimization

• Identify which AMOEBA parameters are suitable for Q2MM optimization (valence terms? multipoles? polarizabilities?)
• Add AMOEBA parameter types to ForceField.get_param_vector() / set_param_vector()
• Validate with known AMOEBA parameterization cases

Phase 3: Analytical gradients for AMOEBA

• OpenMM: Use addEnergyParameterDerivative() on AMOEBA CustomForce terms
• Tinker: Investigate if analytical dE/d(param) is feasible for multipole/polarization terms
• JAX: Potentially implement AMOEBA energy functions in JAX (complex but possible)

Relationship to Poltype2

Poltype2 automates AMOEBA parameterization for small molecules from QM data. It handles:

• Atom typing and multipole assignment
• Torsion fitting
• Polarizability assignment

Q2MM could complement Poltype2 by:

• Providing an alternative optimization engine with analytical gradients
• Supporting transition state parameterization (which Poltype2 doesn't do)
• Offering a unified framework that works across AMOEBA, MM3, and OPLSAA

Technical considerations

• AMOEBA's mutual polarization makes the energy a self-consistent function -- differentiating through the SCF iteration requires implicit differentiation or unrolling
• Multipole rotation frames add complexity to parameter gradients
• OpenMM's AMOEBA implementation is well-tested but the parameter derivative API may not cover all AMOEBA force types (needs investigation)

Related issues

• feat: Differentiable MM backend via JAX for analytical gradients #68 -- JaxEngine (analytical gradients infrastructure)
• feat: MM3 functional forms in JAX for differentiable MM3 energy evaluation #91 -- MM3 JAX forms
• Tinker analytical gradients issue
• OpenMM analytical gradients issue