GCMC acceptance: the de Broglie particle count

This note records a deliberate, non-obvious convention in the grand-canonical acceptance test, the discussion behind it, and the change history. It exists so the choice is not silently “fixed” by a future contributor who (correctly) notices it departs from the textbook form.

The acceptance criterion

Insertion and deletion are accepted with the de Broglie (grand-canonical) probabilities (see mcpy.ensembles.GrandCanonicalEnsemble and mcpy.utils.set_unit_constant):

\[p_\text{ins} = \frac{V}{(N+1)\,\Lambda^3}\; e^{-\beta(\Delta E - \mu)}, \qquad p_\text{del} = \frac{N\,\Lambda^3}{V}\; e^{-\beta(\Delta E + \mu)} .\]

V is the cell free volume, \(\Lambda\) the thermal de Broglie wavelength of the exchanged species, \(\mu\) its chemical potential, and N the particle count fed to the combinatorial factor. The question this note settles is: which ``N``?

Two conventions for N

total atom count (the convention mcpy uses)

N = len(atoms) before the move: every atom of every species, including the fixed substrate. This is what do_gcmc_step and BatchedReplicaExchange pass today (self.n_atoms / r.n_atoms).

per-species exchangeable count (textbook)

N = number of atoms of the exchanged species inside the insertion region (the move’s cell). This is the physically standard choice.

The physically correct quantity is the per-species count. The de Broglie factor is the ideal-gas combinatorial term: the \(1/N!\) indistinguishability correction applies only among identical, interchangeable particles, and only to the population that shares the insertion volume V. The total atom count mixes in distinguishable species and the fixed substrate, which do not belong in that factor. The interaction with all those atoms enters the acceptance through \(\Delta E\) (the total potential energy change), not through N.

Two related conventions to keep in mind:

  • mu is the full chemical potential, including the ideal/\(\Lambda\) term. So mu must be referenced on the calculator’s absolute energy scale (e.g. an O2 gas reference at the target T and p).

  • mcpy uses a free volume (cell_volume * (1 - occupied_fraction)) in the combinatorial factor while still sampling insertion points over the full cell box, so the proposal and acceptance volumes differ at high density.

Why mcpy keeps the total atom count

Reproducibility. The group’s entire body of GCMC work (including the forthcoming Ag-oxidation study) was produced with the total-atom-count convention. Switching to the per-species count shifts the effective chemical potential by

\[\Delta\mu_\text{eff} = k_BT \,\ln\!\frac{N_\text{total}+1}{N_\text{species}+1},\]

which for a few-thousand-atom substrate with tens of exchanged atoms is ~0.1-0.2 eV toward insertion at typical temperatures. Concretely, after the per-species change an oxide phase boundary moved from roughly \(\Delta\mu \approx -0.5\) eV to \(\approx -0.3\) eV, and oxygen kept inserting well into the reducing regime. Keeping the original convention keeps every simulation on one consistent footing; the per-species count would require re-running and re-calibrating mu across all prior work.

This is a reproducibility choice, not a claim that the total count is more correct. If you start a fresh study with no need to match prior runs, prefer the per-species count (and recalibrate mu against a gas reference).

Side effect that the total count happens to mask

With the per-species count there is a runaway failure mode for species that dissolve below the cell floor. CustomCell.get_atoms_specie_inside_cell excludes atoms with z < bottom_z (subsurface, “absorbed”), so such atoms are neither counted nor eligible for deletion. Under the per-species convention this undercount inflates \(V/((N+1)\Lambda^3)\) and insertion stays favored no matter how negative mu is — one-way accumulation decoupled from mu.

The total-atom-count convention sidesteps this: as oxygen accumulates anywhere (surface or subsurface), N_total grows, which suppresses further insertion. The system is self-limiting. The buried-O irreversibility still exists (buried O is never a deletion candidate), but it no longer drives runaway uptake.

Change history

  • 668e732 / earlier — both ensembles born using the total atom count.

  • adcc15b — switched GrandCanonicalEnsemble to the per-species count.

  • 9a15425 — switched BatchedReplicaExchange to the per-species count.

  • this change — reverted both back to the total atom count for reproducibility, and recorded the discussion here.

Where this lives in the code and tests

  • mcpy/ensembles/grand_canonical_ensemble.py — do_gcmc_step passes self.n_atoms to _acceptance_condition.

  • mcpy/ensembles/batched_replica_exchange.py — _batched_single_move passes r.n_atoms.

  • tests/test_acceptance.py — test_do_gcmc_step_feeds_total_atom_count guards the convention; test_total_vs_perspecies_count_shifts_effective_mu and test_perspecies_count_inserts_where_total_rejects quantify the gap.

  • tests/test_custom_cell_region.py — test_subsurface_oxygen_is_excluded_from_deletion_candidates documents the buried-O caveat.

Molecular moves

Rigid-molecule insertion/deletion (mcpy.moves.MoleculeInsertionMove, mcpy.moves.MoleculeDeletionMove) do not use the total-atom convention above. They use the textbook rigid-molecule form: N is the number of molecules of the exchanged species whose center of mass lies inside the move’s cell, reported by the move itself through MoveSelector.get_exchange_count(). \(\Lambda\) is computed from the total molecular mass, and delta_particles remains \(\pm 1\) (one molecule per move).

Orientations are sampled uniformly, so the rotational partition function is absorbed into \(\mu\): the chemical potential passed for a molecular species must be the full molecular chemical potential (translational, rotational, and internal contributions of the reference reservoir included).

Atomic moves are unaffected: for them get_exchange_count() returns None and the ensembles fall back to the total-atom count documented above.

The textbook form above holds for min_insert=None. When min_insert is set, MoleculeInsertionMove retries the random position/orientation draw (up to 1000 times) against the cell’s species_radii atoms until it clears that minimum separation, which effectively restricts the proposal to the non-overlapping volume; the acceptance test still divides by the full cell volume V. This is the same proposal/acceptance mismatch already noted above for atomic moves: at high density it over-accepts insertions by roughly \(V/V_\text{accessible}\).

Atomic and molecular species of the same element can coexist (e.g. atomic O alongside O2 molecules): an atomic mcpy.moves.InsertionMove tags its inserted atom as free (molecule_id = -1, guarding against ASE’s zero-fill when extending arrays), and an atomic mcpy.moves.DeletionMove only picks free atoms, never molecule members. Displacement-type moves are not molecule-aware: they can drag a member atom individually, which is thermodynamically valid with a relaxing calculator but departs from the rigid-molecule picture.

The MPI mcpy.ensembles.ReplicaExchange does not support molecular species either: its per-species swap bookkeeping counts atoms by symbol (atoms.symbols.count(specie)), which is always 0 for a molecular name, so its constructor raises NotImplementedError when the wrapped GCMC ensemble has molecular species configured; mcpy.ensembles.BatchedReplicaExchange is the supported replica-exchange path for molecular GCMC.