Special Quasirandom Structures (SQS and GQS)

SQS and GQS are methods for identifying configurations that best represent random disorder in a solid solution, by comparing short-range ordering statistics to ideal random-mixing values.

Concept

For a disordered solid solution, ideal random mixing produces characteristic pair probabilities. The sqssod program scores and ranks all configurations by comparing species-resolved pair probabilities against independent-random targets. For a pair orbit connecting targets t and u, the target for species channel alpha,beta is x(t,alpha) * x(u,beta). This target-aware form covers binary, multinary, multi-target, and multinary multi-target substitution without allowing species to move onto forbidden sublattices.

  • SQS (sqssod): identifies the best-matching configuration at a fixed composition (0 K weighting).

  • GQS (gqssod): extends SQS to finite temperature by computing Boltzmann-weighted thermal averages of the multi-site cluster correlation functions (orders 1 to MaxOrder) from calculated energies.

Ensemble source: enumeration or random sampling

sqssod and gqssod score whatever configurations are listed in ENSEMBLE — they do not enumerate or search the configurational space themselves. The SQS is simply the best-scoring member of the ensemble you provide. That ensemble can come from either route:

  • Full enumeration (sod_comb.sh → nXX/ENSEMBLE): the symmetry-inequivalent configurations at the composition. This guarantees the true best SQS within the supercell is present, but is only feasible when the space is small enough to enumerate.

  • Uniform random sampling (sod_random.sh → nXX/random/ENSEMBLE): a large random sample of configurations. This is the practical route when the full space is too large to enumerate — generate a big uniform ensemble and extract the best SQS from it. Point sqssod at the random/ directory (which holds the sampled ENSEMBLE) while EQMATRIX, supercell.cif, and INSOD remain in the parent folder. Random sampling needs no reference energies, so this works before any DFT is run; for GQS, supply ENERGIES for the sampled configurations.

With random sampling the result is the best SQS found in the sample, not provably the global optimum — enlarge the sample to improve it.

Note

Future direction. SOD does not yet perform a directed search for the SQS (e.g. simulated annealing or basin-hopping that minimises the score Q as a pseudo-energy to drive the configuration toward ideal randomness). At present the SQS can only be selected from a pre-generated ensemble (enumerated or randomly sampled). A directed-search mode is a planned enhancement.

Workflow

From a full enumeration (sod_comb.sh → nXX/ENSEMBLE):

Prerequisites: sod_comb.sh must have been run to generate EQMATRIX, supercell.cif, and nXX/ENSEMBLE.

  1. Create an INSQS file in SODPROJECT/ (or in nXX/ for a per-composition override).

  2. Run SQS scoring:

    sod_sqs.sh        # from SODPROJECT/: scores all nXX/ compositions
sod_sqs.sh        # from nXX/:        scores that composition only

  3. Optionally run GQS (requires ENERGIES and TEMPERATURES):

    sod_gqs.sh        # from SODPROJECT/ or nXX/

From a random sample (sod_random.sh → nXX/random/ENSEMBLE):

Prerequisites: sod_comb.sh must have been run (for EQMATRIX and supercell.cif); sod_random.sh must have produced nXX/random/ENSEMBLE.

  1. Create INSQS in nXX/random/ (alongside ENSEMBLE).

  2. Run SQS scoring from that directory:

    cd nXX/random/
sod_sqs.sh        # reads ENSEMBLE and INSQS here; writes OUTSQS here

  3. Generate calculator input files for the best configuration (rank 1 is nearest to ideal randomness):

    sod_gener.sh -choose bestSQS   # still from nXX/random/
                               # writes cYY/ under nXX/random/

    -choose takes either explicit configuration indices or a selection label naming a rule, optionally followed by how many to take (default 1):

    sod_gener.sh -choose bestSQS 10   # the ten best SQS, in rank order

    bestSQS reads OUTSQS from the same directory as the ENSEMBLE being generated, so the ranking always belongs to that ensemble, and it fails rather than falling back if no OUTSQS is there. Labels are case-insensitive. The count is bounded by how many rows sqssod wrote: OUTSQS lists the top n_top_sqs configurations, 10 by default, so asking for more is an error telling you to raise n_top_sqs in INSQS or set it to 0 to rank the whole ensemble. lowestENERGY and highestENERGY read ENERGIES from that same directory and are bounded only by the number of entries in it.

    To generate some other configuration, take the Config column of the row you want and pass it explicitly:

    sod_gener.sh -choose <index>

    The first OUTSQS column is the rank and the second is the configuration index. They are rarely equal, so -choose 1 almost never generates the best SQS – it silently produces a valid but unremarkable configuration. That is the mistake the label removes.

    Note

    There is no bestGQS label. gqssod prints its ranking to the terminal only – OUTGQS holds thermal averages of the cluster correlations, not a ranked list of configurations – and it ranks once per temperature, so “the best GQS” would not name a single structure. Adding the label would mean giving gqssod a ranked output file and the label a temperature.

Input file: INSQS

INSQS controls the pair cutoff and scoring parameters. Example:

# Maximum cluster order (2-6)
4

# Cutoff radii (Angstroms) for orders 2..MaxOrder
8.0 6.0 4.0

# Weights for orders 2..MaxOrder
1.0 1.0 1.0

# omega eps_tol
10  1.0E-8

# n_top_sqs: number of top configurations listed in OUTSQS (0 = rank and list all)
10

Key parameters:

  • MaxOrder: accepted for backward compatibility. The generalized sqssod scorer currently uses pair correlations only, so order 2 controls the SQS score and entries for orders 3+ are ignored by sqssod.

  • Cutoff radii: one per order from 2 to MaxOrder, in Å. sqssod uses the order-2 value as the pair cutoff.

  • Weights: one per order from 2 to MaxOrder. sqssod uses the order-2 weight in the pair-family error normalization.

  • Scoring: van de Walle matched-diameter scoring, Q = -omega * L + Error, where Error is the normalized mean absolute deviation of species-resolved pair probabilities from their target random probabilities.

  • n_top_sqs: number of best-ranked configurations listed in OUTSQS; 0 sorts and lists the whole ensemble. Optional line; defaults to 10 if absent.

Output files

sqssod writes two files in each composition or sample folder:

  • OUTSQS: compact ranked list of the n_top_sqs best configurations (default 10; 0 = all) with matched distance, total error, score, and target-pair family errors such as E11 and E12. Equal-score configurations rank by configuration index.

  • SQS_CORRELATIONS: detailed species-resolved pair-channel probabilities for the best-ranked configuration, including P_cell, P_random, and Delta for each pair orbit/channel.

Rank 1 is the best SQS.

gqssod writes two files:

  • OUTGQS: thermal averages of the cluster correlation functions at each temperature in TEMPERATURES, plus the T → ∞ (equiprobable) limit.

  • wc_parameters.dat: the corresponding Warren-Cowley short-range order parameters αn for each symmetrically distinct pair shell. These are reported as an SRO diagnostic; the SQS/GQS ranking itself is done on the correlation functions, not on αn.

See the repository README.md for full format descriptions of OUTSQS, SQS_CORRELATIONS and OUTGQS.

Selecting a SQS

  • Rank 1 is closest to ideal randomness. Prefer the configuration with the lowest weighted van de Walle score, Q.

  • Degeneracy: higher-degeneracy configurations may be thermodynamically preferred; use judgement when multiple configurations score similarly.

  • Supercell size: larger supercells generally yield better SQS. For small cells, no configuration may achieve exact π = target.

Further information

The repository README.md contains the full INSQS/OUTSQS/OUTGQS format specifications and a worked example using example01/FILER1_gulp.