# d-Band Center Descriptor Benchmark

**Snapshot**: 2026-04-14
**Engine**: FluxMateria physics engine (production stack)
**Prediction basis**: Flux Physics
**Benchmark use**: published references used for validation scoring

---

## Abstract

The d-band center is the central descriptor for transition-metal
catalysis (Hammer-Norskov framework). The FluxMateria physics engine predicts this quantity from atomic quantum numbers and surface coordination using the Flux Physics d-band route.

Against a 100-case multi-source literature benchmark, the engine
achieves a combined mean absolute error of **0.197 eV** — within the
method-to-method spread reported across independent DFT studies.

## Headline numbers

| Set                     |   n | MAE (eV) | RMSE (eV) | ≤0.2 eV | ≤0.3 eV | ≤0.5 eV |
|-------------------------|----:|---------:|----------:|--------:|--------:|--------:|
| **Combined**            | 100 | **0.197**|     0.278 |    71% |    81% |    90% |
| Pure transition metals  |  27 |    0.223 |     0.299 |    63% |    74% |    89% |
| Facet-specific surfaces |  41 |**0.154** |     0.206 |    83% |    90% |    93% |
| Binary alloys           |  32 |    0.230 |     0.334 |    62% |    75% |    88% |

## Coverage

- **3d / 4d / 5d** transition metals, plus group 3 f-block boundary
  elements (Y, La, Ce) — 27 pure elements total.
- **Facets**: (111) close-packed, (100) open, (110), (211),
  (0001) HCP basal, and step-edge geometries — 41 surface cases.
- **Binary alloys**: Pt-, Pd-, Au-, Ag-, Ni-, Co-, Rh-, Mo-, Fe-based
  compositions including skin alloys and intermetallics — 32 cases.

## Error breakdown

| Period |   n | MAE (eV) |
|--------|----:|---------:|
| 3d     |  32 |    0.141 |
| 4d     |  34 |    0.181 |
| 5d     |  34 |    0.266 |

| Facet  |   n | MAE (eV) |
|--------|----:|---------:|
| (0001) |   2 |    0.076 |
| (211)  |   6 |    0.115 |
| step   |   1 |    0.113 |
| (110)  |  12 |    0.135 |
| (100)  |  12 |    0.186 |
| (111)  |   8 |    0.187 |

## Comparison to ML baselines (5-fold cross-validation)

The same atomic feature set — Z, period, group, d-electron count,
filling fraction, metallic radius, work function — was fed to three
standard ML regressors with 5-fold cross-validation on the pure-TM
set (27 samples). The FluxMateria route is evaluated directly on the same held-out descriptor rows.

| Model                           | CV MAE (eV) | CV RMSE (eV) |
|---------------------------------|------------:|-------------:|
| Linear regression               |       0.432 |        0.556 |
| k-Nearest Neighbors (k=3)       |       0.388 |        0.657 |
| Random Forest (20 × depth 4)    |       0.295 |        0.470 |
| **FluxMateria physics engine**         |   **0.223** |    **0.299** |

The physics-derived predictor outperforms every ML baseline trained on the same atomic descriptors.

## Comparison to state-of-the-art

| Method                          | Type                       | MAE (eV)    |
|---------------------------------|----------------------------|------------:|
| **FluxMateria - pure TMs**      | Flux Physics               | **0.223**   |
| **FluxMateria - facets**        | Flux Physics               | **0.154**   |
| **FluxMateria - combined**      | Flux Physics               | **0.197**   |
| PBE-DFT (single method)         | DFT                        | 0.15 – 0.30 |
| SISSO surrogate                 | ML on DFT features         | 0.20 – 0.30 |
| Neural-network surrogate        | ML on DFT features         | 0.15 – 0.25 |
| CGCNN graph neural network      | GNN on DFT + structure     | 0.15 – 0.20 |

## Literature sources used for cross-validation

- Hammer & Nørskov, *Chem. Rev.* **114**, 4259 (2014)
- Kitchin, Nørskov, Barteau, Chen, *J. Chem. Phys.* **120**, 10240 (2004)
- Greeley, Stephens, Bondarenko et al., *Nat. Chem.* **1**, 552 (2009)
- Hammer & Nørskov, *Surf. Sci.* **343**, 211 (1995)
- Chen & Mavrikakis, *Chem. Rev.* **121**, 1007 (2021)

## Known limitations

- Group 3 f-block boundary elements (Y, La, Ce) show larger residuals
  (~0.5 eV) due to 4f/5d orbital mixing.
- Pt-3d skin alloys (Pt₃Ni, Pt₃Co, Pt₃Fe) under-predict the
  ligand-induced deepening by 0.6–0.9 eV.
- Period-6 (5d) surfaces show larger residuals than 3d/4d series due
  to stronger spin–orbit and relativistic contributions.
- Single-source elements (Sc, Cr, Mn, Hf, Ta, Nb, Zr, Tc, La, Ce) are
  cross-validated against only one literature entry.

## Bottom line

- 100-case benchmark across three system types
- Combined MAE 0.197 eV
- Within DFT method-to-method spread on every category
- Outperforms linear, k-NN and random-forest baselines trained on the
  same descriptors
- Flux Physics descriptor route with no per-case DFT input