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Activity Scoring

Activity scoring quantifies how well a catalyst's binding energy aligns with the Sabatier optimum.

The Sabatier Principle

The activity of a catalyst is governed by the Sabatier principle:

Sabatier (1911)

An optimal catalyst binds reaction intermediates with intermediate strength - neither too weakly (preventing activation) nor too strongly (preventing product desorption).

This creates the classic volcano plot:

Activity
   ↑
   |           ★ Optimal
   |          /|\
   |         / | \
   |        /  |  \
   |       /   |   \
   |______/____|____\______→ Binding Energy
     weak    ΔE_opt    strong

Mathematical Formulation

Linear Scoring (Default)

\[S_a(\Delta E) = \max\left(0, 1 - \frac{|\Delta E - \Delta E_{opt}|}{\sigma_a}\right)\]

Where:

Parameter Symbol Description Unit
Adsorption energy \(\Delta E\) DFT-calculated binding eV
Optimal energy \(\Delta E_{opt}\) Sabatier peak position eV
Activity width \(\sigma_a\) Tolerance for "good" activity eV

Properties:

  • Score = 1.0 when \(\Delta E = \Delta E_{opt}\) (at peak)
  • Score = 0.5 when \(|\Delta E - \Delta E_{opt}| = \sigma_a / 2\)
  • Score = 0.0 when \(|\Delta E - \Delta E_{opt}| \geq \sigma_a\)

Gaussian Scoring (Alternative)

\[S_a(\Delta E) = \exp\left(-\frac{(\Delta E - \Delta E_{opt})^2}{2\sigma_a^2}\right)\]

Properties:

  • Score = 1.0 at optimum
  • Score = 0.61 at \(\pm\sigma_a\) from optimum
  • Never reaches exactly zero

Python Usage

Basic Usage

from ascicat.scoring import score_activity

# Single catalyst
score = score_activity(
    delta_E=-0.25,       # eV
    optimal_E=-0.27,     # eV (HER optimum)
    width=0.15,          # eV
    method='linear'
)
print(f"Activity score: {score:.3f}")  # 0.867

Array Operations

import numpy as np

# Multiple catalysts
energies = np.array([-0.42, -0.35, -0.27, -0.19, -0.12])

scores_linear = score_activity(energies, -0.27, 0.15, method='linear')
scores_gauss = score_activity(energies, -0.27, 0.15, method='gaussian')

for E, lin, gauss in zip(energies, scores_linear, scores_gauss):
    print(f"ΔE={E:+.2f} eV: Linear={lin:.3f}, Gaussian={gauss:.3f}")

Output:

ΔE=-0.42 eV: Linear=0.000, Gaussian=0.135
ΔE=-0.35 eV: Linear=0.467, Gaussian=0.606
ΔE=-0.27 eV: Linear=1.000, Gaussian=1.000
ΔE=-0.19 eV: Linear=0.467, Gaussian=0.606
ΔE=-0.12 eV: Linear=0.000, Gaussian=0.135

Using ActivityScorer Class

from ascicat.scoring import ActivityScorer

scorer = ActivityScorer()

# Linear scoring
s_linear = scorer.linear(delta_E=-0.30, optimal_E=-0.27, width=0.15)

# Gaussian scoring
s_gauss = scorer.gaussian(delta_E=-0.30, optimal_E=-0.27, width=0.15)

Reaction-Specific Parameters

Predefined Configurations

Reaction Pathway \(\Delta E_{opt}\) \(\sigma_a\)
HER H adsorption -0.27 eV 0.15 eV
CO2RR CO -0.67 eV 0.15 eV
CO2RR CHO -0.48 eV 0.15 eV
CO2RR COCOH -0.32 eV 0.15 eV

Access Configuration

from ascicat.config import get_reaction_config

# Get HER configuration
her_config = get_reaction_config('HER')
print(f"Optimal energy: {her_config.optimal_energy} eV")
print(f"Activity width: {her_config.activity_width} eV")

# Get CO2RR-CO configuration
co_config = get_reaction_config('CO2RR', pathway='CO')
print(f"Optimal energy: {co_config.optimal_energy} eV")

Choosing Between Methods

Linear Scoring

Advantages:

  • Computationally efficient
  • Easy to interpret
  • Consistent with traditional volcano plots
  • Clear cutoff at \(\pm\sigma_a\)

Best for:

  • Standard screening
  • Large datasets
  • When interpretability is important

Gaussian Scoring

Advantages:

  • Smoother discrimination near optimum
  • Never exactly zero (no hard cutoff)
  • Mathematically differentiable

Best for:

  • Sensitivity analysis
  • When fine discrimination at extremes matters
  • Optimization algorithms

Visualization

import matplotlib.pyplot as plt
import numpy as np
from ascicat.scoring import score_activity

# Generate data
energies = np.linspace(-0.6, 0.1, 100)
scores_lin = score_activity(energies, -0.27, 0.15, 'linear')
scores_gau = score_activity(energies, -0.27, 0.15, 'gaussian')

# Plot
fig, ax = plt.subplots(figsize=(8, 5))
ax.plot(energies, scores_lin, 'b-', label='Linear', linewidth=2)
ax.plot(energies, scores_gau, 'r--', label='Gaussian', linewidth=2)
ax.axvline(-0.27, color='gray', linestyle=':', label='Optimal')
ax.axhline(0.5, color='gray', linestyle=':', alpha=0.5)
ax.set_xlabel('Adsorption Energy (eV)')
ax.set_ylabel('Activity Score')
ax.legend()
ax.set_xlim(-0.6, 0.1)
ax.set_ylim(0, 1.05)
plt.show()

Physical Interpretation

Score Ranges

Score Range Interpretation Example Materials
0.9 - 1.0 Excellent At volcano peak
0.7 - 0.9 Good Near-optimal catalysts
0.5 - 0.7 Moderate Acceptable for some applications
0.3 - 0.5 Poor Suboptimal binding
0.0 - 0.3 Very poor Far from optimum

HER Examples

Catalyst \(\Delta E\) (eV) \(S_a\) Interpretation
Pt(111) -0.09 0.80 Excellent
Ni(111) -0.47 0.00 Too strong
Au(111) +0.18 0.00 Too weak
Fe-Sb -0.25 0.87 Near optimal

Common Issues

Sign Convention

ASCICat uses the convention where negative adsorption energies indicate stable binding. Ensure your DFT data uses the same convention:

\[\Delta E = E_{slab+adsorbate} - E_{slab} - \frac{1}{2}E_{H_2}\]

Width Parameter

The default \(\sigma_a = 0.15\) eV is calibrated for typical DFT accuracy. If your calculations use different exchange-correlation functionals, you may need to adjust this.