SrTiO₃ · Pt cocatalyst · constrained DFT

How to add electrons safely to interfacial Pt6: measuring Q0 first, then setting Q = Q0 − Δn

GPAW CDFT fixed-geometry scan + CDFT geometry optimization · updated 9 October 2026

Abstract. Extra electrons on Pt are imposed with GPAW CDFT by constraining the Pt6 net charge Q = Z − Ne. The only number that must be measured and then written into the input is the unconstrained charge Q0. Targets are then Q = Q0 − Δn. An incorrect Q0 (inconsistent CDFT weight) left residuals of ~17–54 e− already at Δn = 0 and mostly crashed; the corrected protocol gives λ(0) = 0 and a complete Δn = 0→2 scan for all six models. Geometry optimization under the constraint (Δn = 0.1–2.0) is shown in an interactive trajectory viewer (§ CDFT geometry optimization).

What to put in the code after Q0 is measured

DO set the CDFT charges list to the target net charge on Pt6. DO NOT paste λ into the input — λ is optimized by CDFT, not a user parameter.

Qtarget = Q0 − Δn

Example: pristine vacuum, add 1 electron → charges = [34.598857 − 1.0] → charges = [33.598857]. 

# After q0.json is written for a given structure:
Q0 = 34.598857          # from work/

Table 1 | Per-system Q0 to copy into charges = [Q0 − Δn].

Use the same GPAW CDFT weight (default Rc/μ) for measuring Q0 and for the constrained run. Do not mix a hand-rolled weight with CDFT.

Failure versus success of Q0

Round 1 estimated Q0 with a homemade weight integral that did not match GPAW CDFT. At Δn = 0 the constraint residual was already ~17 e− (vacuum) or ~51–54 e− (H2O/KOH), so SCF mostly failed. Round 2 measures Q0 with CDFT itself (λ = 0 probe); residuals at Δn = 0 fall to ~10−4 e− and λ = 0.

Fig. 1 | Constraint residual |Error| at Δn = 0 (λ = 0). Same y-axis range as Fig. 2 (0–60) for direct visual comparison. Red: wrong Q0. Black: CDFT-consistent Q0 (bars near zero).
Fig. 2 | Net charge used as target at Δn = 0. Same y-axis as Fig. 1 (0–60). Open red: inconsistent estimate. Filled black: true CDFT Q0 (use this).

Table 2 | Head-to-head Q0 diagnostics.

Successful Δn scan (after correct Q0)

Fig. 3 | Relative CDFT free energy ΔE = E(Δn) − E(0). Vacuum costs ~8 eV at Δn = 2; aqueous/KOH ~6 eV.
Fig. 4 | Lagrange multiplier λ(Δn). Output of CDFT, not an input. λ(0) = 0 for all six systems.

Table 3 | Full successful scan.

CDFT geometry optimization with extra electrons on Pt6

Each structure was relaxed (BFGS, fmax < 0.05 eV Å−1, PBE+U+D3, bottom layers fixed) while CDFT held the Pt6 charge at Q = Q0 − Δn. Each Δn starts from the optimized geometry of the previous (smaller) Δn.

Table 4 | Optimization status. ✓ = converged (number of BFGS steps). Italic = still running (steps so far, current fmax in eV Å−1).

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Fig. 5 | Interactive optimization trajectories. Drag to rotate, scroll to zoom, right-drag (or Ctrl-drag) to pan. Choose a system and either one Δn optimization (frames = BFGS steps) or the Δn series (frames = neutral relaxed structure followed by the optimized structure at each converged Δn). The trace below the viewer shows energy relative to the first frame and fmax; click it to jump to a frame. Pt is drawn enlarged in gold.

Key observations.

(i) Relaxation removes most of the cost of adding electrons. At fixed geometry, Δn = 2 costs 6–8 eV and needs |λ| ≈ 6–8 eV. After relaxation, the cost is below ~0.8 eV and |λ| < 0.9 eV, with Pt moving only ~0.1–0.2 Å in most systems. Because the CDFT weight that defines “the charge on Pt6” depends on atomic positions, part of this drop may come from the geometry changing the partition rather than from electrons moving. This can be tested by repeating the λ = 0 Q0 probe on an optimized geometry.

(ii) Vo-interface, vacuum: the Pt6 cluster reshapes at Δn ≥ 1. One apex Pt atom drops by ~1.2 Å and shifts sideways (2.0 Å total), flattening the cluster onto the surface; the energy falls ~1 eV below the neutral reference. Watch the Δn = 1.0 trajectory.

(iii) Pristine, H2O: the water network rearranges. Pt hardly moves (< 0.1 Å) but solvent atoms move by 1.0–1.5 Å and the energy drops 0.4–0.8 eV below the neutral reference. This means the neutral water configuration was not the lowest minimum; such drops should not be read as an effect of the extra electrons.

Fig. 6 | Energy cost of adding Δn electrons, relative to the neutral relaxed structure (DFT+D3). Solid: after CDFT geometry optimization. Faint dashed: same Δn at the fixed neutral geometry (Fig. 3, D3 added for consistency).
Fig. 7 | Lagrange multiplier λ needed to hold the charge. Solid: after optimization. Faint dashed: fixed geometry (Fig. 4).
Fig. 8 | Largest Pt displacement from the neutral relaxed structure (log scale). Values ≳ 0.5 Å indicate a change in cluster shape rather than a small relaxation.

Table 5 | Converged CDFT optimizations. ΔE is relative to the neutral relaxed structure (DFT+D3). Max |d|: largest displacement from the neutral structure for Pt and for solvent atoms.

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