Two-stage formation-energy correction (NbZr, TaZr, VZr)

This bundle contains the scripts, the raw and corrected per-structure data, and the manuscript plots for the NbZr / TaZr / VZr BCC binary formation energies and the associated RMSDs.

Why a two-stage correction is necessary: The "raw" formation energy of every relaxed VASP configuration is computed
in the usual way,

FE_raw(c) = E_alloy(c) - sum_i x_i * E_pure_i ,

where E_pure_i are the per-atom total energies of the pure-element reference
structures (Nb, Ta, V, Zr in the same BCC supercell, with identical INCAR /
KPOINTS / PAW choices). With perfectly consistent reference runs the raw FE
should vanish at the two pure-element endpoints (x = 0 and x = 1) by
construction.

In practice this does not hold for two reasons that are present in our
dataset:

1. Reference-energy inconsistency (composition-dependent bias).
Even with identical input parameters, the pure-element runs (stored in
`corrected_DFT_pure_element_runs/`) differ slightly from the values that
would be implied by the alloy runs at near-pure compositions (a few
meV/atom). This bias is approximately linear in concentration, because
the residual error in E_pure_Nb (or E_pure_Ta / E_pure_V) propagates
into FE_raw(c) as (1 - x) * dE_pure_1, and the corresponding error in
E_pure_Zr propagates as x * dE_pure_2. Left uncorrected, this produces
a non-physical "tilt" of FE_raw(x) and shifts the entire FE-vs-x cloud
away from zero at the endpoints.

2. Endpoint anchoring against the audited true endpoints.
The strict endpoint values (FE_x0_meVatom, FE_x1_meVatom in
`corrected_fe_strict_endpoints_20260518/strict_endpoint_check_20260518.csv`)
were re-derived from an independent cross-check of the pure-element
runs. After stage 1 removes the linear bias, the near-pure compositions
in the alloy dataset still extrapolate to values that differ slightly
from these audited endpoints — because stage 1 is fit from a few
near-end alloy bins, not from the audited pure-element references
themselves.

The README.txt file discusses how these issues are addressed by the two-stage correction, and describes folder layout, pipeline summary, and how to re-run.