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8.1 NEB diffusion barriers

These are unexecuted teaching inputs and starting settings to test. Original figures are schematics, not computed results. Use licensed VASP and PAW data, replace every placeholder, record the executable version and validate convergence.

8.1.1 Model, units and provenance

Use eV for energy, Å for length and eV/Å for force; 1 kbar = 0.1 GPa. State normalization per atom, molecule, primitive cell or simulation cell. Record PAW identifiers, release, ZVAL, ENMAX and permitted hashes; never redistribute POTCAR. SCF convergence addresses the chosen electronic problem; convergence of the target property requires separate tests.

NEB diffusion barriers — original conceptual schematic

Original schematic. Curves explain concepts; blank data areas await verified learner results. No calculation is claimed.

\[ E_m^{\mathrm{forward}}=E^\ddagger-E_i,\quad E_m^{\mathrm{reverse}}=E^\ddagger-E_f,\quad E_m^{\mathrm{forward}}-E_m^{\mathrm{reverse}}=E_f-E_i \]

8.1.2 Worked case: procedure, interpretation and checks

What the calculation finds

Two stable sites can have similar energies but a large barrier between them. The nudged elastic band (NEB) method places a chain of intermediate structures between relaxed endpoints, then optimizes the chain toward a minimum-energy path. A spring force maintains spacing along the path while the physical force perpendicular to the local path relaxes the geometry. The spring energy is an optimization device, not part of the physical migration barrier.

The forward barrier is Em,f=E‡−Ei; the reverse barrier is Em,r=E‡−Ef. Therefore Em,f−Em,r=Ef−Ei. This identity is a useful bookkeeping check. The highest sampled image is only a discrete estimate of the saddle energy until the path and transition state are adequately refined.

Prerequisites

Choose one concrete hop, such as a vacancy exchanging with a nearest-neighbor host atom or an interstitial moving between adjacent sites. Both endpoints must have the same atoms, ordering, cell, charge state, magnetic model, and fixed-atom constraints. For a vacancy hop, the same actual atom moves into the empty site; do not independently sort endpoint coordinates so that identities silently change. Standard fixed-cell NEB does not describe an arbitrary variable-cell phase transition.

Stepwise workflow

  1. Independently relax both endpoints at the same fixed cell and numerical settings. Run final static endpoint energies; do not assume the NEB run recalculates frozen endpoint energies for you.
  2. Map every atom between endpoints, choose the correct periodic minimum-image displacement, and unwrap the migrating coordinate consistently. A coordinate crossing a periodic boundary should not force an atom across the entire box.
  3. Generate an initial band, for example five intermediate images. Linear interpolation is a starting guess, not guaranteed safe: visualize all images and inspect minimum interatomic distances. Use a better geometric interpolation if atoms collide.
  4. Prepare image directories 00 through 06, where 00 and 06 are fixed endpoints and 01–05 are moving images. Place shared inputs in the root as required by VASP's NEB layout.
  5. Run ordinary NEB with a supported optimizer and inspect the maximum appropriate NEB force over every moving image. A job reaching NSW is not automatically converged.
  6. Refine the neighborhood of the maximum and test more images. If using a climbing-image implementation, confirm the actual VASP/VTST build and supported tags; do not assume an undocumented LCLIMB line works in every stock executable. A dimer refinement is another route when supported.
  7. Validate the candidate transition state through a vibrational/curvature analysis and connectivity checks where feasible. A first-order saddle has one unstable mode associated with the reaction coordinate in the relevant unconstrained subspace.
  8. Repeat critical parts with tighter forces, larger supercell, and denser electronic sampling. Compare alternative plausible paths; a converged band is not proof of the globally lowest barrier.

The negative SPRING convention activates the nudged-band treatment; the magnitude is a numerical spring parameter, not a material elastic constant. Optimizer behavior and suitable POTIM depend on the case. Start from the installed official tutorial and adjust using actual convergence behavior. Parallel resources must be compatible with VASP's image decomposition and the site's build; a generic MPI command is deliberately not supplied.

Reading the barrier and avoiding overclaims

Plot physical total energy relative to Ei against cumulative atomic path length, with actual images marked. A smooth spline may overshoot sparse points; report whether the saddle estimate comes from a sampled image, an interpolation, or a refined saddle. Examine force convergence on all images, especially the maximum. If symmetry-equivalent endpoints have different energies beyond the target tolerance, fix endpoint consistency first.

A migration barrier is not a diffusion coefficient. A rate approximation might use k≈ν0 exp(−ΔG‡/kBT), requiring a prefactor and activation free energy. A diffusion coefficient additionally needs hop distances, dimensionality, connectivity, correlation factors, and defect concentration where relevant. For vacancy-mediated self-diffusion, equilibrium vacancy formation and migration both matter; for a fixed pre-existing defect population the activation interpretation differs. Light atoms may need zero-point/tunneling effects; finite-temperature entropy can change barriers. Charged pathways need consistent electron reservoirs and finite-size treatment, with attention to localization changing along the path.

Exercise

8.1.3 Unexecuted inputs and analysis scaffolds

These are unexecuted teaching inputs and starting settings to test. Original figures are schematics, not computed results. Use licensed VASP and PAW data, replace every placeholder, record the executable version and validate convergence.

8.1.3.1 Input block 1

# Ordinary fixed-cell VASP NEB starting template; five moving images
IMAGES = 5
SPRING = -5
IBRION = 1
POTIM  = 0.1
NSW    = 250
ISIF   = 2
EDIFF  = 1E-7
EDIFFG = -0.02
ISYM   = 0
LREAL  = .FALSE.
# Add tested ENCUT, KPOINTS, occupations, spin, and functional settings.
# This is not a universal VTST climbing-image input.

8.1.5 Technical sources