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5.1 Converge slab thickness and vacuum

These are unexecuted teaching templates, with no computed ABACUS results. The baseline is ABACUS 3.9.0; verify your executable version and data files before use.

Original model and balanced energy workflow schematic

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5.1.1 Scientific question

When is a repeated slab a model of an isolated surface rather than a stack of interacting films? Separate the number of atomic layers from the empty distance between periodic images. The two controls address different errors and cannot be replaced by one large total cell height.

5.1.2 Model and prerequisites

Use a nonpolar, inversion-symmetric Si(001) slab as a deliberately simplified numerical model. It has two equivalent faces. This is not a prediction of the equilibrium reconstructed Si surface. Start with the bulk lattice from first LCAO silicon, repeat it in the surface plane, and cut equal terminations. Keep a bulk-like central region fixed if the study concerns surface relaxation; document every STRU movement flag. Count layers by actual atomic planes, not by primitive-cell repeats. Begin with an even sequence of slab thicknesses compatible with the chosen termination, and test vacuum independently.

5.1.3 Worked input delta

# Replace these keys in a copy of the converged silicon INPUT.
calculation    scf
suffix         slab_vacuum_test
cal_force      1
cal_stress     1
symmetry       0

5.1.4 Physics and units

The vacuum is the distance between the outermost occupied atomic planes across the periodic boundary, not the full third lattice-vector length. A slab can change its surface geometry as its thickness grows, so compare both the central interlayer distances and the outer-layer forces. A vanishing average cell stress is not a surface convergence criterion: adding empty volume dilutes the stress reported per cell volume. Keep the in-plane lattice fixed at the selected bulk value during this diagnostic. A symmetric slab avoids a net dipole normal to the surface; an asymmetric slab requires a separately verified electrostatic treatment and is outside this baseline.

\[ d_{\mathrm{vac}}=L_z-(z_{\max}-z_{\min}),\qquad A=|\mathbf a\times\mathbf b| \]

5.1.5 Workflow and outputs

  1. Save a thickness series with unchanged in-plane vectors, termination, pseudopotential and orbital family. Write down the number of Si atoms and the fractional coordinates before each calculation.
  2. At one intermediate thickness, use trial empty gaps of 12, 16 and 20 Å. These are proposed tests, not demonstrated adequate values. Increase the third lattice vector while keeping Cartesian atomic separations unchanged; merely editing the cell with fixed fractional coordinates stretches the slab.
  3. Use a two-dimensional KPT mesh such as 6 6 1 only as a starting proposal. Test the in-plane density and retain one point normal to an adequately isolated slab.
  4. Read the converged total energy and all atomic forces from OUT.slab_vacuum_test/running_scf.log. Confirm the echoed cell and species counts. After electronic convergence, repeat selected geometries with stricter SCF and a larger LCAO orbital set.
  5. Return to the thickness series at a vacuum that has passed the independent test. Relax allowed layers using the ionic-relaxation case, then compare central geometry and the surface-energy difference introduced in the next case.

5.1.6 Convergence and acceptance

Build a matrix of layer counts versus vacuum lengths; do not compare raw total energies of cells containing different numbers of atoms. At each layer count, energy differences between vacuum variants have identical composition and are meaningful. Across layer counts, compare an excess energy relative to the same bulk reference. Record maximum force on mobile atoms and the average separation of the two innermost planes. Inspect a side view to detect atoms crossing the periodic boundary or an accidentally compressed vacuum gap. Larger orbitals must remain compatible with the same Si pseudopotential, and their finite radii must not be mistaken for evidence that electrostatic image interactions have disappeared.

5.1.7 Pitfalls

A one-point normal k mesh does not remove periodic electrostatics. A short LCAO orbital can make the energy appear insensitive to vacuum while charge rearrangement remains unconverged. Do not use cell-relax to optimize the empty direction: collapsing vacuum changes the model. Keep any frozen-layer protocol identical when comparing thicknesses.

5.1.8 Exercises with guidance

  1. Increase vacuum by 4 Å using fixed fractional coordinates, then using fixed Cartesian coordinates. Explain why the two generated structures differ. Guidance: reconstruct every Cartesian position from the lattice matrix.
  2. Design the smallest crossed thickness/vacuum test that can distinguish image coupling from insufficient bulk-like interior. Guidance: at least two thicknesses and two vacuum gaps are needed; examine force and excess-energy changes separately.

5.1.9 Primary sources