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1.3 First silicon calculation with plane waves

Version and execution status. Core syntax is checked against the official ABACUS v3.9.0 source documentation and example files, reviewed on 2026-10-05. The release list also contains stable v3.10.1 and 3.11 beta tags; v3.9.0 is this course’s reproducible baseline, not a claim about the newest release. These are unexecuted teaching templates. Initial numerical values are candidates for testing, not certified results. No energies, timings or convergence traces below are measurements.

Reciprocal lattice vectors retained within a plane-wave kinetic-energy cutoff

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1.3.1 The physical question and controlled model

Can a plane-wave calculation reproduce a well-defined static silicon reference that can later test LCAO? We begin with a two-atom primitive diamond cell, non-spin-polarized occupations and the source-example LDA pseudopotential. This is not an equilibrium-lattice prediction: the lattice constant is held at a specified teaching value. The task is to obtain an electronically converged energy for that exact geometry, then demonstrate separately that discretization errors are acceptable.

Silicon’s tetrahedral bonding makes it useful for learning the distinction between a physical approximation and a numerical approximation. LDA and the pseudopotential specify the model; plane-wave cutoff, k-mesh and SCF controls specify how well the model is solved. A small residual cannot repair a too-small basis. A converged energy also does not make an LDA band gap equal to an experimental excitation gap.

1.3.2 Complete PW input

INPUT 

INPUT_PARAMETERS
suffix si_pw
calculation scf
basis_type pw
ntype 1
pseudo_dir ../data/pseudo
ecutwfc 60
scf_thr 1e-9
scf_nmax 150
nspin 1
smearing_method fixed
mixing_type broyden
mixing_beta 0.4
ks_solver cg
out_chg 1 10

STRU 

ATOMIC_SPECIES
Si 28.085 Si.pz-vbc.UPF

LATTICE_CONSTANT
10.2

LATTICE_VECTORS
0.0 0.5 0.5
0.5 0.0 0.5
0.5 0.5 0.0

ATOMIC_POSITIONS
Direct
Si
0.0
2
0.00 0.00 0.00 0 0 0
0.25 0.25 0.25 0 0 0

KPT 

K_POINTS
0
Gamma
4 4 4 0 0 0

The names Si.pz-vbc.UPF and Si_lda_8.0au_50Ry_2s2p1d are taken from the official silicon example: an LDA pseudopotential and an LDA numerical orbital set. PW requires only the UPF; LCAO requires both. These are external assets, not bundled downloads. Obtain the source-matched files needed for your basis, inspect their headers, save their hashes, and place them in ../data/pseudo/ and, for LCAO, ../data/orbitals/. A file with the same name is not proof of the same content. The input intentionally omits dft_functional, allowing the functional recorded in the UPF to be used. Replacing this model with PBE requires a separately documented, mutually consistent PBE pseudopotential/orbital pair for LCAO; setting dft_functional PBE alone does not recreate that pair.

1.3.3 Read the numerical contract

The plane-wave expansion retains reciprocal vectors satisfying

\[\frac{\hbar^2|\mathbf{k}+\mathbf{G}|^2}{2m_e}\le E_{\mathrm{cut}}.\]

ecutwfc 60 means 60 Ry, not 60 eV. At fixed cell volume the number of plane waves grows approximately as \(V E_{\mathrm{cut}}^{3/2}\); this explains why careless cutoff increases become expensive. The input chooses ks_solver cg, a PW solver supported in the baseline. Do not replace it with an LCAO generalized eigensolver merely because another tutorial uses that solver.

smearing_method fixed is appropriate here only if the actual occupied manifold is separated from unoccupied bands. Check the computed occupations and spectrum rather than relying on the chemical name. nspin 1 imposes a non-spin-polarized solution. For a four-valence-electron silicon UPF, two atoms require eight electrons and four doubly occupied bands; verify the UPF and logged electron count instead of treating that number as universally true for silicon datasets.

The initial 4-by-4-by-4 mesh is a test value. Primitive-cell reciprocal vectors differ from conventional-cell reciprocal vectors, so the same integers do not imply the same sampling resolution in the two cells. The fixed coordinate flags deliberately prevent accidental relaxation; forces can be evaluated later without changing this reference geometry.

1.3.4 Workflow: parse, converge, then collect

Prepare a new case directory and retain the three original inputs. On an already configured local Linux installation, a possible launch is shown below. Adapt the executable and MPI launcher to the installed build; this command is an instruction for the learner, not something run in preparing this course. Never submit an unfamiliar example directly to a production queue.

OMP_NUM_THREADS=1 mpirun -np 2 abacus > screen.log 2>&1

Read screen.log and OUT.<suffix>/running_scf.log. Confirm the actual version, basis, cell, species, pseudopotential, occupation method and electronic iteration status before collecting a final energy. In this baseline the final-energy marker is !FINAL_ETOT_IS, reported in eV. Its presence alone is insufficient: an unsuccessful SCF can still leave an energy-like value. Retain the complete electronic trace and any warnings. out_chg 1 10 requests a real-space density cube at ten-digit output precision; it does not tighten the electronic convergence. For the non-spin-polarized examples, inspect SPIN1_CHG.cube only after convergence has been established.

Use one directory per numerical choice: for example si_pw_e60_k4 and si_pw_e80_k4. Before each comparison, check that STRU and the UPF hash are identical. Tighten only one parameter family at a time. If you choose cal_force 1 as a separate diagnostic, keep positions fixed and record the force units from the versioned reference. An ideal symmetric structure may have tiny forces by symmetry even when energy and stress are not converged, so zero force alone is a weak cutoff test.

After a successful run, record the final energy, number of electrons, k-point reduction, SCF iteration count and elapsed resources. The density cube is useful for visualizing periodic covalent charge but is not a bond-order measurement. Check its cell and integrated electron count before choosing an isovalue; different isovalues can make the same density appear qualitatively different.

1.3.5 Separate convergence requirements

Test Vary Hold fixed Proposed acceptance
Electronic solve tighter scf_thr cell, UPF, cutoff, mesh target energy changes negligibly
PW basis 40, 60, 80, 100 Ry candidates converged mesh and electronic controls below chosen property tolerance
Brillouin integration 4, 6, 8, 10 mesh candidates converged cutoff below chosen property tolerance
Cross-check rerun adopted point all physical/numerical inputs reproduce within stated tolerance

For a teaching energy target one might choose 1 meV per atom, explicitly as a proposed tolerance, not a achieved accuracy. A stress-sensitive study needs a stress criterion as well. Compare the last two or three scan points; do not accept a single coincidental small difference. If the energy scan is irregular, check that every SCF actually converged and that the number of occupied states is stable before extending the scan.

1.3.6 Interpretation, traps and exercises

Absolute pseudopotential energies are reference-dependent. Compare values only within the same model and use balanced differences for later material properties. Changing the pseudopotential while increasing the cutoff mixes physical and numerical changes. Increasing output precision does not improve convergence. Copying a number from an official example is not a result for this template: its cell, data files and settings may differ.

  1. Scaling exercise. If the cutoff increases from 40 to 80 Ry in a fixed volume, estimate the PW count ratio. Answer guidance: \((80/40)^{3/2}=2^{3/2}\), approximately 2.83. Runtime need not scale by the same ratio because diagonalization, FFTs and parallel communication contribute differently.
  2. Error-budget exercise. Suppose a real future run changes by 0.8 meV/atom on increasing the cutoff and 1.5 meV/atom on refining the mesh. Can it meet a 1 meV/atom numerical budget? Answer guidance: not on this evidence; the mesh effect alone exceeds the budget. These supplied numbers are hypothetical arithmetic data, not silicon results. Improve the mesh and repeat a coupled high-cutoff check.

Use the cutoff lesson to design the full basis scan and the LCAO lesson to compare representations without changing the Hamiltonian.

1.3.7 Official references and provenance

These references establish file grammar and available options. The explanations, comparison designs, algebraic exercises and figures are original teaching material; no published example energy is presented as a result of this course. Additional learning resources are collected in the course hub references section. The university-linked 2024 guide is supplementary teaching, not a substitute for the pinned input reference.