2.2 Converge numerical orbital quality

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.

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2.2.1 The question: which basis changes improve the observable?

An LCAO energy may have a small SCF residual while the finite orbital space is still inadequate. This case asks whether adding radial flexibility or spatial range changes the target property relative to a converged PW reference under the same Hamiltonian. Start from the complete LCAO silicon parent. Audit its source-matched UPF/orbital pairing first; a basis scan is meaningful only if it is not also a pseudopotential or functional scan.

There are at least three independent numerical controls: orbital multiplicity/angular channels, orbital radial cutoff, and the integration cutoff ecutwfc. A fourth is k-point sampling. The beginner’s mistake is to treat all of them as a single “basis size” dial. Establish an electronic threshold and integration mesh tight enough to resolve the orbital differences before ranking candidate orbital sets.

2.2.2 Scoped orbital and grid deltas

Retain all parent files. For the source-set integration check, change only the INPUT cutoff and suffix:

suffix si_lcao_grid80
ecutwfc 80

For the orbital check, keep the accepted integration cutoff and change only the orbital line under NUMERICAL_ORBITAL, plus a distinguishing suffix. The following is a symbolic placeholder, not the name of an available file:

NUMERICAL_ORBITAL
VERIFIED_Si_LDA_MATCHED_ORBITAL_FILE

Replace the placeholder with a real documented orbital file generated consistently with the same UPF and functional. This course does not invent a downloadable TZDP file or claim an arbitrary family exists. Copy the actual header into the provenance ledger. Compare radial ranges at fixed multiplicity where such a documented family is available, and multiplicities at fixed range where available. If candidates change both, report the coupled change rather than attributing it to one variable.

2.2.3 Radial flexibility, locality and transferability

For each angular momentum \(l\), multiple radial functions allow a local environment to be described more flexibly. A polarization channel provides angular shapes absent from the minimal occupied atomic configuration. Neither guarantees a predictable numerical correction in every environment. Enlarging the radial cutoff adds overlap with more neighbors and usually increases matrix connectivity, which affects storage and matrix construction as well as diagonalization.

Count the basis from the actual header using

\[N_{\mathrm{orb}}=N_{\mathrm{atoms}}\sum_l n_l(2l+1).\]

For the verified two-atom 2s2p1d source set this is 26 unpolarized functions. A candidate with more functions may lower the variational energy if the spaces are nested and all other approximations are controlled, but independently generated sets need not be strictly nested. Do not enforce monotonicity by hand or interpret a small absolute offset as a guarantee of chemical transferability.

Test a second geometry relevant to the final application: mild hydrostatic strain for an equation of state, a displaced atom for force validation, or a low-coordination configuration for surfaces. An orbital fit may describe bulk bonding well while performing less well in another environment. A “best” basis is therefore defined against a validation domain and accuracy/cost requirement, not against a single lowest energy.

2.2.4 Measured table and error analysis

Set	Header/hash	Radial range, bohr	Functions/atom	Integration cutoff, Ry	E/atom error vs PW	Force error	Cost
Source set	fill from real file	verify	verify	scan first	—	—	—
Candidate A	real documented file	verify	verify	fixed accepted	—	—	—
Candidate B	real documented file	verify	verify	fixed accepted	—	—	—

Compute errors only from successful runs. For force validation use matched geometries and components, for example the maximum component difference, and clearly state eV/angstrom. For energy-difference validation define two geometries A and B and calculate

\[\delta\Delta E=(E_B-E_A)_{\mathrm{LCAO}}-(E_B-E_A)_{\mathrm{PW}}.\]

This measures an environment-dependent error more directly than comparing a single absolute energy. Ensure the PW reference is itself tighter than the intended orbital tolerance. A poor reference can falsely make every orbital family appear to agree. Keep both signed and absolute errors: signs can reveal systematic behavior, while absolute magnitudes support the acceptance threshold.

2.2.5 Outputs and acceptance

Inspect the orbital readout and the matrix dimension in running_scf.log as well as the final energy. Retain overlap or diagonalization warnings. If a larger basis creates numerical instability, first check whether it is the intended file and whether the overlap problem is ill-conditioned; increasing SCF iterations alone may not resolve it.

Accept the least expensive candidate that passes all predeclared property tolerances across the validation geometries, after grid and k-mesh checks. Do not label the result “complete-basis” without an established extrapolation and defensible nesting assumptions. In a finite local basis, surface adsorption can also involve environment-dependent basis overlap effects; consistent fragments and basis-superposition diagnostics deserve separate attention rather than assuming a bulk energy benchmark settles them.

Report the validated domain in plain language: for example, fixed-cell nonmagnetic silicon and modest strains with this UPF. Do not extrapolate that claim automatically to oxides, transition-metal magnetism or highly compressed structures.

2.2.6 Pitfalls and exercises

Avoid mixing orbitals generated with different UPFs merely because filenames share a functional label. Do not rank sets at different integration cutoffs. Remember that more diffuse orbitals can alter cost through neighbor connectivity even if the number of functions is unchanged. A density grid that looks smooth is not a convergence test.

Two-axis exercise. A candidate has the same 2s2p1d multiplicity but a longer verified radial cutoff. What changes and what does not? Answer guidance: the function count stays the same, but radial functions, overlap range and matrix connectivity can change; do not describe this as adding zeta functions.
Cancellation exercise. Hypothetical absolute LCAO errors are +0.20 eV for A and +0.21 eV for B. What is the error in B-minus-A? Answer guidance: +0.01 eV, provided both refer to the same Hamiltonian, atom count and successful PW reference. These are invented arithmetic inputs, not simulated results. Whether 0.01 eV is acceptable depends on the target property and size normalization.

Proceed to metal integration, where occupied states near the Fermi level add another coupled numerical control.

2.2.7 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.