Why solid-state calculations use `pob-TZVP`¶

Standard molecular basis sets, the Pople and Dunning families, were designed for isolated molecules. In a crystal, their small-exponent diffuse primitives overlap with periodic images, producing a near-singular overlap matrix and a quietly unreliable SCF. The pob family (Peintinger-Vilela-Oliveira-Bredow 2013) fixes this by re-deriving triple- and double-zeta bases with solid-state overlap in mind.

The one-liner¶

vibe-qc ships three pob bases in basis_library/custom/ and exposes them by name:

from vibeqc import BasisSet
basis = BasisSet(sysp.unit_cell_molecule(), "pob-tzvp")

Every solid-state tutorial in this series (periodic HF, periodic KS-DFT, band structure) uses pob-tzvp by default. If you copy a tutorial input and substitute a molecular basis, expect to see a conditioning warning from the SCF driver, or a linear-dependency abort.

Why a molecular basis goes wrong in a crystal¶

A typical molecular triple-zeta basis, say def2-tzvp on oxygen, includes a smallest-exponent \(s\) primitive around \(\alpha \approx 0.1\) bohr⁻². For an isolated molecule, that \(\alpha\) gives a Gaussian centered on O with a radius \(1/\sqrt{\alpha} \approx 3\) bohr

fine, doesn’t overlap meaningfully with the next O atom typically \(\sim 5\) bohr away.

In a crystal the same primitive does overlap with its counterpart on the image atom. For NaCl with Na-Cl distance \(\sim 5.3\) bohr, the overlap between two \(\alpha = 0.1\) primitives separated by 5.3 bohr is

\[ \langle \chi_\alpha(\mathbf{r}) | \chi_\alpha(\mathbf{r} - \mathbf{T}) \rangle = \bigl( \tfrac{2\alpha}{\pi} \bigr)^{3/2} \cdot \bigl( \tfrac{\pi}{2\alpha} \bigr)^{3/2} \, e^{-\alpha |\mathbf{T}|^2 / 2} = e^{-\alpha |\mathbf{T}|^2 / 2} \approx e^{-1.4} \approx 0.24, \]

a quarter of the self-overlap. Summed over the infinite lattice, the total inter-cell contribution to every basis function’s norm is substantial, and every near-duplicate small-exponent primitive gets a near-zero eigenvalue in \(\mathbf{S}\). The SCF canonical-orthogonalisation step projects those out, but you’re left with effectively fewer independent basis functions than you asked for, and results that depend on the linear-dependency threshold rather than the chemistry.

What pob does¶

Peintinger-Vilela-Oliveira-Bredow (2013, Peintinger, Michael F., Vilela Oliveira, Daniel, and Bredow, Thomas (2013). *Consistent Gaussian basis sets of triple-zeta valence with polarization quality for solid-state calculations.* Journal of Computational Chemistry, **34**(6), 451–459. doi:[10.1002/jcc.23153](https://doi.org/10.1002/jcc.23153)) took the def2-TZVP / def2-DZVP molecular bases and, for every element from H through Br, systematically trimmed or replaced the primitives with \(\alpha < \alpha_\text{min}^\text{solid-state}\), where \(\alpha_\text{min}^\text{solid-state}\) is chosen so that the inter-cell overlap integral for typical crystalline densities stays below a tolerance. The pruning is not only about numerical conditioning. In a molecule the diffuse tail of a basis function describes the decay of the density into vacuum, a region that carries little energy but must be present for the wavefunction to be normalisable. In a dense crystal there is no vacuum: every point in space is already inside the tails of several atoms, so the same diffuse primitive describes nothing new, its Bloch sum \(\sum_{\mathbf T} e^{i\mathbf k\cdot\mathbf T}\chi_\alpha(\mathbf r - \mathbf T)\) becomes nearly \(\mathbf k\)-independent and nearly proportional to the Bloch sums of its neighbours, and the overlap matrix \(\mathbf S(\mathbf k)\) acquires eigenvalues that approach zero. The variational freedom the function was supposed to add is instead spent describing linear dependence. Removing it costs a small, element-specific amount of atomic energy (which is why pob bases are re-optimised rather than merely truncated) and buys a basis whose \(\mathbf S(\mathbf k)\) stays well conditioned at every crystalline density the family was fitted on. The result is:

pob-TZVP, triple-zeta valence + polarization, the most commonly used solid-state basis.
pob-TZVP-rev2, a revised version with tighter polarization functions for specific element subsets.
pob-DZVP-rev2, double-zeta equivalent, cheaper, for screening work.

The bundled pob-tzvp covers H through Br (rare gases excepted) from the 2013 set plus Rb through I from the fifth-period extension (Laun, Joachim, Vilela Oliveira, Daniel, and Bredow, Thomas (2018). *Consistent gaussian basis sets of double- and triple-zeta valence with polarization quality of the fifth period for solid-state calculations.* Journal of Computational Chemistry, **39**(19), 1285–1290. doi:[10.1002/jcc.25195](https://doi.org/10.1002/jcc.25195)), 48 elements at the all-electron level; the rev2 revisions (Vilela Oliveira, Daniel, Laun, Joachim, et al. (2019). *BSSE-correction scheme for consistent Gaussian basis sets of double- and triple-zeta valence with polarization quality for solid-state calculations.* Journal of Computational Chemistry, **40**(27), 2364–2376. doi:[10.1002/jcc.26013](https://doi.org/10.1002/jcc.26013)) re-optimise the family. For elements 4d/5d/5f+ the bundled pob-TZVP-REV2 runtime basis carries ECP-based definitions (the Z+200 convention) that vibe-qc resolves and applies via libecpint, validated against PySCF.pbc for PBE/pob-TZVP-REV2 + ECP.

What you see in practice¶

Running a tutorial’s H₂ molecular crystal (input-h-chain-uniform.py, a 15 bohr lattice parameter in a dim=3 cell with 15 bohr of transverse vacuum, k-point GDF on an 8 x 1 x 1 mesh) with three bases and tracking SCF convergence (vibe-qc 0.17.14):

Basis	nbf / atom	n_iter	Conv.
STO-3G	1	3	✓ (too small to matter)
6-31G*	2	6	✓
pob-TZVP	6	7	✓

All three converge here because each H₂ sits 13.6 bohr from its neighbours along the chain: the crystal is close to the molecular limit, where image overlap is negligible (see the next section). The case for pob-TZVP is the compact cell, where diffuse molecular functions overlap their periodic images and the overlap matrix approaches linear dependence; pob-TZVP is built to stay clear of that.

When you can skip pob¶

Large unit cells where the molecular-limit approximation is exact (30+ bohr separation between relevant atoms): any molecular basis works because image overlap is negligible.
Molecular clusters with PBC only for bookkeeping (a single water in a 50 bohr cubic cell): use def2-tzvp or cc-pvtz happily.

The threshold is roughly: if the smallest-atom separation is \(\geq 6\) bohr and nothing special is going on, a standard molecular basis is fine. Below that, switch to pob.

Parsing your own solid-state basis¶

vibe-qc’s basis_crystal module parses CRYSTAL-format per-element files and emits libint-compatible .g94 text:

from vibeqc.basis_crystal import parse_crystal_atom_basis_file, emit_g94

atom = parse_crystal_atom_basis_file("my-basis/06_C")
with open("my-basis.g94", "w") as f:
    f.write(emit_g94([atom]))

Drop the resulting my-basis.g94 into basis_library/custom/ and run ./scripts/setup_basis_library.sh to pick it up. See user_guide/basis_sets.md for the full CRYSTAL-format reference.

Resources¶

Depends sharply on system. The simple-Mg example in this tutorial is ~150 MB peak RAM, ~3 s on one core (Apple M2 baseline). Real metal-oxide / metal-hydride targets at pob-TZVP (8-20 atoms / cell, [4,4,4] mesh) typically run ~1-10 minutes per SCF on 4-8 cores; defect supercells push into the hour-per-SCF regime where C1a level shifting pays for itself.

References¶

pob-TZVP / pob-DZVP, original paper. M. F. Peintinger, D. V. Oliveira, and T. Bredow, “Consistent Gaussian basis sets of triple-zeta valence with polarization quality for solid-state calculations,” J. Comput. Chem. 34, 451 (2013).
pob-TZVP-rev2 revision. D. V. Oliveira, J. Laun, M. F. Peintinger, and T. Bredow, “BSSE-correction scheme for consistent Gaussian basis sets of double- and triple-zeta valence with polarization quality for solid-state calculations,” J. Comput. Chem. 40, 2364 (2019).
Linear-dependency canonical orthogonalisation. P.-O. Löwdin, “On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals,” J. Chem. Phys. 18, 365 (1950).
CRYSTAL code, for which pob was designed. A. Erba et al., “CRYSTAL23: a program for computational solid state physics and chemistry,” J. Chem. Theory Comput. 19, 6891 (2023).

Next¶

User guide, linear dependence + screening in periodic SCF, the chapter the pob-* family was designed against. Walks through the v0.7 diagnostic + fix stack (vq.eigs_preflight, vq.disambiguate_critical_overlap, auto_optimize_truncation, vq.make_basis(..., exp_to_discard=...)).
Basis-set convergence for the molecular side of the basis-set question, Dunning’s cc-pVXZ hierarchy and the CBS extrapolation.
The Peierls dimerisation tutorial uses pob-tzvp on a 1D H-chain where the diffuse-basis problem is especially acute.

Why solid-state calculations use pob-TZVP¶