Pseudopotentials Explained: Norm-Conserving, Ul…

Pseudopotentials are the quiet workhorses of plane-wave DFT. They are the reason you can run a calculation on a transition metal without describing every core electron explicitly, and the reason your plane-wave cutoff is 40 Rydberg instead of 40,000. Yet many practitioners treat them as a black box and pick whichever file happens to be lying in a directory. That is a mistake: the pseudopotential choice sets both the accuracy floor and the cost of nearly every calculation you will run.

This post explains why pseudopotentials exist, how the three dominant families — norm-conserving, ultrasoft, and PAW — differ, what those differences mean for your plane-wave cutoff, and how to validate a pseudopotential before trusting it.

The Core Electron Problem

Consider a single tungsten atom: 74 electrons, most of them tightly bound core states with wavefunctions that oscillate rapidly near the nucleus. Two facts make these core electrons expensive and pointless to compute repeatedly.

First, core electrons barely participate in bonding. The chemistry — bond formation, band structure, magnetism — is carried almost entirely by the valence electrons. Second, representing the sharp core oscillations in a plane-wave basis requires enormous cutoffs, because a plane-wave basis resolves features down to a wavelength set by the highest-energy wave.

$$ \lambda_{\min} \sim \frac{2\pi}{\sqrt{2\, E_{\mathrm{cut}}}} \quad (\text{atomic units}) $$

Sharp core wiggles demand huge $E_{\mathrm{cut}}$ and an enormous basis.

The Frozen-Core Approximation

The way out is the frozen-core approximation: assume the core electrons are chemically inert and do not respond to the environment. Freeze them in the configuration they had in the isolated atom, and never solve for them again. This is an excellent approximation for the deep core states of most elements, and it is the conceptual foundation of every pseudopotential.

Having frozen the core, we go one step further. We replace the true, strongly attractive nuclear-plus-core potential with a smooth, weaker effective potential — the pseudopotential — engineered so that the valence electrons behave correctly outside a chosen core radius $r_c$, while inside $r_c$ the sharp oscillations are smoothed away.

flowchart TD
  A[All-electron atom] --> B[Freeze core electrons]
  B --> C[Replace nuclear-plus-core potential]
  C --> D[Smooth pseudopotential]
  D --> E["r > r_c: psi_pseudo = psi_AE"]
  D --> F["r < r_c: smooth nodeless psi_pseudo"]
  E --> G[Much lower plane-wave cutoff]
  F --> G

$$ \begin{aligned} r > r_c &\colon\ \psi_{\mathrm{pseudo}}(\mathbf{r}) = \psi_{\mathrm{AE}}(\mathbf{r}) \\ r < r_c &\colon\ \psi_{\mathrm{pseudo}}\ \text{is smooth and nodeless} \end{aligned} $$

Because the pseudo valence wavefunctions are smooth and nodeless in the core region, they need far fewer plane waves. That single trade — physical fidelity inside a small sphere in exchange for a dramatically smaller basis — is what makes plane-wave DFT tractable.

Norm-Conserving Pseudopotentials

The oldest robust family imposes an extra condition: the integrated charge of the pseudo-wavefunction inside $r_c$ must equal that of the true wavefunction.

$$ \int_0^{r_c} \lvert\psi_{\mathrm{pseudo}}(r)\rvert^2\, dr = \int_0^{r_c} \lvert\psi_{\mathrm{AE}}(r)\rvert^2\, dr $$

This norm-conservation guarantees correct electrostatics and, crucially, correct scattering properties over a range of energies, which is what makes the pseudopotential transferable between chemical environments.

The cost is hardness. To conserve the norm inside a small radius, the pseudo-wavefunction cannot be too smooth, so norm-conserving pseudopotentials — especially for first-row elements (O, N, F) and 3d transition metals — demand relatively high plane-wave cutoffs, often 60-120 Ry. Their virtue is simplicity and reliability: the charge density is just the square of the wavefunctions, with no extra terms, which makes forces and stresses straightforward. Modern ONCV (optimized norm-conserving Vanderbilt) pseudopotentials have narrowed the cost gap considerably and are an excellent default.

Ultrasoft Pseudopotentials

Vanderbilt’s ultrasoft (USPP) scheme relaxes norm-conservation deliberately to make the pseudo-wavefunctions as smooth as possible, slashing the required cutoff. The missing charge is added back through augmentation charges localized in the core region.

The consequence is a generalized eigenvalue problem with an overlap operator $S$, and a charge density that has two parts:

$$ n(\mathbf{r}) = \sum_i \lvert\psi_i(\mathbf{r})\rvert^2 + Q(\mathbf{r}) $$$$ H\psi = \varepsilon\, S\psi \quad (S \neq \mathbb{1}) $$

Ultrasoft pseudopotentials can run at cutoffs as low as 25-40 Ry even for oxygen and 3d metals. The catch: the augmentation charge is sharp, so it needs its own denser grid, and the charge-density cutoff ecutrho must be set much higher than the wavefunction cutoff ecutwfc — typically 8-12x rather than the default 4x. Get that ratio wrong and forces become noisy.

The Projector Augmented-Wave Method

Blochl’s PAW method is best understood not as a pseudopotential but as an exact linear transformation between smooth pseudo-wavefunctions and the true all-electron wavefunctions. It keeps the smoothness advantage of ultrasoft (low cutoffs) while retaining access to the full all-electron density near the nucleus.

$$ \lvert\psi_{\mathrm{AE}}\rangle = \lvert\psi_{\mathrm{pseudo}}\rangle + \sum_i \bigl(\lvert\phi_i\rangle - \lvert\tilde{\phi}_i\rangle\bigr)\langle p_i\lvert\psi_{\mathrm{pseudo}}\rangle $$

Because PAW reconstructs the true wavefunction, it is generally the most accurate of the three for a given cutoff, and it is essential when you need properties that depend on the wavefunction near the core: hyperfine parameters, NMR shifts, EFG tensors, core-level spectroscopy. It has effectively become the default in much of the community. The bookkeeping is more complex, but for the user it behaves much like an ultrasoft calculation.

Semicore States and Nonlinear Core Corrections

Two refinements decide whether a pseudopotential is trustworthy for a given element, and both are easy to overlook.

Semicore states. The frozen-core line is not always obvious. In many transition metals and post-transition elements, shallow “semicore” states — the 3s and 3p of early 3d metals, or the d states of Ga, In, and Sn — sit close in energy to the valence and do respond to bonding. Freezing them into the core causes real errors in lattice constants, phonon frequencies, and magnetism. Well-made pseudopotentials therefore include these states explicitly in the valence, at the cost of more electrons and a higher cutoff. When you see a pseudopotential labelled Fe.pbe-sp or a valence configuration listing 3s 3p 3d 4s, that “sp” is telling you semicore states are treated as valence. For high-pressure work or accurate magnetism, prefer these.

Nonlinear core correction (NLCC). The exchange-correlation energy is a nonlinear function of the density, so the usual assumption that valence and core densities can be treated separately breaks down where they overlap — again, most seriously for the compact 3d metals and for spin-polarized systems. The nonlinear core correction stores a smoothed model of the core charge in the pseudopotential and includes it when evaluating $E_{\mathrm{xc}}$, restoring the coupling.

$$ E_{\mathrm{xc}}[n_{\mathrm{valence}}] \;\rightarrow\; E_{\mathrm{xc}}[n_{\mathrm{valence}} + n_{\mathrm{core}}^{\mathrm{model}}] $$

NLCC is essentially free and substantially improves magnetic moments and structural properties for 3d elements. Most modern libraries include it by default; if you are hand-picking older files for a magnetic transition metal, check that it is present.

Comparison of the Three Families

flowchart LR
  A[Choose family] --> B{Need core properties?}
  B -->|Yes NMR EFG| C[PAW]
  B -->|No| D{Lowest cutoff?}
  D -->|Yes| E[Ultrasoft or PAW]
  D -->|Simplicity| F[ONCV norm-conserving]

Property	Norm-conserving	Ultrasoft	PAW
Norm conserved	Yes	No	Effectively (reconstructed)
Typical `ecutwfc`	High (60-120 Ry)	Low (25-40 Ry)	Low (30-50 Ry)
`ecutrho` / `ecutwfc`	~4x	~8-12x	~8x
All-electron density near core	No	Approximate	Yes
Overlap operator $S$	Identity	Non-trivial	Non-trivial
Core-level / NMR properties	Limited	Limited	Excellent
Implementation complexity	Low	High	High
Good default?	ONCV: yes	Legacy	Yes

There is no single winner. PAW and ONCV norm-conserving are the two strong modern defaults; ultrasoft remains common and fast. Match the choice to what you need to compute.

Cutoff Implications in Practice

The two cutoffs in a plane-wave calculation control cost directly:

ecutwfc — the kinetic-energy cutoff for wavefunctions. Set by the hardest element in your cell.
ecutrho — the cutoff for the charge density and potentials. For norm-conserving, ecutrho = 4 * ecutwfc suffices; for ultrasoft and PAW you must raise this ratio to capture the augmentation charge.

Two rules save a great deal of grief. First, in a multi-element cell, the required cutoff is dictated by the hardest pseudopotential present — one oxygen atom forces the whole calculation up. Second, always converge total energy and forces against ecutwfc; forces converge more slowly than energy, and an energy-converged cutoff can still give bad geometries.

Pseudopotential Libraries

You should almost never build your own. Curated libraries exist precisely so you do not have to:

SSSP (Standard Solid-State Pseudopotentials) — a benchmarked, curated set distributed with Quantum ESPRESSO, offering “efficiency” and “precision” variants with recommended cutoffs for every element. This is the sane default for most users.
PSlibrary — the broad official Quantum ESPRESSO generation project, from which many SSSP entries are drawn.
PseudoDojo — a rigorously tested ONCV norm-conserving set with extensive transferability data, including relativistic variants.
GBRV — a high-throughput ultrasoft library tuned for solid-state accuracy.

Never mix pseudopotentials generated with different exchange-correlation functionals in the same calculation. A PBE pseudopotential and an LDA pseudopotential in the same cell is a silent, serious error.

How to Validate a Pseudopotential

Trust, but verify. Three checks catch the large majority of problems:

Cutoff convergence. Plot total energy and the maximum force against ecutwfc (and ecutrho) until both are flat to your tolerance — roughly 1 meV/atom for energies.
The Delta-factor / equation-of-state test. Compute the energy-volume curve and compare the equilibrium volume and bulk modulus against all-electron reference data. The community Delta-factor benchmark quantifies exactly this agreement and is the gold standard for transferability.
Ghost-state check. Inspect the logarithmic derivatives or simply watch for unphysical states in the band structure. A pseudopotential with a ghost state produces spurious low-lying bands that ruin everything downstream.

For a broader discussion of how basis-set and sampling choices interact with accuracy, see our overview of the computational engines behind these calculations.

Run it on Simatra

Choosing a well-converged PAW or ONCV set is only half the battle — the other half is affording the cutoffs it demands, especially for large supercells with hard first-row elements. Simatra runs DFT on GPU-accelerated clusters (instance GPU-Opt-V2) with up to 5x faster convergence, handling supercells up to ~2,000 atoms across two computational engines: the open-source Quantum ESPRESSO and Simatra’s native KRONOS. Run your cutoff-convergence and equation-of-state validation sweeps in parallel instead of overnight, and start with a free trial and $100 in credits at app.simatra.io.