Rydberg Atomic Units in Plane-Wave DFT

All equations in KRONOS are written in Rydberg atomic units — the same convention used by Quantum ESPRESSO and most plane-wave codes in the pseudopotential tradition. This page defines both flavors of atomic units, explains where the factor-of-2 difference comes from, lists every conversion you will need, and shows the exact operator forms that appear in the code. If you have ever debugged a kinetic energy that was exactly half what it should be, this page is for you.

Atomic units: the natural scale

Ordinary SI units are inconvenient for atomic-scale calculations. Energies of interest are ~10⁻¹⁸ J; electron–electron distances are ~10⁻¹⁰ m. Working in SI forces every equation to carry large powers of 10, and it hides the physics behind conversion constants.

Atomic units remove this clutter by choosing the four fundamental constants of one-electron quantum mechanics as the unit basis:

Constant	Symbol	Role
Electron mass	$m_e$	mass unit
Elementary charge	$e$	charge unit
Reduced Planck constant	$\hbar$	action unit
Bohr radius	$a_0 = \hbar^2/(m_e e^2)$	length unit

In any atomic unit system these constants are each set to a small integer (1 or 2). The choice of which integer determines which flavor of atomic units you get.

Hartree vs Rydberg: the factor of two

Two conventions dominate the DFT literature. They share the same length unit (the bohr) but differ in the energy scale.

Hartree atomic units (used by VASP, CP2K, Gaussian) set:

$\hbar = 1,\quad m_e = 1,\quad e^2 = 1,\quad a_0 = 1$

With these definitions the Schrödinger kinetic operator for a free particle becomes:

$\hat{T}_\mathrm{Ha} = -\nabla^2 / 2$

and the ground-state energy of hydrogen is $-1/2\ \mathrm{Ha}$.

Rydberg atomic units (used by Quantum ESPRESSO, KRONOS, many pseudopotential codes) set:

$\hbar = 1,\quad m_e = \tfrac{1}{2},\quad e^2 = 2,\quad a_0 = 1$

Halving $m_e$ and doubling $e^2$ leaves the Bohr radius $a_0 = \hbar^2/(m_e e^2)$ unchanged — so lengths are identical in both systems. But the energy unit doubles:

$E_\mathrm{Ry} = \frac{m_e e^4}{2\hbar^2}\Bigg|_\mathrm{Ry\ defs} = \frac{(\tfrac{1}{2})(2)^2}{2 \cdot 1^2} = 1\ \mathrm{Ry}$

which equals $\tfrac{1}{2}$ Hartree. The kinetic operator becomes:

$\hat{T}_\mathrm{Ry} = -\nabla^2$

(the $m_e = 1/2$ absorbs the factor of 2 that would otherwise sit in the denominator).

Summary of the factor-of-two origin: Rydberg units halve the electron mass. This removes the $1/2$ from the kinetic operator and doubles the Hartree potential prefactor because $e^2 = 2$ in Rydberg units. Every difference between Hartree and Rydberg operator forms traces back to these two substitutions.

Conversion table

Quantity	Rydberg AU	Hartree AU	SI / conventional
Energy	$1\ \mathrm{Ry}$	$\tfrac{1}{2}\ \mathrm{Ha}$	$13.6057\ \mathrm{eV}$
Energy	$2\ \mathrm{Ry}$	$1\ \mathrm{Ha}$	$27.2114\ \mathrm{eV}$
Length	$1\ \mathrm{bohr}$	$1\ \mathrm{bohr}$	$0.529177\ \mathrm{Å}$
Force	$1\ \mathrm{Ry/bohr}$	$\tfrac{1}{2}\ \mathrm{Ha/bohr}$	$25.7110\ \mathrm{nN}$
Pressure	$1\ \mathrm{Ry/bohr^3}$	$\tfrac{1}{2}\ \mathrm{Ha/bohr^3}$	$14710.507\ \mathrm{GPa}$
Velocity (time unit: $\hbar/\mathrm{Ry}$)	$1\ \mathrm{bohr \cdot Ry}/\hbar$	—	$1.09 \times 10^6\ \mathrm{m/s}$
$m_e$	$1/2$	$1$	$9.10938 \times 10^{-31}\ \mathrm{kg}$
$e^2$	$2$	$1$	$1.43996\ \mathrm{eV \cdot nm}$

The energy conversion $1\ \mathrm{Ha} = 2\ \mathrm{Ry}$ is the single most important number to memorize. Force and pressure conversions follow immediately because the length unit is shared.

Why KRONOS uses Rydberg

KRONOS targets compatibility with Quantum ESPRESSO (QE) benchmarks as its primary validation reference. QE, PWscf, and most pseudopotential codes in the norm-conserving and PAW traditions were written in Rydberg units because the earliest plane-wave codes (notably those of Arias, Payne, and Gonze) followed the convention of Hamann, Schlüter, and Chiang (1979) for norm-conserving pseudopotentials, which uses Rydberg units.

Concretely, UPF pseudopotential files (the universal format used by QE and KRONOS) store local potentials in Rydberg units. Reading a UPF file and immediately comparing energies against a Hartree-unit code requires multiplying by 2. KRONOS avoids this entire class of off-by-two bugs by staying in Rydberg throughout:

• UPF files read in → no conversion needed
• QE reference energies → direct comparison, no factor of 2
• All KRONOS test baselines are in Ry, matching QE output directly

If you are porting formulas from a Hartree-unit paper or code, multiply every energy by $\tfrac{1}{2}$ and every potential (energy/charge) by $\tfrac{1}{2}$ before using it in KRONOS.

Operator forms in Rydberg units

The following table shows the three operators where Hartree and Rydberg units differ most consequentially. These are the forms implemented in src/hamiltonian/ and src/potential/.

Operator	Hartree atomic units	Rydberg atomic units
Kinetic (G-space)	$\hat{T} \psi_\mathbf{G} = \tfrac{1}{2}\|\mathbf{k}+\mathbf{G}\|^2 \psi_\mathbf{G}$	$\hat{T} \psi_\mathbf{G} = \|\mathbf{k}+\mathbf{G}\|^2 \psi_\mathbf{G}$
Hartree potential (G-space)	$V_H(\mathbf{G}) = \dfrac{4\pi\, n(\mathbf{G})}{\|\mathbf{G}\|^2}$	$V_H(\mathbf{G}) = \dfrac{8\pi\, n(\mathbf{G})}{\|\mathbf{G}\|^2}$
Bare Coulomb	$V(\mathbf{r}) \to -Z/r$	$V(\mathbf{r}) \to -2Z/r$

Kinetic energy

In reciprocal space (plane-wave basis) the kinetic operator is diagonal. For a wavefunction expanded as $\psi_{n\mathbf{k}}(\mathbf{r}) = \sum_\mathbf{G} c_{n\mathbf{k}\mathbf{G}} e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}$:

$\hat{T}\psi_{n\mathbf{k}} = \sum_\mathbf{G} |\mathbf{k}+\mathbf{G}|^2\, c_{n\mathbf{k}\mathbf{G}}\, e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}$

There is no $1/2$ prefactor. The plane-wave cutoff is therefore ecutwfc in Ry, defined as $|\mathbf{k}+\mathbf{G}|^2 \leq E_\mathrm{cut}$ — not $|\mathbf{k}+\mathbf{G}|^2/2 \leq E_\mathrm{cut}$.

Hartree potential

The Poisson equation in G-space, with $e^2 = 2$ in Rydberg units:

$V_H(\mathbf{G}) = \frac{8\pi\, n(\mathbf{G})}{|\mathbf{G}|^2}$

The $\mathbf{G} = \mathbf{0}$ component (average electrostatic) is set to zero for charge-neutral periodic systems (the uniform background cancels it).

Local pseudopotential Fourier transform

The local part of the pseudopotential has a Coulomb tail $V_\mathrm{loc}(r) \to -2Z/r$ as $r \to \infty$ (Rydberg convention). In G-space this tail contributes an analytic term:

$V_\mathrm{loc}(\mathbf{G}) = V_\mathrm{loc}^\mathrm{short}(\mathbf{G}) + \frac{8\pi Z}{\Omega |\mathbf{G}|^2}$

where $\Omega$ is the unit cell volume. The short-range part is the smooth remainder after subtracting the Coulomb tail in real space.

Stress and pressure

The stress tensor $\sigma_{\alpha\beta}$ in KRONOS is computed in units of $\mathrm{Ry/bohr^3}$. To convert to GPa:

$P\ [\mathrm{GPa}] = \sigma\ [\mathrm{Ry/bohr^3}] \times 14710.507\ \mathrm{GPa\cdot bohr^3/Ry}$

The conversion factor follows from:

$1\ \mathrm{Ry} = 2.17987 \times 10^{-18}\ \mathrm{J}, \quad 1\ \mathrm{bohr} = 5.29177 \times 10^{-11}\ \mathrm{m}$

$1\ \mathrm{Ry/bohr^3} = \frac{2.17987 \times 10^{-18}}{(5.29177 \times 10^{-11})^3}\ \mathrm{Pa} \approx 1.4711 \times 10^{13}\ \mathrm{Pa} = 14710.5\ \mathrm{GPa}$

Pressure is reported as the negative one-third trace of the stress:

$P = -\frac{1}{3}\,\mathrm{tr}(\sigma)$

(positive pressure = compression, following the geophysics/thermodynamics sign convention used by QE and most DFT codes).

Common pitfalls

Reading Hartree-unit output and forgetting the factor of 2. If a VASP or CP2K calculation reports a total energy of $-4.12\ \mathrm{Ha}$, the equivalent KRONOS value is $-8.24\ \mathrm{Ry}$. The numbers look wildly different even though they represent the same physical energy. Always check the unit label in output files.

ecutwfc in Hartree vs Rydberg. The cutoff $E_\mathrm{cut} = 40\ \mathrm{Ry}$ is equivalent to $20\ \mathrm{Ha}$. VASP ENCUT is in eV ($40\ \mathrm{Ry} \approx 544\ \mathrm{eV}$). Getting this wrong produces a basis set that is either half the size or twice the size intended, with order-of-magnitude errors in convergence.

Applying the 1/2 kinetic prefactor from a Hartree paper. If you copy the kinetic energy formula from a textbook written in Hartree units and add a $1/2$, your kinetic energy will be half the correct value. KRONOS has caught this class of bug repeatedly — see the Rydberg call-out at the top of docs/physics_notes.md.

ecutrho must be at least 4× ecutwfc (norm-conserving) or 12× (PAW). These factors are the same in both unit systems because they relate two Rydberg cutoffs. No unit conversion is needed here, but they are easy to violate when translating input files from Hartree-unit codes that express the charge density cutoff differently.

Force sum rule in mixed-unit codes. If forces from one layer of a code are accumulated in Ry/bohr but a second layer computes them in Ha/bohr, the force sum (Newton's third law self-consistency) will appear violated by a factor of 2. The KRONOS force validation tests (test_forces) catch this by comparing to finite differences.

References

• Martin, R. M. Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, 2004 — Appendix on units
• Quantum ESPRESSO documentation — Units conventions
• NIST CODATA recommended values (2018): physics.nist.gov/cuu/Constants
• Hamann, D. R., Schlüter, M. & Chiang, C. (1979). Norm-Conserving Pseudopotentials, Phys. Rev. Lett. 43, 1494 — original NCP paper establishing the Rydberg convention