vibe-qc GPW/GAPW vs GPAW Program – Architectural Comparison¶

1. Orbital basis¶

	vibe-qc GPW/GAPW	GPAW Program
Basis type	Gaussian (atom-centered, contracted GTOs via libint)	Real-space grid (uniform 3D finite-difference grid) + PAW projectors
AO evaluation	libint C++17 `evaluate_ao()` with 27-image periodic-image folding per grid point	Wavefunctions represented directly on the real-space grid; no AO basis
Consequence	Sparse analytic integrals (overlap, kinetic, nuclear attraction, ERI) are exact for the Gaussian basis; only the Coulomb J is done on the FFT grid	ALL operators (kinetic, potential, Hartree, XC) are finite-difference stencils on the same grid – uniform treatment but finite-difference error

Fundamentally different philosophies: vibe-qc is a Gaussian-orbital code borrowing the FFT-Poisson trick from GPW for the long-range Coulomb; GPAW is a pure grid code with PAW for all-electron accuracy.

2. Coulomb / Hartree solver¶

	vibe-qc GPW/GAPW	GPAW Program
Grid density	Gaussian density ρ(r) collocated onto uniform FFT grid via AO evaluation at each grid point: `ρ(r_g) = Σ_μν D_μν χ_μ(r_g) χ_ν(r_g)`	Electron density computed directly on the FD grid from wavefunctions ψ_n(r)
Poisson solve	FFT → `4π/G²` → inverse FFT (same algorithm). C++ `solve_poisson_coulomb` kernel.	FFT → `4π/G²` → inverse FFT. Written in C with `_gpaw` C extension.
AO projection	Grid potential V_H(r_g) projected back to AO J matrix: `J_μν = Σ_g χ_μ(r_g) χ_ν(r_g) V_H(r_g) dV`	N/A – no AO basis. Potential applied directly to wavefunctions on the grid.
Collocation cache	`GpwCollocationCache` stores χ(r_g) table built once per SCF, reused across iterations (95%+ of runtime saved)	No equivalent – wavefunctions ARE the grid representation

3. All-electron treatment¶

	vibe-qc GAPW	GPAW Program
Method	GAPW (Gaussian-based): AO density on per-atom radial×Lebedev grids, augmented with atomic corrections for the hard (core+nodal) density	PAW (Blöchl 1994): Frozen atomic partial waves + projector functions. PAW datasets pre-computed for each element.
Atomic data	No frozen datasets – uses the same Gaussian basis to model both hard (ρ_a) and soft (ρ̃_a) densities on atomic grids. Soft basis built by removing tight primitives (exponent > α_cut).	Pre-computed PAW datasets (setups) containing partial waves φ_i, pseudo partial waves φ̃_i, projector functions p̃_i, and compensation charges. Downloaded from gpaw-data.
Correction form	`E_H = E_H[ρ̃+ρ₀] + Σ_a (E_H[ρ_a] − E_H[ρ̃_a+ρ₀,a])` – Hartree correction on atomic sphere	`E = E_pseudo + Σ_a Σ_ij Δρ_ij D_ij` – PAW correction via atomic density matrix in the partial-wave basis
Frozen core	All-electron by construction (no pseudopotential). The Gaussian basis already describes the core.	Uses frozen-core PAW by default – core states are pre-computed atomic orbitals, frozen during SCF

4. SCF convergence¶

	vibe-qc GPW/GAPW	GPAW Program
Eigensolver	DIIS (Pulay) + EDIIS/ADIIS on the Fock matrix. C++ SCF host (`periodic_gapw_cpp_host`).	Davidson iterative diagonalisation by default; also supports direct minimisation (etdm-fdpw), CG, and LCAO mode.
Mixing	Anderson/Pulay mixing of the density matrix. `auto_optimize_truncation` for convergence.	Pulay density mixing by default (`Mixer`); also `no-mixing` for direct-min.
Smearing	Fermi-Dirac, Gaussian, Marzari-Vanderbilt. `SmearingOptions` class.	Fermi-Dirac, Methfessel-Paxton, Marzari-Vanderbilt, cold smearing.
Open-shell	UHF/UKS/ROHF/ROKS available (post-M3). `GAPWExperimentalWarning` gates production use.	Spin-polarized DFT and HF available throughout. Hund’s rule support for free atoms.

5. GPU acceleration¶

	vibe-qc	GPAW
GPU	No GPU support (CPU-only C++17 with Eigen/FFTW3)	Yes – NVIDIA GPU support via CuPy (Python GPU arrays). Most operations run on GPU if available.

6. Exchange (exact HF exchange)¶

	vibe-qc GPW/GAPW	GPAW Program
Hybrid functionals	Yes – libint analytic 4c-ERIs for HF exchange at Γ-point. BIPOLE route for multi-k.	Yes (PW mode) – but expensive: O(N³) scaling. Usually combined with LCAO mode for hybrids.
Range-separated	Experimental (`periodic_gapw_range_sep`, behind `GAPWExperimentalWarning`).	Supported through libxc range-separated functionals.
COSX (RIJCOSX)	Γ-point COSX-K experimental; BIPOLE for multi-k.	No direct COSX equivalent; uses LCAO or pure grid modes.

7. Functional coverage¶

	vibe-qc GPW/GAPW	GPAW Program
LDA/GGA	✓ Full libxc (500+ functionals)	✓ Full libxc
meta-GGA	✓ r²SCAN and others (M3c). Requires τ on the real-space grid.	✓ Full libxc including r²SCAN. Requires τ.
Hybrids	✓ (analytic ERI at Γ; BIPOLE at multi-k)	✓ (but expensive in PW mode)

8. Production readiness¶

	vibe-qc GPW/GAPW	GPAW Program
GPW	Production for closed-shell Γ-point. Multi-k pure-DFT available. Analytic forces wired.	Production since 2005. Thousands of publications.
GAPW	Experimental (`GAPWExperimentalWarning`). Correct for molecular-limit systems; tight-cell ionic crystals have known issues.	PAW is default and production. Mature PAW datasets.
Tests	Growing test suite. GAPW parity against CP2K targeted.	Extensive test suite. Validated against VASP, ABINIT, QE.

9. Method family overview¶

vibe-qc ships several periodic SCF routes covering different accuracy/cost trade-offs:

Route	Basis	Coulomb	Exchange	All-electron	Status
BIPOLE	Gaussian	Ewald / Multipole (L≥0)	Analytic 4c-ERI (exact)	Yes (all-electron GTOs)	Production (RHF/UHF/RKS/UKS)
GDF	Gaussian + density-fitting auxiliary basis	Ewald / FFT	Analytic 3c-ERI (DF-approximate)	Yes	Production (closed+open shell, multi-k)
GPW	Gaussian → real-space grid collocation	FFT-Poisson	Analytic 4c-ERI	No (smooth density only – PP-like)	Production (Γ-point, closed-shell)
GAPW	Gaussian + per-atom radial grids	FFT-Poisson + atomic-sphere correction	Analytic 4c-ERI	Yes (all-electron via augmentation)	Experimental
CCM	Gaussian (INDO/MNDO parameterised)	Ewald (Madelung)	INDO Fock terms	No (valence-only, semiempirical)	Production (MSINDO)

10. Summary¶

Dimension	vibe-qc	GPAW
Basis	Gaussian (atom-centered)	Real-space grid (FD)
Core treatment	All-electron (GAPW augments GPW) or valence-only (BIPOLE/GDF/GPW/CCM)	PAW (frozen-core, pre-computed datasets)
Hartree	FFT-Poisson on collocated density (GPW/GAPW) or Ewald/multipole (BIPOLE) or DF (GDF)	FFT-Poisson on grid density
Exchange	Analytic 4c-ERI (exact for Gaussians) or DF-approximate (GDF)	FD stencil on grid (or LCAO mode)
Overall approach	Gaussian quantum chemistry + GPW acceleration for Coulomb	Pure real-space DFT from the ground up
Maturity	GPW: production · GAPW: experimental · BIPOLE: production · GDF: production	Production (20 years)

TL;DR: vibe-qc’s GPW is a Gaussian code accelerated by FFT, converging the traditional quantum-chemistry world (libint, contracted GTOs, analytic ERI) with the plane-wave world (FFT-Poisson). GPAW is a real-space finite-difference code that happens to use PAW for all-electron accuracy. They share the FFT-Poisson solver and libxc, but differ in almost everything else. vibe-qc additionally offers BIPOLE and GDF routes that are entirely Gaussian-native – no real-space grid involved – which GPAW has no equivalent for.