RPA and GW

The part of the response claim that is actually demonstrated.

PDEP

The dielectric matrix is the density–density response function. Conventional RPA and GW evaluate it through an explicit sum over empty orbital states — expensive, and slow to converge with basis size.

PDEP — projective dielectric eigenpotentials — instead builds a low-rank basis from the dominant eigenmodes of the dielectric matrix. The spectrum decays quickly, so a modest number of modes captures the physics and the empty-state sum disappears.

This low-rank compression works and is used in production paths. It is the demonstrated half of the design premise; the real-space locality half is not (see Electronic response).

RPA

  • PDEP-RPA — RPA correlation via a low-rank W basis in Gaussians
  • U-PDEP-RPA — open-shell, over a spin-summed dielectric
  • Attenuated RPA — short-range correlation via erfc

The solver defaults to Lanczos; a dense path is used for small problems. Note that the eigensolve is serial by design — that is a deliberate choice, not an oversight, and RPA here is already faster than the PySCF reference.

GW

  • G0W0, COHSEX, evGW0, evGW
  • U-GW — unrestricted

G0W0@HF matches MOLGW to roughly 5 meV.

The quasiparticle solve runs a Newton root-find on the self-energy, which is fragile near \( \Sigma_c \) poles. That fragility has a consequence worth recording: the frequency-quadrature loop inside the self-energy is a sequential floating-point accumulation, so it cannot be parallelized without changing summation order — and reordering would perturb quasiparticle energies in a thread-count-dependent way. The loop is deliberately left serial.

TDDFT

Linear response in both the Tamm–Dancoff approximation (TDA/CIS) and the full Casida equations, closed-shell references.

Important limitation. The \( (ia|f_{xc}|jb) \) XC-kernel response is not implemented. With a pure Hartree–Fock reference (\( c_{HF} = 1 \)) that term is identically zero and the result is exactly CIS/TDHF. With any DFT reference it is not zero, and the excitation energies omit it — they are approximate.

The code warns on stderr when \( c_{HF} \neq 1 \) rather than returning silently incomplete numbers.

Double hybrids

B2PLYP and DSD-PBEP86, plus wB97X-L-V — see Coupled cluster.

Dispersion and polarizability

Static and atom-partitioned polarizabilities, Casimir–Polder \( C_6 \) coefficients, and many-body dispersion.

TDHF/RPAx is a measured negative for dispersion: the static α it produces is reasonable, but the \( C_6 \) stays roughly 60% low regardless of gap. It is a polarizability tool, not a dispersion one.

Dynamic dRPA@PBE α, by contrast, gives \( C_6 \) roughly 3× better than the static Tkatchenko–Scheffler (TS) model.