Standalone Faddeev¶

Experimental implementation. FTDA and FRPA are research methods under development.

Enable canonical RHF and one of FTDA or FRPA in input/methods. The standard restricted driver dispatches the standalone calculation. FTDA uses forward pair propagation. FRPA includes backward pair amplitudes. The physical charged spaces contain 2h1p and 2p1h configurations.

Run the bundled small example with python docs/examples/run.py ftda.

Options¶

The optional standalone record in input/options has seven positional fields. Keep its header, which the reader uses to find the row.

# Faddeev: reference(0=HF,1=GW) pairs(0=bare,1=BSE,2=CCD) coupling(2=ADC2,3=ADC3,4=dressed,5=CCD) static3 external_GW Davidson n_IP_roots
0 0 3 T F T 1

Field	Values and meaning
reference	0 uses HF energies for internal propagation. 1 uses frozen GW energies
pairs	0 bare, 1 BSE, 2 CCD-amplitude pair interactions
coupling	2 ADC(2), 3 ADC(3), 4 static dressed source, 5 CCD-amplitude coupling
static3	Include the third-order static self-energy correction
external_GW	Use GW energies in the external one-particle diagonal
Davidson	T targets principal IP roots. F uses dense diagonalization
n_IP_roots	Positive root count. In Davidson mode, the count is bounded by the number of active occupied orbitals

For FRPA with pairs=2, QuAcK solves CCD at the HF reference and reshapes the converged doubles amplitudes into each pair channel. For a pair Hessian S = [[A, B], [B^T, C]] and amplitude matrix T, it projects onto P = [I; T], giving the pair Hamiltonian P^T S P and metric I - T^T T. The removal channel interchanges the forward and backward pair spaces. This projection recovers the RPA pair pencil for exact RPA amplitudes. Using CCD amplitudes defines an approximation. QuAcK checks that the projected pair metric and Hessian are positive definite. FTDA retains its static Hermitian closure.

For example, 0 2 3 T F T 1 selects CCD pairs, ADC(3) couplings and Davidson for one IP root. Use 5 in the coupling column to reuse the same CCD amplitudes for the external couplings.

BSE pairs require reference=1. CCD pairs are available with both FTDA and FRPA and reject coupling 4. The reader uses the displayed defaults if the standalone record is absent. Select one standalone flag per calculation. At this revision, the common Faddeev driver uses the CC convergence settings for CCD amplitudes, the GW settings for frozen GW energies, and reg_PA, reg_1b, and reg_2b from the Parquet options record for regularization.

Upfolding and spectral weights¶

In the Hermitian FTDA formulation, the one-particle block F is coupled through U to a charged configuration block K. Eliminating the charged amplitudes yields a frequency-dependent contribution to the one-particle self-energy.

Σ(ω) = Σ(static) + U† (ωI − K)⁻¹ U

The position of a pole and its one-particle weight answer different questions. For a normalized eigenvector of the Hermitian upfolded matrix, the weight is the norm squared of its one-particle component. The Davidson example also reports the residual of the selected root in hartree.

Frozen GW/BSE inputs¶

Set reference=1 and pairs=1 in the Faddeev record to use frozen GW energies and BSE pair kernels. The standalone reader takes regularization controls from the common Parquet options row.

For a reproducible comparison, report the reference energies, pair kernel, endpoint coupling, static correction, regularization, solver, and root count. When comparing different endpoints, keep the pair kernel and external one-particle block fixed.