Variational generalized Kohn-Sham approach combining the random-phase-approximation and Green's-function methods

A generalized Kohn-Sham (GKS) scheme which variationally minimizes the random phase approximation (RPA) ground-state energy with respect to the GKS one-particle density matrix is put forward. We introduce the notion of functional self-consistent schemes, which vary the one-particle Kohn-Sham (KS) potential entering an explicitly potential-dependent exchange-correlation (XC) energy functional for a given density, and distinguish them from orbital-self-consistent (OSC) schemes, which vary the density, or the orbitals, density matrix, or KS potential generating the density. It is shown that, for explicitly potential-dependent XC functionals, existing OSC schemes such as the optimized effective potential method violate the Hellmann-Feynman theorem for the density, producing a spurious discrepancy between the KS density and the correct Hellmann-Feynman density for approximate functionals. A functional self-consistency condition is derived which resolves this discrepancy by requiring the XC energy to be stationary with respect to the KS potential at fixed density. We approximately impose functional self-consistency by semicanonical projection (sp) of the Perdew, Burke, and Ernzerhof KS Hamiltonian. Variational OSC minimization of the resulting GKS-spRPA energy functional leads to a nonlocal correlation potential whose off-diagonal blocks correspond to orbital rotation gradients, while its diagonal blocks are related to the RPA self-energy at a real frequency. Quasiparticle $GW$ energies are a first-order perturbative limit of the GKS-spRPA orbital energies; the lowest-order change of the total energy captures the renormalized singles excitation correction to RPA. GKS-spRPA orbital energies are found to approximate ionization potentials and fundamental gaps of atoms and molecules more accurately than semilocal density functional approximations (SL DFAs) or $G_0{W}_0$ and correct the spurious behavior of SL DFAs for negative ions. GKS-spRPA energy differences are uniformly more accurate than the SL-RPA ones; improvements are modest for covalent bonds but substantial for weakly bound systems. GKS-spRPA energy minimization also removes the spurious maximum in the SL-RPA potential energy curve of ${\mathrm{Be}}_2$ and produces a single Coulson-Fischer point at $\sim 2.7$ times the equilibrium bond length in ${\mathrm{H}}_2$. GKS-spRPA thus corrects most density-driven errors of SL-RPA, enhances the accuracy of RPA energy differences for electron-pair-conserving processes, and provides an intuitive one-electron GKS picture yielding ionization potentials energies and gaps of $GW$ quality.

Figure 1

Errors ($\mathrm{computed}-$reference [77]) in binding energies for the dimers of S22 data set. The PBE XC potential and aug-cc-pwCV(T,Q)Z [78] basis set extrapolation were used.

Figure 2

Computed PECs of noble gas dimers compared to reference data [79, 80, 81, 82]. aug-cc-pV6Z [83] basis sets were used for ${\mathrm{He}}_2$, ${\mathrm{Ne}}_2$, and ${\mathrm{Ar}}_2$, and aug-cc-pV5Z [66, 84] basis sets were used for ${\mathrm{Kr}}_2$; an additional set of atomic basis midbond functions was employed to speed up basis set convergence. The PBE XC potential was used.

Figure 3

Computed GKS-spRPA aug-cc-pwCV5Z [95] PECs of ${\mathrm{Be}}_2$ molecule compared to OEP-RPA (using plane-wave basis sets with 50 Ry cutoff) [15], experiment [87], and complete basis-set limit CCSD(T) [96].

Figure 4

PECs of ${\mathrm{H}}_2$ using unrestricted and restricted PBE, RPA-PBE, and GKS-spRPA approaches. aug-cc-pV5Z [66] basis set and PBE potential were used.