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arXiv:2603.19112

Derivative Discontinuity in Many-Body Perturbation Theory and Chemical Potentials in Random Phase Approximation

Jiachen Li, Weitao Yang

Published Mar 19, 2026• 0 citations
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Abstract

We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the $GW$ energy functional evaluated using non interacting Green's functions ($G_s$). The chemical potential is obtained using two formally equivalent approaches: a direct derivative of the total energy with respect to particle number, and a functional derivative via the chain rule through $G_s$, both validated with finite difference benchmarks. We show that the functional derivative of the $GW$ correlation energy$\unicode{x2013}$i.e., the $GW$ correlation self energy$\unicode{x2013}$exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate $GW$ quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard $GW$ self energies neglect this nonanalytic behavior. Our results suggest that derivative discontinuities are a fundamental feature of correlation energy functionals, analogous to the known discontinuity in the exact exchange correlation energy.

DOI
10.48550/arxiv.2603.19112

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We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the GWGWGW energy functional evaluated using non interacting Green's functions (GsG_sGs​). The chemical potential is obtained using two formally equivalent approaches: a direct derivative of the total energy with respect to particle number, and a functional derivative via the chain rule through GsG_sGs​, both validated with finite difference benchmarks. We show that the functional derivative of the GWGWGW correlation energy\unicodex2013\unicode{x2013}\unicodex2013i.e., the GWGWGW correlation self energy\unicodex2013\unicode{x2013}\unicodex2013exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate GWGWGW quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard GWGWGW self energies neglect this nonanalytic behavior. Our results suggest that derivative discontinuities are a fundamental feature of correlation energy functionals, analogous to the known discontinuity in the exact exchange correlation energy.
We generalize the exact strong-interaction limit of the exchange-correlation energy of Kohn-Sham density functional theory to open systems with fluctuating particle numbers. When used in the self-consistent Kohn-Sham procedure on strongly interacting systems, this functional yields exact features crucial for important applications such as quantum transport. In particular, the steplike structure of the highest-occupied Kohn-Sham eigenvalue is very well captured, with accurate quantitative agreement with exact many-body chemical potentials. While it can be shown that a sharp derivative discontinuity is present only in the infinitely strongly correlated limit, at finite correlation regimes we observe a slightly smoothened discontinuity, with qualitative and quantitative features that improve with increasing correlation. From the fundamental point of view, our results obtain the derivative discontinuity without making the assumptions used in its standard derivation, offering independent support for its existence.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
In principle, the Luttinger-Ward Green's-function formalism allows one to compute simultaneously the total energy and the quasiparticle band structure of a many-body electronic system from first principles. We present approximate and exact expressions for the correlation energy within the GWGWGW-random-phase approximation that are more amenable to computation and allow for developing efficient approximations to the self-energy operator and correlation energy. The exact form is a sum over differences between plasmon and interband energies. The approximate forms are based on summing over screened interband transitions. We also demonstrate that blind extremization of such functionals leads to unphysical results: imposing physical constraints on the allowed solutions (Green's functions) is necessary. Finally, we present some relevant numerical results for atomic systems.
We calculate the interacting bandgap energy of a solid within the random-phase approximation (RPA) to density functional theory (DFT). The interacting bandgap energy is defined as ${E}_{g}={E}^{\mathrm{RPA}}(N+1)+{E}^{\mathrm{RPA}}(N\ensuremath{-}1)\ensuremath{-}2{E}^{\mathrm{RPA}}(N)$, where ERPA(N){E}^{\mathrm{RPA}}(N)ERPA(N) is the total DFT-RPA energy of the NNN-electron system. We compare the interacting bandgap energy with the Kohn-Sham bandgap energy, which is the difference between the conduction and valence band edges in the Kohn-Sham band structure. We show that they differ by an unrenormalized ``G0W0{G}_{0}{W}_{0}G0​W0​'' self-energy correction (i.e., a GWGWGW self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to G0W0{G}_{0}{W}_{0}G0​W0​ quasiparticle bandgap calculations, but questions the physics behind the renormalization factors in the expression of the bandgap energy. We also separate the kinetic from the Coulomb contributions to the DFT-RPA bandgap energy, and discuss the related problem of the derivative discontinuity in the DFT-RPA functional. Last we discuss the applicability of our results to other functionals based on many-body perturbation theory.
For every electronic many-body Hamiltonian with a one-body potential that goes to zero at infinity, there is a maximum number of bound electrons (Jmax{J}_{\mathrm{max}}Jmax​) that can be sustained in the system with an infinite lifetime. At this integer Jmax{J}_{\mathrm{max}}Jmax​, the derivative discontinuity in the exact energy of ensemble density functional theory (DFT) is ill defined. However, we investigate the derivative discontinuity of the energy within the framework of density functional resonance theory [D. L. Whitenack and A. Wasserman, Phys. Rev. Lett. 107, 163002 (2011)], which reduces to ground-state DFT as a coordinate scaling parameter is taken to zero, and find that the exact exchange-correlation potential experiences discontinuous jumps at integer particle numbers, including Jmax{J}_{\mathrm{max}}Jmax​. For integers below Jmax{J}_{\mathrm{max}}Jmax​ the jump is purely real because of the real shift in the chemical potential. At Jmax{J}_{\mathrm{max}}Jmax​, the jump has a nonzero imaginary component reflecting the metastability of the system upon addition of one more electron. In addition, the magnitude of the derivative discontinuity at Jmax{J}_{\mathrm{max}}Jmax​ is larger than would have been expected by simply setting the affinity to zero.
We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle random phase approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illustrate the potential of including pairing interactions within a density functional framework.