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

Permalink https://escholarship.org/uc/item/7gf3h1h9

Journal PHYSICAL REVIEW A, 99(1)

ISSN 2469-9926

Authors Voora, Vamsee K Balasubramani, Sree Ganesh Furche, Filipp

Publication Date 2019-01-28

DOI 10.1103/PhysRevA.99.012518

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California Variational Generalized Kohn–Sham Approach Combining Random Phase Approximation and Green’s Function Methods

Vamsee K. Voora, Sree Ganesh Balasubramani, and Filipp Furche∗ University of California, Irvine, Department of Chemistry, 1102 Natural Sciences II, Irvine, CA 92697-4095, USA (Dated: 12/11/2018) 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 introduced. We introduce the notion of functional-selfconsistent (FSC) 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-selfconsistent (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 selfconsistency 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 selfconsistency by by semicanonical projection (sp) of the PBE 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 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 G0W0 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 Be2, and produces a single Coulson–Fischer point at ∼ 2.7 times the equilibrium bond length in H2. 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.

I. INTRODUCTION AND SUMMARY negative ions, and small-gap compounds, producing large “density-driven errors” in energies and other properties Electronic structure methods based on the random- [10, 11]. For example, SL DFAs produce qualitatively phase approximation (RPA) [1,2] are rapidly gaining incorrect densities and ionization potentials for negative popularity in solid-state and molecular applications [3– ions [12], and overly delocalized states for correlated ma- 5]. As opposed to semilocal (SL) density functional ap- terials [13]. proximations (DFAs), RPA-based methods capture non- Unlike SL energy functionals, the RPA energy ex- covalent interactions [6], which have recently moved into plicitly depends on the unknown KS potential, making the focus of research in soft matter, nanomaterials, and straightforward minimization with respect to the den- catalysis [7]. RPA-based methods are comparable in cost sity or KS orbitals impossible. In this paper, we distin- with but more robust than perturbative approaches for guish density- or orbital-selfconsistent (OSC) RPA ap- small-gap systems and offer a way out of the functional proaches, which minimize the energy after choosing an inflation dilemma faced by SL DFAs [8]. approximate KS potential, from functional selfconsistent The vast majority of today’s RPA calculations are per- (FSC) approaches, which aim to determine the KS poten- formed in a “post Kohn–Sham” fashion, i.e., by evaluat- tial functional self-consistently by requiring its exchange- ing the RPA energy functional using Kohn–Sham (KS) correlation (XC) part to coincide with the functional orbitals generated from a variational SL DFA calculation derivative of the RPA energy. The optimized effective [9]. Apart from the lack of a variationally stable energy, potential (OEP) approach [14–17] has been claimed to a major limitation of this SL-RPA approach is that SL achieve “fully selfconsistent” RPA results [16]. However, densities are relatively inaccurate for open-shell systems, while OEP-RPA produces accurate KS orbital energies, OEP-RPA results for bond energies and noncovalent interactions are less accurate than their SL-RPA counter- ∗ Author to whom correspondence should be addressed. Email: parts [16]. This result is puzzling, because the OEP-RPA fi[email protected] KS potentials are considerably more accurate than SL Phys. Rev. A, accepted ones [18], and recent orbital-optimized RPA approaches where the KS orbitals and occupation numbers were improve upon SL RPA energetics [19]. gathered in the column vectors |φi , n. Observables such as quasiparticle spectra and ioniza- Kohn and Sham[42] showed that variational minimization potentials (IPs) have traditionally been the do- tion of the total energy as a functional of ρ leads to the main of many-body Green’s function theory (GFT) [20– KS equations 22]. For example, quasiparticle self-consistent GW the- H [ρ] |φ i = |φ i , (2) ory [23, 24] yields highly accurate IPs and band gaps 0 p p p for a wide variety of materials, but accurate total en- where H0[ρ] = T + Vs[ρ] is the noninteracting KS one- ergy differences and related properties such as structures particle Hamiltonian, T is the one-particle kinetic energy or thermodynamic quantities remain elusive. GW ap- operator, and proaches starting from SL DFAs [25–27] and even fully ext H XC self-consistent GW [28] face a similar dilemma, producing Vs[ρ](x) = V (x) + V [ρ](x) + V [ρ](x) (3) high quality quasiparticle spectra and excitation ener- is the KS one-particle potential. Vext(x) is the external gies, but energy differences generally inferior to SL-RPA H [29, 30]. one-particle potential, and V [ρ](x) is the Hartree potential. The XC potential is the (total) functional derivative Here we generalize the SL-RPA energy functional us- of the XC energy with respect to the density, ing a simple semicanonical projection (sp) of the SL KS Hamiltonian. The resulting spRPA energy is a func- δEXC[ρ] VXC[ρ](x) = . (4) tional of the KS one-particle density matrix that may δρ(x) variationally be minimized using an OSC generalized KS (GKS) scheme [31]. Semicanonical approaches have suc- The OSC minimizing density is given by cessfully been used to devise perturbative corrections to X 2 SL-RPA in the past [32]. The present variational GKS ρ(x) = np|φp(x)| , (5) spRPA method is designed to (i) recover SL-RPA for SL p densities; (ii) reduce density-driven error by determin- with occupation numbers normally [43] chosen according ing the density from the stationary point of the RPA to the Aufbau principle. While this section focuses on rather than a SL energy functional; (iii) systematically density-based “proper KS” schemes, analogous consider- improve SL-RPA energetics; (iv) approximate FSC RPA ations apply to the GKS framework [31], if the local XC without sacrificing the variational principle; (v) yield a potential in Eq. (4) is replaced by the nonlocal one de- complete GKS effective one-particle Hamiltonian, unlike fined by the functional derivative of the XC energy with orbital optimized [19] or Brueckner [33] RPA, providing respect to the KS density matrix. an intuitive one-electron picture and GKS orbital ener- However, Eqs. (1)-(5) do not completely determine the gies which accurately approximate quasiparticle spectra energy as a functional of the density: Obtaining the KS [34]; (vi) establish a straightforward connection to GFT potential via Eq. (4) requires knowledge of EXC[ρ], which and GW theory; (vii) eliminate the need for KS inversion in turn is defined in terms of the KS potential. Two dif- or OEP approaches, which can be ill conditioned [35] and ferent avenues have been used to bypass this quandary: require cumbersome regularization [36–38]. The first uses the density (or, in GKS framework, the orbitals and occupation numbers) as independent variable. In this case, an additional condition specifying the II. VARIATIONAL MINIMIZATION OF KS potential as a functional of the density (or the or- POTENTIAL-DEPENDENT FUNCTIONALS bitals and occupation numbers) is required. The second approach considers the density (or the orbitals and oc- A. Statement of the Problem cupation numbers) as dependent variable(s), and the potential as independent variable. This potential-functional approach requires specification of the density (or the or- Variational minimization of a functional with respect bitals and occupation numbers) as functional of the po- to the density or the KS orbitals requires knowledge at tential. In the following, we discuss different choices for least of the energy functional and its functional derivative either functional(s), which lead to qualitatively different for a given trial density. Here we consider functionals XC energy functionals, labeled by subscripts a-d. that depend on the density ρ(x) through the KS orbitals and occupation numbers |φpi and np and explicitly on the KS potential Vs[ρ](x); x = (r, σ) denotes space-spin B. KS Potentials and Energy Functionals coordinates. This large class includes functionals derived from many-body perturbation theory [39–41] as well as a. Semilocal Potentials. Since SL KS potentials are RPA. The resulting XC energy thus takes the general readily available, a straightforward choice for the XC en- form ergy functional is

XC XC XC XC SL E [ρ] = E [|φ[ρ]i , n[ρ], Vs[ρ]], (1) Ea [ρ] = E [|φ[ρ]i , n[ρ], Vs [ρ]]. (6)

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Here, the SL XC potential is the functional derivative of is an explicit functional of the KS orbitals and orbital a SL XC energy functional such as PBE [44], energies for a given potential Vs [16, 40], and Eq. (11) becomes the OEP integral equation [51]. While grid- δEXC SL[ρ] VXC SL[ρ](x) = , (7) based OEP approaches are fairly straightforward, basis- δρ(x) set OEP approaches can be ill-posed similar to KS inver-

SL XC sion [35] and require additional regularization [36–38]. and Vs [ρ] is obtained by replacing V [ρ] with XC SL OEP approaches to potential-dependent XC function- V [ρ] in Eq. (3). als have been dubbed “fully selfconsistent” [16, 52]. Even The main drawback of this approach is that it violates though they use the “exact” KS potential, however, they Eq. (4), lack functional selfconsistency, causing the KS density ρs δEXC[ρ] to differ from the functional derivative of the energy with a 6= VXC SL[ρ](x). (8) respect to the external potential: δρ(x) In potential functional theory, the density as a func- As a result, there are two KS systems, one related to tional of the KS potential is defined by minimization of EXC[ρ], and the other generated by a δE[V ] VXC SL[ρ], whose density is generally different from the s ext = ρ[Vs](x), (12) orbital density ρ. The XC energy is defined in terms δV (x) of the relatively inaccurate SL potential, giving rise to where E[Vs] is the total energy potential functional density-driven error. In explorative calculations using [49, 53]. Eq. (12) is the equivalent of Eq. (4) in den- the RPA energy functional, the total energy differences sity functional theory and thus fundamental; it may be we obtained from such schemes were not significantly viewed a consequence of the Hellmann-Feynman theo- more accurate than the post-KS semilocal ones. rem, and as such is commonly used to define the density b. “Exact” Potentials Via KS Inversion or OEP. and all related properties even for approximate energy For a noninteracting v-representable trial density, the functionals. For inexact XC functionals with explicit de- “exact” KS potential Vs[ρ] (and thus H0[ρ]) may be dependence on the KS potential, however, the KS density, termined, up to a constant, by inversion of the KS equa- defined by Eq. (10), generally differs from the exact one, tions. One thus might wonder if this choice results in defined by Eq. (12): Evaluating the functional deriva- better properties than the use of SL potentials. While tive of the total energy expression defined by the poten- the uniqueness of Vs[ρ] is guaranteed by the Hohenberg– XC tial functional Eb [Vs] in Eq. (9) using the Hellmann- Kohn theorem [45], general trial densities may not be Feynman theorem and Eq. (12) yields pure-state noninteracting v-representable; see, e.g., Ref. XC 46 for examples. In common finite basis sets, this condi- δEb [Vs] tion is rarely satisfied [47], and KS inversion procedures ρs[Vs] + = ρb[Vs](x), (13) δVs(x) can be ill-posed [48]. |φ[Vs]i,n[Vs] The OEP approach claims to bypass some of these where the functional derivative is a partial derivative at difficulties [14–16, 40] by using the local KS potential fixed orbitals and occupation numbers, and ρb[Vs](x) = ext Vs(x) as independent variable[49, 50] and the KS orbitals δEb[Vs]/δV (x) is the Hellmann-Feynman density gen- XC φp[Vs] and occupation numbers np[Vs] as dependent vari- erated by Eb [Vs]. The quantity ables; the functionals φp[Vs] and np[Vs] are defined by XC the requirement that they satisfy the KS equations (2) δE [Vs] ρXC[V ](x) = b , (14) and minimize the total KS kinetic energy. The resulting b s δV (x) s |φ[Vs]i,n[Vs] XC energy thus becomes a potential functional, is zero for the exact XC functional by construction, XC XC Eb [Vs] = Eb [|φ[Vs]i , n[Vs], Vs]. (9) but generally nonzero for the approximate explicitly potential-dependent XC functionals discussed here (see, To make a connection to density functionals, OEP meth- e.g., Ref. 16 and Sec.IIIB for examples). An immediate ods consider the density generated by the KS orbitals consequence is that and occupation numbers, XC XC X 2 δEb [ρs] δEb [ρb] ρs[Vs](x) = np[Vs]|φp[Vs](x)| . (10) 6= , (15) δρs(x) δρb(x) p i.e., the OEP XC potential produces the functional By the HK theorem, V is a functional of ρ , and thus s s derivative of the XC energy with respect to the nonin- the XC potential can be obtained from the chain rule, teracting KS density ρs rather than ρb[Vs], which is the XC Z XC 0 correct density by Eq. (12). δEb [Vs] 0 δEb [ρs] δρs[Vs](x ) = dx 0 . (11) In conclusion, even using the “exact” KS potential pro- δVs(x) δρs(x ) δVs(x) duces the paradoxical result of two different densities cor- 0 Since ρs[Vs] is the KS density, δρs[Vs](x )/δVs(x) is the responding to two different KS potentials, and the less ac- well-known KS density-density response function, which curate of the two (according to the Hellmann-Feynman

3 Phys. Rev. A, accepted theorem) is used to obtain the total energy. This con- causing problems such as initial state dependence and firms that lacking functional selfconsistency is a funda- multiple solutions familiar from selfconsistent GFT ap- mental problem of explicitly potential-dependent func- proaches [54]. tionals which, contrary to previous suggestions [16, 40], d. Semicanonical KS. The above analysis suggests cannot be remedied by the OEP approach. that lack of functional selfconsistency of SL and OEP c. Functional-Selfconsistent Potentials. The para- KS potentials critically limits the accuracy of these ap- doxical result of two different densities and KS potentials proaches. We take a step towards a fully FSC solution by are caused by inconsistency of the KS potentials defining constructing an approximation to the FSC KS Hamilto- the XC energy functional with the functional derivative nian, which leads to an explicit energy functional of the thereof. A resolution is possible if the KS potential sat- orbitals and occupation numbers. Using a GKS frame- isfies the functional selfconsistency condition work to achieve orbital selfconsistency allows us to by-

XC pass the numerical challenges of OEP methods and ob- ρ [|φi , n, Vs](x) = 0. (16) tain orbital energies corresponding to physical ionization By Eq. (13), the orbitals and occupation numbers gen- potentials and electron affinities. erated by such an FSC KS potential yield the correct Rather than imposing the full functional selfconsis- Hellmann-Feynman density, and thus Eq. (16) is an ex- tency condition, which implies that the KS potential entering the XC energy functional is identical to the func- act constraint for the KS potential Vs. Eq. (16) is equivalent to requiring the XC energy functional to be stable tional derivative thereof, see Eq. (17), we only require with respect to variations of the KS potential at fixed that the KS potential defining the XC energy generate orbitals and occupation numbers. Thus, the chain rule the same eigenstates (up to unitary equivalence) as the yields for the FSC XC potential one obtained from functional differentiation. This weaker condition is readily imposed by semicanonical projection XC SL δE [|φi , n, Vs] of a readily available SL KS Hamiltonian H . In the VXC[|φi , n, V ](x) = 0 s δρ(x) most general case, the KS density matrix takes the form XC (17) δE [|φi , n, Vs] X = , D = Pλnλλ0 Pλ0 , (19) δρ(x) 0 Vs λλ since the partial derivative with respect to Vs vanishes where Pλ denotes orthogonal projectors belonging to according to Eq. (16). For potential-independent XC blocks of KS orbitals with degenerate occupation num- functionals, the partial derivative is a total derivative, bers, and nλλ0 = nλδλλ0 is diagonal, with nλ denoting and Eq. (17) reduces to the conventional definition of the occupation number matrices [55], see AppendixA. For XC potential. For potential-dependent XC functionals, example, for integer KS occupations 1, 0, there are two Eq. (17) is a statement of the functional selfconsistency distinct matrices nλ with eigenvalues nλ = 1, 0. The sp condition. KS Hamiltonian is defined by It may be possible to define a FSC XC potential ˜ X SL Vs,c[|φi , n] as a functional of the orbitals and occupa- H0 = PλH0 Pλ, (20) tion numbers implicitly by Eqs. (16) or (17). The con- λ ditions under which the functional selfconsistency con- 0 SL and contains only the diagonal (λ = λ ) blocks of H0 . straint uniquely determines Vs,c and the resulting XC ˜ energy functional H0 commutes with D by construction; thus, one may find a common “semicanonical” basis in which both H˜ 0 and XC XC ˜ Ec [|φi , n] = E [|φi , n, Vs,c[|φi , n]] (18) a given KS density matrix are diagonal. Moreover, H0 is invariant under unitary transformations of KS orbitals depend on the specific form of the XC energy functional with degenerate occupation numbers, since the projectors (1). If such a unique potential Vs,c exists, then the den- SL ˜ Pλ are invariant. If D is generated by H0 , then H0 = sity (or KS orbitals and occupation numbers) minimiz- HSL; perturbation theory implies that the deviation of XC 0 ing the energy functional associated with Ec are gen- the sp orbital energies from the SL ones is quadratic in erated by it, because, by Eq. (17), Vs,c is obtained from HSL − H˜ . The nonlocal sp XC potential may thus be XC 0 0 the functional derivative of Ec . At the same time, Eq. defined as (16) guarantees that the resulting KS density equals the ˜ XC XC SL ˜ SL Hellman-Feynman one. In this sense, the FSC KS poten- V [|φi , n] = V [ρ] + H0[|φi , n] − H0 [ρ], (21) tial is optimal for a given explicitly potential-dependent energy functional. and the corresponding sp XC energy functional, It is important to distinguish orbital and functional EXC[|φi , n] = EXC[|φi , n, V˜ XC[[|φi , n]], (22) selfconsistency in such a scheme: The latter determines d the KS potential and thus the energy functional for fixed is an explicit functional of the KS orbitals and occupa- density, while the former determines the minimizing den- tion numbers only, which may be subject to OSC opti- sity (or the KS orbitals and occupation numbers). Mix- mization using GKS methodology. We will thus focus on XC ing the two may produce ill-defined energy functionals, functional of type Ed [|φi , n] in the following.

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C. spRPA Energy Functional III. VARIATIONAL GKS MINIMIZATION OF SEMICANONICAL PROJECTED RPA For a given KS determinant Φ, the RPA total energy, A. Energy Lagrangian and Euler Equations ERPA = hΦ|H|Φi + EC RPA, (23) Within the GKS-spRPA formalism, the ground state equals the expectation value of the physical Hamiltonian energy is obtained as the minimum of the spRPA energy H plus the RPA correlation energy, [56, 57] functional with respect to D, subject to the Fermion N- representability constraint that D have eigenvalues be- 1 Z 1 Z ∞ dω tween 0 and 1 whose sum is the total electron num- EC RPA = − dα= hΠRPA(ω) − Π (ω) Vi. 2 2π α 0 ber N. We explicitly impose the latter by the substi- 0 −∞ † (24) tution n = m m, where the Hermitian matrices m satisfy † V is the bare electron-electron Coulomb interac- hm mi = N. The GKS-spRPA energy Lagrangian is thus RPA −1 tion, Πα (ω) = (1 − αΠ0(ω)V) Π0(ω) denotes the time-ordered RPA polarization propagator at coupling LspRPA[|φi , m, η, µ] = EspRPA[D[|φi , m]] strength α and real frequency ω, and Π0(ω) is its nonin- − hη(hφ|φi − 1)i − µ hm†mi − N . (28) teracting KS equivalent; brackets stand for traces. The rank four operator Π0(ω) factorizes as [58] Here, the GKS orbitals were gathered in the transpose vectors |φi, i.e., hφ|φi is a matrix with respect to GKS Z ∞ 0 dω 0 0 orbital indices (but a scalar with respect to the one- Π0(ω) = G0(ω ) ⊗ G0(ω − ω), (25) −∞ 2πi particle Hilbert space). In this notation, the GKS density matrix becomes into a convolution product of one-particle KS Green’s † functions G0(ω). D[|φi , m] = |φi m m hφ| . (29) The KS Green’s function corresponding to the sp KS the Hermitian Lagrange multiplier matrix η enforces or- Hamiltonian H˜ 0 is bital orthonormality, and the real scalar Lagrange mul- 1/2 ˜ + −1 1/2 tiplier µ accounts for normalization of D. The present G0(ω) = n (ω − H0 − i0 ) n approach is a special case of variational density matrix 1/2 + −1 1/2 + (1 − n) (ω − H˜ 0 + i0 ) (1 − n) ; (26) functional minimization [55, 61, 62]. A necessary condition for a minimum of the GKS-spRPA energy subject this symmetrized form remains Hermitian even for den- to the above constraints is that the first partial deriva- sity matrix variations causing off-diagonal occupation tives of LspRPA with respect to all variational parameters number matrices. By construction, G0(ω) reproduces D, vanish. Requiring stationarity with respect to the GKS or- Z ∞ dω iω0+ bitals leads to the GKS self-consistent field (SCF) equa- D = e G0(ω). (27) −∞ 2πi tions spRPA On the other hand, the quasiparticle spectrum of the sp H0 [D] n |φi = η |φi , (30) Green’s function is given by the semicanonical KS eigen- where the effective one-particle GKS-spRPA Hamilto- values of H˜ , which approximate the semilocal ones in a 0 nian is defined as the functional derivative of the spRPA perturbative sense. energy with respect to the GKS density matrix, Eqs. (20)-(26) define the spRPA energy as a functional spRPA of the KS density matrix, E [D], or, equivalently, the δEspRPA[D] spectral projectors P and occupation number matrices HspRPA[D] = . (31) λ 0 δD nλ – clearly, a functional of type d according to Sec.IIB. EspRPA[D] depends on the SL XC potential DFA enter- Eq. (30) is equivalent to ing H˜ 0, for which we choose PBE [44]. This introduces spRPA a dependence on the choice of SL potential; however, H0 n = η. (32) this dependence is less strong than for SL-RPA, see Sec- tion IV of the Supplemental Material (SM) [59], since the The Hermiticity of all three matrices in the previous spRPA current scheme is OSC and partially FSC, and even for equation then implies that H0 and n commute, and spRPA SL-RPA, the dependence of energy difference on the spe- since n is block-diagonal, so must be H0 , i.e., ma- cific choice of SL potential is moderate [3]. The spRPA trix elements of the spRPA Hamiltonian between or- energy functional is conveniently evaluated by expressing bitals belonging to different occupation number blocks G0 in the semicanonical basis, factorizing the interaction must vanish (“Brillouin’s Theorem”). Evaluating nλ as V, and performing the frequency integration along the nijλ = nλδij, where indices i, j label orbitals with identi- imaginary axis, in analogy to SL-RPA [60]. cal occupation numbers nλ, and defining the Hermitian

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matrices The functional derivative δG0(ω)/δD is a rank-four tensor operator whose properties are analyzed in Appendix 0 0 λ 6= λ , B; λλ0 = 0 (33) ηλλ/nλ λ = λ , δEC spRPA ΣC(ω) = (40) Eq. (30) takes a more familiar form, δG0(ω)

spRPA X is the correlation part of the RPA self-energy [18]. Eqs. H [D] |φiλi = ijλ |φjλi . (34) 0 (39) and (40) hold for any correlation energy functional j of the GKS Green’s function, and reveal the close con- Eq. (34) is form-invariant under unitary transformations nection of the GKS correlation potential and the corre- of orbitals belonging to degenerate occupation numbers sponding self-energy. If the correlation energy further depends on G through Π only, then the functional chain nλ. A unique “canonical” GKS-spRPA orbital basis may 0 0 rule may be used once more to show that the correlation be defined by requiring ijλ to be diagonal with elements self-energy has the form[20, 21] iλ, in which case takes the form of the canonical GKS equations, Z ∞ 0 C dω C 0 0 Σ (ω) = W (ω )G0(ω − ω). (41) spRPA −∞ πi H0 [D] |φiλi = iλ |φiλi , (35) The correlation part of the effective interaction which need to be solved iteratively along with the orthonormality and normalization constraints. C RPA C δE The occupation numbers are determined by the sta- W (ω) = (42) δΠ0(ω) tionarity condition for m, is also a rank four tensor operator. Within RPA, spRPA (H0 − µ)m = 0. (36) C −1 W (ω) = V(1 − Π0(ω)V) ; (43) In the canonical basis, Eq. (36) simplifies to ( − iλ thus, ΣC(ω) is identical to the correlation part of the 1/2 spRPA µ)nλ = 0. The second variation of L with respect GW self-energy [63, 64], evaluated at the spGKS Green’s to m is nonnegative for bound states with an Aufbau function. occupation, i.e., for all i, To gain further insight into the physical meaning of the nonlocal RPA correlation potential, one may decompose 1; iλ < µ nλ = , (37) the total density matrix derivative into a sum of three 0; iλ > µ partial derivatives, where the ionization potential µ is chosen such that the VC spRPA[D] = VC,1[D] + VC,2[D] + VC,3[D]. (44) normalization condition hni = N is satisfied. If iλ = µ, then any nλ with 0 ≤ nλ ≤ 1 yielding correct normaliza- The first term corresponds to the partial density matrix ˜ tion is permissible[55]. derivative at fixed sp Hamiltonian H0. Using Appendix B and denoting (anti)commutators by (curly) brackets, this part of the potential is found to be

B. One-Particle GKS-spRPA Hamiltonian ( 1 C 0 Z G0(ω), Σ (ω) 0 , λ 6= λ , C,1 dω nλ−nλ0 λλ Vλλ0 = n C o 0 2π πi δ(ω − H˜ 0), Σ (ω) , λ = λ . The eﬀective one-particle GKS-spRPA Hamiltonian λλ0 may be analyzed by decomposing the functional deriva- (45) tive in Eq. (31) according to The oﬀ-diagonal (λ 6= λ0) blocks of VC,1 reduce to the gradient of the RPA energy with respect to orbital rota- spRPA HF C spRPA H0 [D] = H0 [D] + V [D]. (38) tions, thus establishing a link to orbital-optimized RPA approaches [19]. The diagonal (λ = λ0) blocks, on HF Here, H0 [D] is the well-known Hartree-Fock one- the other hand, result from variations corresponding to particle Hamiltonian, and VC spRPA[D] denotes the non- changes in the occupation numbers and cannot be ob- local RPA correlation potential resulting from the func- tained in an orbital optimization framework. In the sem- tional derivative of the spRPA correlation energy. Since icanonical basis, the λ blocks of H˜ 0 are diagonal, the spRPA correlation energy depends on D only through ˜ the sp Green’s function, Eq. (26), the functional chain H0ijλ = δij˜iλ, (46) rule yields and thus the diagonal blocks of VC,1 spRPA take the form Z ∞ dω δG0(ω) VC spRPA[D] = ΣC(ω) . (39) C,1 i C C V = Σλij(˜iλ) + Σλji(˜jλ) . (47) −∞ 2π δD ijλ 2

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Perturbative expansion of the GKS-spRPA orbital energies around the SL ground state solution reveals the TABLE I. Comparison of highest occupied molecular orbital (HOMO) energies (in eV) to experimental vertical ionization physical significance of Eq. (47): potentials for negatively charged atoms. G0W0 calculations were carried out using PBE orbitals. aug-cc-pVTZ basis sets SL XC SL C,3 XC SL iλ = iλ + iΣiiλ (iλ ) + Viiλ − Viiλ [66, 67] were used. spRPA SL 2 PBE HF G0W0 GKS-spRPA Ref. + O(kH0 − H0 k ). (48) H− +1.5 −1.4 +0.3 −0.9 −0.75a XC − b Here, Σ (ω) is the XC self-energy within the G0W0 ap- Li +0.8 −0.4 −0.0 −0.8 −0.68 XC SL proximation, and V is the SL XC potential. The F− +1.5 −4.9 −1.9 −3.2 −3.40c GKS-spRPA orbital energies hence reduce to the quasi- Au− +0.5 −1.2 −1.7 −2.4 −2.31d particle GW energies with unit normalization factor [21] a C,3 Ref. [68] in a perturbative first-order SCF sense apart from V , b C,1 Ref. [69] which is typically small compared to V . c Ref. [70] The remaining parts of the GKS-spRPA correlation d Ref. [71] potential arise from changes in H˜ 0, see AppendixC. The second term accounts for the semicanonical projection and its diagonal part vanishes, TABLE II. Mean absolute (MAE), mean signed (MSE, computed−reference) and maximum absolute (Max AE) er- 1 SL 0 [H , T]λλ0 , λ 6= λ , C,2 nλ−n 0 0 rors (eV) for highest occupied molecular orbital energies of V 0 = λ (49) λλ 0, λ = λ0, GW27 testset [27]. For reference values, negative of the ionization potentials from CCSD(T) were used [72]. def2- Here, the RPA unrelaxed difference density matrix [65] TZVPP basis-sets [73] was used for CCSD (T), PBE, HF, or first-order density matrix [64] G0W0 and GKS-spRPA. For comparison, OEP-RPA results from Ref. [16] using aug-cc-pCVTZ basis-sets [74] are also δEC spRPA shown. For G0W0 and GKS-spRPA, the exchange-correlation T = (50) potential was approximated using the PBE functional. δH˜ 0 a b c Error PBE HF G0W0 OEP-RPA GKS-spRPA familiar from RPA analytical derivative theory has been Measure introduced. Since the spGKS RPA Lagrangian is sta- MAE 3.83 0.65 0.64 0.26 0.30 tionary at the converged spGKS density matrix D, there MSE 3.83 −0.31 0.64 −0.08 −0.29 are no additional “orbital relaxation” terms, and the cor- Max AE 6.49 2.22 1.29 0.84 0.68 responding total interacting spRPA one-particle density matrix is simply D + T; an explicit expression for T is a Ref. [72] b Ref. [72] provided in AppendixC. The density generated by T c Ref. [16] equals ρXC(x), Eq. (14). Since GKS-spRPA is not fully FSC, ρXC(x) does not vanish, but is found to be small in most circumstances. The remaining part of the correlation potential, For ionization potentials of neutral molecules in the GW27 benchmark [27, 72], OEP-RPA and GKS-spRPA C,3 HXC V = F T (51) reduce the errors by ∼50% compared to the G0W0 method, see TableII. Both OEP-RPA and GKS-spRPA accounts for changes in the density entering the SL have similar mean absolute errors of ∼0.3 eV for the Hamiltonian due to ρXC(x); FHXC is the SL Hartree-, GW27 testset. The mean signed errors show that GKS- exchange-, and correlation (HXC) kernel. spRPA ionization potentials are systematically too large, whereas the OEP-RPA HOMO energies can be too small or too large, as is also reflected in higher maximum ab- IV. RESULTS solute deviations. OEP-RPA and GKS-spRPA quasiparticle energies dif- A. Ionization Potentials and Fundamental Gaps fer substantially for HOMO-LUMO gaps, see TableIII: OEP-RPA HOMO-LUMO gaps do not contain derivative Comparison of the computed highest occupied molecu- discontinuities since they come from a local KS potential lar orbital energies to the experimental first IPs of nega- [75], and substantially underestimate fundamental gaps. tively charged atoms, see TableI, suggests that GKS- G0W0 and GKS-spRPA, on the other hand, yield funda- spRPA is substantially more accurate than G0W0 for mental gaps within 1 eV of the coupled cluster reference negative ions, where G0W0 suffers from density-driven er- results for unpolar molecules. For the highly polar tetra- ror. The GKS-spRPA highest occupied molecular orbital cyanoethylene molecule, G0W0 and GKS-spRPA under- (HOMO) energies also improve greatly upon SL DFAs, estimate the fundamental gap by 3.3 and 1.7 eV, respec- and negative ions such as H− are correctly bound. tively, which may indicate residual self-interaction error.

7 Phys. Rev. A, accepted

TABLE III. Gaps (in eV) obtained as diﬀerences between the 0.01 0.02 He Ne lowest unoccupied and highest occupied molecular orbital en- 2 0 2 0 ergies, and reference fundamental gaps computed using diﬀer- −0.02 ences of CCSD(T) total energies (see SM for molecular struc- −0.01 −0.04 ture details [59]). CCSD(T), GKS-spRPA and G0W0 calcula- −0.06 RPA −0.02 GKS−spRPA tions use aug-cc-pVTZ basis-sets while OEP-RPA results [16] −0.08 CCSD(T) were obtained using aug-cc-pCVTZ basis-sets [74]. For GKS- Binding energy (kcal/mol) 2 3 4 5 6 7 8 2 3 4 5 6 7 8 spRPA, the exchange-correlation potential is approximated using the PBE functional. a 0.2 0.2 OEP-RPA G0W0 GKS-spRPA CCSD(T) Ar Kr 0.1 2 2 Li 1.23 4.43 5.28 4.76 0 2 0 Na2 1.18 4.35 4.98 4.48 −0.1 −0.2 LiH 2.94 6.92 7.66 7.67 −0.2

CH3NO2 9.82 11.48 11.41 −0.3 −0.4 Binding energy (kcal/mol) 3 4 5 6 7 8 3 4 5 6 7 8 C2(CN)4 6.96 8.48 10.29 Bond length (Å) Bond length (Å) a Ref. [16] FIG. 2. Computed PECs of noble gas dimers compared to reference data [79–82]. aug-cc-pV6Z[83] basis sets were used B. Noncovalent Interactions for He2, Ne2 and Ar2, and aug-cc-pV5Z[66, 84] basis sets were used for Kr2; an additional set of atomic basis mid-bond functions was employed to speed up basis set convergence. The GKS-spRPA substantially reduces the underbinding PBE XC potential was used. error observed in SL-RPA calculations of weakly interacting noble gas dimers and the dimers of S22 dataset [76], see Figs. S1 and S2. spRPA SL total energy change in kH0 − H0 k is

SL spRPA SL SL 2 (2) X | hφiλ |H0 − H0 |φjλ0 i | 3 E = (n − n 0 ) . RPA λ λ SL SL 0 iλ − jλ0 GKS−spRPA ijλλ (52) spRPA SL The exchange portion of H0 − H0 gives rise to the 2 “single excitations” correction to SL-RPA, which were shown to be important for noncovalent bonding by Ren and co-workers [86]. Replacing the SL orbitals and orbital energies in Eq. (52) with the GKS-spRPA ones 1 Error (kcal/mol) obtained in the first iteration corresponds to the “renormalized single excitations” (rSE) correction [32] plus additional correlation-relaxation contributions. While the 0 rSE approach also improves considerably upon SL-RPA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 for noble-gas dimers, it spuriously overbinds in cases such as Ne2 [32], whereas GKS-spRPA remains accurate, re- S22 flecting the additional stability resulting from variational optimization. Similar to orbital optimized RPA [19], FIG. 1. Errors (computed−reference [77]) in binding energies GKS-spRPA thus implicitly accounts for singles correc- for the dimers of S22 dataset. The PBE XC potential and tions to all orders. A comparison of equilibrium proper- aug-cc-pwCV(T,Q)Z[78] basis set extrapolation were used. ties for Ar2 and Kr2 and mean average errors for binding energies of S22 dataset shows that the GKS-spRPA im- While RPA based on PBE [44] orbitals (RPA-PBE) proves upon both SL-RPA and OEP-RPA, see Table.IV. produces hardly any binding for He2 and substantially underbinds Ne2, the nuclear potential energy curves obtained from GKS-spRPA closely resemble those obtained C. Covalent Bonding from the coupled cluster singles doubles with perturbative triples (CCSD(T)) [85] method, which is nearly exact Compared to SL-RPA, GKS-spRPA reduces errors in for these systems. atomization energies marginally but systematically for Perturbative expansion of the GKS-spRPA energy nearly all molecules contained in the G21 [88] and HEAT around the SL ground state provides a rationale for the [89] atomization energy benchmarks. The mean absolute significant accuracy gains upon variational optimization errors of SL-RPA and GKS-spRPA for binding energies of for weak intermolecular interactions: The lowest-order covalently bound molecules are within ∼ 0.1 kcal/mol of

8 Phys. Rev. A, accepted

correlation between low-lying excited 1P states of the TABLE IV. Comparison of SL-RPA, OEP-RPA and GKS- isolated Be atoms [90]. The HF PEC is repulsive, spRPA results for molecular equilibriumproperties (bond second-order Møller-Plesset perturbation (MP2) theory lengths, re, in A˚ and binding energies, De, in kcal/mol), as well as mean absolute errors (MAEs) in binding energies of produces binding energies nearly 3 times too small com- dimers in the S22 benchmark [76]. For SL-RPA and GKS- pared to experiment, while those from local and SL den- spRPA, the exchange-correlation potential was approximated sity functional theory are 3-5 times too large [91, 92]. using the PBE functional. Negative binding energies indicate Some of the SL functionals also display a small unphysical bound states. maximum. RPA-PBE produces too shallow well depth Reference and an unphysical repulsive barrier at intermediate bond- SL OEP a GKS sp Expt. ing distances, see Fig.3, which neither the perturbative rSE correction nor second order screened exchange (SO- Ar 2 SEX) correct entirely [4, 93, 94]. However, the combina- ˚ re (A) 3.84 3.85 3.79 3.775 [81] tion of these corrections, i.e. RPA-PBE+rSE+SOSEX, De (kcal/mol) −0.201 −0.193 −0.247 −0.284 [81] removes this unphysical barrier [4]. OEP-RPA does not Kr2 remove the barrier and produces a positive well-depth, re (A)˚ 4.12 4.13 4.07 4.06 [82] i.e., Be2 is unbound within OEP-RPA, see Fig.3 and Ta- De (kcal/mol) −0.293 −0.299 −0.343 −0.393 [82] bleIV. GKS-spRPA not only removes the unphysical bar-

Be2 rier, but also considerably improves the well-depth, yielding results close to the CCSD(T) ones. As CCSD(T) cal- re (A)˚ 2.40 2.52 2.45 2.45 [87] culations are approximately 2 orders of magnitude more D (kcal/mol) −1.018 +0.498 −1.911 −2.67 [87] e costly than GKS-spRPA calculations for many applications, this is a signiﬁcant result, even though both meth- S22 ods fall short of capturing the strong static correlation at MAE (kcal/mol) 0.83 1.08 0.64 short bond distances. a Refs. [15, 16] 2

TABLE V. Mean absolute (MAE), mean signed (MSE, 1 computed−reference), and maximum absolute (Max AE) errors in kcal/mol for SL-RPA and GKS-spRPA using the G21 AE [88] and HEAT [89] atomization energy benchmarks. RPA 0 and GKS-spRPA correlation energies were obtained using cc- pV(Q-5)Z[66] extrapolation for the G21 AE set, and using −1 aug-cc-pV(Q-5)Z[66, 84] extrapolation for HEAT. Core or- RPA bitals with energies below −3 Hartree were frozen, and the GKS−spRPA −2 OEP−RPA exchange-correlation potential was approximated using the

Binding energy (kcal/mol) CCSD(T)/extrap PBE functional. Expt. −3 Benchmark Error Measure RPA GKS-spRPA 2 3 4 5 6 7 MAE 9.08 8.92 Bond length (Å) G21 AE MSE −9.08 −8.84 Max AE 25.39 25.43 FIG. 3. Computed GKS-spRPA aug-cc-pwCV5Z [95] PECs of MAE 10.09 10.01 Be2 molecule compared to OEP-RPA (using plane-wave basis HEAT MSE −10.09 −10.01 sets with 50 Ry cutoff) [15], experiment [87] and complete Max AE 25.81 24.77 basis-set limit CCSD(T) [96]. each other, see TableV and SM [59]. This result reflects the accuracy of SL orbitals for covalent bonds. Orbital E. Stability of the GKS-spRPA Solution optimization using OEP-RPA, on the other hand, tends to worsen errors in reaction energies [16]. The Coulson–Fischer point [97] (CFP) denotes the crit- ical internuclear separation of a diatomic molecule where an approximate SCF solution spuriously breaks spin sym- D. Beryllium Dimer metry. The position of the CFP for a given method is an important measure of its ability to describe static cor- The potential energy curve (PEC) of beryllium dimer is relation. For H2, for example, the CFP occurs at ∼ a stringent test for approximate electron correlation the- 1.64 times the bond distance for HF, and ∼ 2.14 times ories, because it requires an accurate description of long- the bond-length for most SL DFAs [98], which reflects range dispersion interactions and strong near-degeneracy the increased robustness of SL DFAs for small-gap sys-

9 Phys. Rev. A, accepted tems such as transition metal compounds. The SL-RPA havior can be mitigated to some extent by level shifting method using PBE orbitals inherits the instability from techniques familiar from OSC perturbation theory[102], the PBE reference and displays spin symmetry break- it may be necessary to go consider fully FSC schemes ing at the same internuclear distance, see Fig.4. How- beyond the present sp approximation to further improve ever, the energy of the spin-symmetry broken solution is the stability of RPA. higher than the one of the spin-restricted solution between 3 and 3.8 Bohr, and crosses the spin-restricted curve at a second CFP at 3.8 Bohr. This highly un- V. CONCLUSIONS physical behavior reflects the lack of orbital stationarity of SL-RPA. GKS-spRPA, on the other hand, produces OSC optimization of approximate XC energy function- only one CFP at an internuclear distance of 3.8 Bohr als with an explicit dependence on the KS potential re- which is ∼ 2.71 times the bond-length. Our current requires specification of the KS potential as functional of sults show no visually discernible first-order derivative the density and/or the KS orbitals and occupation num- discontinuity in the GKS-spRPA PEC at the CFP (see bers. However, most choices, including the so-called “ex- also Sec. III of the SM [59]), which is an undesirable act” KS potential, produce an inconsistency between the feature of unregularized orbital-optimized MP2 theory KS potential and the functional derivative of the thus [99]; however, a more detailed investigation is necessary defined energy expression. Equivalently, the KS den- to determine the analytical behavior of the two solutions sity used to evaluate the energy functional differs from at the CFP with certainty. While specialized density the presumably more accurate Hellmann-Feynman den- functionals for strong correlation without any instabil- sity. This paradox appears to have been overlooked pre- ities have been devised [100], the present result under- viously, and casts doubt onto OSC schemes for explicitly lines that GKS-spRPA yields a significant stability gain potential-dependent functionals such as OEP-RPA and compared to existing SL functionals without sacrificing OEP-based many-body perturbation theory. RPA’s broad appeal for diverse molecular and condensed This paradox is resolved by imposing the exact con- matter applications. Further improvement may require straint on the KS potential that it must stationarize a energy functionals beyond RPA. given approximate XC energy functional at fixed KS orbitals and occupation numbers. If this condition uniquely determines the KS potential, this FSC potential is opti- PBE RPA GKS−spRPA mal in the sense that it is consistent with the energy −0.9 UPBE URPA GKS−UspRPA functional it defines, and the resulting KS density equals the Hellmann-Feynman density. The FSC condition is a −1 constraint on the KS potential and as such distinct from the OSC variational principle or OEP methods. If this −1.02 condition gives rise to a unique KS potential, it provides −1.1 −1.04 a more consistent definition of the mapping between the KS potential and the best available approximation to the Energy (a.u.) −1.06 interacting density than the “exact” KS potential. −1.2 −1.08 The GKS-spRPA scheme implements the FSC con- −1.1 3.2 3.6 4 dition approximately, by requiring the GKS potentials defining the energy and obtained from its functional −1.3 1 2 3 4 5 6 7 8 9 10 derivative to have the same ground state density matrix (and, equivalently, GKS determinant). Our results Distance (Bohrs) show that GKS-spRPA overcomes major limitations of SL DFAs and SL-RPA resulting from density-driven er- FIG. 4. PECs of H2 using unrestricted and restricted PBE, ror: Negative ions are bound when they should be, RPA-PBE, and GKS-spRPA approaches. aug-cc-pV5Z [66] and noncovalent interaction energies are significantly im- basis set and PBE potential were used. proved. GKS-spRPA covalent binding energies are only slightly more accurate than the SL-RPA ones, but the Despite the good local stability properties of GKS- improvement is very systematic, as expected for a well- spRPA solutions, it is unclear whether the spRPA en- defined OSC variational approach. Errors previously at- ergy functional is globally well-defined. In rare cases, tributed to missing beyond-RPA correlation, such as the the unoccupied eigenvalues of the semicanonical Hamil- spurious maximum in the Be PEC [4, 94] or inaccurate tonian drop below the occupied ones, and the RPA corre- energy differences [52] vanish upon GKS optimization, lation energy may become ill-defined. In such cases, the suggesting that they may be primarily caused by lacking spRPA energy may not have a minimum as a functional variational stability and functional selfconsistency rather of D. Indeed, in the GFT framework, the RPA energy than inherent errors of the RPA method. as a functional of G0 generated by nonlocal potentials is The GKS-spRPA orbital energies match experimental unbounded from below [101]. While this undesirable be- ionization potentials and electron affinities of atoms and

10 Phys. Rev. A, accepted molecules within a few tenths of an eV, surpassing the acknowledged. This material is based upon work sup- popular G0W0 method in accuracy. Analysis of the non- ported by the National Science Foundation under CHE- local GKS-spRPA correlation potential supports these 1464828 and CHE-1800431. observations, showing that quasiparticle GW energies are a first-order perturbative limit of canonical GKS-spRPA orbital energies. Unlike KS band gaps, the GKS-spRPA Appendix A: Functional Derivatives for Degenerate band gaps contain the energy derivative discontinuity at KS Density Matrices integer particle numbers, and thus accurately approximate the observable fundamental gap; this is of particular The spRPA energy functional depends implicitly on interest for infinite systems such as periodic solids, where the KS density matrix through the occupation numbers IPs cannot be obtained from total energy differences. nλ and the spectral projectors Pλ. However, the com- The GKS-spRPA GKS solution is considerably more putation of functional derivatives with respect to D us- resilient to symmetry breaking than OSC SL KS solu- ing the chain rule is not straightforward, because the Pλ tions, as suggested by the location of the Coulson–Fischer are invariant under unitary transformations of orbitals point for the H2 ground state. Moreover, GKS-spRPA with degenerate occupation numbers; as a result, the eliminates the spurious second Coulson–Fischer point ob- partial derivatives δD/δPλ can become singular, causing served in non-OSC SL-RPA approaches. In conjunction unphysical singularities in the functional derivative for with the above results, these improved stability charac- degenerate D. Since noninteracting Fermion density ma- teristics provide additional evidence for the viability and trices at zero temperature are usually highly degenerate, robustness of the GKS-spRPA energy functional. In par- the possibility of degeneracy must be explicitly accounted ticular, the high stability of the GKS-spRPA solution for. bodes well for applications to response properties, whose According to Eq. (19), D is a direct sum of degenerate accuracy can sensitively depend on the stability of the blocks reference state. From a computational viewpoint, the cost of a GKS- nλλ0 = δλλ0 nλ, (A1) spRPA is on the order of that of a SL-RPA calculation where nλ are dλ × dλ Hermitian matrices [55], and dλ times the number of GKS iterations; thus, GKS-spRPA is the eigenvalue multiplicity. For a given D, the nλ are calculations for systems with hundreds of atoms are multiples of the dλ × dλ identity, i.e., nλ = nλ1, where within reach. KS-spRPA is considerably less costly than nλ is a dλ-fold degenerate occupation number. However, OEP-RPA, because it does not require ad-hoc regular- certain variations of D associated with changes in elec- ization or special basis sets, and it is relatively straight- tron numbers can lift the degeneracy. To account for forward to implement starting from RPA analytical gra- such variations, we do not require nλ to be diagonal or dients. While GKS-spRPA does not achieve complete degenerate in general. functional selfconsistency, its considerably improved per- The remaining degrees of freedom correspond to uni- formance for energy differences and density-related prop- tary transformations between orbitals with different oc- erties compared to both SL RPA and OEP-RPA suggests cupation numbers. A convenient orbital parameteriza- that semicanonical projection provides a simple yet relation is tively accurate approximation to the FSC RPA potential. κ The GKS scheme with semicanonical projection pre- |φiλ(κ)i = e |φiλi , (A2) sented here may be applied beyond RPA to turn any conserving [103–105] energy functional of the KS one-particle where Green’s function into a density functional, and provides X κ = κ 0 , (A3) a solution to the conundrum of how to obtain accurate λλ λλ0 energy differences and choose the non-interacting system, e.g., in GW theory [106]: All observables, such as self- and unitarity implies that the dλ ×dλ0 blocks κλλ0 satisfy energies or response properties, are obtained as deriva- † tives of a single, variationally stable energy functional κλλ0 = −κλ0λ. (A4) of the KS density matrix, combining and enhancing the As a result, κ = 0, i.e., transformations between de- accuracy of RPA for energetics with the one of GW for λλ generate orbitals are excluded. The spectral projectors quasiparticle spectra. Thus, GKS-spRPA is a step to- X wards accurate, efficient, and universal electronic struc- P = |φ i hφ | (A5) ture methods sharing favorable characteristics of both λ iλ jλ i,j=1 DFT and GFT. transform as P (κ) = eκP e−κ. (A6) ACKNOWLEDGMENTS λ λ The total number of independent variables contained in Discussions with Kieron Burke and Georg Kresse are n and κ equals that of D.

11 Phys. Rev. A, accepted

Consider a variation of D around a given D(0) corre- The chain rule ﬁnally yields (0) sponding to nλ = nλ = nλ1, κ = 0. The rank four partial derivative tensor ( ∂ 1 ∂ , λ 6= λ0, = nλ0 −nλ ∂κλλ0 (A11) ∂Dλiλ0j ∂ 0 0 ∂Dλλ0 , λ = λ , = δλλ δλµδikδjl (A7) ∂nλ ∂nµkl D(0) is invertible for each λ = λ0 = µ diagonal block. Simi- where the partial derivatives have been evaluated at D = larly, D(0).

∂Dλiλ0j 0 = δλµδλ0µ0 δikδjl(nλ0 − nλ), λ 6= λ (A8) ∂κµkµ0l D(0)

0 is invertible for each λ 6= λ, because nλ0 −nλ 6= 0 by construction. Adopting a short-hand matrix notation which Appendix B: Density Matrix Derivative of the sp implies the subindices of each λ block, Eq. (A7) may be Green’s Function written

∂Dλλ0 = 1λµ ⊗ 1µλ0 , (A9) Eq. (26) expresses G0(ω) as an explicit functional of ∂n µ D(0) the occupation number matrices nλ and the spectral projectors Pλ, as well as the sp Hamiltonian H˜ 0. Parame- and Eq. (A8) becomes terizing the orbitals according to Eq. (A2) and using Eq. (A11), the partial density matrix derivative of G0(ω) for ∂Dλλ0 0 ˜ (0) = (nλ0 − nλ) 1λµ ⊗ 1µ0λ0 , λ 6= λ . (A10) H0, evaluated at D = D , is straightforward, ∂κµµ0 D(0)

 (1) 1µλ⊗G0λ0µ0 (ω)−G0µλ(ω)⊗1λ0µ0 0 0 , λ 6= λ , ∂G0µµ (ω)  n 0 −n = λ λ (B1) 0 ˜ ˜ 0 ∂Dλλ  πi 1µλ ⊗ [δ(ω − H0λ)]λ0µ0 + 1λ0µ0 ⊗ [δ(ω − H0λ)]µλ , λ = λ .

Here, the identity we obtain two partial derivatives of the sp Hamiltonian blocks at D = D0, 1/2 ∂[n ]kl 1 (2) SL SL λ ! ( 1 ⊗H −H ⊗1 0 = δikδjl (B2) ˜ µλ 0λ0µ 0µλ λ µ 0 1/2 ∂H0µµ[D] , λ 6= λ , ∂nλij n −n 0 n(0) 2nλ = λ λ λ ∂Dλλ0 0, λ = λ0, was employed to evaluate the occupation number matrix (C1) derivatives. and (3) ˜ ! SL ∂H0µµ[D] ∂H0µµ[D] HXC = = Fµµλλ0 . (C2) ∂Dλλ0 δDλλ0 Appendix C: Density Matrix Derivative Arising HXC ˜ Here, F denotes the rank-four tensor operator repre- from Changes in H0 senting the SL Hartree-, exchange, and correlation kernel. In keeping with the main text, superscript (2) denotes for According to Eq. (20), the sp Hamiltonian depends on partial derivatives arising from the semicanonical projec- D through variations of the orbitals as well as through the tion, and superscript (3) denotes partial derivatives aris- SL SL SL Hamiltonian H0 [D]. Using the techniques of Sec.A, ing from changes of the SL Hamiltonian H0 [D].

By Eq. (26), the total derivative of the sp Green’s function with respect to the sp Hamiltonian is

∂G0µµ0 (ω) 1/2 ˜ + −1 ˜ + −1 1/2 = δµµ0 nµ (ω − H0λ − i0 )µλ ⊗ (ω − H0λ − i0 )λµ0 nµ0 ∂H˜ 0λλ 1/2 ˜ + −1 ˜ + −1 1/2 +(1 − n)µ (ω − H0λ + i0 )µλ ⊗ (ω − H0λ + i0 )λµ0 (1 − n)µ0 . (C3)

12 Phys. Rev. A, accepted

Combining Eqs. (C1)-(C3) by the chain rule, the remaining partial derivatives of G0 are

 ˜ + −1 SL ˜ + −1 nλ (ω − H0 − i0 )µλ ⊗ [H0 (ω − H0 − i0 ) ]λ0µ0  nλ−nλ0  ˜ + −1 SL ˜ + −1 1−nλ  +(ω − H0 + i0 ) ⊗ [H0 (ω − H0 + i0 ) ]λ0µ0 (2)  µλ nλ−nλ0 ∂G0µµ0  ˜ + −1 SL ˜ + −1 nλ0 = δµµ0 −[(ω − H − i0 ) H ] ⊗ (ω − H − i0 ) 0 0 (C4) 0 0 µλ 0 λ µ n −n 0 ∂Dλλ0 λ λ  ˜ + −1 SL ˜ + −1 1−nλ0 0  +[(ω − H + i0 ) H ] ⊗ (ω − H + i0 ) 0 0 , λ 6= λ ,  0 µλ 0 λ µ n −n 0  λ λ  0, λ = λ0.

The HXC kernel part is

(3) ∂G0kµlµ0 h ˜ + −1 HXC ˜ + −1 = δµµ0 (ω − H0 − i0 ) Fλλ0 (ω − H0 − i0 ) nµ ∂Diλjλ0 ˜ + −1 HXC ˜ + −1 i +(ω − H0 + i0 ) Fλλ0 (ω − H0µ + i0 ) (1 − nµ) , (C5) µµ0 where matrix multiplication is implied for the first two indices of FHXC. The chain rule also affords an explicit expression for the unrelaxed RPA difference density matrix,

Z ∞ dω h + −1 C + −1 Tλλ0 = δλλ0 (ω − H˜ 0 − i0 ) Σ (ω)(ω − H˜ 0 − i0 ) nλ −∞ 2π + −1 C + −1 i +(ω − H˜ 0 + i0 ) Σ (ω)(ω − H˜ 0 + i0 ) (1 − nλ) , (C6) λλ0 which follows from Eqs. (50), (40), and (C3).

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