Exchange-Correlation Energy from Green's Functions

This is the accepted manuscript made available via CHORUS. The article has been published as: Exchange-Correlation Energy from Green’s Functions Steven Crisostomo, E. K. U. Gross, and Kieron Burke Phys. Rev. Lett. 133, 086401 — Published 21 August 2024 DOI: 10.1103/PhysRevLett.133.086401 Exchange-correlation energy from Green’s functions Steven Crisostomo,1E.K.U. Gross,2and Kieron Burke1, 3 1Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA∗ 2Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel 3Department of Chemistry, University of California, Irvine, CA 92697, USA DFT calculations yield useful ground-state energies and densities, while Green’s function techniques (such as GW) are mostly used to produce spectral functions. From the Galitskii-Migdal formula, we extract the exchange-correlation of DFT directly from a Green’s function. This spectral representation provides an alternative to the fluctuation-dissipation theorem of DFT, identifying distinct single-particle and many-particle contributions. Results are illustrated on the uniform electron gas and the two-site Hubbard model. Ground-state density functional theory [1,2] (DFT) is used to great effect in modern molecular and materials calculations, limited only by the approximation for the exchange-correlation (XC) energy [3]. But there is also interest in the response properties of a system, such as its spectral function and associated gap [4]. While range-separated hybrids can yield moderately accurate gaps within the generalized KS scheme [5–7], the standard method for gaps remains GW calculations [8–10]. For strongly-correlated materials, dynamical mean field theory has become a very useful tool [11–15] Such Green’s function (GF) methods come from traditional many-body theory. KS-DFT does not, and it is difficult to relate the two [16–19]. In practical calculations, DFT is used to find some set of KS orbitals, which form the starting point of GF calculations [20]. The choice of initial orbitals can be important [21–23], while a self-consistent GF can eliminate this dependence [24–29] . The XC energy of DFT is traditionally analyzed in terms of static quantities extractable from the ground-state wave-function, such as the XC hole, the dearth of conditional electronic density around an electron [30–32]. If a frequency decomposition is performed, it is in terms of the density-density response function [33]. We provide a bridge between these two distinct approaches by extracting the contribution to the XC energy directly from an interacting GF. Our GXC formula can be used to (a) extract approximate XC energies from approximate GF, (b) provide a novel decomposition of XC energies into single-particle and many-particle contributions, (c) provide models for existing DFT approximations in terms of GF, (d) relate the quality of spectral functions to their performance for XC energies. We illustrate our results on the uniform electron gas and the two-site Hubbard model. Consider Hamiltonians of the form (in Hartree atomic units) ˆ H= N X j ˆ h(ri) + N X i>j ˆ Vee(|ri−rj|),(1) where ˆ h(r) = ˆ t(r) + v(r)is the single-particle Hamiltonian, with kinetic energy operator ˆ t(r) = −∇2/2, multiplicative spin-independent external potential v(r), and ˆ Vee(u)=1/u is the Coulomb repulsion between pairs of electrons. We work within the Born-Oppenheimer approximation in the non-relativistic limit, with no external magnetic fields. We use x= (r, σ)as a space-spin coordinate, and n(x)is the ground-state spin density. Within spin-DFT, the corresponding KS system consists of fictitious non-interacting electrons with the same groundstate density as the interacting system. The exact groundstate energy is then E=TS+V+EHXC ,(2) where TSis the kinetic energy of non-interacting electrons, Vis the external potential energy, and EHXC is the sum of the Hartree (electrostatic) and XC energies. The KS potential is vS(x) = v(r) + vHXC(x),(3) where vHXC (x) = δEHXC [n]/δn(x), and is generally spindependent. The KS equations are solved self-consistently, yielding the exact density and energy given the exact XC [2]; this was demonstrated with exact calculations in Ref. [34]. The time-ordered Green’s function is G(xt, x′t′) = −iTˆ ψ(xt)ˆ ψ†(x′t′),(4) where Tis the time-ordering operator and ⟨·⟩ denotes the expectation value over the N-electron ground-state |Ψ0⟩. The fermionic operators, ˆ ψ†(x)and ˆ ψ(x), create and destroy a particle of spin σat rand are time-evolved in the Heisenberg picture according to ˆ H. For time-independent ˆ H, we denote the Fourier transform G(x,x′, ω). We define Tr{F}=−iZdω 2πtr{F},(5) where a convergence factor exp(iωδ)with δ→0+is implied, to respect time-ordering (t′→t+). Here tr denotes tr{F}=Zd3rX σ F(x,x′=x, ω).(6) 1 Exchange-correlation energy from Green’s functions Changed definition in Eq. 6. The Galitskii-Migdal (GM) formula [35] yields the total interacting ground-state energy from the GF E=1 2Tr{(ω+ˆ h(r))G}.(7) To isolate XC, apply GM to the KS system, ES=1 2Tr{(ω+ˆ hS(x))GS},(8) where ˆ hS(x) = ˆ h(r) + vHXC (x)and GS(x,x′)are the KS single-particle Hamiltonian and Green’s function. Subtraction yields GXC, the GF XC contribution GXC =EXC −⟨vXC ⟩ 2=1 2Tr ω+ˆ t(r)∆G,(9) where ∆G=G−GS. More explicitly GXC =−i 2Zd3rX σZ∞ −∞ dω 2π ×ω−∇2 u 2∆Gσ(r,r+u, ω)u=0 ,(10) where ∆Gσ(r,r+u, ω)is an analog of the XC hole. Like the XC hole, ∆Ghas analogous sum rules. Without the ω term, Eq. (10) yields TC/2=(T−TS)/2, half the correlation kinetic energy. Moreover −iZ∞ −∞ dω 2π∆G(x,x, ω)=0,(11) because both Green’s functions have the same density. An early version appeared [36] even before the adiabatic connection formula of DFT was precisely defined [33,37]. Godby and Garc´ıa-Gonz´alez have also derived closely related expressions [18]. First, Eq. (9) provides a method for extracting an XC contribution directly from any GF. At the end of any GF calculation, the density of Gcan be extracted, a KS inversion [38–42] performed and the corresponding GS constructed. But GXC is not EXC. Given an explicit approximation for EXC, it is easy to construct GXC, but not vice versa. Thus, for a non-self-consistent GW calculation (the vast majority), a measure of inconsistency would be GXC of the original DFT calculation versus that of Eq. (9). A self-consistent calculation would presumably satisfy Eq. (9) on its final iteration. An important feature of GXC is that it can be used to construct the total energy from the sum of KS orbital energies ES: E=ES+GXC −⟨vHXC⟩ 2.(12) Since ESand ⟨vHXC ⟩are given by a KS calculation, GXC corrects the KS energy into the true many-body energy. This is analogous to (but not the same as) double-counting corrections in the literature [43,44] . Now we use Eq. (9) to create a novel decomposition of XC energies in DFT. For finite systems, we define the energy difference ωJ=E0(N)−EJ(N−1),(13) where EJ(M)is the energy of the Jth interacting eigenstate of the M-electron system; J= 0 denotes the ground-state. Then the Lehmann representation is G(x,x′, ω) = X J ρJ(x,x′) ω−ωJ−iδ +X J′ ¯ρJ′(x,x′) ω−¯ωJ′+iδ ,(14) where the spectral weights are ρJ(x,x′) = ⟨Ψ0|ˆ ψ†(x′)|J⟩⟨J|ˆ ψ(x)|Ψ0⟩.(15) Here {|J⟩} are the interacting eigenstates of the N-1electron system, while ¯ρJ′(x,x′)is defined analogously, with fermionic operators in Eq. (15) swapped, and {|J′⟩} enumerating eigenstates of the N+1-electron system. The sum over Jof ρJ(x,x′)is simply the first-order density matrix. Likewise, we write GS(x,x′, ω) = X j ρs,j(x,x′) ω−ϵj−iδ +X j′ ¯ρs,j′(x,x′) ω−¯ϵj′+iδ ,(16) where jruns over KS orbitals and we consider only the occupied orbitals. Inserting into Eq. (9) yields GXC =1 2X J (TJ+ωJfJ)−X j (Ts,j +ϵjfs,j),(17) TJare the kinetic contributions from Jenumerating the interacting N-1-electron eigenstates and fJ= tr{ρJ}are the ’occupations’, with analogous contributions from the KS system. They satisfy the sum rule that each, when summed over all occupied states, yields N. To make further progress, we relate the two sums in Eq. (17) via the adiabatic connection of DFT. Multiply Vee in Eq. (1) by λwhile choosing a λ-dependent one-body potential to keep the density fixed [45]. As λ→0, a subset of Japproach KS Slater determinants, while all other eigenstates vanish. We call the non-vanishing eigenstates the single-particle contributions (SP), the rest many-particle (MP). This is analogous to how we define single-particle excitations in TDDFT [46]. Thus GXC =GSP XC +GMP C,(18) where the SP contribution is GSP XC =1 2X j (TC,j +ωjfj−ϵjfs,j),(19) and TC,j =Tj−Ts,j = tr{ˆ t(r)(ρj−ρs,j)}is the contribution to the correlation kinetic energy from each SP state. The 2 Exchange-correlation energy from Green’s functions many-particle contributions, enumerated by K, are purely correlation GMP C=1 2X K (TK+ωKfK).(20) In the SP contribution, the highest occupied KS level is special, as KS-DFT guarantees that the HOMO KS eigenvalue is exactly minus the ionization potential for finite systems, i.e., ω0=−I=ϵ0. Labelling it as zero, we write GSP XC,0=1 2(TC,0−I(f0−fs,0)).(21) The exchange contribution is found by using the exchange self-energy, GX=X j tr{(ΣX[GS]−vX)ρs,j}=GSP X,(22) with a detailed description in the Supplemental Materials (SM). For a two-electron singlet GXvanishes. The ground-state density matrix is ρ(x,x′) = Zµ −∞ dω A(x,x′, ω),(23) where µis the chemical potential and the spectral function is A(x,x′, ω) = −1 πIm{G(x,x′, ω)}sgn(ω−µ).(24) Thus ρ(and its diagonal, the ground-state density) can be uniquely decomposed into its spectral contributions. Decomposing ⟨vXC⟩via the density, and using Eq. (9), EXC =ESP XC +EMP C,(25) i.e. individual peaks in the spectral function yield individual contributions to the EXC energy. An alternative decomposition uses ωj=−I−∆Ej,where ∆Ej=Ej(N−1) −E0(N−1) ≥0are the transition frequencies. Then GXC =GSPI XC +GMPI C,(26) with different SP and MP contributions GSPI XC =1 2X j [Tc,j −(∆Ejfj−∆Es,jfs,j)] ,(27) GMPI C=1 2X K [TK−∆EKfK].(28) In particular, GSPI xc,0=Tc,0/2. Each decomposition is useful in different circumstances. The asymmetric two-site Hubbard model [47,48] is particularly well-suited as an illustration, because of its extremely truncated Hilbert space. We have two fermions in the Hamiltonian, −tX σˆc† 1σˆc2σ+h.c.+U 2 X j=1 ˆnj↑ˆnj↓+ 2 X j=1 vjˆnj,(29) where tis the hopping term, Uthe on-site interaction strength, ˆnjthe site-occupation operator, and ˆc† jσ,ˆcjσ are the fermionic operators associated to each site. The groundstate is a singlet, and its density is characterized by one number, ∆n=n2−n1, and we define ∆v=v2−v1. KS-DFT applies [47] and the KS system is simply the tightbinding dimer. 3210123 0.0 0.2 0.4 0.6 0.8 1.0 1.2 U = 2 v = 1 n 1= 1.147 exact KS FIG. 1: Hubbard dimer local spectral functions A11(ω) (black) and As,11(ω) (red). Here µ=U/2 and t= 1/2. The spectral weight below 0 sums to n1= 1.147, and above to n2. Figure 1shows a typical spectral function [47] with bars whose height is the pole weight. Ghas 4 poles, two for removal and two for gain, but GShas only one of each. The difference between the two central peaks is the gap, with the KS gap famously being smaller than the true gap [49, 50]. Only the removal peaks contribute to XC. The larger exact removal peak is the only single-particle contribution, and the smaller is a many-particle. By Eq. (23), they sum to n1, and the KS peak must be higher. Because GCis negative and its SP contribution is positive, the behavior of GCrequires understanding both SP and MP contributions simultaneously. Moreover, for large U, both SP and MP contributions grow linearly with U, just as ECdoes, but such contributions cancel in GC(and the corresponding SPI/MPI terms also remain constant). The first use of our formula is to identify how much each peak contributes to XC. Table Ishows values of various quantities for a range of U.GXC is comparable to ECfor weak correlation (as GX= 0 for N= 2), but 3 Exchange-correlation energy from Green’s functions U EXC ECGCGSP CGMP CGSPI CGMPI C 0.5 -0.339 -0.01062 -0.013 0.011 -0.024 0.00338 -0.0164 1 -0.643 -0.0676 -0.0524 0.0516 -0.104 0.0189 -0.0713 2 -1.39 -0.3666 -0.139 0.224 -0.363 0.0847 -0.224 4 -3.23 -1.224 -0.194 0.689 -0.883 0.188 -0.381 10 -9.1 -4.098 -0.206 2.16 -2.36 0.27 -0.476 20 -19. -9.05 -0.207 4.64 -4.85 0.297 -0.504 TABLE I: XC of dimer with t= 1/2 and ∆v= 1. significantly smaller for strong correlation, with cancellation between SP and MP contributions (see SM). The alternative decomposition yields terms of much smaller magnitude, especially for large U. Z EXC ⟨vXC⟩/2ECTCGXC 1 -0.4229 -0.3562 -0.04199 0.02788 -0.06667 2 -1.067 -1.01 -0.04211 0.03664 -0.05691 3 -1.695 -1.638 -0.04352 0.03983 -0.05627 4 -2.321 -2.265 -0.04427 0.04148 -0.05604 6 -3.572 -3.516 -0.04506 0.04318 -0.05584 10 -6.073 -6.017 -0.04569 0.04456 -0.0557 20 -12.32 -12.27 -0.04618 0.0456 -0.0556 TABLE II: XC for the two-electron ions [51]. For realistic DFT calculations, as the local density approximation yields ELDA X[n] = −AXRn4/3(r), typically GXC ≈EXC/3, especially for large N. For N= 2, we cannot assume Hubbard results are typical, but (essentially) exact DFT calculations have been performed for twoelectron ions (Table II) and Hooke’s atom (quadratic confining potential in the SM), where GCis comparable to ECin magnitude. The LDA can be understood as a local approximation to the XC hole [52,53], as the systemand spherically averaged LDA XC hole reflects the accuracy of LDA [54]. The real-space construction of the GGA from the gradient expansion for the hole underlies both the PW91 and PBE XC approximations [31,55]. XC holes are also behind some of the most popular functionals in chemistry [30,32,55]. Now we derive GLDA XC from an ansatz for the Green’s function. Define ∆GUEG(n, u, ω)as the difference between exact and KS Green’s functions of a uniform gas of density n, separation u=|r−r′|, and frequency ω. Approximating ∆Gwith ∆GUEG(n(r),|r−r′|, ω)in Eq. (9) directly yields GLDA XC =Zd3r gUEG XC (n(r)),(30) where gUEG XC (n) = (2 −d/dn)eUEG XC (n)/2is the GXC energy density, Tr{(ω+ˆ t(u))∆GUEG}/V , and Vis the volume. System-averaged and frequency-integrated quantities should agree to the extent that LDA yields reasonably accurate energies, but are there major cancellations of errors inside the frequency integral? And do approximate GF calculations improve this frequency dependence? These are the sorts of questions that can be explored with our formula. Our final point concerns the approximate GF calculations that our formula is designed to analyze, illustrated for GW calculations on the Hubbard dimer. We use the Hartree-Fock GF to generate the initial GW self-energy and iterate until the total energy converges. Each iteration generates extra poles, but we retain only a few (see SM). If the self-energy has (correctly) only two poles, the next GGW generally has six, instead of the correct four. But if we reduce the 6 poles of Gto the correct 4, the selfenergy has 4. We chose the former scheme with two-pole approximation to the self-energy, denoted by ”2pscGW”. To aid convergence, for cases with moderate to strong correlation we generated the nth GF according to: G(n)= γG(n−1) + (1 −γ)G(n−2), with γ= 0.67. Our one-shot results are identical to Ref [56] when ∆v= 0. Both the six-pole and four-pole GW GF yield the same density and vXC, but they differ considerably. Both satisfy the Sham-Schl¨uter equation at each iteration [50], −iZdω 2πGSΣGii =−iZdω 2πGSvHXC Gii,(31) but have different kinetic and XC energies. For the dimer GXC =Z0 −∞ dω [ω(∆A11(ω)+∆A22(ω)) −2t∆A12(ω)] , (32) where we have summed over spins and ∆Aij is the difference in retarded spectral functions. Due to Eq. (11), the frequency integral of ∆A11(ω) + ∆A22(ω)up to the chemical potentials yields 0. Fig. 2shows ∆A11, which comprises a majority of GXC, as ∆A22 gives a negligible contribution. From Fig. 2, the SP pole positions and heights for oneshot GW are more accurate than 2pscGW. Table III also shows a poorer approximation to EC, worsening with increasing U. However, full self-consistency produces better total energies for weak correlation, while in the stronglycorrelated case, neither flavor of GW produces accurate energies (see SM). Practical GF calculations are performed on solids and molecules, with a variety of approximations, such as GW and dynamical mean field theory [19,57–60]. Once a KS inversion can be performed [40–42] on the density of the approximate GF, a value for GXC can be extracted which can be compared with a DFT counterpart, especially when standard XC functionals are known to fail. However, molecules have both bound and continuum states, while all states are in continuua in solids, complicating the identification of SP and MP contributions. For GW , there are many different recipes yielding distinct spectral features [22,25,61,62], but one must always use the KS GF of 4 Exchange-correlation energy from Green’s functions 0.4 0.2 0.0 0.2 0.4 U = 2 v = 1 nMP 1 nMP 1 2 E 0(1) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.4 0.2 0.0 0.2 0.4 U = 1 v = 1 nMP 1 nMP 1 2 E 0(1) FIG. 2: Spectral function differences ∆A11(ω) with µ=U/2 and t= 1/2. Exact (black), one-shot (orange), and 2pscGW (green). The separation of the SP and MP poles is twice the energy of the one-particle groundstate, but GW introduces an erroneous interaction dependence. Exact one-shot 2pscGW U GCGSPI CGMPI CGCGSPI CGMPI CGCGSPI CGMPI C 0.25 -3.05 0.67 -3.72 -4.88 1.12 -6. -4.95 1.15 -6.1 0.5 -13. 3.38 -16.4 -15.9 4.4 -20.3 -16.9 4.69 -21.6 1 -52.4 18.9 -71.3 -41.9 14.4 -56.3 -54.8 16.2 -71. 2 -139. 84.7 -224. -98.1 33.8 -132. -186. 42. -228. 4 -194. 188. -381. -232. 57.5 -289. -583. 88. -671. Exact one-shot 2pscGW U ECESPI CEMPI CECESPI CEMPI CECESPI CEMPI C 0.25 -1.95 1.76 -3.71 -3.09 2.88 -5.97 -3.83 2.25 -6.08 0.5 -10.6 5.62 -16.2 -13.8 6.29 -20.1 -19.3 2.18 -21.5 1 -67.6 1.43 -69. -68.1 -13.3 -54.9 -87.9 -16.5 -71.4 2 -367. -142. -225. -288. -153. -135. -292. -56.3 -236. 4 -1220. -627. -597. -798. -463. -335. -761. -60. -701. 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