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Ensemble Generalized Kohn-Sham Theory: the Good, the Bad, and the Ugly

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Ensemble Generalized Kohn-Sham Theory: the Good, the Bad, and the Ugly Ensemble generalized Kohn-Sham theory: the good, the bad, and the ugly Tim Gould1 and Leeor Kronik2 1)Qld Micro- and Nanotechnology Centre, Griffith University, Nathan, Qld 4111, Australia 2)Department of Materials and Interfaces, Weizmann Institute of Science, Rehovoth 76100, Israel Two important extensions of Kohn-Sham (KS) theory are generalized KS theory and ensemble KS theory. The former allows for non-multiplicative potential operators and greatly facilitates practical calculations with advanced, orbital-dependent functionals. The latter allows for quantum ensembles and enables the treatment of, e.g., open systems and excited states. Here, we combine the two extensions, both formally and practically, first via an exact yet complicated formalism, then via a computationally tractable variant that involves a controlled approximation of ensemble \ghost interactions" by means of an iterative algorithm. The resulting formalism is illustrated using selected examples. This opens the door to the application of generalized KS theory in more challenging quantum scenarios and to the improvement of ensemble theories for the purpose of practical and accurate calculations. I. INTRODUCTION tures of GKS and EKS theories, though some ad hoc solutions39,41 and approximations.54,59 have been re- Kohn-Sham1,2 (KS) density functional theory (DFT) ported. This hampers the application of ensemble theo- has proven to be an indispensable tool for first principles ries, which must be solved non-self-consistently, or with calculations across a wide range of disciplines.3,4 Many less-sophisticated approximations, or using OEPs. widely used density functional approximations (DFAs), Here, we demonstrate how to combine EKS and GKS e.g. Refs. 5{10, are \hybrid functionals", i.e., they mix theories into an ensemble GKS (EGKS) theory. We show, exact (Fock) exchange with KS exchange. These are al- however, that the resulting approach is ill-suited to exist- most always applied using the non-multiplicative Fock- ing GKS implementations in its direct form. We then in- exchange operator, which places them outside the realm troduce simple, formally-motivated approximations that of KS theory, but well within generalized Kohn-Sham make EGKS tractable without losing its good features. (GKS) theory.11{14 The popularity of hybrid function- als reflcts three useful properties of hybrids: first, the inclusion of exact Fermionic exchange allows for a sys- tematically higher predictive accuracy;15 second, this is II. BACKGROUND THEORY done in a numerically straightforward fashion, whereas use of Fock exchange within \pure" KS theory must A. Ensemble Kohn-Sham theory rely on difficult to calculate optimized effective poten- tials (OEPs);16{21 third, the non-multplicative potential allows for a reduction and in some circumstances even the We first briefly introduce EKS theory before turning elimination of the KS derivative discontinuity,22,allowing to GKS and EGKS theory. For simplicity, we use a spin- one to overcome notoriously difficult problems for KS unpolarized formalism throughout. To begin, we intro- theory such as the bandgap problem23,24 or the charge duce the universal \Levy" ensemble density functional, transfer excitation problem.25{27 λ ^ ^ ^ Recent years have seen renewed interest in ensemble F [n; w] = min Tr[(T + λW )Γw] ; (1) Γ^ !n KS (EKS) theory, which broadens the scope of traditional w KS theory to include mapping to an ensemble of refer- ence non-interacting electron states, and thereby extends where λ 2 [0; 1] is an adjustable interacting-strength pa- the reach of DFT to a more diverse range of quantum rameter. Here, n is the density, w is a set of statis- tical weights that define the ensemble, T^ is the kinetic scenarios, e.g., systems with non-integer electron num- ^ ber22,28{34 and mixtures of ground- and low-lying excited energy operator and W is the electron-electron inter- 35{37 action energy operator. The ensemble density matrix, states. It can also improve the treatment of quantum ^ P systems that are technically within the reach of KS the- Γw = κ wκjκihκj, involves a weighted sum over or- ^ ory, such as those with significant spin-contamination.38 thonormal wave functions jκi. Γw ! n is short-hand for 28,33,35{59 Excited-state EDFT, In particular, it has at- Tr[^n(r)Γ^w] = n(r), wheren ^(r) is the electron-density tracted significant recent attention as a route to low cost operator. By varying the weights, w, we can obtain use- prediction of charge transfer40,56 and double40,50,53 exci- ful properties (e.g. excitation energies) of the system that tations, which are hard to predict using traditional time- are inaccessible to ground-state DFT. dependent KS theory. The Levy functional F λ provides a versatile framework To date, we are not aware of any formal theory that for defining key functionals in EKS52,55,57, specifically has been able to combine rigorously the attractive fea- the KS kinetic energy (Ts), Hartree-exchange (Hx), and 2 correlation (c), functionals, respectively: where v is the external potential and vR is a multiplica- tive effective potential, can then be used to find orbitals 0 Ts[n; w] =F [n; w] ; (2) f'g and density n. We obtain different operatorsv ^S and λ F [n; w] − Ts[n; w] potentials vR for different orbital functionals S[f'g] that EHx[n; w] = lim ; (3) are invariant to unitary transformations of the orbitals. λ!0+ λ 1 We use f'g to indicate orbital solutions of Eq. (8), as Ec[n; w] =F [n; w] − Ts[n; w] − EHx[n; w] : (4) opposed to fφg of Eq. (7). Each is a functional of the electron density n and weights The existence of the GKS equation is deduced by defin- 1 w, denoted by [n; w]. ing: S[n] = minf'g!n S[f'g], and RS[n] = F [n] − S[n], where we use non-calligraphic F 1 to represent the special It has recently been shown that Ts[n; w] ≡ Tr[Γ^s;wT^] 55 case of Eq. (1) for pure ground states (w0 = 1, wκ6=0 = 0). and EHx[n; w] ≡ Tr[Γ^s;wW^ ], for some non-interacting Then, the ground-state energy, E0 = minnfS[n]+RS[n]+ ensemble density matrix Γ^s;w, which is formed on con- V [n]g = minf'gfS[f'g] + Rs[n'] + V [n']g, is minimized figuration state functions. Ts and EHx are thereby func- for the density n ≡ n' ≡ n0. Uniqueness (in non- tionals of a set of orbitals, fφg: degenerate systems) follows from the external potential Z v being uniquely defined by the ground state density n0. X w ∗ ^ X w R 1 Ts[fφg] = fi drφi (r)tφi(r) ≡ fi ti,φ ; (5) Here V [n] = drn(r)v(r) and we used S + RS = F . i i Importantly, almost all practical hybrid approxima- 1 X J;w W K;w W tions can be re-expressed as functionals of the first- EHx[fφg] = F (ijjij)φ + F (ijjji)φ ; (6) 2 ij ij 0 ij order reduced density matrix (1-RDM), γ1(r; r ) = P ∗ 0 iσ2occ 'iσ(r)'iσ(r ). This includes all Hartree-Fock- W R 0 involving electron-repulsion integrals (ijjkl)φ = drdr like expressions, S[f'g] := hΦjT^ + W^ SjΦi, where Φ is 0 ∗ 0 ∗ 0 1 60 ^ W (r; r )φi (r)φj(r )φk(r)φl (r ) with W = jr−r0j . The a Slater determinant formed from f'g, and WS is a w coefficients fi 2 [0; 2] are average occupancies of orbitals (modified) interaction term involving interactions of form J;w 0 0 0 < WS(jr − r j) ≤ W (jr − r j), which can be a simple i across the ensemble. The pair-coefficients, Fij and fraction W = αW of the full interaction term, or a more F K;w, encode all information about the non-interacting S ij complex range-separated 8,61 expression.12,62 Then, ensemble and are ensemble-specific. They are more com- plicated to obtain { some examples are provided in ap- Z 1 2 0 pendix A. Importantly, these pair-coefficients are not 0 S[γ1] ≡ − 2 r γ1(r; r )jr=r dr necessarily (scaled) products of f w, i.e., generally F J 6= i ij Z K 0 0 1 0 2 drdr0 −2Fij 6= fjfj. Recent work has used the fluctuation- + WS(r; r ) n(r)n(r ) − 2 γ1(r; r ) 2 : (9) dissipation theorem to show that separated EH and Ex also yield functionals of similar form to Eq. (6).58 For pedagogical simplicity we shall not consider the separated We restrict ourselves to this broad and popular class of terms further. hybrids, though many results derived below are general. Just as in KS theory, the EKS orbitals, φi[n; w](r), are Our first step toward EGKS theory is to recognise that eigen-solutions of the KS equation, Eq. (9) can also accept more general 1-RDMs, ^ ft + v + vHxc[n; w]gφi[n; w] =i[n; w]φi[n; w] : (7) 0 X ∗ 0 γ1(r; r ) = fi'i (r)'i(r ) ; (10) δEHx[n;w] δEc[n;w] i Here, vHxc[n; w] = δn + δn is the multiplica- tive Hxc effective potential of EKS theory, and v is the ex- in which f 2 [0; 2] can be non-integer [note, n(r) = ternal potential.36 The effective potential, v := v +v , i s Hxc γ (r; r)]. Such 1-RDMs appear naturally in ensemble applied to the electrons thus depends both on the overall 1 theories, e.g., the 1-RDM for a lithium cation with a ensemble density, n, and the set of weights, w.
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