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 ˆ ˆ ˆ Recent years have seen renewed interest in ensemble F [n, w] = min Tr[(T + λW )Γw] , (1) Γˆ →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 ∈ [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, Tˆ is the kinetic scenarios, e.g., systems with non-integer electron num- ˆ 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 ˆ P systems that are technically within the reach of KS the- Γw = κ wκ|κihκ|, involves a weighted sum over or- ˆ ory, such as those with significant spin-contamination.38 thonormal wave functions |κi. Γw → n is short-hand for 28,33,35–59 Excited-state EDFT, In particular, it has at- Tr[ˆn(r)Γˆw] = n(r), wheren ˆ(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) {ϕ} and density n. We obtain different operatorsv ˆS and λ F [n, w] − Ts[n, w] potentials vR for different orbital functionals S[{ϕ}] that EHx[n, w] = lim , (3) are invariant to unitary transformations of the orbitals. λ→0+ λ 1 We use {ϕ} to indicate orbital solutions of Eq. (8), as Ec[n, w] =F [n, w] − Ts[n, w] − EHx[n, w] . (4) opposed to {φ} 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] = min{ϕ}→n S[{ϕ}], 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[Γˆs,wTˆ] 55 case of Eq. (1) for pure ground states (w0 = 1, wκ6=0 = 0). and EHx[n, w] ≡ Tr[Γˆs,wWˆ ], for some non-interacting Then, the ground-state energy, E0 = minn{S[n]+RS[n]+ ensemble density matrix Γˆs,w, which is formed on con- V [n]} = min{ϕ}{S[{ϕ}] + Rs[nϕ] + V [nϕ]}, is minimized figuration state functions. Ts and EHx are thereby func- for the density n ≡ nϕ ≡ n0. Uniqueness (in non- tionals of a set of orbitals, {φ}: degenerate systems) follows from the external potential Z v being uniquely defined by the ground state density n0. X w ∗ ˆ X w R 1 Ts[{φ}] = 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[{φ}] = F (ij|ij)φ + F (ij|ji)φ , (6) 2 ij ij 0 ij order reduced density matrix (1-RDM), γ1(r, r ) = P ∗ 0 iσ∈occ ϕiσ(r)ϕiσ(r ). This includes all Hartree-Fock- W R 0 involving electron-repulsion integrals (ij|kl)φ = drdr like expressions, S[{ϕ}] := hΦ|Tˆ + Wˆ S|Φi, where Φ is 0 ∗ 0 ∗ 0 1 60 ˆ W (r, r )φi (r)φj(r )φk(r)φl (r ) with W = |r−r0| . The a Slater determinant formed from {ϕ}, and WS is a w coefficients fi ∈ [0, 2] are average occupancies of orbitals (modified) interaction term involving interactions of form J,w 0 0 0 < WS(|r − r |) ≤ W (|r − r |), 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 ∇ γ1(r, r )|r=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, ˆ {t + v + vHxc[n, w]}φi[n, w] =i[n, w]φi[n, w] . (7) 0 X ∗ 0 γ1(r, r ) = fiϕi (r)ϕi(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 ∈ [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. 0 ∗ 0 partial charge +q is γ1(r, r ) = 2ϕ1s(r)ϕ1s(r ) + (1 − ∗ 0 q)ϕ2s(r)ϕ2s(r ). Importantly, Eq. (9) can be obtained by the sum of Eqs. (5) and (6), for the special case of B. Generalized Kohn-Sham theory J K the product form Fij = −2Fij = fifj. Conveniently, the 11 self-consistent orbitals from Eq. (9) always obey Eq. (8). We now turn to GKS theory. Seidl et el rigorously This is because the functional chain rule ensures that or- showed that the usual KS equation can be modified to in- δS[{ϕ}] δS[γ1] δγ1 bital derivatives, ∗ = ? ∗ = fi(tˆ+v ˆS)ϕi, δϕ δγ1 δϕ clude a non-local operatorv ˆS, which is usually of Hartree- i i δ(S−Ts) attain the same form for all i. Here,v ˆSϕ := ? ϕ, Fock-like form, without changing the fundamental prop- δγ1 erties of density functionals. The generalized KS equa- where ? indicates any necessary integrals. Similarly, δRS δER δn tion, ∗ := ∗ ≡ fivRϕi, and normalisation leads to δϕ δn δϕi εifiϕi because the resulting operator equation is Hermi- [tˆ+ v +v ˆ + v [n]]ϕ [n] =ε [n]ϕ [n] , (8) S R i i i tian. Division by fi yields Eq. (8). 3
∗ III. ENSEMBLE GENERALIZED KOHN-SHAM THEORY respect to ϕi (which can be treated as independent of ϕi) to obtain, A. Fundamental theory δSw δRe X + f ϕ (r) S − f ε ϕ (r) =0 . (15) δϕ∗(r) i i δn(r) i ij j We are now ready to tackle EGKS. The properly en- i j e semblized version of Eq. (9) is S [{ϕ}, w] := Ts[{ϕ}, w]+ Finally, rewrite (15) as a series of coupled orbital equa- EHx,S[{ϕ}, w], which uses the ensemble functionals given tions, in Eqs. (5) and (6) [but with W → WS in (6)]. We define the ensemble density functionals, ˆ ˆ e e X hiϕi ≡ t + v +v ˆS,i + vR ϕi = εijϕj , (16) j Se[n, w] := min Se[{ϕ}, w], (11) {ϕ}→n for all i, j with fi, fj > 0. Eq. (16) must be simulta- e 1 e RS[n, w] :=F [n, w] − S [n, w], (12) neously solved to obtain the minimum energy, E[n, w]. Importantly, because where {ϕ} → n is short-hand for P f |ϕ(r)|2 = n(r). i i δEe [{ϕ}] It is straightforward to show that Eqs. (11) and (12) are e Hx,S vˆS,iϕi := ∗ , (17) well-defined functionals: both are unique, bounded below fiδϕi (T [n] ≤ Se[n], F 1[n] − T [n] − E [n] ≤ Re [n]), and s s Hx,S S varies non-trivially with i, a separate one-body problem have at least one valid solution ({φ} = {ϕ}). needs to be solved for each orbital, and Lagrange multi- Eq. (11) is maximally free55 of “ghost-interactions” pliers ε are required to deal with normality and orthog- (GI), namely spurious interactions between electrons and ij onality (i.e., ε [R ϕ∗ϕ dr − δ ], with ε = ε∗ ). their counterparts in different ensemble members63 that ij i j ij ij ji Let us now consider (16) in the titular framework. are analogous to self-interactions in pure states. In other A solution provably exists and may thus, formally, be words, Eq. (11) will always yield the lowest possible en- found. In this sense (16) is a good result. But it loses ergy for states that are consistent with the weights, w. many of the appealing features (simple non-local opera- In contrast, S[γw] of (9) generally exhibits GI errors63. 1 tor, guaranteed orthogonal orbitals) that make GKS use- Therefore, it has intrinsic positive energy errors even at ful. Furthermore, it cannot easily be implemented using the exact Hartree-exchange level, and is inconsistent with existing GKS infrastructure, although more expensive in- Eq. (6). frastructure, e.g., that used in complete active space self We therefore define SGI[{ϕ}, w] := Se[{ϕ}, w] − consistent field (CASSCF) calculations, may be useful. S[γw] ≡ E [{ϕ}, w] − E [γw] < 0; or equivalently, 1 Hx Hx 1 It is thus a bad result, in the sense of offering few obvi- ous practical advantages over existing approaches [using GI 1 X J,w w w WS S [{ϕ}, w] := 2 (Fij − fi fj )(ij|ij)ϕ Eq. (7) or (8)]. ij In the following we show that an approximation that K,w 1 w w WS is amenable to solution using existing numerical ap- + (Fij + 2 fi fj )(ij|ji)ϕ . (13) proaches, while still maintaining the gains from both This energy term accounts for differences between the GI- EKS and GKS theories, can be obtained by combining a free orbital functional [Eq. (11)] and the 1-RDM energy non-multiplicative potential with an iterative algorithm. functional [Eq. (9)]. It is worth noting that Eq. (13) Such a theory could be viewed, in comparison to the par- typically only has a small number of non-zero terms, as ent EKS or GKS theory, as ugly. However, we show that J,w w w K,w 1 w w it offers formal and practical advantages over existing ap- typically Fij = fi fj and Fij = − 2 fi fj for most combinations of i and j. Explicit dependence on w is proaches. henceforth dropped for brevity. Owing to the GI correction term [SGI, Eq. (13)], Eq. (11) cannot be written as a 1-RDM functional. B. More amenable EGKS Therefore, the series of steps leading to Eq. (8) cannot be reproduced in full for Se. Instead, one must carry out In anticipation of further approximation, we first recast ∗ the exact result in a slightly different form. As discussed explicit differentiation with respect to orbitals ϕi . First w write, previously, S[γ1 ] lends itself to standard GKS treatment. We therefore “generalize” the KS formalism on this term X Z separately from the difficult, SGI [Eq. (13)]. The total C =Se[{ϕ}] + Re [n] − f ε [ ϕ∗ϕ dr − δ ] (14) S i ij i j ij energy may be written as, ij E[{ϕ}] =S[γ ] + SGI[{ϕ}] + Re [n] + V [n] . (18) e e 1 S as our functional to be minimized, where S and RS are energies, the final term is the constraint that the orbitals This is an exact expression that partitions the straight- e GI be orthonormal, {fi}, are average occupation factors and forward 1-RDM part, S[γ1], in S = S + S from the GI {εij} are Lagrange multipliers. Next, differentiate C with more difficult ghost interaction term, S . 4
Our goal is to find the orbitals that minimize E, i.e. GKS theory implemented in quantum chemistry codes. solutions of (16). The partitioning is preserved via differ- We thus propose two approximations that can be solved entiation, which lets us similarly partition the non-local by using existing machinery directly, supplemented by potential operator, simple linear algebra routines that are present in any quantum chemistry code. δSGI[{ϕ}] e GI A first, crude approximation is to assume thatv ˆGI is vˆS,iϕi =v ˆSϕi + ∗ := {vˆS +v ˆi }ϕi . (19) i fiδϕi of little consequence to the obtained EGKS orbitals so Here, can be set to zero. We therefore introduce the “1-RDM” approximation, in which the orbitals {ϕ}1-RDM are found vˆSϕi =ˆvHF[γ1]ϕi := vHϕi +v ˆxϕi using: Z 0 0 Z 0 0 0 n(r )dr 1 γ1(r, r )ϕi(r )dr = ϕi(r) − (20) ˆ 1-RDM 1-RDM 1-RDM |r − r0| 2 |r − r0| h1-RDMϕi =εi ϕi , (25) is the usual 1-RDM expression for Hartree-Fock theory, but energies are found by using {ϕ}1-RDM in Eq. (18). that can be used in Eq. (8) without modification by as- Here, the off-diagonal terms disappear naturally because ˆ signing it an ensemble 1-RDM. h1-RDM is Hermitian. The ensemble energy from this ap- J J Our next step is to recognise that ∆Fij = Fij − fifj proximation provides an upper bound to the true EGKS K K 1 and ∆Fij = Fij + 2 fifj are zero for almost all combi- energy. nations of i and j in all common types of ensembles, per On the positive side, the 1-RDM approximation is sim- δSGI ple and convenient. On the negative side it is uncon- Appendix A. Therefore, δφ∗ is also zero for most values i trolled, misses key ensemble physics, and even leads to of i – including all doubly occupied orbitals with fi = 2. asymptotically incorrect potentials – see Li+q example in We define a “doubly occupied space”, D, for all orbitals Section IV. Nonetheless, it is a decent approximation in with fi = 2, and a “frontier space”, F, for remaining or- some cases, such as for the cation of HCN (shown later bitals with fi > 0. Furthermore, we can eliminate most in Figure 1) of the off-diagonal terms, εi6=j [in (16)] by recognising that The second, “diagonal” approximation represents the titular ugly approach. It involves setting all off-diagonal ε δ , i, j ∈ , i ij D terms in Eq. (21) to zero, not just the ones for i, j ∈ D. ω¯ , i ∈ , j ∈ That is, it involves solving, ε = j D F (21) ij ω¯∗, i ∈ , j ∈ i F D ˆ diag diag diag h1-RDMϕi =εi ϕi , i ∈D , (26) εij, i, j ∈ F, ˆ GI diag diag diag {h1-RDM +v ˆi }ϕi =εi ϕi , i ∈F , (27) which follows from the fact thatv ˆS is Hermitian, that ∗ 1-RDM εij = εji, and that γ1 and S [γ1] are invariant to and algorithmically imposing orthogonality. The result- unitary transformations amongst i ∈ D. ing solution has better properties than the 1-RDM so- It finally follows from Eqs. (13), (19) and (21) that, lution and in almost all tested cases gives lower ener- X gies. The orthogonalisation and need to solve multiple hˆ ϕ =ε ϕ + ω¯ ϕ , i ∈ (22) 1-RDM i i i j j D equations nonetheless makes it rather ugly compared to j∈F standard GKS theory of Eq. (8). ˆ GI X ∗ X {h1-RDM +v ˆi }ϕi = εijϕj +ω ¯i ϕj , i ∈F (23) The difference between the 1-RDM and diagonal ap- j∈F j∈D proximations is best illustrated using a finite M-element real basis set, so that ϕ → C are the orbitals and ˆ i i where h1-RDM := tˆ+ v +v ˆS + vR and P ∗ 0 P T γ1 = i fiϕi (r)ϕi(r ) → D = i fiCiCi is the 1- J Z 0 0 ˆ GI X ∆Fij nj(r )dr RDM. Then, h1-RDM[γ1] → F 1-RDM[D] andv ˆi [{ϕ}i] → vˆiϕi = f 0 ϕi(r) GI i |r − r | V [{Ci}]i are symmetric matrix representations of effec- j tive Hamiltonian terms, expressed in the basis. We also ∗ 0 0 0 K Z ϕ (r )ϕ (r )dr X ∆Fij j i introduce S as the M × M overlap matrix of our basis, + ϕj(r) , (24) fi |r − r0| to deal with non-orthogonal choices. j For both approximations we first solve, Simultaneously solving Eqs. (22) (for doubly occupied 1-RDM 1-RDM orbitals) and (23) (separately for each frontier orbital) F 1-RDMCi =εiSCi (28) yields the orbitals that minimize Eq. (18). 1-RDM for all orbitals, such that Ci are guaranteed to be orthogonal. The 1-RDM approximation then involves up- 1-RDM C. Practical approximations dating F 1-RDM using D from these orbitals and solving to self-consistency. Similarly, in the diagonal ap- Solving Eqs. (22),(23) is difficult in general, and cannot proximation, the solutions, Cdiag := C1-RDM are used i∈D i be trivially done using standard iterative approaches to for our doubly occupied orbitals. The difference comes 5
1-RDM s because the solutions, Ci6∈ are used to expand the which the resulting orbitals, Ci , minimize the total en- D s frontier orbitals, Cdiag, which are then solved separately. ergy, E , including the GI term. SEGKS provides an i∈F upper bound to the true EGKS energy. To solve for Cdiag we must first impose an order on i ∈ i∈F diag F, e.g. via εi from a previous step or the ground state. i 1-RDM 1-RDM For the first i ∈ F, we define P = [Ci , Ci+1 , ··· ] E. A very slow route to exact solutions (i.e. a matrix formed out of the frontier and virtual or- bitals expressed as column vectors) that projects solu- Before concluding this section, we return to exact the- tions onto the unoccupied space of Eq. (28). We then o diagonalise, ory, where our goal is to find the orbitals, Ci , that minimize Eq. (18), to provide the best possible en- i,T GI i i ergy, Eo := E[{Co}] ≤ E[{C }] within the given fi- P {F 1-RDM + V i }P xk = εkxk , (29) i i diag nite basis. In all tested cases we found that Ci i i diag o to obtain Ck := P xk, ordered for k ≥ 1. Then, Ci := forms a good starting point for finding Ci , in the sense i o P diag Ck=1 is the next orbital for the diagonal approximation. that we may write, Ci = j UijCj , where the uni- i+1 i i We then set P = [C2, C3, ··· ] and solve Eq. (29) tary matrix U (in our finite basis) is nearly an iden- again for i + 1, until we have obtained all i ∈ . Once we Q F tity. Consequently, we may write, U ≈ i>j uij(θij) have the full set of orbitals we can use them to update where, uij(θij), are matrices that leave all orbitals un- diag GI diag F 1-RDM[D ] and V i [{Ci }]; and then use the new changed except, Ci → cos(θij)Ci −sin(θij)Cj and Cj → matrices to repeat the process starting from Eq. (28). sin(θij)Ci + cos(θij)Cj. diag o The approach thus gives us a set of orbitals, Ci We are thus able to inefficiently find E by individually that are guaranteed to be orthogonal to one another. minimzing the energy with respect to rotations, θij, for GI Furthermore, the use ofv ˆi in the effective Hamiltonian each combination of j > i (including doubly occupied, diag frontier and virtual orbitals but excluding double-double for Ci means that they, prima facie, have the cor- rect asymptotics,64 unlike the 1-RDM solutions. They and virtual-virtual rotations as these leave the energy may thus be expected to be better approximations to unchanged). This brute force approach leads to O(N 2) the real solutions. Results (see Section IV) are almost calculations of O(N 3) energies, for a very poor scaling of always better than the 1-RDM, sometimes substantially O(N 5) that is unsuitable for large systems. In practice, so. Nevertheless, they can only form an upper bound to we repeat the process several times until energies can the true energy since εi6=j = 0 is imposed via the diagonal no longer be reduced. We are thus able to find exact approximation, rather than found from a true variational solutions, to within a few percent of an eV, for the small solution. systems reported in the next section. Of final note, it is obvious that the frontier orbitals, Cdiag 6= C1-RDM, differ from the 1-RDM approxima- i∈F i tion because they obey different effective Hamiltonians. IV. RESULTS However, it is also true that the doubly occupied or- bitals, Cdiag 6= C1-RDM, are different despite both com- i∈D i Having established both exact theory and useful ap- ing from Eq. (28). The difference comes from different proximate forms, we turn to examples. Throughout, we 1-RDMs,the different D, Ddiag 6= D1-RDM, being used perform EGKS calculations corresponding to the ensem- 55 to form F 1-RDM. ble exact exchange theory approximation in EKS the- ory, i.e., we set Re = 0, S1-RDM ≡ SHF and SGI = HF Ts + EHx − S . As mentioned above, we use a spin- D. Simplified EGKS theory unpolarized formalism throughout so that potentials and orbitals are independent of spin, i.e. ϕi↑ = ϕi↓ and We next turn briefly to a less rigorous approximation Ci↑ = Ci↓. recently introduced by one of the authors – the “simpli- For the purpose of the present theoretical study, exact fied EGKS” (SEGKS) scheme.59 – that is designed to ef- exchange represents an effective choice of density func- ficiently approximate the EGKS solution of excited state tional approximation as it lets us clearly delineate errors ensembles. This approximation is based on an ansatz, caused by the orbital approximations (1-RDM or diag) F ≈ (1 − µ)F gs + µF h→l, for the Fock matrix that inter- from those caused by the functional approximation. We polates between the ground state Fock matrix, F gs, and are thus able to gain insights into the quality of approxi- a different Fock matrix, F h→l, formed by double promo- mations to the effective Hamiltonians (and thus orbitals), tion of the HOMO to the LUMO. separate from other considerations. These calculations This physical intuition behind this ansatz is the idea are denoted “exchange-only” throughout, and the un- that F gs screens the HOMO more than the LUMO, derlying orbital approximation (EGKS, diag, 1-RDM or whereas F h→l does the reverse which gives the model unrestricted Hartree-Fock) are mentioned separately. sufficient flexbility to optimize the orbitals. The precise We begin with an analytic example: the fractional ion amount of linear of mixing is found by seeking, µs, for Li+q, with 0 < q < 1. This system is represented as an 6 ensemble Γˆ ≡ q|1s2ih1s2| + (1 − q)/2[|1s22s↑ih1s22s↑| + |1s22s↓ih1s22s↓|]. It is straightforward to show (see Ap- TABLE I. Errors (in kcal/mol = 0.043 eV) for various EGKS approximations, with respect to exact EGKS energies, within pendix A) that the density is, n = 2|φ |2 + (1 − q)|φ |2, 1s 2s exchange-only theory. Also shown are energies from UHF i.e. f1s = 2 and fh ≡ f2s = 1 − q. Furthermore, theory, for triplet state (ts) and doublet states (ds). Singlet the 2s orbital never interacts with itself, which gives states are identical in all theories. J K F2s2s = F2s2s = 0 as the only pair-coefficient that con- tributes to C(ts) O(ts) B(ds) F(ds) CO(ts) Mean 1-RDM 11.6 15.6 5.2 8.3 14.1 11.0 2 GI GI f2s Diag 4.0 7.7 0.0 0.1 6.3 3.6 S [{ϕ}] :=S [φ2s] = − EH[n2s] . (30) 2 Exact 0.0 0.0 0.0 0.0 0.0 0.0 We are interested in the asymptotic behaviour of the UHF -3.1 -3.9 -2.7 -2.9 -8.9 -4.3 ∗ 0 0 0 R ϕi (r )ϕj (r )dr δij 2s orbital. By recognising that |r−r0| → r 0 1-RDM R n(r ) 0 0.6 we see [Eq. (20)] that vH ϕ2s = |r−r0| dr ϕ2s(r) → 1RDM 0 0.3 f1s+f2s 1-RDM R γ1(r,r ) 0 0 Diag r ϕ2s andv ˆx ϕ2s = − 2|r−r0| ϕ2s(r )dr → 0.0 EGKS f2s − 2r ϕ2s, so that the 1-RDM approximation [Eq. (25)] yields, 0.3
0.6 f2s 1−q f1s + 2 2 + 2 Electron Aff. [eV] lim vˆSϕ2s → ϕ2s ϕ2s . (31) 0.9 F r→∞ r r F F 1/2 F Including the GI term [Eq. (30) in (24)], whether in Charge [unitless] the exact theory or via the diagonal approximation 0.6 2 1RDM GI f 0.5 [Eqs. (26),(27)], gives an additionalv ˆ → − 2s = Diag 2s 2f2sr 0.4 −f2s EGKS 2r , to yield, 0.3
GI f2s 2 0.2 lim [ˆvS +v ˆ2s ]ϕ2s → ϕ2s = ϕ2s . (32) r→∞ r r 0.1 Ion. Potential [eV]
0.0 HCN Note that we have assumed limr→∞ ϕ1s(r)/ϕ2s(r) = 0. + The outermost 2s electron thus “feels” the charge of HCN HCN+1/2 HCN Charge [unitless] the two 1s electrons only (i.e., no self interaction) when the GI term is included, a result that is physical and con- FIG. 1. Fractional anion energy curve of F (top) shown rel- sistent with previous findings of ensemble theories yield- 65 ative to the neutral atom energy of EGKS theory; and frac- ing the correct asymptotic behaviour in ensembles. . By tional cation curve of HCN (bottom), shown relative to a contrast, the 1-RDM approximation leads to an unphys- straight line fit between the neutral and cation EGKS re- ical self interaction in the outermost electron, with an sults. Here and in subsequent figures we show 1-RDM in teal q−1 effective charge 2 , and thus underbinds the 2s orbital. dash-dots, diag in orange dashes , and exact EGKS results in We now turn to numerical examples, where we com- navy solid lines. Dots, where shown, indicate UHF values. pare our two approximations (1-RDM and diag) against benchmark exchange-only solutions found by minimizing against unitary transformations of the orbitals, as de- We first notice that the diagonal approximation of- scribed in Section III E. All calculations were performed fers significant improvements over the 1-RDM approxi- using Gaussian type orbitals in the def2-tzvp basis set,66 mation in all tested cases – an unsurprising success that and were implemented in a customized Python3 code us- is replicated in almost all tests reported here. Perhaps ing Psi4/numpy.67,68 Note that in all examples we have more surprisingly, we see that even exact EGKS the- allowed spatial symmetries to break. ory can yield substantially larger energies than UHF, of Table I reports errors for a selection of simple systems up to 8.9 kcal/mol (0.38 eV) for the CO excited triplet with degenerate ground states (triplet states, ts; and dou- state. Whether this should be considered an “error” of blet states, ds), shown relative to exact EGKS theory. In EGKS (which has the correct ground state degeneracy the case of CO, we show the excited triplet state. These but higher energy) or UHF (which lowers energies via systems have the advantage that not only can we com- an unphysical breaking of fundamental symmetries) is pare exchange-only EGKS approximations against ex- a matter of taste that dates back to at least 1963 as act results, we can also compare exact and approximate L¨owdin’sclassic “symmetry dilemma”.69 EGKS against broken-symmetry unconstrained Hartree- A more comprehensive example is provided by the full Fock (UHF) theory. UHF is guaranteed to have an energy fractional ionization curve for the anion of the fluorine that is less than or equal to EGKS by virtue of having atom, which represents the zero temperature limit of a additional degrees of freedom. grand canonical ensemble. At net charge q the ensemble 7 describing this process is, 2.0 Be
ˆ 1+q − − 1.5 Γq = 2 [|F↑ihF↑| + |F↓ihF↓|] − q|F ihF | , (33) which mixes equal amounts of the ↑ and ↓ dominant de- 1.0 1RDM generate F doublet ground states with the singlet F− SEGKS 0.5 ground state, to achieve the correct net charge. Results Diag EGKS are shown in the top panel of Figure 1. Excitation energy [eV] 0.0 In a correlated theory, this curve should look like a 0.0 0.2 0.4 0.6 0.8 1.0 straight line.22 Exchange-only theory introduces a sub- Fraction of triplet [unitless] 28,70,71 0.0 stantial curvature as a function of charge. It is 1RDM nonetheless clear that the diagonal approximation is 0.5 SEGKS nearly exact for all charges, whereas the 1-RDM approx- Diag 1.0 ination is only accurate when more than half an electron EGKS − is added to the system – note that F is the closed shell 1.5 pure state case for which standard 1-RDM theories apply so that exact, 1-RDM and diag EGKS all become exactly 2.0
Excitation energy [eV] C equivalent to standard Hartree-Fock theory. 2.5 The bottom panel is similar to the top, but for the 0.0 0.2 0.4 0.6 0.8 1.0 Fraction of triplet [unitless] cation of HCN: 0.0 1RDM ˆ q + + + + 0.5 Γq = 2 [|HCN↑ ihHCN↑ | + |HCN↓ ihHCN↓ |] SEGKS 1.0 + (1 − q)|HCNihHCN| . (34) Diag 1.5 EGKS 2.0 Rather than showing full energies, it reports errors rel- 2.5 ative to a straight line fit between exchange-only EGKS 3.0 results for the cation and neutral molecule, i.e., it shows Excitation energy [eV] 3.5 O the curvature, so that difference between approximations 0.0 0.2 0.4 0.6 0.8 1.0 are visible at the scale of the plot. Again diag is nearly Fraction of triplet [unitless] exact, whereas 1-RDM has issues for positive charges of more than half an electron – here the netural system is FIG. 2. Energies (in eV) of mixtures of singlet and triplet the closed shell case. states for Be (top), C (middle) and O (bottom). EDFT can also be used to predict excitations energies. This is done by forming an ensemble in which the excited 9 HCN states have weights less than or equal to the weights for 8 the states that are lower in energy. We therefore next 7 1 = 7.4, 2 = 16.4 [eV] = 7.6, = 16.9 [eV] consider a relevant ensemble for predicting singlet-triplet 6 1 2 1 = 7.4, 2 = 17.1 [eV] 5 excitations, 1 = 7.4, 2 = 17.1 [eV] 4 1RDM 3 ˆ w X SEGKS Γw =(1 − w)|S0ihS0| + 3 |T0,Mz ihT0,Mz | , (35) 2 Diag 1 Mz EGKS Excitation energy [eV] 0 which mixes different fractions of the singlet ground state gs gs/sx gs/sx/dx Mixed excitation [unitless] (gs, S0) with an equal mixture of the three lowest energy triplet states (ts, T ,M ∈ {−1, 0, 1}). Here, w = 0.0 1 = 1.2, 2 = 0.9 [eV] 0,Mz z = 1.2, = 0.9 [eV] 1RDM 0.1 1 2 0 indicates a pure ground (singlet) state while w = 1 1 = 1.3, 2 = 1.0 [eV] SEGKS 0.2 represents the state average of the triplets. Results are 1 = 1.4, 2 = 1.2 [eV] Diag 0.3 shown in Figure 2 for Be, C and O. EGKS In all cases the diag approximation is substantially bet- 0.4 ter than the 1-RDM, except for the pure singlet ground 0.5 0.6 state where both approximations are exact. The diag re- 0.7 sults are nearly exact for mixings up to 60% of triplet, Excitation energy [eV] C2 gs gs/sx gs/sx/dx but then become poorer. We also include results from Mixed excitation [unitless] the SEGKS scheme.59 In all cases SEGKS outperforms 1-RDM slightly, especially for mid-range weights, but is FIG. 3. Energies (in eV) for mixtures of singlet excitations for worse than the diag approximation. HCN (top) and C2 (bottom). The Ω values show extrapolated The same scheme described above for triplet states can excitation energies in the different approaches, using the same also be used to describe excitations between states of the colours and order as the legend. same fundamental symmetry, which are not accessible to 8
standard DFT. Ensembles that provide access to the first C2 h: 1RDM and second excited states (for single and double excita- h: diag 2 h: EGKS tions from h → l) are, l: 1RDM ]
Å l: diag [
r 4 10 l: EGKS ˆ 2 Γ 1 =(1 − w)|S ihS | + w|S ihS | , (36) 1 4 10 0 0 1 1 10 2 w≤ 2 10 ˆ 2−w Γ 1 = 3 |S0ihS0| + |S1ihS1| 2 ] 1 LUMO 2w−1 1 2 + |S2ihS2| . (37) 3 Å [ ) z 1 ( HOMO Here, setting w ≤ 2 gives mixtures of S0 and S1 only, 1 0 whereas 2 < w ≤ 1 mixes S2 with an equal mixture of 3 2 1 0 1 2 3 z [Å] S0 and S1. It thus provides a means of obtaining both single and double excitation energies. Ensembles of the 1 FIG. 4. HOMO and LUMO densities for C2. We retain the form proposed for w ≤ 2 were recently used by one of the authors to obtain accurate singlet-singlet excitation colours of previous figures (1-RDM in teal, diag in orange and energies of small molecules, using SEGKS together with EGKS in navy) for the HOMO. LUMO are coloured as cyan 59 (1-RDM), red (diag) and blue (EGKS). In the lower panel we standard density functional approximations. also change the line styles (to dots, dashes, solid, respectively) Figure 3 shows results for excitations of HCN (top) to improve clarity. and C2 (bottom), the latter of which has very small gaps. In both cases we see that diag outperforms 1-RDM for all values of w, with the diagonal approximation being It is clear that the diagonal approximation yields bet- nearly exact for HCN but a little poorer for double exci- ter densities than the 1-RDM, as expected from the bet- tations of C2. SEGKS once again slightly out-performs ter energies. What is interesting is that neither approx- 1-RDM, but is worse than diag. For HCN, the curvature imation manages to shift sufficient charge from outside is rather small compared to the overall energy, whereas the molecule to the bond, in the HOMO. This can be for C2 it is significant. Note, the categorisation of S0, S1 seen best by comparing the approximations against ex- and S2 here is based on pure state orbital ordering, not act EGKS for z ≈ 0, in the bottom panel. The diag does energies. For C2 in the exchange-only approximation the a better, albeit imperfect, job than the 1-RDM. energies are reversed, as seen by the negative energies in the figure. The above ensembles can be used further to estimate Be 0.1 excitation gaps. This is done via extrapolation, by using a quadratic fit to the energy to approximate the energy 1RDM 38,59 Diag of a desired excited pure state. The energy, ES0 , of the ground state S0 is found by setting w = 0. We ex- EGKS 1 trapolate results for w ≤ 2 to w = 1 to obtain the energy, 1 ES1 , of the S1 state; and extrapolate 2 ≤ w < 1 to w = 2 Excitation energy [eV] 0.0 to obtain the energy, ES2 , of S2. The excitation energies, gs gs/sx gs/sx/dx Mixed excitation [unitless] Ω1 = ES1 − ES0 and Ω2 = ES2 − ES0 are included in the curves, for each method. Keep in mind that these values are rather poor, due to our choice to use an exchange-only FIG. 5. Energy errors [eV] for singlet excitations of Be. The approximation that completely neglects any correlations. EGKS error is zero, by definition. For this reason, we do not include exact values but rather compare against the exchange-only benchmark value pro- Finally, we turn to a rare example where the diag ap- vided by EGKS theory. proximation is higher in energy than the 1-RDM: the first BEcause the energy functional employed in all calcula- singlet-excitation of Be. Figure 5 shows the errors of the tions reported here is the same, differences between ap- diag and 1-RDM approximations for the same excitation proximations and with exact theories come from the or- process as in Fig. 3. When the amount of S0 and S1 is bitals. Thus, to further understand differences between 1 approximately equal (w → 2 ) the 1-RDM approximation the approximations, we show in Figure 4 the highest oc- is actually better than the diag, although it is worth not- cupied and lowest unoccupied molecular orbital densities ing that both approximations are within a few kcal/mol (HOMO and LUMO, h and l) of C2 under the two ap- of the exact theory. Once S2 is mixed in this discrepancy proximations and in exact EGKS. We set w = 1 to ob- disappears. Although surprising, such a result is not in tain an equal mixing of S0, S1 and S2. The top panel contradiction to theory, since the diagonal approximation −4 −2 shows contours (10 and 10 ) of angular integrals, is not a guaranteed lower bound to the 1-RDM approxi- 1 R 2π ρ(z, r) = 2π 0 ρ(r)dθ, while the bottom panel shows, mation. However, as expected from the improved physics R ∞ ρ(z) = 2π 0 rρ(z, r)dr. in the diag approximation, this was the only case where 9 it wasn’t a lower bound in practice. We speculate that Appendix A: Pair-coefficients it might be caused by an increased breaking of spatial symmetry in 1-RDM compared to diag. The appendix first derives pair-coefficients for the com- mon ensembles and pure states considered in this work, including which orbitals belong to the double occupied space, , and the frontier space, . It then shows how V. CONCLUSIONS D F these coefficients combine linearly to produce general en- semble formulae. To conclude, we showed that ensemble KS and gener- alized KS theories can be unified rigorously into EGKS theory – a good result. However, in doing so directly, 1. Pair coefficients for singlet ground states one must optimize an energy functional [Eq. (13)] that is not expressible as a 1-RDM functional, and thus not A singlet ground state represents a pure state, amenable to a standard GKS treatment – a bad result. ˆ 2 2 2 2 2 2 We then showed that solutions to EGKS can be defined ΓS0 =|S0ihS0| = |1 2 ··· h ih1 2 ··· h | , (A1) in its usual operator form [Eq. (16)], provided an orbital- dependent correction term is introduced to the GKS so- in which all orbitals are doubly occupied or unoccupied. lutions for a small number of “frontier” orbitals that con- Thus, one trivially finds, fi = 2, ∀i ≤ h, and, tribute “ghost interactions” – an ugly result. Still, the F J =f f = 4,F K = − 1 f f = −2, (A2) ghost interaction term can be dealt with via a “diagonal ij i j ij 2 i j approximation” (Sec. III C) that is reasonably accurate for the pair-coefficients. These systems therefore have no and more amenable to efficient evaluation. ghost interactions. All occupied orbitals thus belong to The approach was demonstrated on several analyti- the doubly occupied space, D = {i ≤ h}, and the frontier cal and numerical examples based on exchange-only ap- space, F = {}, is empty. Any coefficients not specified proximations. The diagonal approximation was shown to here, or in the remainder of the appendix, are zero. successfully reproduce exact results (within an exchange- only formalism) and to consistently outperform a simpler 1-RDM approximation, except in one notable case. Our 2. Pair coefficients for doublets work thus demonstrates that not only is ensemble den- sity functional theory formally amenable to a generalized The next simplest ground state is a doublet, in which Kohn-Sham framework, but that the approach is both all electrons air paired except one, i.e. the state is, practical and advantageous. |1222 ··· h↑i or ↑ → ↓. Since our ensemble formalism Table I and Figures 1 to 3 illustrate that both EGKS treats all electrons in the same effective potential, these theory (relative to unrestricted Hartree-Fock) and the di- two states are degenerate (not so in unrestricted Hartree- agonal approximation are much better for the two dou- Fock theory). The relevant ensemble is thus, blet cases than for other cases. We suspect that this re- ˆ 1 2 2 ↑ 2 2 ↑ flects the fact that in doublets only one electron (h) needs ΓD0 = 2 |1 2 ··· h ih1 2 ··· h | a GI correction, which is therefore easier to approximate + |1222 ··· h↓ih1222 ··· h↓| . (A3) than the other cases which require that two electrons (h and l) be corrected. The general conditions in which the We may, without any loss of generality, evaluate the den- 1-RDM and/or diag approximations perform well should sity of the ↑ state to obtain the coefficients, fi 3. Pair coefficients for triplets 5. 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