A Self-Interaction-Free Local Hybrid Functional: Accurate Binding

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(OEP) of potential energy effective the optimized on Kohn- based Self-consistent calculations ex- Sham energy. the exchange defines Kohn-Sham orbitals Kohn-Sham act exact the with uated exchange prxmtost ihrIswe hyaecalculated [35–40]. are potentials they xc good when accurate are IPs from eigenvalues higher valence impor- to upper equally approximations that purposes condition the practical tant nor for [31–34], (IP) but potential approximate ionization first the match iinta h ihs cuideigenvalue occupied con- highest exact the the that neither fulfill dition typically approx- mentioned functionals functionals the above imate semi-local 30], the that of [29, eigenvalues discontinuity fact Kohn-Sham derivative the Due the and DFAs. over” these error “average for SI so do the not to does but system, electron E 2, x ex hl h frmnindfntoasi aycases many in functionals aforementioned the While † satr,adi ecos ootmz their optimize to choose we if and isfactory, eto efitrcin epcsconstraints respects self-interaction, lectron igeege n ffrsohradvantages, other affords and energies ding = mtri band eptti nigin finding this put We obtained. is ameter eo Kronik, Leeor rfntoasadpsil ietosfor directions possible and functionals er − iainptnil.Tssfrastof set a for Tests potentials. nization binitio ab smttcbhvo ftex energy xc the of behavior asymptotic t ihacmail o-oa correlation non-local compatible a with ucinl ln iha improvement an with along functional, oee,tecrepnec fthe of correspondence the However, . omacrt idn nriscnbe can energies binding accurate form 2 1 fSine eooh76100,Israel Rehovoth Science, of e σ i,j X = N =1 ↑ σ , ↓ eain(c nryo Kohn- of energy (xc) relation eivsiaewehri is it whether investigate We . ZZ 2 E uh Germany euth, ϕ n tpa Kümmel Stephan and xc iσ ∗ idn energies binding e + ( r E ) ϕ H jσ eigenvalues hudvns o n one- any for vanish should ( | r r ) − ϕ iσ r ′ ( | r ′ ) ϕ jσ ∗ 1 ( r ε ′ ) 1 ho s the use ) d 1 3 should eval- ) r d 3 (1) r ′ , 2

ex r Although the interpretation of occupied eigenvalues mated with (semi-)local expressions, whereas ex ( ) rep- even with the exact xc potential is approximate (except resents the EXX energy density per particle deduced from for εho), it is of great practical importance. For example, Eq. (1). The function f[n](r) is the local mixing func- the band-structure interpretation of Kohn-Sham eigen- tion (LMF). It is a functional of the density and a decisive values has had a great impact on solid-state physics and part of the local hybrid concept. materials science [41]. In recent years the interpretation Eq. (2) can be viewed as a generalization of the com- of eigenvalues has become particularly important in the mon (global) hybrids. Instead of a fixed amount of EXX, field of molecular semiconductors and organic electronics. the local hybrid can describe different spatial regions of Efforts to understand, e.g., photoemission experiments, a system with varying combinations of EXX and semi- have revealed severe shortcomings of traditional DFAs local xc, by means of f[n](r) (where 0 f 1). For that go considerably beyond a spurious global shift of the example, whereas one-electron regions are≤ supposed≤ to eigenvalue spectrum [37–39, 42–49]. A similar problem is be well-described using EXX, regions of slowly varying witnessed also in solid state systems [50–55]. We empha- density are expected to be captured appropriately by size that these problems of interpretation arise already (semi-)local xc functionals. The idea of local hybrids can for the occupied eigenvalues, i.e., the issues are separate also be understood in terms of the adiabatic connection from the well known band-gap problem [28, 29, 56, 57] theorem [69], because f[n](r) may offer further flexibil- of Kohn-Sham theory. The KS EXX potential leads to ity in an accurate construction of the coupling-constant- band structures and eigenvalues that match experiments dependent xc energy [66], especially for small coupling much better than the eigenvalues from (semi-)local ap- constant values. proximations [26, 43, 58–61]. The local hybrid form was pioneered by Jaramillo et A comparison to hybrid functionals is more involved, al. [67] with a focus on reducing the one-electron SI-error because already the occupied eigenvalue spectrum de- in single-orbital regions. Numerous further local hybrid pends sensitively on whether one uses the hybrid func- constructions followed [68, 70–78]. They proposed var- tional in a KS or a generalized KS calculation [45]. For ious LMFs with different one-electron-region indicators, well understood reasons [29, 56], the differences between suggested several (semi-)local exchange and correlation the KS and the generalized KS eigenvalues become yet functionals to be used in the construction, and followed larger for unoccupied eigenvalues (see, e.g., the review different procedures to satisfy known constraints and de- [57]).This article’s focus is on Kohn-Sham theory, there- termine remaining free parameters. fore we do not discuss the comparison to hybrid function- In the present manuscript we propose a new local- als used in the generalized KS scheme in detail. We note, hybrid approximation that combines full exact exchange however, that in particular range-separated hybrid func- with a compatible correlation functional. The develop- tionals used in the generalized KS approach can predict ment is guided by the philosophy of fulfilling known con- gaps and band structures quite accurately, as discussed, straints [79]: Our xc energy density per particle, exc, is e.g., in [57, 62], but global hybrid functionals tend to one-electron SI-free, possesses the correct behaviour un- yield a less reliable density of states for complex systems der uniform coordinate scaling, and has the right asymp- than self-interaction free Kohn-Sham potentials [39, 45]. totic behaviour at large distances. It includes one free Besides these practical benefits, EXX also appears as parameter that is not determined uniquely from these a natural component of DFAs because it may be con- constraints. sidered attractive to treat as many energy components In difference to earlier work, our emphasis is not on im- as possible exactly, and including EXX has shown to be proving further the accuracy of binding energies beyond beneficial for, e.g., describing ionization, dissociation and charge transfer processes [28]. However, bare EXX is a the one that was achieved with global hybrids. Instead, we focus on whether it is possible to construct an ap- very poor approximation for binding energies, (see, e.g., Refs. [63],[27], and [5], chapter 2). Combining EXX with proximation that yields binding energies of at least the same quality as established hybrids and at the same time a (semi-)local correlation term in many situations leads to results of inferior quality compared to pure EXX or affords other advantages, notably KS eigenvalues that approximate IPs reasonably well. We find that if we choose semi-local DFAs, because of an imbalance between the delocalized exchange hole and the localized correlation the parameter in our functional by optimizing the prediction of binding energies, the latter are obtained with hole [8, 28, 64, 65]. One promising approach for combining EXX with ap- an accuracy that is similar to the one reached with usual global hybrids. At the same time, we achieve a significant propriate correlation in a balanced way is the local hybrid form [66–68] improvement in prediction of dissociation energy curves for selected systems. Improvement in prediction of the r r ex r r sl r sl r ionization energy via the highest occupied KS eigenvalue exc( )=(1 f[n]( ))ex ( )+ f[n]( )ex ( )+ ec ( ). (2) − is also observed. It is especially large for alkali atoms. Here, exc(r) is the xc energy density per particle that However, the quality of the ionization energy prediction 3 yields the xc energy via Exc[n] = n(r) exc(r) d r. The is not yet satisfactory, and if we aim to optimize the sl r sl r quantities ex ( ) and ec ( ) denote exchange and correla- prediction of the latter, a rather different value for the tion energy densities per particle,R respectively, approxi- functional’s free parameter is obtained. We put this find- 3 ing in a larger context by discussing similar observations where a0 is the Bohr radius, Φ(ζ(r)) = 1 2/3 2/3 for other functionals. (1 + ζ) + (1 ζ) and ζ(r) = (n↑(r) 2 − − The paper is organized as follows: Section II is devoted n↓(r))/(n↑(r) + n↓(r)) is the spin polarization, is a to the description of the new local hybrid functional. Sec- natural ingredient to be used to construct a f[n](r) tion III (and the Appendix) provide methodological and that aims at enforcing the uniform coordinate scaling, computational details. Sec. IV presents and discusses the because in the high density limit t2 γ. We make use results, and Sec. V offers conclusions and a summary. of this quantity as described in detail∼ below. Property (ii) is reflected in the equation EH [niσ] + E [n ] = 0, where n (r) = ϕ (r) 2 are one-spin- xc iσ iσ | iσ | II. CONSTRUCTION OF THE FUNCTIONAL orbital densities, with ϕiσ(r) denoting the i-th KS-orbital in the spin-channel σ. One can attempt to realize such a In the construction of our functional, we choose to one-spin-orbital condition by detecting regions of space concentrate on satisfying the following exact properties: in which the density is dominated by just one spin-orbital (i) use the concept of full exact exchange, as defined and making sure that full exact exchange and zero cor- by the correct uniform coordinate scaling [80, 81] (see relation is used there. Previous works have discussed elaboration below); (ii) freedom from one-electron self- the use of iso-orbital indicators [83–86] for similar tasks. interaction [11]; (iii) correct asymptotic behavior of the Here, we define a one-spin-orbital-region indicator by xc energy density per particle at r [82]; (iv) re- | | → ∞ r production of the homogeneous electron gas limit. In τW ( ) d(r)= ζ2(r), (6) addition, we wish to maintain an overall balanced non- τ(r) locality of exchange and correlation [68]. 2 Regarding property (i), under the uniform coordi- where τW (r) = n(r) /(8n(r)) is the von Weizsäcker r r r |∇ | 1 Nσ 2 nate scaling γ the density transforms as nγ ( ) = kinetic energy density and τ(r)= ϕiσ (r) 3 r → 2 σ i=1 γ n(γ ), with its integral, N, unchanged and the ex- is the Kohn-Sham kinetic energy density. |∇ | r r change scales as Ex[nγ ( )] = γEx[n(γ )] [5, 28], which For one-spin-orbital densities of ground-stateP P charac- ex r ex r implies ex [nγ ( )] = γex [n(γ )]. This scaling relation is ter, d(r) 1, because τ(r) τ (r) and ζ2(r) 1. For LSDA → → W → fulfilled, e.g., by ex , the exchange energy density per regions with slowly varying density, however, d(r) 0 particle in the local spin-density approximation (LSDA). → because τW (r) tends to zero, whereas τ(r) does not. In For the correlation functional, Ec[n], no such simple contrast to expressions suggested in the past [67], Eq. (6) r scaling rule exists: the correlation scales as Ec[nγ( )] = does not classify a region of two spatially identical or- 2 (1/γ) γ Ec [n(r)], where the superscript (1/γ) indicates a bitals with opposite spins as a one-orbital region. It also system with an electron-electron interaction that is re- avoids introducing [68, 70–74, 78] any parameters in d(r). duced by a factor of γ [5]. Additional scaling results for Despite our use of Eq. (6) and the frequent use of the correlation energy can be found in, e.g., Refs. [80, 81]. similar indicators in the past, we wish to point out two Here, we concentrate on the limiting case of high elec- caveats before proceeding. First, it should be noted that tron densities, i.e. γ , where the xc energy should formally there exists a difference between one-electron → ∞ be dominated by Ex[n], [81] and one-spin-orbital regions. The former correspond to spatial regions in the interacting-electrons system where E [n ] lim xc γ =1. (3) the probability density is such that one finds just one →∞ ex γ Ex [nγ ] electron. The latter, however, correspond to spatial regions in the KS system dominated by a single KS spin- A functional is said to use full exact exchange if it obeys orbital [87]. There is no guarantee that these two re- Eq. (3) [68]. gions coincide, because, strictly speaking, the interacting With this definition in mind, we return to Eq. (2). r ex r r system and the KS system have only the total electron Using exc( )= ex ( )+ ec( ), we obtain density in common. sl ex sl Our second caveat refers to the fact that orbital den- ec(r)= f[n](r) e (r) e (r) + e (r). (4) x − x c sities are typically not of ground-state character. There- r r We now see that when f[n] scales in the high density limit fore the equivalence of τ( ) and τW ( ) is not guaran- as γa with a< 0, then it is clear that the first term on the teed for these over all space. It is reached, however, in RHS of Eq. (4) is a correlation contribution rather than the energetically relevant asymptotic region. We further sl r b an exchange term [134]. Assuming ec ( ) scales as γ with note that it has recently been pointed out [88] that also b < 1, the functional exc(r) that fulfills this condition the Perdew-Zunger SI correction [11] may have problems can therefore justly be viewed as a combination of EXX, because of orbital densities not being ground-state den- ex r r namely, ex ( ), and a compatible correlation term, ec( ). sities. This may indicate that the question of how to The reduced density gradient [17] associate orbitals with electrons for the purposes of eliminating self-interaction is a fundamental one, affecting all 1/3 2 π a0 n(r) of the presently used concepts for self-interaction correc- t2(r) := |∇ | , (5) 3 16Φ2(ζ(r)) n7/3(r) tion that we know of. 4

With the aim of fulfilling conditions (i) to (iv) we There remains one important point to be discussed. propose the following approximate form for our EXX- In Eq. (7) we are left with one undetermined parame- compatible correlation energy density per particle, ec(r): ter, c. Unfortunately, we presently do not know of an ab initio constraint that would allow us to fix this parame- r 1 τW ( ) ζ2(r) ter uniquely, although we do not rule out the possibility τ(r) LSDA ex ec(r)= − e (r) e (r) + that future work may achieve this. The value of c affects 1+ ct2(r) x − x the amount of EXX that is used in a calculation and is τ (r) + 1 W ζ2(r) eLSDA(r). (7) therefore expected to have an influence in practical ap- − τ(r) c plications. One can therefore argue that not having c determined from first principles is a disadvantage. How- In other words, we approximate the LMF function of ever, with c being a free parameter, the functional form Eq. (4) by contains some freedom which allows one to adjust it to specific many-electron systems. One can therefore argue τW (r) 2 r 1 d(r) 1 τ(r) ζ ( ) that our yet undetermined c is in line with the principle f[n](r)= − = − , (8) 1+ ct2(r) 1+ ct2(r) of reducing (but not eliminating) empiricism in DFT [7]. In this first study, the freedom of varying c will be the semi-local exchange energy density per particle by used deliberately to explore the properties of the pro- sl r LSDA r posed functional. We perform fitting of c per system for its LSDA form [3] ex ( ) = ex ( ), and the semi-local correlation energy density per particle by a representative test set to observe how much its optimal value varies between the different systems, and whether τ (r) a global fitting procedure, i.e. fitting for all systems com- esl(r)= 1 W ζ2(r) eLSDA(r), (9) c − τ(r) c bined, is at all justified. In particular, we wish to eluci- date the question of whether good binding energies and which is the LSDA correlation energy density per parti- good eigenvalues can be achieved with the suggested local cle, multiplied by (1 d(r)). hybrid functional form. As an aside we note that when c The proposed functional− is one-electron SI-free, has the is a fixed, system-independent parameter, the proposed required asymptotic behavior for exc(r) at r , be- functional is fully size-consistent and complications that haves correctly under uniform coordinate scaling,| | → and∞ re- are known to occur with system-specific adjustment pro- duces to the LSDA for regions of slowly varying density. cedures [89] are avoided. One-electron self-interaction is addressed via d(r). When d(r) tends to 1, ec(r) vanishes and the only re- ex r maining term is ex ( ), which then cancels the Hartree re- III. METHODS pulsion. Note that the semi-local correlation part, which is the last term in Eq. (7), also vanishes for one-spin- The proposed functional was implemented and tested orbital regions. This is assured by introducing the pref- using the program package DARSEC, [90, 91] an all- r LSDA actor (1 d( )) in front of ec . Otherwise, for one- electron code, which allows for electronic structure cal- − orbital regions one would get the undesired, unbalanced culations of single atoms or diatomic molecules on a real- combination of EXX and local correlation. space grid represented by prolate spheroidal coordinates. The correct uniform scaling is achieved due to the de- We therefore avoid possible uncertainties associated with nominator in f[n](r), which scales as γ, and cancels the the use of pseudopotentials or complicated basis sets in γ-dependence of the exchange terms that multiply it. In OEP calculations [28] – an advantage for accurate func- addition, eLSDA scales as ln(γ) (see Eq. (10) in Ref. [10], tional testing. c − Sec. II of Ref. [81]), which is slower than γ. Therefore, DARSEC allows the user to solve the KS equations the limit in Eq. (3) is satisfied. self-consistently for density- as well as orbital-dependent r r For slowly varying densities, f[n]( ) 1 and τW ( ) functionals (ODFs), for example the proposed functional. r LSDA r LSDA→ r → 0, which yields exc( ) e ( )+e ( ), reproducing For ODFs, the xc potential is constructed by using either → x c the LSDA limit as required. the full optimized effective potential formalism (OEP) Finally, note that the proposed exc approaches the [26, 28] via the S-iteration-method [92, 93] or, with re- known exact limit at r . Since EXX already duced computational effort, by employing the Krieger- | | → ∞ ex has the right asymptotic decay of e (r) 1/(2r) Li-Iafrate (KLI) approximation [94]. We note that other x ∼ − [82], it suffices to verify that ec(r) of Eq. (7) decays ways of defining approximations to the OEP exist [95– faster. Indeed, because the orbitals asymptotically tend 97]. However, for pure exchange earlier works have shown −αiσ r to ϕiσ e , where αiσ = √ 2εiσ, the density that total energies and eigenvalues are obtained with very ∼ − is dominated by the highest occupied orbital, ϕho, and high accuracy in the KLI approximation [26, 94, 95], and 2 −2αhor tends to n ϕho e . Because asymptotically for our local hybrid we explicitly compare KLI results to ∼ | | ∼2 2 τ /τ 1 and t2 e 3 αhor, one finds f t−2 e− 3 αhor, full OEP results in Sec. IV A and find very good agree- W ≈ ∼ ∼ ∼ which makes ec(r) decay exponentially. Therefore, the ment. correct asymptotic behavior at r is achieved. In DARSEC , all computations were converged up to | | → ∞ 5

0.001 Ry in the total energy, Etot, as well as in the highest as well as atoms (no bonding). occupied KS eigenvalue, εho, by appropriately choosing the parameters of the real-space grid and by iterating the self-consistent DFT cycle. For full OEP calculations, IV. RESULTS applying the S-iteration method to the KLI xc potential typically resulted in a reduction of the maximum value of A. Comparison of KLI and OEP the S-function [92] by a factor of 100. The spin and the axial angular momentum of the systems were taken as in While good agreement between the KLI and OEP experiment. Note that to this end, for some systems it scheme has been demonstrated before for ground-state was necessary to force the KS occupation numbers. energy calculations using EXX [101], the accuracy of the Numerical stability of self-consistent computations us- KLI approximation needs to be checked anew when it ing ODFs, in the KLI- or OEP-scheme, mainly depends is applied to a previously untested functional. Table I on the numerical realization of the functional derivative provides this check for our functional. It compares the total energy and the highest occupied KS eigenvalue as r 1 δExc[ ϕjτ ] uiσ( )= ∗ { } , (10) obtained with the OEP and the KLI approximation for ϕ (r) δϕiσ(r) iσ different values of the parameter c (cf. Eq. (7)) for differ- which “conveys” the special character of the correspond- ent systems, and lists the corresponding differences for ing xc functional into the calculation of the xc po- EXX for comparison. tential. Because our functional approach results in a Table I shows that the requirement EOEP EKLI [28] tot ≤ tot rather complicated function uiσ(r) (see Appendix A, is fulfilled independent of the value of c employed. Un- Eq. (A41)), careful analytical restructuring was neces- like for the total energy, there is no theorem stating that sary in order to avoid diverging and unstable calcula- the highest occupied KS eigenvalue found in the OEP tions. In particular, an explicit division by the KS or- scheme must be below its KLI counterpart. For example, bitals or the electron density should be avoided, be- for the C atom and the N2 molecule, we observe the op- cause their exponential decay [98] leads to instabilities posite. Furthermore, because the suggested local hybrid at outer grid points. A numerically stable u (r) was with c = 0 for spin-unpolarized systems (ζ(r)=0 r) iσ ∀ gained by such considerations, for example by replacing is exactly equivalent to the purely semi-local constituent 2 τW (r) = n(r) /(8n(r)) in Eq. (7) with the equiva- functional, one would expect the KLI and OEP results |∇ | 1 r 1 2 r 2 to coincide. This is indeed fulfilled within numerical ac- lent expression τW ( )= 2 n ( ) , or, in case division by the density cannot be|∇ avoided,| by equally balancing curacy. A detailed listing of the total energies and eigen- density terms of the same power in numerator and de- values of the highest occupied KS states obtained by the nominator (for details see Appendix A). KLI approximation in comparison to full OEP can be All results using (semi-)local functionals (LSDA [10], found in Appendix B, Tables IV and V. PBE [17]) or the B3LYP hybrid functional [23] (eval- With increasing c, a larger amount of EXX is employed uated within the generalized KS scheme [99]) were ob- and the functional gains more non-local character, lead- tained with the Turbomole program package [100], using ing to greater deviations between KLI and OEP results. the def2-QZVPP basis set. The pure EXX calculations Note that, within the considered c-range, the deviations were performed in DARSEC by employing the functional with our functional are consequently lower than those derivative uiσ(r) originating from Eq. (1) (as derived in obtained for EXX. The last statement applies to both E and ε . Furthermore, in agreement with Ref. [27] Appendix A, Eq. (A6)). tot | ho| When evaluating a new functional, it is reasonable to (p. 255), we observe an increasing difference between concentrate on a class of relatively simple systems to KLI and OEP results with growing number of electrons keep computational costs low and to refrain from ad- in the system. ditional sources of error beyond the xc approximation, To summarize, using the KLI approximation for our e.g., searching for an optimal geometry in systems with functional is as justified as it is for pure EXX. This ob- many degrees of freedom. However, the systems should servation is in agreement with the fact that EXX is the limiting case of the suggested functional for c . not be too simple, so as to pose a significant challenge → ∞ for the proposed functional. The class of systems has to be large enough, as success or failure for one particular system has very limited meaning. It should also be rich B. Fitting the parameter c for each system enough to try to represent other systems that are not in- cluded. Previous work [17] has shown that a limited set The proposed functional has one unknown parameter, of well selected small molecules can allow for meaningful c. We aim to define a global value for c, relying on fitting exploration of a functional’s properties. For these reasons it such that for a group of selected systems, some prede- we focus on a set of 18 light diatomic molecules: H2, LiH, fined quantity is optimally predicted (possible choices are Li2, LiF, BeH, BH, BO, BF, CH, CN, CO, NH, N2, NO, discussed in detail below). As a prerequisite, we obtain OH, O2, FH, F2, and their constituent atoms. The sys- individual c-values by optimizing the parameter for each tems include single-, double-, and triple-bond molecules of the systems separately. As a test for whether a global 6

System cD cE TABLE I: Comparison of total energy, E, and highest occu- H2 0.552 ± 0.002 0.537 ± 0.012 pied KS eigenvalue, εho, obtained with the suggested local LiH 0.642 ± 0.005 0.556 ± 0.004 hybrid functional and with pure EXX, within both the KLI Li2 1.50 ± 0.06 0.571 ± 0.002 KLI OEP and OEP schemes, as a function of c (∆E = E − E , LiF 0.141 ± 0.006 0.976 ± 0.003 KLI OEP ∆ε = εho − εho ). All values are in Hartree. BeH 0.746 ± 0.025 0.648 ± 0.004 suggested functional EXX BH 0.590 ± 0.010 0.685 ± 0.004 system c = 0 c = 0.5 c = 2.5 BO 0.288 ± 0.007 0.916 ± 0.002 C ∆E 0.0000 0.0002 0.0003 0.0004 BF 0.578 ± 0.027 0.943 ± 0.002 ∆ε -0.0005 0.0001 0.0003 0.0007 CH 0.672 ± 0.028 0.741 ± 0.003 BH ∆E 0.0000 0.0002 0.0005 0.0006 CN 0.146 ± 0.005 0.908 ± 0.002 ∆ε 0.0000 0.0003 0.0004 0.0010 CO 0.283 ± 0.009 0.916 ± 0.002 Li2 ∆E 0.0000 0.0001 0.0002 0.0002 NH 0.667 ± 0.027 0.811 ± 0.004 ∆ε 0.0000 0.0002 0.0005 0.0006 N2 0.107 ± 0.009 0.908 ± 0.003 NH ∆E 0.0001 0.0005 0.0008 0.0011 NO 0.329 ± 0.009 0.960 ± 0.002 ∆ε 0.0007 0.0013 0.0025 0.0055 OH 1.20 ± 0.07 0.942 ± 0.004 N2 ∆E 0.0000 0.0009 0.0017 0.0023 O2 0.472 ± 0.009 1.004 ± 0.002 ∆ε 0.0000 -0.0010 -0.0019 0.0018 FH 0.075 ± 0.011 1.105 ± 0.004 F2 0.356 ± 0.006 1.206 ± 0.003 H — any fitting procedure is meaningful, we verify that these indi- Li — 0.543 ± 0.005 Be — 0.644 ± 0.005 vidual c-values are clustered within a reasonable numer- B — 0.698 ± 0.003 ical range. C — 0.757 ± 0.002 In the following, we present two ways to fit c. One N — 0.848 ± 0.005 possibility is fitting the dissociation energy: To find c O — 0.925 ± 0.004 for the molecule AB, the total energies of the molecule F — 1.067 ± 0.003 and its constituent atoms have to be calculated, with the same c. Then, the dissociation energy D(c) = TABLE II: The parameter c optimized for various systems, using the D- and E-fitting procedures. EA(c) + EB(c) EAB(c) is fitted to its experimental value [102], Dexp−, by varying c. Alternatively, one can compute the total energy of the system for various val- taining the c-dependent average relative errors ues of c and fit it to the experimental total energy. The latter is obtained for atoms as Eexp = Iexp - the atom i i M exp 2 exp − 1 Am(c) A sum of all its experimental IPs, Ii ; For molecules as δ (c)= − . (11) exp exp exp exp P A vM Aexp EAB = EA + EB D . Unless explicitly stated u m=1 − u X otherwise, here and throughout molecular properties are t calculated at their experimental bond lengths [102]. Here, A can refer to the dissociation energy, D, the total Table II presents optimized c-values for various sys- energy, E, or the ionization potential I evaluated via I = ε , the IP-theorem for the exact functional. The tems, obtained from both the D-fitting and the E-fitting − ho procedures. The numerical uncertainty reported for the index m runs over all the systems calculated [135]. c-values is due to the 1 mRy numerical accuracy in the The functions δD(c) and δE(c) are plotted in Figs. 1 total energy. The table confirms that the chosen numeri- and 2, respectively, accompanied by the average relative cal accuracy for the total energy is indeed sufficient. We errors for commonly used functionals: the LSDA, PBE, note that in the E-fitting there is a tendency for c to in- and B3LYP. As mentioned previously, the B3LYP results crease with the electron number, which reflects a larger here and in the following were obtained in the generalized contribution of exact exchange. We attribute this to the KS approach, which we, based on previous experience fact that the energy of the core electrons (which is less [28], expect to yield total energies that are very similar important in D-fitting) is more strongly dominated by ex- to the ones from the KS approach for the systems studied change. For our purposes, the most important conclusion here. For completeness, results obtained with pure EXX to be drawn from Table II is that for all systems exam- evaluated in the KLI approximation are also reported. ined in both approaches, optimal values for c lie between In both figures we observe clear minima for the pro- 0 and 1, and are never larger than 2. This observation posed functional at the values of c0 =0.4 for δD and 0.6 justifies our pursuit of a global value of c . for δE, with minimal error values of 5.3% and 0.09%, respectively. These error values are close to those achieved with the B3LYP functional, and are significantly better than the PBE and LSDA results. Because optimiz- C. Determining a global value for the parameter c ing δD(c) and δE(c) demands almost the same value for c, a satisfying description of both properties is possi- Following the conclusion that the parameter c can in- ble using a common parameter of c = 0.5. For this deed be fitted, we performed a series of calculations, ob- c, the relative error in the dissociation energy ∆Dm = 7

35 PBE 40 LSDA 30 LSDA 25 B3LYP 30 20 (%) (%) I D

δ 20 δ 15

10 PBE 10 EXX 5 B3LYP 0 0 0 1 2 3 4 5 0 1 2 3 4 5 c c

FIG. 1: Average relative error of the dissociation energy, δD, FIG. 3: Average relative error of the IP predicted via the high- as a function of the parameter c (solid line). Relative errors for est occupied KS eigenvalue, εho, as a function of the parame- the LSDA (dashed), PBE (dash-dotted) and B3LYP (dotted) ter c (blue solid line). Relative errors for the LSDA (dashed), functionals are given for comparison. Pure EXX reaches an PBE (dash-dotted) and B3LYP (dotted) functionals, as well EXX error of δD (c) = 66% and exceeds the scale we chose here. as pure EXX(KLI) (purple solid line) are given for comparison.

1.5 LSDA ing the parameter c from εho = ε3↓ to ε5↑ at approxi- mately c =0.7 for NH, and from ε6↓ to ε7↑ at c =1.6 for the BO molecule. Such systems could therefore be good 1 EXX candidates for checking the functional’s ability to predict physically meaningful orbitals in the sense of Ref. [39]. (%) E

δ Last, we checked the previously made assumption that 0.5 experimental bond lengths can be used, assuming they are not very diﬀerent from those obtained by relaxation. PBE To this end, all 18 molecules in the reference set were re- B3LYP laxed, and the obtained bond lengths Lm were compared exp 0 to the experimental values, Lm [102]. It was found that 0 1 2 3 4 5 exp for most systems Lm lies within the computational er- c exp ror for Lm and the diﬀerence Lm Lm is below 0.02 Bohr [137], except F , where |L − Lexp | 0.08 Bohr. 2 | m − m | ≈ FIG. 2: Average relative error of the total energy, δE , as a [138] function of the parameter c (blue solid line). Relative errors for the LSDA (dashed), PBE (dash-dotted) and B3LYP (dotted) functionals, as well as pure EXX(KLI) (purple solid line) are given for comparison. D. Achievements of the suggested functional

In the following, we examine some of the proposed (D Dexp)/Dexp is lowest for the BF molecule (0.7%) functional’s properties at the value of c = 0.5, which m − and highest for Li2 and F2 (14% and 17%, respectively). was determined in Sec.IVC. The relative error in the total energy is more evenly As the functional is one-electron SI-free (see Sec. II), spread around 0.12%. it is important to investigate its behavior in systems The function δI (c) shown in Fig. 3 exhibits a different that are known to suffer from a large self-interaction er- behavior, reaching its minimum of 6 % at a higher value ror when described by standard DFAs. First, for one- of c 4.5. [136] When evaluated at c =0.5, the average electron systems, like H, He+, H+, etc. the functional ≈ 2 relative error is δI (c = 0.5) = 26%. Although lower reduces analytically to the EXX functional, as can be than 31% for B3LYP and 42% for both LSDA and PBE, seen from Eq. (7). Therefore, all the properties of these such a deviation is rather significant. Therefore, Fig. 3 systems are obtained, by construction, exactly. This ad- suggests that in order to reach good agreement between vantage is not shared by (semi-)local or most hybrid func- the experimental IP and εho, a larger amount of EXX tionals. is required. − In particular, Fig. 4 presents the dissociation curve of + Interestingly, when calculating NH and BO, we ob- H2 , obtained with various functionals. It can be seen served that the highest occupied state changes with vary- that, as expected, the curve obtained with the proposed 8

−0.4 −4.4 EXX CCSD(T) LSDA LSDA −4.5 PBE PBE −0.45 B3LYP B3LYP this work −4.6 EXX this work −0.5 −4.7

−4.8 E (Hartree) −0.55 E (Hartree)

−4.9 −0.6 −5

−0.65 0 2 4 6 8 10 0 2 4 6 8 L (Bohr) L (Bohr)

+ + FIG. 4: Dissociation curve of the H2 molecule, for the LSDA FIG. 5: Dissociation curve of the He2 molecule, for the LSDA (squares), PBE (x’s), B3LYP (dots), EXX (solid line) and the (squares), PBE (x’s), B3LYP (dots), EXX (rhombi), and suggested functional (circles). the suggested functional with c = 0.5 (circles), compared to CCSD(T) results from Ref. [104] (dashed line) functional perfectly agrees with the EXX curve, which provides the exact result in this case. In particular, our do not possess the spurious maximum, which appears in the conventional approximations – LSDA, PBE and local hybrid does not exhibit a spurious maximum in the + curve, which appears in conventional approximate func- B3LYP around 4 Bohr. Unlike for the H2 system, here tionals at bond lengths around 5-6 Bohr [103–107] and the proposed functional does not automatically reduce whose electrostatic origin has recently been discussed to the exact expression, and therefore the accurate prediction for He+ in Fig. 5 can be seen as a consequence [108]. The dissociation of neutral H2 is a special chal- 2 lenge for most density functionals and is closely con- of the strong reduction of SI-errors, in agreement with a nected to the question of how static correlation is ac- previous study. [111] counted for [109]. Our local hybrid for H2 yields a bind- We further investigated how well εho corresponds to ing curve that is qualitatively similar to the one obtained the experimental IP [102] for the atoms Li, Na, and K. in pure exchange calculations, i.e., for large internuclear These atoms can be considered as quasi-one-electron sys- separation the lowest energy is obtained with a spin- tems, consisting of electrons arranged in closed shells, polarized atomic density centered around each nucleus, which screen the charge of the nucleus, and one addi- which yields a total energy of 1 Hartree. Quantitatively, tional electron in the last open shell. Table III shows there are differences with respect to the EXX solution: that the εho obtained from our functional evaluated with The point from which on the spin-polarized solution has a c =0.5 is closer to the experimental IP than the εho from lower energy than the spin-unpolarized one lies at about LSDA, PBE, and B3LYP. Note that for these systems a 3.6 Bohr with our local hybrid, and the minimum energy remarkable improvement is achieved, as one would expect is -1.173 Hartree as compared to -1.134 Hartree obtained from their strong one-electron character. with EXX. Generally, delocalization in stretched molecular bonds is conceptually connected to the SI-error of DFAs [105]. V. CONCLUSIONS Therefore, reduction of this error marks a first step to- wards enhancing the description of dissociation processes In this article we presented the construction of a lo- and chemical reactions [110]. cal hybrid functional that combines full exact exchange To examine this, the dissociation curve of the 3- with compatible correlation. The functional respects + electron molecule He2 is shown in Fig. 5. The curve the homogeneous electron gas limit and addresses the achieved with the suggested functional is the closest to one-electron self-interaction error via a one-spin-orbital- the reference result obtained with a highly accurate wave- region indicator. The qualitative improvement that is functions method (see Ref. [104] and references therein). achieved with this construction is reflected in, e.g., dis- + + Here again, only our local hybrid and the EXX curves sociation energy curves for H2 and He2 that are much 9

A third example is provided by the various self- TABLE III: The highest occupied eigenvalue compared to the interaction correction schemes. Different forms of self- experimental IP for Li, Na, and K, computed with four different functionals (LSDA, PBE, B3LYP and our suggested interaction correction greatly improve the interpretabil- functional using c = 0.5). The table contains the absolute ity of the eigenvalues when compared to semi-local func- numbers in Hartree as well as the relative error in %. tionals [11, 37, 111, 125–128], but binding energies are IP −εho not accurately predicted [128, 129] unless the correction system Exp. LSDA PBE B3LYP suggested functional is “scaled down” [126, 130]. Li 0.1981 0.1163 0.1185 0.1311 0.1797 This rather universal difficulty to achieve accurate (-41%)(-40%)(-34%) (-9%) eigenvalues and accurate binding energies at the same Na 0.1886 0.1131 0.1116 0.1251 0.1647 time may indicate that combining these two properties (-40%)(-41%)(-34%) (-13%) may require a type of physics that all present day func- K 0.1595 0.0961 0.0930 0.1038 0.1334 tionals lack [131]. (-40%)(-42%)(-35%) (-16%) Recent work provides two new and interesting perspec- tives on this problem. On the one hand, it has been noted that even a semi-local functional can yield eigenvalues more realistic than the ones obtained from (semi-) local that are qualitatively similar to the ones obtained from functionals and global hybrids. We investigated different bare EXX when the asymptotic properties of a GGA are conditions for fixing the undetermined parameter of the carefully determined [132]. On the other hand, it has functional. When the parameter is fit to minimize bind- recently been shown that an ensemble perspective offers ing energy errors or total energy errors, with respect to new and improved ways of interpreting eigenvalues and experiment, then our local hybrid reaches an accuracy extracting information from semi-local functionals [133]. that is better than LSDA or PBE and similar to the one Exploring in particular this latter option, i.e., combin- afforded by the B3LYP global hybrid. Predicting the first ing the ensemble approach with the present local hybrid ionization energy via the highest occupied eigenvalue ε ho functional, will be the topic of future work and may shed is more accurate with our functional than with LSDA, further light on the question of how to obtain accurate PBE or B3LYP, but still not satisfactorily accurate. binding energies and Kohn-Sham eigenvalues from the When the parameter is fit such that ε should be ho same functional. as close as possible to the experimental− first ionization potential, then the local hybrid functional achieves much smaller errors in this quantity than, e.g., B3LYP. How- ever, the value obtained for the free parameter differs Acknowledgments considerably from the one that was obtained by fitting to binding or total energies. As a result, prediction of these S.K. gratefully acknowledges discussions with energies considerably differs from their experimental val- J. P. Perdew on local hybrids in general and on an ues. Therefore, our local hybrid does allow for reaching early version of this functional in particular. We thank accurate binding energies or accurate highest eigenvalues, Baruch Feldman for fruitful discussions. Financial sup- but not with the same functional parametrization. port by the DFG Graduiertenkolleg 1640, the European Looking at this from a more general perspective, we Research Council, the German-Israeli Foundation, and note that many functionals can achieve good accuracy the Lise Meitner center for computational chemistry on one of the aforementioned properties or the other, is gratefully acknowledged. E.K. is a recipient of the but not on both properties at the same time. Levzion scholarship. T.S. acknowledges support from A first example are global hybrids. With the usual the Elite Network of Bavaria (“Macromolecular Science” 0.2 to 0.25 fraction of exact exchange they yield good program). binding energies, but highest eigenvalues that are considerably too small in magnitude. Increasing the aforementioned fraction to 0.75 leads to improved gap pre- Appendix A: Derivation of the correlation potential diction [112–114]. However,∼ such a large fraction of exact exchange can compromise significantly the accuracy In order to employ the OEP formalism [26, 28] (or in thermochemical [24, 25, 115] or electronic structure its KLI approximation [94]), one has to provide an an- properties. [45, 48, 57] alytical expression for the functional derivative of the A second example are range-separated hybrid function- explicitly orbital-dependent exchange-correlation energy, als. When combined with a tuning procedure based on Exc[ ϕiσ ], with respect to the orbitals ϕiσ . For the the IP theorem [57, 116–123], they allow for obtaining functional{ } proposed in the present contribution,{ } eigenvalues that reflect ionization potentials very accu- ex iso sl rately by construction. However, the typical value of the Exc[ ϕiσ ]= Ex [ ϕiσ ]+ Ec [ ϕiσ ]+ Ec [ ϕiσ ], range-separation parameter that is reached by such tun- { } { } { } { (A1)} ex ing is quite different from the one that is reached when where Ex [ ϕiσ ] is the exact exchange defined in Eq.(1), sl { } atomization energy errors are minimized via the range- Ec [ ϕiσ ] is the semi-local correlation energy, whose en- { } sl r separation parameter [124]. ergy density per particle, ec ( ), is given in Eq. (9), and 10

Eiso[ ϕ ] equals b) Von Weizs¨acker kinetic energy density c { iσ} 2 n(r) 1 1 iso ′ ′ LSDA ′ ex ′ 3 ′ r 2 r 2 E [ ϕ ]= f(r ) n(r ) e (r ) e (r ) d r , τW ( )= |∇ | = n ( ) ; (A11) c { iσ} x − x 8n(r) 2|∇ | Z (A2) c) spin polarization with the LMF function, f(r), being deﬁned in Eq. (8). Due to the additive structure of Eq. (A1), the func- n↑(r) n↓(r) ζ(r)= − tional derivative n↑(r)+ n↓(r) . r 1 δExc[ ϕiσ ] uiσ( )= ∗ r { r } (A3) Taking the functional derivative based on Eq. (A7) re- ϕiσ( ) δϕiσ ( ) sults in two parts: can be split in three terms: r′ ∗ r sl r δg( ) r′ 3 ′ ϕiσ( )uiσ ( ) = r Q( ) d r r exx r iso r sl r δϕiσ ( ) uiσ( )= uiσ ( )+ uiσ ( )+ uiσ( ), (A4) Z δQ(r′) + g(r′) d3r′ (A12) which are considered separately in the following. δϕ (r) Z iσ By denoting the constituent functions of the function g(r) 1. The exact exchange contribution by ψ1(r)= τ(r), ψ2(r)= τW (r) and ψ3(r)= ζ(r), chain rule arguments lead to the following expression:

The ﬁrst term on the RHS of Eq. (A4) can be computed ′ 3 ′ ′ δg(r ) δψl(r ) δg(r ) directly from the exact-exchange expression (Eq. (1)) = ′ (A13) δϕiσ (r) δϕiσ (r) δψl(r ) l=1 Nσ ϕ∗ (r)ϕ (r)ϕ (r′)ϕ∗ (r′) X ex 1 iσ jσ iσ jσ 3 3 ′ Here we explicitly took into account the fact that g(r) Ex = d rd r −2 r r′ depends on ψ (r) locally. i,j=1 Z Z | − | l σX=↑,↓ In order to obtain an analytical expression for ′ (A5) δg(r )/δϕiσ (r), which can then be inserted into and simply reads Eq. (A12), one has to consider the three constituent functions ψ (r) separately: Nσ l ϕ∗ (r′)ϕ (r′) ϕ∗ (r) uexx(r)= ϕ∗ (r) iσ jσ d3r′. iσ iσ − jσ r r′ l = 1 : j=1 Z X | − | r′ (A6) δτ( ) 1 r r′ ′2 ∗ r′ ′ ∗ r′ ′ = δ( ) ϕiσ ( ) + ( ϕiσ( )) δϕiσ(r) −2 − ∇ ∇ · ∇ (A14) 2. The semi-local correlation contribution δg(r′) τ (r′)ζ2(r′) = W (A15) δτ(r′) τ 2(r′) The self-interaction-free semi-local correlation energy sl contribution Ec [ ϕiσ ] is deﬁned by l = 2 : { } ′ ∗ ′ δτW (r ) ϕ (r ) 1 iσ r r′ ′2 2 r′ sl r′ r′ 3 ′ = 1 δ( ) (n ( ))+ Ec [ ϕiσ ]= g( )Q( ) d r , (A7) δϕ (r) − 2 r′ − ∇ { } iσ 2n ( ) Z 1 ( ′n 2 (r′)) ′ (A16) where ∇ · ∇ r δg(r′) ζ2(r′) τW ( ) 2 = (A17) g(r)=1 ζ (r) (A8) r′ r′ − τ(r) δτW ( ) − τ( ) l = 3 : and r′ r′ δζ( ) ∗ r′ r r′ δσ ζ( ) r r LSDA r = ϕiσ ( )δ( ) − (A18) Q( )= n( )ec ( ). (A9) δϕ (r) − n(r′) iσ δg(r′) 2τ (r′)ζ(r′) For completeness, we list out all the quantities that = W (A19) are required to construct the function g(r): δζ(r′) − τ(r′) Here the operator ′ denotes a gradient relative to the a) kinetic energy density ′ ∇ coordinate (r ) and the quantity δσ distinguishes between

Nσ the two spin channels by 1 τ(r)= ϕ (r) 2; (A10) 2 |∇ iσ | 1 if σ = i=1 δσ = ↑ (A20) σX=↑,↓ 1 if σ = . (− ↓ 11

It is noted that the functional derivatives above have the with presented form with respect to the occupied orbitals only; 1 2 2 Φ(r)= (1 + ζ(r)) 3 + (1 ζ(r)) 3 (A26) derivatives with respect to unoccupied orbitals equal 2 − zero. h i The derived relations (A14) - (A19) now have to be and 1 1 2 inserted via Eq. (A13) into Eq. (A12). By further em- π 6 a ploying a chain rule argument for the second term on the a = 0 = const. 3 4 RHS of Eq. (A12) The exact relation r′ r′ r′′ δQ( ) δQ( ) δnτ ( ) ′′ 8τ (r) = d3r 2 r W r r′′ r tn( )= 4 (A27) δϕ ( ) δn ( ) δϕ ( ) 3 r iσ ↑ ↓ τ iσ n ( ) τX= , Z r′ ∗ r δQ( ) ∗ r LSDA r′ r r′ will be useful for later derivations. = ϕiσ ( ) = ϕiσ( )vc,σ ( )δ( ), δnσ(r) − Analogously to Eq. (A12), the application of the func- (A21) tional derivative with respect to the KS-orbitals leads to two contributions: one arrives at the final expression of the functional deriva- r′ ∗ r iso r δf( ) r′ 3 ′ sl ϕiσ( )uiσ ( ) = P ( ) d r tive of Ec [ ϕiσ ]: δϕ (r) { } Z iσ r′ ∗ r sl r ′ δP ( ) 3 ′ ϕiσ ( )uiσ( )= + f(r ) d r (A28) δϕ (r) 1 δg(r) δg(r) Z iσ 2ϕ∗ (r) Q(r)+ ϕ∗ (r) Q(r) − 2 ∇ iσ δτ(r) ∇ iσ · ∇ δτ(r) Moreover, an analogous relation to Eq. (A13) helps to ∗ rewrite the first part of this equation, only that now one ϕ (r) 1 δg(r) iσ 2 2 r r r 2 r 1 n ( ) Q( )+ has to consider also the functions ψ4( ) = tn( ) and 2 r r − 2n ( ) ∇ δτW ( ) ψ (r)=Φ(r): 5 1 δg(r) n 2 (r) Q(r) r′ 5 r′ r′ r δf( ) δψl( ) δf( ) ∇ · ∇ δτW ( ) = (A29) r r r′ r δϕiσ ( ) δϕiσ ( ) δψl( ) ∗ r r δg( ) LSDA r l=1 + ϕiσ ( ) (δσ ζ( )) ec ( ) X − δζ(r) We evaluate each term separately and obtain: ∗ r r LSDA r + ϕiσ ( ) g( ) vc,σ ( ) (A22) l = 1 : ′ τW (r ) 2 ′ r′ 2 ′ ζ (r ) r′ r′ Equation (A22) corresponds to the functional derivative δf( ) τ (r ) δf( ) τW ( ) ′ = 2 ′ = ′ ′ (A30) the way it was implemented into the KLI/OEP-routine δτ(r ) 1+ c t (r ) −δτW (r ) τ(r ) DARSEC · in the program package . l = 2 : δf(r′) ζ2(r′) = (A31) iso r′ r′ 2 r′ 3. The contribution E [{ϕ }] δτW ( ) −τ( )(1+ c t ( )) c iσ · l = 3 : iso The correlation term E [ ϕ ] is defined in Eq. (A2). r′ c iσ τW ( ) r′ { } δf(r′) 2 r′ ζ( ) Let us denote = τ( ) (A32) δζ(r′) −1+ c t2(r′) P (r)= n(r) eLSDA(r) eex(r) (A23) · x − x l = 4 : r 2 r′ r′ r′ ∗ r′ and recall that the LMF function f( ) equals δtn( ) δτW ( ) 1 32 τW ( )ϕiσ ( ) r r′ r =8 r 4 7 δ( ) δϕiσ ( ) δϕiσ( ) n 3 (r′) − 3 n 3 (r′) − τW (r) 2 r 1 r ζ ( ) (A33) f(r)= − τ( ) (A24) 1+ c t2(r) δf(r′) ca2f(r′) · = (A34) 2 r′ 2 r′ 2 r′ δtn( ) −Φ ( )(1+ c t ( )) In addition to the quantities τ, τW and ζ introduced · above, the function f(r) additionally employs the so- l = 5 : called reduced density gradient [17] δΦ(r′) δΦ(r′) δζ(r′) = ′ 1 δϕiσ(r) δζ(r ) δϕiσ (r) 1 2 π 6 a n(r) 1 1 r 0 1 − − t( ) = r |∇7 | = (1 + ζ(r′)) 3 (1 ζ(r′)) 3 3 4Φ( ) n 6 (r) 3 − − · t (r) h δ ζ(r′) i := a n (A25) ϕ∗ (r′)δ(r r′) σ − (A35) Φ(r) iσ − n(r′) 12

To avoid numerical instability due to the negative powers While the ﬁrst term contributes simply via the regular 1 1 r′ 3 of 3 , we multiply the above relation by (1+ζ( )) (1 density-dependent LSDA exchange potential (similar to − 1 − ζ(r′)) 3 and then divide by the same term expressed in Eq. (A21)), requires the second term explicit evaluation terms of the spin-densities. We then obtain of the exact exchange energy density

δΦ(r′) 1 1 1 r′ 3 r′ 3 r = (1 + ζ( )) (1 ζ( )) δϕiσ ( ) −3 − − · Nυ ∗ r′ r′ r′′ ∗ r′′ 1 ϕ ( )ϕqυ( )ϕkυ ( )ϕqυ( ) h2 i r′ ex r′ kυ 3 ′′ 3 r′ r′ n( )ex ( )= d r . n ( ) ∗ ′ ′ δσ ζ( ) −2 r′ r′′ 1 ϕ (r ) δ(r r ) 2 iσ −r′ k,q=1Z | − | 3 r′ r′ 3 · − n( ) X↑ ↓ 2 (n↑( )n↓( )) υ= , (A36) (A39) Therefore, Eq. (A38) results in δf(r′) 2ct2(r′)f(r′) = (A37) δΦ(r′) Φ(r′)(1+ c t2(r′)) · r′ In order to compute the ﬁrst term of Eq. (A28), one δP ( ) ∗ r LSDA r′ r r′ = ϕiσ ( )vx,σ ( )δ( ) now has to evaluate all the contributions originating from δϕiσ(r) − the diﬀerent ψ (Eqs. (A14), (A16), (A18), (A30), (A31), Nσ l 1 ϕ (r′′)ϕ∗ (r′′) (A32), (A33), (A34), (A36), (A37)) via the chain rule + δ(r r′)ϕ∗ (r′) kσ iσ d3r′′ kσ r′ r′′ argument (A29). 2 − Xk=1 Z | − | Similar considerations are now used for the second Nσ ∗ ′ ′ ∗ 1 ϕ (r )ϕqσ (r )ϕ (r) term on the RHS Eq. (A28). By applying chain rule + iσ qσ . (A40) 2 r′ r arguments only to the semi-local energy density part of q=1 | − | P (r), one arrives at the following equation: X

r′ δ n(r′)eLSDA(r′) r′′ δP ( ) x δnτ ( ) 3 ′′ Evaluating this expression with the corresponding inte- = ′′ d r δϕiσ(r) δnτ (r ) δϕiσ (r) τ=↑,↓ gral in Eq. (A28) and adding the previously derived ﬁrst X Z r′ ex r′ term, one arrives at the ﬁnal expression for the functional δ (n( )ex ( )) iso (A38) derivative of Ec [ ϕ ] with respect to the KS orbitals: − δϕiσ (r) { }

Nσ f(r) 1 ϕ∗ (r′)ϕ (r′) ϕ∗ (r)uiso(r)= ϕ∗ (r) uexx(r)+ ϕ∗ (r) f(r′) iσ jσ d3r′ + ϕ∗ (r)f(r)vLSDA(r) iσ iσ − 2 iσ iσ 2 jσ r r′ iσ x,σ j=1 X Z | − | ∗ 1 δf(r) δf(r) ϕ (r) 1 δf(r) 2 ∗ r r ∗ r r iσ 2 2 r r ϕiσ( ) r P ( )+ ϕiσ ( ) r P ( ) 1 n ( ) r P ( )+ − 2 ∇ δτ( ) ∇ · ∇ δτ( ) − 2n 2 (r) ∇ δτW ( ) 1 δf(r) n 2 (r) P (r) ∇ · ∇ δτW (r) 2 1 1 ∗ r 3 r r 1 3 3 ϕiσ ( )n ( ) δf( ) LSDA ex (1 + ζ(r)) (1 ζ(r)) 1 (δ ζ(r)) e (r) e (r) 2 σ r x x − 3 − − 2 3 (n (r)n (r)) 3 − δΦ( ) − h i ↑ ↓ ∗ r r r 2ϕiσ( ) 2 r LSDA r ex r δf( ) 28 r LSDA r ex r δf( ) 4 n( ) ex ( ) ex ( ) 2 τW ( ) ex ( ) ex ( ) 2 + − 3 r ∇ − δt (r) − 3 − δt (r) n ( ) n n δf(r) n(r) eLSDA(r) eex(r) ∇ · ∇ x − x δt2 (r) n 2ct2(r) δf(r) + ϕ∗ (r) f(r) eLSDA(r) eex(r) + ϕ∗ (r) (δ ζ(r)) eLSDA(r) eex(r) (A41) iσ · 1+ ct2(r) x − x iσ σ − δζ(r) x − x

r′ Finally, we note that when numerically implementing δΦ( ) and the quantity δϕiσ (r) as in Eq. (A36) is highly advan- such complex expressions, questions of numerical stabil- tageous. In addition, we store τW (r)/τ(r) as a separate ity may emerge. We found that implementing the von 1 quantity, enforcing the exact condition that it is never Weizs¨acker kinetic energy density as τ (r)= 1 n 2 (r) 2 W 2 |∇ | larger than 1. We also store separately the quantity 13

(1 + ct2(r))−1 and express ct2(r)/(1 + ct2(r)) in terms of the former, to avoid the divergence of t(r) at large TABLE V: Comparison of highest occupied orbital energy εho using the suggested local hybrid functional in both the distances. KLI and OEP schemes, as a function of c. All values are in Hartree.

Appendix B: OEP/KLI comparison KLI OEP system c KLI OEP εho − εho C 0 -0.2740 -0.2736 -0.0005 This Appendix reports detailed numerical results for 0.5 -0.3067 -0.3068 0.0001 the total energies, Etot, as well as the eigenvalues of the 2.5 -0.3688 -0.3691 0.0003 BH 0 -0.2031 -0.2031 0.0000 highest occupied KS state, εho, using the proposed local hybrid functional for selected systems: the BH, Li , 0.5 -0.2412 -0.2415 0.0003 2 2.5 -0.3043 -0.3047 0.0004 NH, and the N2 molecules, as well as the C atom. A Li2 0 -0.1189 -0.1189 0.0000 multiplicative, local KS potential was obtained by em- 0.5 -0.1286 -0.1289 0.0002 ploying the functional derivative of Eq. (A41) either in 2.5 -0.1522 -0.1527 0.0005 the full OEP scheme or by the KLI approximation. Ta- NH 0 -0.3157 -0.3164 0.0007 ble IV lists the absolute values of Etot for KLI and OEP, 0.5 -0.3770 -0.3783 0.0013 as well as the diﬀerences between results obtained with 2.5 -0.4581 -0.4607 0.0025 both schemes. Table V provides the same comparsion for N2 0 -0.3825 -0.3825 0.0000 εho. 0.5 -0.4456 -0.4447 -0.0010 2.5 -0.5463 -0.5444 -0.0019 Note that the systems BH, Li2, and N2 are spin- unpolarized. Therefore, for c = 0 the functional reduces to the LSDA xc functional (cf. Eq. (7)) and thus no diﬀer- ence between KLI and OEP should occur. This is indeed the case, within numerical accuracy.

TABLE IV: Comparison of total energy, Etot, using the suggested local hybrid functional in both the KLI and OEP schemes, as a function of c. All values are in Hartree.

KLI OEP system c KLI OEP Etot − Etot C 0 -37.4804 -37.4804 0.0000 0.5 -37.8108 -37.8110 0.0002 2.5 -37.9494 -37.9497 0.0003 BH 0 -24.9768 -24.9768 0.0000 0.5 -25.2612 -25.2614 0.0002 2.5 -25.3983 -25.3988 0.0005 Li2 0 -14.7244 -14.7244 0.0000 0.5 -14.9809 -14.9810 0.0001 2.5 -15.1245 -15.1247 0.0002 NH 0 -54.7769 -54.7770 0.0001 0.5 -55.1769 -55.1774 0.0005 2.5 -55.3555 -55.3563 0.0008 N2 0 -108.6958 -108.6958 0.0000 0.5 -109.4464 -109.4474 0.0009 2.5 -109.7593 -109.7609 0.0017

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[136] When calculating δI (c), the vertical experimental extremely shallow E(L) minimum. The uncertainty of ionization potentials were used (see Ref. [102] and 1 mRy in the total energy translates into a numerical http://webbook.nist.gov) uncertainty in the bond length of 0.16 Bohr and 0.26 exp [137] The numerical error in Lm is governed by the accuracy Bohr, respectively. Therefore, the diﬀerence |Lm −Lm |, of 1 mRy in the total energy, rather than by the con- being 0.04 Bohr and 0.22 Bohr, although large, has no vergence of the relaxation process. actual meaning due to the large numerical uncertainty. [138] Two exceptional cases are LiH and Li2, which have an