Jellium-with-gap model applied to semilocal kinetic functionals
aa r X i v : . [ c ond - m a t . o t h e r] M a y Jellium-with-gap model applied to semilocal kinetic functionals
Lucian A. Constantin, Eduardo Fabiano,
2, 1
Szymon ´Smiga,
3, 4, 1 and Fabio Della Sala
2, 1 Center for Biomolecular Nanotechnologies @UNILE,Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy Institute for Microelectronics and Microsystems (CNR-IMM),Via Monteroni, Campus Unisalento, 73100 Lecce, Italy Istituto Nanoscienze-CNR, Italy Institute of Physics, Faculty of Physics, Astronomy and Informatics,Nicolaus Copernicus University, Grudziadzka 5, 87-100 Torun, Poland (Dated: April 16, 2018)We investigate a highly-nonlocal generalization of the Lindhard function, given by the jellium-with-gap model. We find a band-gap-dependent gradient expansion of the kinetic energy, whichperforms noticeably well for large atoms. Using the static linear response theory and the simplestsemilocal model for the local band gap, we derive a non-empirical generalized gradient approximation(GGA) of the kinetic energy. This GGA kinetic energy functional is remarkably accurate for thedescription of weakly interacting molecular systems within the subsystem formulation of DensityFunctional Theory.
PACS numbers: 71.10.Ca,71.15.Mb,71.45.Gm
I. INTRODUCTION
Density Functional Theory (DFT) is the most usedcomputational method for electronic structure calcula-tions of molecular and extended systems, providing thehighest accuracy/computational cost ratio. In the con-ventional DFT formalism, the Kohn-Sham (KS) scheme ,the ground-state electronic density n ( r ) is determinedfrom a set of auxiliary KS orbitals ( φ i ( r )): the KS-DFT method is exact but for the approximations of theexchange-correlation (XC) functional. However, for largescale calculations, the computational cost of KS-DFT be-comes unaffordable, as one needs to compute all the oc-cupied KS orbitals in order to construct the density as n ( r ) = P occ.i f i | φ i ( r ) | , where f i is the occupation num-ber (2, for closed-shell systems).Among other linear scaling methods , two DFTmethods are attracting strong interest: i) In the orbital-free version of DFT (OF-DFT) , n ( r ) can be com-puted directly via the Euler equation , without the needof KS orbtials; ii) In the subsystem version of DFT (Sub-DFT) , also known as Frozen-Density-Embedding(FDE), n ( r ) is computed as the sum of the electronic den-sities of several (smaller) subsystems in which the totalsystem is partitioned, which can be computed simultane-ously. Both approaches allow in principle calculations oflarge systems, but the final accuracy depends directly onthe approximations of the non-interacting kinetic energy(KE) functional T s (which are definitely more importantthan the ones for the XC energy, that are also present instandard KS calculations). We recall that the exact KSKE functional is: T exacts = 12 occ. X i Z f i |∇ φ i ( r ) | d r . (1)Thus the KE is explicitly known only as a function of φ i but not as a functional of n . On the other hand, in Sub-DFT the interaction be-tween the subsystems is taken into account via the socalled embedding potentials , which depends on thenon-additive-KE: in the case of just two subsystems (Aand B) it is T nadds [ n A ; n B ] = T s [ n A + n B ] − T s [ n A ] − T s [ n B ].The development of an accurate approximation of T s [ n ] (and/or T nadds [ n A ; n B ]) is one of the biggestDFT challenges . Nowadays, the most sophisti-cated KE approximations have been constructed to beexact for the linear response of jellium model, by in-corporating the Lindhard function in their fully non-local expressions . We recall that the Lindhardfunction F Lind = (cid:18)
12 + 1 − η η ln (cid:12)(cid:12)(cid:12)(cid:12) η − η (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − , (2)where η = k/ (2 k F ) is the dimensionless momentum( k F = (3 π n ) / being the Fermi wave vector of the jel-lium model with the constant density n ), is related to theJellium density response χ Jell via − χ Jell. = π k F F Lind . (3)The non-local KE functionals based on the Lindhardfunction are accurate for simple metals where the nearly-free electron gas is an excellent model but they can notdescribe well semiconductors and insulators, where thedensity response function behaves as − χ Semic. ( k ) −→ k → bk , (4)with b being positive and material-dependent. SeveralKE functionals have been constructed to improve the de-scription of semiconductors , but Eq. (4) has not beenexplicitly used in their expressions due to the lack of asophisticated analytical form that can recover both theLindhard function and Eq. (4).In this article, we will investigate the generalizationof the Lindhard function for the jellium-with-gap modelwhich satisfies Eq. (4).The jellium-with-gap model , was developed outsidethe KS framework, using perturbation theory to takeinto account the band gap energy. This model wasused to have qualitative and quantitative insight forsemiconductors , to develop an XC kernel for theoptical properties of materials , and to construct accu-rate correlation energy functionals for the ground-stateDFT . We will show that the Lindhard functionfor the jellium-with-gap model ( F GAP ), previously in-troduced by Levine and Louie in a different context(dielectric constant and XC potential), may be seen asa sophisticated analytical form suitable for KE approxi-mations.The article is organized as follow:In Section II we discuss the properties of F GAP , wederive its (band-gap-dependent) KE gradient expansion,and we assess it for large atoms. By using a local gapmodel, we propose a simple KE gradient expansion thatit is very accurate for the semiclassical atom theory.In Section III we discuss the implications of this re-sult in DFT by constructing a simple KE functional atthe Generalized Gradient Approximation (GGA) level oftheory based on the gradient expansion of the jellium-with-gap model. GGA KE functionals are computation-ally very efficient and play a key role for the simulationof large systems. We mention that the development ofsemilocal KE functionals is nowadays an active field .Finally, in Section IV we summarize our results.
II. THEORYA. Properties and gradient expansions for thejellium model
For the conventional infinite jellium model, the Lind-hard function behaves as: F Lind → η + 845 η + O ( η ) , for η → , (5) F Lind → η − − η + O ( η − ) , for η → ∞ . (6)Equation (5) contains important physics that has beenused in the construction of semilocal KE densityfunctionals . Thus, the KE gradient expansion whichrecovers the first three terms in the right hand side ofEq. (5) can be easily derived (see also Eqs. (15) and(16) in section II-C and the corresponding discussion). Itis T Lind s [ n ] = Z d r τ T F (cid:18) s + 881 q (cid:19) , (7) where τ T F = (3 π ) / n / is the Thomas-Fermi KEdensity , which is exact for the jellium model, and s = |∇ n | / [2 k F n ], q = ∇ n / [4(3 π ) / n / ] are the re-duced gradient and Laplacian, respectively. Equation (7)resembles the second-order gradient expansion (GE2) T GE s [ n ] = Z d r τ T F (1 + 527 s )(derived also within the linear response of the jel-lium model), as well as the fourth-order gradientexpansion of the KE T GE s [ n ] = Z d r τ T F (1 + 527 s + 881 q − s q + 8243 s ) , with the exception of the terms ∝ s q , ∝ s , which arebeyond the linear response.Note that F Lind ( η = 0) = 1 is the leading termin the expansion of Eq. (5) and it corresponds tothe Thomas-Fermi local density approximation, whoselinear response in the wave vector space is just theFourier transform of the second-functional derivative, i.e. δ T T Fs /δn ( r ) δn ( r ′ ) ∼ k − F δ ( r − r ′ ). We recall that thelimit η = 0 is very powerful, being also used in the con-struction of the adiabatic local density approximation(ALDA) XC kernel of the linear response time-dependentDFT . B. Properties of the Lindhard function for thejellium-with-gap model
Levine and Louie proposed the density-responsefunction χ GAP ( k, ω ) of the jellium-with-gap model, andthe corresponding [i.e. from Eq. (3)] Lindhard functionfor jellium-with-gap model is1 /F GAP = 12 − ∆(arctan( η +4 η ∆ ) + arctan( η − η ∆ ))8 η ++( ∆ η + 18 η − η + (4 η + 4 η ) ∆ + (4 η − η ) ) , (8)where ∆ = 2 E g /k F and E g is the gap.For a given ∆, a series expansion of F GAP for η → F GAP −→ η + 95 + 3175 175∆ − η + − − η + O ( η ) when η → . (9)Thus, for any system with ∆ > F GAP ∝ ∆ η − . This term is correct (see Eq. (4)) and it hasbeen also used in the jellium-with-gap XC kernel , whichgives accurate optical absorption spectra of semiconduc-tors and insulators. On the other hand, if we first performa series expansion for ∆ →
0, and then a series expansionfor η → F GAP −→ (cid:20) η + 845 η + ... (cid:21) +∆ (cid:20) π η + π η + 7 π η + ... (cid:21) + ∆ " π −
464 1 η + 3 π − (cid:18) − π (cid:19) η + (cid:18) − π (cid:19) η + ... + ... . (10)Equation (10) confirms that, by construction, we have F GAP = F Lind , when ∆ = 0 . (11)Inspection of Eqs. (9) and (10) clearly shows thatlim ∆ → lim η → F GAP = ∞ , (12)lim η → lim ∆ → F GAP = 1 , (13)meaning that F GAP has an “order of limits problem”.Such a situation is common in DFT. For example, we re-call that several meta-GGA XC functionals (e.g. TPSS ,revTPSS , BLOC , SA-TPSS , VT { } ) suf-fer of such a order of limits problem. Nonetheless, theyare accurate for many systems and properties, showingrealistic system-averaged XC hole models .In the opposite limit, i.e. for η → ∞ , we have F GAP → η −
35 +( − ) 1 η + O ( 1 η ) . (14)Therefore, in this limit, F GAP always behaves as F Lind for ∆ = 0.In the upper panel of Fig. 1, we show 1 /F GAP forseveral values of ∆. The plots are all smooth. At large η , F GAP recovers the Lindhard function [see Eq. (14)],while at small η it is driven by the term ∝ η − . The plotof the linear response of T Lind s [ n ] (Eq. (5)) is also givenfor comparison. In the lower panel of Fig. 1 we reportthe accuracy of Eq. (10), considering only the termsexplicitly indicated in the equation, for ∆ = 0 . , . , and1. Even for the case ∆ = 1, this expansion is still veryaccurate for η ≤ C. Kinetic energy gradient expansions from thelinear-response of the jellium-with-gap model
Next we proceed to build the linear-response jellium-with-gap KE gradient expansion, that should recover Eq.(7) when ∆ = 0. To this purpose, we consider the GAP4expansion, with the general form of the KE fourth-ordergradient expansion T GAP s [ n ] = Z d r τ T F ( a s + a s + a + a s + a q ++ a s + a sq + a s + a s + a q + a s q ) . (15) / F ∆ =0 (Lindhard) ∆ =0.1 ∆ =1 ∆ =10Eq. (3) η / F ∆ =0.1 ∆ =0.5 ∆ =1 FIG. 1. Upper panel:1 /F GAP versus η for various values of∆. Also shown is the small- η expansion of Eq. (5). Lowerpanel: Comparison between F GAP (solid-lines) and the ex-pansion (dashed-lines) of Eq. (10), for ∆ = 0 .
1, 0.5, and 1,respectively.
Performing the linear response of such a functional F ( η ) = k F π F (cid:18) δ T s [ n ] δn ( r ) δn ( r ′ ) | n (cid:19) , (16)where F represents the Fourier transform, we can find thecoefficients a i , by comparing term-by-term with Eq. (10).Nevertheless, the straightforward calculation of Eq. (15)requires a tedious and long algebra . Instead, a moreelegant and simpler way to obtain the linear response ofa given semilocal functional has been proposed in Ref.67: consider a small pertubation in density at r = , ofthe form n = n + n k e i kr , such that ∇ n = n k i k e i kr , and ∇ n = − n k k e i kr , with n k ≪ n . Thus, at r = , theseexpressions are simply n = n + n k , ∇ n = n k i k , and ∇ n = − n k k . Inserting them in the functional expres-sion, the linear response is obtained as twice the second-order coefficient of the series expansion with respect to n k /n . After some algebra, the KE gradient expansionwhich gives the linear response of Eq. (10) is found to be T GAP s [ n ] = Z d r τ T F [∆ π −
464 1 s + ∆ 5 π
72 1 s + 1 +∆ ( π −
112 ) + ∆ 5 π s + ( 527 + ∆ ( − π s +∆ − π sq + ( 881 + ∆ ( − π q ] . (17)The terms ∝ s − and ∝ s − account for the terms ∝ η − and ∝ η − of Eq. (10). These terms contribute only for anon-zero gap, i.e. in semiconductors and insulators, butnot in metals. At ∆ = 0, T GAP s [ n ] correctly recovers T Lind s [ n ].To test T GAP s [ n ], we perform calculations for no-ble atoms, up to Z = 290 electrons, using LDA or-bitals and densities, in the Engel code . We consider∆ = 2 E g /k F ( r ) with E g being the KS band gap of theatoms. Because the gradient expansion is well definedonly at small gradients and small-∆, we perform all theintegrations over the volume V defined by the conditions − ≤ q ≤ ≤
1, in a similar manner as in Ref.70. The results are reported in Table I. For small atoms(Ne and Ar), the GE2 is more accurate than T Lind s [ n ]and T GAP s [ n ]. However, we recall that in the case ofa small number of electrons, the semiclassical and sta-tistical concepts beyond the gradient expansions do nothold. In fact, for larger atoms (Kr to the noble atomwith 290 electrons), both T GAP s and T Lind s outperformGE2. In particular, T GAP s shows the best performance,improving over T Lind s [ n ] and proving that, due to theinclusion of the gap, F GAP contains important physicsbeyond F Lind . D. Local band-gap
In order to use Eq. (17) in semilocal DFT, we need toreplace the true band gap E g , with a density dependentlocal band gap. There are several models for the localband gap , constructed from the exponentially decay-ing density behavior or from conditions of the correla-tion energy . In the slowly-varying density limit, theybehave as E g ∼ |∇ n | m , with m ≥
2. However, none ofthem can be considered accurate in this density regime.On the other hand, under a uniform density scaling n λ ( r ) = λ n ( λ r ), the local band gap should behave as E g ∼ λ . This condition is fulfilled by the general for-mula E g ( r ) = a |∇ n ( r ) | m /n ( r ) m − / , m ≥ , a ≥ . (18)Because other exact conditions of the local gap in theslowly varying density limit are not known, we use Eq.(18) in the expression of T GAP s , considering the casewith m = 2. We fix the parameter a requiring that thegradient expansion should recover the first two terms ofthe kinetic energy asymptotic expansion for the large,neutral atom T s = c Z / + c Z + c Z / + ..., (19)where Z is the number of electrons. The first term inEq. (19) is the Thomas-Fermi one , the second is theScott correction due to the atomic inner core , and thelast term accounts for quantum oscillations . Theexact coefficients are shown in the first line of Table II.As in Ref. 77, we assume that any gradient expansionthat is exact for the uniform electron gas, should have theexact c coefficient. The calculation of c and c has beendone using the method proposed in Ref. 77. We recallthat the semiclassical atom theory has been often usedin the development of exchange functionals and occasionally also for kinetic energy functionals . Finally,we mention that these gradient expansions are models forthe total KE, and not for the KE density, where the useof the reduced Laplacian q (which is not present in linearresponse of the jellium model) is essential .Using the procedure described above, we find a =0 . T LGAP − GEs = R d r τ T F [1 + a π s + ( + a π − ) s + a π s + O ( |∇ n | )]= R d r τ T F [1 + 0 . s + 0 . s + 0 . s ] . (20)Note that in Eq. (20) only terms to up s are considered(terms in Eq. (17) proportional to q or q are neglected,as these terms will correspond to s ).As shown in Table II, LGAP-GE gives a very accuratelarge- Z expansion, having the c coefficient close to ex-act. The results for noble atoms are reported in Table I.LGAP-GE is reasonably accurate for all atoms and, asexpected due to the inclusion of the semiclassical atomtheory, the accuracy increases with the number of elec-trons.One additional observation is that LGAP-GE containsodd powers of the reduced gradient, in contrast with F Lind . Nevertheless, Ou-Yang and Mel Levy have al-ready shown that using non-uniform coordinates scal-ing requirements , the GE4 terms in the KE gradi-ent expansion can be replaced by an s -only dependentterm , whose coefficient must be positive (and was fit-ted to the Xe atom). The resulting simple KE func-tional, that behaves better than GE4 for the non-uniformdensity scaling, has the following enhancement factor( F s = τ approx /τ T F ): F OL s = 1 + 527 s + cs, (21)with c = 0 . F s ) has the general form δT s δn = ∂τ T F ∂n F s ( s )+ τ T F ∂F s ∂s ∂s∂n −∇· [ 1 s ∂F s ∂s · ∇ nn / ] , (22)a necessary condition for it to be well defined is | s ∂F s ∂s | < ∞ . This is not satisfied by the LGAP-GE (and OL1).Thus, the term ∝ s gives a diverging kinetic potential( δT s /δn → ∞ ) at s = 0. This is due to the high non-locality of Eq. (17), which was not fully suppressed bythe local gap model of Eq. (18) with m = 2. Note thatthis divergence is a direct consequence of the jellium-with-gap theory. Nevertheless, for molecular systems s = 0 only at the middle of bonds, and it has beenfound that this divergence is not important in real cal-culations of weakly-bounded molecular systems . Infact, the same problem is shared by other well-knownKE functionals . TABLE I. Comparison of several linear-response KE gradient expansions. All integrations are performed over the volume V ,defined by − ≤ q ≤ ≤
1. We show the exact KE ( T exacts ) and the errors E approxs = T approxs − T exacts (in Hartree).The GAP4 and LGAP functionals are defined in Eq. (17) and Eq. (20), respectively. The best result of each line is shown inbold style. We use LDA orbitals and densities.atom T exacts E GE s E Lind s E GAP s E LGAP − GEs
Ne 125.8 -0.2 -8.9
Uuo 46259.6 -298.4 -162.3 -139.6 e − e − e − TABLE II. The coefficients c , c , and c of the large- Z ex-pansion of the kinetic energy (see Eq. (19)). c c c Exact 0.768745 -0.500 0.270GE2 0.768745 -0.536 0.336LGAP-GE 0.768745 -0.500 0.283
III. KINETIC ENERGY FUNCTIONALCONSTRUCTED FROM THE LGAP GRADIENTEXPANSIONA. The LGAP GGA
To show the importance of the LGAP-GE, we con-struct a simple GGA functional (named LGAP-GGA orsimply LGAP) that recovers the LGAP-GE in the slowly-varying density regime. We consider the RPBE exchangeenhancement factor form , F RP BEx = 1+ κ (1 − e − µs /κ ),and we fix κ = 0 . , usingthe approximate link between the kinetic and exchangeenergies (i.e. the conjointness conjecture ). Notethat, to our knowledge, the RPBE functional form hasnot been yet used in the development of kinetic function-als. The LGAP kinetic enhancement factor is thereforedefined as F LGAPs = 1 + κ (cid:16) − e − µ s − µ s − µ s (cid:17) , (23)where µ = b /κ , µ = b /κ + µ /
2, and µ = b /κ + µ µ − µ /
6, such that it recovers the LGAP-GE in theslowly-varying density limit. Here b = 0 . b =0 . b = 0 . s F s LGAP-GGArevAPBEKLC94GE2OL1 s F s LGAP-GGALGAP-GEGE2
FIG. 2. Comparison of kinetic enhancement factors
B. The kinetic energy benchmark
In order to assess the LGAP KE functional, we con-sider several known tests.For total KE : • The benchmark set of atoms and ions . Allcalculations employed analytic Hartree-Fock or-bitals and densities ; • The Na jellium clusters ( r s = 3 .
93) set with magicelectron numbers 2, 8, 18, 20, 34, 40, 58, 92, and106, used in Refs. 42, 44, and 93. We use exactexchange orbitals and densities; • The set of two interacting jellium slabs at differentdistances . Each jellium slab has r s = 3 and athickness of 2 λ F . Here λ F = 2 π/k F is the Fermiwavelength. The calculations were performed usingthe orbitals and densities resulting from numerical TABLE III. Mean absolute relative errors (MARE) of the non-self-consistent benchmark tests, and mean absolute errors (MAEin mHa) of FDE self-consistent tests, given by several KE functionals. The best result of each group is highlighted in boldstyle. GE2 revAPBEk OL1 LC94 LGAPTotal KE (non-self-consistent calculations)Atoms and ions 1.1 1.2 1.1
Jellium slabs 0.6 0.5 0.5 0.5
Molecules 0.9 0.4 0.7 0.5
KE differences (non-self-consistent calculations)Jellium cluster DKE 27.2 23.1 28.9
Jellium slabs dKE 5.0 3.5 4.7 4.1
Molecules AKE 184 155 185 a (self-consistent calculations)Weak-interactions (WI) 2.46 Hydrogen bonds (HB) 10.68
Charge transfer (CT) 5.05 2.61 6.94 2.79
MAE FDE 5.66 1.72 6.31 1.82 a Embedding energy errors ∆ E = E FDE − E KS (mHa) for different KE functionals and complexes. In the last line, the mean absoluteerror (MAE) is reported. Kohn-Sham calculations within the local densityapproximation for the XC functional; • The set of molecules (H , HF, H O, CH , NH ,CO, F , HCN, N , CN, NO, and O ) used in Refs.44, 93, and 95. The noninteracting kinetic ener-gies of test molecules were calculated using thePROAIMV code . The required Kohn-Sham or-bitals were obtained by Kohn-Sham calculationsperformed with the uncontracted 6-311+G(3df,2p)basis set, the Becke 1988 exchange functional ,and Perdew-Wang correlation functional .For KE differences : • The disintegration kinetic energy (DKE) of a jel-lium cluster ; • The jellium surfaces test with bulk parameter r s =2,4, and 6 into the liquid drop model (LDM), as inRefs. 42, 44, and 93; • The dissociation KE (dKE) of a jellium slab intotwo pieces (as in Ref. 44); • The atomization KE (AKE) of molecules .For non-additive KE :We employ the LGAP functional in subsystemDFT calculations, using the TURBOMOLE pro-gram, together with FDE script . The FDE cal-culations have been performed with a supermolecular def2-TZVPPD basis set and the Perdew-Burke-Ernzerhof XC functional. Five weakly interactinggroups of molecular complexes are considered as abenchmark : WI : weak interaction (He-Ne, He-Ar, Ne , Ne-Ar, CH -Ne, C H -Ne, (CH ) ); DI : dipole-dipole interaction ((H S) , (HCl) , H S-HCl, CH Cl-HCl,CH SH-HCN, CH SH-HCl); HB : hydrogen bond ((NH ) , (HF) , (H O) , HF-HCN,(HCONH ) , (HCOOH) ); DHB : double hydrogen bond (AlH-HCl, AlH-HF, LiH-HCl, LiH-HF, MgH -HCl, MgH -HF, BeH -HCl,BeH -HF); CT : charge transfer (NF -HCN,C H -F ,NF -HCN,C H -Cl , NH -F , NH -ClF, NF -HF, C H -ClF,HCN-ClF, NH -Cl , H O-ClF, NH -ClF). C. Results
We compare our results with revAPBEk andLC94 GGAs, which are considered state-of-the-art KEfunctionals for FDE , as well as with GE2 andOL1 . The KE enhancement factors of the consideredfunctionals are reported in Fig. 2. In the inset of Fig.2, we show that LGAP and LGAP-GE are identical (byconstruction) at relatively small values of the reducedgradient (0 ≤ s ≤ . F LGAPs ≥∼ F revAP BEks ≥∼ F LC s ) when s ≤ . . On the other hand, the LGAP enhancementfactor recovers its maximum value F s → κ at s ≈ s = 0 of theLGAP kinetic potential is not important for calculationsof weakly-bounded molecular systems. Moreover, resultsindicate that the LGAP-GE gradient expansion can besuccessfully used in the kinetic energy functional con-struction, which perform relatively well in FDE theory. IV. CONCLUSIONS
In conclusion, we have investigated the linear responseof the jellium-with-gap model, in the context of semilocalkinetic functionals. We have shown that the Levine andLouie analytical generalization of the Lindhard func- tion ( F GAP ) contains important physics beyond jelliummodel, and in particular we mention the following prop-erties:( i ) F GAP recovers the Lindhard function when theband gap is zero (i.e. E g = 0);( ii ) F GAP has the correct behavior (see Eq. (4)) atsmall wave vectors, expressing the material-dependentconstant b in terms of the band gap;( iii ) In the regime of small band gap energy (i.e. E g ≤ E F , with E F being the Fermi energy), F GAP givesthe GAP4 gradient expansion of the kinetic energy (seeEq. (17)), which is band-gap-dependent, and performsremarkably well in the atomic regions where the densityvaries slowly, improving over T Lind s of Eq. (7) (see TableI).These features show that F GAP should be further in-vestigated and exploited in the field of non-local kineticfunctionals , and we will like to addressthis important issue in further work.Finally, by considering a local band gap model, anda simple enhancement factor form, we have constructedthe non-empirical LGAP GGA kinetic energy functional,derived from the linear response of the jellium-with-gapmodel (a.i. the GAP4 gradient expansion). Thisfunctional showed the best performance in the contextof FDE theory. Thus, it can be further used in realapplications.
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