Hartree potential dependent exchange functional
aa r X i v : . [ c ond - m a t . o t h e r] A ug Hartree potential dependent exchange functional
Lucian A. Constantin, Eduardo Fabiano,
2, 1 and Fabio Della Sala
2, 1 Center for Biomolecular Nanotechnologies @UNILE,Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy Istituto Nanoscienze-CNR, Euromediterranean Center for NanomaterialModelling and Technology (ECMT), via Arnesano, Lecce 73100, Italy (Dated: June 25, 2018)We introduce a novel non-local ingredient for the construction of exchange density functionals: thereduced Hartree parameter, which is invariant under the uniform scaling of the density and representsthe exact exchange enhancement factor for one- and two-electron systems. The reduced Hartreeparameter is used together with the conventional meta-generalized gradient approximation (meta-GGA) semilocal ingredients (i.e. the electron density, its gradient and the kinetic energy density) toconstruct a new generation exchange functional, termed u-meta-GGA. This u-meta-GGA functionalis exact for the exchange of any one- and two-electron systems, is size-consistent and non-empirical,satisfies the uniform density scaling relation, and recovers the modified gradient expansion derivedfrom the semiclassical atom theory. For atoms, ions, jellium spheres, and molecules, it shows a goodaccuracy, being often better than meta-GGA exchange functionals. Our construction validates theuse of the reduced Hartree ingredient in exchange-correlation functional development, opening theway to an additional rung in the Jacob’s ladder classification of non-empirical density functionals.
PACS numbers: 71.10.Ca,71.15.Mb,71.45.Gm
I. INTRODUCTION
Kohn-Sham (KS) ground-state density functional the-ory (DFT) is one the most used methods in electroniccalculations of quantum chemistry and condensed-matterphysics. Its practical implementation is based on approx-imations of the exchange-correlation (XC) energy ( E xc ),which is a subject of intense research .The simplest functionals, beyond the local densityapproximation (LDA), are those based on the gener-alized gradient approximation (GGA), which are con-structed using the electron density ( n ) and its reducedgradients (e.g. s in Eq. (9)). These functionalscan achieve reasonable accuracy for various energeticaland/or structural properties of molecules and/or solids,at a moderate computational cost . However, becauseof their simplicity, GGA functionals also show severalimportant limitations, especially in terms of broad ap-plicability. Moreover, they are based on a heavy errorcancellation betweeen exchange and correlation parts .To improve over GGAs, meta-generalized-gradient-approximations (meta-GGAs) can be considered .These are the most sophisticated semilocal functionalsand use, as additional ingredient with respect to theGGA ones, the positive-defined kinetic energy density τ = (1 / P Ni =1 |∇ φ i | (with φ i being the KS orbitalsand N being the number of occupied KS orbitals). Thisquantity enters in the expansion of the angle-averaged ex-act exchange hole , being thus a natural and importanttool in the construction of XC approximations. Meta-GGA functionals incorporate important exact conditionsand have an improved overall accuracy with respect tothe GGA functionals. Moreover, because the kinetic en-ergy density can be easily computed at any step of the KSself-consistent scheme, the meta-GGA functionals have almost the same attractive computational cost as anyGGA.Further improvements, beyond the meta-GGA levelof theory, are usually realized abandoning the semilo-cal framework. Here we mention the so-called 3.5Rung functionals , that incorporate a linear de-pendence on the nonlocal one-particle density matrix,and non-local functionals based on the properties andmodelling of the exchange-correlation hole . More-over, popular tools in computational chemistry are thehybrid functionals , which mix a fraction of non-local Hartree-Fock exchange with a semilocal XC func-tional. Alternatively, even more complex possibilities canbe considered, such as hyper-GGA functionals ororbital-dependent functionals . In this way, a sig-nificant increase of the accuracy can be achieved. Nev-ertheless, because of the need to compute non-local con-tributions (e.g. the Hartree-Fock exchange), the compu-tational cost of such methods is considerably larger thanthe one of semilocal functionals.In this paper, we consider an alternative strategy to in-troduce non-local effects into a density functional, with-out affecting too much the final computational cost. Theidea is to consider, as additional ingredient beyond theconventional meta-GGA level of theory, the Hartree po-tential u ( r ) = Z d r ′ n ( r ′ ) | r − r ′ | . (1)The Hartee potential appears to be a natural input in-gredient in the construction of exchange functionals forseveral reasons: • For one- and two-electron systems, the exact ex-change energy is E x [ n ] = − Z d r n ( r ) u ( r ) , for N = 1 , (2) E x [ n ] = − Z d r n ( r ) u ( r ) , for N = 2 , (3)where N is the number of electrons. Note thatEq. (2) is the basis of the self-interaction correctionapproach of Perdew and Zunger . • The asymptotic decay of the Hartree potentiallim r →∞ u ( r ) = N/r (4)is proportional to that of the exact exchange perparticle and potential :lim r →∞ ǫ x ( r ) = − / (2 r ) (5)lim r →∞ v x ( r ) = − /r. (6)In fact the Fermi-Amaldi potential whichequals u ( r ) /N , has been largely used to con-struct exchange and exchange-correlation function-als, see e.g. Refs. 87–90. However, the Fermi-Amaldi potential depends on N , thus it is not size-consistent .In this work we consider the construction of an exchangefunctional of the general form E u − MGGAx [ n ] = Z nǫ u − MGGAx ( n, ∇ n, τ, u ) d r = (7)= Z nǫ LDAx ( n ) F x ( n, ∇ n, τ, u ) d r , where ǫ LDAx = − (3 / π )(3 π ) / n / is the local densityapproximation for exchange and F x is the exchange en-hancement factor. The functional of Eq. (7) constitutesthe prototype for a new class of functionals, that we name u-meta-GGA (u-MGGA in short). The construction ofa correlation u-meta-GGA functional is also conceivablebut it is a more complex task and it is left for futurework. The u-meta-GGA exchange functional is expectedto have higher accuracy than conventional meta-GGAs,thanks to the inclusion of non-local effects via the Hartreepotential. At the same time, because the Hartree poten-tial must be anyway computed at every step of any KScalculation (even at the LDA level), it bears no essen-tial additional computational cost with respect to meta-GGAs. II. CONSTRUCTION OF THE U-META-GGAEXCHANGE FUNCTIONALA. The reduced Hartree ingredient
To start our work, we consider the construction of aproper reduced ingredient that depends on the Hartree potential and has the correct features to be usefully em-ployed in the construction of density functionals. This isthe
Hartree reduced parameter η u = u /π ) / n / . (8)This ingredient is invariant under the uniform scaling ofthe density ( n γ ( r ) = γ n ( γ r ), with γ > η uγ ( r ) = η u ( γ r ). This invariance is a key property forany input ingredient to be used in the development ofsemilocal DFT functionals. In fact, it is satisfied by allthe semilocal ingredients: s = |∇ n | k F n , z = τ W τ , α = τ − τ W τ unif , (9)where k F = (3 π n ) / is the Fermi wavevector, τ unif = (3 π ) / n / is the Thomas-Fermi kinetic energydensity , and τ W = τ unif s / .Additional important formal properties of η u canbe obtained considering its behavior under the coordi-nate and particle-number density scaling ( n ( β ) γ ( r ) = γ β +1 n ( γ β r ), with γ > β being a parameter),which defines a whole family of scaling relations. Un-der this scaling, we have η u → γ / η u , i.e. the Hartreereduced ingredient scales as N / , with N being the num-ber of electrons. This result indicates that, unlike thesemilocal parameters, η u is a size-extensive quantity, in-creasing with the number of electrons. We note that,in spite of this feature, the reduced Hartree parameteris anyway behaving in a proper size-consistent way asshown in Appendix A.Moreover, we can note that the scaling propertiesof η u do not depend on the value of the parameter β . Thus, for the uniform-electron-gas and the Thomas-Fermi scalings , where the γ → ∞ limit is important,large values of η u are relevant; on the opposite, for the ho-mogeneous and fractional-particle scalings , where γ →
0, small values of η u are important. These consid-erations will be significant to analyze the behavior of an η u -dependent enhancement factor in different conditions.In Fig. 1, we compare the behavior of η u with thatof the other semilocal ingredients for some atoms anddimers. It can be seen that η u behaves rather differentthan the other conventional semilocal reduced parame-ters ( s , z , and α ), being in general more shallowed andaveraged: η u is in fact a non-local ingredient and thusit contains, at every point of space, information on thewhole system. Moreover, it is larger than zero at anypoint in space (unlike s , for example); in the density tailasymptotic region we always have η u → + ∞ , (10)like s but in contrast to α which vanishes for iso-orbitaldensity tails (e.g. for Be).Thus, η u appears to be an interesting tool for the con-struction of advanced functionals both to complement the r (au) V a l u e ( a u ) s αη u r (au) V a l u e ( a u ) r (au) V a l u e ( a u ) Position (au) V a l u e ( a u ) Position (au) V a l u e ( a u ) Position (au) V a l u e ( a u ) He Be NeBe Ne N FIG. 1: Plot of the reduced gradient ( s ), the α meta-GGA ingredient, and the Hartree reduced parameter η u , for some atoms(top) and dimers (bottom), as functions of the position. The green cross in the bottom part of each dimer plot denotes theposition of the atom (note that only half of the dimer is plotted). information available from standard semilocal reducedingredient and to add information on the shape of theexchange enhancement factor.The most important feature of η u is that the simpleexchange enhancement factor F x = η u (11)yields immediately Eqs. (2) and (3) [for the former,note that the exchange energy satisfies the spin-scalingrelation E x [ n ↑ , n ↓ ] = ( E x [2 n ↑ ] + E x [2 n ↓ ]) / η u represents the exact exchange enhancement factor forany one- and two-electron system .Finally, it is also useful to define the bounded ingredi-ent υ u = 11 + η u , (12)that is small everywhere for large systems, but also in the tail of the density (where η u → ∞ ). In Fig. 2, we show υ u for the noble atoms of the periodic table. One can seethat in most of the space the curves are not intersecting,such that υ u can be considered a good atomic indicator,being of interest for functional development. B. The u-meta-GGA exchange functional
In the previous subsection we introduced the Hartreereduced parameter and we showed that it posses interest-ing properties that suggest its utility as input quantity inthe construction of advanced density functionals. On theother hand, we have observed that in general η u is alwayslarge in magnitude. Thus, the proper use of this quantityin functional development is not trivial and the construc-tion of a good u-meta-GGA functional represents insteada challenge. r/R v u HeNeArKrXeRnUuo
FIG. 2: The bounded ingredient υ u versus the scaled radialdistance r/R for noble atoms (He-Uuo). Here R is the atomicradius. For He-Rn we use the atomic radii of Ref. , while forUuo we extrapolate the data of Ref. , finding R = 2 . A . To attempt to fulfill this task, we consider the followingansatz for the u-meta-GGA exchange enhancement factor F u − MGGAx = A F x (13)where A = β + η u β /η u η u (14) β = b √ s (15) b = (1 − z ) a , (16) F x = 1 + b h µ z + a s π √ √ α i ba s p ln(1 + α ) , (17) µ = µ MGE + a υ u , (18)with µ MGE = 0 .
26 being the coefficient of the mod-ified second-order gradient expansion (MGE2) ,and the a = 1 / a = 0 .
05, and a = 0 .
08 beingnon-empirical parameters fitted to a class of four-electronmodel systems described in Section II C.The function b = b ( z ) controls the transition from pureu-meta-GGA behavior ( F u − MGGAx = η u ), which is exactfor iso-orbital regions ( z = 1), to a meta-GGA like behav-ior ( F u − MGGAx = F x ), which is appropriate for slowly-varying density limit ( z ≈ a is a param-eter which helps to tune the value of the second-ordercoefficient in the Taylor expansion at slowly-varying den-sities for each atom. This is done using the parameter υ u as an atomic indicator. Note that for small atoms µ > µ MGE (here the gradient expansion is less mean-ingful, and the results are very sensitive to the functionalform) , whereas in the semiclassical limit (with an infinitenumber of electrons) µ = µ MGE . Finally, a modulatesthe behavior of the functional in the tail of the density. The u-meta-GGA exchange functional has beencostructed satisying the following properties:- For one and two electron systems z = 1, so that b = β = 0, yielding A = η u and F X = 1, therefore F u − MGGAx = η u is exact (see Eq. (11));- Under the uniform density scaling n γ ( r ) = γ n ( γ r ), with γ >
0, it behaves correctly as E u − MGGAx [ n γ ] = γE u − MGGAx [ n ];- It is size-consistent (see Appendix A);- For many-electron systems, we can distinguish dif-ferent regions:– In the slowly-varying density limit ( s → z → s / O ( |∇ n | ), α → O ( |∇ n | )) wehave b → β →
1, thus A → F u − MGGAx → F x → µs . (19)Note that in the limit of large atoms µ → µ MGE , such that the semiclassical atomtheory is correctly recovered.– In the density tail asymptotic region with va-lence orbitals having a non-zero angular mo-mentum quantum number ( s → ∞ , z <
1) sothat β →
0. We have also that η u → ∞ and A → β (1 /η u ) → F u − MGGAx → F x → π √ α √ p ln(1 + α ) . (20)Equation (20) is an exact meta-GGA con-straint for metallic surfaces , making asym-totically exact both the exchange energy perparticle and the potential, while for finite sys-tems we found (see Appendix B) that the ex-change energy per particle decays as ǫ x →− C/r / , and the exchange potential decaysas v x → − C/ (2 r / ), with C being a con-stant dependent on the angular momentumquantum number of the outer shell, if α →∞ . Thus, for finite systems Eq. (20) isnot an exact constraint . Nevertheless,this behavior is definitely more realistic thanthe usual exponential decay behavior of mostsemilocal functionals. In any case, we un-derline that the Hartree potential is not usedto describe the asymptotic region, in contrastto functionals based on the Fermi-Amaldi po-tential: instead, the meta-GGA expression inEq. (20) is used. Moreover, in this workwe are only considering non self-consistent re-sults, which are thus quite unaffected by thechoice of the functional in the asymptotic re-gion. Z E rr o r ( % ) u-MGGATPSSrevTPSSBLOCMS2MVSSCAN FIG. 3: Percent error ( E exactx − E approxx ) /E exactx ×
100 versus Z for the model systems described in Eq. (21). We note that the u-meta-GGA enhancement factordiverges for η u → + ∞ and/or s → + ∞ (i.e. in thetail of the density). Therefore, unlike other function-als, it does not respect the local form of the Lieb-Oxfordbound . We note that this feature is anyway not anexact constraint and it is indeed also strongly violated bythe conventional exact exchange energy density . How-ever, we recall that the global Lieb-Oxford bound, whichis the true exact condition, is not tight, being usuallyfulfilled for all known physical systems, by most of thefunctionals . As shown in the next section, the u-meta-GGA functional is accurate for atoms and molecules.Thus, it implicitly satisfies the Lieb-Oxford bound forthese systems. C. Parametrization of the functional
Because the u-meta-GGA is exact for any one- andtwo-electron systems, we require it to be as accurate aspossible also for four-electron systems. To this purpose,we consider the four-electron hydrogenic-orbital model(1 s s ), with the following one-electron wavefunctions( ψ nlm with n , l , and m being the principal, the angular,and the azimuthal quantum numbers respectively) ψ ( r ) = q π Z / e − Z r ,ψ ( r ) = q π Z / e − Z r/ (2 − Z r ) , (21)with Z and Z being the nuclear charges seen by the1 s and 2 s electrons, respectively. Note that for the realberyllium atom, Z ≈
4, and Z ≈
2. This model systemis analytical and simple, and can cover important physicsby varying Z and Z . In Appendix C we show in detailthe case Z = Z = Z . We also recall that the hydrogenicorbitals are important model systems in DFT, having r (au) F x Exactu-MGGA TPSS
FIG. 4: Exchange enhancement factor F x versus the radialdistance r , for Be atom. been used to find various exact conditions andto explain density behaviors .The parameters a , a and a have been fitted by fix-ing Z = 4 (as in the beryllium case) and varying Z between 1 and 4. In Fig. 3 we show the resulting percenterror (i.e. 100 × ( E exactx − E approxx ) /E exactx ) as a func-tion of Z and we compare the u-meta-GGA results withthose of other popular functionals. All the consideredmeta-GGA exchange functionals (TPSS , revTPSS ,BLOC , MGGA MS2 , MVS , and SCAN ) per-form similarly, while u-meta-GGA improves considerably,showing errors below 1.5 %.In Fig. 4, we report the u-meta-GGA exchange en-hancement factor for the Be atom, comparing it to theexact one (obtained as the ratio of the conventional ex-act exchange and the LDA exchange energy densities)and the popular TPSS meta-GGA. This is a difficult andimportant example for the u-meta-GGA, because in theatomic core the density varies rapidly, the 1 s and 2 s or-bitals overlap strongly, showing a significant amount ofnon-locality. Thus, s and α are large ( s ≈ r = 0 . α ≈ r = 1), while z is relatively small ( z ≈ . r = 1). See also Fig. 1. F u − MGGAx is smooth and morerealistic than the TPSS one, at every point in space. Re-markably, the u-meta-GGA can also describe well theatomic core. Using the PBE orbitals and densities, thetotal exchange energies for Be atom are: E exactx = − . E T P SSx = − .
673 Ha, and E u − MGGAx = − .
655 Ha.
III. COMPUTATIONAL DETAILS
All calculations for spherical systems (atoms, ions, andjellium clusters) have been performed with the numericalEngel code , using PBE orbitals and densities.All calculations for molecules have been performedwith the TURBOMOLE program package using
TABLE I: Relative errors (10 × ( E approxx − E exactx ) /E exactx ) forthe exchaneg energy of the H, G, and C one-electron densities.Functional H G Cu-meta-GGA 0.0 0.0 0.0TPSS 0.0 0.3 -3.6revTPSS 0.0 1.5 -3.3MS2 0.0 -9.4 -7.0MVS 0.0 -5.9 -5.5SCAN 0.0 -3.5 -4.8 PBE orbitals and densities and a def2-TZVPP basisset . Similar results (not reported) have been foundusing LDA and Hartree-Fock orbitals and densities.Following a common procedure in DFT calculations,we have set a minimum threshold (10 − ) for the electrondensity in order to avoid divide-by-zero overflow errorsin tail regions and one-electron systems. All results arecompletely insensible to the value of the threshold. IV. RESULTSA. One- and two-electron systems
For one- and two-electron systems, the u-meta-GGAfunctional satisfies the exact condition in Eq. (11).This is a very powerful exact constraint, that cannotbe achived at the GGA and meta-GGA levels of the-ory. In fact, even if some meta-GGAs have been fit-ted to the exchange energies of the hydrogen atom (e.g.TPSS , revTPSS , BLOC , and Meta-VT { } ),they are not exact for many other interesting one- andtwo-electron densities. On the contrary, the u-meta-GGAfunctional is exact, not only for total exchange energies,but also for exchange energy densities and potentials, byconstruction, in all cases.To make this point more clear, we consider briefly somerelevant examples of one- and two-electron densities. Thefirst case concerns the hydrogen (H), Gaussian (G), andcuspless hydrogen (C) one-electron densities, that are de-fined as n H ( r ) = e − r π , n G ( r ) = e − r π / , n C ( r ) = (1 + r ) e − r π . (22)These densities are models for atomic, bonding, andsolid-state systems . They have analytical ex-change energies E H = − / E G = − / √ π , and E C = − / distance (au) -0.6-0.5-0.4-0.3-0.2 E n er gy ( H a ) Exactu-meta-GGAPBETPSSM06L
FIG. 5: Dissociation curve of the H +2 molecule as computedwith different functionals. λ -0.7-0.6-0.5-0.4-0.3 E x ( H a ) Exact/u-MGGATPSS/BLOCrevTPSSMS2MVSSCAN
FIG. 6: Exchange energy (Ha) versus the scaling parame-ter λ , for the non-uniformly scaled hydrogen atom in onedirection . two- electron densities.Another example is shown in Fig. 5, where we plotthe dissociation curve of the H +2 molecule, which is thesimplest possible molecule. This is a notoriously difficultproblem for semilocal functionals , being related to thedelocalization error. Nevertheless, because the u-meta-GGA is exact for any one-electron density, it yields theexact description for this difficult case.Finally, we report in Fig. 6 the exchange energy com-puted for the non-uniformly scaled hydrogen atom versusthe scaling parameter λ . This is a model for quasi-two-dimensional systems and to study the three-dimensionalto two-dimensional crossover . All functionals, includ-ing meta-GGAs, are very accurate at λ = 1 (i.e. theconventional three-dimensional hydrogen atom). How-ever, for larger values of the confining parameter onlyu-meta-GGA is exact (by construction). The meta-GGAfunctionals instead fail badly even for mild and moder-ately large values of λ . -1.0-0.50.00.51.0 ( E x e xa c t - E xa pp r ox ) / E x L DA HeNeKrRn u-MGGATPSSrevTPSSBLOCMS2MVSSCAN Z -1/3 -1.0-0.50.00.51.0 FIG. 7: Upper panel: Percent exchange energy error(100( E exactx − E approxx ) /E LDAx ) versus Z − / for all periodictable atoms (2 ≤ Z ≤ E exactx − E approxx ) /E LDAx ) versus Z − / , fornoble atoms (2 ≤ Z ≤ Other examples of two-electron densities of interestin DFT are the Hooke’s atom , the Loos-Gillmodel , and the strictly-correlated two-electronsmodel . In all these cases, the u-meta-GGA func-tional yields, by construction, an exact description of ex-change.
B. Atoms
Computing the absolute energies of atoms can be ex-pected to be quite a hard task for the u-meta-GGAfunctional. In fact, the functional is exact for one- andtwo-electron system (i.e. H and He atoms) but for in-creasingly large atoms the Hartree reduced parameterbecomes soon very large (see Fig. 1). Therefore, a par-ticular care is required to balance the contribution of thisingredient in different cases.To check this issue, we have calculated the exchangeenergy of all periodic table atoms (2 ≤ Z ≤ E exactx − E approxx ) /E LDAx ) for noble atoms with 2 ≤ Z ≤ ≤ Z ≤ C. Isoelectronic series and Jellium clusters
We consider the first 17 ions of the isoelectronic seriesof Beryllium (4 ≤ Z ≤ ≤ Z ≤ ≤ Z ≤ ≤ Z ≤ Z values it soon becomes very accurate.We also tested the u-meta-GGA for magic jellium clus-ters with 2, 8, 18, 20, 34, 40, 58, and 92 electrons for bulkparameters r s = 1 and r s = 4. The error statistics arereported in Table II. In both cases u-meta-GGA is accu-rate, being in line with the best semilocal functionals. D. Molecules
In Table III we report the exchange atomization en-ergies of the systems constituting the AE6 test set ,as computed with several methods. One can see thatthe u-meta-GGA functional performs quite well in thiscase, being often superior to meta-GGA functionals andyielding overall the best MAE. This result shows that theu-meta-GGA functional provides a well balanced descrip-tion of atoms and molecules, at the exchange level. Wenote that this success goes beyond the exactness of thisfunctional for one- and two-electron systems, since in thepresent case this feature concerns only the computationof the H atom energy, which is exact also for all the othertested meta-GGAs.As additional test, we consider in Table IV theexchange-only barrier heights and reaction energies ofthe systems defining the K9 test set . This is a hardertest than the previous one, since transition-state struc-tures display rather distorted geometries and are there-fore characterized by a different density regime than or-dinary molecules. Inspection of the table shows that theerrors on reaction energies display a similar trend as forthe atomization energies, even though the differences be-tween the functionals are smaller because of the smallermagnitude of the computed energies. Instead, for barrierheights no clear trend can be extracted. Nevertheless, theu-meta-GGA functional shows a reasonable performancebeing similar to meta-GGAs. This finding supports therobustness of the construction presented in Section II B.
TABLE II: Mean absolute errors (mHa) for various systems and properties.System Property TPSS revTPSS BLOC MS2 MVS SCAN u-MGGAAtoms (2 ≤ Z ≤ E x /Z ≤ Z ≤ E x /Z e − -ions (4 ≤ Z ≤ E x e − -ions (7 ≤ Z ≤ E x e − -ions (10 ≤ Z ≤ E x e − -ions (29 ≤ Z ≤ E x r s = 4 (2 ≤ Z ≤ E x /Z r s = 1 (2 ≤ Z ≤ E x /Z Z -0.10-0.050.000.050.10 E x e xa c t - E xa pp r ox ( H a ) Z -0.12-0.09-0.06-0.030.000.03 E x e xa c t - E xa pp r ox ( H a ) TPSSrevTPSSBLOCMS2u-MGGAMVSSCAN
10 12 14 16 18 20 22 24 26 Z -0.3-0.2-0.10.00.1 E x e xa c t - E xa pp r ox ( H a )
28 30 32 34 36 38 40 42 44 46 Z -1.0-0.50.00.5 E x e xa c t - E xa pp r ox ( H a ) Be NNe Cu
FIG. 8: Exchange errors ( E exactx − E approxx ) versus nuclear charge Z , for Beryllium (top-left panel), Nitrogen (top-right panel),Neon (bottom-left panel), and Copper (bottom-right panel) isoelectronic series. V. COMPATIBILITY OF THE U-META-GGAWITH SEMILOCAL CORRELATIONFUNCTIONALS
In this section we investigate the possibility to com-bine the u-meta-GGA exchange with an existing semilo-cal correlation functional. Thus, we consider the per-formance of different combinations of the u-meta-GGAexchange with an existing semilocal correlation func- tional, for the description of molecular properties, namelythe AE6 and K9 test sets. In more de-tail, we consider the following correlation function-als: PBE , PBEloc , GAPloc , TCA , vPBE (the semilocal correlation of the MGGA-MS func-tional), PBEsol , LYP [GGA functionals], TPSS ,revTPSS , BLOC , JS [meta-GGA functionals].In Fig. 9 we report the MAE on the AE6 test versusthe MAE for the K9 test as obtained by the different TABLE III: Errors (kcal/mol) and error statistics for the exchange atomization energies of the AE6 test. The best result ofeach line is highlighted in bold style.TPSS revTPSS BLOC MS2 MVS SCAN u-MGGACH -3.1 -1.9 -2.8 -3.0 3.8 5.6 -15.6SiO 31.8 30.5 27.8 26.7 38.4 34.3 S C H C H O C H -2.8 -8.5 -11.2 21.3 34.2 48.4 -47.6MAE 22.6 21.9 20.8 26.1 37.0 38.6 MARE 13.8 13.6 12.1 12.8 19.4 16.0
TABLE IV: Errors (kcal/mol) and error statistics of several exchange functionals for the K9 representative test. The bestresult of each line is highlighted in bold style.System TPSS revTPSS BLOC MS2 MVS SCAN u-MGGAForward barriersOH+CH → CH +H O -12.5 -11.6 -10.5 -10.2 -11.1 -12.1 -1.0
H+OH → O+H -2.7 -8.1 -6.8 -2.2 -5.5 -3.0 -8.4H+H S → H +HS -1.0 -3.6 -4.1 -3.4 -4.5 -5.0 -3.1MAE 8.0 7.8 7.1 5.3 7.0 6.7 Backward barriersOH+CH ← CH +H O -7.6 -8.1 -6.8 -10.3 2.1 -5.5
H+OH ← O+H -10.4 -9.9 -12.2 -15.7 -13.6 -16.6 -16.9H+H S ← H +HS -2.2 -0.9 -1.6 -8.2 -2.9 -7.0 -3.7MAE 7.3 -CH +H O) -4.9 -3.5 -3.7 -13.2 -6.5 -4.6∆(H+OH-O+H ) 7.7 S-H +HS) -2.0 -2.7 -2.5 -0.5 3.7 2.1 MAE 3.9 functionals. The best performance is found for PBEloc,GAPloc, and BLOC. These are indeed the only corre-lation functionals that allow to achieve for both testsresults that are better than the simple PBE XC ones(13.4 kcal/mol for AE6 and 7.5 kcal/mol for K9), whichwe have used here as a reference. This result indicatesthat a more localized correlation energy density may fa-vor the compatibility with the u-meta-GGA in finite sys-tems. This conclusion can be traced back to the fact thatthe localization constraint in the PBEloc, GAPloc, andBLOC correlation functionals has been introduced to en-hance the compatibility of the semilocal correlation withexact exchange , thus it also improves the compatibil- ity with the u-meta-GGA exchange which is rather closeto the exact one.Nevertheless, we find that none of the semilocal corre-lation functionals can yield highly accurate results, whenused with the u-meta-GGA exchange. This is not muchsurprising since the usual error cancellation that occursat the semilocal level between exchange and correlationcontributions cannot work properly in this case becausethe u-meta-GGA functional is exact for one- and two-electron systems. This suggests the need for the con-struction of a proper u-meta-GGA correlation functionalbeing able to include the non-local effects on equal foot-ing with the exchange part. Such a development is any-0
10 15 20 25 30 35
MAE AE6 (kcal/mol) M A E K ( k c a l/ m o l ) vPBEPBEsolPBEloc PBETPSSBLOC revTPSSGAPloc TCALYPJS SG4APBE FIG. 9: Mean absolute error (MAE) on the AE6 test versusMAE on the K9 test for the combination of the u-meta-GGAexchange with different semilocal correlation functionals. Thegrey-shaded area highlights the combinations that performbetter than the PBE XC functional. way not trivial, since it requires the development of ahighly accurate correlation functional for two-electronsystems, including also static correlation effects, that are(correctly) not accounted for by the u-meta-GGA ex-change (in contrast to simple semilocal exchange func-tionals). Such functionals are usually developed at thehyper-GGA level of theory and they include ex-act exchange as a basic input ingredient. However, theuse of exact exchange as an ingredient would make theu-meta-GGA construction of the exchange term mean-ingless. A possible strategy to solve this dilemma canbe to consider a smooth interpolation of a hyper-GGAexpression for one- and two-electron cases (where z = 1and the exact exchange is given by the Hartree potential)with a more traditional semilocal correlation expressionfor many-electron cases. Anyway, this very challengingtask will be the subject of other work. VI. CONCLUSIONS
The success of semilocal DFT is mainly based on thecorrectness of the semiclassical physics that it incorpo-rates (e.g. gradient expansions derived from small per-turbations of the uniform electron gas), and on the sat-isfaction of several formal exact properties (e.g. densityscaling relations). However, it also relays on a heavyerror cancellation between the exchange and correlationparts. Thus, semilocal DFT can often achieve good ac-curacy for large systems, where the semiclassical physicsis relevant, but not for small systems, that are usuallytreated with hybrid functionals.Using the reduced Hartree parameter η u ( r ) [Eq. (8)]as a new ingredient in the construction of DFT function-als, can guarantee the exactness of the exchange func-tional for any one- and two-electron systems. This is an important exact condition, also related to the ho-mogeneous density scaling , the delocalization andmany-electron self-interaction errors , and it can boostthe accuracy of the functional.Hence, we have constructed a prototype u-meta-GGAexchange functional, showing that it is possible and use-ful the use of the reduced Hartree parameter η u ( r ). Notethat even if η u ( r ) is non-local, we have shown that it iscompatible with the semilocal quantities. The u-meta-GGA has been tested for a broad range of finite systems(e.g. atoms, ions, jellium spheres, and molecules) beingbetter than, or comparable with, the popular meta-GGAexchange functionals.Nevertheless, we have showed that η u ( r ) is a size-extensive quantity, increasing with the number of elec-trons. This fact represents a real challenge for functionaldevelopment, limiting the applicability of the present for-malism to periodic (infinite) systems. This limitation canbe removed only by a large screening. Such a screeningis given, in the present work, [ Eqs. (13)-(17)] by thefunction β ( s, z ). An alternative way will be the use ofthe screened reduced Hartree potential x u ( r ) , definedby x u ( r ) = 13(3 n ( r ) /π ) / Z d r ′ n ( r ′ ) | r − r ′ | e − aα ( r ′ ) b k F ( r ′ ) β | r − r ′ | β , (23)where a , b , and β are other positive constants. Note that x u ( r ) = η u ( r ) for any one- and two-electron systems,and x u ( r ) is realistic at the nuclear region . However,such an approach, which is theoretically more powerful,is significantly more complex. In addition, it is also com-putationally more expensive since the bare Hartree po-tential is computed at every step of the Kohn-Sham self-consistent method, and thus its use does not affect thespeed of the calculation, whereas the screened Hartreepotential should be calculated separately for the onlypurpose of constructing the functional.We also note that the bounded ingredient υ u of Eq.(12), can by itself be of interest for the development ofexchange-correlation and even kinetic functionals, sinceit is a powerfull atomic indicator. In this sense, a fur-ther investigation of this issue may be worth. Construc-tion of the exchange enhancement factors of the form F x ( s, υ u ) should be much simpler, because υ u is bounded,and should reveal the importance of the non-locality con-tained in this ingredient.In any case, the u-meta-GGA exchange functional de-fined in Eqs. (13)-(18) is just a first attempt, and othersimpler and/or better functional forms could possibly bedeveloped. Thus, the class of u-meta-GGA functionalsmay represent a new semi-rung on the Jacob’s ladder: itis above the third one as it includes the Hartree potentialto describe exactly the exchange for any one- and two-electron systems, but with a computational cost lowerthan functionals dependent on exact exchange. In thiswork, all calculations are non-self consistent. In a futurework we will consider the functional derivative of the u-meta-GGA functionals.1 Acknowledgments.
We thank TURBOMOLEGmbH for the TURBOMOLE program package.
Appendix A: Size consistency
Because the Hartree reduced parameter η u is a sizeextensive quantity, it is important to prove that the u-meta-GGA functional is properly size consistent. Thatis, given two systems, A and B , separate by an infinitedistance and whose densities are not overlapping, we have E u − MGGAx [ A + B ] = E u − MGGAx [ A ] + E u − MGGAx [ B ] , (A1)where E u − MGGAx = R nǫ LDAx F u − MGGAx d r . To show this,we can use the fact that the integrand is finite every-where ( nǫ LDAx decays exponentially, while in the evanes-cent density regions F u − MGGAx behaves according to Eq.(20)), to write E u − MGGAx [ A + B ] = R Ω A nǫ LDAx F u − MGGAx d r + R Ω B nǫ LDAx F u − MGGAx d r , (A2)where Ω A and Ω B are the space domains where n A and n B , respectively, are not zero. Then, considering any r ∈ Ω A (analogous considerations hold for Ω B ), we have η uA + B ( r ) = R n A ( r ′ )+ n B ( r ′ ) | r − r ′ | d r ′ /π ) / ( n A ( r ) + n B ( r )) / (A3)= R Ω A n A ( r ′ ) | r − r ′ | d r ′ + R Ω B n B ( r ′ ) | r − r ′ | d r ′ /π ) / ( n A ( r ) + n B ( r )) / . Now, because r ∈ Ω A , we have that n B ( r ) = 0; moreover,because the two systems lay at infinite distance from eachother, | r − r ′ | = ∞ for any r ′ ∈ Ω B . Hence, η uA + B ( r ) = R Ω A n A ( r ′ ) | r − r ′ | d r ′ +3(3 /π ) / ( n A ( r )) / = η uA ( r ) . (A4)In the same way, for r ∈ Ω B we have η uA + B ( r ) = η uB ( r ).At this point, since all the other input quantities aresemilocal, Eq. (A2) immediately yields Eq. (A1). Appendix B: Asymptotic behavior
In case of spherical systems in a central potential (e.g.atoms, jellium spheres), the following equation holds τ − τ W = l ( l + 1)2 nr , (B1)in the asymptotic region. Here l is the angular momen-tum quantum number of the outer shell, and the densitydecays exponentially n ∼ e − br , when the radial distanceis large ( r → ∞ ). Here b = 2 √− µ , with µ being theionization potential. Then, for any l = 0, α diverges as α = l ( l + 1)2 C s n / r , (B2) where C s = (3 π ) / . Considering the enhancementfactor of Eq. (20), i.e. F MGGAx ( α ), the exchange energyper particle ǫ x = − C x n / F MGGAx ( α ) with C x = 34 ( 3 π ) / , (B3)decays as ǫ x → − √ p l ( l + 1) √ b r / + O ( 1 r / ) . (B4)Concerning the exchange potential, we consider the gen-eralized Kohn-Sham framework to write v x φ i = ∂ ( nǫ x ) ∂n φ i − ∇ · (cid:20) ∂ ( nǫ x ) ∂ ∇ n φ i + 12 ∂ ( nǫ x ) ∂τ ∇ φ i (cid:21) + (cid:18) ∂ ( nǫ x ) ∂ ∇ n (cid:19) · ∇ φ i . (B5)Using the following equations ∂ ( nǫ x ) ∂n = − C x n / F MGGAx ( α ) − C x n / dF MGGAx ( α ) dα ∂α∂n ,∂ ( nǫ x ) ∂ ∇ n = − C x n / dF MGGAx ( α ) dα ∂α∂ ∇ n ,∂ ( nǫ x ) ∂τ = − C x n / dF MGGAx ( α ) dα ∂α∂τ ,∂α∂n = |∇ n | n τ unif − αn ,∂α∂ ∇ n = − ∇ n nτ unif ,∂α∂τ = 1 τ unif ,τ unif = C s n / , (B6)we obtain after some simple algebra v x φ = − C x n / ( 43 F MGGAx − dF MGGAx ( α ) dα α ) φ − C x n / dF MGGAx ( α ) dα ∇ n nτ unif · ( ∇ n n φ − ∇ φ ) −∇ · [ C x n / dF MGGAx ( α ) dα τ unif ( ∇ n n φ − ∇ φ )] , (B7)where φ is the highest occupied orbital. Then, theasymptotic density is n = f φ (with f being the oc-cupation number) and ∇ n n φ − ∇ φ = 0 . (B8)The final formula for the exchange potential is v x = − C x n / (cid:18) F MGGAx − dF MGGAx ( α ) dα α (cid:19) , (B9)which is valid for any exchange enhancement factor thatdepends only on the α ingredient. Then, the exchangepotential of F MGGAx defined in Eq. (20) behaves at r →∞ as v x → − √ p l ( l + 1) √ b r / + O ( 1 r / ) . (B10)2 Appendix C: Hydrogenic orbitals
The system of Eq. (21), with Z = Z = Z has thefollowing density n ( r ) = 2 Z (cid:0) e − r Z (cid:1) π + Z (cid:0) e − / r Z (cid:1) (2 − r Z ) π , (C1)kinetic energy density τ ( r ) = Z (cid:0) e − r Z (cid:1) π + 1128 Z e − r Z ( − r Z ) π , (C2)Hartree potential u ( r ) = 14 r (16 − ve − v − e − v − v e − v − ve − v − e − v − v e − v ) , (C3)and exchange energy density e x ( r ) = n ( r ) ǫ x ( r ) = − Z πr (864 e − v − ve − v +6048 e − v + 216 v e − v + 216 v e − v + 1024 v e − v − v e − v − e − v − ve − v + 1024 ve − v +216 ve − v − v e − v − v e − v + 54 v e − v ) , (C4) where v = Zr . The Hartree, exact exchange and LDAexchange energies are U = 4956520736 Z = 2 . Z,E x = − Z = − . Z,E
LDAx = − . Z. (C5)Note that all the exchange ingredients ( s , z , α , η u ) areonly functions of v = Zr , such that for any exchange en-hancement factor F x ( s, α, z, η u ), the total exchange en-ergy will be E x = − constant Z . W. Kohn and L. J. Sham, Phys. Rev. , A1133 (1965). J. F. Dobson, G. Vignale, and M. P. Das,
Electronic Den-sity Functional Theory (Springer, 1998). R. G. Parr and W. Yang,
Density-Functional Theory ofAtoms and Molecules (Oxford University Press, 1989). J. M. Seminario, ed.,
Recent Developments and Appli-cations of Modern Density Functional Theory (Elsevier,1996). D. Sholl and J. A. Steckel,
Density Functional Theory: APractical Introduction (Wiley, 2009). R. O. Jones, Rev. Mod. Phys. , 897 (2015). K. Burke, J. Chem. Phys. , 150901 (2012). G. E. Scuseria and V. N. Staroverov,
Progress in the de-velopment of exchange-correlation functionals (2005). D. C. Langreth and M. J. Mehl, Phys. Rev. B , 1809(1983). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,Phys. Rev. Lett. , 136406 (2008). L. A. Constantin, E. Fabiano, S. Laricchia, and F. DellaSala, Phys. Rev. Lett. , 186406 (2011). E. Fabiano, L. A. Constantin, and F. Della Sala, J. Chem.Theory Comput. , 3548 (2011). R. Peverati and D. G. Truhlar, J. Chem. Theory Comput. , 2310 (2012). Y. Zhao and D. G. Truhlar, J. Chem. Phys. , 184109(2008). V. Tognetti, P. Cortona, and C. Adamo, Chem. Phys.Lett. , 536 (2008). V. Tognetti, P. Cortona, and C. Adamo, J. Chem. Phys. , 034101 (2008). A. D. Becke, Phys. Rev. A , 3098 (1988). C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B , 785(1988). J. Carmona-Esp´ındola, J. L. G´azquez, A. Vela, andS. Trickey, J. Chem. Phys. , 054105 (2015). R. Armiento and A. E. Mattsson, Phys. Rev. B , 085108(2005). E. Fabiano, L. A. Constantin, and F. Della Sala, Phys.Rev. B , 113104 (2010). L. A. Constantin, E. Fabiano, and F. Della Sala, Phys.Rev. B , 233103 (2011). M. Swart, A. W. Ehlers, and K. Lammertsma, Mol. Phys. , 2467 (2004). L. C. Wilson and S. Ivanov, Int. J. Quantum Chem. ,523 (1998). A. J. Thakkar and S. P. McCarthy, J. Chem. Phys. ,134109 (2009). E. Fabiano, P. E. Trevisanutto, A. Terentjevs, and L. A.Constantin, J. Chem. Theory Comput. , 2016 (2014). L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem.Phys. , 194105 (2012). L. Chiodo, L. A. Constantin, E. Fabiano, and F. DellaSala, Phys. Rev. Lett. , 126402 (2012). L. A. Constantin, A. Terentjevs, F. Della Sala, P. Cortona,and E. Fabiano, Phys. Rev. B , 045126 (2016). L. A. Constantin, E. Fabiano, and F. Della Sala, Phys.Rev. B , 035130 (2012). F. Della Sala, E. Fabiano, and L. A. Constantin, Int. J.Quantum. Chem. (2016); (doi: 10.1002/qua.25224). J. P. Perdew, S. Kurth, A. Zupan, and P. Blaha, Phys.Rev. Lett. , 2544 (1999). J. M. del Campo, J. L. G´azquez, S. Trickey, and A. Vela,Chem. Phys. Lett. , 179 (2012). J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria,Phys. Rev. Lett. , 146401 (2003). J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Con-stantin, and J. Sun, Phys. Rev. Lett. , 026403 (2009). J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Con-stantin, and J. Sun, Phys. Rev. Lett. , 179902 (2011). L. A. Constantin, E. Fabiano, and F. Della Sala, J. Chem.Theory Comput. , 2256 (2013). L. A. Constantin, E. Fabiano, J. Pitarke, and F. DellaSala, Phys. Rev. B , 115127 (2016). J. Sun, B. Xiao, Y. Fang, R. Haunschild, P. Hao,A. Ruzsinszky, G. I. Csonka, G. E. Scuseria, and J. P.Perdew, Phys. Rev. Lett. , 106401 (2013). J. Sun, R. Haunschild, B. Xiao, I. W. Bulik, G. E. Scuse-ria, and J. P. Perdew, J. Chem. Phys. , 044113 (2013). J. Sun, B. Xiao, and A. Ruzsinszky, J. Chem. Phys. ,051101 (2012). A. Ruzsinszky, J. Sun, B. Xiao, and G. I. Csonka, J.Chem. Theory Comput. , 2078 (2012). J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. , 036402 (2015). J. Sun, J. P. Perdew, and A. Ruzsinszky, Proc. Nat. Ac.Sc. , 685 (2015). J. Wellendorff, K. T. Lundgaard, K. W. Jacobsen, andT. Bligaard, J. Chem. Phys. , 144107 (2014). Y. Zhao and D. G. Truhlar, Theor. Chem. Acc. , 215(2008). R. Peverati and D. G. Truhlar, J. Phys. Chem. Lett. ,117 (2011). R. Peverati and D. G. Truhlar, Philosophical Transactionsof the Royal Society of London A: Mathematical, Physicaland Engineering Sciences , 20120476 (2014). A. Becke and M. Roussel, Phys. Rev. A , 3761 (1989). Y. Zhao and D. G. Truhlar, Acc. Chem. Res. , 157(2008). R. Peverati and D. G. Truhlar, Phys. Chem. Chem. Phys. , 16187 (2012). A. D. Becke, J. Chem. Phys. , 2092 (1998). L. A. Constantin, E. Fabiano, and F. Della Sala, Phys.Rev. B , 125112 (2013). A. Becke, Int. J. Quantum Chem. , 1915 (1983). B. G. Janesko and A. Aguero, J. Chem. Phys. , 024111(2012). B. G. Janesko, Int. J. Quantum Chem. , 83 (2013). B. G. Janesko, J. Chem. Phys. , 104103 (2010). B. G. Janesko, J. Chem. Phys. , 224110 (2012). O. Gunnarsson, M. Jonson, and B. Lundqvist, Phys. Lett.A , 177 (1976). O. Gunnarsson, M. Jonson, and B. Lundqvist, Solid StateComm. , 765 (1977). J. Alonso and L. Girifalco, Phys. Rev. B , 3735 (1978). O. Gunnarsson, M. Jonson, and B. Lundqvist, Phys. Rev.B , 3136 (1979). Z. Wu, R. Cohen, and D. Singh, Phys. Rev. B , 104112(2004). K. J. Giesbertz, R. van Leeuwen, and U. von Barth, Phys.Rev. A , 022514 (2013). J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. , 9982 (1996). K. Burke, M. Ernzerhof, and J. P. Perdew, Chem. Phys. Lett. , 115 (1997), ISSN 0009-2614. M. Marsman, J. Paier, A. Stroppa, and G. Kresse, J.Phys. : Cond. Mat. , 064201 (2008). Y. Zhao, , and D. G. Truhlar, J. Phys. Chem. A ,6908 (2004). A. D. Becke, J. Chem. Phys. , 5648 (1993). A. D. Becke, J. Chem. Phys. , 1372 (1993). R. Baer, E. Livshits, and U. Salzner, Ann. Rev. Phys.Chem. , 85 (2010). E. Fabiano, L. A. Constantin, P. Cortona, and F. DellaSala, J. Chem. Theory Comput. , 122 (2015). E. Fabiano, L. A. Constantin, and F. Della Sala, Int. J.Quantum Chem. , 673 (2013), ISSN 1097-461X. J. P. Perdew, A. Ruzsinszky, J. Tao, V. N. Staroverov,G. E. Scuseria, and G. I. Csonka, J. Chem. Phys. ,062201 (2005). J. P. Perdew, V. N. Staroverov, J. Tao, and G. E. Scuseria,Phys. Rev. A , 052513 (2008). R. Haunschild, M. M. Odashima, G. E. Scuseria, J. P.Perdew, and K. Capelle, J. Chem. Phys. , 184102(2012). A. D. Becke and E. R. Johnson, J. Chem. Phys. ,124108 (2007). S. K¨ummel and L. Kronik, Rev. Mod. Phys. , 3 (2008). R. J. Bartlett, V. F. Lotrich, and I. V. Schweigert, J.Chem. Phys. , 062205 (2005). I. Grabowski, E. Fabiano, and F. Della Sala, Phys. Rev.B , 075103 (2013). I. Grabowski, E. Fabiano, A. M. Teale, S. ´Smiga, A. Buk-sztel, and F. Della Sala, J. Chem. Phys. , 024113(2014). J. P. Perdew and A. Zunger, Phys. Rev. B , 5048(1981). P. W. Ayers, R. C. Morrison, and R. G. Parr, Mol. Phys. , 2061 (2005). E. Fermi and E. Amaldi, Accad. Ital. Rome , 117 (1934). R. G. Parr and S. K. Ghosh, Phys. Rev. A , 3564(1995). A. Cedillo, E. Ortiz, J. L. G´azquez, and J. Robles, J.Chem. Phys. , 7188 (1986). W. Yang and Q. Wu, Phys. Rev. Lett. , 143002 (2002). P. W. Ayers, R. C. Morrison, and R. G. Parr, Mol. Phys. , 2061 (2005). N. Umezawa, Phys. Rev. A , 032505 (2006). S. B. Trickey and A. Vela, J. Mex. Chem. Soc. , 105(2013), ISSN 1870-249X. L. H. Thomas, in
Mathematical Proceedings of the Cam-bridge Philosophical Society (Cambridge Univ Press,1927), vol. 23, pp. 542–548. E. Fermi, Rend. Accad. Naz. Lincei , 32 (1927). C. F. von Weizs¨acker, Zeitschrift f¨ur Physik A Hadronsand Nuclei , 431 (1935). E. Fabiano and L. A. Constantin, Phys. Rev. A , 012511(2013). P. Elliott, D. Lee, A. Cangi, and K. Burke, Phys. Rev.Lett. , 256406 (2008). A. Borgoo, A. M. Teale, and D. J. Tozer, J. Chem. Phys. , 034101 (2012). A. Borgoo and D. J. Tozer, J. Chem. Theory Comput. ,2250 (2013). G. L. Oliver and J. P. Perdew, Phys. Rev. A , 397(1979). M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, and D. G. Truhlar, J. Phys. Chem. A , 5806 (2009). P. Elliott and K. Burke, Can. J. Chem. , 1485 (2009). F. Della Sala, E. Fabiano, and L. A. Constantin, Phys.Rev. B , 035126 (2015). L. A. Constantin, E. Fabiano, and F. Della Sala, Compu-tation , 19 (2016), ISSN 2079-3197. E. H. Lieb and S. Oxford, Int. J. Quantum Chem. , 427(1981). L. A. Constantin, A. Terentjevs, F. Della Sala, andE. Fabiano, Phys. Rev. B , 041120 (2015). D. V. Feinblum, J. Kenison, and K. Burke, J. Chem. Phys. , 241105 (2014).
J. Vilhena, E. R¨as¨anen, L. Lehtovaara, and M. Marques,Phys. Rev. A , 052514 (2012). M. M. Odashima and K. Capelle, J. Chem. Phys. ,054106 (2007).
O. J. Heilmann and E. H. Lieb, Phys. Rev. A , 3628(1995). E. Engel and S. Vosko, Phys. Rev. A , 2800 (1993). E. Engel, in
A primer in density functional theory (Springer, 2003), pp. 56–122.
TURBOMOLE V6.2, 2009, a development of Universityof Karlsruhe and Forschungszentrum Karlsruhe GmbH,1989-2007, TURBOMOLE GmbH, since 2007; availablefrom . F. Furche, R. Ahlrichs, C. H¨attig, W. Klopper, M. Sierka,and F. Weigend, Wiley Interdisciplinary Reviews: Com-putational Molecular Science , 91 (2014). F. Weigend, F. Furche, and R. Ahlrichs, J. Chem. Phys. , 12753 (2003).
F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys. , 3297 (2005). A. J. Cohen, P. Mori-S´anchez, and W. Yang, Science ,792 (2008).
S. Kurth, J. Mol. Str.: THEOCHEM , 189 (2000).
C. Filippi, C. J. Umrigar, and M. Taut, J. Chem. Phys. , 1290 (1994).
J. Sun, J. P. Perdew, Z. Yang, and H. Peng, J. Chem.Phys. , 191101 (2016).
P. F. Loos and P. M. W. Gill, Phys. Rev. Lett. ,123008 (2009).
M. Seidl, S. Vuckovic, and P. Gori-Giorgi, Mol. Phys. ,1076 (2016).
G. Buttazzo, L. De Pascale, and P. Gori-Giorgi, Phys.Rev. A , 062502 (2012). B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A ,8996 (2003).
B. J. Lynch and D. G. Truhlar, J. Phys. Chem. A ,3898 (2003).
R. Haunschild and W. Klopper, Theor. Chem. Acc. ,1 (2012), ISSN 1432-2234.
L. A. Constantin, L. Chiodo, E. Fabiano, I. Bodrenko,and F. Della Sala, Phys. Rev. B , 045126 (2011). A. D. Becke, The Journal of Chemi-cal Physics , 064101 (2005), URL http://scitation.aip.org/content/aip/journal/jcp/122/6/10.1063/1.1844493 . A. V. Arbuznikov, M. Kaupp, V. G. Malkin, R. Reviakine,and O. L. Malkina, Phys. Chem. Chem. Phys.4