Julien Toulouse

François Colonna

Andreas Savin

Exchange-correlation potentials and local energies per particle along non-linear adiabatic connections

We study non-linear adiabatic connection paths in density-functional theory using modified electron-electron interactions that perform a long-range/short-range separation of the Coulomb interaction. These adiabatic connections allows to define short-range exchange-correlation potentials and short-range local exchange-correlation energies per particle that we have calculated accurately for the He and Be atoms and compared to the corresponding quantities in the local density approximation (LDA). The results confirm that the LDA better describes exchange-correlation potentials and local exchange-correlation energies per particle when the range of the interaction is reduced.

I. INTRODUCTION

The adiabatic connection procedure of density functional theory (DFT) [1] connects the fictitious noninteracting Kohn-Sham (KS) [2] system to the physical one by continuously switching on the electron-electron interaction. The connection is carried out at constant density and is controlled by a interaction parameter µ. The partially interacting N -electron system for a given value of µ along this connection is described by the Hamiltonian

Ĥµ = T + V µ ee + V µ , (1)
where T is the kinetic energy operator, V µ ee = i<j v µ ee (r ij ) is the modified electron-electron interaction operator and V µ = i v µ (r i ) is the corresponding local external potential which ensures that this fictitious system has the same ground-state density n than that of the physical system. The ground-state wave function of this partially interacting system is denoted by Ψ µ .

In the more common adiabatic connection (see, e.g., Ref. 3), the electron-electron interaction is switched on linearly. However, other non-linear paths are possible [4]. In particular, in view of constructing a multideterminantal extension of the KS scheme of DFT, it has been proposed [START_REF] Stoll | Density Functional Method in Physics[END_REF][START_REF] Savin | Recent Advances in Density Functional Theory[END_REF][START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF][9][10][11] to choose a modified interaction v µ ee (r) representing the long-range part of the Coulomb interaction. In previous works [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF][9][10], this long-range interaction has been defined by

v µ ee,erf (r) = erf(µr) r , (2)
referred to as the erf interaction. More recently [12], a sharper long-range/short-range separation has been achieved by taking for the long-range interaction

v µ ee,erfgau (r) = erf(cµr) r - 2cµ √ π e -1 3 c 2 µ 2 r 2 , (3)
* Electronic address: savin@lct.jussieu.fr referred to as the erfgau interaction. This modified interaction was first introduced by Gill and Adamson [13] in another context. In Eq. ( 3), c = 1 + 6 √ 3 1/2 ≈ 3.375 is a constant chosen so that 1/µ roughly represents the range of the modified interaction as for the erf interaction [12]. For both the erf and erfgau interactions, the modified interaction v µ ee vanishes at µ = 0 and reduces to the full Coulomb interaction 1/r as µ → ∞. Consequently, the Hamiltonian Ĥµ of Eq. ( 1) is the KS noninteracting Hamiltonian at µ = 0 and the physical Hamiltonian as µ → ∞.

With these modified interactions, Eq. ( 1) defines a fictitious system with long-range interactions from which one can define a universal long-range density functional

F µ = Ψ µ | T + V µ ee |Ψ µ , (4)
and its short-range complement F µ = F -F µ where F is the usual Coulombic universal functional. This shortrange universal functional F µ can be decomposed as

F µ = Ū µ + Ēµ xc , (5)
where Ū µ = 1/2 n(r 1 )n(r 1 )v µ ee (r 12 )dr 1 dr 2 is the complement short-range Hartree energy functional and Ēµ xc is the unknown complement short-range exchangecorrelation energy functional. Finally, the ground-state energy of the physical system is given by

E = Ψ µ | T + V µ ee |Ψ µ + v ne (r)n(r)dr + Ū µ + Ēµ xc , (6)
where v ne (r) is the nuclei-electron potential. Accordingly, the potential v µ appearing in Eq. ( 1) can be decomposed as

v µ (r) = v ne (r) + v µ h (r) + v µ xc (r), (7)
where v µ h (r) = δ Ū µ /δn(r) is the complement short-range Hartree potential and v µ xc (r) = δ Ēµ xc /δn(r) is the complement short-range exchange-correlation potential. As in the usual Kohn-Sham case, the short-range exchangecorrelation energy can be decomposed into exchange and correlation contributions, Ēµ xc = Ēµ x + Ēµ c . Correspondingly, the short-range exchange-correlation potential is decomposed as v µ xc (r) = v µ x (r) + v µ c (r). In this approach, the central quantity to approximate is the short-range exchange-correlation energy functional Ēµ xc . Once an approximation has been chosen for Ēµ xc , the potential v µ appearing in Eq. ( 1) is deduced by Eq. ( 7), the multi-determinantal wave function Ψ µ is calculated and the ground-state energy E is deduced by Eq. ( 6). Previous applications of the method to small atomic and molecular systems [START_REF] Leininger | [END_REF]9,14] show that, for a reasonable value of µ, Ψ µ | T + V µ ee |Ψ µ can be efficiently approximated by standard wave function methods using a fewdeterminantal wave function Ψ µ .

In practice, the short-range exchange-correlation energy is conveniently expressed as

Ēµ xc = n(r)ε µ xc (r)dr. ( 8
)
where εµ xc (r) is a short-range local exchange-correlation energy per particle that is not uniquely defined. The local density approximation (LDA) for εµ

xc is εµ,LDA xc (r) = εµ,unif xc (n(r)), (9)
where εµ,unif xc (n) is the short-range exchange-correlation energy per particle of a uniform electron gas with modified interaction [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]15]. The LDA short-range exchangecorrelation potential is

v µ,LDA xc (r) = d n εµ,unif xc (n) dn n=n(r) . (10)
The performance of LDA for Ēµ xc has previously be investigated [START_REF] Leininger | [END_REF]9,12,14]. It has been found in particular on atomic systems that the LDA is very accurate for sufficiently large µ [12]. For example, for the He atom with µ 2, the LDA error on both Ēµ x and Ēµ c is less than 1 mH. However, good accuracy on integrated quantities does not always ensure a similar accuracy on local quantities. In this work, we analyze further the LDA for short-range exchange-correlation effects by performing a detailed local analysis. Indeed, for both the erf and erfgau interactions, we have computed for the He and Be atoms accurate exchange-correlation potentials v µ xc and accurate local exchange-correlation energies per particle εµ xc in two possible definitions and compared to the corresponding LDA quantities.

The paper is organized as follows. In Sec. II, the calculation of accurate potentials v µ is explained and the corresponding exchange-correlation potentials are compared to the LDA ones. In Sec. III, accurate local exchangecorrelation energies per particle are calculated in two possible definitions and compared to the LDA local energy per particle. Finally, Sec. IV contains concluding remarks.

Atomic units will be used throughout this work.

II. SHORT-RANGE EXCHANGE-CORRELATION POTENTIALS

The method we use for calculating accurate external potentials by optimization is explained in Refs. 16 and 10. We rapidly recall here the principal aspects.

The potentials v µ (r) are determined by using the Legendre transform formulation of the universal functional [17,18]

F µ [n] = sup ṽµ E µ [ṽ µ ] -ṽµ (r)n(r)dr , (11)
where E µ [ṽ µ ] is the ground-state energy of the Hamiltonian T + V µ ee + i ṽµ (r i ). If n is chosen to be the physical density of the system, and if n is assumed to be v-representable in the presence of the interaction V µ ee , then the supremum in Eq. ( 11) is a maximum reached for the desired v µ and F µ . In practice, an accurate density n is computed by multi-reference configurationinteraction calculation with single and double excitations (MRCISD) [19,20] and the potential to optimize ṽµ (r) for atomic systems is expanded as

ṽµ (r) = n i=1 c i r pi e γir 2 + C r , (12)
where c i are the optimized coefficients, p i are some fixed integers (-1 or 2), γ i are fixed exponents chosen so as to form an even-tempered basis set (typically, γ i ∈ [10 -3 , 5.10 4 ]), and C is a constant enforces the correct asymptotic behavior for r → ∞. For the Kohn-Sham case (µ = 0), this asymptotic behavior is determined by the nuclei-electron, Hartree and exchange contribution to the potential, giving C = -Z + N -1 (N and Z are the electron number and nuclear charge, respectively). For finite µ, the short-range Hartree and exchange potentials are exponentially decreasing at infinity, so that it remains only the nuclei-electron contribution, giving C = -Z. Actually, the maximizing potential v µ (r) in Eq. ( 11) is defined only up to an additive constant. The potential of Eq. ( 12) is the one which goes to 0 as r → ∞.

The maximization of Eq. ( 11) is carried out with the Simplex method [START_REF] Press | Numerical Recipes[END_REF]. For a given potential, E µ [ṽ µ ] is computed at MRCISD level using the Molpro program [START_REF] Amos | Molpro, a package of ab initio programs designed[END_REF] with modified two-electron integrals (see Appendix A of Ref. 12). Beside the asymptotic behavior for r → ∞, v µ (r) ∼ C/r, the behavior of the potential at the nucleus r = 0, v µ (r) ∼ -Z/r, is also imposed during the optimization. Large one-electron even-tempered Gaussian basis sets are used for all systems (see Refs. 16 and 10 for more details). The quality of the obtained potential is assessed using the Zhao-Parr criterion [START_REF] Zhao | [END_REF]

∆ = 1 2 (ñ(r 1 ) -n(r 1 )) (ñ(r 2 ) -n(r 2 )) |r 1 -r 2 | dr 1 dr 2 , ( 13
)
where ñ is the density given by the approximate potential ṽµ . The largest values of ∆ obtained with our potentials are of order 10 -8 . For two-electron systems, the exchange-correlation potential can be easily decomposed into exchange and correlation contributions. Indeed, the exchange potential is known from the Hartree potential, v µ x (r) = -v µ h (r)/2, and the correlation potential is simply obtained by difference v µ c (r) = v µ xc (r) -v µ x (r). Figure 1 shows the exchange potentials of the He atom with the erf interaction for µ = 0, 0.5 and 2, together with the corresponding LDA potentials. To better visualize the asymptotic behavior as r → ∞, rv µ x (r) is plotted instead of v µ x (r). In the KS case (µ = 0), the LDA exchange potential largely differs from the accurate one. In particular, the LDA potential has not the correct Coulombic asymptotic behavior v µ=0 x (r) ∼ -1/r as r → ∞. When µ increases, the range of the accurate potentials decreases and rv µ x (r) goes to 0 as r → ∞. When µ increases, the asymptote is reached at smaller and smaller values of r. The important observation is that the accuracy of the LDA potentials increases with µ. At µ = 2, at the scale of the plot, the LDA potential is nearly identical to the accurate one.

The accurate correlation potential for the He atom with the erf interaction for µ = 0, 1 and 3 is reported in figure 2, together with the LDA correlation potentials. In the KS case (µ = 0), the correlation potential calculated by Umrigar and Gonze [24] by inversion of the KS (10), dashed curves]. For µ = 0, the correlation potential calculated by Umrigar and Gonze [24] (dotted curve) is also shown.

equation with a very accurate density is also reported. This potential agrees well with our potential except at very small r where the precision of our calculations (using Gaussian basis sets) does not allow to extract the correlation potential. The LDA correlation potential for µ = 0 is a poor approximation to the accurate, structured potential. As for the exchange potential, the range of correlation potential is reduced when µ increases. At large µ, the correlation potential has less structure which enables the LDA to perform better but on average only.

The potentials obtained for the Be atom with the erf interaction are plotted in figure 3. At µ = 0, the exchange-correlation potential calculated by Umrigar and Gonze [START_REF] Umrigar | High Performance Computing and its Application to the Physical Sciences[END_REF] using quantum Monte Carlo methods is also reported. In comparison to the He atom, the accurate potentials for small µ now exhibit a shell structure with a jump separating the core region (r 1) and the valence region (r 1). The LDA potentials does not reproduce well this jump. When µ is large enough (µ 1) so that the valence region is cut off, the shell structure disappears in the potential and the quality of the LDA potentials is improved.

The potentials with the erfgau interaction are qualitatively similar to the potentials with the erf interaction FIG. 3: Exchange-correlation potentials rv µ xc (r) for Be with the erf interaction with µ = 0, 0.5 and 1.5: the potentials obtained in the present work (full curves) are compared to the LDA potentials [Eq. (10), dashed curves]. For µ = 0, the exchange-correlation potential calculated by Umrigar and Gonze [START_REF] Umrigar | High Performance Computing and its Application to the Physical Sciences[END_REF] (dotted curve) is also shown. and will not be shown here. We just mention that a careful comparison shows that the potentials with the erfgau interaction are more short-ranged, confirming than the erfgau interaction achieves a better long-range/shortrange separation than the erf interaction [12].

III. SHORT-RANGE LOCAL EXCHANGE-CORRELATION ENERGIES PER PARTICLE

The short-range local exchange-correlation energy per particle εµ xc (r) associated to the exact Ēµ xc is not uniquely defined. However, one can define and calculate some "physically realistic" εµ xc (r) which can be compared to approximations like the LDA. We discuss now two such possible choices: local energies per particle obtained from integration of pair densities along adiabatic connections and local energies per particle obtained directly from potentials.

A. Short-range local exchange-correlation energies per particle from pair densities

It is natural to define a short-range local exchange energy per particle by εµ,pd

x (r 1 ) = 1 2 n x (r 1 , r 2 )v µ ee (r 12 )dr 2 . ( 14
)
where n x (r 1 , r 2 ) is the exchange hole calculated from the one-particle KS density matrix:

n x (r 1 , r 2 ) = -|n KS 1 (r 1 , r 2 )| 2 /(2n(r 1 )
). The superscript pd stands for pair density. Similarly, the exact representation of Ēµ c by integration over the non-linear adiabatic connection

Ēµ c = 1 2 ∞ µ dξ n(r 1 )n ξ c (r 1 , r 2 ) ∂v ξ ee (r 12 ) ∂ξ dr 1 dr 2 , (15)
where n ξ c (r 1 , r 2 ) is the correlation hole at interaction parameter ξ, suggests a natural definition for the shortrange local correlation energy per particle (see also Ref. 26) εµ,pd

c (r 1 ) = 1 2 ∞ µ dξ dr 2 n ξ c (r 1 , r 2 ) ∂v ξ ee (r 12 ) ∂ξ . (16)
Note that even for the KS system (µ = 0), the local correlation energy per particle εµ=0,pd c (r) can depend on the adiabatic connection path followed.

We have calculated these local energies per particle from accurate exchange and correlation holes associated to the system of Eq. ( 1) using the accurate potentials of Sec. II and the programs Molpro [START_REF] Amos | Molpro, a package of ab initio programs designed[END_REF] and CASDI [27].

B. Short-range local exchange-correlation energies per particle from potentials

We now define another short-range local exchangecorrelation energy per particle εµ,local xc (n) as a function of n, by requiring that it yields of course the exact shortrange exchange-correlation energy

Ēµ xc = n(r)ε µ,local xc (n(r))dr, (17)
but also the exact short-range exchange-correlation potential

d nε µ,local xc (n) dn n=n(r) = v µ xc (r) + C. ( 18
)
The potential is defined only up to an additive constant C and v µ xc (r) is the exchange-correlation potential which goes to 0 at infinity (calculated in Sec. II). For a given system, εµ,local xc (n) defines an exact local short-range exchange-correlation functional in the sense that it gives the exact exchange-correlation potential and energy. This approach has already been applied for the Kohn-Sham scheme [28]; we briefly recall here how εµ,local xc (n) is calculated. The condition of Eq. ( 18) implies

∇ nε µ,local xc (n) = d nε µ,local xc (n) dn ∇n = (v µ xc (r) + C)∇n, (19)
which becomes for spherically symmetric systems

d n(r)ε µ,local xc (r) dr = dn(r) dr (v µ xc (r) + C). (20)
Integration of Eq. ( 20) leads to

n(r)ε µ,local xc (r) -n(r)ε µ,local xc (r) r→∞ = - ∞ r dn(r ′ ) dr ′ (v µ xc (r ′ ) + C)dr ′ . ( 21
)
To avoid divergence of the integral in Eq. ( 17), we must have n(r)ε

(r) = -1 n(r) ∞ r dn(r ′ ) dr ′ v µ xc (r ′ )dr ′ + C, (22)
where the constant C is fixed by requiring that

Ēµ xc = ∞ 0 n(r)ε µ,local xc (r)4πr 2 dr = - 4π 3 ∞ 0 d n(r)ε µ,local xc (r) dr r 3 dr = - 4π 3 ∞ 0 dn(r) dr v µ xc (r)r 3 dr + CN, (23)
with the number of electrons N = n(r)4πr 2 dr. Similarly, using only the exchange or correlation parts of v µ xc (r) and Ēµ xc in Eqs. ( 22) and ( 23) leads to the local exchange energy per particle εµ,local x (r) and local correlation energy per particle εµ,local c (r), respectively. We have calculated these local energies per particle using the accurate potentials of Sec. II and accurate values of Ēµ

x and Ēµ c obtained from F µ of Eq. (11).

C. Results

For monotonically decaying spherical densities, the map r → n(r) can be inverted and the local exchangecorrelation energy per particle can therefore be expressed as a function of the density n or, equivalently, as a function of 1/r s = (4πn/3) 1/3 . The reason to use 1/r s is that the exchange energy per particle of the Coulombic uniform electron gas εµ=0,unif

x is proportional to 1/r s . For the He atom with the erf interaction, the two accurate short-range local exchange energies per particle εµ,pd

x (1/r s ) and εµ,local

x (1/r s ) are compared in figure 4, together with the LDA local energy per particle. The difference between the two accurate local energies per particle, both giving the same correct short-range exchange energy Ēµ

x , illustrates the arbitrariness in the definition of local quantities from global ones. However, the differences soften when µ is increased, i.e. when long-range interactions are removed. The LDA local exchange energy per particle is significantly different from both accurate local exchange energies per particle at the KS end of the adiabatic connection (µ = 0), but as µ increases the LDA local energy per particle better and better agrees with the accurate local energies per particle. At low densities (1/r s < 0.5), the LDA local energy per particle is close to the accurate εµ,pd

x (1/r s ) while at high densities (for 1/r s > 1.5) the LDA local energy per particle is closer to the accurate εµ,local

x (1/r s ). In figure 5, again for the He atom with the erf interaction, the accurate local correlation energies per particle εµ,pd c (1/r s ) and εµ,local c (1/r s ) are represented and compared to the LDA. The two accurate local energies per particle give the same correct short-range correlation energy Ēµ c but have rather different shapes, although the differences partly soften when µ increases. The LDA generally overestimates the accurate local correlation energies per particle at µ = 0, but again when µ increases the LDA is improved on average. The LDA local energy per particle is much more comparable to εµ,pd c (1/r s ) than εµ,local c (1/r s ) especially for large µ and at low densities. In general, similar behaviors are obtained with the erfgau interaction. However, the attractive character of this interaction for small values of µ (see Ref. 15) can lead to some significant differences compared to the erf interaction. Figure 6 shows these differences in the accurate local correlation energy per particle εµ,pd c (1/r s ) at µ = 0. In comparison to the erf interaction, the local energy per particle obtained with the erfgau interaction exhibits in particular a positive contribution at very low densities. This emphasizes now the arbitrariness of the definition εµ,pd c (1/r s ) with respect to the choice of the adiabatic connection along which the pair density is integrated.

Figure 7 show the evolution of the local exchangecorrelation energies per particle with the erf interaction for the Be atom. In comparison to the He atom, both εµ,pd c (1/r s ) and εµ,local c (1/r s ) exhibit a shell structure for small values of µ, the contributions from the valence region (r s 0.5) and the core region (r s 0.5) being easily identifiable. The LDA local exchange-correlation energy per particle differs significantly from the two accurate ones, especially in the core region which constitutes the major contribution to the exchange-correlation energy. Also, in this representation in term of 1/r s , the LDA does not reproduce at all the intershell jump. However, when µ increases, the contribution from the valence region is progressively cut off and the shell pattern finally disappears. For large µ, when only the contribution from the core remains, the LDA local energy is close to the accurate local energies.

Finally, the accurate local correlation energy per particle εµ,pd c (1/r s ) for the Be atom is reported in figure 8, together with the LDA. The shell structure in this plot is even clearer. At µ = 0, the LDA overestimates the accu- rate local correlation energy per particle at all densities. When µ increases, the LDA starts to better reproduce the valence region but still overestimate the accurate local energy in the core region. Keeping on increasing µ, the contribution from the valence region vanishes, and the LDA starts to reproduce well the core region.

IV. CONCLUDING REMARKS

Non-linear adiabatic connections have been used to decompose the energy of an electronic system into a longrange wave function part and short-range density functional part, the position of the frontier of this decomposition being determined by a single parameter µ.

When the interaction range of the short-range energy component is reduced, corresponding to a rise of µ, the short-range LDA becomes an increasingly accurate approximation to the short-range exchange-correlation density functional. Indeed, as it has been verified in this work for a few atomic systems, the short-range LDA gives, for a sufficiently large µ, good short-range exchange- 16)] with respect to 1/rs for He with the erf (solid curve) and erfgau (dotted curve) interactions for µ = 0. correlation potentials and local exchange-correlation energies per particle.

More formally, one can define an optimal local interaction parameter, µ opt (r), so that for µ > µ opt (r) the LDA reproduces, at a given precision, the exact short-range potentials or local energies at the considered point of space. Physically, 1/µ opt (r) represents the maximum lo- cal interaction range over which the exchange-correlation effects are well transferable from the uniform electron gas to the inhomogeneous system of interest. In general, the larger the density at point r is, the larger µ opt (r) is. However, it is not conceivable in the method to use a rdependent interaction parameter µ because of the wave function part of the calculation. In a simple system like the He atom, the improvement of the LDA upon increasing µ is quite uniform in space and one can basically define an global average optimal interaction parameter, μopt , and use it to fix the frontier of the decomposition, i.e. µ = μopt .

More generally, in strongest inhomogeneous systems like the Be atom, at least two regions of space are identifiable, the core and valence shells, and it is meaningful to define separate averages of µ opt (r) over each regions, μcore opt and μvalence opt . As the density is higher in the core than in the valence, we have μcore opt > μvalence opt . If one fixes the frontier of the long-range/short-range decomposition at the optimal value for the core shell, i.e. µ = μcore opt , the short-range functional part of the calculation can be well treated in LDA but of course the comple- ment part of the core region and the entire valence region are assigned to the wave function part which can result in the need of a long expansion into Slater determinants.

On the contrary, if one fixes the frontier of the decomposition at the optimal value for the valence shell, i.e. µ = μvalence opt , the short-range LDA functional is not enable to treat well the core region but can describe well that part of the valence region assigned to it. Moreover, the wave function part of the calculation has only to handle with the remaining part of the valence region which allows in general the use a short expansion into Slater determinants.

Obviously, one would like to re-conciliate these two alternatives. For practical applications of the method to molecular systems of chemical interests, it is desirable to keep the wave function expansion minimal and thus to chose the frontier of the decomposition inside the valence region, i.e. µ = μvalence opt . For properties essentially depending on the valence shell only, such as atomization energies, the error of the LDA in the core shell can then be ignored, or the core electrons can be replaced by a pseudo-potential. Otherwise, one must then find better approximations to the short-range exchange-correlation functional which extend the domain of accuracy of the LDA toward larger interaction ranges (small µ). We hope that the presented results will help the construction of such approximations.

Data for the atomic systems presented in this paper (and other systems) are available from the authors upon request.

FIG. 1 :

 1 FIG. 1: Exchange potentials rv µx (r) for He with the erf interaction with µ = 0, 0.5 and 2: the potentials obtained in the present work (full curves) are compared to the LDA potentials [Eq.(10), dashed curves].

FIG. 2 :

 2 FIG.2: Correlation potentials v µ c (r) for He with the erf interaction with µ = 0, 1 and 3: the potentials obtained in the present work (full curves) are compared to the LDA potentials [Eq.(10), dashed curves]. For µ = 0, the correlation potential calculated by Umrigar and Gonze[24] (dotted curve) is also shown.

FIG. 4 :

 4 FIG.4: Short-range local exchange energies per particle with respect to 1/rs for He with the erf interaction for µ = 0, 1, 3: εµ,LDAx

FIG. 5 :FIG. 6 :

 56 FIG.5: Short-range local correlation energies per particle with respect to 1/rs for He with the erf interaction for µ = 0, 1, 3. εµ,LDA c (long-dashed curve) is compared to the accurate εµ,pd c

FIG. 7 :

 7 FIG.7: Short-range local exchange-correlation energies per particle with respect to 1/rs for Be with the erf interaction for µ = 0, 1, 3: εµ,LDA xc [Eq. (9), long-dashed curve] is compared to the accurate εµ,pd xc [Sec. (III A), solid curve] and εµ,localxc

FIG. 8 :

 8 FIG.8: Short-range local correlation energies per particle with respect to 1/rs for Be with the erf interaction for µ = 0, 1, 3: εµ,LDA c (long-dashed curve) is compared to the accurate εµ,pd c

Acknowledgments

We thank D. Maynau (Université Paul Sabatier, Toulouse, France) for providing us the CASDI program. We are also grateful to C. Umrigar (Cornell University, USA) for the data of Refs. 24, 25.


Julien Toulouse

François Colonna

Andreas Savin

Short-range exchange and correlation energy density functionals: beyond the local density approximation

We propose approximations which go beyond the local density approximation for the short-range exchange and correlation density functionals appearing in a multi-determinantal extension of the Kohn-Sham scheme. A first approximation consists in defining locally the range of the interaction in the correlation functional. Another approximation, more conventional, is based on a gradient expansion of the short-range exchange-correlation functional. Finally, we also test a short-range generalized-gradient approximation by extending the Perdew-Burke-Ernzerhof exchange-correlation functional to short-range interactions.

I. INTRODUCTION

The Kohn-Sham (KS) scheme [1] of density functional theory (DFT) (see, e.g., [START_REF] Parr | Density-Functional Theory of Atoms and Molecules[END_REF][START_REF] Dreizler | Density Functional Theory[END_REF]) can be extended to handle multi-determinantal wave functions [START_REF] Savin | [END_REF]. This enables to describe near-degeneracy correlation effects by explicitly taking into account several Slater determinants. The method is based on a decomposition of the true Coulomb electron-electron interaction as 1 r = v µ ee (r) + vµ ee (r),

where v µ ee (r) is a long-range interaction and vµ ee (r) is the complement short-range interaction. This separation is controlled by the parameter µ. In previous studies [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF][7], the error function has been used to define the long-range interaction v µ ee,erf (r) = erf(µr) r ,

referred to as the erf interaction. More recently [8], we have proposed a sharper long-range/short-range separation with the erfgau interaction

v µ ee,erfgau (r) = erf(cµr) r - 2cµ √ π e -1 3 c 2 µ 2 r 2 , (3)
where c = 1 + 6 √ 3 1/2 ≈ 3.375 is a constant chosen to facilitate the comparison with the erf interaction. Indeed, with this choice, the parameter µ has roughly the same physical meaning for the erf and erfgau interactions: 1/µ represents the distance beyond which the interaction reduces to the usual Coulomb long-range tail. (Note that in Ref. 8 the constant c was absorbed into the parameter µ). Both modified interactions define a generalized adiabatic connection [9] between the noninteracting KS system corresponding to µ = 0 and the physical interacting system corresponding to µ → ∞. * Electronic address: savin@lct.jussieu.fr

The ground-state energy of an electronic system can then be (exactly in principle) calculated by the sum of a wave function part associated to the long-range electronelectron interaction V µ ee = i<j v µ ee (r ij ) and a density functional part describing the remaining interactions

E = Ψ µ | T + V µ ee + Vne |Ψ µ + Ū µ [n] + Ēµ xc [n], ( 4
)
where T is the kinetic energy operator, Vne = i v ne (r i ) is the nuclei-electron interaction, Ū µ is the short-range Hartree energy and Ēµ xc is the short-range exchangecorrelation functional defined as the difference between the standard KS exchange-correlation energy E xc and the long-range exchange-correlation energy E µ xc associated to the interaction v µ ee

Ēµ xc = E xc -E µ xc . (5)
In Eq. ( 4), Ψ µ is the multi-determinantal ground-state wave function of a fictitious system with long-range electron-electron interaction V µ ee

T + V µ ee + V µ |Ψ µ = E µ |Ψ µ , (6)
where V µ = i v µ (r i ) is the external local potential ensuring that this fictitious system has the same ground state density n than the physical system. The potential v µ is simply related to the functional derivative of the short-range functionals through

v µ (r) = v ne (r) + δ Ū µ δn(r) + δ Ēµ xc δn(r) . ( 7
)
Previous applications of the method [START_REF] Leininger | [END_REF]7] show that, for a reasonable long-range/short-range separation (µ not too small) and if the few most important (nearly degenerate) configurations are included in the wave function Ψ µ , good results are obtained for total atomic and molecular energies, including near-degenerate systems. This is remarkable since, in these previous studies, only the simple local density approximation (LDA) was used for the short-range functional Ēµ xc [n].

The purpose of the present work is to further improve the accuracy of the method by proposing approximations for Ēµ xc [n] which go beyond the LDA. We will not present in this paper total energies E (or equivalently total exchange-correlation energies E xc ), but will focus only on short-range exchange-correlation energies Ēµ xc . Indeed, for a chosen µ, it is the approximation for Ēµ xc [n] that limits in practice the accuracy of the method, not the other long-range contributions to the energy that can be calculated by a traditional ab initio wave function method. We will test our proposed approximations for Ēµ xc by comparison to accurate data obtained for small atomic systems.

The paper is organized as follows. In Sec. II, we discuss the limitations of the LDA for Ēµ xc [n]. In Sec. III, we introduce a local interaction parameter µ to improve the LDA correlation functional. We then discuss the extension of traditional gradient corrections to the shortrange functional Ēµ xc [n]. Indeed, the short-range versions of the gradient expansion approximation (GEA) is presented in Sec. IV, while Sec. V contains the extension of a generalized-gradient approximation (GGA), namely the PBE functional [10], to a modified interaction. Finally, Sec. VI contains our concluding remarks.

Atomic units will be used throughout this work.

II. LIMITATIONS OF THE LDA

In the LDA, the short-range exchange-correlation energy per particle εµ,unif xc (n) of a uniform electron gas with modified interaction [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]11] is transferred locally to the inhomogeneous system of interest

Ēµ xc [n] = n(r) εµ,unif xc (n(r))dr. ( 8
)
To underline the dependence on the interaction parameter, we will refer to this approximation as µ-LDA.

In Figs. 1 and2, we have plotted the short-range LDA exchange and correlation energies of the Be atom with respect to µ for the erf and erfgau interactions. An accurate density obtained from a multi-reference configuration interaction calculation with single and double excitations (MRCISD) [12,13] has been used. For comparison, accurate calculations of the exchange and correlations energies along the adiabatic connections are also reported. In these accurate calculations, we start from an accurate reference density and, for each µ, numerically optimize the external potential v µ (r) appearing in Eq. ( 6) so as to recover the reference density. The ground-state wave function Ψ µ is then computed according to Eq. ( 6) by MRCISD and the various energy components like the short-range exchange and correlation energies are deduced. (For further details, see Refs. [14][15][16].

For both the erf and erfgau interactions, the µ-LDA is very accurate for large µ but fails near the KS end (µ = 0) of the adiabatic connection. In particular, the exchange energy is underestimated and the correlation energy is overestimated. Thanks to a sharper separation of longrange and short-range electron-electron interactions, the erfgau interaction provides a slight improvement over the erf interaction. In fact, Fig. 1 shows that the erfgau µ-LDA exchange energy curve reaches the exact one for a larger energy (≈ -0.75 Hartree) and a smaller µ (≈ 2.5) than in the erf case (≈ -0.5 Hartree at µ ≈ 3), meaning that the erfgau µ-LDA is able to correctly describe a larger part of the exchange energy. For the correlation energy, the two interactions lead to similar results. We note in passing that the unimportant little bump on the curve of the accurate correlation energy for small µ with the erfgau interaction in Fig. 2 (and in other following figures) is a manifestation of the non-monotonicity of this interaction with respect to µ (being in turn responsible for its attractive character for very small µ [11]). In the remaining of the paper, we will present results only for the erfgau interaction.

The performance of the µ-LDA can be further analyzed by looking, for instance, at the correlation energy density εµ c (r). There is not unique definition of this quantity; we used the definition of Ref. 17 or 16 based on the integration over the adiabatic connection

εµ c (r 1 ) = 1 2 ∞ µ dξ dr 2 n ξ c (r 1 , r 2 ) ∂v ξ ee (r 12 ) ∂ξ , (9)
where n ξ c (r 1 , r 2 ) is the correlation hole for the interaction parameter ξ. In Fig. 3, this correlation energy density accurately computed for the Be atom is compared with the µ-LDA with the erfgau interaction for a series of µ's. For µ = 0 (KS system), the correlation energy density of Eq. ( 9) is largely overestimated by the LDA over the whole range of r. When µ is increased (Fig. 3 with µ = 0.21 and µ = 1.20), the µ-LDA starts to better reproduces the accurate energy density in the valence region (r 1) but still overestimates it in the core region (r 1). For the µ-LDA to also well reproduce the core region, larger values of µ are required (Fig. 3 with µ = 3.00).

Having in mind that the choice of a large value of µ means an important computational effort for the part of the energy treated by wave function methods, it is important to improve the short-range exchange-correlation functional toward the small µ region of the adiabatic connection.

III. LDA CORRELATION WITH LOCAL INTERACTION PARAMETER

In inhomogeneous systems the electrons are correlated on a shorter distance than in the uniform electron gas. Thus, within the KS scheme, the LDA transfers spurious long-range correlation contributions from the electron gas to the finite system of interest. This point have long been understood and have guided important developments in density functional approximations. For instance, Langreth and Mehl [18,19] proposed a gradient correction to the LDA based on a cut-off in momentum-space for small k which basically corresponds to removing the long-range interaction effects.

In the method proposed here, for small µ, i.e. when a large range of the electron-electron interaction is retained in the functional part, the µ-LDA also transfers spurious long-range interaction effects from the uniform electron gas to finite systems and leads to an overestimation of the correlation energy. A simple way to eliminate this incorrect long-range interaction effects, is to use in the µ-LDA correlation functional an effective interaction parameter, larger than the one used in the wave function part of the calculation. Of course, in the uniform electron gas, the range of the interaction is relative to the density (the relevant scaled variable is r/r s with r s = (4πn/3) -1/3 ) and consequently one has to actually consider an effective local interaction parameter, µ l (r), defined from the inhomogeneous density profile n(r). However, the interaction parameter used in the functional must not be smaller that the global µ used in the wave function part of the calculation to avoid a double counting of correlation effects. We therefore take in the correlation µ-LDA functional an effective local interaction parameter given by µ eff (r) = max(µ l (r), µ). (10) Thus, this approximation corresponds to simply drop off some long-range correlation from the LDA in regions of space where µ l (r) > µ.

Several choices are possible for the local interaction parameter µ l (r). The first natural length scale in real space for the electron-electron interaction is provided by the Wigner-Seitz radius r s , representing the radius of the sphere centered in r containing one electron. Equivalently, the same length scale can be expressed in momentum space by the Fermi wave vector k F = 1/(αr s ) where α = (9π/4) -1/3 . Following a previous work [7], we thus take as a local interaction parameter

µ l,a (r) = 1 r s (r) = αk F (r). (11)
However, studies of the uniform electron gas indicate that the relevant length scale for correlation effects in momentum space is given by the Thomas-Fermi wave vector k s = 4k F /π describing the screening of the Coulomb interaction. Similarly to Eq. ( 11), we can thus take as local interaction parameter

µ l,b (r) = αk s (r) = 2 α/π r s (r) . (12)
More elaborated dependences of µ l on r s are possible.

For example, Ferrell [20] has determined a momentum cut-off in the uniform electron gas corresponding to the shortest wavelength at which plasma oscillations can be sustained by the electron gas. It turns out however that results obtained with a local interaction parameter set to Ferrel's momentum cut-off are very similar to those obtained with the local interaction parameter of Eq. ( 12) and will not be discussed here.

A much more interesting possibility is to choose a local interaction parameter depending on the gradient of the density. In fact, as summarized by Svendsen and von Barth [21] from the work of Langreth and Mehl [18,19] and Perdew et al. [22], "distances over which electrons are correlated in realistic inhomogeneous systems are determined more by actual variations in the density profile than by the screening of an electron gas at the local value of the density". We use a simple geometrical argument similar of that of Langreth and Mehl [18] to determine a local characteristic distance d(r) over which the density varies in an inhomogeneous system. Consider a density distribution with constant density gradient consisting of an isosceles triangle as a prototype of an atom. The extension of this density distribution is d = 4n/|∇n| where n is the density at the middle of one side of the triangle, i.e. a typical value of the density. We thus define a (semi)local interaction parameter by FIG. 3: Short-range correlation energy density εµ c (r) with respect to r for the Be atom with the erfgau interaction for µ = 0, 0.21, 1.20, 3.00. Accurate data computed from integration over the adiabatic connection (Eq. 9, full curve) are compared with the µ-LDA (dashed curve).

µ l,c (r) = |∇n(r)| 4n(r) . (13)
The constant 4 in Eq. ( 13) is somehow arbitrary; experience shows that this is reasonable.

Fig. 4 shows the correlation energy is greatly improved by the local µ approach. However, for the choices µ l,a = αk F or µ l,b = αk s , a more detailed analysis through the correlation energy density in Fig. 5 indicates the µ-LDA functional with a local µ is only an average of the "exact" correlation energy density. Of course, the µ-LDA functional with a local µ improves on increasing of µ (not shown) but it is clear that both choices are inadequate for describing the shell structure. On the contrary, one sees on the same Fig. 5 that the choice µ l,c = |∇n|/(4n) enables to recover well the shell structure in Be. We therefore consider this last choice as the more appropriate local parameter µ l for improving the LDA correlation functional.

Finally, we note that this local µ approach cannot be directly applied to the exchange energy. Indeed, the LDA underestimates the exchange energy and thus choosing a large effective interaction parameter in the µ-LDA would only deteriorate the exchange energy.

IV. GRADIENT EXPANSIONS

We now consider more conventional approaches to correct the local density approximation at small interaction parameter, namely gradient corrections. In the gradient expansion approximation (GEA), the short-range ex- change energy density is expanded in term of the reduced gradient s = |∇n|/(2k F n) and the reduced Laplacian

q = ∇ 2 n/((2k F ) 2 n) εµ,GEA-Lap x (n) = εµ,unif x (n) 1 + a(μ)s 2 + d(μ)q , ( 14
)
where the reduced interaction parameter μ = µ/(2k F ) has been introduced. The gradient and Laplacian coefficients a(μ) and d(μ) can be calculated numerically for all µ's except for µ = 0 where there are not defined (see Appendix A). Alternatively, one can perform an integration by parts and obtain the more conventional GEA exchange density εµ,GEA

x (n) = εµ,unif x (n) 1 + b(μ)s 2 , (15)
where the exchange second-order gradient coefficient b(μ) has been calculated analytically for the erf and erfgau interactions [see Appendix A, Eqs. (A18) and (A19)]. The approximations of Eqs. ( 14) and ( 15) will be referred to as µ-GEA-Lap and µ-GEA, respectively. Fig. 6 compares the µ-GEA-Lap and µ-GEA shortrange exchange energy densities for the Be atom with the erfgau interaction at µ = 0.21. The exchange energy density in the µ-LDA and calculated from the accurate exchange hole n x (r 1 , r 2 ),

εµ x (r 1 ) = 1 2 dr 2 n x (r 1 , r 2 )v µ ee (r 12 ), (16)
are also reported. To avoid the Laplacian divergence at the nucleus in µ-GEA-Lap, radial energy densities 4πr 2 n(r)ε µ x (r) are actually represented. One sees that the integration by parts in µ-GEA does not change qualitatively the shape of the curve and it is meaningful to compare both the µ-GEA-Lap and µ-GEA exchange energy densities to the accurate and µ-LDA ones. Besides, for larger values of µ, µ-GEA-Lap and µ-GEA become nearly identical (not shown). Of course, µ-GEA-Lap and µ-GEA give the same short-range exchange energy Ēµ x and we will used in practice µ-GEA because of its greatest simplicity.

Similar to Eq. ( 15), the short-range correlation energy density is expanded in term of the reduced gradient t = |∇n|/(2k s n)

εµ,GEA c (n) = εµ,unif c (n) + β(r s , µ)t 2 . ( 17
)

FIG. 6:

Radial short-range exchange energy density 4πr 2 n(r)ε µ x (r) with respect to r for the Be atom with the erfgau interaction for µ = 0.21. Accurate data (Eq. 16, full curve) are compared with the µ-LDA (long-dashed curve), the µ-GEA (short-dashed curve) and the µ-GEA-Lap (dotted curve).

The correlation second-order gradient coefficient β(r s , µ) can be estimated by imposing that the correlation gradient expansion cancels the exchange gradient expansion

β(r s , µ) = -ε µ,unif x (n)b(μ)(k s /k F ) 2 . ( 18
)
This is based on the observation that, for µ = 0, the exchange-correlation contribution to the linear response of the uniform electron gas to a slowly oscillating perturbation (small momenta) almost vanishes [23]. This shortrange GEA functional approximation will be referred to as µ-GEA. Our interest is to evaluate the improvement brought by µ-GEA over µ-LDA for the exchange and correlation energies separately. Notice that, in our construction, if we consider exchange and correlation together µ-GEA is identical to µ-LDA.

The µ-GEA exchange energy is plotted in Fig 7 . For large µ (µ 3), the range of the interaction is too small to feel the slowly oscillating gradient correction and the µ-GEA functional reduces to the µ-LDA functional. The µ-GEA curve reaches the exact one at a smaller value of µ (µ ≈ 2) than the µ-LDA curve (µ ≈ 3). For small µ, the µ-GEA reduces the µ-LDA error by about a factor two. For the correlation energy (Fig. 8), the µ-GEA also extends the domain of validity of the µ-LDA when µ is decreased down to µ ≈ 4. For smaller µ when long-range correlation effects are introduced in the functional, the gradient expansion breaks down.

Whereas the GEA for the KS scheme (µ = 0) often disappointingly deteriorates the LDA in real inhomogeneous systems [24,25], it can be consider as a systematic way to extend the domain of validity of the µ-LDA for the short-range part only of the exchange and correlation functionals. These results are in agreement with the wave vector analysis of the GEA by Langreth and Perdew [26] who found that the GEA works well for large momenta (corresponding to short-range density fluctuations).

V. SHORT-RANGE PBE FUNCTIONAL

Starting from the gradient expansions of the previous section, it is possible to construct generalized-gradient approximations (GGA) for the short-range exchange and correlation energy functionals which cure the main shortcomings of the µ-GEA. We have actually constructed a modified version of the PBE functional [10] for the modified interactions. All of the details of the construction of this functional that we will call µ-PBE are given in Appendix B. We simply mention here that we have use the same ansatz than PBE and impose similar theoretical constraints. However, a difference with the PBE approach is that we impose the second-order gradient coefficient for exchange and approximate that of correlation from it while PBE did the opposite. Consequently, our µ-PBE functional does not reduce to the original PBE for µ = 0. The exchange and correlation energies of the Be atom obtained with this µ-PBE functional along the erfgau adiabatic connection are compared to the µ-LDA and µ-GEA functionals in Figs. 7 and8. For a simple system like the Be atom, the µ-GEA and µ-PBE functionals are nearly identical for the exchange energy. For the correlation part, one sees that µ-PBE greatly improved the µ-LDA and µ-GEA results. The µ-PBE correlation energy is indeed very accurate along the entire adiabatic connection, except near the KS end (µ = 0) where a significant error remains. This inaccuracy of (semi)local functionals at µ = 0 is in favor of the strategy consisting in escaping the KS scheme by using a short-range functional at finite µ and treating the remaining part of the energy by other, more appropriate methods. In Fig. 9, it is shown that the correlation energy density of µ-PBE functional qualitatively describes the shell structure of the Be atom for small µ.

Finally, the difficult case of the Ne 6+ atom, presenting strong near-degeneracy correlation effects, is reported in Fig. 10. The correlation energy given by the µ-PBE functional actually over-corrects that of the µ-LDA. Again, this result suggests that for small µ semilocal approximations like the µ-PBE functional are inappropriate, especially when near-degeneracy correlation effects play an important role.

VI. CONCLUDING REMARKS

In this paper, we have considered exchange and correlation energy functionals associated to short-range electron-electron interactions. We have proposed new approximations which partially correct the local density approximation. Indeed, we showed that the short-range LDA correlation energy can be significantly corrected by simply adjusting in the functional the range of the interaction locally from the density and its gradient. We have also constructed generalizations of the GEA and of a GGA (namely, the PBE functional) to the case of short-range interactions. These new short-range functionals treat well a larger range of electron-electron interaction than the short-range LDA. However, better (possibly non-local) approximations are still needed to further extend the range of interaction well treated by the functional.

Appendix A: Gradient expansion of the short-range exchange energy functional

The short-range exchange energy functional writes

Ēµ x = 1 2 n(r)n x (r, r + R)v µ ee (R)drdR, (A1)
where n x (r, r + R) is the exchange hole. In the gradientexpansion approximation (GEA), the exchange hole is expanded to second order in term of the gradients of the density [27,28]

n GEA x (r, r + R) = - 1 2 n(r)y(r, r + R), (A2)
where

y = J + Lk -3 F R • ∇k 2 F + M k -6 F ( R • ∇k 2 F ) 2 +N k -6 F (∇k 2 F ) 2 + zLk -4 F ( R • ∇) 2 k 2 F /6 -z 2 Jk -4 F ∇ 2 k 2 F /48, (A3) with k F (r) = (3π 2 n(r)) 1/3 , R = R/R, z = 2k F R and
J(z) = 72[4 + z 2 -(4 -z 2 ) cos z -4z sin z]/z 6 , (A4) L(z) = 9(2 -2 cos z -z sin z)/z 3 , (A5)
M (z) = 9(-z cos z + sin z)/(16z), (A6) N (z) = 3[8-(8-4z 2 ) cos z-(8z-z 3 ) sin z]/(16z 4 ). (A7)
The spherical average of the GEA exchange hole in term of the reduced gradient s = |∇n|/(2k F n) and the reduced Laplacian q = ∇ 2 n/((2k

F ) 2 n) writes n GEA x (k F , s, q, z) = 1 4π n GEA x (r, r + R)dΩ R = n unif x (k F , z) + n grad x (k F , z)s 2 + n lap x (k F , z)q, (A8)
with the exchange hole of the uniform electron gas

n unif x (k F , z) = 24k 3 F [z cos(z/2) -2 sin(z/2)] 2 π 2 z 6 , (A9)
the gradient coefficient

n grad x (k F , z) = k 3 F [-4(18 + z 2 ) + (72 -32z 2 + 3z 4 ) cos z + 2z(36 -5z 2 ) sin z] 54π 2 z 4 , (A10)
and the Laplacian coefficient

n lap x (k F , z) = k 3 F [12(12 + z 2 ) + 12(-12 + 5z 2 ) cos z + 12z(-12 + z 2 ) sin z] 54π 2 z 4 . (A11)
The exchange energy density is

εGEA-Lap x (n, s, q, μ) = 1 2 n GEA x (r, r + R)v µ ee (R)dR = 1 2(2k F ) 3 ∞ 0 n GEA x (k F , s, q, z)v µ ee ( z 2k F )4π 2 z 2 dz, (A12)
where μ = µ/(2k F ). For convenience, εGEA-Lap

x (n, s, q, μ) can be expressed by εGEA-Lap

x (n, s, q, μ) = εunif x (n, μ) 1 + a(μ)s 2 + d(μ)q , (A13)
where εunif x (n, μ) has been given in Refs. 5, 11. (Note that, for the erfgau interaction, in Ref. 11 the constant c of Eq. ( 3) was absorbed into the parameter µ). The integral in Eq. (A12) diverges for µ = 0 due to the long-range character of the Coulomb interaction and thus εGEA-Lap x (n, s, q, μ = 0) is not defined. (For a recent discussion on the non-analyticity of the inhomogeneous exchange energy density with respect to the electron density, see Refs. 29,30). Nevertheless, for finite µ, the integral does exist and the gradient and Laplacian coef-ficients, a(μ) and d(μ), can be computed numerically.

To eliminate the divergence at µ = 0, one can perform an integration by parts of n GEA x (k F , s, q, z) over r in Eq. (A1) and define a new GEA exchange hole in term of the reduced gradient s only ñGEA

x (k F , s, z) = n unif x (k F , z) + ñgrad x (k F , z)s 2 , (A14)
with the associated gradient coefficient ñgrad

x (k F , z) = - k 3 F [-72 + (72 -36z 2 + z 4 ) cos z -2z(-36 + 5z 2 ) sin z] 54π 2 z 4 , ( A15
)
and the corresponding exchange energy density

εGEA x (n, s, μ) = 1 2(2k F ) 3 ∞ 0 ñGEA x (k F , s, z)v µ ee ( z 2k F )4π 2 z 2 dz. (A16)
Similarly to Eq. (A13), εGEA

x (n, s, μ) is expressed as

εGEA x (n, s, μ) = εunif x (n, μ) 1 + b(μ)s 2 . (A17)
The integral in Eq. ( A16) is now defined for all µ ′ s and can be done analytically for both the erf and erfgau modified interactions. For the gradient coefficient b(μ) with the erf interaction, we obtain b erf (μ) = -c 1 + c 2 e 1/(4μ 2 ) c 3 + 54c 4 e 1/(4μ 2 ) , (A18)

and, with the erfgau interaction, b erfgau (μ) = -3 √ 3c 5 + 2ν 2 c 1 e 1/(2ν 2 ) + c 6 e 3/(4ν 2 ) c 7 -2ν 2 c 3 e 1/(2ν 2 ) + 12ν 2 (-9c 4 + c 8 )e 3/(4ν 2 ) , (A19) with The approximations εGEA-Lap

c 1 = 1 + 22ν 2 + 144ν 4 , c 2 = 2ν 2 (-7 + 72ν 2 ), c 3 = -864ν 4 (-1 + 2ν 2 ), c 4 = ν 2 [-3 -24ν 2 + 32ν 2 + 8ν √ π erf(1/(2ν))], c 5 = 3 + 18ν 2 + 48ν 4 + 64ν 6 , c 6 = 4ν 4 (7 -72ν 2 + 48 √ 3ν 2 ), c 7 = -192 √ 3ν 6 (-3 + 8ν 2 ), c 8 = 8ν 3 [-18 √ 3ν+16 √ 3ν 3 +9 √ π erf( √ 3
x (n, s, q, μ) and εGEA

x (n, s, μ) will be referred to as µ-GEA-Lap and µ-GEA, respectively. They are compared in Sec. IV for the case of the Be atom with the erfgau interaction (Fig. 6). Because of its greatest simplicity, εGEA x (n, s, μ) will be used in practice.

To see the effect the short-range interaction vµ ee (R), the z-integrand of the µ-GEA exchange energy density (Eq. A16) has been plotted with respect to z in Fig. 11 in the case of the erf interaction. One sees that the spurious strong oscillations of the GEA exchange hole at large interelectronic distances are efficiently cut off at finite µ by the short-range interaction. In other words, only the short-range part of the GEA exchange hole is used in the short-range exchange functional.

In order to appreciate the contribution of the gradient term of the µ-GEA with respect to the interaction parameter, we have represented εunif

x (n, μ)b(μ)/ε unif x (n) in Fig. 12 for the erf and erfgau interactions. Not surprisingly, the gradient correction is decreased for a shortrange interaction. Notice however that for small μ, the gradient correction for the erfgau is increased compared to the Coulomb case μ = 0. In the context of Kohn-Sham DFT, Perdew, Burke and Ernzerhof [10] have proposed a gradient-corrected functional where the parameters are determined so as to satisfy theoretical conditions. We propose here to generalize the PBE functional along the erf or erfgau adiabatic connection. We note that for a different purpose Heyd, Scuseria and Ernzerhof [33] has already derived a short-range version of the exchange PBE functional corresponding to the erf interaction by using a model of the exchange hole [34]. (The obtained functional is called ωPBE where ω corresponds to our parameter µ). We shall not follow this procedure but come back instead to the original formulation of the PBE functional.

We thus take for the short-range exchange energy functional the PBE form

Ēµ

x = drn(r)ε unif x (n(r), μ(r))F x (s(r), μ(r)), (B1)

where we have introduced a dependence on the reduced interaction parameter μ = µ/(2k F ). In Eq. (B1), s = |∇n|/(2k F n) is a reduced density gradient, εunif x (n, μ) is the exchange energy per particle of a uniform electron gas with short-range interaction (see Refs. 5, 11), and F µ x (s, μ) is the enhancement factor

F x (s, μ) = 1 + κ(μ) - κ(μ) 1 + b(μ)s 2 /κ(μ) . ( B2
)
Note that this form had also be proposed by Becke [35].

The constant b(μ) is fixed by imposing the correct gradient expansion of F x (s, μ) as s → 0

F x (s, μ) ≈ 1 + b(μ)s 2 + • • • , (B3)
while κ(μ) is connected to the rapidly varying limit (s → ∞) For b(μ) we take the coefficient arising from the gradient expansion of the short-range exchange energy with erf or erfgau interaction (see Appendix A, Eqs. A18 and A19). κ(μ) is determined by imposing the Lieb-Oxford bound [36] which still holds for the short-range exchange functional Ēµ x

F x (s, μ) → 1 + κ(μ). (

Ēµ

x ≥ E x ≥ -C n(r) 4/3 dr, (B5)

since Ēµ

x = E x -E µ x and the long-range exchange energy E µ

x is always negative. The constant C, for which Lieb and Oxford originally found 1.6787, has recently be improved by Chan and Handy to a value C = 1.6358 [37]. A sufficient (but not necessary) condition for this bound to be satisfied is F x (s, μ) ≤ -Cn 1/3 /ε unif x (n, μ). (B6)

We thus take κ(μ) = -Cn 1/3 /ε unif x (n, μ) -1, the largest value insuring condition (B6). Fig. 13 shows κ(μ) for the erf and erfgau interaction. One sees that κ(μ) increases with μ, but the Lieb-Oxford bound is actually of no effect for large μ since the GEA gradient correction vanishes anyway and the enhancement factor reduces to 1.

The short-range correlation energy is given by Ēµ c = drn(r)[ε unif c (r s (r), µ) + H(r s (r), t(r), µ)], (B7)

with the reduced density gradient t = |∇n|/(2k s n). We imposed the same conditions on the gradient correction H(r s , t, µ) as PBE.

(a) The second-order gradient expansion in the slowly varying limit (t → 0) is

H(r s , t, µ) ≈ β(r s , µ)t 2 + • • • . (B8)
where the coefficient β(r s , µ) is estimated from the exchange gradient expansion according to Eq. ( 18). (b) The correlation is set to zero in the rapidly varying limit (t → ∞), thus H(r s , t, µ) → -ε unif c (r s , µ). (B9) (c) Under uniform scaling to the high-density limit, i.e. n(r) → λ 3 n(λr), r s → λ -1 r s , t → λ 1/2 t and λ → ∞, the correlation energy scales to a constant. This well-known property for the Coulomb interaction case [38] is also valid for the short-range correlation functional. Thus, H(r s , t, µ) must cancel the logarithmic divergence of the local short-range correlation energy εunif c (λ -1 r s , µ) ≈ γ ln(λ -1 ) + • • • with γ ≈ 0.031091, appearing as in the Coulomb case. Thus, as λ → ∞, H(λ -1 r s , λ 1/2 t, µ) ≈ γ ln(λ) + • • • .

(B10)

We then take the same ansatz than PBE H(r s , t, µ) = γ ln 1 + β(r s , µ)t 2 γ Note that in the original PBE functional, the secondorder gradient coefficient for correlation was imposed in the high-density limit, β(r s → 0, µ = 0) = β = 0.066725 and the second-order gradient coefficient for exchange was chosen so that to exactly cancel the correlation gradient expansion: b(µ = 0) = β(π 2 /3) = 0.21951. On the contrary, we have used the exact exchange gradient expansion for finite µ and estimate the correlation gradient expansion from it. Therefore, our µ-PBE functional does not reduce to the original PBE functional for µ = 0.

1 + At 2 1 + At 2 + A 2 t 4 , (

FIG. 1 :

 1 FIG.1: Accurate (full curves) and µ-LDA (dashed curves) short-range exchange energy along the erf (thick curves) and erfgau (thin curves) adiabatic connections for the Be atom.

FIG. 2 :

 2 FIG.2: Accurate (full curves) and µ-LDA (dashed curves) short-range correlation energy along the erf (thick curves) and erfgau (thin curves) adiabatic connections for the Be atom.

FIG. 4 :

 4 FIG.4: Short-range correlation energy along the erfgau adiabatic connection for the Be atom. Accurate data (full curve) are compared to the µ-LDA functional with a global µ (thick long-dashed curve) and with (semi)local interaction parameters: µ l,a = αkF (long-dashed curve), µ l,b = αks (shortdashed curve) and µ l,c = |∇n|/(4n) (dotted curve).

FIG. 5 :

 5 FIG.5: Short-range correlation energy density εµ c (r) with respect to r for the Be atom with the erfgau interaction for µ = 0.21. Accurate data (Eq. 9, full curve) computed from integration over the adiabatic connection are compared with the µ-LDA functional with local interaction parameters: µ l,a = αkF (long-dashed curve), µ l,b = αks (short-dashed curve) and µ l,c = |∇n|/(4n) (dotted curve).

FIG. 7 :

 7 FIG.7: Short-range exchange energy along the erfgau adiabatic connection for the Be atom. Accurate data (full curve) are compared to the µ-LDA functional (long-dashed curve), the µ-GEA functional (dotted curve) and the µ-PBE functional (short-dashed curve). The curves corresponding to the µ-GEA and µ-PBE functionals are nearly superimposed.

FIG. 8 :

 8 FIG.8: Short-range correlation energy along the erfgau adiabatic connection for the Be atom. Accurate data (full curve) are compared to the µ-LDA functional (long-dashed curve), the µ-GEA functional (dotted curve) and the µ-PBE functional (short-dashed curve).

FIG. 9 :

 9 FIG.9: Short-range correlation energy density εµ c (r) with respect to r for the Be atom with the erfgau interaction for µ = 0.21. Accurate data (full curve) computed from integration over the adiabatic connection are compared with the µ-PBE functional (dashed curve).

FIG. 10 :

 10 FIG.10: Short-range correlation energy along the erfgau adiabatic connection for the Ne 6+ atom. Accurate data (full curve) are compared to the µ-LDA functional (long-dashed curve) and the µ-PBE functional (short-dashed curve).

1 / 2 ≈

 12 /(2ν))], ν = μ/c and c = 1 + 6 √ 3 3.375. For μ = 0 for both interactions, b(μ) reduces to Sham's coefficient[START_REF] Sham | Computational Methods in Band Theory[END_REF][START_REF] Perdew | [END_REF] of the standard exchange

FIG. 11

 11 FIG.11:z-integrand of the short-range µ-GEA exchange energy density εGEA x (n, s, μ) (Eq. A16) with the erf interaction and with s = 0 and µ = 0 (full curve), s = 1 and µ = 0 (longdashed curve) and s = 1 and µ = 0.2 (short-dashed curve), all for kF = 1.

14 FIG. 12 :

 1412 FIG.12: Ratio of the gradient contribution of µ-GEA shortrange exchange energy density to the LDA exchange energy density with Coulomb interaction εunifx (n, μ)b(μ)/ε unif x (n) with respect to µ/(2kF ) for the erf (full curve) and erfgau (dashed curve) interactions. The horizontal line corresponds to the Coulomb case (µ = 0) where b(µ = 0) = 7/81.

FIG. 13 :

 13 FIG.13: Coefficients κ of the µ-PBE exchange functional with respect to µ/(2kF ) for the erf (full curve) and erfgau (dashed curve) interactions.