Julien Toulouse

François Colonna

Andreas Savin

Long-range-short-range separation of the electron-electron interaction in density functional theory

By splitting the Coulomb interaction into long-range and short-range components, we decompose the energy of a quantum electronic system into long-range and short-range contributions. We show that the long-range part of the energy can be efficiently calculated by traditional wave function methods, while the short-range part can be handled by a density functional. The analysis of this functional with respect to the range of the associated interaction reveals that, in the limit of a very short-range interaction, the short-range exchange-correlation energy can be expressed as a simple local functional of the on-top pair density and its first derivatives. This provides an explanation for the accuracy of the local density approximation (LDA) for the short-range functional. Moreover, this analysis leads also to new simple approximations for the short-range exchange and correlation energies improving the LDA.

I. INTRODUCTION

In the Kohn-Sham (KS) approach [1] of density functional theory (DFT) [2] of inhomogeneous electronic systems, the central quantity is the unknown exchangecorrelation energy functional E xc [n] which encompasses all the many-body effects. The vast majority of approximations for this functional are based on the original local density approximation (LDA) [2], an approximation that turned out to be more accurate and reliable than expected and rather difficult to improve in a systematic way [3].

Actually, it has been realized for long, with the wavevector analysis of E xc [n] by Langreth and Perdew [4], that the LDA describes accurately (but not exactly in general [5]) short wavelength density fluctuations, but is inadequate for long wavelength fluctuations. A dual analysis in real space of the exchange-correlation energy (see, e.g., Ref. 6) leads to the same conclusion that the LDA is accurate at small interelectronic distances but fails at large distances. This observation lead to the development of the first gradient corrected functionals [7][8][9][10][11][START_REF] Perdew | Electronic Structure of Solids '91[END_REF] with the basic objective to cure the wrong long-range contribution to the exchange-correlation energy of the LDA.

However, describing accurately the non-local correlation effects arising from the long-range character of the Coulomb interaction by (semi)local density functional approximations still seems out of reach. This idea in mind, it is has been proposed to use a density functional approximation only for the short-range part of the electronic energy, and treating the long-range part by more appropriate many-body methods [START_REF] Kohn | [END_REF][START_REF] Kohn | [END_REF][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]. This approach was somehow inspired from early calculations of the correlation energy of the uniform electron gas based on a separate treatment of long-range and short-range contri- * Electronic address: savin@lct.jussieu.fr butions (see, e.g., Refs. [START_REF] Nozières | [END_REF][START_REF] Raimes | Many-electron theory[END_REF][START_REF] Gross | Manyparticle theory[END_REF].

For atoms and molecules, this approach leads to a rigorous method for combining traditional ab initio wave function calculations with DFT [START_REF] Leininger | [END_REF]22], and we recall now the formalism. The Coulomb electron-electron interaction is decomposed as

1 r = v µ ee (r) + vµ ee (r), (1)
where v µ ee (r) is a long-range interaction, vµ ee (r) is the complement short-range interaction and µ is a parameter controlling the separation. The universal functional [23] F [n] = min Ψ→n Ψ| T + Vee |Ψ , where T is the kinetic energy operator and Vee = i<j 1/r ij is the Coulomb interaction operator, can then be decomposed as

F [n] = F µ [n] + F µ [n], (2)
where F µ [n] is the universal functional corresponding to the long-range interaction V µ ee = i<j v µ ee (r ij )

F µ [n] = min Ψ→n Ψ| T + V µ ee |Ψ , (3)
and

F µ [n] = F [n]-F µ [n]
is by definition the complement (short-range) part.

The exact ground-state energy of an electronic system in the external local nuclei-electron potential v ne (r) can be written using this short-range functional F µ [n] via application of the variational principle

E = min n F µ [n] + F µ [n] + n(r)v ne (r)dr = min Ψ Ψ| T + V µ ee |Ψ + F µ [n Ψ ] + n Ψ (r)v ne (r)dr = Ψ µ | T + V µ ee |Ψ µ + F µ [n Ψ µ ] + n Ψ µ (r)v ne (r)dr. (4)
In this equation, Ψ µ is given by the Euler-Lagrange equation

Ĥµ |Ψ µ = E µ |Ψ µ , (5)
where Ĥµ = T + V µ ee + V µ . Thus, Ψ µ is the groundstate multi-determinantal wave function of a partially interacting system with interaction V µ ee and external local potential V µ = i v µ (r i ) where v µ (r) = v ne (r) + δ F µ [n Ψ µ ]/δn(r). By virtue of the Hohenberg-Kohn theorem [2], v µ (r) is the unique potential (up to an additive constant) which insures that the ground-state density of this fictitious system is identical to the ground-state density of the physical system n Ψ µ = n.

The short-range functional F µ [n] is further decomposed as

F µ [n] = Ū µ [n] + Ēµ xc [n], (6)
where Ū µ [n] = 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 [n] is the unknown complement short-range exchangecorrelation energy functional. Once an approximation is chosen for Ēµ xc [n], the wave function Ψ µ can be computed by solving self-consistently Eq. ( 5) using ab initio wave function methods like configuration interaction (CI) or multi-configurational self-consistency field (MCSCF) and the total energy is calculated according to the last line of Eq. (4). Notice also that it is possible to separate the functional Ēµ xc [n] into short-range exchange and correlation contributions, Ēµ

xc [n] = Ēµ x [n] + Ēµ c [n], with Ēµ x [n] = Φ| V µ ee |Φ -Ū µ [n], (7)
and

Ēµ c [n] = F µ [n] -Φ| V µ ee |Φ ( 8
)
where Φ is the KS determinant and V µ ee = Vee -V µ ee is the complement short-range interaction operator.

A simple approximation for Ēµ xc [n] is the LDA associated to the modified interaction [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]24] Ēµ,LDA

xc [n] = n(r)ε µ,unif xc (n(r))dr, (9)
where εµ,unif xc (n) is the complement short-range exchangecorrelation energy per particle obtained by difference from the exchange-correlation energies per particle of the uniform electron gas with the standard Coulomb interaction, ε unif xc (n), and with the long-range interaction v µ ee ,

ε µ,unif xc (n), εµ,unif xc (n) = ε unif xc (n) -ε µ,unif xc (n). ( 10
)
The method has been implemented at an experimental level into the quantum chemistry package Molpro, allowing a combination of CI-type wave function and DFT calculations [START_REF] Leininger | [END_REF]. Recently, the method has also been efficiently implemented into the quantum chemistry program Dalton, performing the coupling of MC-SCF wave function calculations with DFT [25]. For a reasonable long-range/short-range separation and using only the LDA for Ēµ xc [n], the approach already yields good results for total energies of atomic and molecular systems [START_REF] Leininger | [END_REF]22,25]. The purpose of this paper is to analyzed in further details the two separate needed approximations of the method: the wave function calculation of Eq. ( 5) and the short-range exchange-correlation functional Ēµ xc [n]. Accurate calculations have been performed on a few small atomic systems to assess the approximations. Concerning the wave function part, we show that the modification of the interaction enables to decrease the effort, or alternatively increase the accuracy, of a CItype calculation. Concerning the functional part, we gain more insights into the short-range functional and explain the performance of the LDA by studying the behavior of Ēµ xc [n] with respect to the interaction parameter µ. In particular, we show that, when the interaction vµ ee is short ranged enough, Ēµ xc [n] can be expressed as a simple local functional of the on-top pair density and its first derivatives, explaining the accuracy of (semi)local functional approximations. This analysis also enables us to propose new approximations for Ēµ xc [n] which correct the LDA in the domain of µ where it fails.

The paper is organized as follows. In Sec. II, we give two possible choices for the long-range interaction v µ ee (r). In Sec. III, technical details concerning the calculations made on atomic systems are given. In Sec. IV, the impact of the modification of the interaction on the performance of wave function calculation is investigated. In Sec. V, we study the behavior of the corresponding short-range exchange and correlation functionals with respect to the interaction parameter µ. In Sec. VI, we test the obtained exact behaviors and compared them with the LDA. In Sec VII, interpolations for the short-range exchange and correlation functionals with respect to µ are proposed and tested. Finally, Sec VIII draws a conclusion of this work.

Atomic units (a.u.) will be used throughout this work.

II. LONG-RANGE/SHORT-RANGE SEPARATION

In a number of previous works [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF]22,26,27], a splitting of the Coulomb interaction based on the error function has been studied to describe the long-range part of the electron-electron interaction

v µ ee,erf (r) = erf(µr) r . (11)
This interaction, referred to as the erf interaction, has also been used by other authors in DFT for various purposes [28][29][30][31][32][33][34][35]. In this work, we introduce another interaction achieving a sharper separation of long-range and short-range interactions by subtracting a Gaussian function from the erf interaction

v µ ee,erfgau (r) = erf(µr) r - 2µ √ π e -1 3 µ 2 r 2 , (12)
where the coefficient and the exponent of the Gaussian have been chosen so that v µ ee,erfgau (r) and its derivative with respect to r vanish at r = 0. This modified interaction is referred to as the erfgau interaction. Notice that this partition of Coulomb interaction has already been proposed by Gill and Adamson [28]. In another context, Prendergast et al. [36] have also used a similar form of long-range interaction. For the sake of completeness, we finally note that another possible form of modified electron-electron interaction based on the Yukawa potential has also been investigated in the past [13-16, 37, 38].

Both interactions (11) and ( 12) enable to define a generalized adiabatic connection [39] between the noninteracting KS system at µ = 0 where the interaction vanishes v µ=0 ee (r) = 0, and the physical system at µ → ∞ where the full Coulomb interaction is recovered v µ→∞ ee (r) = 1/r. Notice that, although we have chosen the same notation for convenience, the parameter µ in Eq. ( 11) is a priori independent of that of Eq. [START_REF] Perdew | Electronic Structure of Solids '91[END_REF]. Actually, in all the plots of the paper involving the erfgau interaction, we apply a scale factor to the interaction parameter of the erfgau interaction: µ → cµ. The constant c is chosen so as to have the same leading term in the distributional asymptotic expansion of the two interactions when µ → ∞ (see Appendix C) which leads to the value c = 1 + 6 √ 3 1/2 ≈ 3.375. This also insures that the leading term in the asymptotic expansion of short-range Hartree, exchange and correlation energies for large µ is the same for the two interactions. In Fig. 1, the erf and erfgau interactions are compared along with the Coulomb interaction. The scale factor on the parameter of the erfgau interaction enables to define a common "cut-off radius" giving the range of the interaction and defined by the inverse of the interaction parameter r c ≈ 1/µ. For interelectronic distances larger than r c , the two modified interactions reproduce the long-range Coulomb tail. Notice that short-range interactions are better removed with the erfgau interaction. We finally note that, for Gaussian basis set calculations, the evaluation of the two-electron integrals corresponding to the erf or erfgau interaction requires only simple modifications of standard algorithms for Coulomb integrals (see Appendix A).

III. DETAILS ON ACCURATE CALCULATIONS

We explain rapidly how the accurate data for atomic systems presented in this work have been obtained.

For each system, accurate potentials v µ (r) needed in Eq. ( 5) and corresponding value of the universal FIG. 1: Coulomb interaction 1/r (dotted curve), erf interaction (dashed curve) and erfgau interaction (solid curve) for µ = 1. In order to compare the two modified interactions, a scale factor is applied on the interaction parameter of the erfgau interaction: µ → cµ with c = 3.375 (see text). It is possible to define a common cut-off radius rc ≈ 1/µ. long-range functional F µ are computed using the Lieb's Legendre transform formulation of the universal functional [40]

F µ [n] = max ṽµ E µ [ṽ µ ] -n(r)ṽ µ (r)dr , (13)
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, the maximum is reached for the desired v µ and F µ . In practice, an accurate density n is computed by multi-reference CI with single and double excitations (MRCISD) [41,42] and the potential to optimize ṽµ (r) is expanded as

ṽµ (r) = n i=1 c i r pi e γir 2 + C r , (14)
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), C = -Z + N -1 whereas for finite µ, C = -Z (N and Z are the electron number and nuclear charge, respectively). The maximization of Eq. ( 13) 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 Molpro [START_REF] Molpro | a package of ab initio programs designed[END_REF] with modified two-electron integrals (see Appendix A). 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. 45 and 26 for more details).

The standard universal functional F and the KS potential v KS (and thus the KS determinant Φ) are obtained as a special case for µ = 0. The complement short-range functional F µ = F -F µ can then be deduced and the short-range exchange and correlation energies, Ēµ

x and Ēµ c , are obtained from Eqs. (7) and (8).

IV. PERFORMANCE OF THE APPROXIMATIONS FOR THE WAVE FUNCTION CALCULATION

We now investigate the effect of the modification of the electron-electron interaction on the wave function part of the calculation. For this purpose, we evaluate the efficiency of approximate resolutions of Eq. ( 5) as follows.

We first construct, for each µ, the Hamiltonian Ĥµ using an accurate potential v µ and compute accurately its ground-state energy,

E µ = Ψ µ | Ĥµ |Ψ µ ,
∆E µ S = E µ S -E µ . ( 15
)
The differences ∆E µ S are plotted along the erf and erfgau adiabatic connections in Fig. 2 for the He atom with the orbital spaces S = 1s, S = 1s2s and S = 1s2s2p. One sees that, in the Coulombic limit µ → ∞, the reduction of the orbital space leads to important errors in the energy. When µ is decreased, i.e. when the interaction is reduced, the errors due to limited orbital spaces get smaller and smaller. For instance, at µ = 1, using only the single-determinant wave function Ψ µ 1s , leads to an error ∆E µ 1s of less than 0.005 Hartree. The erfgau interaction generally gives smaller errors than the erf interaction, except near µ = 0 where the non-monotonicity of the erfgau interaction leads to peculiar behaviors [24]. Anyway, we will not use the erfgau interaction in this region of very small µ.

The case of the Be atom with the orbital spaces S = 1s2s and S = 1s2s2p is reported in Fig. 3. Because of the near-degeneracy of the 2s and 2p levels, the inclusion of 2p configurations in the wave function is important, quite independently of the electron-electron interaction. Indeed, the difference E µ 1s2s -E µ 1s2s2p remains large for almost all µ's. On the contrary, the error of the calculation where the 2p orbitals are included, ∆E µ 1s2s2p , quickly falls off when µ is decreased. Again, for µ = 1 for instance, the error ∆E µ 1s2s2p given by the few-determinant CI-type wave function Ψ µ 1s2s2p is less than 0.005 Hartree. Therefore, the modification of the interaction enables to increase the accuracy of CI-type wave function calculations, or equivalently for a fixed target accuracy, decrease the effort of the calculation by reducing the orbital space. The crucial point for this effect to appear is the Ground-state energy differences

∆E µ S = Ψ µ S | Ĥµ |Ψ µ S -Ψ µ | Ĥµ |Ψ µ
where Ψ µ is an accurate wave function and Ψ µ S are approximate wave functions generated from small orbital spaces S = 1s, S = 1s2s and S = 1s2s2p along the erf (full curves) and erfgau (dashed curves) adiabatic connections for He.

E Μ a.u. 1s2s 1s2s2p FIG. 3: Ground-state energy differences ∆E µ S = Ψ µ S | Ĥµ |Ψ µ S -Ψ µ | Ĥµ |Ψ µ
where Ψ µ is an accurate wave function and Ψ µ S are approximate wave functions generated from small orbital spaces S = 1s2s and S = 1s2s2p along the erf (full curves) and erfgau (dashed curves) adiabatic connections for Be.

reduction of the electron-electron interaction compared to the Coulomb interaction and not really the long-range character of the modified interaction.

V. BEHAVIOR OF SHORT-RANGE EXCHANGE AND CORRELATION FUNCTIONALS WITH RESPECT TO THE INTERACTION PARAMETER

At µ = 0, the short-range exchange-correlation functional reduces to the standard exchange-correlation functional of the KS scheme, Ēµ=0 c with respect to µ constitutes an analysis of the exchange and correlation functionals in term of the range of the interaction. Indeed, the behavior at µ → 0 tells us how the KS exchange-correlation functional responds when very long-range interactions are removed from it, while the asymptotic expansion for µ → ∞ gives the exchange-correlation functional associated to very shortrange interactions. To release the text from mathematical details, the full derivation of the expansions are given in the Appendices.

A. Exchange functional for small µ

In Appendix B, we show that the short-range exchange energy has the following formal expansion around µ = 0

Ēµ x = E x - 1 √ π ∞ n=0 (-1) n a n n! µ 2n+1 × n 2,x (r 1 , r 2 )r 2n 12 dr 1 dr 2 , (16)
where E x is the usual KS exchange energy, n 2,x (r 1 , r 2 ) is the exchange contribution to the pair density, and a n are coefficients depending of the interaction chosen and defined after Eq. (B2). More specifically, for the erf interaction, this expansion writes

Ēµ x,erf = E x + µ √ π N + µ 3 3 √ π n 2,x (r 1 , r 2 )r 2 12 dr 1 dr 2 +• • • , (17)
where the term linear in µ comes from the normalization of the exchange hole [39]. For the erfgau interaction, the expansion is

Ēµ

x,erfgau = E x -

2µ 5 45 √ π n 2,x (r 1 , r 2 )r 4 12 dr 1 dr 2 + • • • . ( 18
)
The exchange energy with the erf interaction varies linearly in µ near the Kohn-Sham end of the adiabatic connection (see Fig. 4). On the contrary, the exchange energy with the erfgau interaction varies very slowly like µ 5 which implies that the corresponding curve is very flat near µ = 0 (see Fig. 5). The latter behavior is not unexpected for a good long-range/short-range separation. Indeed, with the erfgau interaction, when µ increases near µ = 0 only very long-range interaction effects are removed from the functional; they practically do not later the exchange energy of a finite system. With the erf interaction, the long-range/short-range separation is imperfect and thus the short-range part of the interaction is also affected near µ = 0 which is responsible for the linear behavior of the exchange energy in this case. This clearly shows that the erfgau interaction realizes a better separation of long-range and short-range interactions than the erf interaction.

B. Correlation functional for small µ

The general expansion of short-range correlation energy around µ = 0 is derived in Appendix B. It reads

Ēµ c = E c - 1 √ π ∞ n=1 ∞ k=1 (-1) n (2n + 1)a n n!k!(2n + k + 1) µ 2n+k+1 × ∂ k n µ 2,c (r 1 , r 2 ) ∂µ k µ=0 r 2n 12 dr 1 dr 2 . ( 19
)
where E c is the usual correlation energy of the KS scheme and n µ 2,c (r 1 , r 2 ) is the correlation pair density with interaction v µ ee . Actually, several terms of this expansion vanish. For the erf interaction, the expansion writes

Ēµ c,erf = E c + µ 6 36 √ π × ∂ 3 n µ 2,c (r 1 , r 2 ) ∂µ 3 µ=0 r 2 12 dr 1 dr 2 + • • • . (20)
Similarly, the expansion for the erfgau interaction is

Ēµ c,erfgau = E c - µ 10 5400 √ π × ∂ 5 n µ 2,c (r 1 , r 2 ) ∂µ 5 µ=0 r 4 12 dr 1 dr 2 + • • • . (21)
The correlation energy varies much more slowly than the exchange energy near the KS end of the adiabatic connection. Therefore, the curve of the correlation energy with respect to µ is very flat around µ = 0 (see Figs. 6 and7). Again, this is due to the fact that removing very long-range interactions from the functional has no effect in a finite system. It will be seen in Sec VII that this makes the correlation energy difficult to interpolate near µ = 0 from a knowledge of the functional for large µ.

C. Exchange functional for large µ

In Appendix C, we derive the general asymptotic expansion of the short-range exchange energy for µ → ∞

Ēµ x = 2 √ π m n=0 A 2n (2n)!(2n + 2)µ 2n+2 × n (2n) 2,x (r, r)dr + O( 1 µ 2m+3 ), (22)
where n (2n) 2,x (r, r) are the on-top exchange pair density and its spherical-averaged (with respect to r 12 ) derivatives. Simple explicit expressions can be given for the first two terms of this expansion. Indeed, the on-top exchange pair density writes

n 2,x (r, r) = - σ n 2 σ (r), (23)
where the summation is over the two spin states σ = α, β and n σ (r) are the spin-densities; its second-derivative can be expressed by [46] n

2,x (r, r) = -

1 3 σ n σ (r) × ∇ 2 n σ (r) -4τ σ (r) + 1 2 |∇n σ (r)| 2 n σ (r) , (24)
where τ σ (r) are the KS spin kinetic energy densities expressed in term of the KS spin-orbitals φ iσ (r) by

τ σ (r) = 1 2 Nσ i=1 |∇φ iσ (r)| 2 . ( 25
)
where N σ is the number of electrons of spin σ. The Laplacian in Eq. ( 24) can be eliminated by integration by parts

n σ (r)∇ 2 n σ (r)dr = -|∇n σ (r)| 2 dr, (26)
which leads for the leading terms of the expansion when µ → ∞ of the short-range exchange energy

Ēµ x = - √ πA 0 µ 2 σ n σ (r) 2 dr + √ πA 2 12µ 4 σ n σ (r) |∇n σ (r)| 2 2n σ (r) + 4τ σ (r) dr + • • • , (27)
with

A 0,erf = √ π/2, A 0,erfgau = (1 + 6 √ 3)A 0,erf , A 2,erf = 3 √ π/4 and A 2,erfgau = (1 + 36 √ 3)A 2,erf .
Note that this last result has already been demonstrated for the case of the erf interaction [29]. In particular, for spinunpolarized systems, Eq. ( 27) becomes

Ēµ x = - √ πA 0 2µ 2 n(r) 2 dr + √ πA 2 24µ 4 n(r) |∇n(r)| 2 2n(r) + 4τ (r) dr + • • • . (28)
Eq. ( 27) or (28) shows that the first term in the expansion for large µ of the short-range exchange energy is an exact local functional of the density (or, in general, of the spin-densities). Therefore, the local density approximation becomes exact in this limit. An alternative view of this result can be achieved in the framework of the wave-vector analysis of the usual KS exchange functional by considering the short wavelength limit [5]. The next term of expansion (28) involves the gradient of the density |∇n| and the kinetic energy density τ and is therefore of the meta-GGA type. The following higher-order terms involve of course more and more ingredients constructed from higher-order derivatives of the KS orbitals. Note that instead of considering these spin-dependent quantities one can also directly use the on-top exchange pair density and its derivatives.

A similar asymptotic expansion can be derived for the short-range Hartree energy

Ū µ = √ πA 0 µ 2 n(r) 2 dr - √ πA 2 12µ 4 |∇n(r)| 2 dr + • • • . ( 29
)
By carefully comparing Eq. ( 27) and Eq. ( 29), one sees that, for a one-electron system (n α = n, n β = 0, τ α = τ W = |∇n| 2 /(8n), τ β = 0), the Hartree and exchange energies cancel out order by order in the expansion with respect to µ. As the first term (in µ -2 ) of the asymptotic expansion of the exchange energy is a local functional of the density, this cancellation is exactly maintained in the LDA, i.e. there is no self-interaction error in LDA for this first term. For the next term of the expansion (in µ -4 ), the removal of the self-interaction error requires the consideration of a meta-GGA functional depending explicitly on |∇n σ | and τ σ .

In the KS scheme, it has been often argued that (semi)local approximate exchange functionals actually mimics near-degeneracy correlation effects (see, e.g., Ref. 47). It is clear from this above discussion that, for short-range exchange functionals, (semi)local approximations become exact at large µ and therefore do not fortuitously mimic near-degeneracy correlation anymore.

D. Correlation functional for large µ

In Appendix C, we derive the first two terms in the asymptotic expansion of the short-range correlation energy for µ → ∞

Ēµ c = √ πA 0 µ 2 n 2,c (r, r)dr + 2 √ πA 1 3µ 3 n 2 (r, r)dr + • • • , (30)
where n 2,c (r, r) and n 2 (r, r) are the correlation and total pair density, the A 0 coefficient is given after Eq. ( 27) and in addition A 1,erf = 1 and A 1,erfgau = 28.

This result show that for large µ the short-range correlation energy becomes an exact local functional of the on-top pair density n 2 (r, r) (and of the density via n 2,c (r, r) = n 2 (r, r) -n(r)n(r)/2 for closed-shell systems). This emphasizes the importance of this quantity for density functional approximations. It is in the same line of thought of a number of previous studies [48][49][50] stressing the need of including explicitly the on-top pair density in approximate density functionals.

VI. PERFORMANCE OF THE LDA FOR SHORT-RANGE EXCHANGE AND CORRELATION ENERGIES

The asymptotic expansions for µ → ∞ enables to analyze the local density approximation to the short-range exchange-correlation functional. We restrict the discussion to spin-unpolarized systems. Applying the LDA to the expansion of the exchange energy (22) corresponds to transferring the on-top exchange pair density and its derivatives ñ(2n) 2,x (r, r) from the uniform electron gas [51] ñ(2n) 2,x (r, r) ≈ n(r)n(r) g

(2n)
x,0 (n(r)) -1 ,

where g

x,0 (n(r)) are the on-top derivatives of the exchange-only pair-distribution function of the uniform electron gas [52]

g x (r 12 , n) = 1 - 9 2 sin(k F r 12 ) -k F r 12 cos(k F r 12 ) k 3 F r 3 12 2 , (32)
with k F = (3π 2 n) 1/3 . The expansion (28) of the shortrange exchange energy then writes

Ēµ,LDA x = - √ πA 0 2µ 2 n(r) 2 dr + 3 2/3 π 11/6 A 2 20µ 4 n(r) 8/3 dr + • • • . ( 33
)
This last expansion is compared with the exact expansion (28) for the He atom with the erf and erfgau interactions in Figs. 4 and5, respectively. In these expressions, an accurate density of the He atom is used. The LDA for the modified interactions [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]24] evaluated with the same density is also reported, as well as an accurate calculation of the exchange energy along the adiabatic connection [26,53] (see Sec. III). The LDA expansion (33) is exact for the first term and very close to the exact expansion (28) with the first two terms. Consequently, in the domain of validity of this expansion (for µ 2), the LDA is nearly exact. The success of the LDA for large µ is therefore due to the exactness of the exchange ontop pair-density and the good transferability of its first derivatives from the uniform electron gas to the finite system. Actually, it can be remarked that the expansion of LDA up to the µ -4 term performs even slightly better than the exact expansion to the same order. Similarly, the LDA is already very accurate for µ 1 while achieving a comparable accuracy from the slowly-improving expansion (28) of the exact exchange energy would require a rather long expansion.

For the correlation energy, the LDA of expansion (30) consists in transferring the on-top density from the uni- FIG. 4: Accurate short-range exchange energy of He (thick solid curve) along the erf adiabatic connection, local density approximation (thick long-dashed curve), exact asymptotic expansion for µ → ∞ (Eq. 28) with the first term (lower solid curve) and with the first two terms (upper solid curve), and asymptotic expansion in LDA (Eq. 33) with the first two terms (short-dashed curve), the first term in LDA being exact. FIG. 5: Accurate short-range exchange energy of He (thick solid curve) along the erfgau adiabatic connection, local density approximation (thick long-dashed curve), exact asymptotic expansion for µ → ∞ (Eq. 28) with the first term (lower solid curve) and with the first two terms (upper solid curve), and asymptotic expansion in LDA (Eq. 33) with the first two terms (short-dashed curve), the first term in LDA being exact.

form electron gas to obtain

Ēµ,LDA c = √ πA 0 µ 2 n(r)n(r) g 0 (r s (r)) - 1 2 dr + 2A 1 √ π 3µ 3 n(r)n(r)g 0 (r s (r))dr + • • • , (34)
where r s = 3/(4πn) 1/3 is the local Wigner-Seitz radius and g 0 (r s ) is the on-top pair-distribution function of the electron gas for which Burke, Perdew and Ernzerhof have proposed an estimation [54] g FIG. 6: Accurate short-range correlation energy of He (thick solid curve) along the erf adiabatic connection, local density approximation (thick long-dashed curve), exact asymptotic expansion for µ → ∞ (Eq. 30) with the first term (lower solid curve) and with the first two terms (upper solid curve), and asymptotic expansion in LDA (Eq. 34) with the first term (lower short-dashed curve) and with the first two terms (upper short-dashed curve).

0 (r s ) = D (γ + r s ) 3/2 + β e -A √ γ+rs , (35)
exact expansion (30) using an accurate calculation of n 2 (r, r) for the He atom with the erf and erfgau interactions in Figs. 6 and7, respectively. An accurate calculation of the correlation energy computed along the adiabatic connection [26,53] and the LDA for modified interactions [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]24] are also reported. As for the exchange energy, the first terms of the LDA expansion (34) nearly coincide with the exact expansion. This expansion with the first two terms gives a very accurate approximation to the exact correlation energy in the region of accuracy of the full LDA curve (from µ ≈ 2 to µ → ∞). From these results, it is clear that the total on-top pair density n 2 (r, r) have good transferability from the uniform electron gas to the He atom. Actually, this good (while not exact) transferability seems quite general and has already been pointed out for several atomic and molecular systems [50,54,55]. This gives an explanation for the success of the LDA in treating short-range electron-electron interactions (large µ). On the contrary, toward the Kohn-Sham end (µ = 0) of the adiabatic connection, the LDA transfers spurious long-range correlations from the uniform electron gas and therefore poorly extrapolates the exact correlation energy of the finite system which does not contain these long-range correlation effects.

VII. INTERPOLATIONS FOR THE SHORT-RANGE EXCHANGE AND CORRELATION FUNCTIONALS

In the previous section, we have shown that the LDA treats successfully short-range interactions corresponding to large interaction parameters µ but is inaccurate toward the KS end of the adiabatic connection, i.e. for small µ. But for µ = 0 a lot of better estimates of the exchange and correlation energies are available with density functional approximations of the KS scheme which go FIG. 7: Accurate short-range correlation energy of He (thick solid curve) along the erfgau adiabatic connection, local density approximation (thick long-dashed curve), exact asymptotic expansion for µ → ∞ (Eq. 30) with the first term (lower solid curve) and with the first two terms (upper solid curve), and asymptotic expansion in LDA (Eq. 34) with the first term (lower short-dashed curve) and with the first two terms (upper short-dashed curve).

beyond the LDA such as gradient-corrected functionals. A simple idea for improving the short-range exchangecorrelation energy functional along the adiabatic connection is therefore to interpolate between an available density functional approximation (DFA) for µ = 0 and the µ-dependent LDA for µ → ∞. In the spirit of the usual DFT approximations, this interpolation will be done locally, i.e. for the short-range exchange-correlation energy density εµ xc (r) related to the global functional Ēµ

xc [n] via Ēµ xc [n] = dr n(r) εµ xc (r). (36)

A. Rational interpolations

A simple possibility is to interpolate εµ xc (r) along the adiabatic connection using an estimate at µ = 0 and the expansions for large µ (and eventually for small µ) presented in Sec. V. For example, consider the rational approximant for the short-range exchange energy density of the erf interaction εµ

x,erf ≈

ε DFA x 1 + b 1 µ + b 2 µ 2 , (37)
where

b 1 = -1/(ε DFA x √ π) and b 2 = -4ε DFA
x /(πn) are chosen to satisfy the expansion for small µ (Eq. 17) and the expansion for large µ (Eq. 28) to leading order. In Eq. ( 37), ε DFA x is the exchange energy density at µ = 0 estimated by one of the usual density functional approximations of the KS scheme.

For the erfgau interaction, the short-range exchange energy density can also be interpolated between ε DFA x at µ = 0 (no linear term in µ, Eq. 18) and the expansion for large µ (Eq. 28) according to εµ

x,erfgau ≈

ε DFA x 1 + c 2 µ 2 , (38)
with c 2 = -4ε DFA x /((1 + 6 √ 3)πn). In the same spirit, the short-range correlation density along the adiabatic connection can be approximated by

εµ c ≈ ε DFA c 1 + d 1 µ + d 2 µ 2 , (39)
with d 1,erf = -2d 2 2,erf √ πng 0 (r s )/(3ε c ), d 2,erf = 2ε c /(πn(g 0 (r s ) -1/2)) for the erf interaction, and

d 1,erfgau = -56d 2 2,erfgau √ πng 0 (r s )/(3ε c ), d 2,erfgau = d 2,erf /(1+6 √
3) for the erfgau interaction. Again, ε DFA c is the correlation energy density at µ = 0 given by an available density functional approximation of the KS scheme. The rational approximant (39) therefore interpolates between ε DFA c at µ = 0 and the expansion of the LDA for large µ with the first two terms (Eq. 34).

B. Weighted interpolations

The interpolation formulas (37) and (38) only make use of the first term of the asymptotic expansion of the LDA exchange energy for µ → ∞. However, it has been realized in Sec. VI that the LDA works well on a larger domain of µ than the first terms of its asymptotic expansion. In order to take better advantage of the LDA, one can modify it only in the region of small µ by using information from better estimates of the KS exchange energy at µ = 0. We therefore interpolate locally the short-range exchange density along the adiabatic connection by

εµ x ≈ (ε DFA x -εµ=0,unif x )w(µ) + εµ,unif x , (40)
where w(µ) is a weight function acting at small µ only. More precisely, w(µ) must be a positive function satisfying w(µ = 0) = 1 so that εµ=0

x = ε DFA x
, having significant values in the region where the LDA fails and with a fast decay for µ → ∞ so as to recover the correct behavior of εµ,unif x for large µ. For the erf interaction, we found that the local value of µ delimiting the domain where the LDA is inaccurate is well estimated by 1/r s (r) where r s (r) is the local Wigner-Seitz radius. We therefore take w erf (µ) = erfc(µr s ) where the complementary error function erfc ensures that the weight function has significant values only for µ < 1/r s . Eq. ( 40) thus interpolates between ε DFA x at µ = 0 and the LDA at large µ. For the erfgau interaction, we can used the same weight function except that now the local value of µ delimiting the domain where the LDA must be corrected is estimated by µ = c/r s where c = 1 + 6 √ 3 1/2 ≈ 3.375 is the scale factor between the erf and erfgau interactions discussed in Sec. II. We therefore take as weight function w erfgau (µ) = erfc(µr s /c).

Naturally, this interpolation can also be applied to the short-range correlation energy

εµ c ≈ (ε DFA c -εµ=0,unif c )w(µ) + εµ,unif c , (41)
with the same weight function w(µ).

C. Other interpolations

Finally, we mention the approximation for the shortrange exchange energy proposed by Iikura, Tsuneda, Yanai and Hirao [30] based on a modification of the short-range LDA exchange functional [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF]24]. The spinunpolarized version of their approximation for the erf interaction is

εµ x,erf ≈ ε DFA x 1 - 8 3 A √ π erf 1 2A +(2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 , ( 42
)
where

A = µ/(2k), k = ε LDA x /ε DFA x k F and k F = (3π 2 n) 1/3
. The approximation reduces to ε DFA x at µ = 0 and has an asymptotic expansion for µ → ∞ incorporating the correct leading term (cf. Eq. 28). Therefore, Eq. ( 42) provides an interpolation between a density functional approximation at µ = 0, ε DFA x , and the correct limit as µ → ∞. The same approximation can also be derived for the erfgau interaction using the LDA exchange functional associated to this interaction [24]; it reads

εµ x,erfgau ≈ ε DFA x 1 - 8 3 A √ π erf 1 2A +(2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 + 8 3 A √ π erf 1 2B +(2B -16B 3 )e -1/(4B 2 ) -6B + 16B 3 , (43)
where B = A/ √ 3. Notice that in the interpolations of Sec. VII A, VII B and VII C the exchange and correlation energy densities at µ = 0, ε DFA x and ε DFA c , can be estimated by any of the available exchange-correlation functionals of the Kohn-Sham scheme. In this sense, formulas (37) to [START_REF] Press | Numerical Recipes[END_REF] provide extensions of these exchange-correlation functionals over the erf and erfgau adiabatic connections.

D. Results

We now test the interpolations formulas (37) to [START_REF] Press | Numerical Recipes[END_REF] on a few atomic systems. In all the results presented here, we use the PBE functional [56] as the density functional approximation at µ = 0 both for exchange and correlation: ε DFA FIG. 8: Accurate short-range exchange energy of He (solid curve) along the erf adiabatic connection, local density approximation (long-dashed curve) and three interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 37, short-dashed curve), a weighted approximant (Eq. 40, dotted curve) and the approximation of Iikura et al. (Eq. 42, dashed-dotted curve). The last two curves are nearly superimposed. FIG. 9: Accurate short-range exchange energy of He (solid curve) along the erfgau adiabatic connection, local density approximation (long-dashed curve) and three interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 38, short-dashed curve), a weighted approximant (Eq. 40, dotted curve) and the approximation of Iikura et al. (Eq. 43, dashed-dotted curve). The last two curves are nearly superimposed. connections, respectively. An accurate calculation is compared to the LDA and to the three interpolations of Sec. VII A, VII B and VII C. One sees that the rational approximants give an overall reasonable estimate of the exchange energy along both adiabatic connections but are actually less accurate that the LDA in the region of intermediate µ. As already noticed, this reflects the fact that the LDA works well on a larger range of µ that the first terms of its expansion for µ → ∞. The weighted interpolations make better use of the LDA and constitutes an improvement over it for all µ's. Finally, the approximation of Iikura et al. is nearly identical to the weighted interpolation for both the erf and erfgau interactions.

Similar curves for the short-range correlation energy FIG. 10: Accurate short-range correlation energy of He (solid curve) along the erf adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve).

of the He atom with the erf and erfgau interactions are shown in Figs. 10 and11. As for the exchange energy, the rational approximants constitute an overall correction to the LDA but not in a systematic way since the LDA still performs better for intermediate µ. On the contrary, the weighted interpolations always improve the LDA. As one can expect, considering exchange and correlation together further improves the results at small µ.

We now discuss the Be atom. The short-range exchange energy along the adiabatic connections looks very similar to that of the He atom and will not be shown. The short-range correlation energy for which the LDA has more difficulties than for the He case is represented in Figs. 12 and 13 for the erf and erfgau interactions. One can see that both the rational approximant and the weighted interpolation improve the LDA along the whole adiabatic connection.

The He and Be atoms are simple cases where a gradient-corrected functional like PBE give a very accurate correlation energy. On the contrary, the Ne 6+ atom constitutes a much more difficult system for (semi)local functionals because of the presence of strong neardegeneracy correlation effects due to the proximity of the 2s and 2p levels. The short-range correlation energy of this system is reported in Figs. 14 and15 for the erf and erfgau interactions. The PBE functional of the KS scheme (µ = 0) strongly underestimates the correlation energy. One sees that with the rational approximants or the weighted interpolations, the error is rapidly decreased when µ is increased, i.e. when long-range interactions are removed from the functional.

VIII. CONCLUSION

In a quantum electronic system, the long-range/shortrange separation of the Coulomb interaction enables to rigorously decompose the total energy into long-range FIG. 11: Accurate short-range correlation energy of He (solid curve) along the erfgau adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve). FIG. 12: Accurate short-range correlation energy of Be (solid curve) along the erf adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve). FIG. 13: Accurate short-range correlation energy of Be (solid curve) along the erfgau adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve). FIG. 14: Accurate short-range correlation energy of Ne 6+ (solid curve) along the erf adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve). FIG. 15: Accurate short-range correlation energy of Ne 6+ (solid curve) along the erfgau adiabatic connection, local density approximation (long-dashed curve) and two interpolations between the PBE value at µ = 0 and the LDA at µ → ∞ using a rational approximant (Eq. 39, short-dashed curve) and a weighted approximant (Eq. 41, dotted curve). and short-range components which can be calculated by different methods. In particular, a density functional approximation can be used for the short-range part of the energy, while the long-range contribution can be treated by traditional wave function methods. In this work, we have considered two possible long-range/short-range separation of the Coulomb interaction: the erf and erfgau modified interactions. The erfgau interaction achieves a better separation than the erf interaction. We have shown that the use of these modified interactions facilitates wave function calculations. We have also studied the short-range part of the exchange-correlation functional with respect to the range of the associated interaction, and shown that, in the limit of a very short-range interaction, the exchange contribution to this functional can be expressed as a local functional of the density and is therefore exact in the LDA. In the same limit, the correlation contribution can be expressed as a local functional of the on-top pair density which is generally accurate in the LDA. However, when the interaction becomes more and more long-ranged, the LDA is inaccurate. It is nevertheless possible to improve the LDA description of the shortrange exchange-correlation functional by making use of the available gradient-corrected functional of the Kohn-Sham scheme in the limit of the full Coulomb interaction. We have indeed proposed two kind of interpolations for the short-range exchange-correlation functional along the erf or erfgau adiabatic connection which improve the LDA. In order to extend the interaction range well treated by functional approximations in a more systematic way, we are currently investigating gradient cor-rections for the short-range exchange-correlation functional.

+ f y + f z in t 2 P n (t) = 2 ρ π 1/2 κ AB κ CD ρ ζ 3/2 ρ η 3/2 I ′ x (e x , f x , t)I ′ y (e y , f y , t)I ′ z (e z , f z , t) (A5)
where I ′ λ (λ = x, y, z) are 2D integrals obeying the recurrence relation

I ′ λ (e λ + 1, f λ ) = (P λ -A λ ) + ρt 2 ζ (Q λ -P λ ) I ′ λ (e λ , f λ ) + e λ 2ζ 1 - ρt 2 ζ I ′ λ (e λ -1, f λ ) + f λ ρt 2 2ζη I ′ λ (e λ , f λ -1), ( A6
)
with the starting value I ′ λ (0, 0) = 1. Equation (A4) can be evaluated exactly by the Rys quadrature

[e0|1/r 12 |f0] = 2 ρ π 1/2 κ AB κ CD ρ ζ 3/2 ρ η 3/2 nRys α=1 I ′ x (e x , f x , t α )I ′ y (e y , f y , t α )I ′ z (e z , f z , t α )w α , (A7)
where t α and w α = exp{-t 2 α ρ(P -Q) 2 } are the roots and the weights of Rys polynomials and n Rys > n/2. The modified integrals for the erf interaction [e0| erf(µr 12 )/r 12 |f0] can simply be computed with the same scheme by applying the following simple modification everywhere [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF]: 1/ρ → 1/ρ + 1/µ 2 . Thus, with respect to the Coulomb case, in Eq. (A6) the recurrence coefficients are modified, and in Eq. (A7) the prefactor, the weights w α and the roots t α , depending on ρ, are modified.

To calculate the integrals for the erfgau interaction, additional integrals of type [e0|C exp(-ar 2 12 )|f0] are needed. These integrals can be computed like an one-point Rys quadrature with root t a = a 2 /(ρ + a 2 ), weight w a = exp{-t In this appendix, we study the erf and erfgau adiabatic connections near the KS system, i.e. for small interaction parameter µ. The fictitious system along these connections is described by the Hamiltonian [cf. Eq. ( 5)]

Ĥµ = i -1 2 ∇ 2 i + i<j v µ ee (r ij ) + i v µ (r i ), ( B1
)
where v µ ee is the long-range erf or erfgau interaction and v µ is the external local potential associated to the shortrange Hartree, exchange and correlation energy func-

tionals v µ (r) = v ne (r) + δ Ū µ [n]/δn(r) + Ēµ x [n]/δn(r) + Ēµ c [n]/δn(r).
The behavior for µ → 0 of all quantities associated to this fictitious system can be derived from the Maclaurin series of the long-range interaction v µ ee

v µ ee (r 12 ) = 2 √ π ∞ n=0 (-1) n a n n! r 2n 12 µ 2n+1 , ( B2
)
where a n,erf = 1/(2n + 1) for the erf interaction and a n,erfgau = 1/(2n + 1) -1/3 n ( = 0 for n ≤ 1 by construction) for the erfgau interaction.

1. Short-range Hartree energy for µ → 0

Eq. (B2) leads immediately to the expansion of the short-range Hartree energy

Ū µ = 1 2 n(r 1 )n(r 2 )v µ ee (r 12 )dr 1 dr 2 = U - 1 √ π ∞ n=0 (-1) n a n n! µ 2n+1 × n(r 1 )n(r 2 )r 2n 12 dr 1 dr 2 , ( B3
)
where U is the standard KS Hartree energy. To obtain Eq. (B3), the integral and summation signs have been interchanged. This is reasonable for a finite system where r 12 is always bounded since in this case the series (B2) is uniformly convergent and can thus be integrated term by term.

2. Short-range exchange energy for µ → 0

The expansion for µ → 0 of the short-range exchange energy of Eq. ( 7) is

Ēµ x = 1 2 n 2,x (r 1 , r 2 )v µ ee (r 12 )dr 1 dr 2 = E x - 1 √ π ∞ n=0 (-1) n a n n! µ 2n+1 × n 2,x (r 1 , r 2 )r 2n 12 dr 1 dr 2 , (B4)
where E x is the standard KS exchange energy and n 2,x (r 1 , r 2 ) is the exchange contribution to the pair density.

3. Short-range correlation energy for µ → 0

The short-range correlation energy of Eq. ( 8) can be written with the correlation contribution to the pair density n µ 2,c (r 1 , r 2 ) with interaction v µ ee as

Ēµ c = E c - 1 2 n µ 2,c (r 1 , r 2 )v µ ee (r 12 )dr 1 dr 2 , (B5)
where E c is the usual correlation energy of the KS scheme. The expansion of Ēµ c for small µ can be obtained by first studying the derivative of Ēµ c with respect to µ which by the Hellmann-Feynman theorem writes

∂ Ēµ c ∂µ = - 1 2 n µ 2,c (r 1 , r 2 ) ∂v µ ee (r 12 ) ∂µ dr 1 dr 2 , (B6)
and assuming for n µ 2,c (r 1 , r 2 ) the following expansion around µ = 0

n µ 2,c (r 1 , r 2 ) = ∞ k=1 1 k! ∂ k n µ 2,c (r 1 , r 2 ) ∂µ k µ=0 µ k , (B7)
since for the KS system n µ=0 2,c (r 1 , r 2 ) = 0. Therefore, inserting equation (B7) and the derivative of Eq. (B2) into (B6), and assuming the commutativity of summation and integration, leads to

∂ Ēµ c ∂µ = - 1 √ π ∞ n=0 ∞ k=1 (-1) n (2n + 1)a n n!k! µ 2n+k × ∂ k n µ 2,c (r 1 , r 2 ) ∂µ k µ=0 r 2n 12 dr 1 dr 2 , (B8)
where the term n = 0 can be dropped since n µ 2,c (r 1 , r 2 ) integrates to zero. After integration we obtain the expansion of the short-range correlation energy

Ēµ c = E c - 1 √ π ∞ n=1 ∞ k=1 (-1) n (2n + 1)a n n!k!(2n + k + 1) µ 2n+k+1 × ∂ k n µ 2,c (r 1 , r 2 ) ∂µ k µ=0 r 2n
I = f (r)v µ ee (r)dr. (C3)
f can be expanded into its Maclaurin series

f (r) = m n=0 f (n) (0) n! r n + R m (r), (C4)
where the Lagrange remainder R m (r) is

R m (r) = f (m+1) (θr) (m + 1)! r m+1 , (C5)
with 0 ≤ θ ≤ 1. Inserting the first term of the righthand-side of expansion (C4) into (C3) gives the first contribution to integral I

I 1 = m n=0 f (n) (0) n! r n vµ ee (r)dr, (C6)
where the last integral can be easily evaluated

r n vµ ee (r)dr = 4π r n+2 vµ ee (r)dr = 4 √ π (n + 2) µ n+2 A n , (C7) with A n,erf = Γ( n+3
2 ) for the erf interaction and

A n,erfgau = Γ( n+3 2 ) + 3 n+3 2 (n + 2)Γ( n+3
2 ) for the erfgau interaction. Thus, Eq. (C6) becomes

I 1 = 4 √ π m n=0 A n f (n) (0) n!(n + 2) µ n+2 . (C8)
The second contribution to I coming from the remainder writes

I 2 = f (m+1) (θr) (m + 1)! r m+1 vµ ee (r)dr. ( C9
)
We shall assume in addition that, for any m, the (m + 1)th derivative of f is bounded, i.e. |f (m+1) (r)| ≤ M m , then

|I 2 | ≤ M m (m + 1)! r m+1 vµ ee (r)dr = 4 √ πM m A m+1 (m + 1)!(m + 3) µ m+3 , (C10)
meaning that I 2 = O(1/µ m+3 ). Finally, using the definition of the n-th derivative of the three-dimensional Dirac delta distribution δ (n) (r),

f (r)δ (n) (r)dr = (-1) n f (n) (0), (C11)
we obtain the following distributional asymptotic expansion of vµ ee (r) when µ → ∞

vµ ee (r) = 4 √ π m n=0 (-1) n A n n!(n + 2) µ n+2 δ (n) (r) + O( 1 µ m+3 ).
(C12) Note that if we apply a scale factor to the interaction parameter of the erfgau interaction µ → cµ such that c = (A n,erfgau /A n,erf ) = 1 + 6 √ 3 1/2 ≈ 3.375, then the asymptotic expansions of the erf and erfgau interactions are the same to leading order. This provides a criteria for comparison of the two interactions.

Following the same procedure on the derivative of the modified interaction with respect to µ which writes (C15) which is just the derivative of expansion (C12). In the following, we will apply Eqs. (C12) and (C15) assuming that the corresponding test function f (r) related to the pair density satisfies all the required assumptions which are reasonable for finite systems. Using then the distributional asymptotic expansion when µ → ∞ of the short-range interaction (Eq. C12) and noting that ñ(r 12 ) can be expanded as an even series of r 12 around r 12 = 0, we obtain the asymptotic series of the short-range Hartree energy

Ū µ = 2 √ π m n=0
A 2n (2n)!(2n + 2)µ 2n+2 × n(r)n (2n) (r)dr + O( 1 µ 2m+3 ), (C18) where the notation n (2n) (r) = (∂ 2n ñ(r 12 )/∂r 2n

12 ) r12=0 has been used for the density and its spherical-averaged derivatives.

3. Short-range exchange energy for µ → ∞ Similarly, the short-range exchange energy can be written as

Ēµ

x = 1 2 ñ2,x (r 1 , r 12 )v µ ee (r 12 )dr 1 dr 12 , (C19) where ñ2,x (r 1 , r 12 ) is the spherical-average exchange pair density which can be expanded like the density as an even series of r 12 around r 12 = 0 ("no cusp for exchange"). We therefore obtain the asymptotic series of the short-range exchange energy 12 ) r12=0 .

4. Short-range correlation energy for µ → ∞

The asymptotic expansion of the short-range correlation energy when µ → ∞ can formally be found by considering its derivative (cf. Eq. B6)

∂ Ēµ c ∂µ = 1 2 ñµ 2,c (r 1 , r 12 )
∂v µ ee (r 12 ) ∂µ dr 1 dr 12 , (C21)

where ñµ 2,c (r 1 , r 12 ) is the spherical average of n µ 2,c (r 1 , r 2 ), inserting the asymptotic expansion of ñµ 2,c (r 1 , r 12 ) and the distributional asymptotic expansion of ∂v µ ee /∂µ (Eq. C15) and re-integrating with respect to µ. Actually, considering only the first term of the asymptotic expansion of the correlation pair density, i.e. ñµ 2,c (r 1 , r 12 ) = ñ2,c (r 1 , r 12 ) + • • • , is sufficient to find the first term of the asymptotic expansion of the correlation energy. Similarly to the exchange energy, the leading term of the short-range correlation energy for large µ is consequently given by the on-top correlation pair density n 2,c (r, r)

Ēµ c = √ πA 0 µ 2 n 2,c (r, r)dr + • • • . (C22)
Notice that, for the erf interaction, this result has already been derived [26]. Using Eqs. (C18), (C20) and (C22), the external local potential v µ (r) = v ne (r) + δ Ū µ /δn(r) + δ Ēµ xc /δn(r) has the following behavior for large µ v µ (r) = v ne (r) + √ πA 0 µ 2 δ δn(r) n 2 (r ′ , r ′ )dr ′ + • • • , (C23) where n 2 (r, r) is the total on-top pair density. As v µ ee has a similar expansion for µ → ∞ beginning with µ -2 (see Eq. C12)

v µ ee (r) = 1 r - 2 √ πA 0 µ 2 δ(r) + • • • . (C24)
the behavior for large µ of the modified Hamiltonian Ĥµ = T + i<j v µ ee (r ij ) + i v µ (r i ) is of the form

Ĥµ = Ĥ + 1 µ 2 Ĥ(2) + • • • , (C25)
The corresponding wave function Ψ µ and consequently the correlation pair density ñµ where n ′ 2,c (r, r) is the spherical-averaged derivative of the on-top correlation pair density which, according to the electron-electron cusp condition [60], is equal to the total on-top pair density: n ′ 2,c (r, r) = n 2 (r, r). Therefore, for large µ the short-range correlation energy has the exact behavior (C28)

  FIG. 2:Ground-state energy differences ∆E µ

xc=

  E xc , and in the limit µ → ∞, the short-range functional vanishes, Ēµ→∞ xc = 0. Near these two limits, the study of the behavior of Ēµ x and Ēµ

  with D = 32/(3π), A = 3.2581, β = 163.44 and γ = 4.7125. The expansion (34) is compared with the

  and 9 represent the short-range exchange energy of the He atom along the erf and erfgau adiabatic

1 0

 1 dtP n (t)e -t 2 ρ(P-Q) 2 , (A4)with ρ = ζη/(ζ + η) and P n (t) are the Rys polynomials of order n = e x + e y + e z + f x

(- 1 )

 1 n A n n! µ n+3 δ (n) (r) + O( 1 µ m+4 ).

2 .

 2 Short-range Hartree energy for µ → ∞Introducing the trivial variable transformation r 2 → r 12 and the spherical average of the density n(r 12 ) 1 )ñ(r 12 )v µ ee (r 12 )dr 1 dr 12 .(C17)

  (r, r)dr + O( 1 µ 2m+3 ), (C20) with the on-top exchange pair density and its spherical-averaged derivatives n (2n) 2,x (r, r) = (∂ 2n ñ2,x (r, r 12 )/∂r 2n

  2,c (r 1 , r 12 ) has a similar asymptotic expansionn µ 2,c (r 1 , r 12 ) = n 2,c (r 1 , r 12 ) (r 1 , r 12 ) + • • • . (C26)The absence of term in 1/µ in the last expansion implies that the first two terms in the asymptotic expansion of ∂ Ēµ c /∂µ can be determined by considering only n 2,c (r 1 , r 12 ). We find (r, r)dr + • • • , (C27)

n 2

 2 (r, r)dr + • • • .

  2 a ρ(P -Q) 2 } and a modified prefactor depending on t a ×I ′x (e x , f x , t a )I ′ y (e y , f y , t a )I ′ z (e z , f z , t a )w a .

	[e0|C exp(-ar 2 12 )|f0] = C(1 -t 2 a ) 3/2 κ AB κ CD	ζ ρ	3/2	η ρ	3/2
					(A8)
	Appendix B: Adiabatic connection near the KS
	system

  12 dr 1 dr 2 . (B9)

	interaction vµ ee (r) = 1/r -v µ ee (r) for the erf and erfgau interactions,
	vµ ee,erf (r) =	erfc(µr) r	,	(C1)
	vµ ee,erfgau (r) =	erfc(µr) r	+	2µ √ π	e -1 3 µ 2 r 2 .	(C2)
	Let f : R + → R be a test function (i.e., of bounded sup-port and infinitely differentiable) and consider the follow-
	ing integral

Acknowledgments

We are grateful to P. Gori-Giorgi (Université Paris VI, France) for stimulating discussions and in particular for suggesting us Eq. (B13).

Appendix A: Modified two-electron integrals

In this Appendix, we give details of the evaluation of the two-electron integrals over Gaussian basis sets for the modified erf and erfgau interactions. The modified integrals have been implemented into the Seward program [57] available in the Molpro package [START_REF] Molpro | a package of ab initio programs designed[END_REF] and using the Rys quadrature scheme [58]. For other implementations of twoelectron integrals with modified interactions, see Refs. 28,36,59.

The general form of four-center two-electron integrals is

where v ee (r 12 ) is the electron-electron interaction and φ is is an unnormalized primitive Cartesian Gaussian basis function

Actually, it is only required to explicitly compute the integrals of the type For the Coulomb interaction v ee (r 12 ) = 1/r 12 , these integrals can be written as

Actually, several terms of this expansion vanish as it will be seen below.

Ground-state wave function for µ → 0

Using the expansion of the interaction v µ ee (Eq. B2) and taking the functional derivatives of the expansions of the short-range Hartree, exchange and correlation energies (Eqs. B3, B4 and B9) to obtain the expansion of the potential v µ , we arrive to the following formal expansion for the Hamiltonian of the fictitious system

In this equation, ĤKS is the KS Hamiltonian, v reflect the fact that the terms in the expansion of the correlation energy (Eq. B9) come from two sources: the interaction v µ ee and the pair density n µ 2,c . The ground-state wave function Ψ µ of this Hamiltonian can also be expanded with respect to µ

where Φ is the KS determinant. We now show that several terms in the expansion of Ψ µ actually vanish.

For the erf interaction, v

ee and v

(1)

hx are constants (cf. Eqs. B2, B3 and B4), and consequently v (1,k) c = 0 for all k ≥ 1, thus Ψ (1) = 0. As there is no term in µ 2 in (B10), we also have Ψ (2) = 0. It implies in turn that in Eq. (B7) the terms k ≤ 2 vanish, and thus by Eq. (B9) v (2n+1,k) c = 0 for k ≤ 2 and all n ≥ 1. In particular, v

= 0 which leads to the expansion of the Hamiltonian

hx µ 5 +v (3,3)

We therefore have the following expansion for the wave function

Therefore, the terms k = 1, 2, 4 in Eqs. (B7) and (B9) vanish for for the erf interaction.

For the erfgau interaction, v

ee = 0, and consequently v = 0 for all k ≥ 1, thus Ψ (1) = Ψ (2) = Ψ (3) = Ψ (4) = 0. It implies in turn that in Eq. (B7) the terms k ≤ 4 vanish, and thus by Eq. (B9) v (2n+1,k) c = 0 for k ≤ 4 and all n ≥ 1. In particular, v

= 0 which leads to the expansion of the Hamiltonian

hx µ 5 + v (7) ee + v

hx µ 5 + v (9) ee + v

hx µ 9 + v (5,5) c

We therefore have the following expansion for the wave function

Therefore, the terms k = 1, 2, 3, 4, 6, 8 in Eqs. (B7) and (B9) vanish for the erfgau interaction.

Appendix C: Adiabatic connection near the physical system

In this appendix, we study the erf and erfgau adiabatic connections near the physical system, i.e. for large interaction parameter µ.

1. Short-range interaction for µ → ∞

We start by deriving a distributional asymptotic expansion for large µ of the short-range electron-electron


Julien Toulouse

Andreas Savin

Heinz-Jürgen Flad

Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction

We calculate the short-range exchange-correlation energy of the uniform electron gas with two modified electron-electron interactions. While the short-range exchange functionals are calculated analytically, Coupled-Cluster and Fermi-hypernetted chain calculations are carried out for the correlation energy and the results are fitted to an analytical parametrization. These data enable to construct the local density approximation corresponding to these modified interactions.

I. INTRODUCTION

In electronic structure calculations using density functional theory (DFT) [1] in the Kohn-Sham (KS) scheme [2], the central quantity that needs to be approximated is the exchange-correlation energy functional. The vast majority of approximations for this functional originates from the local density approximation (LDA) [2] consisting in locally transferring the exchange-correlation energy of the uniform electron gas to the inhomogeneous system of interest. Actually, it is been realized for long that the LDA can describe accurately short-range correlation effects but is inadequate for long-range correlation effects (see, e.g., [3]). This observation lead to the development of the first gradient corrected functionals [4][5][6][7][8][START_REF] Perdew | Electronic Structure of Solids '91[END_REF] with the basic idea that the long-range contribution to the exchange-correlation energy of the uniform electron gas must not be transferred to inhomogeneous systems.

Guided by the same idea, it has been proposed [START_REF] Savin | [END_REF][START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF][START_REF] Leininger | [END_REF][13][14][START_REF] Stoll | Density Functional Method in Physics[END_REF] to describe only the short-range electronic correlations effects by a density functional, and leaving the remaining long-range correlations effects to a more appropriate method like Configuration Interaction. Concretely, 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][13], the error function has been used to define the longrange interaction v µ ee,erf (r) = erf(µr) r ,

(2) * Electronic address: savin@lct.jussieu.fr referred to as the erf interaction. More recently [START_REF] Toulouse | [END_REF], we have used a sharper long-range/short-range separation with the erfgau interaction

v µ ee,erfgau (r) = erf(µr) r - 2µ √ π e -1 3 µ 2 r 2 . ( 3
)
The method then consists in finding the ground-state multi-determinantal wave function Ψ µ of a fictitious system containing only the long-range part of the electronelectron interaction V µ ee = i<j v µ ee (r ij ) and having the same density n than the physical system. The total ground-state electronic energy of a physical system is then given by

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 is the nucleielectron interaction, Ū µ is the short-range Hartree energy and Ēµ xc is the short-range exchange-correlation 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)
Eq. ( 4) provides an exact decomposition of the total energy into a long-range component written in a wave function formalism and a remaining short-range component expressed as a density functional. In particular, there is no double counting of correlation effects. The only unknown quantity in this approach is the short-range exchange-correlation functional Ēµ xc which is not the usual exchange-correlation functional of the KS scheme E xc .

For a reasonable long-range/short-range separation (µ not too small), Ēµ xc essentially describes short-range interactions, and it is therefore expected to be well approximated by the LDA corresponding to the modified interaction

Ēµ xc [n] = n(r)ε µ xc (n(r))dr. (6)
In Eq. ( 6), εµ xc is the short-range exchange-correlation energy per particle obtained by difference from the exchange-correlation energies per particle of the uniform electron gas with the standard Coulomb ε xc and with the erf or erfgau interaction ε µ xc

εµ xc (n) = ε xc (n) -ε µ xc (n). (7)
As for the original LDA in the KS scheme with the Coulomb interaction, knowledge of εµ xc is crucial to apply the LDA to the short-range exchange-correlation functional. In this paper, we give the expressions of this short-range exchange-correlation energy per particle of the uniform electron gas with the erf and erfgau modified interactions. While the exchange part can be calculated analytically, the correlation are derived from Coupled-Cluster and from Fermi-hypernetted chain calculations.

Atomic units will be used throughout this work.

II. SHORT-RANGE EXCHANGE ENERGY

The short-range exchange energies per particle εµ x (r s ) of the uniform electron gas for the Wigner-Seitz radius r s = (3/(4πn)) 1/3 with the erf and erfgau interactions are calculated analytically (see Eq. A9 and A12 of Appendix A). The inverse of the interaction parameter, 1/µ, represents the range of the modified interaction and has to be compared with r s which is the characteristic length for exchange. Thus, the relevant variable for the exchange energy is actually µr s . Fig. 1 shows the ratio of the short-range exchange energy per particle with the erf and erfgau interactions to the exchange energy per particle with the Coulomb interaction εµ x (r s )/ε x (r s ). In order to compare the two interactions, a scale factor is applied on the parameter of the erfgau interaction µ → (1 + 6 √ 3) 1/2 µ ≈ 3.375µ so that the erf and erfgau exchange energies have the same asymptotic behavior for µr s → ∞ (see below).

It has been shown that for a finite system where the exchange contribution to the second-order density matrix n 2,x (r 1 , r 2 ) decays exponentially with r 12 , the shortrange exchange energy Ēµ

x can formally be expanded around µ = 0 into an odd series in µ [START_REF] Toulouse | [END_REF] Ēµ

x = E x - 1 √ π ∞ n=0 (-1) n a n n! µ 2n+1 × n 2,x (r 1 , r 2 )r 2n 12 dr 1 dr 2 , (8)
where E x is the usual KS exchange energy, a n,erf = 1/(2n + 1) for the erf interaction and a n,erfgau = 1/(2n + 1) -1/3 n ( = 0 for n ≥ 2) for the erfgau interaction. Except for the term linear in µ, the expansion for µr s → 0 FIG. 1: Ratio of the erf (solid line) or erfgau (dashed line) short-range exchange energy per particle to the exchange energy per particle with Coulomb interaction in the uniform electron gas εµ x (rs)/εx(rs) with respect to µrs. In order to compare the two interactions, a scale factor has been applied on the interaction parameter of the erfgau interaction: µ → 3.375µ.

of the erf short-range energy per particle of the uniform electron gas does not exhibit the same behavior

εµ x,erf (r s ) ≈ ε x (r s ) + 1 √ π µ - 3 2π 4 1/3 r s µ 2 + 2 9π 2 r 3 s µ 4 + exponential terms . (9)
Similarly, the expansion corresponding to the erfgau interaction is

εµ x,erfgau (r s ) ≈ ε x (r s ) + 2 √ 3 -3 (18π 4 ) 1/3 r s µ 2 + 2(9 -4 √ 3) 81π 2 r 3 s µ 4 + exponential terms . ( 10
)
These different behaviors of the short-range exchange energy in the uniform electron gas and in a finite system is consistent with the important LDA error arising at µ = 0, i.e. for the standard DFT within the Kohn-Sham scheme.

The short-range exchange energy of a finite system can also be formally expanded for µ → ∞ into the asymptotic series [17]

Ēµ x = 2 √ π ∞ n=0 A 2n (2n)!(2n + 2)µ 2n+2 n (2n) 2,x (r, r)dr, (11)
where n (2n) 2,x (r, r) are the exchange contribution to the on-top second-order density matrix and its derivatives, A n,erf = Γ( n+32 ) for the erf interaction and

A n,erfgau = Γ( n+3 2 )-3 n+3 2 Γ( n+3 2 )+ 2 × 3 n+3 2 Γ( n+5
2 ) for the erfgau interaction. The asymptotic expansions of the short-range energies per particle of the uniform electron gas for large µ do have the same form εµ

x,erf (r s ) ≈ -

3 16 1 r 3 s µ 2 + 3π 2 2 1/3 27 640 1 r 5 s µ 4 + • • • , (12) εµ
x,erfgau (r s ) ≈ -

3(1 + 6 √ 3) 16 1 r 3 s µ 2 + 3π 2 2 1/3 27(1 + 36 √ 3) 640 1 r 5 s µ 4 + • • • . (13)
Again, this is consistent with the quality of the local density approximation for large µ [14,17].

III. SHORT-RANGE CORRELATION ENERGY

The long-range correlation energy per particle ε µ c (r s ) with the erfgau interaction has been computed for several values of r s (from r s = 0.2 to 10) and µ (from µ = 0 to 25). Coupled-Cluster calculations with double excitations (CCD), according to a method introduced by Freeman [18], have been performed (see Appendix B), as well as Fermi-hypernetted-chain (FHNC) calculations (see Appendix C). Data for the erf interaction is already available [START_REF] Savin | Recent Developments of Modern Density Functional Theory[END_REF].

Once the long-range correlation energy per particle ε µ c (r s ) is obtained, the short-range correlation energy per particle εµ c (r s ) is expressed as

εµ c (r s ) = ε c (r s ) 1 - ε µ c (r s ) ε µ→∞ c (r s ) , (14)
where ε c (r s ) is the correlation energy per particle of the uniform electron gas with Coulomb interaction taken from the usual parametrization of Vosko, Wilk and Nusair (VWN) [19]. According to Eq. ( 14), εµ c (r s ) correctly reduces to the VWN value for µ = 0 and vanishes for µ → ∞.

The erf and erfgau short-range correlation energies per particle with respect to µ for r s = 0.5 and r s = 2 are plotted in Fig. 2 and3. For erfgau, the differences between the results from the CCD and FHNC calculations are visible only for a high density (r s = 0.5). Surprisingly, both methods diverge with the erfgau interaction when µ √ r s 1, explaining the absence of points between µ = 0 and µ ≈ 0.7 for r s = 2, or between µ = 0 and µ ≈ 1.4 for r s = 0.5 in Fig. 3. We connect this behavior to the attractive character of the erfgau interaction for small µ (see Appendix D). In practice, the lack of accuracy of the LDA correlation functional for small µ because of these missing points does not represent a serious problem since the LDA exchange functional produces large errors anyway in finite systems in this domain of µ, as suggested by its incorrect expansion as µ → 0 (Eq. 10).

It has been shown [14,17] that the leading term in the expansion of the short-range correlation energy for large

µ is Ēµ c ≈ Cπ 2µ 2 n 2,c (r, r)dr + • • • , (15)
where n 2,c (r, r) is the on-top correlation pair density for the full Coulomb interaction, C = 1 for the erf interaction and C = (1 + 6 √ 3) for the erfgau interaction. For FIG. 2: Short-range correlation energy per particle (dots) of the uniform electron gas for the erf interaction with respect to the interaction parameter µ for rs = 0.5 and rs = 2 computed with the CCD method. The analytical parametrization (Eq. 18) is represented by a dashed line. FIG. 3: Short-range correlation energy per particle of the uniform electron gas for the erfgau interaction with respect to the interaction parameter µ for rs = 0.5 and rs = 2, computed with the CCD method (dots) and with the FHNC method (triangles). The analytical parametrizations (Eq. 18) using the CCD data and the FHNC data are represented by the long-dashed and short-dashed lines, respectively. the uniform electron gas, n 2,c (r, r) can be expressed in term of the on-top pair-distribution function g 0 (r s ) so that the short-range correlation energy per particle has the following exact behavior for µ → ∞

εµ c (r s ) ≈ 3C 8µ 2 r 3 s g 0 (r s ) - 1 2 + • • • . ( 16
)
An estimation of g 0 (r s ) which includes the correct limits for r s → ∞ and r s → 0 was given by Burke, Perdew and Ernzerhof [20] g c (rs) computed from the CCD method (dots) and from the FHNC method (triangles) for the erfgau interaction and rs = 2. The horizontal line is the exact limit for µ → ∞.

0 (r s ) = D (γ + r s ) 3/2 + β e -A √ γ+rs , (17)
g 0 (r s ) ≤ 1/2. In Fig. 4, we have plotted µ 2 εµ c (r s ) computed with the CCD and FHNC methods with respect to µ for r s = 2. This plot actually illustrates a general trend: for large values of r s , the correlation energy per particle computed from the CCD method does not exhibit the correct behavior for µ → ∞. On the contrary, the FHNC method seems to perform better in this limit in spite of an important numerical noise. However, for small r s (r s 1), the CCD method becomes exact, since it reduces to the Random Phase Approximation (See Appendix B), and thus respects the µ → ∞ limit.

The short-range correlation energies per particle for the erf and erfgau interactions are represented by the analytical parametrization

εµ c (r s ) = ε c (r s ) 1 + c 1 (r s )µ + c 2 (r s )µ 2 , ( 18
)
where c 1 (r s ) is determined by a fit

c 1 (r s ) = u 1 r s + u 2 r 2 s 1 + v 1 r s , (19)
with u 1 = 1.0271, u 2 = -0.2302, v 1 = 0.6197 for erf, u 1 = 0.3916, u 2 = 0.0223, v 1 = 0.9105 for erfgau using the CCD data and u 1 = 0.4795, u 2 = 1.0094, v 1 = 10.1247 for erfgau using the FHNC data, and c 2 (r s ) is imposed by the exact limit for µ → ∞

c 2 (r s ) = 8r 3 s ε c (r s ) 3C(g 0 (r s ) -1/2) . ( 20
)
These analytical parametrizations for erf and erfgau interactions are represented in Fig. 2 and 3 for r s = 0.5 and 2. The two parametrizations for the erfgau interaction differ only at small r s where the imposition of the exact µ → ∞ limit make both expressions close to the CCD data.

The uniform electron gas can be considered as a system of N electrons in a box of volume Ω with a uniform background of positive charge to ensure neutrality, studied in the thermodynamic limit (i.e N → ∞ and Ω → ∞ such that the density n = N/Ω remains constant). This system is described by the electronic Hamiltonian

Ĥ = Ĥ0 + Ĥint , (A1)
where Ĥ0 = T is the kinetic energy operator and Ĥint is the electron-electron interaction which can be expressed by its Fourier expansion

Ĥint = 1 Ω i<j k =0 v ee (k)e ik.rij , ( A2
)
where v ee (k) is the Fourier transform of the (modified) electron-electron interaction v ee (r). The term k = 0 corresponding to the Hartree energy has been removed since it cancels out with the background energy and the electron-background interaction energy, provided that the same modified interaction has been applied to all these terms. The exchange energy corresponds to the first-order correction

E x = Φ| Ĥint |Φ , ( A3
)
where Φ is the ground-state wave function of Hamiltonian Ĥ0 (a Slater determinant of plane-waves). It has been shown [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF] that the exchange energy reduces to

E x = - 1 12π 4 k 3 F ∞ 0 q 2 v ee (q) 1 - 3 2 x + 1 2 x 3 θ(1-x)dq, (A4) with k F = (3π 2 n) 1/3 , x = q/(2k F ), θ(y) = 1 if y > 0 and θ(y) = 0 if y < 0.
The Fourier transform of the Coulomb interaction is v ee,coul (q) = 4π q 2 , (A5) so that Eq. (A4) leads after integration over x the well known exchange energy per particle

ε x = E x /N ε x,coul (r s ) = - 3 8 18 π 2 1/3 1 r s , (A6)
with r s = 1/(αk F ) and α = (4/(9π)) 1/3 . For the erf interaction, inserting the Fourier transform v µ ee,erf (q) = 4π q 2 e -q 2 /(4µ 2 ) (A7) into Eq. (A4) leads to the long-range exchange energy per particle

ε µ x,erf (r s ) = - 18 π 2 1/3 1 r s A √ π erf 1 2A + (2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 , (A8)
where A = µ/(2k F ). The short-range exchange energy per particle is then

εµ x,erf (r s ) = ε x (n) -ε µ x,erf (n) = - 18 π 2 1/3 1 r s 3 8 -A √ π erf 1 2A + (2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 . (A9)
Similarly, the Fourier transform of the erfgau interaction writes v µ ee,erfgau (q) = 4π q 2 e -q 2 /(4µ 2 ) -6 √ 3π µ 2 e -3q 2 /(4µ 2 ) , (A10) so that the long-range exchange energy per particle is

ε µ x,erfgau (r s ) = - 18 π 2 1/3 1 r s A √ π erf 1 2A + (2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 -A √ π erf 1 2B + (2B -16B 3 )e -1/(4B 2 ) -6B + 16B 3 , (A11)
and the short-range exchange energy per particle is εµ x,erfgau (r s ) = -

18 π 2 1/3 1 r s 3 8 -A √ π erf 1 2A + (2A -4A 3 )e -1/(4A 2 ) -3A + 4A 3 +A √ π erf 1 2B + (2B -16B 3 )e -1/(4B 2 ) -6B + 16B 3 , ( A12
)
where B = µ/(2 √ 3k F ).

Appendix B: Coupled-Cluster calculations of the uniform electron gas with modified interaction

For the Coulomb interaction, Freeman [18] has calculated the correlation energy of the uniform electron gas by summing the ring and screened exchange diagrams using the Coupled-Cluster method with double excitations (CCD). In this appendix, we rapidly give the corresponding equations for an arbitrary electron-electron interaction v ee (r).

The CCD wave function is constructed from the non-interacting determinant of plane waves Φ through

|Ψ = e T2 |Φ , (B1)
where the excitation operator T2 is expressed in second quantization notation as

T2 = ki,kj ,q t q (k i , k j )a † ki+q a † kj -q a kj a ki . (B2)
Retaining only the ring diagrams, the amplitudes t q (k i , k j ) are solutions of the equations (with momentum in k F units)

t q (k i , k j ) = v ee (q) 3π 2 k F D q (k i , k j ) 1 + 6π 2 dk (2π) 3 (t q (k i , k) + t q (k j , k))θ(1 -k)θ(|k + q| -1) +18π 4 dk (2π) 3 dk ′ (2π) 3 (t q (k i , k)t q (k j , k ′ ) + t q (k i , k ′ )t q (k j , k)) ×θ(1 -k)θ(1 -k ′ )θ(|k + q| -1)θ(|k ′ + q| -1) , (B3) with D q (k i , k j ) = -(q 2 + q.(k i + k j ))
. Compared to the original work of Freeman, the only modification appears in the Fourier transform v ee (q) of the arbitrary electronelectron interaction v ee (r). Once the amplitudes have been computed, the correlation energy per particle can be calculated by

ε c = ε c,dir + ε c,ex , (B4)
where ε c,dir and ε c,ex are the direct and exchange contributions given by

ε c,dir = 18π 4 k F dq (2π) 3 dk i (2π) 3 dk j (2π) 3 v ee (q)t q (k i , k j ) ×θ(1 -k i )θ(1 -k j )θ(|k i + q| -1)θ(|k j + q| -1), (B5) ε c,ex = -9π 4 k F dq (2π) 3 dk i (2π) 3 dk j (2π) 3 v ee (|k i + k j + q|)t q (k i , k j ) ×θ(1 -k i )θ(1 -k j )θ(|k i + q| -1)θ(|k j + q| -1). ( B6
)
The direct contribution, corresponding to the ring diagrams, is the usual correlation energy within the Random Phase Approximation (RPA). The exchange contribution includes additional screened exchange diagrams. In the high-density limit (r s → 0), the method reduces to the RPA and thus becomes exact.

In practice, it is convenient to introduce the intermediate quantity

T q (k i ) = dk (2π) 3 t q (k i , k)θ(1 -k)θ(|k + q| -1), (B7)
and to perform the integration by Gauss-Legendre quadrature. Equation (B3) is then equivalent to

j A ij T j = B i , ( B8
)
with T j = T q (k j ) and

A ij = δ ij 1 - 2 k F m W m D im - 2 k F W j D ij , (B9) B i = m W m D im 3π 2 k F + 12π 2 k F T i T m D im , ( B10
)
where D ij = v ee (q)/D q (k i , k j ) and W m are the quadrature weights. As B i actually depends on the T i 's coefficients, Eq. (B8) have to be solved iteratively.

Appendix C: Fermi-hypernetted chain theory for homogeneous systems

Similar in spirit to the CCD approach (B1), the Fermihypernetted chain (FHNC) method [START_REF] Clark | Progress in Nuclear and Particle Physics[END_REF] is based on an approximate product ansatz for the wave function

Ψ (r 1 , r 2 , . . . , r N ) = exp   i<j u 2 (r i , r j )   Φ (r 1 , r 2 , . . . , r N ) , (C1)
where the correlation factor, called Jastrow factor, acts on a single Slater determinant Φ. For homogeneous systems the pair-correlation function u 2 depends only on the inter-electron coordinate r ij . The close relation-ship between CCD and FHNC methods is not restricted to a purely formal analogy between the pair-correlation function u 2 and the CCD excitation operator T2 . This topic has been extensively discussed in a review article by Bishop [START_REF] Bishop | [END_REF]. For bosonic systems, both methods are actually equivalent on a certain level of approximation.

It is an important feature of the Jastrow ansatz that the exact short-and long-range asymptotic behavior of a homogeneous system can be expressed as simple functions of the inter-electron coordinate. In the case of a Coulomb potential, Kato's cusp condition for electrons with antiparallel spin imposes a constraint on the first derivative of the pair-correlation function

du 2 (r 12 ) dr 12 r12=0 = 1 2 , (C2)
which can be exactly represented by a Jastrow factor. We discuss below how the modified interaction affects the short-range behavior of the Jastrow factor. The longrange asymptotic behavior of electron correlations is well described by the RPA approximation [START_REF] Fulde | Correlations in Molecules and Solids[END_REF]. It provides an explicit asymptotic expression for the pair-correlation function

lim r12→∞ u 2 (r 12 ) = - 1 ω pl r 12 , (C3)
where the plasmon frequency ω pl = √ 4πn of the electron gas enters into the denominator. This asymptotic behavior can be reproduced by the FHNC method [START_REF] Krotscheck | [END_REF].

For a given pair-correlation function, the FHNC equations represent a nonlinear system of equations between "nodal" N (r 12 ), "non-nodal" X(r 12 ) and "elementary" E(r 12 ) functions. Each of these functions can be expressed as an infinite sum of certain types of diagrams build up from the pair-correlation function and the oneparticle density matrix of the noninteracting system. Some of these equations are conveniently expressed in coordinate space, the others in momentum space. The system of equations is underdetermined and requires an a priori knowledge of the "elementary" diagrams in order to get a unique solution. In a series of papers Krotscheck developed a consistent approximation scheme for the FHNC equations [26][27][28], which preserves the correct asymptotic behavior on each level of approximation. We have used the FHNC//0 method which corresponds to the lowest level of approximation, where "elementary" diagrams are neglected altogether. The FHNC//0 equations are given by Γ dd (r 12 ) := X dd (r 12 ) + N dd (r 12 )

= exp [2 u 2 (r 12 ) + N dd (r 12 )] -1, (C4)

Ñdd (k) = Xdd (k) S F (k) Γdd (k), ( C5
)
where S F is the liquid structure function of the noninteracting system. We have used the dimensionless Fourier transform f (k) = n dr f (r) exp(ik • r).

(C6)

The "nodal" and "non-nodal" functions provide a link between the Jastrow factor and the liquid structure function of the interacting system This connection enables an approximate variational treatment of the Jastrow factor within FHNC theory.

S(k) = S F (k) + S F (k) 2 Γdd (k), (
In the following we want to give a brief outline of the FHNC//0 optimization cycles following essentially Krotscheck's paper [28]. Starting point is an effective particle-hole potential

V ph (r 12 ) = [1 + Γ dd (r 12 )] v ee (r 12 ) + ∇ [1 + Γ dd (r 12 )] 1/2 2
+Γ dd (r 12 ) ω I (r 12 ), (C9) which depends, beside diagrammatic contributions, on the bare (modified) interaction potential v ee and an induced interaction ω I . In momentum space the induced interaction

ωI (k) = - k 2 4 1 + 2 S(k) S F (k) 1 S(k) - 1 S F (k) 2 , (C10)
can be expressed in terms of the liquid structure functions of the interacting and noninteracting system. Within the high density regime, v ee can be taken as an initial guess for V ph . Performing FHNC calculations at successively lower densities it is possible to reach the low density regime by taking V ph from a slightly higher density as an initial guess in the optimization process. The particle-hole potential is related to the liquid structure function

S(k) = S F (k) 1 + (4/k 2 ) S 2 F (k) Ṽph (k) 1/2 . (C11)
In the first step of the optimization cycle Eq. (C11) is used to get an improved approximation of the liquid structure function. Using Eqs. (C7) and (C10) it is now possible to obtain improved approximations for the induced interaction ω I and the diagrammatic quantity Γ dd (r 12 ). These can be used in the second step to calculate an improved approximation of the particle-hole potential V ph (Eq. C9). The two steps provide a selfconsistent optimization cycle, which can be repeated until convergence has been achieved. Finally we have used the FHNC Eqs. (C4) and (C5) in order to obtain the optimized FHNC//0 Jastrow factor. Jastrow factors for the Coulomb interaction and the long-range erfgau interaction are shown in Fig. 5. With decreasing value of the interaction strength µ, the shortrange part of the Jastrow factors is modified; it changes from a cusp at r 12 = 0 for µ → ∞ into a smooth behavior for any finite µ. For small values of µ, a local minimum appears at an intermediate distance. As expected, the long-range behavior of the Jastrow factor is not affected by the modified interaction.

Appendix D: Divergence of calculations on the uniform electron gas with modified interaction With the erfgau interaction, CCD and FHNC calculations of the uniform electron gas diverge for small values of µ and r s . This is due to the particular form of the erfgau interaction. In fact, whereas the Fourier transform of the Coulomb or erf interaction is always positive, the Fourier transform of the erfgau interaction v ee,erfgau (q) = 4π q 2 e -q 2 /(4µ 2 ) -6 √ 3π µ 2 e -3q 2 /(4µ 2 ) (D1) can be negative (see Fig. 6). For small µ, the negative part of v ee,erfgau (q) is not negligible, introducing an attractive contribution to the electron-electron interaction.

It is possible to estimate the domain of µ and r s for which CCD calculations do not converge because of this attractive interaction. Let's consider the "state-average" approximation to the CCD equations proposed by Bishop and Lührmann [29]. This model consists in neglecting the exchange contribution to the correlation energy and averaging over the occupied momentum k i and k j the equations given in Appendix B. The correlation energy per particle of the uniform electron gas is then written as (with momentum in k F units)

ε c = k F 4π 2
∞ 0 dqq 2 v ee (q)P (q) t q , (D2)

where P (q) = 3q/4 -q 3 /16 if q ≤ 2, P (q) = 1 if q > 2, v ee (q) is the Fourier transform of the electronelectron interaction and t q is the average of the amplitude t q (k i , k j ) which is solution of the equation

t q = 1 3π 2 k F v ee (q) D -1 q 1 + t q P (q) 2 , (D3)
where D -1 q is the average of the inverse of D q (k i , k j ), introduced in Appendix B.

The general solution of (D3) is t q = 1 -A(q) + 1 -2A(q) A(q)P (q) , (D4)

with A(q) = (2 D -1 q v ee (q)P (q))/(3π 2 k F ). Using the additional approximation D -1 q ≈ D q -1 = -P (q)/q 2 , one sees immediately that this solution breaks down (more precisely, becomes imaginary) if v ee (q) < -3π 2 k F q 2 4P (q) 2 . (D5)

Let's evaluate this inequality in the worst situation where v ee (q) and P (q) take their minimum values. The interaction v ee (q) reaches its minimum v min ≈ -5.6k 2 F /µ 2 for q ≈ µ/k F , and in this domain P (q) ≤ 3q/4 so that condition (D5) roughly gives

µ √ r s 1, ( D6
)
where r s = 1/(αk F ) with α = (4/(9π)) 1/3 has been used. In Fig. 7, we have reported the values of µ and r s at the limit of convergence for the calculation of the correlation energy of the uniform electron gas with the erfgau interaction, together with the divergence condition (D6). Obviously, the domain of divergence is well approximated by this condition.

The divergence of both CCD and FHNC methods for these values of µ and r s where the attractive part of electron-electron interaction becomes important is reminiscent of the situation happening in a superconductor where the ordinary perturbation expansion breaks down for the superconducting phase.

FIG. 4 :

 4 FIG.4: µ 2 εµ c (rs) computed from the CCD method (dots) and from the FHNC method (triangles) for the erfgau interaction and rs = 2. The horizontal line is the exact limit for µ → ∞.

  C7) which is essentially the Fourier transform of the pairdensity n 2 (r 12 ) S(k) = 1+n dr 12 n 2 (r 12 )/n 2 -1 exp(ik•r 12 ). (C8)

FIG. 5 :

 5 FIG.5: Jastrow factors at rs = 1 for Coulomb interaction (solid line), and erfgau interactions at interaction parameters µ = 20 (dashed line), µ = 10 (dotted line) and µ = 5 (dotteddashed line).

FIG. 6 :

 6 FIG.6: Fourier transforms of the Coulomb interaction (dotted line), erf interaction (dashed line) and erfgau interaction (solid line), each plotted with a interaction parameter µ = 0.5.

6 ΜFIG. 7 :

 67 FIG. 7: Points at the limit of convergence for calculation of the correlation energy of the uniform electron gas with erfgau interaction. The domain of divergence is well approximated by µ √ rs 1 (solid line).

Acknowledgments

We are grateful to P. Gori-Giorgi (University "La Sapienza", Roma) for stimulating discussions.