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## CONVERGENCE IN ONE-ELECTRON ATOMIC BASIS SETS

In this section, we study the convergence of the long-range wave function and correlation energy with respect to the size of the one-particle atomic basis. This problem is closely related to the convergence of the partial-wave expansion studied in the previous section. Indeed, for a two-electron atom in a singlet S state, it is possible to use the sphericalharmonic addition theorem to obtain the partial-wave expansion in terms of the relative angle between two electrons by products of the spherical harmonic part Y`,m of the oneparticle atomic basis functions P`(cos ) = 4 2` + 1 X` m=−` (−1)mY`,m(1, 1)Y`,−m(2, 2), (24) where cos = cos 1 cos 2+sin 1 sin 2 cos(1−2) with spherical angles 1,1 and 2,2. The partial-wave expansion can thus be obtained from a one-particle atomic basis, provided that the basis saturates the radial degree of freedom for each angular momentum `. In practice, of course, for the basis sets that we use, this latter condition is not satisfied. Nevertheless, one can expect the convergence with the maximal angular momentum L of the basis to be similar to the convergence of the partial-wave expansion. For this study, we have analyzed the behavior of He, Ne, N2, and H2O at the same experimental geometries used in Ref. 44 (RN−N = 1.0977 °A, RO−H = 0.9572 °A and [HOH = 104.52). We performed all the calculations with the program MOLPRO 2012 [69] using Dunning correlation-consistent cc-p(C)VXZ basis sets for which we studied the convergence with respect to the cardinal number X, corresponding to a maximal angular momentum of L = X − 1 for He and L = X for atoms from Li to Ne. We emphasize that the series of Dunning basis sets does not correspond to a partial-wave expansion but to a principal expansion [70, 71] with maximal quantum number N = X for He and N = X + 1 for Li to Ne. The short-range exchange-correlation PBE density functional of Ref. 18 (which corresponds to a slight modification of the one of Ref. 72) was used in all range-separated calculations.

### Convergence of the wave function

We start by analyzing the convergence of the FCI ground-state wave function of the He atom with respect to the cardinal number X of the cc-pVXZ basis sets. We perform a FCI calculation with the long-range Hamiltonian in Eq. (3) using a fixed RSH density, calculated from the orbitals obtained in Eq. (4), in the short-range Hartree–exchange-correlation potential. To facilitate the extraction of the wave function from the program, we use the L¨owdin-Shull diagonal representation of the spatial part of the FCI wave function in terms of the spatial natural orbitals (NO) {‘μ i } [76, 77] lr,μ(r1, r2) = X i1 cμ i ‘μ i (r1)’μ i (r2), (25) where the coefficients cμ i are related to the NO occupation numbers nμ i by the relation nμ i = 2|cμ i |2. As the signs of cμ i are undetermined we have chosen a positive leading coefficient cμ 1 = p nμ 1/2, and we assumed that all the other coefficients are negative cμ i = − p nμ i /2 for i 2 [78]. Even though it has been shown that, for the case of the Coulomb interaction, there are in fact positive coefficients in the expansion in addition to the leading one, for a weakly correlated system such as the He atom, these positive coefficients appear only in larger basis sets than the ones that we consider here and have negligible magnitude [79–81]. In Figure 3 we show the convergence of the FCI wave function lr,μ(r1, r2) with the cardinal number X for μ ! 1 which corresponds to the Coulomb interaction (left) and for μ = 0.5 (right). The first electron is fixed at the Cartesian coordinates r1 = (0.5, 0., 0.) (measured from the nucleus) and the position of the second electron is varied on a circle at the same distance of the nucleus, r2 = (0.5 cos , 0.5 sin , 0.). For the Coulomb interaction, we compare with the essentially exact curve obtained with a highly accurate 418-term Hylleraastype wave function [73–75]. The curve of lr,μ(r1, r2) as a function of reveals the angular correlation between the electrons [82]. Clearly, correlation is much weaker for the longrange interaction. Note that a single-determinant wave function (r1, r2) = ‘1(r1)’1(r2), where ‘1 is a spherically symmetric 1s orbital, does not depend on , and the HF and RSH single-determinant wave functions indeed just give horizontal lines in Figure 3.

#### Convergence of the correlation energy

We also study the basis convergence of the long-range MP2 correlation energy, given in Eq. (7), calculated with RSH orbitals for He, Ne, N2 and H2O. In Table I we show the valence MP2 correlation energies and their errors as a function of the cardinal number X of the cc-pVXZ basis sets for X 6. We compare the longrange MP2 correlation energies Elr,μ c,X at μ = 0.5 and the standard Coulomb MP2 correlation energies Ec,X corresponding to μ ! 1. For the case of the Coulomb interaction, the error is calculated as Ec,X = Ec,X − Ec,1 where Ec,1 is the MP2 correlation energy in the estimated CBS limit taken from Refs. 44 and 84. For the range-separated case we do not have an independent estimate of the CBS limit of the long-range MP2 correlation energy for a given value of μ. Observing that the difference between the long-range MP2 correlation energies for X = 5 and X = 6 is below 0.1 mhartree for μ = 0.5, we choose the cc-pV6Z result as a good estimate of the CBS limit. Of course, the accuracy of this CBS estimate will deteriorate for larger values of μ, but in practice this is a good estimate for the range of values of μ in which we are interested, i.e. 0 μ 1 [85]. The error on the long-range correlation energy is thus calculated as Elr,μ c,X = Elr,μ c,X − Elr,μ c,6 .

The first observation to be made is that the long-range MP2 correlation energies only represent about 1 to 5 % of the Coulomb MP2 correlation energies. Although the long-range correlation energy may appear small, it is nevertheless essential for the description of dispersion interactions for instance. The errors on the long-range MP2 correlation energies are also about two orders of magnitude smaller than the errors on the Coulomb MP2 correlation energies.

**COMPUTATIONAL DETAILS**

The 1DH calculations have been performed with a development version of MOLPRO 2015 [62], and the OEP-1DH ones with a development version of ACES II [63]. In all calculations, we have used the B exchange [23] and the LYP correlation [24] density functionals, for EDFA x and EDFA c , respectively. This choice was motivated by the fact that 1DH-BLYP was found to be among the one-parameter double-hybrid approximations giving the most accurate thermochemistry properties on average [22, 28, 64]. We expect however that the effect of the OEP self consistency to be similar when using other density functional approximations. The performance of both DH methods has been tested against a few atomic (He, Be, Ne, and Ar) and molecular (CO and H2O) systems. For the latter, we considered the following equilibrium geometries: for CO d(C–O) = 1.128°A, and for H2O d(H–O) = 0.959°A and a(H–O–H) = 103.9. In all cases, core excitations were included in the second-order correlation term.

In our OEP calculations, for convenience of implementation, the same basis set is used for expanding both the orbitals and the exchange-correlation potential. To ensure that the basis sets chosen were flexible enough for representation of orbitals and exchange-correlation potentials, all basis sets were constructed by full uncontraction of basis sets originally developed for correlated calculations, as in Refs. 65 and 66. In particular, we employed an even tempered 20s10p2d basis for He, and an uncontracted ROOS–ATZP basis [67] for Be and Ne. For Ar, we used a modified basis set [68] which combines s and p basis functions from the uncontracted ROOS–ATZP [67] with d and f functions coming from the uncontracted aug–cc–pwCVQZ basis set [69]. In the case of both molecular systems, the uncontracted cc–pVTZ basis set of Dunning [70] was employed. For all OEP calculations standard convergence criteria were enforced, corresponding to maximum deviations in density-matrix elements of 10−8. In practice, the use of the same basis set for expanding both the orbitals and the exchange-correlation potential leads to the necessity of truncating the auxiliary function space by the TSVD method for constructing the pseudo-inverse of the linear-response function. The convergence of the potentials with respect to the TSVD cutoff was studied. Figure 1 shows the example of the convergence of the exchange-correlation and correlation potentials of the Be atom and the CO molecule. For Be, the potentials obtained with the 10−4 and 10−6 cutoffs are essentially identical, while for the 10−8 cutoff the exchangecorrelation potential has non-physical oscillations and the correlation potential diverges. For CO, the potentials obtained with the 10−4 cutoff are significantly different from the potentials obtained with the 10−6 cutoff, while no difference can be seen between the potentials obtained with the 10−6 and 10−8 cutoffs. A cutoff of 10−6 was thus chosen for all systems to achieve a compromise between convergence and numerical stability.

**Table of contents :**

Introduction

**1 Review of density-functional theory and hybrid methods **

1.1 Schrodinger equation

1.2 Density-functional theory

1.2.1 Hohenberg-Kohn theorems

1.2.2 Kohn-Sham approach

1.2.3 Some approximated functionals

1.3 Range-separated hybrid approximations

**2 Basis convergence of range-separated density-functional theory **

**3 Self-consistent double-hybrid density-functional theory using the optimized- eective-potential method **

**4 Study of the short-range exchange-correlation kernel **

4.1 Introduction

4.2 Review on time-dependent density-functional theory

4.2.1 Time-dependent Schrodinger equation for many-electron systems

4.2.2 Time-dependent density-functional theory: the Kohn-Sham formalism

4.2.3 Linear-response

4.2.4 Range-separated TDDFT

4.3 Study of the short-range exchange kernel

4.3.1 Range-separated time-dependent exact-exchange method

4.3.2 Short-range exact-exchange kernel

4.4 Asymptotic expansion with respect to the range-separation parameter of the short-range exchange kernel

4.4.1 Leading-order contribution

4.4.2 Next-order contribution

4.4.3 Examples of H2 and He

4.5 Study of the exact frequency-dependent correlation kernel

4.5.1 FCI of H2 in minimal basis set

4.5.2 Calculation of the linear-response function

4.5.3 Derivation of the exact short-range correlation kernel

4.5.4 Calculations on H2 in STO-3G basis

4.6 Conclusion

Conclusion

**A Additional results for the basis-set convergence of the long-range correlation energy **

A.1 Convergence of the correlation energy including core electrons

A.2 Extrapolation scheme

**B Derivation of the exact-exchange kernel **

**Resume en francais**