Curing basis-set convergence of wave-function theory using density-functional theory: a systematically improvable approach

Curing basis-set convergence of wave-function theory using density-functional theory: a systematically improvable approach

Emmanuel Giner emmanuel.giner@lct.jussieu.fr Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France    Barthélemy Pradines Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France Institut des Sciences du Calcul et des Données, Sorbonne Université, F-75005 Paris, France    Anthony Ferté Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France    Roland Assaraf Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France    Andreas Savin Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France    Julien Toulouse Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France
August 23, 2018
Abstract

The present work proposes to use density-functional theory (DFT) to correct for the basis-set error of wave-function theory (WFT). One of the key ideas developed here is to define a range-separation parameter which automatically adapts to a given basis set. The derivation of the exact equations are based on the Levy-Lieb formulation of DFT, which helps us to define a complementary functional which corrects uniquely for the basis-set error of WFT. The coupling of DFT and WFT is done through the definition of a real-space representation of the electron-electron Coulomb operator projected in a one-particle basis set. Such an effective interaction has the particularity to coincide with the exact electron-electron interaction in the limit of a complete basis set, and to be finite at the electron-electron coalescence point when the basis set is incomplete. The non-diverging character of the effective interaction allows one to define a mapping with the long-range interaction used in the context of range-separated DFT and to design practical approximations for the unknown complementary functional. Here, a local-density approximation is proposed for both full-configuration-interaction (FCI) and selected configuration-interaction approaches. Our theory is numerically tested to compute total energies and ionization potentials for a series of atomic systems. The results clearly show that the DFT correction drastically improves the basis-set convergence of both the total energies and the energy differences. For instance, a sub kcal/mol accuracy is obtained from the aug-cc-pVTZ basis set with the method proposed here when an aug-cc-pV5Z basis set barely reaches such a level of accuracy at the near FCI level.

I Introduction

The development of accurate and systematically improvable computational methods to calculate the electronic structure of molecular systems is an important research topic in theoretical chemistry as no definitive answer has been brought to that problem. The main difficulty originates from the electron-electron interaction which induces correlation between electrons, giving rise to a complexity growing exponentially with the size of the system. In this context, the two most popular approaches used nowadays, namely wave-function theory (WFT) and density-functional theory (DFT), have different advantages and limitations due to the very different mathematical formalisms they use to describe the electronic structure.

The clear advantage of WFT relies on the fact that, in a given one-electron basis set, the target accuracy is uniquely defined by the full-configuration-interaction (FCI) limit. Therefore, there exists many ways of systematically improving the accuracy by refining the wave-function ansatz, and ultimately by enlarging the basis set. In particular, perturbation theory is a precious guide for approximating the FCI wave function and it has given birth to important theoremsGoldstone (1957); Lindgren (1985) and many robust methods, such as coupled clusterBartlett and Musiał (2007) or selected configuration interaction (CI)Bender and Davidson (1969); Huron, Malrieu, and Rancurel (1973); Buenker and Peyerimholf (1974); Buenker, Peyerimholf, and Bruna (1981); Evangelisti, Daudey, and Malrieu (1983); Harrison (1991); Holmes, Tubman, and Umrigar (2016). Despite these appealing features, the main disadvantages of WFT are certainly the slow convergence of many important physical properties with respect to the size of the one-particle basis set and the rapidly growing computational cost when one enlarges the basis set. Such behavior very often prohibits the reach of the so-called complete-basis-set limit which is often needed to obtain quantitative agreement with experiment. At the heart of the problem of slow convergence with respect to the size of the basis set lies the description of correlation effects when electrons are close, the so-called short-range correlation effects near the electron-electron cuspKato (1957). To cure this problem, explicitly correlated () methods have emerged from the pioneering work of HylleraasHylleraas (1929) and remain an active and promising field of research (for recent reviews, see Refs. Hättig et al., 2012; Kong, Bischoff, and Valeev, 2012; Grüneis et al., 2017). One possible drawback of the methods is the use of a rather complex mathematical machinery together with numerically expensive quantities involving more than two-electron integrals.

An alternative formulation of the quantum many-body problem is given by DFT which, thanks to the Hohenberg-Kohn theoremsHohenberg and Kohn (1964), abandons the complex many-body wave function for the simple one-body density. Thanks to the so-called Kohn-Sham formalism of DFTKohn and Sham (1965) and the development of practical approximations of the exchange-correlation density functional, DFT is nowadays the most used computational tool for the study of the molecular electronic problem. Despite its tremendous success in many areas of chemistry, Kohn-Sham DFT applied with usual semilocal density functional approximations generally fails to describe nonlocal correlation effects, such as strong correlation or dispersion forces. To overcome these problems ingredients from WFT have been introduced in DFT, starting from Hartree-Fock (HF) exchangeBecke (1993) to many-body perturbation theoryGoerigk and Grimme (2014). Nevertheless, the lack of a scheme to rationally and systematically improve the quality of approximate density functionalsJones and Gunnarsson (1989) remains a major limitation of DFT.

A more general formulation of DFT has emerged with the introduction of the so-called range-separated DFT (RS-DFT) (see Ref. Toulouse, Colonna, and Savin, 2004 and references therein) which rigorously combines WFT and DFT. In such a formalism the electron-electron interaction is split into a long-range part which is treated using WFT and a complementary short-range part treated with DFT. The formalism is exact provided that full flexibility is given to the long-range wave function and that the exact short-range density functional is known. In practice, approximations must be used for these quantities and the splitting of the interaction has some appealing features in that regard. As the long-range wave-function part only deals with a non-diverging electron-electron interaction, the problematic cusp condition is removed and the convergence with respect to the one-particle basis set is greatly improvedFranck et al. (2015). Regarding the DFT part, the approximate semilocal density functionals are better suited to describe short-range interaction effects. Therefore, a number of approximate RS-DFT schemes have been developed using either single-reference WFT approaches (such as Møller-Plesset perturbation theoryÁngyán et al. (2005), coupled clusterGoll, Werner, and Stoll (2005), random-phase approximationsToulouse et al. (2009); Janesko, Henderson, and Scuseria (2009)) or multi-reference WFT approaches (such as multi-reference CILeininger et al. (1997), multiconfiguration self-consistent fieldFromager, Toulouse, and Jensen (2007), multi-reference perturbation theoryFromager, Cimiraglia, and Jensen (2010), density-matrix renormalization groupHedegård et al. (2015)). These mixed WFT/DFT schemes have shown to be able to correctly describe a quite wide spectrum of chemical situations going from weak intermolecular interactions to strong correlation effects. Nonetheless, these methods involve a range-separation parameter, often denoted by , and there is no fully satisfying and systematic scheme to set its value, even if some interesting proposals have been madeKronik et al. (2012); Krukau et al. (2008); Henderson et al. (2009).

The main goal of the present work is to use a DFT approach to correct for the basis-set incompleteness of WFT. The key idea developed here is to make a separation of the electron-electron interaction directly based on the one-particle basis set used and to express the remaining effects as a functional of the density. In practice, we propose a fit of the projected electron-electron interaction by a long-range interaction, leading to a local range-separation parameter which automatically adapts to the basis set. This is done by comparing at coalescence a real-space representation of the Coulomb electron-electron operator projected in the basis set with the long-range interaction used in RS-DFT. Thanks to this link, the theory proposed here can benefit from pre-existing short-range density functionals developed in RS-DFT.

The present paper is composed as follows. We present the general equations related to the splitting of the electron-electron interaction in a one-particle basis set in sections II.1 and II.2. In section II.3 we point out the similarities and differences of this formalism with RS-DFT. A real-space representation of the electron-electron Coulomb operator developed in a one-particle basis set is proposed in section II.4 (with details given in Appendix A and B), which leads to the definition of a local range-separation parameter that automatically adapts to the basis set. This allows us to define in section II.5 a short-range local-density approximation (LDA) correcting FCI energies for the basis-set error. The formalism is then extended to the selected CI framework in section II.6. In section III we test our theory on a series of atomic systems by computing both total energies and energy differences. We study the basis-set convergence of the DFT-corrected FCI total energy in the case of the helium atom in section III.1. We then investigate the basis-set convergence of DFT-corrected selected CI for both total energies and ionization potentials (IPs) of the B-Ne series in section III.2. In the case of the IPs, we show that chemical accuracy is systematically reached for all atomic systems already from the aug-cc-pVTZ basis set within our approach, whereas an aug-cc-pV5Z basis set is needed to reach such an accuracy at near FCI level. In order to better understand how the DFT-based correction acts for both total energies and energy differences, a detailed study is performed in section III.2.3 for the oxygen atom and its first cation. Finally, we summarize the main results and conclude in section IV.

Ii Theory

ii.1 Finite basis-set decomposition of the universal density functional

We begin by the standard DFT formalism for expressing the exact ground-state energy:

(1)

where

(2)

is the nuclei-electron interaction energy, and is the Levy-Lieb universal density functional

(3)

where the minimization is over -electron wave functions with density equal to , and and are the kinetic-energy and Coulomb electron-electron interaction operators, respectively. The Levy-Lieb universal functional only depends on the density , meaning that, given a density , one does not in principle needs to pass through the minimization over explicit -electron wave functions to obtain the value . Provided that the search in equation (1) is done over -representable densities expanded in a complete basis set, the minimizing density will be the exact ground-state density , leading to the exact ground-state energy .

First, we consider the restriction on the densities over which we perform the minimization to those that can be represented within a one-electron basis set , which we denote by . By this we mean all the densities that can be obtained from any wave function expanded into -electron Slater determinants constructed from orbitals expanded on the basis . Note that this is a sufficient but not necessary condition for characterizing these densities, as these densities can in general also be obtained from wave functions not restricted to the basis set. Therefore, the restriction on densities representable by a basis is much weaker than the restriction on wave functions representable by the same basis . With this restriction, there is a density, referred to as , which minimizes the energy functional of Eq. (1) and give a ground-state energy :

(4)

Therefore, provided only that the exact ground-state density is well approximated by this density ,

(5)

the exact ground-state energy will be well approximated by ,

(6)

Considering the fast convergence of the density with the size of the basis set, we expect the approximation of equation (6) to be very good in practice for the basis sets commonly used.

Next, we consider the following decomposition of the Levy-Lieb density functional for a given density :

(7)

where are wave functions restricted to the -electron Hilbert space generated by the basis , and is a complementary density functional

(8)

It should be pointed out that, in contrast with the density functionals used in DFT or RS-DFT, the complementary functional is not universal as it depends on the basis set used to describe a specific system. As the restriction to the basis set is in general much more stringent for the -electron wave functions than for the densities , we expect that the complementary functional gives a substantial contribution, even for basis sets for which the approximation of equation (5) is good.

By using such a decomposition in equation (4), we obtain now

(9)

or, after recombining the two minimizations,

(10)

where is the density of . By writing the Euler-Lagrange equation associated to the minimization in equation (10), we find that the minimizing wave function satisfies the Schrödinger-like equation

(11)

where , , , and are the restrictions to the space generated by the basis of the operators , , , and , respectively, and is the density operator. The potential ensures that the minimizing wave function gives the minimizing density in equation (4). It is important to notice that the accuracy of the obtained energy depends only on how close the density of is from the exact density: .

ii.2 Approximation of the FCI density in a finite basis set

In the limit where is a complete basis set, equation (10) gives the exact energy and . When the basis set is not complete but sufficiently good, can be considered as a small perturbation. Minimizing in equation (10) without simply gives the FCI energy in a given basis set

(12)

where we have introduced the ground-state FCI wave function which satisfies the eigenvalue equation:

(13)

Note that the FCI energy is an upper bound of in equation (10) since . By neglecting the impact of on the minimizing density , we propose a zeroth-order approximation for the density

(14)

which leads to a first-order-like approximation for the energy

(15)

The term constitutes a simple DFT correction to the FCI energy which should compensate for the incompleteness of the basis set . The next sections are devoted to the analysis of the properties of and to some practical approximations for this functional.

ii.3 Qualitative considerations for the complementary functional

The definition of [see equation (8)] is clear but deriving an approximation for such a functional is not straightforward. For example, defining an LDA-like approximation is not easy as the wave functions used in the definition of are not able to reproduce a uniform density if the basis set is not translationally invariant. Nonetheless, it is known that a finite one-electron basis set usually describes poorly the short-range correlation effects and therefore the functional must recover these effects. Therefore, a natural idea is to find a mapping between this functional with the short-range functionals used in RS-DFT. Among these functionals, the multi-determinant short-range correlation functional of Toulouse et al.Toulouse, Gori-Giorgi, and Savin (2005) has a definition very similar to the one of :

(16)

where the wave function is defined by the constrained minimization

(17)

where is the long-range electron-electron interaction operator

(18)

with

(19)

and the pair-density operator . By comparing equation (16) to the definition of in equation (8), one can see that the only difference between these two functionals relies in the wave functions used for the constrained minimization: in one uses whereas is used in . More specifically, is determined by using a non-diverging long-range electron-electron interaction defined in a complete basis set [equation (18)], whereas the diverging Coulomb electron-electron interaction expanded in a finite basis set is involved in the definition of . Therefore, as these two wave functions qualitatively represent the same type of physics, a possible way to link and is to try to map the projection of the diverging Coulomb interaction on a finite basis set to a non-diverging long-range effective interaction.

ii.4 Effective Coulomb electron-electron interaction in a finite basis set

This section introduces a real-space representation of the Coulomb electron-electron operator projected in a basis set , which is needed to derive approximations for .

ii.4.1 Expectation values over the Coulomb electron-electron operator

The Coulomb electron-electron operator restricted to a basis set is most naturally written in orbital-space second quantization:

(20)

where the sums run over all (real-valued) orthonormal spin-orbitals in the basis set , and are the two-electron integrals. By expanding the creation and annihilation operators in terms of real-space creation and annihilation field operators, the expectation value of over a wave function can be written as (see Appendix A for a detailed derivation):

(21)

where we introduced the function

(22)

and is the two-body density matrix of

(23)

and collects the space and spin variables.

(24)

From the properties of the restriction of an operator to the space generated by the basis set , we have the following equality

(25)

which translates into

(26)

where is the pair density of . Therefore, by introducing the following function

(27)

one can rewrite equation (26) as

(28)

One can thus identify as an effective interaction, coming from the restriction to the basis set . This can be seen as a generalization of the exchange potential of SlaterSlater (1951). It is important to notice that all the quantities appearing in the integrals of equation (28) can be considered as functions and not operators or distributions, and therefore they can be compared pointwise. Of course, the function is not defined when vanishes, but we leave this for a future study.

Equation (28) means that the two integrands have the same integral, but it does not mean that they are equals at each point . Of course, one could argue that there exists an infinite number of functions of satisfying

(29)

which implies that the effective interaction is not uniquely defined, and that the choice of equation (27) is just one among many and might not be optimal. For instance, the definition of the effective electron-electron interaction of equation (27) implies that it can depend on the spin of the electrons, whereas the exact Coulomb electron-electron interaction does not. Nevertheless, one can show (see Appendix B) that, in the limit of a complete basis set (written as “”), correctly tends to the exact Coulomb interaction:

(30)

In particular, in this limit, does not depend on or on the spins of the electrons.

Figure 1: Effective electron-electron interaction in the helium atom for different cc-pVZ basis sets () as a function of . The two upper curves are for a reference point at the helium nucleus and moving along the diagonal of the plane, and the two lower curves are for a reference point at bohr from the helium nucleus and moving along the diagonal of the plane with bohr. Two types of wave functions have been used: HF and FCI in the corresponding basis set. The exact Coulomb interaction is also reported for comparison.

ii.4.2 Effective electron-electron interaction for opposite spins and its properties

The fact that tends to the exact Coulomb electron-electron interaction in the complete-basis-set limit supports the choice of this effective interaction. Nevertheless, it is also important to analyze a few properties of in the finite basis sets used in actual quantum chemistry calculations, and to understand how it differs from the true interaction.

We will consider the effective electron-electron interaction between electrons of opposite spins ( and )

(31)

since the interaction between same-spin electrons is normally not the limiting factor for basis convergence. The first thing to notice is that, because in practice is composed of atom-centered basis functions, the effective interaction is not translationally invariant nor isotropic, which means that its does not depend only on the variable

(32)

Thus, the quality of the representation of the Coulomb electron-electron operator (and therefore of the electron correlation effects) are not expected to be spatially uniform. Nevertheless, is symmetric in and :

(33)

A simple but interesting quantity is the value of the effective interaction at coalescence at a given point in space

(34)

In a finite basis set, is finite as it is obtained from a finite sum of bounded quantities [see equation (22)]. Therefore, provided that the on-top pair density does not vanish, , is necessarily finite in a finite basis set:

(35)

As mentioned above, since the effective interaction is not translationally invariant, the value has no reason to be independent of .

ii.4.3 Illustrative examples of on the helium atom

In order to investigate how behaves as a function of the basis set, the wave function , and the spatial variables , we performed calculations using Dunning basis sets of increasing sizes (from aug-cc-pVDZ to aug-cc-pV5Z) using a HF or a FCI wave function for and different reference points . We report these numerical results in figure 1.

From figure 1, several trends can be observed. First, for all wave functions and reference points used here, the value of at coalescence is finite, which numerically illustrates equation (35). Second, the value at coalescence increases with the cardinal of the basis set, suggesting that the description of the short-range part of the interaction is improved by enlarging the basis set. Third, the global shape of the is qualitatively modified by changing the reference point , which illustrates the lack of transitional invariance of . In particular, the values of and at coalescence are much larger when the reference point is on the He nucleus, which is a signature that the atom-centered basis set does not uniformly describe the Coulomb interaction at all points in space. Fourth, the difference between the and is almost unnoticeable for all basis sets and for the two reference points used here.

Figure 2: Effective electron-electron interaction and long-range electron-electron interaction for different cc-pVZ basis sets () as a function of . The two upper curves are for a reference point at the helium nucleus and moving along the diagonal of the plane, and the two lower curves are for a reference point at bohr from the helium nucleus and moving along the diagonal of the plane with bohr. Two types of wave functions have been used: HF and FCI in the corresponding basis set.

ii.4.4 Link with range-separated DFT: Introduction of a local range-separated parameter

From the numerical illustration of the properties of given in section II.4.3, it appears that the development of approximations for the density functional seems rather complicated since the effective interaction is system- and basis-dependent, non translationally invariant, and non isotropic. Nevertheless, as it was numerically illustrated, the effective interaction typically describes a long-range interaction which is finite at coalescence. Therefore, a possible way to approximate is to locally fit by the long-range interaction of equation (19) used in RS-DFT. To do so, we propose here to determine a local value of the range-separation parameter such that the value of the long-range interaction at coalescence is identical to the value of the effective interaction at coalescence at point . More specifically, the range-separation parameter is thus determined for each and by the condition:

(36)

with given by equations (34) which, since , simply gives

(37)

Therefore, defining the function as

(38)

we make the following approximation:

(39)

One can notice that the definition of in equation (37) depends on the choice of , and therefore the approximation of equation (39) depends also on . Nevertheless, in the limit of a complete basis set the dependence on vanishes.

Figure 3: Local range-separated parameter for the helium atom for different aug-cc-pVZ basis sets () as a function of the position along the diagonal of the plane. The curve on the left is when using the HF wave function for , and the curve on the right is when using the FCI wave function for .

In order to illustrate how compares to , we report in figure 2 these two functions for several basis sets, for different reference points , and for two different wave functions . From these plots it appears that the approximation of equation (39) is reasonably accurate when the reference point is on the helium nucleus and becomes even more accurate when the reference point is farther away from the helium nucleus.

In figure 3, we report the local range-separation parameter , as determined by equation (37), for different basis sets and when is the HF or FCI wave function. It clearly appears that the magnitude of increases when the size of the basis set increases, which translates the fact that the electron-electron interaction is better described by enlarging the basis set. Also, for all basis sets, the maximal value of is reached when is at the nucleus, which demonstrates the non-homogeneity of the description of the electron-electron interaction with atom-centered basis functions. Finally, one can notice that the values of are very similar when using the HF or FCI wave function for , but nevertheless slightly larger for the FCI wave function which reflects the fact that the corresponding effective interaction is slightly stronger.

ii.5 Practical approximations for the complementary functional : a short-range LDA-like functional with a local

A proper way to define an LDA-like approximation for the complementary density functional would be to perform a uniform-electron gas calculation with the function as the electron-electron interaction. However, such a task would be rather difficult and ambiguous as is not translationally invariant nor isotropic, which thus questions how a uniform density could be obtained from such an interaction. Instead, by making the approximation of equation (39), one can define for each point an effective interaction which only depends on . For a given point in space one can therefore use the multi-determinant short-range correlation density functional of equation (16) with the range-separation parameter value corresponding to a local effective interaction at [see equation (37)]. Therefore, we define an LDA-like functional for as

(40)

where is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref. Paziani et al., 2006. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case. One can interpret equation (40) as follows: the total correction to the energy in a given basis set is approximated by the sum of local LDA corrections obtained, at each point, from an uniform electron gas with a specific electron-electron interaction which approximatively coincides with the local effective interaction obtained in the basis set. Within the LDA approximation, the final working equation for our basis-correction scheme is thus

(41)

We will refer to this approach as FCI+LDA where indicates the wave function used to define the effective interaction within the basis set employed in the calculation.

ii.6 Basis-set-corrected CIPSI: the CIPSI+LDA approach

Equation (41) requires the calculation of the FCI energy and density whose computational cost can be very rapidly prohibitive. In order to remove this bottleneck, we propose here a similar approximation to correct the so-called CIPSI energy which can be used to approximate the FCI energy in systems where the latter is out of reach.

ii.6.1 The CIPSI algorithm in a nutshell

The CIPSI algorithm approximates the FCI wave function through an iterative selected CI procedure, and the FCI energy through a second-order multi-reference perturbation theory. The CIPSI algorithm belongs to the general class of methods build upon selected CIBender and Davidson (1969); Huron, Malrieu, and Rancurel (1973); Buenker and Peyerimholf (1974); Buenker, Peyerimholf, and Bruna (1981); Evangelisti, Daudey, and Malrieu (1983); Harrison (1991); Holmes, Tubman, and Umrigar (2016) which have been successfully used to converge to FCI correlation energies, one-body properties, and nodal surfaces.Evangelisti, Daudey, and Malrieu (1983); Rubio, Novoa, and Illas (1986); Cimiraglia and Persico (1987); Angeli and Persico (1997); Angeli, Cimiraglia, and Malrieu (2000); Giner, Scemama, and Caffarel (2013); Scemama et al. (2014); Giner, Scemama, and Caffarel (2015); Giner, Assaraf, and Toulouse (2016) The CIPSI algorithm used in this work uses iteratively enlarged selected CI spaces and Epstein–NesbetEpstein (1926); Nesbet (1955) multi-reference perturbation theory. The CIPSI energy is

(42)

where is the variational energy

(43)

where the reference wave function is expanded in Slater determinants I within the CI reference space , and is the second-order energy correction

(44)

where denotes a determinant outside . To reduce the cost of the evaluation of the second-order energy correction, the semi-stochastic multi-reference approach of Garniron et al. Garniron et al. (2017) was used, adopting the technical specifications recommended in that work. The CIPSI energy is systematically refined by doubling the size of the CI reference space at each iteration, selecting the determinants with the largest . The calculations are stopped when a target value of is reached.

ii.6.2 Working equations for the CIPSI+LDA approach

The CIPSI algorithm being an approximation to FCI, one can straightforwardly apply the DFT correction developed in this work to correct the CIPSI energy error due to the basis set. For a given basis set and a given reference wave function , one can estimate the FCI energy and density by the following approximations:

(45)
(46)

with

(47)

Assuming these approximations, for a given choice of to define the effective interaction and within the LDA approximation of equation (40), one can define the corrected CIPSI energy as

(48)

Note that the reference wave function can be used for the definition of the effective interaction through its two-body density matrix [see equation (22)], but we leave that for further investigation and for the rest of the calculations we use the HF wave function for in the definition of the effective interaction, and we denote the method by CIPSI+LDA.

Figure 4: Convergence of the total energy of the helium atom for FCI and FCI+LDA, where is either the HF or FCI wave function, as a function of the inverse of the cardinal number of the AVZ basis sets (). The exact non-relativistic (NR) energy is also reported.
FCI FCI+LDA FCI+LDA
Total energy Error Total energy Error Total energy Error
AV2Z
AV3Z
AV4Z
AV5Z
AV6Z
Exact non-relativistic total energy
-2.90372
Table 1: Total energies (in Hartree) of the helium atom and errors (in mH) with respect to the exact non-relativistic energy for FCI, FCI+LDA, and FCI+LDA with the AVZ basis sets ().

Iii Numerical tests: Total energy of He and ionization potentials for the B-Ne atomic series

For the present study, we use the LDA approximation of equation (40) and investigate the convergence of the total energies and energy differences as a function of the basis set. All calculations were performed with Quantum PackageScemama et al. (2016) using the Dunning aug-cc-pVZ basis sets which are referred here as AVZ.

iii.1 FCI+DFT: Total energy of the helium atom

We report in figure 4 and table 1 the convergence of the total energies computed for the helium atom in the Dunning basis sets AVZ () using FCI and FCI+LDA where is either the HF or FCI wave function. The first striking observation from these data is that the FCI+LDA energies rapidly converge to the exact energy as one increases the size of the basis set and that FCI+LDA is systematically closer to the exact energy than the FCI energy. Also, one can observe that overestimates the correlation energy (in absolute value) for the AV3Z basis and the larger ones, which is consistent with the fact that LDA is known to give too negative correlation energies in regular Kohn-Sham DFT or in RS-DFT. Interestingly, is almost independent of the choice of the wave function used for the definition of the effective interaction within , as the FCI+LDA and FCI+LDA energies are overall very close and get closer as one increases the size of the basis set. This last point is the numerical illustration that, in the limit of a complete basis set, the effective interaction is independent of the wave function [see equation (30)]. Nevertheless, one observes that the correction obtained using the FCI wave function for is systematically smaller in absolute value than the one obtained with the HF wave function for . This result can be qualitatively understood by noticing that the introduction of the HF two-body density matrix in equation (22) reduces the number of two-electron integrals involved in the definition of [see equation (27)]. This reduction implies that the effective interaction misses a part of the interaction within the basis set, namely the repulsion between electrons in virtual orbitals. However, the fact that and are close suggests that misses only a small part of the interaction. This statement can be intuitively understood by noticing that some two-electron integrals involved in the definition of are of the type (where and run over the occupied and virtual orbitals, respectively) which are the ones giving rise to the dominant part of the MP2 correlation energy in a given basis set.

iii.2 CIPSI+LDA: Total energies and energy differences for atomic systems

iii.2.1 Convergence of the CIPSI+LDA total energy with the number of determinants

We report in figure 5, in the case of the oxygen ground state using the AV4Z basis set, the convergence of the variational energy , the CIPSI energy, the CIPSI+LDA energy, and the LDA correction as a function of the number of Slater determinants in the reference wave function. The behavior of and reported in figure 5 are typical of a CIPSI calculation: a rapid convergence of the variational energy and an even faster convergence of the CIPSI energy. In this case, with a reference wave function including and determinants provides an estimation of the FCI energy with an error smaller than 1 mH and 0.1 mH, respectively, whereas the size of the FCI space of this system for this basis set is approximatively of determinants. Regarding