Emmanuel Giner

Anthony Scemama

Michel Caffarel

Using perturbatively selected configuration interaction in quantum Monte Carlo calculations

Keywords: Perturbatively Selected Configuration Interaction, quantum Monte Carlo, Fixed-Node Diffusion Monte Carlo

Defining accurate and compact trial wavefunctions leading to small statistical and fixed-node errors in quantum Monte Carlo (QMC) calculations is still a challenging problem. Here, we propose to make use of selected configuration interaction (CI) expansions obtained by selecting the most important determinants through a perturbative criterion. A major advantage with respect to truncated CASSCF wavefunctions or CI expansions limited to a maximum number of excitations (e.g, CISD) is that much smaller expansions can be considered (many unessential determinants are avoided), an important practical point for efficient QMC calculations. The most important determinants entering first during the selection process (hierarchical construction) the main features of the nodal structure of the wavefunction can be expected to be obtained with a moderate number of determinants. Thanks to this property, the delicate problem of optimizing in a Monte Carlo framework the numerous linear/nonlinear parameters of the determinantal part of the trial wavefunction could be avoided. As a first numerical example, the calculation of the ground-state energy of the oxygen atom is presented. The best DMC value reported so far is obtained.

I. INTRODUCTION

Quantum Monte Carlo (QMC) methods are known to be powerful stochastic approaches for obtaining accurate ground-state properties of quantum systems. For bosonic systems the results obtained are essentially exact, up to some statistical error as in any Monte Carlo approach (see, for example the reference simulations for the He 4 quantum liquid [1]). In contrast, for Fermi systems where the antisymmetry of the wavefunction is to be imposed, the situation is different. As known, it has not been possible so far to devise a QMC algorithm for fermions that would be both stable (controlled fluctuations of the wavefunction sign) and exact (no systematic error in the limit of infinite simulation time). In practice, this difficulty known as the "fermionic sign problem" is circumvented by using the so-called "Fixed-Node"(FN) approach where the sign instability is removed at the expense of a small systematic (fixed-node) error. In short, the FN approach consists in defining a juxtaposition of bosonic-type simulations defined independently within each nodal pocket (domains of constant sign) of an approximate Fermi (antisymmetric) trial wavefunction, Ψ T . When the nodes (or zeroes) of ψ T coincide with the exact nodes, the algorithm is exact. If not -which is the general case except for some elementary systems-a fixed-node error is introduced. [

2]

Using standard trial wavefunctions, this error is in general small, [3] typically a few percent of the correlation energy for total energies. However, this remarkable accuracy on total energies can still be insufficient when calculating the energy differences at the heart of quantitative chemistry:[4] Binding energies, energy variations along a reaction path, forces (viewed as infinitesimal differences), electronic affinities, electronic transition energies, etc. Indeed, on the total energy scale these differences are extremely small, at least of the same order of magnitude, and most often smaller, than the already very small fixed-node error. The precision on energy differences is thus directly related to the quality of the nodal hypersurfaces of the trial wavefunction and on the way the fixed-node error compensates or not when computing differences of large total energies. [5], [START_REF] Toulouse | Advances in the Theory of Quantum Systems in Chemistry and Physics[END_REF] For this reason, defining trial wavefunctions with accurate nodes is still one of the important issues of quantum Monte Carlo (QMC) approaches for chemistry.

Besides this important aspect, the trial wavefunction is also directly related to the efficiency of QMC simulations (defined here as a good convergence of the estimators as a function of the simulation time and a low level of statistical fluctuations). The QMC meth-ods being highly CPU-intensive ("number crunching" approaches) this aspect is of particular importance. In the important case of the energy, the crucial role of the trial wavefunction is directly seen from the fact that the closest from the exact wavefunction the trial wavefunction is, the smallest the statistical error on the total energy is (zero-variance property, see e.g. [3]).

In view of the importance of the trial wavefunction, much effort has been made to propose and optimize functional forms that lead to accurate nodes and low level of fluctuations.

In addition, the trial wavefunction must be compact enough to be rapidly evaluated at each of the millions and more Monte Carlo steps. The standard expression employed in QMC is the Jastrow-Slater form expressed as a short expansion over a set of Slater determinants multiplied by a global Jastrow factor describing explicitly the electron-electron and electron-electron-nucleus interactions and, in particular, imposing the electron-electron CUSP conditions associated with the zero-interelectronic distance limit of the true wavefunction. A variety of alternative forms aiming at better describing the exact wavefunction have been introduced. Without entering into the details, let us cite the geminal wavefunction of Sorella and coworkers, [7] the Pfaffian wavefunction of Mitáš and collaborators, [8], the backflow trial wavefunction of Rios et al. [9], the generalized valence bond (GVB) of Anderson and Goddard [10], the linear scaling GVB version of Fracchia et al. [11], the Jastrowvalence-bond wavefunction of Braida et al. [12], or the multi-Jastrow trial wavefunction of Bouabça et al. [13] Once a trial wavefunction has been chosen, its various parameters (Jastrow parameters, determinantal coefficients, molecular orbital coefficients, basis set exponents, and so on) are in general optimized. The criterion employed can be either the minimization of the variational energy associated with the trial wavefunction, Ψ T |H|Ψ T , its

variance Ψ T |H 2 |Ψ T -Ψ T |H|Ψ T 2
, or a combination of both. Note that optimizing hundreds/thousands of linear and nonlinear parameters within a framework where the energy and variance are subject to a statistical noise is not an easy task. A number of solutions have been proposed, let us just mention here the recent approach of Umrigar and collaborators [14] and references therein.

In this work we propose to exploit the multi-determinant expansions of the post-HF approaches of quantum chemistry. So far, multi-determinant expansions in QMC have been mainly limited to the use of CASSCF-like or CI-like wavefunctions truncated at a moderate number of determinants or configurations (say, a maximum of a few thousands) selected by using some threshold in the expansion coefficients (see, e.g.

[15], [START_REF] Flad | Quantum Monte Carlo calculations with multi-reference trial wave functions Recent Advances in Quantum Monte Carlo Methods[END_REF], [START_REF] Caffarel | [END_REF], [18], [19], [20], [11], [21]). To the best of our knowledge, the only DMC calculation using a complete full CI wavefunction (expansion over all possible determinants in a given set of molecular orbitals) has been made in Ref. [START_REF] Monari | In Remote Instrumentation for eScience and Related Aspects[END_REF] for the Li 2 molecule. However, this calculation involving about 16 000 determinants has been possible only because the number of active electrons for such a molecule is very small. As well-known, due to the exponential increase of the dimension of the FCI space, to consider much larger systems is not realistic.

Here, we propose to use trial wavefunctions based on truncated expansions containing the most "important" determinants while remaining compact enough to be evaluated at each Monte Carlo step (say, expansions involving at most a few hundreds of thousands of determinants). In quantum chemistry there is a long history of developing approximate schemes based on various approximations: Configuration Interaction expansion truncated to a certain level of excitation (single: CIS, double: CISD, etc.), exponential ansatz: CCSD, CCSD(T) etc., perturbative approaches: Møller-Plesset MP2, etc., CASSCF approaches, and so on.

However, such schemes still generate too many determinants to be tractable in QMC and, furthermore, the choice of the criterion employed for truncation may be questionable. Here, we shall follow a different route introduced in the last decades by a number of authors (see, e.g., [23][24][25][26][27][28][START_REF] Buenker | In Computational Theoretical Organic Chemistry[END_REF][START_REF] Evangelisti | [END_REF][31]). In a few words, the approach consists in building the multi-determinantal expansion iteratively by selecting at each step one determinant (or a group of determinants) according to a perturbative criterion. In short, a determinant D i (or a group of determinants) is added to the current wavefunction if its (their) energetic contribution(s) is (are) sufficiently large. In this work the formalism employed is close to the one adopted in the CIPSI algorithm. [24,[START_REF] Evangelisti | [END_REF] Finally, let us note that some time ago Koch and Dalgaard [32] proposed a method having some similar aspects (add one determinant at a time and try to keep the multi-determinant expansion as compact as possible) but resting on a completely different approach for building the expansion.

The contents of this paper is as follows. In Section II, the algorithm used for building the selected configuration interaction expansion (CIPSI-like algorithm) is presented. In Section III a few words about the Fixed-Node Diffusion Monte Carlo method employed here are given. In Section IV the first application to the calculation of the 3 P ground-state energy of the oxygen atom is presented. The various aspects related to the implementation of the CIPSI algorithm and the fixed-node calculations are described. Finally, the main ideas and

|Ψ 0 = i c i |D i (1)
where c i are the ground-state coefficients obtained by diagonalizing the matrix,

H ij = D i |H|D j , within the full orthonormalized set, D i |D j = δ ij , of determi- nants |D i .
In its simplest form, our multi-determinant wavefunction is iteratively built as follows:

• Step 0: Start from a given determinant (e.g. the Hartree-Fock determinant) or set of determinants, thus defining an initial reference subspace: S 0 = {|D 0 , ...}. Diagonalize H within S 0 and get the ground-state energy E (0) 0 and eigenvector:

|Ψ (0) 0 = i∈S 0 c (0) i |D i (2)
Here and in what follows, a superscript on various quantities is used to indicate the iteration number.

Then, do iteratively (n = 0, ...):

• Step 1: Collect all different determinants |D ic connected by H to |Ψ (n) 0 , namely Ψ (n) 0 |H|D ic = 0 (3)
•

Step 2: Compute the second-order perturbative energetic change of the total energy resulting from each connected determinant:

δe(|D ic ) = - Ψ (n) 0 |H|D ic 2 D ic |H|D ic -E (n) 0 (4)
• Step 3: Add the determinant |D i * c associated with the largest |δe| to the reference subspace:

S n → S n+1 = S n ∪ {|D i * c } •
Step 4: Diagonalize H within S n+1 to get:

|Ψ (n+1) 0 = i∈S n+1 c (n+1) i |D i with E (n+1) 0 (5)
• Go to step 1 or stop if the target size for the reference subspace has been reached.

Denoting N dets the final number of determinants in |Ψ 0 (N dets ) and E 0 (N dets ) the final energy, a perturbative second-order estimate of the exact energy -the so-called CIPSI energycan be obtained as

E 0 (CIPSI) = E 0 (N dets ) - i∈M Ψ 0 (N dets )|H|D i 2 D i |H|D i -E 0 (N dets ) (6)
where M denotes the set of all determinants not belonging to the reference space and connected to the reference vector |Ψ 0 (N dets ) by H (single and double excitations). In what follows the wave function |Ψ 0 (N dets ) that will be used for QMC calcaultations will be referred to as the reference wave function.

At this point a number of remarks are in order:

i.) Although the selection scheme is presented here for computing the ground-state eigenvector only, no special difficulties arise when generalizing the scheme to a finite number of states (see, e.g. [START_REF] Evangelisti | [END_REF])

ii.) The decomposition of the Hamiltonian H underlying the perturbative second-order expression introduced in step 2 is given by

H = H 0 + D ic |H|D ic |D ic D ic |
where H 0 is the restriction of H to the reference subspace. This decomposition known as the Epstein-Nesbet partition [33,34] is not unique, other possible choices are the Møller-Plesset partition [35] or the barycentric one, [24] see discussion in [START_REF] Evangelisti | [END_REF].

iii.) Instead of calculating the energetic change perturbatively, expression (4), it can be preferable to employ the non-perturbative expression resulting from the diagonalization of H into the two-dimensional basis consisting of the vectors |Ψ

δe(|D ic ) = [ D ic |H|D ic -E 0 (N dets )] 1 -1 + 4 Ψ (n) 0 |H|D ic 2 [ D ic |H|D ic -E 0 (N dets )] 2 2 (7)
In the limit of small transition matrix elements, Ψ

0 |H|D ic , both expressions (4) and ( 7) coincide. In what follows the non-perturbative formula will be used. iv.) In step 3 a unique determinant is added at each iteration. Adding a few of them simultaneously is also possible, a feature particularly desirable when quasi-degenerate low-lying determinants are showing up. In the applications to follow this possibility has been systematically used by keeping at each iteration all determinants associated with an energetic change whose absolute value is greater than a given threshold. v.) The implementation of this algorithm can be performed using limited amount of central memory. On the other hand, the CPU time required is essentially proportional to N dets N 2 occ N 2 virt where N occ is the number of occupied molecular orbitals and N virt the number of virtual orbitals.

III. THE FIXED-NODE DIFFUSION MONTE CARLO

In this work the Fixed-Node Diffusion Monte Carlo (FN-DMC) method -the standard quantum Monte Carlo electronic-structure approach for molecules-is employed. For a detailed presentation of its theoretical and practical aspects, the reader is referred to the literature, e.g [START_REF] Hammond | Monte Carlo Methods in Ab Initio Quantum Chemistry[END_REF][START_REF] Towler | Quantum Monte Carlo, or, how to solve the many-particle Schrdinger equation accurately whilst retaining favourable scaling with system size[END_REF][START_REF] Caffarel | Quantum Monte Carlo Methods in Chemistry[END_REF]. Here, we just recall that the central quantity of such approaches is the trial wavefunction Ψ T determining both the quality of the statistical convergence (good trial wavefunctions = small statistical fluctuations) and the magnitude of the fixed-node bias resulting from the approximate nodes of the trial wavefunction. The computational cost of FN-DMC is almost entirely determined by the evaluation at each Monte Carlo step of the value of Ψ T and its first (drift vector) and second derivatives (Laplacian needed for the local energy). In view of the very large number of MC steps usually required (typically at least millions and often much more) to be able of computing such quantities very rapidly is essential. In contrast with most implementations of FN-DMC where compact forms for Ψ T are used (typically, at most a few hundreds of determinants) quite lengthy multi-determinantal expansions are considered here (up to 200 000 determinants in the numerical application presented below). As a consequence, some care is required when computing such expansions.

At first glance, the CPU cost is expected to be proportional to the number of determinants N dets involved in the expansion of the trial wavefunction. Actually, it is not true since in the spin-free formalism used in QMC (Ref. [START_REF] Matsen | [END_REF] and also [START_REF] Hammond | Monte Carlo Methods in Ab Initio Quantum Chemistry[END_REF][START_REF] Towler | Quantum Monte Carlo, or, how to solve the many-particle Schrdinger equation accurately whilst retaining favourable scaling with system size[END_REF][START_REF] Caffarel | Quantum Monte Carlo Methods in Chemistry[END_REF]) each Slater determinant expressed in terms of spin-orbitals decomposes into a product of two determinants, each of them corresponding to a given occupation of a set of purely spatial molecular orbitals.

The number of different determinants to be computed is thus of the order of √ N dets and not N dets . Another point having a significant impact on the computational cost is the order with which determinants are calculated. Indeed, avoiding to re-compute successive determinants differing from preceding ones only by a single or double molecular orbital substitution can be efficient. In practice, this is done by re-actualizing the determinants using a Sherman-Morisson-type formula. [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF] Now, to determine which ordering is the most effective is a difficult problem of combinatorial complexity. Here, we employ a simple strategy based on a preliminary preconditioning step. More involved strategies for treating bigger systems and larger expansions will be presented elsewhere.

IV. NUMERICAL RESULTS

In this section CIPSI and FN-DMC calculations of the 3 P ground-state energy of the oxygen atom are presented. Among the first-row atoms O is known to be the atom whose nodes are the most difficult to describe (fixed-node error of about 6% of the correlation energy when using Hartree-Fock nodes [START_REF] Brown | [END_REF]).

A. CIPSI wavefunctions

Selected configuration interation wave functions are constructed using Hartree-Fock canonical atomic orbitals obtained with various Dunning basis sets, [42] cc-pVXZ (denoted as VXZ), aug-cc-pVXZ (denoted as AVXZ), and cc-pCVXZ basis sets (denoted as CVXZ) with X=D,V,T,Q,5. All-electron calculations are performed and the full set of atomic orbitals are active in the excitation process, except when freezing the 1s innermost atomic orbital as indicated (FC for Frozen Core).

In figure 1 the convergence of the variational E 0 (N dets ) and CIPSI energies (variational + PT2 correction) for the VDZ, VTZ, VQZ, and V5Z basis sets as a function of the number of determinants kept in the selection process is shown. The exact non-relativistic ground-state energy -75.0674 evaluated by Chakravorty et al. [43] is also indicated. As seen a striking feature is the extremely rapid convergence of total energies. To better visualize this convergence, particularly at small number of determinants, the abscissa has been scaled logarithmically.

As it should be the variational energy associated with the reference wavefunction converges to the asymptotic limit from above. In contrast, the CIPSI energy including second-order correction is found to systematically converge from below. Remarkably, in all cases the convergence is attained for a tiny fraction of the total number of determinants of the FCI space. Quantitatively, it is interesting to determine, for each basis set, the minimum number of determinants needed to reach one percent or less of the total correlation energy computed within the basis considered. For the DZ basis set this level of accuracy is attained with about 500 determinants for E 0 (N dets ) and about 80 determinants for the CIPSI energy. For TZ, QZ, and 5Z, these numbers are (∼10 000, ∼ 40), (∼30 000, ∼ 90), and (∼50 000, ∼ 300), respectively. It is seen that the CIPSI energy convergence depends weakly on the basis set size, a remarkable result illustrating that the second-order correction is able to recover most of the missing part. In contrast, for the variational energy the minimum number of determinants needed increases as a function of the basis set size. However, this increase is moderate when compared with the exponential increase of the FCI space dimensionality.

Quantitatively, the target accuracy of less than 1% of the correlation energy is attained at the variational level for a fraction of determinants given by f = N dets N FCI ∼ 7.0 10 -4 , ∼ 2.0 10 -5 , ∼ 3.3 10 -7 , and ∼ 9.1 10 -9 for the DZ, TZ, QZ, and 5Z basis sets, respectively. However, note that in the context of building compact trial wave functions for QMC the key quantity is the total number of contributing determinants (associated with the computational cost)

and not the fraction. To better visualize how the energies converge, Figure 2 presents a blow-up of figure 1 in the region of the large number of determinants for the two biggest basis sets, QZ and 5Z. Very similar behaviors are obtained for the DZ and TZ smaller basis sets. Once more the very rapid convergence of the full energy including the second-order energy correction is clearly seen. Regarding the variational energy, the convergence is much less impressive but still very satisfactory. It is reasonable to expect the convergence of both variational and CIPSI energies to a common asymptotic value as it should be.

A key aspect of CIPSI is that the multi-determinantal expansion is built by selecting the determinants according to their contribution to the wavefunction and total energy and not according to some fixed maximum level of excitations (single, double, triple,..) as it is usual in most post-Hartree-Fock schemes. In figure 3 this aspect is illustrated by showing the evolution of the number of selected determinants corresponding to a given excitation level as a function of the total number of determinants kept during the selection process up to the first 50 000 determinants (TZ basis basis set). Starting from the unique HF determinant with no excitation (determinant #1), the first determinants to enter are of double-excitation type (from determinant #2 to determinant #19), the 20-th is a single excitation determinant, etc.

Table I reports when a determinant of a given excitation level class appears for the first time

("occurence" time) together with the total number of determinants in each class of excitation level at the end of the selection process. It is remarkable to see how far we are from a selection process where the derminants would be chosen according to their level of excitation (lower numbers of excitations first). In particular, high-level excitations enter quite soon into the wavefunction: At N = 1188 and N = 8159 for 5-uple and 6-uple excitations, respectively (here, no 7-and 8-uple excitations are observed). Single and double-excitation are found to saturate while the other types of excitations are still increasing in number. After 50 000 selected determinants, the most numerous determinants are of quadruple-and tripleexcitation types.

Table II collects the set of results obtained with various basis sets and truncation levels

(number of determinants kept in the reference wavefunction). The data are presented for the VDZ, AVDZ, VTZ, AVTZ, VQZ, AVQZ, CVQZ, and V5Z basis sets. In the case of the VDZ, VTZ, and VQZ basis set results obtained by freezing the 1s core orbital are also given. When available (VDZ and VTZ basis sets) the Full CI (FCI) energy is indicated. We also report the FCI-QMC results of Booth and Alavi [44] for the AVDZ, AVTZ, AVQZ, and AV5Z basis sets. In each case the total number of determinants of the FCI space is given.

For each total energy the percentage of the exact correlation energy is given in brackets.

As already seen qualitatively on the figures 1 and 2 the convergence of the variational and CIPSI energies to the FCI limit is very statisfactory for all basis sets. For the DZ basis set, the FCI, variational, and CIPSI energies coincide with seven digits in the VDZ-FC and VDZ cases and with about six digits for AVDZ. For the larger basis sets, the differences are always smaller than the milliHartree (chemical accuracy). Finally, it is interesting to note that using a basis set adapted to the core region (CVQZ) an energy value of -75.054 38 has been obtained here, much lower than the best value of Booth and Alavi obtained with the V5Z basis set, namely E 0 = -75.037 49 [START_REF] Toulouse | Advances in the Theory of Quantum Systems in Chemistry and Physics[END_REF]. This latter result illustrates the quantitative importance of the core contribution to the total atomic energy. However, when computing energy differences for molecular properties such a purely atomic contribution is expected to cancel almost entirely.

B. QMC calculations using selected configuration interaction wavefunctions

In figures 4 and 5 the fixed-node DMC ground-state energies for the oxygen atom as a function of the number of determinants kept for the VDZ, VTZ, and VQZ basis sets are presented. Figure 4 shows the entire curve using a logarithmic scale for the abscissa, while the region at small number of determinants is presented in figure 5 with a standard linear scale.

As seen, except in one case that shall be discussed below (VQZ at N dets = 50), the fixednode energy is found to decrease regularly with the number of determinants. Such a result shows that a simple and systematic control of the nodal quality of the trial wavefunction through determinantal selection is possible. Although such a remarkable behavior is not easy to justify mathematically, it is coherent with the fact that the selection process realizes a hierarchical build up of the main features of the wavefunction and, thus, of its nodal structure. It is also coherent with the fact that the second-order perturbative correction was able to recover most of the missing part of the FCI energy very rapidly as a function of the number of determinants kept, thus confirming that the most important features enter quickly the reference wavefunction upon selection.

In most QMC works a prelimary optimization step of the trial wavefunction is performed before running DMC calculations. It is done because the introduction of the various N-body terms aiming at better describing the physical properties of the trial wavefunction (Jastrow factor, backflow transformation, geminal functions, etc.) has a strong impact on the initial determinantal part optimized in the absence of such terms. To decrease the fixed-node error, it is thus necessary to re-optimize all parameters of the trial wavefunction including those of the determinantal part (all molecular orbitals and determinantal coefficients). In practice, it can be a particularly tedious task, although much effort has been produced to make it as automatic as possible. The fact that the objective function to be minimized (total energy, energy variance, or a combination of both) is calculated with a statistical noise and that most parameters are non-linear are the two main difficulties for the optimization.

Here, the situation is different since the optimization of the determinantal part is overcome when using perturbatively selected interaction configuration expansions. Eventually, the only optimizable parameters left are those not changing the nodes (typically the Jastrow parameters). Here, we have chosen to compute the FN-DMC energies using the pure CIPSI wave functions without Jastrow term, so that no optimization at all was necessary. Actually, the deterministic construction of a reference function through the diagonalization of a truncated Hamiltonian matrix must be considered as an optimization step (minimization of the total energy with respect to the coefficients of the multi-determinantal expansion).

However, in sharp contrast with what happens in QMC, such a step is simple and automatic (no noise and linear parameters). This aspect will be particularly interesting when more complex systems will be considered. Finally, a few words of caution are in order. As seen above, in one case (VQZ basis set and small number of determinants, see Figs. 4 and5) the fixed-node energy is found to go up instead of decreasing. This result may indicate that the behavior of the fixed-node error is not so simple. However, it is also possible to interpret it as a transient effect related to the small number of determinants and large basis set regime.

In Table III the Fixed-Node DMC energies obtained using several basis sets and various sizes of the reference wave function are presented. The correlation energy recovered in the FN-DMC calculations varies from 90% to nearly 100% depending on the nodal structure of the reference wave function. As was the case at the FCI level, to use a basis set adapted to the core region (CVQZ) is quantitatively important when highly-accurate total FN-DMC energies are searched for. This result illustrates the fact that the nodes in the nucleus region play a significant role. Our best total energy is obtained with the CVQZ basis set and 200 000 determinants. The value obtained is -75.065 8 ± 0.0001 recovering 99.4 ± 0.1% of the correlation energy. To the best of our knowledge it is the best FN-DMC value reported so far for the oxygen atom. Note that it is slighter lower than the value of -75.065 4 ± 0.0001 obtained very recently by Seth et al. [20] with a fully optimized multideterminant-Jastrow-backflow trial wave function. For comparison Table III reports also some of the most accurate energies obtained for this atom by different methods. At the FCI level the best result we know is that of Booth and Alavi. [44] At the FN-DMC level, it is that of Seth et al. just mentioned. Finally, to the best of our knowledge the best energy reported up now is the value obtained by Gdanitz using the r12-MR-ACPF method. [45] V. SUMMARY

In this work we have proposed to use trial wave functions for QMC based on selected configuration interaction (CI) expansions. The expansion is built iteratively by selecting at each step the most important determinants through a perturbative criterion. In short, a determinant D i (or a group of determinants) is added to the reference wavefunction if its (their) energetic contribution(s) is (are) sufficiently large (CIPSI-like algorithm). In practice, the expansion is stopped for the maximum number of determinants compatible with a rapid evaluation of the trial wavefunction at each step of the QMC calculation (here, a few hundreds of thousands of determinants for the oxygen atom). The main advantages of such wave functions are as follows.

i.) In constrast with truncated CASSCF wavefunctions or CI expansions limited to a maximum number of excitations (e.g, CISD), near FCI quality for the reference function can be reached with a much smaller number of determinants (many determinants with negligible weight in the exact wavefunction are avoided). For the case of the oxygen atom, the error with respect to the FCI limit for the DZ and TZ basis sets is systematically smaller than one milliHartree with a number of determinants not exceeding 10 5 . For the largest QZ and 5Z and a comparable number of determinants, the error does not exceed a few milliHartree.

ii.) Because the reference wave function is built hierarchically (most dominant determinants first), it is reasonable to expect that its overall quality improves as the number of determinants is increased and, in particular, its nodal structure. In this work, this point has been verified for the oxygen atom but its validity for more complex systems remains to be investigated.

iii.) Taking for granted the nodal quality of the multi-determinantal wavefunction when the CI expansion is sufficiently large, the tedious and delicate task of re-optimizing in a QMC framework the numerous linear and nonlinear parameters of the determinantal part can be avoided. Such a possibility of constructing optimal nodes in a purely deterministic and automatic way is a very appealing feature for future applications.

Finally, let us insist that the various aspects just discussed need to be investigated for more realistic systems beyond the oxygen atom. A study of the potential energy curve of the first-row diatomics molecules is presently underway. • Fig. 4 Fixed-node DMC ground-state energies for the oxygen atom as a function of the number of determinants kept for the VDZ, VTZ, and VQZ basis sets. A logarithmic scale for the abscissa is used.

• Fig. 5 Fixed-node DMC ground-state energies in the small number of determinants region.

  results of this work are summarized in Sec. V II. PERTURBATIVELY SELECTED CONFIGURATION INTERACTION In multi-determinantal expansions the ground-state wavefunction |Ψ 0 is written as a linear combination of Slater determinants {|D i }, each determinant corresponding to a given occupation by the N α and N β electrons of N = N α + N β orbitals among a set of M spinorbitals {φ 1 , ..., φ M } (restricted case). When no symmetries are considered the maximum number of such determinants is M Nα M N β , a number that grows factorially with M and N. The best representation of the exact wavefunction in the determinantal basis is the Full Configuration Interaction (FCI) wave function written as

  |D ic . Simple algebra shows that the energetic change is given by

FIGURE CAPTIONS • Fig. 1

 1 FIGURE CAPTIONS

FIG

  FIG. 2:

TABLE I :

 I First occurence and total number of determinants of each class of excitation during the selection process up to the first 50 000 determinants (VTZ basis set). 2%] 100 000 -74.985 19[68.1%] -9.0 10 -5 -74.985 28[68.2%] AVTZ ∼ 2.1 10 10 -74.990 77(4) a [70.3%] 100 000 -74.990 13[70.1%] -2.9 10 -4 -74.990 41[70.2%] QZ VQZ FC b ∼ 4.5 10 8 -50 000 -74.993 15[71.2%] -4.3 10 -4 -74.993 57[71.4%] VQZ ∼ 9.1 10 10 -100 000 -75.022 69[82.7%] -3.2 10 -4 -75.023 00[82.8%] AVQZ ∼ 2.0 10 12 -75.025 34(4) a [83.7%] 200 000 -75.022 69[82.7%] -2.1 10 -3 -75.024 76[83.5%] CVQZ ∼ 2.7 10 12 -250 000 -75.051 67[93.9%] -2.7 10 -3 -75.054 38[95.0%] 5Z V5Z ∼ 5.6 10 12 -200 000 -75.027 40[84.5%] -9.9 10 -4 -75.028 39[84.9%] AV5Z ∼ 8.5 10 13 -75.037 49(6) a [88.4%]

	Excitation class	First occurence	Total number of determinants
	Basis set N dets	None (HF) E 0 (FCI)	1	N dets	E 0 (N dets )	1	E PT2	E 0 (CIPSI)
	Single Double VDZ FC b ∼ 5.6 10 4 -74.910 15 DZ Triple VDZ ∼ 7.3 10 5 -74.911 75 Quadruple AVDZ ∼ 5.9 10 7 -74.927 72(2) a [45.9%] 50 000 -74.927 67[45.9%] -9.9 10 -6 -74.927 68[45.9%] 20 34 2 [39.1%] 50 000 -74.910 15[39.1%] 0.0 -74.910 15[39.1%] 2 205 194 [39.7%] 50 000 -74.911 75[39.7%] -3.5 10 -9 -74.911 75[39.7%] 16 870 413 29 618
	Quintuple Sextuple VTZ FC b ∼ 9.6 10 6 -74.974 24 TZ Septuple VTZ ∼ 5.8 10 8 -74.985 28	1188 8159 [63.9%] 50 000 -74.974 15[63.9%] -1.5 10 -5 -74.974 17[63.9%] 1 184 88 0 0 [68.
		Octuple	0			0
		Total			50 000

TABLE II :

 II Variational total energy, E 0 (N dets ), and CIPSI total energy, E 0 (CIPSI)= E 0 (N dets ) + E PT2 , for the 3 P ground-state of the oxygen atom using various basis sets and numbers of determinants in the reference wavefunction. For each energy the percentage of the exact correlation energy recovered is given in brackets. FCI-QMC calculations of Booth and Alavi[44] are given for comparison.

		E 0 (FN-DMC) Correlation energy (%)
	cc-pVDZ 1 (HF) -75.041 8(5)	90.1(2)
	cc-pVDZ 5 000	-75.051 9(4)	94.0(2)
	cc-pVTZ 1 (HF) -75.045 7(4)	91.6(2)
	cc-pVTZ 2 000	-75.059 5(4)	96.9(2)
	cc-pVQZ 20 000 -75.064 2(2)	98.8(1)
	cc-pCVQZ 100 000 -75.065 8(1)	99.4(1)
	Other works
	FCIQMC a	-75.037 49(6)	88.40(2)
	FN-DMC b	-75.065 4(1)	99.2(1)
	r12-MR-ACPF c	-75.066 960	99.83
	Exact NR d	-75.067 4	100.0

a Values from Ref.

[44]

. b FC refers to a (1s 2 ) Frozen-Core.

TABLE III :

 III FN-DMC energies and percentages of the correlation energy recovered for the 3 P ground-state of the oxygen atom using various basis sets and truncation levels for the reference wavefunction. Comparison with accurate values available in the literature.

a Ref.

[44]

b Ref.

[20]

c Ref.

[45]

d Ref.

[43]

Acknowledgments. One of us (MC) would like to dedicate this work with sincere admiration and friendship to Dennis Salahub who was very early interested in emerging quantum Monte Carlo techniques for computational chemistry. [46] We would also like to acknowledge interesting discussions with Nathalie Guihery and Jean-Paul Malrieu (Toulouse). AS and MC thank the Agence Nationale pour la Recherche (ANR) for support through Grant No ANR 2011 BS08 004 01. This work has been made through generous computational support from CALMIP (Toulouse) under the allocation 2012-0510, GENCI, and CCRT (CEA).


A Scemama

T Applencourt

E Giner

M Caffarel

Accurate nonrelativistic ground-state energies of 3d transition metal atoms

We present accurate nonrelativistic ground-state energies of the transition metal atoms of the 3d series calculated with Fixed-Node Diffusion Monte Carlo (FN-DMC). Selected multi-determinantal expansions obtained with the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) method and including the most prominent determinants of the full configuration interaction expansion are used as trial wavefunctions. Using a maximum of a few tens of thousands determinants, fixed-node errors on total DMC energies are found to be greatly reduced for some atoms with respect to those obtained with Hartree-Fock nodes. To the best of our knowledge, the FN-DMC/(CIPSI nodes) ground-state energies presented here are the lowest variational total energies reported so far. They differ from the recently recommended non-variational values of McCarthy and Thakkar [J. Chem. Phys. 136, 054107 (2012)] only by a few percents of the correlation energy. Thanks to the variational property of FN-DMC total energies, our results provide exact lower bounds for the absolute value of all-electron correlation energies, |E c |.

I. INTRODUCTION

An accurate knowledge of nonrelativistic ground-state energies of atoms is known to be of great interest for computational chemistry. Atomic total energies are indeed routinely used to calibrate theoretical studies in electronic structure theory. For example, let us cite the search for more accurate exchange-correlation energy functionals in Density Functional Theory (DFT), the calibration of various approximations in wavefunction-based approaches (finite basis set effects, truncation at a given order in multi-particle excitations, etc.), the study of the fixed-node approximation in quantum Monte Carlo (QMC), the definition of alternative/exotic electronic approaches, etc. Furthermore, by combining experimental results and accurate nonrelativistic values, some valuable information about relativistic effects can also be obtained.

Here, accurate nonrelativistic, clamped nucleus, allelectron ground-state energies for the metal atoms of the 3d series (from Sc to Zn) are reported. Calculations are performed using the Fixed-Node Diffusion Monte Carlo (FN-DMC) approach, a QMC method known to be particularly powerful for computing ground-state energies. [START_REF] Hammond | Monte Carlo Methods in Ab Initio Quantum Chemistry[END_REF]2 An overwhelming number of works have been devoted to the calculation of accurate atomic ground-state energies using various highly correlated approaches; here, we shall only restrict ourselves to give the typical accuracies presently achievable. For small atoms (say, less than 10 electrons, that is, from H to Ne for neutral atoms), very accurate values with errors smaller than 10 -4 -10 -5 a.u. (or much smaller for the lightest atoms) can be obtained. For heavier atoms up to Ar (18 electrons), the accuracy reduces to the millihartree level (∼ chemical accuracy). For even bigger atoms to obtain a precision close to the millihartree is problematic and only a small number of results have been published. Regarding quantum Monte Carlo studies using FN-DMC or a closely related QMC variant, most of the works have been concerned with atoms from Li to Ne; for the most recent ones, see, e.g., Refs. 3-6. For heavier atoms, most calculations have been performed using pseudo-potentials to remove core electrons (see, e.g., Refs. 7-9). At the all-electron level, very little has been done. We can essentially cite the FN-DMC calculations by Ma et al. 10 for the Ar, Kr, and Xe atoms; calculations for the Cu atom and its cation; 11,12 and two studies by Buendia and collaborators for 3d transition metal atoms. 4,6 It is fair to say that FN-DMC is presently the most accurate method for computing total ground state energies for large enough electronic systems. Potentially, diffusion Monte Carlo allows an exact stochastic solution of the Schrödinger equation. Several sources of error make, in practice, FN-DMC simulations non-exact. However, most of the errors are not of a fundamental nature and can be easily kept under control (mainly, the statistical, finite time-step, and population control errors). In contrast, the fixed-node error resulting from the use of trial wavefunctions with approximate nodes is much more problematic since, up to now, no simple and systematic scheme to control this error has been devised. Note that the fixed-node approximation is variational, E FN ≥ E 0 , a convenient property to get upper and lower bounds for total energies and absolute values of correlation energies, respectively [in contrast, e.g., with the non-variational character of the commonly used Coupled Cluster with Singles, Doubles and Triples perturbatively (CCSD(T)) or Møller-Plesset approaches].

To decrease the fixed-node error, the common strategy is to use high-quality trial wavefunctions and to resort to multi-parameter stochastic optimization techniques to get the best parameters (usually via minimization of the total energy and/or its variance). A great variety of functional forms 24 The key ideas are (i) to avoid the use of a Jastrow factor, (ii) to rely on standard multi-determinant configuration interaction (CI) expansions, and (iii) to avoid any stochastic optimization of the wavefunction. To build CI expansions compact enough to make QMC feasible in practice, we employ a selected CI method (CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm, see Refs. 26 and 32) allowing to extract the most prominent determinants of the Full Configuration Interaction (FCI) expansion. Using an efficient implementation of the calculation of the multideterminant part 34,35 expansions containing a large number of determinants can be used in QMC. In the present application to 3d atoms, expansions up to a few tens of thousands of determinants have been employed. A nice feature of using CIPSI wavefunctions is that simulations can be performed in a fully black-box way: The CI program is first run, then the QMC code is started using as input the CI wavefunction directly as it comes from the output of the CI program. In particular, the preliminary multi-parameter Monte Carlo optimization of the trial wavefunction done in standard FN-DMC approaches (see, e.g., Ref. 36) is avoided. A number of recent applications 24,[START_REF] Giner | Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule F 2 using selected configuration interaction trial wavefunctions[END_REF][START_REF] Giner | Méthodes d'interaction de configurations et Monte Carlo quantique : Marier le meilleur des deux mondes (configuration interaction and QMC: The best of both worlds)[END_REF] have illustrated that CIPSI trial wavefunctions are effective in improving nodes. Furthermore, the fixed-node energy is found to decrease monotonically as a function of the number of determinants or the size of the basis set. This remarkable property allows to define a simple control of the fixed-node error. The price to pay for such attractive properties is that Jastrow-free CI expansions have (much) larger variances and thus require longer simulation times. However, this aspect is in practice not as critical as it may appear. By using our efficient implementation of the calculation of the multi-determinant part and by running our QMC code on medium size computational platforms with a few thousands of compute cores, it has been possible to realize the entire set of calculations presented here for the 3d series in less than two days. From a more general perspective, we would like to emphasize that the possibility of having a simple, automatic, and systematic way of decreasing and controlling the fixed-node bias is of utmost practical importance. It clearly remains true even at the price of more central processing unit-intensive simulations that will anyway become more and more easy to perform in view of the embarrassingly parallel property of QMC and the rapid development of ever more powerful parallel machines.

The FN-DMC/(CIPSI nodes) total ground-state energies of metal atoms of the 3d series obtained here are compared to the very recent results of Buendia et al. 6 using a FN-DMCtype approach with a single-determinant trial wavefunction. A systematic improvement in FN total energies is obtained for all atoms, except for Zn for which similar results are found. For the lightest 3d atoms, the total energy values are much improved (a gain up to about 4.5 × 10 -2 a.u. for Sc).

To the best of our knowledge, the data presented here are the lowest variational total ground-state energies reported so far for the 3d transition metal atoms. They differ from the recently recommended non-variational values of McCarthy and Thakkar [START_REF] Mccarthy | [END_REF] (denoted as McCT in what follows) obtained by an hybrid approach combining MP2 results at completebasis-set (CBS) limit and accurate CCSD(T) values, only by a few percents of the correlation energy, namely, between 5.6% and 8% while these authors estimate their errors to be about ±3%. Thanks to the variational property of FN-DMC total energies, our results also provide exact lower bounds for the absolute value of all-electron correlation energies, |E c |.

II. METHODS AND COMPUTATIONAL DETAILS

A. CIPSI

The CIPSI method, and similar approaches closely related, have been introduced and developed a long time ago by a number of authors (see Refs. 25-33). In a few words, the approach consists in building the multi-determinantal expansion iteratively by selecting at each step one determinant (or a group of determinants) according to a perturbative criterion. A determinant D i (or a group of determinants) is added to the current wavefunction if its (their) energetic contribution(s) calculated by second-order perturbation theory is (are) sufficiently large. In this way, the wavefunction is built hierarchically, the most important determinants of the FCI solution entering first in the expansion. Such a construction must be contrasted with standard approaches (configuration interaction with singles, configuration interaction with singles and doubles, etc.), where the contributions at a given order are calculated by considering all possible particleexcitations with respect to some reference wavefunction (usually, the Hartree-Fock (HF) solution). The CIPSI multideterminantal expansion is thus much more compact than standard expansions, an important practical point for FN-DMC where the trial wavefunction and its derivatives must be computed a very large number of times during the simulations. Let us now briefly summarize the algorithm. More details can be found in Ref. 24 and references therein.

In multi-determinantal expansions, the ground-state wavefunction |Ψ 0 ⟩ is written as a linear combination of Slater determinants {|D i ⟩}, each determinant corresponding to a given occupation by the N α and N β electrons of N = N α + N β electrons among a set of M spin-orbitals {φ 1 ,...,φ M } (restricted case). The best representation of the exact wavefunction in the entire determinantal basis is the FCI wavefunction written as

|Ψ 0 ⟩ =  i c i |D i ⟩, (1)
where c i are the ground-state coefficients obtained by diagonalizing the Hamiltonian matrix,

H i j = ⟨D i |H |D j ⟩, within the orthonormalized set, ⟨D i |D j ⟩ = δ i j , of determinants |D i ⟩.
In its simplest form, the multi-determinant wavefunction is iteratively built as follows. Let us call

|Ψ (n) 0 ⟩ =  i ∈S n c (n)
i |D i ⟩ the current wavefunction at iteration n, where S n is the set of selected determinants at iteration n. Typically, at the initial step n = 0, a mono-determinantal HF-type or a short completeactive-space-self-consistent-field-type wavefunction is used. The first step consists in collecting all different determinants |D i c ⟩ connected by H to |Ψ (n) 0 ⟩, that is, ⟨Ψ (n) 0 |H |D i c ⟩ 0. Then, the second-order correction to the total energy resulting from each connected determinant is computed,

δe(|D i c ⟩) = - ⟨Ψ (n) 0 |H |D i c ⟩ 2 ⟨D i c |H |D i c ⟩ -E (n) 0 (2)
and the determinant (or group of determinants) |D i * c ⟩ associated with the largest |δe| (or greater than a given threshold) is (are) added to the reference subspace

S n → S n+1 = S n ∪ {|D i * c ⟩}.
Finally, the Hamiltonian matrix is then diagonalized within S n+1 to obtain the new wavefunction at iteration n + 1 and the process is iterated until a target size N dets for the reference subspace is reached. The CIPSI wavefunction issued from this selection process is the trial wavefunction used here for FN-DMC.

B. FN-DMC

For a detailed presentation of the theoretical and practical aspects of FN-DMC, the reader is referred to the literature, e.g., Ref. 40. Here, let us just emphasize that the central quantity of such approaches is the trial wavefunction Ψ T determining both the magnitude of the fixed-node error through its approximate nodes and the quality of the statistical convergence (good trial wavefunctions imply small statistical fluctuations). The computational cost of FN-DMC is almost entirely determined by the evaluation at each Monte Carlo step of the value of Ψ T and its first (drift vector) and second derivatives (Laplacian needed for the local energy). In view of the very large number of MC steps usually required (typically at least billions and often much more), it is essential to compute such quantities very rapidly. In the present work, the typical size of the expansion considered is a few tens of thousands of determinants. Some care is thus required when computing such expansions to keep the computational cost reasonable. The various aspects regarding this problem are presented in Ref. 35.

C. Computational details

The atomic basis sets used for the calculations are the Slater-type orbitals of Bunge 41 supplemented with four additional 4 f and three 5g functions (a total of 112 atomic basis functions). All the CIPSI calculations have been performed using Hartree-Fock molecular orbitals using the code developed in our group (quantum package), and all the FN-DMC calculations have been performed using QMC = Chem. 42 For each atom, the CIPSI calculation was stopped when more than 10 6 determinants were selected in the variational wave function. This wave function was then truncated such that the least significant determinants contributing to 0.5% of the norm of the wave function were discarded: 10 4 -5 × 10 4 determinants were kept. This wave function was used without any modification as the trial wave function for the FN-DMC calculations (no Jastrow factor).

For the FN-DMC calculations, we have employed the algorithm described in Ref. 43 allowing us to use a small constant number of walkers. A block consisted of 30 walkers performing 5000 steps with a time step of 10 -5 a.u., a value chosen such that the time-step error was smaller than the statistical error. Long enough simulations have been performed to make the statistical error negligible with respect to the fixed-node one: depending on the atom, a number of blocks between 7 × 10 4 and 1.5 × 10 5 were calculated (∼10 10 MC steps).

III. RESULTS

In Table I, the variational energy, the number of determinants in the CIPSI expansion, an estimate of the percentage of the total CE recovered, and the average value of the S 2 total spin operator for each trial wavefunction Ψ T used in FN-DMC are given. The CE's reported are calculated from the recommended values of McCarthy and Thakkar. [START_REF] Mccarthy | [END_REF] We emphasize that, in contrast with the present work, these values have not been computed directly from a unique (very) accurate energy calculation but have been obtained indirectly by combining Møller-Plesset (MP2) correlation energies extrapolated at the CBS limit and CCSD(T) calculations using Dunning's basis sets of various sizes. The percentages of correlation energies retrieved at the CIPSI variational level are found to be around 60%. It is important to note that the correlation effects recovered in such CI calculations are mostly valence correlation effects (valence basis set) and non-dynamical correlation effects resulting from the use of multi-determinant expansions including the most prominent determinants. In particular, most of the energetic core contributions are not taken into account in these valence CI calculations. Some caution is thus needed when comparing the total energies obtained with the Slater-Jastrow wavefunctions usually employed in FN-DMC and the valence CI wavefunctions proposed here. Finally, note that the total In Table II, the fixed-node DMC total energies obtained using standard Hartree-Fock nodes and newly proposed CIPSI nodes are reported. For the sake of comparison, we also give the very recent results of Buendia et al. 6 that were up to now the lowest variational fixed-node energies reported for these atoms. In their study, the trial wavefunctions employed are written as the product of a nodeless correlation factor and a so-called model function obtained within the parametrized Optimized Effective Potential (OEP) approximation. The model function determining the nodal structure is written as a linear combination of a few Configuration State Functions (CSFs), mainly to take into account 4s-4p near-degeneracy effects. For the Cr and Cu atoms with a singly occupied 4s shell, the model function is represented by a single CSF, while for the other atoms 4s 2 3d n and 4p 2 3d n configurations are mixed. For each type of nodes used, an estimate of the percentage of the correlation energy is also reported. The percentages retrieved by all FN-DMC calculations presented are important and range between 89% and 94%. A first observation is that energies resulting from HF and OEP nodes are of comparable quality, while CIPSI nodes may lead to significantly lower fixed-node energies. The gain in energy with the new nodes is found to decrease almost uniformly with Z. For the lightest elements (Sc, V, and Ti), a maximum gain of about 0.045 a.u. is achieved; for the intermediate atoms (Cr-Ni), about 0.02-0.03 a.u. is obtained, while for the two heaviest elements (Cu and Zn), no gain is observed within statistical fluctuations. The fact that CIPSI performs better for lightest elements is not surprising since Hartree-Fock nodes are known to be well-adapted to atoms with spherical symmetry. In the extreme case of the Cu and Zn atoms having a totally filled and spherically symmetric 3d shell, HF and CIPSI nodes give similar results. In the opposite case of light atoms, the CIPSI wavefunctions, that have many more degrees of freedom than the single-configuration HF solution to describe non-symmetrical electronic configurations, lead to much improved results. In Table III, the correlation energies resulting from our FN-DMC simulations are reported and compared to the recommended values of McCT. As already noted, these latter results have been obtained with a mixed approach including MP2-CBS and CCSD(T) calculations. According to the authors, the errors in these values are estimated to be ±3%. The relative differences between FN-DMC/[HF nodes] or FN-DMC/[OEP nodes] and the McCT values go from 8% to 11%. Using CIPSI nodes, the differences are reduced and range between 6% and 8%. Note that the typical statistical error on these percentages is small and about 0.2%. Although our final values for correlation energies are slightly less accurate than the estimates made by McCT, we would like to emphasize and conclude on three important points: (i) In contrast with what has been done by McCT, our correlation energies have been directly computed with a unique highly correlated electronic structure method. No hybrid scheme mixing results of two different approaches has been employed. To the best of our knowledge, the FN-DMC values presented here are the most accurate (lowest) nonrelativistic total energies ever reported for the 3d transition metal atoms. (ii) As a consequence of the variational property of FN-DMC total energies and also in contrast with McCT's results, the absolute values of our correlation energies are exact lower bounds of the unknown CE's. (iii) Finally, in view of the great versatility of FN-DMC/CIPSI, there is no reason why improved lower bounds would not be achieved in the near future.

TABLE I .

 I Number of determinants, N dets , of the CIPSI expansion used as FN-DMC trial wavefunction. Variational energy, E var (CIPSI), estimated percentages of the correlation energy (CE) recovered, and eigenvalue of the S 2 spin operator. Energy in hartree.

	Atom	N dets	E var (CIPSI)	(CE in %)	S 2
	Sc 2 D(d 1 s 2 )	11 389	-760.32556	(66.5)	0.75
	Ti 3 F(d 2 s 2 )	14 054	-849.02624	(66.9)	1.99
	V 4 F(d 3 s 2 )	12 441	-943.53667	(64.9)	3.74
	Cr 7 S(d 5 s 1 )	10 630	-1044.03692	(63.6)	11.97
	Mn 6 S(d 5 s 2 )	11 688	-1150.57902	(63.0)	8.73
	Fe 5 D(d 6 s 2 )	13 171	-1263.21805	(62.5)	5.99
	Co 4 F(d 7 s 2 )	15 949	-1382.24964	(62.8)	3.74
	Ni 3 F(d 8 s 2 )	15 710	-1507.74694	(62.3)	1.99
	Cu 2 S(d 10 s 1 )	48 347	-1639.96605	(63.3)	0.75
	Zn 1 S(d 10 s 2 )	44 206	-1778.82784	(60.5)	0.00

TABLE II .

 II FN-DMC total energies for the 3d series of transition metal atoms together with the percentage of correlation energy recovered for different nodal structures. Energy in hartree.

								FN energy gain with
	Atom	HF nodes	(CE in %)	OEP nodes a	(CE in %)	CIPSI nodes	(CE in %)	CIPSI nodes b
	Sc	-760.5265(13)	[89.2(2)]	-760.5288(6)	[89.50(7)]	-760.5718(16)	[94.4(2)]	-0.0453(21)
	Ti spin values S 2 obtained from the truncated CIPSI expansions
	show that the trial wavefunctions built are almost pure spin-
	eigenstates.

TABLE III .

 III Fixed-node DMC correlation energies, -E c , in hartree using HF and CIPSI nodes. Comparison with the recommended values of McCarthy and Thakkar (McCT).[START_REF] Mccarthy | [END_REF]

	Atom	HF nodes	OEP nodes a	CIPSI nodes	McCT b
	Sc	0.7900(13)	0.7923(6)	0.8353(16)	0.8853
	Ti	0.8454(14)	0.8541(7)	0.8890(19)	0.9433
	V	0.9000(13)	0.9039(6)	0.9390(16)	1.0049
	Cr	0.9728(16)	0.9725(6)	1.0039(17)	1.0695
	Mn	1.0218(17)	1.0235(7)	1.0495(20)	1.1304
	Fe	1.1122(19)	1.1140(6)	1.1401(21)	1.2343
	Co	1.2016(21)	1.2055(8)	1.2216(24)	1.3270
	Ni	1.3043(23)	1.3141(7)	1.3299(25)	1.4242
	Cu	1.4634(26)	1.4629(7)	1.4691(29)	1.5842
	Zn	1.4890(26)	1.4944(8)	1.4905(31)	1.6202

a Reference 6. b Reference 39.

ACKNOWLEDGMENTS

A.S. and M.C. thank the Agence Nationale pour la Recherche (ANR) for support through Grant No. ANR 2011 BS08 004 01. This work has been possible thanks to the computational support of CALMIP through a Meso-Challenge on their new Eos supercomputer (www.calmip.univtoulouse.fr).