Andreas Heßelmann
email: andreas.hesselmann@theochem.uni-stuttgart.de

Emmanuel Giner
email: emmanuel.giner@lct.jussieu.fr

Peter Reinhardt
email: peter.reinhardt@lct.jussieu.fr

Peter J Knowles
email: knowlespj@cardiff.ac.uk

Hans-Joachim Werner
email: werner@theochem.uni-stuttgart.de

Julien Toulouse
email: toulouse@lct.jussieu.fr

A density-fitting implementation of the density-based basis-set correction method

Keywords: basis-set convergence, Møller-Plesset perturbation theory, density-functional theory, reaction energies

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

I. INTRODUCTION

One of the main goals of quantum chemistry is the accurate prediction of molecular properties, which requires to tackle the electron correlation problem. For this, there are two main families of computational electronic-structure methods: wave-function theory (WFT) [1] which targets the complicated many-electron wave function, and density-functional theory (DFT) [2] which uses the simpler one-electron density. While DFT has become the workhorse of quantum chemistry thanks to its appealing balance between computational cost and accuracy, the lack of a systematic scheme to improve the quality of density-functional approximations has renewed the interest in the development of WFT methods in the last few decades.

A serious limitation of WFT methods is their slow convergence of the correlation energy with the size of the oneelectron basis set. This slow convergence originates from the short-range singularity of the Coulomb electron-electron repulsion which induces a derivative discontinuity in the exact eigenstate wave functions, known as the electron-electron cusp condition [3]. There are two main approaches for dealing with this problem. The first approach consists in extrapolating the results to the complete-basis-set (CBS) limit by using increasingly large basis sets [4,5]. The second approach consists in using explicitly correlated R12 or F12 methods which incorporate in the wave function a correlation factor reproducing the electron-electron cusp (see, e.g., Refs. [6][7][8][9][10][11][12][13][14][15][16]).

An alternative approach to accelerate basis-set convergence was recently proposed, which we will refer as the densitybased basis-set correction (DBBSC) method [17]. It consists in correcting the energy calculated by a WFT method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the CBS limit. In practice, this basisset correction density functional is constructed from rangeseparated DFT [18] by defining a basis-dependent local rangeseparation parameter which provides a local measure of the incompleteness of the basis set. This DBBSC method was validated for configuration-interaction and coupled-cluster calculations of atomization energies [19][20][21], excitation energies [22], dissociation energy curves [23], and dipole moments [24,25]. It was also extended to GW calculations [26] and to linear-response theory [27], and some mathematical aspects of the method were studied in detail on a onedimensional model system [28].

In this work, we report an efficient implementation of the DBBSC method in the MOLPRO software [29][30][START_REF] Werner | version 2023, a package of ab initio programs[END_REF] in which density fitting [START_REF] Werner | [END_REF] is used to alleviate the computational bottleneck of the method, namely the calculation of the local range-separation parameter. This allows us to use the DBBSC method on larger molecular systems than what was previously possible. We thus apply the DBBSC method for correcting the basis-set errors in the molecular reaction energies of the FH51 benchmark set [33,34] at the second-order Møller-Plesset (MP2) level. We also test different basis-set correction density-functional approximations, as well as the addition of a single-excitation correction for one-electron basis-set errors. Finally, we compare the performance of the DBBSC method with the explicitly correlated MP2-F12 method [11].

The paper is organized as follows. In Section II, we explain the theory of the present implementation of the DBBSC method. Section III provides computational details for the cal-culations on the FH51 benchmark set. In Section IV, we give and discuss our results. Finally, Section V contains our conclusions.

II. THEORY

For simplicity, we give the equations for closed-shell states and we assume real-valued HF spatial orbitals {ϕ p }.

A. The DBBSC method at the MP2 level Given the MP2 total energy E B MP2 in a basis set B, we apply the non-self-consistent basis-set correction [17,19] as

E B MP2+DFT = E B MP2 + ĒB [n B HF ], (1)
where ĒB [n B HF ] is the basis-correction density functional evaluated at the active HF density n B HF (i.e., excluding core orbitals in case of frozen-core calculations). In order not to affect the CBS limit, this functional ĒB [n] must be such that it vanishes when the basis set B is complete. Moreover, provided a good enough approximation is used for ĒB [n], the basis-set corrected MP2 energy, referred to as "MP2+DFT", is expected to converge faster to the MP2 CBS limit than the uncorrected MP2 energy.

B. Local range-separation parameter

The dependence on the basis set of the basis-correction density functional ĒB [n] comes from the local range-separation parameter µ B (r). It is defined as [17,19]

µ B (r) = √ π 2 W B (r), (2)
where W B (r) is the on-top value of the effective interaction localized with the HF wave function

W B (r) =          f B HF (r) n B 2,HF (r) , if n B 2,HF (r) 0, ∞, otherwise. (3)
In Eq. (3), n B 2,HF (r) is the HF on-top pair density

n B 2,HF (r) = n B HF (r) 2 2 , (4)
with the active HF density n B HF (r) = 2 act i ϕ i (r) 2 , and f B HF (r) has the expression

f B HF (r) = 2 all p,q act i, j ϕ p (r)ϕ i (r)(ϕ p ϕ i |ϕ q ϕ j )ϕ q (r)ϕ j (r), (5)
where p and q run over all (occupied + virtual) HF spatial orbitals, i and j run over active HF spatial orbitals, and (ϕ p ϕ i |ϕ q ϕ j ) are the two-electron Coulomb integrals in chemists' notation. We recall that by active orbitals we mean occupied orbitals without the frozen-core orbitals, in case of frozen-core calculations. The local range-separation parameter µ B (r) provides a local measure of the incompleteness of the basis set. A straightforward calculation of f B HF (r) in Eq. ( 5) requires to first calculating the molecular-orbital two-electron integrals (ϕ p ϕ i |ϕ q ϕ j ) with a dominant scaling of O(N act N 4 all ), and then performing the sums at each grid point which scales as O(N 2 act N 2 all N grid ), where N act is the number of active orbitals, N all is the total number of orbitals in the basis, and N grid is the number of spatial grid points. This is the computational bottleneck of the basis-set correction calculation.

This scaling can be reduced by density fitting [START_REF] Werner | [END_REF]35]. Introducing an auxiliary fitting basis set {χ A }, the orbital product is approximated as

ϕ p (r)ϕ i (r) ≈ fit A d pi A χ A (r), (6)
where d pi A are the Coulomb-fitting coefficients

d pi B = fit A (ϕ p ϕ i |χ A )[J -1 ] AB , (7)
with

J AB = χ A (r 1 )χ B (r 2 ) ||r 2 -r 1 || dr 1 dr 2 , (8) and
(ϕ p ϕ i |χ A ) = ϕ p (r 1 )ϕ i (r 1 )χ B (r 2 ) ||r 2 -r 1 || dr 1 dr 2 . ( 9
)
Orthonormalizing the auxiliary fitting basis functions with respect to the metric J,

χA = fit B [J -1/2 ] AB χ B , (10)
we can approximate the two-electron integrals as

(ϕ p ϕ i |ϕ q ϕ j ) ≈ fit A (ϕ p ϕ i | χA )( χA |ϕ q ϕ j ), (11)
and the quantity f B HF (r) in Eq. ( 5) as

f B HF (r) ≈ 2 fit A         all p act i ϕ p (r)ϕ i (r)(ϕ p ϕ i | χA )         2 . ( 12
)
Thus, with density fitting, there is no need to build explicitly the two-electron integrals anymore and the calculation of f B HF (r) in Eq. ( 12) now scales as O(N act N all N fit N grid ) where N fit is the number of auxiliary fitting basis functions. In practice, the same auxiliary fitting basis sets optimized for density fitting in MP2 can be used here.

C. Approximate basis-correction density functional

We approximate the basis-correction density functional with the local form [19] ĒB [n] ≈ ēsr c,md (n(r), ∇n(r), µ B (r))dr, (13) where ēsr c,md (n, ∇n, µ) is the complementary multi-determinant short-range correlation functional energy density [19,36] ēsr c,md (n, ∇n, µ B ) = e c (n, ∇n)

1 + e c (n,∇n) c n 2 (n) µ 3 , ( 14
)
where c = (2 √ π(1 - √ 2
))/3 and n 2 (n) is a model of the ontop pair density. In Eq. ( 14), e c (n, ∇n) is a standard Kohn-Sham correlation functional energy density. As in previous works, the default choice is the PBE correlation functional [37]. In this work, we also test using the LDA [38], LYP [39], TPSS [40], and SCAN [41] correlation functionals. Note that the TPSS and SCAN functionals are meta-GGA functionals, i.e. they depend also on the non-interacting positive kinetic energy density τ(r) = (1/2) act i |∇ϕ(r)| 2 , and thus constitute a slight extension of Eqs. ( 13) and ( 14).

The default choice [19] for n 2 (n) is to use the on-top pair density of the uniform-electron gas (UEG)

n UEG 2 (n) = n 2 g 0 (n), (15)
where the on-top pair-distribution function g 0 (n) is parametrized in Eq. ( 46) of Ref. 42. In this work, we also explore two other on-top pair-density models. The first one is the Colle-Salvetti (CS) model [43][44][45]]

n CS 2 (n) = n 2 2 Φ CS (n) 2 , (16)
where

Φ CS (n) = √ π β(n) 1 + √ π β(n) , (17)
and

β(n) = q n 1/3 , ( 18
)
where q is an empirical parameter. The second one is the Hollett-Pegoretti (HP) model [46] n

HP 2 (n) = n 2 2 Φ HP (n), (19)
where

Φ HP (n) = 2 √ π β(n) 2 2β(n)e -1 4β(n) 2 + √ π 1 + 2β(n) 2 1 + erf 1 2β(n)
.

(20) We may choose the value of the parameter q, e.g., by imposing that the integral of the model on-top pair density equals the integral of the exact on-top pair density, n model 2 (n(r))dr = n exact 2 (r)dr, in the helium atom. Estimating n exact 2 (r) with a highly accurate 418-term Hylleraas-type wave function [47][48][49], we find q = 1.88 for the CS model and q = 2.05 for the HP model. When these on-top pair-density models are used with the PBE correlation functional in Eq. ( 14), we call the resulting basis-set correction functionals PBE-CS and PBE-HP, respectively.

D. CABS single-excitation correction

For small basis sets B, the HF energy can have a substantial basis-set error. This HF basis-set error is not corrected by the approximate basis-set correction functionals in Section II C since they only correct for missing short-range correlation. The HF basis-set error can however be easily corrected by using the complementary auxiliary basis set (CABS) (see, e.g., Refs. [10,12,[50][51][52][53][54]) used in explicitly correlated R12/F12 methods. In this approach, a large orthonormal basis set is formed by the occupied+virtual HF orbitals obtained in the normal basis set B and an additional set of virtual orbitals obtained from the CABS. The HF energy correction due to the addition of the CABS is estimated by second-order perturbation theory, leading to the expression, in a closed-shell formalism, [12,50,51]

∆E B,CABS HF = 2 act i vir α t i α f α i , (21)
where i runs over active HF orbitals and α runs over all virtual orbitals (obtained in the normal basis set B and from the CABS). In Eq. ( 21), f α i are Fock matrix elements and t i α are single-excitation coefficients found by solving the first-order perturbation equations

f i α = act j t j α f i j - vir β f β α t i β . ( 22
)
The correction is often referred to as the CABS singleexcitation correction. Note that a similar correction is used in the dual basis-set approach proposed by Wolinski and Pulay [55] for improving HF energies and by Liang and Head-Gordon [56] for Kohn-Sham DFT energies. The total basis-set corrected MP2 energy is thus

E B MP2+CABS+DFT = E B MP2 + ∆E B,CABS HF + ĒB [n B HF ], (23)
and will be referred to as "MP2+CABS+DFT". For comparison, we will also present MP2 results only corrected by the CABS single-excitation correction, which will referred to as "MP2+CABS".

III. COMPUTATIONAL DETAILS

The DBBSC method with density fitting has been implemented in the MOLPRO software [29][30][START_REF] Werner | version 2023, a package of ab initio programs[END_REF]. We have performed tests on the FH51 benchmark set. The FH51 set [33,34] is a set of 51 reaction energies for various organic molecules. It is included in the GMTKN55 database [57]. The FH51 set contains a large variety of molecules of different sizes (from 2 to 29 atoms). It is thus suitable to test the DBBSC method over systems of different sizes. Moreover, many systems are large enough so that the present densityfitting implementation has a large impact on the performance of the method. As regards the basis set B, we use the augcc-pVnZ basis sets [58] for first-row atoms and the aug-cc-pV(n+d)Z basis sets [59] for second-row atoms, which we jointly abbreviate as avnz, for n = 2 (d), 3 (t), 4 (q), and 5.

We perform canonical-orbital density-fitting HF [60] and density-fitting MP2 [START_REF] Werner | [END_REF] calculations with the frozen-core approximation. We calculate the basis-set correction with different functionals evaluated at the active HF density, and including the CABS single-excitation correction [12,50,51]. The basis-set correction is consistently calculated in the frozencore approximation, corresponding to using only active orbitals in Eq. ( 5) and in the HF density used in Eq. ( 1). For the n = 2, 3, for comparison, we also perform canonicalorbital density-fitting MP2-F12 (in the default 3C(F) variant with a Slater-type geminal exponent of 1 bohr -1 ) [11] calculations, implicitly including the CABS single-excitation correction. The MP2/CBS reference values are estimated as the MP2-F12 values with the av5z basis set.

For a given basis set B, the density-fitting basis sets used are the corresponding B/JKFIT and B/MP2FIT basis sets of Weigend et al. [61,62] (and their extensions [51]) for the HF and MP2 calculations, respectively. The B/JKFIT basis set is also used as CABS for the CABS single-excitation correction. We have checked the density-fitting errors and found them to be insignificant. For large systems, density-fitting calculations of the basis-set correction can be more than an order of magnitude faster than non-density-fitting calculations (computation times can be found in the Supplementary Material).

IV. RESULTS AND DISCUSSION

As a first test, we compare in Fig. 1 the different basis-set correction density-functional approximations for the basis-set convergence of the ground-state MP2 correlation energy of the tetramethylpentane molecule (C 9 H 20 ). We see that all the density-functional approximations lead to a quite similar acceleration of the convergence of MP2 correlation energy toward its CBS limit. At the level of the correlation energy, all the proposed density-functional approximations thus provide a reasonable basis-set correction with quite a substantial acceleration of the basis-set convergence, albeit not as impressive as the one obtained with MP2-F12.

The errors on the reaction energies of the FH51 set with respect to MP2/CBS calculated with MP2, MP2+CABS, MP2+CABS+PBE, and MP2-F12 are reported in Fig. 2. With the avdz basis set, MP2 can have quite large basis errors for some reaction energies, up to about 13 kcal/mol. Obtaining MP2 reaction energies with all basis errors below 1 kcal/mol requires the use of the av5z basis set. The CABS singleexcitation correction is crucial to reduce the largest basis er- rors on MP2 reaction energies obtained with the avdz basis set. Even with larger basis sets, the CABS single-excitation correction still helps to reduce the basis errors for some reaction energies. Adding the PBE-based basis-set correction further reduces the basis errors, albeit not always in a systematic way since there are a few cases where the basis error increases.

It is noteworthy that the basis errors of the MP2+CABS+PBE reaction energies are all smaller than 1 kcal/mol with the avtz basis set and larger basis sets. MP2-F12 globally outperforms MP2+CABS+PBE, giving reaction energies with basis errors below about 1 kcal/mol already with the avdz basis set.

In Table I, we report the mean absolute errors (MAEs) on the reaction energies of the FH51 set with respect to MP2/CBS obtained with the methods already discussed, as well as with additional basis-set correction functionals, namely LDA, LYP, TPSS, SCAN, PBE-CS (q = 1.88), and PBE-HP (q = 2.05). For the methods already discussed, the mean errors are consistent with the observations made previously. For the avdz basis set, we go from a MAE of 2.08 kcal/mol for uncorrected MP2 to a MAE of 0.62 kcal/mol for MP2+CABS+PBE and a MAE of 0.33 kcal/mol for MP2-F12. For the avtz basis set, we go from a MAE of 0.72 kcal/mol for uncorrected MP2 to a MAE of 0.21 kcal/mol for MP2+CABS+PBE and a MAE of 0.13 kcal/mol for MP2-F12. For the avqz and av5z basis sets, the PBE-based basis-set correction is still effective in reducing the basis errors, as we go from MAEs of 0.25 and 0.13 kcal/mol, respectively, for uncorrected MP2 to MAEs of 0.10 and 0.04 kcal/mol, respectively, for MP2+CABS+PBE. Thus, MP2+CABS+PBE with an avnz basis set globally gives uncorrected MP2 reaction energies with slightly higher av(n+1)z quality, whereas MP2-F12 with an avnz basis set roughly gives uncorrected MP2 reaction energies with slightly lower av(n + 2)z quality.

With the other basis-set correction functionals tested, the MAEs are very similar, except for the LYP correlation functional which gives much larger basis errors. We have also tested optimizing the parameter q in the CS and HP on-top pair density-density models in Eq. ( 18) and the parameter c in front of the on-top pair density in Eq. ( 14), but we did not 0.66 0.23 0.10 0.04 MP2+CABS+PBE-CS (q = 1.88) 0.62 0.22 0.09 0.04 MP2+CABS+PBE-HP (q = 2.05) 0.64 0.22 0.10 0.04 MP2-F12 0.33 0.13 0.03 0.00 obtain significant improvements. Thus, if we set aside LYP, we find a rather small sensitivity of the method to the underlying correlation functional for calculating reaction energies.

In the Supplementary Material, we report additional statistical indicators which confirm this conclusion. Finally, as regards the computational cost of the DFT-based basis-set correction in comparison with MP2-F12, we consistently observe, for all basis sets, that MP2+CABS+PBE is approximately 10 times faster than MP2-F12 in the default 3C variant. However, we note that MP2-F12 can be made faster using the 3*A approximation [11] without losing much accuracy in most cases, and MP2+CABS+PBE is only approximately 3 to 4 times faster than this cheaper MP2-F12 variant. Of course, the relative gains in computational cost would be much less for more expensive wave-function methods such as CCSD(T).

V. CONCLUSION

We have reported an efficient density-fitting implementation of the DBBSC method in the MOLPRO software using different basis-set correction density-functional approximations and including the CABS single-excitation correction. We have tested the method on the FH51 benchmark set of reaction energies at the MP2 level and provided a comparison with the explicitly correlated MP2-F12 method.

For the smallest basis sets, the CABS single-excitation correction provides an important correction on reaction energies which is not included in the basis-set correction densityfunctional approximations. The basis-set corrected reaction energies are quite insensitive to the choice of the basis-set correction density-functional approximation, with the notable exception of the LYP functional which gives much worse results. This point should be further analyzed in the future. Overall, the basis-set corrected MP2 reaction energies calculated with a n-zeta basis set are of slightly higher quality than uncorrected MP2 reaction energies calculated with (n+1)-zeta quality. However, the explicitly correlated MP2-F12 method is consistently more accurate, with reaction energies calculated with a n-zeta basis set being of slightly lower quality than uncorrected MP2 reaction energies calculated with (n + 2)-zeta quality. We believe that the DBBSC method is still valuable for accelerating the basis convergence of MP2 due to the fact that it has a lower computational cost than MP2-F12.

Finally, let us mention that the present implementation of the DBBSC method can be applied to any other wave-function methods, such as CCSD(T), with expected similar gains in accuracy. After completion of the present work, we became aware of a very similar independent work that has just been published [63].

FIG. 1 :

 1 FIG.1: Basis-set convergence of the ground-state MP2 correlation energy of the tetramethylpentane molecule (C 9 H 20 ) with different basis-set correction density-functional approximations (evaluated at the HF density) and with MP2-F12 using avnz basis sets.

FIG. 2 :

 2 FIG.2: Errors in reaction energies of the FH51 set with respect to MP2/CBS calculated with MP2, MP2+CABS, MP2+CABS+PBE, and MP2-F12 with avnz basis sets. The order of reactions is the one from Refs.33, 34.

TABLE I :

 I Mean absolute errors (in kcal/mol) in reaction energies of the FH51 set with respect to MP2/CBS with avnz basis sets.

		avdz avtz avqz av5z
	MP2	2.08 0.72 0.25 0.13
	MP2+CABS	0.94 0.55 0.22 0.12
	MP2+PBE	1.73 0.34 0.13 0.04
	MP2+CABS+PBE	0.62 0.21 0.10 0.04
	MP2+CABS+LDA	0.67 0.19 0.09 0.04
	MP2+CABS+LYP	1.12 0.46 0.22 0.21
	MP2+CABS+TPSS	0.63 0.21 0.10 0.04
	MP2+CABS+SCAN

Acknowledgments

It is a pleasure to dedicate the present paper to Carlo Adamo on the occasion of his 60th birthday.

Supplementary Material

See the Supplementary Material for computation times and further statistical measures.

Conflict of interest

None of the authors have a conflict of interest to disclose.


Roger Soliva

Juliette Lamarche

P Labaume

L Ducarme

Jean-Pierre Masse

Fault

Fault Zone Structure and Diagenetic Evolution in Porous Carbonate Rocks, Provence, France

We provide an analysis of normal fault zones exceptionally well exposed in porous calcarenites from La Fare les Oliviers (Provence, France). The fault zones crop out along their entire length due to their higher resistance to erosion compared to the porous host rock, and provide excellent exposures for detailed study. This high resistance to erosion is due to significant diagenetical and lithological modifications of the fault damage zone (i.e. pore cementation and sediment infiltration from the surface). The fault core zone is characterized by cataclastic shear and constitutes a vertical path for late gravity-driven oxidant fluids assisting karstification. The fault zone composition and structure is therefore the result of the interplay between the fault zone mechanical behaviour and the the Earth surface environement (marine vs continental).

Introduction

Because of the progressive exhaustion of hydrocarbon resources and the increasing exploitation of faulted and fractured reservoirs, the optimization of productivity depends nowadays on the understanding of faults and fractures architecture and composition. Many works based on bore hole data and seismic survey have shown that fault zones may constitute drains or permeability barriers to fluid flow in natural reservoirs. The knowledge about fault zones in siliciclastic context (porous sandstones or layered sandstone/clays units) gave the opportunity to estimate their role on fluid migration or trapping. However in carbonate rocks, the structure and petrophysical properties of fault zones, as well as the mechanisms governing the development of normal faults are not well known although they become important issues to understand fluid migration in such reservoirs.

We provide an analysis of normal fault zones exceptionally well exposed in porous calcarenites from La Fare les Oliviers (Provence, France). The fault zones crop out along their entire length due to their higher resistance to erosion compared to the porous host rock, and provide excellent exposures for detailed study (Figure 1). The normal fault zones formed during the middle Cretaceous as the result of a pre-Pyrenean extension ("phase durancienne").

Field work

The « Belvédère fault » exposure (see Figure 1b) has been inspected in detail in the field and in the laboratory on ~30 macroscopic samples and more than 50 thin-sections. The analysis reveals changes both in lithology and compartmentalisation of the deformation. Outside the fault zone, the host grainstone ranges from 5 to 15% of porosity, whereas in the 2 nd International Conference on Fault and Top Seals -From Pore to Basin Scale Montpellier, France, 21 -24 September 2009 damage zone (see Figure 1b), these pores are sealed by calcite cement. In addition to these diagenetic changes, the damage zone is characterized by non-porous marine wackestones and packstones in-filling, infiltred in opened fractures (tens of cm opening) affecting the host grainstone. Cathodoluminescence analysis suggests that the sparite cement sealing the pores of the damaged grainstone is in chemical equilibrium with the wackestone -packstone infiltred sediment. The damage zone is also affected by late secondary faulting allowing the formation of an isotropic dilation breccia in extensive quadrants. This dilation breccia is formed by coalesced mode I fractures, which created voids sealed by mudstones laminations and coarse sparite. The gravity-driven distribution of the mudstones and desiccation cracking observed in the laminations suggest a vadose context of deposition/precipitation in the dilated volumes. The core zone is a shear and compactant (solution seams) cataclastic breccia composed of micrometric to centimetric size elements of various material from the damage zone. The core zone also shows post-faulting alteration related to micro-karstification. X-ray diffraction analysis reveals that the mineralogy of the micro-karst deposits has similar composition than the iron oxides observed along the regional mid-Cretaceous unconformity lying above the studied formation. This reveals that the cataclastic fault core constituted a preferential vertical path for post-faulting oxidizing fluids incoming from the earth surface in a continental arid environment.

Discussion

The observed variations of fault rock diagenesis and lithology partly reflects changes in fault zone mechanical behaviour. The fault zone reveals a shallow deformation depth (maximum 400 m) progressively evolving from (1) an early stage of opening deformation associated to sediment infiltration, to (2) an intermediate dilational-shear stage, and (3) to a final cataclastic compactant faulting associated to solution seams. Therefore, non-isochore deformation in association with various diagenetic processes (sediment infiltration, cementation and pressure-solution) characterizes the fault zone behaviour from its initiation to the last faulting stages. The early opening deformations require the rock to be brittle and tensile stresses to be applied. This is a reasonable hypothesis which may be inherent to the initiation of fault zones in a shallow context in early-lithified carbonate rocks. In contrast, later stages of fault development evolve as frictional shear, then probably under a compressive state of stresses allowing fault zone compaction without evidences of fluid transfers from the earth surface.

The observed variations of fault rock diagenesis and lithology are also dependent on the surface environment (marine vs continental), since the fault zone has been in connection with the surface. Opening fractures inherent to the early stages are conducts for gravity-driven sediment infiltration from the sea bottom, which provides strong diagenitic and lithologic modification of the fault rock in a wide part of the fault damage zone. In contrast, the cataclastic core zone constitutes vertical path for late, gravity-driven circulation of continental fluids assisting the formation of micro-karsts.

The diagenetic and lithological changes observed in space and time are the results of the interplay between fault zone mechanical behaviour, diagenesis and earth surface environment. These variations may imply strong petrophysical modifications of the fault rock and potentially provide traps or barriers for fluid migrations within the porous host grainstone.

Figure 1

 1 Figure 1 (a) Overview of the Castellas fault cropping out as a resistant relief along its entire length compared to the eroded porous host grainstone. The background shows the village of La Fare les Oliviers and the Etang de Berre. (b) Scheme of the studied Belvedere fault zone showing the compartmentalisation of the deformation.


Katherine Johnson

Kaisu Pitkälä

Laura Rautiainen

Ashby Sachs

Uwe Siebert

Jill Litt

Keywords: Loneliness, Ecosystem, Interorganizational Networks, Nature Based Social Prescribing

Background: Loneliness is increasingly reported as a growing problem throughout society and is insu ciently addressed through traditional health care systems. Current evidence highlights the physical, emotional, and mental health implications of loneliness. Further, the impact of loneliness as a global public health issue is evidenced by connections to increased obesity, depression, weakened immune systems, and high blood pressure. Interventions aimed at addressing reducing loneliness are being developed and despite their promises, limited evidence of their e cacy exists. In this paper, we look at the cross-sector networks being developed across six cities as a way to develop and implement a speci c intervention (Nature Based Social Prescribing) to address social isolation and loneliness.

Methods: A network analysis of participating countries was launched with the goal of knowing more about potential networks in partner cities that the RECETAS consortium could draw on when designing and implementing NBSP interventions. We were able to visualize and measure the existing networks and create a place-based understanding of how NBSP can be promoted and sustained locally.

Results: The social network analysis reveals similarities and differences across six international cities, and their experiences of establishing networks and collaborating between organizations and their perceptions of and responses to nature-based prescribing and outcomes. These maps are indicative of the number and scope of relational networks that exist in each study site. Our analysis shows that the number of members varies greatly by country, but that quite a few have a large number of relationships and activities. All cities reported an active network in these areas. The membership of these networks was expected, although there are members from unusual sectors that are encouraged to participate in order to diversify the networks.

Conclusions: The RECETAS project will use these network data and results to understand how stakeholders across the six cities are working to address mental health, well-being, and loneliness through social prescribing and nature-based activities with the ultimate goal to systematically improve mental health and well-being, reduce loneliness, promote vibrant socially-connected communities, contribute to the sustainability of cities, and reduce health inequities by connecting diverse populations to nature in meaningful ways.

Background

Loneliness is increasingly reported as a growing problem throughout society and is insu ciently addressed through traditional health care systems. Current evidence highlights the physical, emotional, and mental health implications of loneliness. Further, the impact of loneliness as a global public health issue is evidenced by connections to increased obesity, depression, weakened immune systems, and high blood pressure [1]. Interventions aimed at addressing reducing loneliness are being developed and include support groups, social activities, skill development and short courses. Despite their promises, limited evidence of their e cacy exists [START_REF] Helen | Non-clinical community interventions: a systematised review of social prescribing schemes[END_REF]. However, increasingly approaches that involve increasing social connections between people, organizations, and resources are developed as a mechanism to strengthen and scale proposed solutions to address the population's health. In this paper, we look at the cross-sector networks being developed across six countries as a way to develop and implement a speci c intervention (Nature Based Social Prescribing) to address social isolation and loneliness.

Nature based social prescribing as an intervention to decrease loneliness.

NBSP is access to natural environments and outdoor activities associated with better health and wellbeing [START_REF] Astell-Burt | Green space and loneliness: A systematic review with theoretical and methodological guidance for future research[END_REF]. Living near green and blue areas may encourage physical activity and socialization with friends and neighbors [START_REF] Smyth | Increased Wellbeing following Engagement in a Group Nature-Based Programme: The Green Gym Programme Delivered by the Conservation Volunteers[END_REF]. Accessible solutions to the problem of loneliness are needed within an individual's sociocultural and geographical context. Social prescribing (SP) offers health and social care workers nonmedical referral to community-based activities (e.g., arts-based activities, choirs, gardening) which facilitate social connectedness -a key aspect of physical and mental health -and complement rather than replaces medical treatment [START_REF] Bickerdike | Social prescribing: less rhetoric and more reality[END_REF].

NBSP is an attempt to harness both the therapeutic potential of social connections and health bene ts of the natural environment. NBSP involves organized social activities connected to the natural environment

[6] such as outdoor "green gyms," [START_REF] Smyth | Increased Wellbeing following Engagement in a Group Nature-Based Programme: The Green Gym Programme Delivered by the Conservation Volunteers[END_REF] "park prescriptions," [START_REF] James | A historical and critical analysis of park prescriptions[END_REF], backcountry adventures [8] and farmers market referrals [START_REF] Sadler | Participation in a Fruit and Vegetable Prescription Program for Pediatric Patients is Positively Associated with Farmers' Market Shopping[END_REF]. NBSPs motivate contact with nature and others, therefore supporting connections between wellbeing and the natural world.

Cross-sector community networks as public health interventions to address loneliness.

Given the complex and contextual factors driving loneliness, formulating and implementing effective solutions (such as NBSP) is unlikely to be successful by any one group, organization, or solo effort. While some organizations may not have traditionally seen themselves as part of the solution to ending loneliness, interventions that leverage diverse resources from multiple types of organizations and providers present viable alternatives to the challenges of any one organization solving the issue alone. Not only does this bring additional resources and people together who can help, but it also sparks new ideas and the ability to test a variety of solutions. The bene ts of working collaboratively across sectors to address intractable problems can include leveraging resources, sharing knowledge, joint programming, and covering more ground than any one organization can do alone [START_REF] Hogg | Insights Into Collaborative Networks of Nonpro t, Private, and Public Organizations That Address Complex Health Issues[END_REF]. To successfully implement NBSP at the population level, a muti-level, cross-sector approach can increase knowledge of the approach and elevate feasibility of implementing. Project Context. RECETAS (Re-imagining Environments for Connection and Engagement: Testing Actions for Social Prescribing in Natural Spaces), a multi-country initiative, was designed and implemented to devise, validate, and exploit solutions that address loneliness through nature-based social prescribing (NBSP) interventions. As part of the RECETAS initiative, a cross-sector interorganizational strategy was planned which builds on existing partnerships in local settings, attempts to identify new partners, and maps points of connection in a collective attempt to implement NBSP as an innovative solution to the problem of loneliness. The assumption is that relationships between providers and outdoor social programs present a holistic option for addressing persistent health inequities, confronting unmet psychosocial needs, and reducing burden on the health care system [START_REF] Helen | Non-clinical community interventions: a systematised review of social prescribing schemes[END_REF]. The project began by generating a baseline understanding of local understandings and de nitions of loneliness and NBSP, ascertain what connections /networks exist, and identify similar interventions and approaches. This was an important step because recognition of loneliness as a public health issue and NBSP as a potential solution is relatively new [1]. This study presents one of the only comparative, multi-country assessments of interventions happening at scale to address loneliness and isolation as a public health issue.

The RECETAS strategy to increase NBSP in community begins with mapping the ecosystem of organizations related to this strategy, understanding who is connected to whom, and will leverage that information to create, implement, and scale a NBSP intervention as a pathway to reduce social isolation. Cities participating in the project include Marseille (France), Prague (Czech Republic), Helsinki (Finland), Cuenca (Ecuador), Barcelona (Spain), and Melbourne (Australia). For example, in Finland, forest walks have been organized in primary health care to support the treatment of chronic diseases. However, NBSP in some countries such as Prague, Marseille and Ecuador is a very new concept and its practice is recently emerging. See Fig. 1. Figure 1. Participating Sites

Drawing on Network Science to Understand the Ecosystem of Stakeholder Networks

A growing response to address emerging problems such as loneliness are efforts to build partnerships between organizations [START_REF] Mays | Preventable Death Rates Fell Where Communities Expanded Population Health Activities Through Multisector Networks[END_REF]. While this is an increasingly viable strategy in some countries, others are only starting to think about how building cross-sector relationships can be used to leverage a greater variety of resources and solutions. The development of cross-sector networks to address social issues and deliver programs and services has a long history [START_REF] Provan | A Preliminary Theory of Interorganizational Network Effectiveness: A Comparative Study of Four Community Mental Health Networks[END_REF]. Broadly speaking, networks are "structures of interdependence involving multiple organizations or parts thereof, where one unit is not merely the formal subordinate of the others in some larger hierarchical arrangement" [START_REF] O'toole | Treating networks seriously: Practical and research-based agendas in public administration[END_REF]. However, bringing together diverse partners from diverse sectors presents many challenges. Organizations have their own missions and priorities and may not have the capacity to create new partnerships. Some have worried that increasing networking among organizations takes time away from providing direct services to people [START_REF] Varda | Community Carrying Capacity to Address Health System Referrals: A Networked Approach[END_REF]. The newness of NBSP and developing evidence-base means organizations may be reluctant to contribute to the development of a network addressing loneliness.

Previous research on how heterogeneity (organizations from different sectors and elds) in uences the outcomes of projects was addressed by Varda and Retrum [START_REF] Varda | Collaborative Performance as a Function of Network Members' Perceptions of Success[END_REF] where they found that "some of this complexity-balancing increased diversity while managing varying missions and expectations-could be managed by appropriately leveraging qualities of network members, such as those with high perceptions of trust and those that have many resources to contribute (647)." When asking organizations to create network strategies to address complex problems like mental health, wellness, and speci cally loneliness, it is critical to assess how individual organizations understand the nature of the problem, potential solutions, and established partnerships, including how they work with existing partners and perceptions of trust and value [16].

A revised model of network effectiveness (measured as perceptions of network success) based on an analysis of 98 interorganizational cross-sector networks was developed by Varda and Retrum [START_REF] Varda | Collaborative Performance as a Function of Network Members' Perceptions of Success[END_REF], see Fig. 2. In contrast to the original model which links trust, value, resources, and diversity to perceptions of success, the revised model emphasizes two factors -an increased number of resource contributions and greater diversity among organizational types. Varda and Retrum [START_REF] Varda | Collaborative Performance as a Function of Network Members' Perceptions of Success[END_REF] found that increased contributions from individual organizations and greater diversity within the network resulted in more disagreement over whether a networked approach to a speci ed problem was successful. "[O]n average, higher perceptions of network success are reported by networks that have members with higher perceived trust toward their partners and also have more resources to contribute" [START_REF] Varda | Collaborative Performance as a Function of Network Members' Perceptions of Success[END_REF].

The RECETAS initiative proposed that organizations in communities across six cities around the world collaborate across diverse sectors to address loneliness using NBSP. A key objective is the development of supportive infrastructure and tailoring of NBSP to local contexts through the leveraging of existing assets and resources, and identi cation of opportunities to develop and expand strategic partnerships.

Speci cally, we addressed the following questions:

1. What is the current knowledge of stakeholders on the subject of loneliness, social isolation, and/or NBSP?

2. Who is working on issues related to loneliness, social isolation, and/or NBSP in each city?

3. What activities do organizations engage in? Which population groups do they work with? How do they discuss the issue of loneliness, social isolation, and/or NBSP?

4. What connections exist between organizations working within similar areas? What are the outcomes of those connections?

5. What are the characteristics and nature of these relationships? How do stakeholders currently work together?

. How did COVID-19 impact these efforts?

Methods

A network analysis of participating countries was launched with the goal of knowing more about potential networks in partner cities that the RECETAS consortium could draw on when designing and implementing NBSP interventions. Using network analysis, we were able to visualize and measure the existing networks and create a place-based understanding of how NBSP can be promoted and sustained locally.

In collaboration with local research teams in each of the six pilot study sites: Helsinki, Melbourne, Cuenca, Barcelona, Marseille and Prague, Visible Network Labs (VNL) conducted a cross-sector interorganizational social network analysis. VNL is a social enterprise committed to helping people, organizations, and communities use network science to increase their capacity to engage in collaborative innovation. Social Network Analysis (SNA) is a method that looks at who is connected to whom and how those connections vary and change [START_REF] Wasserman | Social network analysis: Methods and applications[END_REF]. SNA often looks for patterns in those connections to understand things like outcomes or perceptions of network interventions. The goal of the social network analysis was to better understand how stakeholders within individual study sites were working to facilitate social connections and improve mental health and well-being using either social prescribing and / or naturebased activities.

To begin, the teams modi ed the PARTNER CPRM™ Community Network Survey (www.partnertoo.net) and included additional questions ranging from which populations the stakeholder organizations serve to the extent to which they are familiar with the term "nature-based social prescribing". The PARTNER survey includes validated dimensions to determine the overall quality of relationships among organizations in each city based on aspects of value and trust. Four dimensions were used in this study. These include power and in uence, level of involvement, resource contributions, and support of mission (see de nitions below in Fig. 3). Survey participants assessed each of their reported relationships on these four dimensions according to a 4-point scale, with 1 = Not at all, 2 = A Small Amount, 3 = A Fair Amount, and 4 = A great deal. Scores over 3 are considered the most positive. Understanding network relationships is important in leveraging the different ways in which members contribute to the network.

A unique component of the survey is that respondents are asked to share which organizations with whom they currently have a working relationship around addressing mental health, well-being and loneliness. In subsequent questions they are asked to describe their relationship with each of the partners they identi ed. In this network analysis, respondents were asked to select everyone with whom they have an established working relationship around addressing mental health, well-being and loneliness (See Image 6). In subsequent questions, respondents are asked to describe their relationship with each of the partners they selected. These questions cover topics such as what kind of relationship you have with these partners or what is the outcome or intensity of your relationship.

Developing the list of stakeholders who would receive the survey in each of the six cities involved exploratory work and collaboration across all pilot study sites. All sites agreed to include organizations that fell into four different types of categories, which included some boundaries and de nitions of who might be included in this type of work:

1. Organizations Committed to the Mission of Nature-based Solutions and Mental Health: Organizations whose work focuses on this mission in some way.

2. VIP Organizations: VIPs have power and in uence in the community and/or the ecosystem around nature-based solutions and mental health. They are important power brokers and are strategically bene cial to the mission of the network.

3. Not the Usual Suspects: Organizations that have a stake in the ecosystem mission and may facilitate out-of-the box thinking.

4. Voices Representing the Lived Experience: Organizations or individuals that can speak to the lived experience of the social problem or solution the ecosystem is trying to address.

The PARTNER survey was customized for this project and shared with all project leads in each study site.

In countries where English is not predominantly used, the survey was translated into native languages.

Local research teams were encouraged to adjust language to ensure the survey was culturally sensitive and appropriate to the local context.

Data Collection. In October and November 2021, 253 organizations across ve cities (Melbourne, Australia; Helsinki, Finland; Barcelona, Spain; Prague, Czech Republic; and Cuenca, Ecuador) were invited to participate in the survey using VNL's PARTNER platform (www.partnertool.net) to record their current organizational partnerships as part of the RECETAS research project. Within each of the survey email invitations was a link to a RECETAS infographic (which was also translated into local languages) which contained further information about the project. The timing of each survey launch varied across study sites, with various country holidays taken into consideration. The Helsinki, Cuenca, Barcelona, and Melbourne surveys launched in October of 2021 and the Marseille and Czech surveys launched in November of 2021. Of these, 160 organizations responded to the survey, for a 63% response rate.

Another Social Network Analysis was carried out in Marseille, France using a different survey distribution method because Marseille survey respondents were recruited differently than other survey respondents. Due to privacy concerns, a generic (rather than personalized) survey link was distributed to identi ed stakeholder organizations, who then forwarded the survey on to additional stakeholders unknown to the research team in a snowball sample design. A total of 66 organizations responded to the survey, and through their survey responses, 392 organizations were identi ed as working in Marseille to address loneliness, mental health, and well-being.

For response rates speci c to each city please refer to Fig. 4.

Summary of Data Collection Methods:

Social Network Analysis using PARTNER (www.partnertool.net) launched in October and November 2021 across 6 cities.

253 organizations across ve cities (Melbourne, Australia; Helsinki, Finland; Barcelona, Spain; Prague, Czech Republic; and Cuenca, Ecuador) invited to participate.

63% response rate with 160 organizations completing the survey.

In addition, another Social Network Analysis was carried out in Marseille, France using a different survey distribution method. A total of 66 organizations responded to the survey, and through their survey responses, 392 organizations were identi ed as working in Marseille to address loneliness, mental health, and well-being.

Results

Composition & Structure of Stakeholder Networks. The goal of this part of the analysis was to map (visualize) the local landscape and identify who is currently working to address mental health, well-being, and loneliness through social prescribing and nature-based activities, along with where and how stakeholders worked together. By identifying the structure of how these organizations are connected, we can focus on key members (e.g., those with a lot of connections, or those with positions that can act as bridging connections). The network maps below (Fig. 5) provide a visual representation of these partnerships and provides a baseline understanding of these relationships to help us to better understand how organizations work together, including what they do and the outcomes of that work.

Network maps show each organization represented as a circle, or a node (the network science term). Lines between nodes represent all relationships reported by survey respondents within study sites. Nodes are sized by the number of connections they have with one another; larger nodes have more connections. The larger the node, the more connections an organization has. The Helsinki, Cuenca, and Melbourne network maps show a core periphery structure, with more connected organizations located at the center of each network, whereas in the Barcelona, Prague, and Marseille networks, there are no dominant organizations based on number of connections.

Network Composition-Sector & Industry. Across all 6 cities, the majority of stakeholders working in the areas of mental health, well-being and loneliness were associated with the elds of environment/ecology, social services, health care, and public health sectors (see Fig. 6).

Quality of Relationships. Understanding network relationships is important in leveraging the different ways in which members contribute to the network. As mentioned above, four dimensions of relationship quality were captured in the survey. The column chart below depicts the average relationship scores within the network (Fig. 7).

We found similar variation amongst these relationships, with some of the highest scores reported on mission congruence. Interestingly, Marseille has one of the biggest differences between perceptions of respondent organizations as having power and in uence, and mission congruence, which could be due to the different approach used to recruit respondents.

Relationship Outcomes. Respondents reported that 36% of their network relationships involved an exchange of resources, 30% contributed to an increased public awareness and understanding of mental health and well-being, and collaboration resulted in a 26% improvement in organizational capacity (see Fig. 8).

Nature Based Solutions Used. Survey respondents were also asked whether they used nature-based solutions to address mental health, well-being and loneliness and indicate the nature of these activities their organization uses to address mental health, well-being and loneliness. The most used nature-based solutions were green space (47%), outdoor recreation (45%), and nature walks and/or cycling (39%). Less commonly used solutions were Built environments, Pet Meetings, and Farmers Markets (see Fig. 9).

Familiarity with Nature-based Social Prescribing & Solutions. Survey respondents were also asked about their familiarity with some of the terms and words used within the project. In this case we asked people To what extent are you aware of the term "nature-based social prescribing"? As many as 46% of respondents indicate they were either "somewhat" or "very" aware of the term "nature-based social prescribing." Fig. 10 illustrates the average understanding across all sites at the top of the image and then outlines the awareness for individual cities. We found signi cant variation in understanding between study sites. For instance, 64% of respondents in Helsinki were somewhat aware of the term natural-based social prescribing, whereas in Cuenca 44% of respondents, were very aware of the term (note: the pro le of Cuenca respondents was related to educational sector (professors and researchers), who tend to be more aware of NBSP more than others). Marseille respondents were the least familiar with the term nature-based social prescribing, with 66% of respondents answering, "Not at all aware".

Community Impact. Perceptions of Success. Respondents across all sites thought their cities were effective at increasing green space and green infrastructure; and increasing public awareness of problems with mental health, well-being, or loneliness. Respondents likewise expressed the most certainty when sharing their opinion on increased green space and green infrastructure, with as few as 9% reporting "not sure." See Fig. 12.

Impact of Covid-19. As this study was conducted during the COVID-19 Pandemic, we asked how the pandemic impacted the problem mental health, well-being, and loneliness. Most respondents believe that the COVID-19 pandemic increased public awareness of the problem. Respondents were evenly split on how the pandemic effected funding, and signi cantly uncertain on both the question of funding (41%) and the number of people receiving services and treatment (46%) (see Fig. 13).

Discussion

The social network analysis reveals similarities and differences across six international cities, and their experiences of establishing networks and collaborating between organizations and their perceptions of and responses to nature-based prescribing and outcomes. Figure 2 highlights distinct differences in the shape and size of the network in each city. Most have the expected core/periphery structure of most networks, with a large number of organizations linked together at the center and fewer at the edges (e.g., Barcelona and Marseille). However, others demonstrate very decentralized structures (e.g., Helsinki and Prague), and still others show a more centralized structure (e.g., Melbourne and Cuenca). Much of these structural differences could be indicative of variations in response rates, and in one case (Marseille), approach to data collection. However, these maps are indicative of the number and scope of relational networks that exist in each study site. What we learned here is that the number of members varies greatly by country, but that quite a few have a large number of relationships and activities. All cities reported an active network in these areas. The membership of these networks was expected, although there are members from unusual sectors that are encouraged to participate in order to diversify the networks.

Increasing the diversity of networks is considered a strength of a network, which leads to both hard work for the RECETAS project, but also quite a bit of room for improvement. We learned that most organizations identify with the Environment, Ecology or Sustainability sectors or industries, followed by Social Services, Health care, Public Health and Education. We see far fewer organizations that identify with sectors like Technology, Economic Development and Landscape design. This data demonstrates sectors/industries where the network might be interested in growing the presence of these industries in this type of work. To encourage additional cross-sector collaboration, RECETAS may want to consider recruiting network members and building relationships from these less represented industries and sectors to diversify partners involved in this work. And a limitation to note is that many of the local RECETAS consortium partners, who come from a scienti c background, could not be completely aware or updated about the presence of these industries at the start of the study.

In terms of outcomes, it was not surprising that most organizations said that working in partnership leads to an exchange of resources. Many respondents felt that partnerships contributed to greater public awareness and understanding of the issue of loneliness, social isolation, and/or NBSP. In contrast, respondents within most study sites did not feel as strongly about the e cacy of their relationships in reducing city-wide health disparities or contributed to systems change. Overall, some commonality exists across all of the different cities on what respondents felt were the most common results of these relationships and this gives us a baseline for understanding where we can target interventions and identify areas to focus on and increase awareness of the issue of loneliness and potential of NBSP as a possible solution .

At the beginning of the project, it was unclear to what degree networks of organizations in each study site shared a common nomenclature and understanding of many topics informing RECETAS, which SNA suggests are impactful ways to increase effectiveness of interventions. Signi cant variation was found in people's understanding and use of these terms across the different cities. This is also a challenging and urgent task for the project -getting people to understand and adopt a common language about the problem and how to solve it.

Similar to the familiarity of the term nature-based social prescribing, there was signi cant variation in perceptions of the way NBSP was used to address loneliness between cities. In Prague, Helsinki and Melbourne 50% or more of respondents selected "a great deal" or a "fair amount", whereas in Marseille 80% of respondents selected "not at all". This may be representative of which cities have embraced more of this type of work and provides a baseline of where each city is at the start of this project in terms of their understanding of the term nature-based social prescribing and their adoption of this practice to address mental health, well-being, or loneliness.

In terms of community impact, the survey asked respondents whether public health interventions within their city successfully addressed mental health, well-being and loneliness. Systems level change such as increasing green space and green infrastructure and increasing public awareness of problems with mental health, well-being, or loneliness were felt to be more successful. Individual-level treatments like improving access to mental health services or increasing the number of people in hard-to-reach populations who receive treatment were considered less successful. Interestingly, this nding indicates consensus across the cities in the belief that system level work and programs are succeeding but more work is needed to increase access to individual-level programs. This could present an opportunity to see where speci c interventions could make an impact in increasing success in delivering individual-level services to people in these cities.

Finally, as this project was launched during the COVID-19 pandemic, it was an opportunity to better understand how the impact of Covid-19 affected their community's progress in addressing mental health, well-being and loneliness. Most respondents believed that Covid-19 pandemic increased public awareness of the problem of mental health, well-being, and loneliness. Less consensus was found amongst responses to the question whether Covid-19 increased funding to address mental health, wellbeing and loneliness. In fact, respondents' answers were split when it came to funding to address the problem and 41% of respondents were unsure. In addition, almost half of respondents were unsure about whether the Covid-19 pandemic had impacted the number of people receiving services and treatment for mental health, well-being, or loneliness. Overall, public awareness and collaboration has increased as a result of the Covid-19 pandemic, however, direct services being delivered to the community such as the number of people receiving services and treatment for the problem might not have changed.

Conclusion

The RECETAS project will use these network survey data to better understand how stakeholders across the six cities are working to address mental health, well-being, and loneliness through social prescribing and nature-based activities with the ultimate goal to systematically improve mental health and well-being, reduce loneliness, promote vibrant socially-connected communities, contribute to the sustainability of cities, and reduce health inequities by connecting diverse populations to nature in meaningful ways.

These data will also be used to help inform the community building and intervention work of upcoming work packages that are part of the RECETAS research project [START_REF] Hidalgo | -Alias, Co-Creating a Nature-Based Social Prescription Intervention in Urban Socioeconomically Deprived Neighborhoods: A Case Study from RECETAS Project in Barcelona, Spain, Health & Social Care in the Community[END_REF].

Implications for Research, Practice, and Policy. A big question that remains is how these networks will change over time as the project continues. Given the effort to coordinate the remaining work of the project and implement and test solutions, we would expect to see the networks grow and strengthen.

While network science tells us that more is not always better, at the start of a movement (or network intervention) there is a certain amount of growing and changing that must occur. As the networks become clearer on the de nition of nature-based prescribing, evidence of success becomes apparent, and instances of loneliness decrease, more partners (and potentially funders) will want to be involved. Ideally, we will see network members develop more integrated relationships made up of increased programing, client sharing, and knowledge transfer. The research draws which relationships they have as a network, but RECETAS is not working on making this network grow and strengthen but these results are the basis for the cocreation process at the local level to implement the intervention. Outcomes Attributed to Network Relationships

Nature-Based Solutions

Familiarity of NBS Term

Extent of NBSP Adoption

Figure 3 Four 4 Site

 34 Figure 3

Figure 8

 8 Figure 8

  The PARTNER Community Network survey was rst published in 2008 and has been used in hundreds of communities to understand the relationships, resources, and outcomes of cross-sector network strategies. Measures included in the survey include who is working with whom, the activities they engage in, the outcomes of those relationships, and perceptions of value and trust among partners. It also can measure attribution of system building to certain interventions and provides a resource inventory to not only see who has what resources, but how they are connected (or not) to each other. Most of these measures were rst published by the authors of the survey in 2008[18].

Competing interests: The authors declare that they have no competing interests Funding: This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no. 945095 and research in Melbourne is additionally funded by the National Health and Medical Research Council (NHMRC) under grant agreement no. GNT2007059. This work is also supported by the Spanish Ministry of Science, Innovation and Universities through the "Centro de Excelencia Severo Ochoa 2019-2023" Program (CEX2018-000806-S), and by the Generalitat de Catalunya through the CERCA Program. These funding institutions have not taken part in any aspect of the research. Acknowledgement: Not Applicable

Availability of data and materials: The datasets generated and/or analysed during the current study are not publicly available due to multi-country permissions required, but may be available from the corresponding author on reasonable request.

Abbreviations