Sylvain ; Chabbal

Hermann Stoll

Hans-Joachim Werner

Thierry Leininger
email: thierry.leininger@irsamc.ups-tlse.fr

Analytic gradients for the

Keywords: short-range/long-range separation, analytical gradients, DFT/MP2, non-bonded complexes long-range separation, analytical gradients, DFT/MP2, non-bonded complexes

come

For theoretical calculations aiming at the treatment of medium to large systems, energy gradients are indispensable as a tool for geometry optimization. The determination of analytical gradients has been pioneered in quantum chemistry by Pulay [1], and gradients are available nowadays in practically all density-functional and ab-initio wavefunction-based program packages, see [START_REF] Helgaker | Modern Electronic Structure Theory[END_REF] for a review.

It is clear that the emergence of new methods should be accompanied by efforts to also code corresponding energy gradients. The present paper is devoted to the development and implementation of a gradient code for the class of recently developed range-separated hybrid methods which feature a coupling between a density-functional (DFT) treatment for short inter-electronic distances r ij and a wavefunction-based one for large r ij , see e.g. [START_REF] Leininger | [END_REF][4][5]. The introduction of such a coupling should help to improve the accuracy of standard DFT which is not reliable for the treatment of long-range correlation effects; it should also help to reduce the heavy computational effort of wavefunction-based post-Hartree-Fock (post-HF) methods which is due to difficulties in describing the inter-electronic cusp of the wavefunction.

When calculating gradients for a hybrid scheme like sr -DFT/lr -MP2 (short-range DFT coupled with long-range Møller-Plesset second-order perturbation theory), it seems natural to make use of the already available machinery for standard DFT and MP2 gradients. However, the coupling leads to some new features to be discussed in the present paper. The paper is organized as follows: Section 2 contains summaries of the range-separated hybrid method and of the MP2 gradient formulae, and then describes the changes to be introduced in the latter ones for an account of short-range DFT. Application of the newly implemented gradient code for geometry optimization at the sr -DFT/lr -MP2 level for a benchmark set of weakly bound systems [6] is discussed in Section 3, followed by an outlook in Section 4.

Theory

Hybrid method

Here we will briefly review the basic equations leading to the electronic energy expression in the range-separated hybrid formalism. The range-separated electron interaction is obtained by splitting the interelectronic potential in two terms [START_REF] Leininger | [END_REF]

V ee = i<j erfc (µr ij ) r ij + i<j erf (µr ij ) r ij = V SR ee + V LR ee ( 1
)
where erf is the standard error function and erfc its complement. The relative weights of the short-range and long-range parts are controlled by the µ (bohr -1 ) parameter which, in a sense, may be considered as an inverse splitting "distance" for the electronic correlation treatment. For µ = 0, the pure Kohn-Sham result is recovered and for µ → ∞, the ab-initio limit.

In the Levy constrained-search formalism [7], the ground-state energy reads

E 0 = min ρ min Ψ→ρ Ψ |T + V ne + V ee | Ψ (2)
with T the kinetic energy operator, V ne the electron-nuclear potential, and ρ the electron density. By replacing the interelectronic potential with equation (1), we have

E 0 = min ρ E SR 0 [ρ] + min Ψ→ρ Ψ T + V ne + V LR ee Ψ , (3)
where E SR 0 [ρ] is a universal short-range energy functional which is defined as

E SR 0 [ρ] = min Ψ→ρ Ψ |T + V ee | Ψ -min Ψ→ρ Ψ T + V LR ee Ψ , (4)
and can be rewritten as

E SR 0 [ρ] = J SR [ρ] + E SR XC [ρ] . (5)
Here J SR is the short-range Coulomb energy expressed as Analytical gradients for the combined sr-DFT/ lr-MP2 method and E SR XC is the exchange-correlation term which in the local-density approximation (LDA) reads

J SR [ρ] = 1 2 ρ (r
E SR XC [ρ] = ρ SR XC (ρ) dr . (7)
The short-range exchange-correlation energy per particle, SR XC , can be obtained from the difference of the exchange-correlation energies per particle of the electron gas with full and long-range interelectronic interaction, respectively. Generalized gradient approximations (GGA) for the short-range functional are also available [5].

If Ψ LR is the wavefunction obtained from the double minimization in equation ( 3), the ground-state energy reads

E 0 = E SR 0 [ρ Ψ LR ] + Ψ LR |T + V ne + V LR ee |Ψ LR (8)
where ρ Ψ LR is the density associated with Ψ LR . If we assume the form of a single Slater determinant for Ψ LR , its orbitals are eigenfunctions of a Fock-like operator F which contains the functional derivative of E SR 0 [ρ] , δE SR 0 /δρ, as well as Coulomb and exchange operators formed with V LR ee , in addition to the usual one-electron operators. We can then improve on Ψ LR , at the MP2 level, by using H 0 = i F (i) as zeroth-order Hamiltonian and H 1 = T +V ne +V LR ee +δE SR 0 /δρ-H 0 as perturbation. The energy gradients relative to the atomic coordinates X = (x 1 , y 1 , z 1 ; x 2 , y 2 , z 2 ; ...) are obtained by differentiation of the energy functional with respect to X. The energy depends on the mono-and bi-electronic integrals in the AO basis, the MO coefficients, and the MP2 amplitudes. (The short-range functional also depends on the AO basis and the MO coefficients via the density ρ.) Thus, the evaluation of ∂E0 ∂X requires partial derivatives with respect to all these quantities (with the exception of the MP2 amplitudes if the MP2 Hylleraas functional is variationally optimized).

The gradients for the hybrid sr -DFT/lr -MP2 case are obtained by combining (and modifying) the standard DFT and MP2 contributions. This is straightforward for the DFT part. For the MP2 part, the code has to be modified by using longrange bi-electronic integrals and introducing short-range contributions to the Fock matrix. In the next section, we give the resulting equations and details concerning the implementation in the Molpro [8] code.

MP2 gradients

The following notation for the orbital indices is used throughout the text: µ, ν, ... refers to atomic orbitals (AO), i, j, ... to occupied molecular orbitals (MO), a, b, ... to virtual unoccupied orbitals and p, q, r, s, . . . to any molecular orbital. The MOs φ are expressed as linear combinations of atomic orbitals χ,ie, φ i = µ C µi χ µ . We will also use the following conventions : x denotes the derivative with respect to the atomic coordinates; a superscript in parenthesis, (x) , denotes the corresponding quantity evaluated with integral derivatives but using the initial SCF coefficients and MP2 amplitudes, while a superscript in square bracket [x] denotes a quantity evaluated with derivatives of SCF coefficient matrices, but using the zeroth-order integrals. Below, we briefly review the basic equations for the MP2 gradients which have to be modified in the sr -DFT/lr -MP2 case. The derivative of the MP2 energy with respect to atomic coordinates is [START_REF] Azhary | [END_REF]10]:

E x HF + E x M P 2 = µν d µν h (x) µν + 1 2 µνρσ D µν,ρσ (µν |ρσ ) (x) + µν X µν S (x) µν (9)
with h (x) and (µν |ρσ ) (x) the one-and two-electron integral derivatives, S (x) the overlap matrix derivative, and d, D the effective first-and second-order density matrices, respectively. The effective one-electron density matrix d is given by:

d µν = d (0) µν + d (2) µν + pq C µp z pq C µq , (10)
where d (0) is the SCF density matrix, d (2) the MP2 correction to it, and z is the Lagrangian multiplier for zeroing out Fock matrix elements between occupied and virtual orbitals. The second-order density matrix is

D µν,ρσ = 4 ij C µi T ij νσ C ρj + 2 d µν - 1 2 d (0) µν d (0) ρσ -d µρ - 1 2 d (0) µρ d (0) νσ , ( 11
)
where T is the MP2 amplitude matrix and T the contravariant amplitude matrix defined as Tij = 2T ij -T ji . Finally, X is defined as

X µν = -2 ij C µi F ij C νj - 1 2 rs C µr x rs C νs , (12)
where F is the Fock matrix and x is the Lagrangian multiplier for keeping orbital orthogonality. The Lagrangian multipliers x and z are determined from the condition that the MP2 Lagrangian is stationary with respect to orbital changes. In the case of z this leads to the well-known coupled-perturbed Hartree-Fock (CPHF) [11][12][13] equations which have to be solved iteratively. Both the CPHF equations and the equations for x depend linearly on derivatives of the Fock operator with respect to the matrix U -1 where U is a unitary matrix of orbital transformations.

Adaptation to the hybrid case

Our implementation is based on the existing code for the DFT and MP2 gradients in Molpro [START_REF] Werner | version 2009.1, a package of ab initio programs[END_REF]. However, the calculation of the analytical gradients in the hybrid sr -DFT/lr -MP2 scheme requires several modifications which can be illustrated with the formulae of the preceding paragraphs. A summary of these changes is given below:

(1) As a first step, all the bi-electronic integrals in the MP2 gradient of equation ( 9) have to be replaced with corresponding ones involving V LR ee instead of V ee . Also, the pre-screening of the bi-electronic integral derivatives has to be changed.

(2) Derivatives of the short-range Coulomb energy, J SR (x) , and short-range exchange-correlation energy, E

SR (x) XC

, must be added to the long-range MP2 gradient.

(3) In the Fock operator F appearing in X and the equations determining the Lagrangian multipliers x and z, its short-range part, F SR , must be taken into account. Analytical gradients for the combined sr-DFT/ lr-MP2 method (4) In the CPHF equations (and the equations for the Lagrangian multiplier x) also orbital derivatives of F SR [x] are needed.

Modification of the pre-screening

The computation of the atomic integral derivatives in Molpro relies on the Alaska program [14] which uses a pre-screening for two-electron evaluation. In the case of long-range bi-electronic integrals, this pre-screening has to be modified to introduce the sr/lr coupling parameter µ. Thus, equation [START_REF] Werner | version 2009.1, a package of ab initio programs[END_REF] in the original paper of Lindh [14] should be replaced by the following expression :

P ζη,abcd = 2π 5 2 κ ab κ cd 1 η + ζ + ζ+η µ 2 × P ηζ,abcd (13)
where ζ, η, κ ab , κ cd and P ηζ,abcd are quantities depending on the Gaussian basis functions (more details can be found in [15]). Of course for the µ → ∞ limit, the standard prescreening is recovered.

2.3.2. Addition of the J SR (x) and E SR (x) XC contributions
The derivative of the short-range Coulomb energy, for fixed density matrices, has to be added to the long-range MP2 gradient. This term can be written as (cf. equation ( 9) and second term in equation ( 11)):

J SR (x) = µνρσ d µν - 1 2 d (0) µν d (0) ρσ (µν |ρσ ) (x) SR , (14)
where (µν |ρσ )

(x)
SR are the short-range two-electron integral derivatives, obtained from the difference between full-range and long-range integrals. Of course, this adds to the corresponding term of the long-range MP2 gradient to yield the gradient of the Coulomb energy with full interelectronic interaction.

With regard to short-range exchange-correlation, we first need the Hartree-Fock contribution for fixed SCF density matrix d (0) . As other DFT contributions, it is calculated by summation over grid points λ with weights ω λ , and reads for the LDA case as follows:

E SR XC [ρ (0) ] (x) = λ ω (x) λ ρ (0) λ SR XC (ρ (0) λ ) + λ ω λ V SR XC (ρ (0) λ )(ρ (0) λ ) (x) = λ ω (x) λ ρ (0) λ SR XC (ρ (0) λ ) + µν d (0) µν λ ω λ V SR XC (ρ (0) λ )(χ λ µ χ λ ν ) (x) , ( 15
)
where

V SR XC = d ρ SR XC (ρ) /dρ is the short-range exchange-correlation potential; ω (x) λ , (ρ (0) λ ) (x)
describe the changes due to the change of the grid points and the values of the basis functions at the grid points, respectively; χ λ µ is the value of basis function χ µ at grid point λ.

We then have a second contribution which originates from the MP2 Hylleraas functional and is related to the Fock operator (cf. equation ( 9) and third term in equation ( 11)):

µν dµν µ|V SR XC (ρ (0) )|ν (x) = λ ω (x) λ V SR XC (ρ (0) λ )ρ λ + λ ω λ V SR XC (ρ (0) λ )(ρ (0) λ ) (x) ρλ + V SR XC (ρ (0) λ )ρ (x) λ (16)
Here, d = d -d (0) , ρ = µν dµν χ µ χ ν , and V SR XC (ρ) = dV SR XC /dρ. The evaluation of (ρ (0) λ ) (x) and ρ(x) λ proceeds as in equation ( 15).

Expression of the hybrid Fock operator

Using the standard MP2 code with bi-electronic integrals formed over V LR leads to a Fock operator where short-range contributions are missing. Adding the contributions from the short-range Coulomb and exchange-correlation potentials leads to a sum of pure ab-initio (F LR ) and DFT (F SR ) contributions :

F = F LR + F SR (17)
with

F LR µν = h µν + ρσ d µν (µν | ρσ) LR - 1 2 (µρ | νσ) LR (18)
F SR µν = ρσ d µν (µν | ρσ) SR + λ ω λ V SR XC ρ λ d µν χ λ µ χ λ ν ( 19
)
where the χ λ µ are basis function values at grid points. In all expressions involving F in the MP2 gradient expression, the F SR have to be supplemented to the F LR , with density matrices d (0) taken from the preceeding sr -DFT/lr -HF calculation.

CPHF equations

As already mentioned above, the determination of the Lagrangian multipliers which add to the MP2 Hylleraas functional requires the calculation of the derivative, with respect to orbital changes, of an expression of the form tr(F d) [x] where F is the Fock matrix and d = d -d (0) . We are concerned here with the extra terms which arise due to the orbital dependence of F .

The extra term from the short-range Coulomb part in F SR is completely analogous to that in F LR and can simply be handled by using integrals with full interelectronic interaction. For the short-range exchange-correlation potential we have

tr(V SR [x] XC d) = µν dµν < µ|V SR XC (ρ (0) )|ν > [x] (20) = λ ω λ ρλ V SR XC (ρ (0) λ )(ρ (0) λ ) [x] = µν (d (0) µν ) [x] λ ω λ ρλ V SR XC (ρ (0) λ )χ λ µ χ λ ν
where ρ = µν dµν χ µ χ ν (as above), and the χ λ µ are the values of the basis functions at the grid points λ. Analytical gradients for the combined sr-DFT/ lr-MP2 method

The short-range terms discussed above appear linearly in the equation for x and also in the equations for the CPHF iterations. The CPHF part is the most timeconsuming step of our implementation. Indeed, all the density dependent quantities are recomputed on the DFT grid at each CPHF iteration, and the added modifications scale as n grid-points ×N 2 , where N is the number of basis functions, n grid-points the number of grid points.

All these modifications have been implemented and are available in the 2009.2 development version of the Molpro package. The code has been checked against numerical calculations (using a 4-point formula), for several molecular test systems using different basis sets. The following functionals are available : ecerf [16][17][18][19] and exerf [START_REF] Savin | Recent developments and applications of modern Density Functional Theory[END_REF], which are based on the LDA approximation, and ecerfpbe and exerfpbe [START_REF] Goll | [END_REF] wich are based on the GGA/PBE approximation.

In all cases, the difference between numerical and analytical gradient values were less than 1.10 -7 au. It is worthwhile noting that as a byproduct of this code it is also possible to perform geometry optimization at the sr -DFT/lr -HF level. The implementation has been done in such a way that the present method can also be applied in the framework of local-correlation formalism [START_REF] Azhary | [END_REF].

Application to weakly bound systems

In this section, we present results obtained for some of the benchmark databases of Zhao and Truhlar [6]. They comprise a hydrogen bonding database (HB6/04), which consists of six dimers:

(N H 3 ) 2 , (HF ) 2 , (H 2 O) 2 , N H 3 • • • H 2 O, (HCON H 2 ) 2 ,
and (HCOOH) 2 . The second one is a charge transfer database (CT7/04), made of the seven systems:

C 2 H 4 • • • F 2 , N H 3 • • • F 2 , C 2 H 2 • • • ClF , HCN • • • ClF , N H 3 • • • Cl 2 , H 2 O • • • ClF , and N H 3 • • • ClF .
The third one considered is a dipole interaction database (DI6/04), with 6 molecules:

(H 2 S) 2 , (HCl) 2 , H 2 S • • • HCl, CH 3 Cl • • • HCl, CH 3 SH • • • HCN, CH 3 SH • • • HCl.
The last one is a weak interaction database (WI9/04) consisting of nine compounds: HeN e, HeAr, N e 2 , N eAr,

CH 4 • • • N e, C 6 H 6 • • • N e, (CH 4 ) 2 , (C 2 H 2 ) 2 and (C 2 H 4 ) 2 .
We chose these databases because they were already used as tests systems for hybrid methods without geometry optimisation by Goll et al. [4]. Three different basis-sets were used for our calculations: DIDZ [22,[START_REF] Hehre | Ab Initio Molecular Orbital Theory[END_REF] (an abreviation for 6-31+G(d,p)), MG3S [START_REF] Clark | [END_REF][25][26][27][28][29] (approximately corresponding to 6-311+G(3df,2p)), and aug-cc-pVTZ [30][31][32]. We compared values from different hybrid calculations to the W1 references obtained by Zhao and Truhlar [6] and by Boese and Martin [33].

All geometry optimisations with the sr -DFT/lr -MP2 hybrid gradients were done starting from the reference geometries (optimized with a MC-QCISD method) available on the Truhlar group website [34]. For all of the above molecular systems, the dissociation energies D e have then been computed for the optimized geometries using the sr -LDA/lr -MP2 and sr -PBE/lr -MP2 hybrid methods. Results for optimized geometries and dissociation energies are available with and without counterpoise correction. In each case the mean absolute deviation (MAD) with respect to the reference values has been calculated. Typical values of the µ parameter (equation ( 1)) used in the literature lie in the range of between 0.3 and 0.5 bohr -1 . (Figure 1) shows the evolution of the dissociation energy for the H 2 O • • • ClF system in the sr -LDA/lr -MP2 case with the hybrid parameter µ. It can be seen that the convergence to the MP2 value is already achieved for µ = 0.5 bohr -1 (the deviation is less than 5%). This type of convergence has been observed for all the calculations presented here. Thus, all the results below are given for this value of µ. The results for this database are summarized in (Table 1). Using the counterpoise correction yields results closer to the reference values. Indeed, the LDA/MP2 MAD starting from 0.63 (kcal/mol) without CP is reduced to 0.43 (kcal/mol). The same trend is observed for the other methods. Therefore, we used the counterpoise correction for all following discussion. For these compounds, the MAD obtained with LDA/MP2 (0.43 kcal/mol) is not very different from the MP2 one (0.26 kcal/mol), ie, our results with the hybrid method are in good agreement with the reference values. As noted in Ref. [4], the slightly better performance of MP2 is to some extent due to an error cancellation between basis-set effects and errors of the method, while the LDA/MP2 values are already close to the basis-set limit (see below). The PBE/MP2 MAD (0.54 kcal/mol) is not quite as good as the LDA/MP2 one, but is still close to the MP2 one. It is important to notice the small difference between the MADs of the two hybrid methods. This means that we obtain nearly the same level of accuracy with different functionals as already pointed out by Goll et al. [4]. This is clearly not the case for standard DFT calculations. For instance, the MAD is 11.28 (kcal/mol) for LDA and 4.97 (kcal/mol) for PBE. We can also compare our hybrid results obtained after geometry optimisation with those of Ref. [4]. In the latter reference, the MAD for PBE/MP2 without geometry optimisation is 0.66 (kcal/mol). The geometry optimisation then improves the results. Indeed, our value is 0.54 (kcal/mol). If one compares with the results obtained by Zhao et al. [6], it can be noticed that in the case of pure density functional calculations, geometry optimisation basically doubles the MAD. This is obviously not the case for hybrid geometry optimisation. This fact is confirmed by (Table 2), which shows the evolution of the binding distance with geometry optimisation. Differences between the reference geometry and the optimized one obtained with hybrid methods are very small, contrary to pure LDA or PBE results. The MADs for hybrid methods are basically the same with and without counterpoise correction. This is not the case for pure DFT and MP2 methods. In (Table 3), we compare the influence of different basis sets on D e . One can notice a small variation (∼0.2 kcal/mol) of the MAD between the different basis sets. The most important binding energy variation between the basis sets (ie, between MG3S and DIDZ results) is of the same order of magnitude as that observed between DFT/MP2 and MP2 for the same basis set [4]. as the hybrid ones. In this database as for the former one, both hybrid MADs are very close to each other. In (Table 5), the differences between the geometries of the hybrid calculations and the reference geometries are small compared to corresponding differences with pure functionals. This is another proof of the better accuracy of the hybrid methods compared to standard DFT. We can also see that the MAD is nearly the same between LDA/MP2 and PBE/MP2. For these reactions, the MAD values for hybrid methods with counterpoise correction are bigger than without (see Table 6). For example, the PBE/MP2 MAD is 0.10 with CP and 0.05 kcal/mol without. This trend is also observed for MP2, and has already been pointed out by Zhao et al [6] . Our hybrid LDA/MP2 (0.10) and PBE/MP2 (0.10 kcal/mol) values are closer to the reference as compared to LDA and MP2 values (0.67 and 0.14 respectively). Besides, the both LDA/MP2 and PBE/MP2 MADs are the same. (Table 9) shows the counterpoise-corrected optimized geometries for different methods. In this case, the MADs are small for every methods, but the MP2 and PBE MADs are the smallest ones. The LDA/MP2 and PBE/MP2 MADs are the same (0.06 kcal/mol).

Conclusion

Analytical gradients for mixed short-range density-functional (sr -DFT) / longrange Hartree-Fock (lr -HF) and sr -DFT / long-range Møller-Plesset second-order perturbation theory (lr -MP2) methods have been implemented into the Molpro package of ab-initio programs. This exclusive new feature has been validated on the Truhlar benchmark set of weakly interacting molecules. It is shown that it is now possible to optimize geometries of at least similar quality than with conventional MP2 for cases where standard functionals usually fail. Due to the flexibility of the implementation, large molecular systems are accessible using local correlation methods and gradient corrected short-range functionals. In addition, smaller basis sets than for conventional MP2 are sufficient due to the weak basis-set dependence of the mixed method. However, there is still room for improvement with respect to the efficiency and the scaling of the code. Work towards this direction is currently in progress. Dear Editor, since the submission of the above manuscript, we found a cut/paste error in one of the table. Actually, in Table 4 (Dissociation enegies for the DI6/04 set), the "LDA/MP2 cp" column was wrong: the values appearing in the submitted paper were these from column "MP2/cp" of Table 1. As you will see this doesn't change the conclusions of the present work as this erroneous substitution must have appeared close to the end of the manuscript preparation.

  et al.

3. 1 .

 1 CT7/04.

  @irsamc.ups-tlse.fr Ref.: Manuscript n°TMPH-2010-0224

I

  send enclosed a corrected version of the manuscript. I am very sorry for this. Please send all my apologies to the referees. With my best regards, Thierry Leininger Page 14 of 14 URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

Table 1 .

 1 Dissociation energies (kcal/mol) for the CT7/04 compounds for different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [6]
		no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	C 2 H 4 ...F 2	17.25	10.32	4.92	4.43	1.16	0.77	1.25	0.87	1.56	0.86	1.06
	N H 3 ...F 2	20.22	19.50	14.3	13.76	1.71	1.28	1.87	1.46	1.99	1.33	1.81
	C 2 H 2 ...ClF	14.67	14.21	7.86	7.55	4.36	4.00	4.71	4.37	4.89	3.94	3.81
	HCN...ClF	14.21	13.55	6.90	6.52	5.81	5.45	5.97	5.65	5.72	4.87	4.86
	N H 3 ...Cl 2	15.51	15.05	9.62	9.22	5.19	4.72	5.51	5.06	5.59	4.65	4.88
	H 2 O...ClF	14.68	13.95	8.78	8.18	6.29	4. 80	6.49	5.88	6.00	4.80	5.36
	N H 3 ...ClF	25.45	24.80	17.98	17.52	12.10	11.34	12.49	11.82	11.95	10.38	10.62
	MAD	12.80	11.28	5.42	4.97	0.63	0.43	0.84	0.54	0.76	0.26

Table 2 .

 2 Intermolecular distances ( A) for CT7/04 compounds with different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction.

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [34]
		no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	C 2 H 4 ...F 2	2.272	2.287	2.490	2.510	3.014	3.095	2.960	3.054	2.912	3.020	3.053
	N H 3 ...F 2	1.859	1.864	1.932	1.937	2.678	2.730	2.629	2.673	2.646	2.743	2.696
	C 2 H 2 ...ClF	2.371	2.381	2.529	2.542	2.827	2.846	2.790	2.806	2.800	2.865	2.876
	HCN...ClF	2.076	2.095	2.308	2.334	2.532	2.554	2.515	2.535	2.548	2.606	2.609
	N H 3 ...Cl 2	2.277	2.283	2.388	2.394	2.644	2.663	2.611	2.624	2.603	2.660	2.688
	H 2 O...ClF	2.246	2.255	2.369	2.380	2.476	2.494	2.461	2.483	2.512	2.564	2.557
	N H 3 ...ClF	2.120	2.127	2.195	2.204	2.2410	2.259	2.237	2.253	2.234	2.276	2.302
	MAD	0.509	0.498	0.367	0.354	0.053	0.042	0.082	0.051	0.075	0.022

Table 3 .

 3 Comparison of dissociation energies (kcal/mol) for CT7/04 compounds with different basis sets using the sr -PBE/lr -MP2 approach with (cp) and without (no-cp) counterpoise correction.

	PBE/MP2	DIDZ	MG3S	aug-cc-pVTZ
		no-cp	cp	no-cp	cp	no-cp	cp
	C 2 H 4 ...F 2	1.39	0.66	1.25	0.87	1.17	1.08
	N H 3 ...F 2	2.91	1.93	1.87	1.46	1.69	1.63
	C 2 H 2 ...ClF	3.72	2.95	4.71	4.37	4.64	4.50
	HCN...ClF	5.32	4.84	5.97	5.65	5.93	5.74
	N H 3 ...Cl 2	6.00	4.87	5.51	5.06	5.31	5.20
	H 2 O...ClF	5.32	5.85	6.49	5.88	6.07	5.96
	N H 3 ...ClF	12.59	11.14	12.49	11.82	12.08	11.90
	MAD	0.80	0.35	0.84	0.54	0.77	0.57
	3.2. DI6/04.
	(Table 4) shows the evolution of D e (kcal/mol) for different methods. Here, the
	MAD of the hybrid LDA/MP2 and PBE/MP2 results (0.33 and 0.36 kcal/mol,
	respectively) are significantly smaller than values obtained with standard LDA
	(3.93 kcal/mol) but only moderately smaller than standard PBE (0.47 kcal/mol).
	In comparison, the MP2 MAD (0.24 kcal/mol) is of the same order of magnitude

Table 4 .

 4 Dissociation energies (kcal/mol) for DI6/04 compounds with different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction.

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [6]
		no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	(H 2 S) 2	4.34	4.09	1.86	1.69	1.90	1.72	1.90	1.73	1.88	1.55	1.66
	(HCl) 2	5.00	4.73	2.19	1.99	2.23	2.01	2.25	2.04	2.24	1.81	2.01
	HCl • • • H 2 S	8.14	7.81	4.45	4.19	3.85	3.57	3.86	3.59	3.75	3.20	3.35
	CH 3 Cl • • • HCl	7.92	7.63	3.58	3.37	3.89	3.63	3.97	3.72	3.90	3.28	3.55
	HCN • • • CH 3 SH	6.91	6.73	3.71	3.60	4.07	4.07	4.10	4.10	3.91	3.44	3.59
	CH 3 SH • • • HCl	11.27	10.93	6.15	5.87	5.61	5.29	5.64	5.32	5.46	4.69	4.16
	MAD	4.21	3.93	0.60	0.47	0.54	0.33	0.57	0.36	0.47	0.24

Table 5 .

 5 Intermolecular distances ( A) for DI6/04 compounds with different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction.

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [34]
		no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	(H 2 S) 2	3.752	3.756	4.054	4.083	4.106	4.136	4.096	4.096	4.110	4.184	4.115
	(HCl) 2	3.464	3.470	3.722	3.742	3.774	3.779	3.767	3.774	3.778	3.742	3.787
	HCl • • • H 2 S	3.438	3.448	3.612	3.622	3.699	3.708	3.694	3.708	3.731	3.779	3.756
	CH 3 Cl • • • HCl	3.330	3.335	3.574	3.566	3.571	3.592	3.558	3.570	3.599	3.644	3.611
	HCN • • • CH 3 SH	3.295	3.297	3.625	3.645	3.563	3.565	3.548	3.549	3.597	3.645	3.616
	CH 3 SH • • • HCl	3.315	3.320	3.489	3.495	3.538	3.550	3.533	3.545	3.571	3.621	3.607
	MAD	0.316	0.311	0.072	0.066	0.040	0.034	0.049	0.041	0.017	0.036
	3.3. WI9/04.

Table 6 .

 6 Dissociation energies (kcal/mol) for WI9/04 compounds with different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction.

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [6]
		no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	N eHe	0.40	0.33	0.13	0.09	0.03	0.01	0.02	0.01	0.05	0.01	0.04
	ArHe	0.39	0.35	0.10	0.07	0.06	0.04	0.05	0.04	0.05	0.03	0.06
	N e 2	0.59	0.42	0.21	0.11	0.09	0.01	0.08	0.01	0.18	0.01	0.08
	ArN e	0.71	0.56	0.23	0.14	0.13	0.04	0.12	0.04	0.21	0.03	0.13
	CH 4 • • • N e	0.84	0.64	0.27	0.15	0.18	0.06	0.17	0.06	0.29	0.04	0.22
	C 6 H 6 • • • N e	1.53	1.13	0.39	0.19	0.51	0.21	0.50	0.22	0.63	0.13	0.47
	(CH 4 ) 2	1.23	1.14	0.16	0.16	0.41	0.36	0.44	0.39	0.41	0.32	0.51
	(C 2 H 2 ) 2	2.54	2.71	1.02	1.23	1.48	1.37	1.53	1.41	1.49	1.27	1.34
	(C 2 H 4 ) 2	3.10	3.03	0.56	0.53	1.52	1.32	1.60	1.51	1.57	1.17	1.42
	MAD	0.78	0.67	0.22	0.20	0.05	0.10	0.06	0.10	0.09	0.14
	(

Table 7 )

 7 shows the geometries for different methods with counterpoise correction. The hybrid LDA/MP2 and PBE/MP2 MADs are the smallest ones, and are basically the same (0.18 and 0.17 kcal/mol respectively).

	June 27, 2010	9:41	Molecular Physics	RSH_gradient_070610
			Analytical gradients for the combined sr-DFT/ lr-MP2 method
			F o
			r
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				e e r
				R e v i e w
				O n
				l y

Table 7 .

 7 Intermolecular distances ( A) for WI9/04 compounds with different approaches using MG3S basis sets with counterpoise correction.For this last subset, the hybrid no-cp MADs are not as good as for the previous ones (seeTable 8). Indeed, in this case the MP2 MAD is 0.27 kcal/mol, while it increases up to 1.55 for LDA/MP2 and 1.37 for PBE/MP2. Using the counterpoise correction leads to better results. With this correction, the LDA/MP2 MAD is reduced to 0.94 kcal/mol (compared to 1.55), the PBE/MP2 MAD to 0.81 kcal/mol (compared to 1.37). These results are close to the counterpoise corrected MP2 MAD (0.91 kcal/mol). However, the PBE MAD (0.29 kcal/mol) is better than the hybrid ones. The difference between the LDA/MP2 MAD and the PBE/MP2 MAD is very small again. The MADs for hybrid methods are less sensitive to the basis set quality than for standard MP2 and DFT methods. For example, the differences of the MADs between the most accurate basis set (aug-cc-pVTZ) and the least accurate one (DIDZ) are 0.64 and 1.95 (kcal/mol) for the MP2 and LDA calculations, respectively, and reduces to 0.06 (kcal/mol) for LDA/MP2. We also note that the difference of D e with/without geometry optimization is weak: in our case, we obtained a MAD of 0.87 for optimized PBE/MP2 calculations compared to 0.97 (kcal/mol) without optimisation[4].

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [34]
	NeHe	2.476	2.911	3.449	3.433	3.059	3.031
	ArHe	2.937	3.495	3.617	3.604	3.708	3.480
	Ne 2	2.583	3.156	3.553	3.544	3.648	3.091
	ArNe	3.016	3.490	3.494	3.493	3.868	3.489
	CH 4 • • • Ne	2.971	3.536	3.683	3.723	3.769	3.486
	C 6 H 6 • • • Ne	3.183	3.784	3.692	3.660	3.950	3.509
	(CH 4 ) 2	3.329	4.040	3.759	3.712	3.788	3.615
	(C 2 H 2 ) 2	3.530	3.867	3.442	3.453	3.477	3.456
	(C 2 H 4 ) 2	3.551	4.154	3.849	3.856	3.892	3.834
	MAD	0.396	0.254	0.187	0.166	0.241
	3.4. HB6/04.

Table 8 .

 8 Dissociation energies (kcal/mol) for HB6/04 compounds with different approaches using MG3S basis sets with (cp) and without (no-cp) counterpoise correction.

		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [6]
		no-cp	cp	co-cp	cp	no-cp	cp	no-cp	cp	no-cp	cp
	(N H 3 ) 2	6.13	5.93	3.34	3.16	3.50	3.25	3.51	3.27	3.33	2.91	3.15
	(HF ) 2	8.59	8.01	5.11	4.67	5.49	4.93	5.31	4.85	4.91	3.96	4.57
	(H 2 O) 2	9.29	8.62	5.58	5.05	6.04	5.47	5.92	5.40	5.47	4.54	4.97
	N H 3 ...H 2 O	11.42	10.99	7.25	6.89	7.40	6.96	7.30	6.90	6.78	5.97	6.41
	(HCON H 2 ) 2	23.27	22.72	14.37	13.99	16.83	16.05	16.66	15.95	14.88	13.25	14.94
	(HCOOH) 2	29.29	28.19	16.94	16.24	20.33	19.19	19.70	18.67	16.29	14.10	16.15
	MAD	6.30	5.71	0.59	0.29	1.55	0.94	1.37	0.81	0.27	0.91

Table 9 .

 9 Intermolecular distances ( A) forHB6/04 compounds with different approaches using MG3S basis sets with counterpoise correction.

	Page 11 of 14			Molecular Physics
		LDA	PBE	LDA/MP2	PBE/MP2	MP2	Ref [34]
	(N H 3 ) 2	3.272	3.257	3.230	3.221	3.247	3.267
	(HF ) 2	2.569	2.750	2.720	2.720	2.820	2.783
	(H 2 O) 2	2.732	2.917	2.883	2.883	2.980	2.937
	N H 3 • • • H 2 O	2.757	2.910	2.902	2.901	2.974	2.967
	(HCON H 2 ) 2	2.697	2.862	2.833	2.833	2.911	2.881
	(HCOOH) 2	2.478	2.633	2.600	2.614	2.720	2.702
	MAD	0.173	0.035	0.061	0.061	0.026

URL: http://mc.manuscriptcentral.com/tandf/tmph

URL: http://mc.manuscriptcentral.com/tandf/tmph Molecular Physics

Analytical gradients for the combined sr-DFT/ lr-MP2 method

Acknowlegdments S.C. thanks the Deutsch-Französische Hochschule (DFH-UFA) and the Ecole Doctorale des Sciences de la Matière of the University Toulouse 3 for financial support. T.L. thanks the Alexander von Humboldt Foundation for visiting grants to the Institut für Theoretische Chemie, University of Stuttgart.


Karine Constant
email: constant.karine@gmail.com.

Carine Nourry
email: carine.nourry@univmed.fr.

Thomas Seegmuller
email: thomas.seegmuller@univmed.fr.

Document de Travail n°2011-43 Polluting Industrialization

Keywords: JEL classification: J13, 044, Q56 Growth, Population dynamics, Pollution, Altruism, Development

Recently, many contributions have focused on the relationship between capital accumulation, growth and population dynamics, introducing fertility choice in macro-dynamic models. In this paper, we go one step further highlighting also the link with pollution. We develop a simple overlapping generations model with paternalistic altruism according to wealth and environmental concerns. One can therefore explain a simultaneous increase of capital intensity, population growth and pollution, namely a polluting industrialization. We show in addition that a permanent productivity shock, possibly associated to technological innovations, promotes such a polluting development process, escaping a trap where the economy is relegated to a low capital intensity, population growth and pollution.

Introduction

According to the hysteretic nature of the Environment,1 contemporary actions condition the wellness of future generations. Consequently, it seems crucial to understand its long term trend. Historically, the development of human activities has generated a huge increase in pollution, starting with the industrialization. Actually, this event represents a break in the effects of mankind on his environment and even the beginning of a new geological era: the Anthropocene. This term, coined by [START_REF] Crutzen | The Anthropocene[END_REF], 2 describes the current geological epoch where the impacts of human activities on earth and atmosphere are become predominant. Still informal, the large diffusion of this expression in the geological literature and its recognition by the Stratigraphic Commission of the Geological Society of London show the importance of the phenomenon and lead us to look into this period of development. 3The awareness of this process is confirmed by the extensive literature in environmental economics (e.g. [START_REF] Howarth | Environmental Valuation under Sustainable Development[END_REF], [START_REF] Gradus | The Trade-off between Environmental Care and Long-term Growth; Pollution in Three Prototype Growth Models[END_REF], [START_REF] Michel | Disutility of Pollution and Endogenous Growth[END_REF] or [START_REF] Xepapadeas | Economic Growth and the Environment[END_REF]). Nevertheless, as emphasized by [START_REF] Brock | Economic Growth and the Environment: a Review of Theory and Empirics[END_REF], "the relationship between economic growth and the environment is and may always remain controversial". While some economists see the pollution-income relationship (PIR) as monoticaly positive, others see it as an inverted U-shaped curve, also known as the environmental Kuznets curve (EKC), discovered by [START_REF] Grossman | Environmental Impacts of a North American Free Trade Agreement[END_REF] and so called by [START_REF] Panayotou | Optimal Tax Schemes and the Environmental Externality[END_REF]. According to the second group, the economic growth increases pollution in the early stages of development, but beyond some level of income per capita, the trend reverses, so that at high income levels, economic growth leads to environmental improvement. Whereas there exists theoretical explanations to this phenomenon: definition of environment as a luxury good, technological progress making able to be less polluting, or sectoral change diminishing the share of industry in favor of services etc. There are also theoretical contestations for each argument: pollution decrease only per unit of output and not in absolute levels, pollution-intensive industry are exported to the least developed countries etc. ; at witch add a lack of empirical proof, since the relation is verified only for some pollutants. 4Here, we will introduce an important missing dimension inside this debate: the demographic one. Indeed, many papers emphasize that population is a key element in the economic development process (e.g. [START_REF] Ehrlich | The Problem of Population and Growth: A Review of the Literature from Malthus to Contemporary Models of Endogenous Population and Endogenous Growth[END_REF], [START_REF] Galor | Population, Technology and Growth: From Malthusian Stagnation to the demographic Transition and Beyond[END_REF] or [START_REF] Galor | From Stagnation to Growth: Unified Growth Theory[END_REF]). Despite these two abounding literatures and the consensus on the existence of a link between them, economists have rarely focused on the relationship between population, economic and pollution growth, as underlined by [START_REF] Chu | Population Dynamics and the Decline in Biodiversity: a Survey of the Literature[END_REF] and [START_REF] Robinson | Long-term Consequences of Population Growth: Technological change, Natural Resources, and the Environment[END_REF]. Those are the reasons why we are interested in the role of population. More precisely, we will study it for the first part of the PIR where pollution and economic growth evolve in the same way. We want to look at the effect of endogenous population growth à la [START_REF] Barro | Fertility Choice in a Model of Economic Growth[END_REF] on factors accumulation during industrialization.

Our motives for focusing on this complex link, at low development levels, stems from several intuitive elements. Concerning growth and population, we have observed empirically a positive adjustment of population to an increase in income per capita due to industrialization. 5 However, the population evolution has allowed a very important increase in production through a heavy increase in demand, favoring cumulative growth process [START_REF] Bairoch | Victoires et Déboires: Histoire Economique et Sociale du Monde du XVIe siècle à nos Jours[END_REF]). About the connection between growth and pollution, we know that the production process often causes environmental damages, but as [START_REF] Dasgupta | Population, Poverty, and the Natural Environment[END_REF] reminded, environment being widely seen as a luxury when a country is richer, environmental concerns are also stronger. With regard to population and pollution, on one hand population growth generates a priori more pollution and resources depletion but on the other hand, pollution, affecting wellness has an influence also on demographics behaviors. The aim of this paper is to develop a simple growth model with endogenous fertility and pollution change being able to understand the evolution of capital intensity (a proxy of GDP per capita), population growth and pollution, during the industrialization process. 6 In order to study the intergenerational aspects, we are going to employ an overlapping generations model, due to [START_REF] Allais | Économie et Intérêt[END_REF], [START_REF] Samuelson | An Exact Consumption-Loan Model of Interest With or Without the Social Contrivance of Money[END_REF] and [START_REF] Diamond | National debt in neoclassical growth model[END_REF]. This standard tool was used more lately in environmental economics by seminal papers as [START_REF] John | An Overlapping Generations Model of Growth and the Environment[END_REF] that pointed out that "such a demographic structure permits analysis of situation where agent's action have consequence that outlive them". So, this kind of model turns out to be well suitable for pollution issues and our long-term environmental analysis, allowing to look at the external effects of finite-life agents' choices on environment. The question, which arises then, concerned the environmental bequest, deriving from a trade-off between capital bequest, consumption's choices and ecological impacts.

We consider an overlapping generations model with pollution in which altruism has a central role as Jouvet, [START_REF] Jouvet | Intergenerational Altruism and the Environment[END_REF]. But contrary to these authors and for more realism, the altruism is paternalistic: parents experience a warm glow from leaving bequest to their offspring, i.e. joy-ofgiving. This legacy is multiple and related to the amount of wealth parents want to leave to heirs and to environmental perspective, illustrating their volition to pass on an favorable environment to their children. The first is classic and called here the wealth's altruism, while the second is new and constitutes an environmental altruism. Regarding this second type of altruism, it is important to notice that households take care about both natural and economic environment. We set, therefore, a perception index of pollution adjusted to development: when households anticipate an increase in standard of living, they tolerate, as an acceptable compensation, a higher level of pollution. This second altruism is thus not a standard green altruism but an index of future generations' well-being according to natural and economic environment preoccupations. Note that it allows the existence of a positive link between wealth and pollution, corresponding to the facts.

Against the prejudice about environment problems recognition in the era studied, the environmental altruism is justified. Indeed, pollution is an old issue: despite its contemporary-looking, this word exists since the 12th century and is a matter of concern for a long time. 7 As nowadays, its definition represented multifaceted realities but was not identical, evolving over time with knowledge and technology. It concerned especially urban sanitary issues, resource depletion (particularly deforestation), but also later, water and air pollution. Studying the first stages of development, we will use this large definition and look at local pollution related to agrarian and urban concerns.

We do not take into account explicitly mortality in our model, meaning that fertility and population growth are represented by the same variable, as in well-known contributions like [START_REF] Galor | Population, Technology and Growth: From Malthusian Stagnation to the demographic Transition and Beyond[END_REF]. This assumption is supported by the stage under consideration where birth rate and population growth experienced a common trend. In addition, at this time, the birth rate increased while the decrease of mortality was weak. 8 Actually, we consider a fertility rate in net terms which represents the desirable number of survival offspring that agents choose. Parents know that there is infant mortality but they can not control it, as medicine was not enough developed. When income increases, there would have an increase in the number of children who will survive to the childhood.

The benefits of having children through altruisms are weighted against a cost. For that, we adopt a constant cost of rearing children in terms of final good, appropriate for the early stages of development on which we are interested. 9 It allows wealth and population to evolve in the same direction, as shown by empirical studies on this era.

These key features of the model allow us to show that if the economy has not too low initial conditions on wealth and population size, it experiences a polluting industrialization, i.e. the convergence to a steady state with high capital intensity, population growth and pollution. Indeed, the constant cost of rearing children in terms of final good promotes that an increase in population size goes with the rise in capital intensity. Moreover, environmental altruism involves that households agree that their children face a less healthy environment if the economy is more developed, i.e. the population growth is larger. On the contrary, if the economy starts with too low initial conditions, it may be relegated to a poverty trap characterized by a low capital intensity, population growth and pollution.

In this case, the wealth level is not large enough to engage a growth process. Finally, we demonstrate that a permanent technological shock promotes a polluting industrialization process, being able to escape the economy from the poverty trap and to converge to the industrial steady state characterized by high capital intensity, population growth and pollution.

The paper is organized as follows. The next section presents the overlapping generations model with paternalistic altruism, endogenous fertility and pollution. Section 3 provides our explanation of polluting industrialization, analyzing steady states, dynamics and the effect of a permanent technological shock. We conclude in Section 4, while several technical details are relegated to an Appendix.

The Model

We present a simple overlapping generations model which allow us to analyze the trend of capital intensity, population growth and pollution during the early stages of the process of development,

i.e. what we call a polluting industrialization. Our explanation is mainly based on the behavior of households, which make choices according to their two motives of altruism, a wealth one and an environmental one. Hence, to have a convenient model, the production sector and the pollution change are kept as simple as possible.

The Environment

Despite its contemporary-looking, environmental deterioration is an old issue. Indeed, the word pollution exists since the 12th century and numerous official reactions against pollution issues have been observed since this time. 10If the environment refers nowadays to multiple concepts, like sustainable development, biodiversity, natural resource, or air, water and ground qualities, it was also the case in the first stages of development. The same form of generality persists when we focus on developing or industrializing countries, only the different facets of definition change (evolving with knowledge and technology).

In pre-industrial societies, pollution can mainly be appreciated from an agrarian point of view with resource depletion (e.g. deforestation) or insufficiency of agricultural production, but also from an urban one referring especially to sanitary issues. Then, with development toward industrial regime, other problems, closer to present one, are added as air and water pollution, because of the inconveniences of the industrial production process. Consequently, we will consider a large definition of Environment, including all these aspects. Environmental quality will be described by an aggregate index, as usual in most of the macro-dynamic models. More precisely, we use an index of pollution stock P t which represents the environmental damages (opposed to environmental quality). 11 Moreover, we privilege a local view of pollution which evolves according to the following law of motion:

P t 1 p1 ¡ αqP t aN t c t (1)
With P 0 the pollution level in t=0 given, N t the population size and c t the individual consumption, at period t.

The accumulation of pollution is due to aggregate consumption N t c t , with a ¡ 0 the rate of pollution flow. We want to center our analysis on households behavior, consequently we choose a polluting consumption. This usual assumption allows us to look at the effect of consumers on environment more directly than with polluting production 12 . The environment regenerates at a rate α p0, 1q, i.e. in the absence of human activity, it can reconstitute itself partly. Such a phenomenon is called ecological resilience and refers to the reversible part of pollution. Note that our formulation is similar to the one of John and Pecchenino (1994), without environmental maintenance, which is done in our model only through consumption choice.

Households

Households' behavior plays a key role in our analysis. We consider a representative family composed to N t adults and their children. The choices are made at the family level such that: the head of the family makes choices, knowing that his relatives has the same preferences than him ; these choices are then followed by all family members of his generation. 13 In order to examine the intergenerational consequences of choices, we develop an overlapping generations model with endogenous fertility and paternalistic altruism. With this form of altruism, parents derive utility not from their children's utilities (corresponding to dynastic altruism), but from the size of the wealth they leave to them. The bequests are related to parental view on what is good for their heir, and to the pleasure they derive from giving. The reason why we choose this paternalistic altruism or impure one as called by [START_REF] Andreoni | Giving with Impure Altruism: Applications to Charity and Ricardian Equivalence[END_REF] rather than a dynastic or pure one à la [START_REF] Barro | Are Government Bonds Net Wealth?[END_REF], is the analytical convenience of this form, allowing a non optimal framework, essential to analyze externalities across generations. Moreover, the joy-of-giving is more realistic than the pure altruism which requires human foresight capacities "that are beyond capacities of the most prescient" as Becker (1993) admits.

We grant a fundamental place to altruism in our model, studying two types of legacy: a wealth one, equivalent to a classic joy of giving (i.e. in term of capital bequest), and an environmental one, taking into account natural and economic environment. Each adult is interested in what he passes down to his children: a capital inheritance and a favorable environment relatively to the development level (embodied by an environmental quality index). Thus, we will shed light on intergenerational 11 Pollution stock is defined as the opposite of environmental quality . 12 The same choice was done by John and Pecchenino (1994), John and al. (1995)), [START_REF] Ono | Optimal Tax Schemes and the Environmental Externality[END_REF]etc. 13 Our formulation is in the spirit of the dynastic framework of choice à la Ramsey (1928) but with finite-lived agents.

transmissions of the capital and the environment and theirs interplays with other private decisions, concerning consumption and children, over agent's life-cycle.

About environmental concerns, we assume that agents take into account a perception index of environmental damages adjusted for family size. Here, family size plays the role of a family prosperity index (the coming family income): larger the family is, larger the workforce is and larger the income is. 14 Thus, households make a trade-off between a lower level of pollution and a higher level of family wealth. When they expect higher standards of living, they tolerate a higher level of pollution as an acceptable compensation. As illustrations of this compromise, [START_REF] Bairoch | Victoires et Déboires: Histoire Economique et Sociale du Monde du XVIe siècle à nos Jours[END_REF] observes that more than a break with the social and family backgrounds, migration toward city associated with the industrialization process of England, was accompanied by urban excess of death and appalling life conditions. [START_REF] Williamson | Did British Capitalism Breed Inequality?[END_REF] and [START_REF] Brown | The Condition of England and the Standard of Living: Cotton Textiles in the Northwest, 1806-1850[END_REF] found also that a large part of the raise in wage in the factory sector, during industrialization, appears to be explained as compensation for poor working and living conditions. At an aggregate level, the family prosperity index is comparable to a development index, considering the important link between economic and demographic growths in the era studied. 15 An increase in population generates a larger number of producer and consumer, which favors the economic growth process. Finally, note that this environmental altruism is not per se a barrier to economic development and will allow the existence of a positive relationship between wealth and pollution, corresponding to the first step of the environmental Kuznets curve. Households live two periods, childhood and adulthood, but take all decisions during their second period of life. The utility of a household, born in t ¡ 1, depends on his consumption level during his second period of life (c t ), on his number of children (n t ), on the amount of capital kept at life-ending with the intention of bequeath it to his children (x t n t k t 1 ), and finally on the index of pollution stock he leaves to future generations (Π t 1 P t 1 {N t 1 ).

We choose to represent preferences through a specified utility function to have a simple model, which captures, however, the mechanisms we want to emphasize. The preferences of an agent born in t ¡ 1 are defined by the following utility function:

ln c t 1 lnpx t q ¡ 2 n t Π 1 µ3 t 1 1 µ 3 (2)
where 1 ¡ 0 and 2 ¡ 0 are the paternalistic altruism factors on wealth and environmental concerns, respectively, and µ 3 ¡ 0. Note that 2 Π 1 µ3 t 1 {p1 µ 3 q can be interpreted as a disutility of relative pollution per child. 16 During their first period of life, agents are children and economically inactive, but they generate an individual cost of rearing, β ¡ 0, to their parents. We assume a constant cost in terms of final good, which allows wealth and population to evolve in the same direction. The wealthier people are, the more children they want to have. Such an assumption is appropriate for the pre-industrial period, more than the alternative in time that would introduce a quality-quantity trade-off in term of child leading to a demographic transition, away from our point. 17 Moreover, this usual assumption, used by [START_REF] Becker | A Reformulation of the Economic Theory of Fertility[END_REF], is justified empirically in this era: the cost of raising a child seems to remained quite stable and weak during this period. Indeed, it corresponds to a simply cost necessary 14 Being in a full employment framework without technological progress, economic growth will be driven by population growth in the long run, as usually in exogenous growth model. 15 See Galor (2005). 16 We choose a utility function specification such that there is "distaste effects" of pollution on consumption (U cP 0) and on the number of children (U nP 0): an increase in pollution implies a decrease in marginal utility to consume and to have children. See [START_REF] Michel | Disutility of Pollution and Endogenous Growth[END_REF]. 17 Our paper deals with first stages of development process. Our study stops before the demographic transition, representing the beginning of the modern economy (current developed countries level).

to ensure children's subsistence, notably because of child labor and the lack of education even with industrialization. 18 During their second period of life, agents become adults and are subject to a budget constraint.

Each agent supplies inelastically one unit of labor remunerated at the wage rate w t and inherits an amount of capital k t from their parents remunerated at the return R t . These incomes are shared between consumption (c t ), bequest (x t ), and children rearing (n t β). Thus, an adult faces the following budget constraint:

c t n t pk t 1 βq R t k t w t (3)
Finally, the number of adults N t in period t is given by the number of adults N t¡1 in t ¡ 1 multiplied by the number of children n t¡1 they choose to have at this period. Therefore, the evolution of population is given by:

N t n t¡1 N t¡1 (4)
with N 0 1 given. We can observe that intergenerational externalities intervene in the equation of pollution: my choice in term of number of children has an effect on my grandchild's pollution.

Maximizing the utility function (2) subject to the budget constraint (3), the pollution change [START_REF] Allais | Économie et Intérêt[END_REF] and the evolution of population (4) lead to two first order conditions. The first one corresponds to the agent's trade-off in term of capital and is given by:

c ¡1 t 1 x ¡1 t 2 a Π µ3 t 1 (5)
Reducing consumption to increase capital bequest has a cost corresponding to the marginal utility (c ¡1 t ) and a direct benefit through the marginal utility (x ¡1 t ) weighted by the altruism factor 1 . However, a second benefit goes through environmental quality: consumption being polluting, increasing bequest generates a marginal gain through the decrease of pollution. Hence, individuals make their choice between consumption and bequest according to the utility provided directly by this choice, but also according to the welfare associated to environmental altruism.

The second trade-off summarizes the choice in term of children:

pk t 1 βq c ¡1 t ¡ 2 a pΠ t 1 q µ3 ¨ 1 n ¡1 t 2 pΠ t 1 q 1 µ3 ¡ pΠ t 1 q 1 µ3 1 µ 3 & (6)
To understand this choice, we have again to interpret the marginal costs and benefits of having children. Concerning the costs of having more children, there is the renunciation to consumption (pk t 1 βqc ¡1 t ) due to the increase of bequest and rearing cost. This effect is, nevertheless, mitigated by the fact that a lower consumption corresponds also to a benefit in term of environmental quality because households pollute less ( 2 a pΠ t 1 q µ3 pk t 1 βq). Concerning the marginal benefits, an adult enjoys to have children through his altruism in terms of wealth ( 1 1 nt ). A second marginal benefit derives from relative environmental altruism. For a given pollution stock, an increase of the fertility rate corresponds to a lower pollution stock per capita, which is the index perceived by agents

( 2 Π 1 µ3 t 1
). This effect is, however, reduced by the increasing weight associated to the pollution index: when adults have more children, they are more affected by pollution because they want a healthy environment for them ( 2 pΠt 1 q 1 µ 3 1 µ3

). 18 Actually, child labor did not stop with industrialization but intensified [START_REF] Horrel | The Exploitation of Little Children": Child Labor and the Family Economy in the Industrial Revolution[END_REF]). Concerning education, although the industrial revolution is a technical revolution, in its first stages it satisfied oneself with an unskilled workforce and was not accompany by majors progress in education [START_REF] Bairoch | Victoires et Déboires: Histoire Economique et Sociale du Monde du XVIe siècle à nos Jours[END_REF] or [START_REF] Galor | From Stagnation to Growth: Unified Growth Theory[END_REF]).

Using the first trade-off (5), equation ( 6) rewrites:

Π t 1 β 1 2 1 µ 3 µ 3 x ¡1 t & 1 1 µ 3 gpx t q (7)
Remark 1 g is a decreasing function, showing that there is a decreasing relation between the pollution index Π t 1 and x t , meaning that there is a positive link between wealth's bequest and the perceived environmental quality.

Substituting ( 7) into (5), we obtain:

c t 1 x ¡1 t 2 a β 1 2 1 µ 3 µ 3 x ¡1 t & µ 3 1 µ 3 ' ¡1 hpx t q (8)
Remark 2 h is an increasing function, revealing the positive relationship between consumption (c t )

and bequest (x t ).

Hence, the optimal behavior of households can be summarized by two equations, which determine the connections between individual consumption (c t ), wealth's bequest (x t ) and pollution index (Π t 1 ). It is important to notice that both forms of altruism are needed for our analysis. Indeed, without wealth's altruism, there is no capital, i.e. no production. Without environmental altruism, the net benefit of having children is lower than the one of investing an adding unit in capital. In this case, the economy degenerates again.

Firms

Since our model is centered on household's behavior, we choose to set an easily understandable production sector and assume then that a unique final good is produced by a representative firm.

The production is given by Y t AK s t N 1¡s t , where K t is aggregate capital, N t aggregate labor, A ¡ 0 the total productivity of factors and s t0, 1u the capital share in total income. The economy is perfectly competitive. From profit maximization, we get the two standard first order conditions of firm program: R t sAk s¡1 t and w t Ap1 ¡ sqk s t [START_REF] Brown | The Condition of England and the Standard of Living: Cotton Textiles in the Northwest, 1806-1850[END_REF] where R t denotes the real return to capital,19 w t the real wage and k t K t {N t the capital intensity.

Intertemporal equilibrium

We end this section with the definition of an intertemporal equilibrium. Substituting [START_REF] Brown | The Condition of England and the Standard of Living: Cotton Textiles in the Northwest, 1806-1850[END_REF] in the budget constraint (3), we get the resource constraint:

c t x t βn t A k s t (10)
Using Π t 1 P t 1 {N t 1 and reminding that N t 1 n t N t , we can rewrite the evolution of pollution (1) as:

n t Π t 1 p1 ¡ αqΠ t ac t (11)
Finally, using functions g and h, an intertemporal equilibrium can be defined as:

Definition 1 Given the initial conditions k 0 ¥ 0 et Π 0 P0
N0 ¥ 0, an intertemporal equilibrium is a sequence (x t¡1 , k t ), for all t ¥ 0, such that the following dynamical system ( 12) is satisfied:

5 hpx t q x t β xt kt 1 A k s t pAq xt kt 1 gpx t q ¡ a
hpx t q p1 ¡ αq gpx t¡1 q pBq [START_REF] Dasgupta | Population, Poverty, and the Natural Environment[END_REF] We notice that, at period t, the two variables x t¡1 n t¡1 k t and k t are predetermined. 20

An explanation of polluting industrialization

Using the simple overlapping generations model with altruism and endogenous fertility developed above, we will be able to exhibit what we call a polluting industrialization. We start with some preliminary results related to the existence of steady states. Then, analyzing dynamics, we show that there is sets of initial conditions such that the economy experiences a polluting industrialization , i.e. an increase of capital intensity, population growth and pollution. Finally, we will look out the effect of a permanent technological shock in the case where the economy is stuck on a pre-industrial trap with low levels of capital intensity, population growth and pollution.

Preliminary results: steady states analysis

As a preliminary study of dynamics, the steady state analysis will allow us to understand some long-term trends of the economy. We will show the existence of two steady states. The first one is characterized by a lower capital intensity, population growth and pollution and will determine a poverty trap. Whereas the second one will represent the outcome of an industrialization process with larger capital intensity, population growth and pollution than the pre-industrial state.

According to the previous section, we define: Definition 2 A steady state equilibrium of the dynamical system ( 12) is defined as a solution, px, kq satisfying:

6 9 9 8 9 9 7 k 1 A βp1 ¡ αq x ¡ 1 βa gpxq © hpxq %% 1 s Ψ 1 pxq k x p1¡αq a hpxq gpxq ¡1 Ψ 2 pxq (13)
To show the existence and characterize the solutions of (13), we begin by exhibiting some properties of Ψ 1 pxq and Ψ 2 pxq: Lemma 1 1. Ψ 1 is a strictly increasing function, where Ψ 1 p0q is strictly positive and lim xÑ V Ψ 1 pxq V. 2. There exists a value x ¦ such that Ψ 2 pxq is increasing for all x ¤ x ¦ and decreasing for all x ¥ x ¦ . Furthermore, Ψ 2 p0q is equal to zero and

lim xÑ V Ψ 2 pxq β 1 p1 µ 3 q{µ 3 ¡ 0. Proof. See Appendix 1.
This lemma allows us to show the coexistence of two steady states as it is illustrated in Figure 1 and shown in the following proposition.

Proposition 1 For β sufficiently weak, there exist two steady states: a pre-industrial one px 1 , k 1 q and an industrial one px 2 , k 2 q. The latter equilibrium is characterized by a larger:

• capital intensity (k 2 ¡ k 1 ) • bequest (x 2 ¡ x 1 ) • individual consumption (c 2 ¡ c 1 ) • population growth (n 2 ¡ n 1 ) • pollution stock (P 2t ¡ P 1t ) Proof. See Appendix 2.
We observe that the industrial steady state has a larger capital intensity, individual consumption, fertility growth21 and pollution stock. Hence, it perfectly corresponds to the long-term state of a regime initiated by industrialization and pointing up a high capital intensity, and therefore, GDP per capita, but also a large population growth and a deterioration of environment quality. Note that, about the relationship between environmental quality and capital intensity, our results contrast with literature (e.g. John and Pecchenino (1994)), being negative. Our model is actually appropriate for developing countries and allow to represent the first part of the environmental Kuznets curve.

From a theoretical point of view, two key mechanisms lead to the industrial equilibrium. Firstly, because there is a constant rearing cost in term of goods, a higher fertility rate is compatible with a higher wage, and therefore capital intensity. Secondly, the environmental altruism tolerates a higher pollution if development goes up too. This explains that a higher capital intensity and population growth are in accordance with a higher pollution, and allows us to reproduce elements observed historically.

Transitional dynamics

In this section, we enrich the study of steady state multiplicity with the analysis of dynamics. This allows us to bring out the difficulty of developing economies. Empirically, we notice important divergences in country development from the beginning with distant dates of priming transition. Indeed, historically, whereas the industrial revolution began in England in the early eighteenth century, the industrialization process came later and more slowly for other countries (in France, it began 50 years after England; in Germany, Canada and United-States of America, 100 years after; in Japan and Russia, 120 years after), 22 not to mention developing countries where it happens only recently. Over the course of history, we have thus observed simultaneously economies in both regimes, the pre-industrial one and the industrial one. We will see that our model allows to explain these features, establishing in addition a link with pollution emissions.

The study of dynamics is carried out in a simple way, using a phase diagram. Considering the dynamic system [START_REF] Dasgupta | Population, Poverty, and the Natural Environment[END_REF], we are able to evaluate two phases lines (∆x t¡1 0 and ∆k t 0) delineating the space, and to compute the sense of variation of the variables in the different regions of the space.

As illustrated in Figure 2, x t¡1 decreases (increases) below (above) the ∆x t¡1 0 locus, while k t increases (decreases) below (above) the ∆k t 0 locus. Therefore, the industrial steady state (E 2 ) is stable, whereas the pre-industrial one (E 1 ) corresponds to an unstable saddle equilibrium, leading to a poverty trap. Proposition 2 For β sufficiently weak, the industrial steady state (E 2 ) is stable and the pre-industrial one (E 1 ) is an unstable saddle point with a stable manifold (SS). Below (SS), the economy is stuck on a poverty trap, whereas above (SS), the economy converges to the industrial steady state.

Proof. See Appendix 3.

For initial conditions such that the economy starts above the border (SS), wealth is high enough to generate an income effect which allows a higher fertility, a more substantial bequest and a larger individual consumption. Therefore, the economy may develop and converge, in the long run, on a balanced growth path identified by the industrial steady state (E 2 ). This occurs because there are restoring forces acting against an infinite progress and leading the economy toward such a steady state. Firstly, the rearing cost of children (β), which is a cost per child, does not stop to grow with child number. Secondly, since adults have a distaste effect of pollution on consumption, they make a trade-off between these two variables and prefer to decrease their consumption growth when pollution becomes too large. Finally, decreasing returns on capital limits directly the income effect and so the increase of wealth.

When the initial conditions are located below the (SS) curve, the economy is relegated to a poverty trap. Indeed, the individual income is too low with respect to rearing cost per child. Hence households have no incentive to increase the number of children and the bequest in terms of wealth, maintaining the economy into the trap. In this case, the economy is characterized by a low capital intensity, population growth and pollution.

Being interested in the development process, we highlight two different scenarios depending on initial conditions. At given weak level of capital intensity k, the economy can experience two different configurations of dynamics. Either n is high enough, i.e. above the poverty trap (green manifold), the economy takes off.

Δx t--1 = 0 k t x t--1
Firstly, the population growth decreases because the rearing cost β is too high relatively to wealth. It has a direct effect on capital per capita k which increases mechanically. Then, the capital accumulation is sufficient to generate an income effect allowing higher consumption, capital intensity and population growth. These increases generate also an increase in pollution, corresponding to the first part of the environmental Kuznets curve where pollution worsen as country's income grows.

In the second scenario, n is too low, and economy is actually stuck in the poverty trap. The fist step is identical, population growth decreases, which induce an increase in capital per capita. But k does not rise sufficiently and so fail to ensure the subsistence of a larger population. In longer run, the lack of population introduces a lack of workforce and makes the economy poorer. The capital accumulation is not enough to generate the economy take-off. Moreover, the limited consumption and population growth bring on a weak pollution level.

The coexistence of the poverty trap and the industrial stable steady state illustrates the heterogeneous experiences of countries during the convergence process. This also explains that at the same time some countries were already engaged in the industrial process while others were still stuck in the pre-industrial trap.

Impact of a technological shock

The question we address now is the following: could a permanent technological shock, that we associate to a major technological innovation, reverse the dynamics of an economy located in the trap, allowing it to converge to the stable industrial steady state?

To keep things as simple as possible , we already specify that we consider a basic production sector.

We interpret a technological change as a permanent improvement of technology, more precisely as a permanent and positive shock on the productivity of factors A. As illustrated in Figure 4, the curve ∆x t¡1 0 shifts down and the stable manifold (SS) is pushed down toward (0,0) when A increases, leading to the two new steady states E I 1 and E I 2 . The increase of productivity facilitates the beginning of the mechanism of growth and convergence towards the industrial steady state, by increasing instantaneously output, and then, by encouraging the development process described previously. • an economy in the low-development trap but sufficiently close to E 1 , will converge to the industrial steady state;

• an economy in the poverty trap may always tend to the industrial steady state if the technological shock is large enough.

As described in Figure 4, following the technological shock, the old unstable steady state E 1 enters the region where one converges to the new industrial steady state E I 2 . Thus, such a shock favors the transition to the industrial long-term equilibrium and the escape from the poverty trap which maintains the economy at a low development level. These results promote technological transfer policies toward the least developed countries, allowing them to escape the trap permanently.

As already emphasized, this technological shock may illustrate a major innovation at the beginning of industrialization. For most of the countries, this process was indeed continuous and progressive, which explains why historians prefer the expression industrialization than industrial revolution. Our

1 1 µ3 1 2 a γ µ 3 1 µ 3 x 1 1 µ 3 % 2
Since Jpxq is strictly increasing for x ¡ 0, Jp0q 0 and lim xÑ V Jpxq V, there exists a unique x ¦ such that Ψ I 2 pxq ¥ 0 for all x ¤ x ¦ , while Ψ I 2 pxq ¤ 0 for all x ¥ x ¦ . Finally, we have Ψ I 2 p0q 1 p1¡αq .

Appendix 2: Proof of Proposition 1

Existence of two steady states

We have lim xÑ V Ψ 1 pxq V and Ψ 2 p0q 0. Furthermore, when β tends to 0 (but stays strictly positive), Ψ 1 p0q and lim xÑ V Ψ 2 pxq become arbitrarily close to 0. Finally, Ψ I 2 p0q 1 p1¡αq ¡ 1, while Ψ I

1 p0q 1 A 1 s ¡ 1 1 1 © 1 A βp1 ¡ αq ¨1¡s s
tends to 0 when β tends to 0. Since Ψ 1 pxq and Ψ 2 pxq are continuous functions, we deduce that there are two solutions satisfying Ψ 1 pxq Ψ 2 pxq (see also Figure 5). By continuity, there exist two steady states for β sufficiently low, namely px 1 , k 1 q and px 2 , k 2 q. To fix ideas, consider that x 1 x 2 . Since Ψ 1 pxq is strictly increasing, we have k 1 k 2 . Recalling that gpxq is decreasing and hpxq is increasing, we deduce that Π 2 Π 1 and c 2 ¡ c 1 .

According to [START_REF] Allais | Économie et Intérêt[END_REF] [START_REF] Galor | From Stagnation to Growth: Unified Growth Theory[END_REF]). We deduce that P 2t ¡ P 1t .

Appendix 3: Proof of Proposition 2

To prove this proposition, we construct the phase diagram associated to the dynamic system [START_REF] Dasgupta | Population, Poverty, and the Natural Environment[END_REF].

See also Figure 2. Equation (12 (B)) is equivalent to xt kt 1 p1¡αqgpxt¡1q a hpxtq gpxtq . Therefore, equation (12 (A)) can be rewritten as:

k t 1 A hpx t q x t β p1 ¡ αqgpx t¡1 q a hpx t q gpx t q & 1 s (16)
i.e. k t can be written as a function of x t and x t¡1 : k t Θ 1 px t , x t¡1 q, where Θ 1 is increasing in the first argument and decreasing in the second one. Note that Θ 1 px t , x t q is the phases line ∆x t¡1 0, and corresponds to the curve Ψ 1 px t q, relevant for the steady state analysis. We also have:

x t ¡ x t¡1 ô Θ 1 px t , x t¡1 q |xt¡xt¡1 ¡ Θ 1 px t , x t¡1 q |xtxt¡1

Therefore, when k t is above (below) Θ 1 px t , x t q, we get x t ¡ x t¡1 (x t x t¡1 ). Now, using ( 16), we implicitly define x t Θ 3 pk t , x t¡1 q, where Θ 3 is increasing in both arguments.

Substituting this in (12 (B)), we obtain: k t 1 Θ 3 pk t , x t¡1 q gpΘ 3 pk t , x t¡1 qq p1 ¡ αqgpx t¡1 q a hpΘ 3 pk t , x t¡1 qq Θ 2 pk t , x t¡1 q

(17)
where Θ 2 is increasing in x t¡1 at least for x t¡1 not too large. When k t 1 k t , equation [START_REF] Gradus | The Trade-off between Environmental Care and Long-term Growth; Pollution in Three Prototype Growth Models[END_REF] becomes: k t Θ 3 pk t , x t¡1 q gpΘ 3 pk t , x t¡1 qq p1 ¡ αqgpx t¡1 q a hpΘ 3 pk t , x t¡1 qq [START_REF] Grossman | Environmental Impacts of a North American Free Trade Agreement[END_REF] which implicitly defines k t as a function of x t¡1 . Using that Θ 2 is increasing in x t¡1 , 23 a raise of x t¡1 above its level defined in [START_REF] Grossman | Environmental Impacts of a North American Free Trade Agreement[END_REF] implies:

θ 2 pk t , x t¡1 q ¡ k t ô k t 1 ¡ k t Thus, when x t¡1 is below the locus k t 1 k t , k t 1 ¡ k t , whereas above this second phase line,

k t 1 k t .
Using all these elements, we can construct the phase diagram drawn in Figure 2 and deduce Proposition 2.

Figure 1 :

 1 Figure 1: Representation of Ψ 1 pxq and Ψ 2 pxq and existence of two steady states

Figure 2 :

 2 Figure 2: Transitional dynamics

Figure 3 :

 3 Figure 3: Two scenarios of developing countries dynamics

Figure 4 :

 4 Figure 4: Impact of a positive technological shock

Figure 5 :

 5 Figure 5: Representation of Ψ 1 pxq et Ψ 2 pxq when β Ñ 0

  , we have: P t 1 P t p1 ¡ αq aN t c t At steady state, Π t is constant, which means that pollution and population grow at the same rate, Since c 2 ¡ c 1 and Π 2 Π 1 , we get n 2 ¡ n 1 . Using N 0 1, N t n t on a steady state. We can therefore rewrite (15) as: n p1 ¡ αq an t c t P t ô P t a n t c t n ¡ 1 α Because Π t and c t are strictly positive, we have n ¡ 1 ¡ α (see

		P t	(14)
	n. Thus, equation (14) rewrites:	n p1 ¡ αq ac t Π t	(15)

The pollution effects continue over time, even after cause end.

Crutzen is the Nobel prize winning atmospheric chemist (1995) for discovering the effects of ozone-depleting compounds.

[START_REF] Bairoch | Victoires et Déboires: Histoire Economique et Sociale du Monde du XVIe siècle à nos Jours[END_REF] See Zaasiewicz and al.(2008

).[START_REF] Barro | Are Government Bonds Net Wealth?[END_REF] This debate is summarized in Van Alstine and Neumayer (2010) for

example.[START_REF] Barro | Fertility Choice in a Model of Economic Growth[END_REF] This is shown in the study directed by[START_REF] Wrigley | The Population History of England 1541-1871: A Reconstruction[END_REF].

In this paper, we are only interested in the first stages of development. So, the period under consideration stops before the demographic transition.

E.g. in France, from the 12th and 13th centuries, monarchs as Philippe Auguste or Louis IX, but also seigneurial authorities or municipal magistrates, denounced the nuisances on their multiple aspects. See[START_REF] Leguay | La Pollution au Moyen Age[END_REF] for further examples.

These two facts are illustrated by the experience of England. See[START_REF] Galor | From Stagnation to Growth: Unified Growth Theory[END_REF] and[START_REF] Wrigley | The Population History of England 1541-1871: A Reconstruction[END_REF].

In this paper, we focus on the industrialization process. Our study stops before the demographic transition, which leads to the modern economy (i.e. the current developed countries level). That is the reason why we choose this cost in good rather than a cost in time, which would introduce a quality-quantity trade-off in terms of child, corresponding to the demographic transition.

E.g. in France, from the 12th and 13th centuries, monarchs as Philippe Auguste or Louis IX, but also seigneurial authorities or municipal magistrates, denounced the nuisances on their multiple aspects. See[START_REF] Leguay | La Pollution au Moyen Age[END_REF] for further examples.

Since the size of one period is quite long, there is full depreciation of capital.

x t¡1 is predetermined because x t¡1 g ¡1 pΠtq and Πt is predetermined.

We can see in Appendix 2 that n ¡ 1 for A sufficiently high, at least at the industrial steady state.

See Bairoch (1997).

This may require that x t¡1 is not too large.

for their helpful comments. This paper benefits from the financial support of French National Research Agency Grant (ANR-09-BLAN-0350-01).

model points up these features: a strong enough technological shock conducts the economy on a new dynamic path converging, in the long run, to an equilibrium with higher capital intensity, population growth and pollution. Hence, it reproduces the economic development, the environmental damage associated and the role of a permanent technological shock in the priming of this mechanism.

However, for a given technological shock, some economies will follow an industrialization path, whereas others, characterized by a lower initial level of economic development, will not escape the trap and will need further innovations to engage in the industrialization process.

Conclusion

In this paper, we develop a simple overlapping generations model, to explain the industrialization process, characterized not only by an increase of capital intensity and population growth, but also a rise of pollution. We bring out the role of finite-lived agents in this process through their altruism and their family choices. A key feature of our explanation is the introduction of an environmental altruism, which stipulates that adults agree to leave a lesser natural environment quality to their children, if they expect an improvement of economic environment. A second important feature of our framework is the introduction of a constant rearing cost per child in terms of the final good, which seems a realistic assumption when one focuses on the period of industrial revolution.

We show that the economy may converge to an industrial steady state with high capital intensity, population growth and pollution. However, when the initial conditions on wealth and population size are too low, the economy is relegated to a poverty trap. Finally, a permanent technological shock that we associate to a major innovation promotes the convergence to the industrial steady state.

Appendix

Appendix 1: Proof of Lemma 1

Substituting gpxq and hpxq, given by ( 7) and ( 8) respectively, in the system (13), we get: 6 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 7

Study of the function Ψ 1

We have Ψ 1 p0q βp1¡αq A % 1 s ¡ 0 and lim xÑ V Ψ 1 pxq V. Computing the first order derivative of Ψ 1 pxq, we obtain:

s .

Study of the function Ψ 2

We have Ψ 2 p0q 0 and lim xÑ V Ψ 2 pxq 2 γ. Using