Dumitru-Claudiu Sergentu

Rémi Maurice
email: remi.maurice@subatech.in2p3.fr

Remco W A Havenith

Ria Broer
email: r.broer@rug.nl

Daniel Roca-Sanjuán
email: daniel.roca@uv.es

Computational determination of the dominant triplet population mechanism in photoexcited benzophenone

In benzophenone, intersystem crossing occurs efficiently between S 1 (nπ*) and the T 1 state of dominant nπ* character, leading to excited triplet states after photoexcitation. The transition mechanism between S 1 (nπ*) and T 1 is still a matter of debate, despite several experimental studies. Quantum mechanical calculations have been performed in order to asses the relative efficiencies of previously proposed mechanisms, in particular the direct S 1 → T 1 and indirect S 1 → T 2 (ππ*) → T 1 ones. Multiconfigurational wave function based methods are used to discuss the nature of the relevant states and also to determine minimum energy paths and conical intersections. It is found that the T 1 state has a mixed nπ*/ππ* character and that the T 2 (ππ*) state acts as an intermediate state between the S 1 and T 1 states. This result is in line with recent experiments, which suggested a two-step kinetic model to populate the phosphorescent state after photoexcitation [Aloïse et al.,

Introduction

After photoexcitation to the S 1 (nπ*) state, benzophenone, C 13 H 12 O, efficiently relaxes radiationless to the T 1 excited state, which is thought to have nπ* character, via a spin-orbit coupling (SOC) driven mechanism. This phenomenon occurs in single-crystals, 1 in solution 2 , and even in isolated matrices. 3 Its existence is then independent on the environment, while its efficiency can be affected by it via a change in the tilting angle between the phenyl planes, 1,4 or via the existence of other (competing) photophysical mechanisms. 1 The present work focusses on the intersystem crossing (ISC) in the gas phase, and will not deal with other phenomena specific to con-densed states.

According to El-Sayed's rules, 5 an ISC process governed by direct SOC interaction is fast if the transition involves a change in the molecular orbital type. Hence, assuming almost pure nπ* characters for both S 1 and T 1 states, the direct ISC rate constant should be low. 4 As evidenced by Yabumoto et al., 6 a T 2 (ππ*) excited state lies almost isoenergetically with S 1 , which could allow a fast S 1 (nπ*) → T 2 (ππ*) transition based on the same reasoning. However, if T 1 has a mixed nπ*/ππ* character, the direct S 1 → T 1 transition may become as efficient as the S 1 → T 2 one, and thus knowledge of the actual nature of the electronic states can be particularly helpful to discuss the efficiency of the main potential mechanisms. The nature of the involved electronic states is actually not the only factor affecting the transition rates. In order to discriminate between the two envisaged mechanisms (direct, i.e., S 1 → T 1 , or indirect, i.e., S 1 → T 2 → T 1 ), a computational study based on an accurate determination of the decay channels using the photochemical reaction path approach [7][8][START_REF] Olivucci | Theoretical and Computational Chemistry[END_REF][START_REF] Giussani | [END_REF][11] is particularly promissing. The aim of the present study is therefore to determine, by means of minimum energy path (MEP) computations, the most favorable decay paths of benzophenone to the emissive T 1 state.

The paper is structured as follows: a review of the main findings and conclusions obtained in previous experimental studies is first done, then the computational details are discussed followed by a presentation and discussion 1-14 | 1 of the results, and finally the conclusions are given.

Survey of previous experimental studies

In the last decades, the fast ISC that occurs in benzophenone as well as in other related aromatic ketones has been intensively studied using different time-resolved techniques. Since some results may be contradictory, we here review the important aspects of these works in order to prepare the discussion of our computational results.

El-Sayed and Leyerle 12 used the low field Zeeman effect in order to elucidate a possible indirect S 1 → T 2 → T 1 mechanism and totally ruled it out. Their results suggest a direct S 1 (nπ*) → T 1 (nπ*) ISC for benzophenone in bis(p-bromophenyl)ether at 1.6 K. Another work that strongly supports direct ISC between the S 1 and T 1 excited states was carried out by the group of Ohmori et al. 3 In this study, the sensitized phosphorescence excitation spectra of jet-cooled benzophenone was measured in the S 0 → S 1 and S 0 → T regions. Following this, an S 1 (nπ*) minimum was encountered at 26.180 cm -1 (3.25 eV) and another one related to the T 1 (nπ*) state at 24.224 cm -1 (3.00 eV). The absence of T(ππ*) signals in the proximity of T 1 (nπ*) led to the idea that this state would lie well above S 1 , in agreement with El-Sayed et al. 12 and also other experiments with benzophenone and 4,4'-diiodobenzophenone crystals. 13 Matushita et al. 14 studied the photochemical reaction of excited benzophenone in the gas phase. After photoexcitation at 337 nm of 1.0 Torr benzophenone -10.0 Torr 1,4-cyclohexadiene system, the absorption spectrum is characterized by a weak S 1 (nπ*) band at ∼ 350 nm and more intense bands starting at ∼310 nm due to population of levels with ππ* character. The kinetic studies on the decay process of triplet benzophenone revealed longliving emission that follows a biexponential decay. This was attributed to a dual phosphorescence, (i) from an unrelaxed benzophenone triplet state and (ii) from the vibrationally relaxed levels of the triplet state.

Katoh et al. 1 analyzed the ISC process in a benzophenone single crystal using the picosecond time-resolved absorption spectroscopy technique. The study revealed an ISC rate of 0.04±0.003 ps -1 which is lower than those measured in solution, 0.11±0.03 ps -1 in acetonitrile and 0.06±0.01 ps -1 in isooctane. 15 Another important feature observed in the benzophenone single crystal was a faster growth of the triplet exciton band compared to the decay of the singlet exciton band. This led to the idea that, besides the population of the triplet exciton through ISC with the lowest singlet exciton, another pro-cess which is more rapid has to be involved. This process was identified as a fission of a polymolecular highly excited state, populated by a biphotonic process, into two triplet excitons.

Yabumoto et al. 6 measured the transient infrared spectra of a carbon tetrachloride solution of benzophenone in the lowest triplet excited state, using the nanosecond time-resolved infrared spectroscopy technique. The S 1 (nπ*) state was populated at the 349 nm photoexcitation. The infrared spectrum of T 1 in the case of benzene and carbon tetrachloride solution of benzophenone lacked the peak related to the CO stretch vibration allowing to conclude the nπ* nature for this state of interest. A band with a broad feature in the wavenumber region of ∼2000 cm -1 was identified and assigned to the T 1 → T 2 transition. The results indicate that the 0-0 energy gap between T 1 and T 2 is lower than 2000 cm -1 , while the gap between S 1 and T 1 is known to be ∼1950 cm -1 . 3 Hence, S 1 and T 2 must be very close in terms of energy. In this scenario, the El-Sayed rules are respected for the S 1 → T 2 ISC and together with an ultrafast T 2 → T 1 internal conversion (IC), it could confer an efficient indirect S 1 → T 1 conversion.

Aloïse et al. The transient absorption spectra of benzophenone in acetonitrile were recorded for two excitation wavelengths, 383 and 267 nm. The peak of the IS of benzophenone in between S 1 and T 1 was found at ∆λ max (IS-T 1 ) = 13 nm and ∆λ max (S 1 -IS) = 41 nm in the case of the 267 nm excitation, and at ∆λ max (IS-T 1 ) = 5 nm and ∆λ max (S 1 -IS) = 40 nm in the case of the 383 nm excitation. The differences between the former values illustrates an excitation wavelength effect: with the 383 nm excitation, S 1 is populated near the ν = 0 vibrational level while more vibrational levels are populated with the 267 nm excitation, through a suggested S 2 (ππ*)-S 1 (nπ*) IC process. The similarities found between the IS and T 1 spectra raised the suggestion that the IS species could be related to one or several vibrational levels of the T 1 (nπ*) manifold, involved in two electronic transitions. The experiment was repeated in different solvents like acetonitrile, methanol, and dichloromethane, and the results confirm that the ISC process is not influenced significantly by the solvent polarity. In contrast, in the substituted molecule 4-methoxybenzophenone the relative position of the lowest two triplet states is switched while the solvent polarity increases, i.e., the lowest triplet state has

2 | 1-14
ππ* character instead of nπ* for polar enough solvents.

In conclusion, whether T 2 (ππ*) is a true spectroscopic state relevant in the photochemistry of benzophenone is still a matter of debate. To the best of our knowledge, no theoretical work has been carried out in order to clearly support one of the mechanisms suggested in the experimental works. In order to elucidate and discuss the efficiency of the main photochemical decay paths for an efficient triplet population in photoexcited benzophenone, we performed quantum-mechanical calculations, as described in the following section.

Computational details

The ground state (S 0 ) geometry of benzophenone was first optimized with density functional theory (DFT) and the def2-TZVP basis set 16 using the TURBOMOLE program package, version 6.4. [START_REF] Turbomole | V6.4 2012, a development of University of Karlsruhe and Forschungszentrum Karlsruhe GmbH[END_REF] The PBE0 [START_REF] Perdew | [END_REF]19 hybrid functional was used. The geometry was also further optimized using the complete active space self-consistent field (CASSCF) [START_REF] Roos | Advances in Chemical Physics; Ab Initio Methods in Quantum Chemistry II[END_REF][START_REF] Roos | [END_REF] method and the L-type atomic natural orbitals double-ζ plus polarization basis set (ANO-L-VDZP) 22,23 with MOLCAS. [START_REF] Andersson | MOLCAS version 7.6[END_REF] MEPs were computed at the same CASSCF/ANO-L-VDZP level of theory by following steepest descents, as described elsewhere. [START_REF] Vico | [END_REF] Mass-weighted coordinates were used. SOCs were computed with the restricted active space state-interaction (RASSI) method 26,27 using the atomic mean-field approximation. 28,29 The CASSCF method introduces static electron correlation and is known to be able to produce a qualitatively correct description of multiconfigurational states, provided that the active space is well chosen. Dynamic correlation is then introduced with the complete active space second-order perturbation theory (CASPT2) method 30,31 at the CASSCF optimized geometries (CASPT2//CASSCF). In order to prevent the occurence of intruder states, an imaginary level-shift 32 of 0.2 a.u., was used. The ionization potential electron affinity (IPEA) shift 33 was set to 0.00 in all the reported calculations.

The relevant orbitals for characterizing the lowest-lying excited states of the (C 6 H 5 ) 2 CO benzophenone molecule are the valence π and π* orbitals plus the n orbital (i.e., the lone pair) of the oxygen atom, which corresponds to 16 electrons and 15 orbitals. In the present study, two pairs of correlated π and π* orbitals, which would have the largest and smallest occupation numbers in a CASSCF(16/15) calculation, were not included in the active space. Hence, geometry optimizations of the states of interest and MEPs were carried out at the CASSCF(12/11) level of theory.

Test calculations were performed at the Franck-Condon (FC) region to evaluate the accuracy and convergence of the CASPT2/ANO-L-VDZP//CASSCF(12/11)/ANO-L-VDZP results with respect to the size of the active space and the size of the basis set. Indeed, the vertical transition energies (∆E V ) at the CASSCF(12/11)/ANO-L-VDZP optimized ground-state geometry were also computed with the valence triple-ζ quality ANO-L-VTZP basis set, while a larger active space, including all the valence π system, was also considered with the valence double-ζ basis. Due to its high associated computational cost, the test calculation with the largest active space was carried out imposing a C 2 symmetry. In addition, transition energies were computed with CASPT2 at the PBE0 S 0 optimized geometry and compared with the corresponding energies obtained with CASPT2//CASSCF. Calculations were also performed with the restrictedactive-space second-order perturbation theory (RASPT2) method. 34,35 In this approach, the active space is divided into three different subspaces: RAS1, RAS2, and RAS3. From the orbitals included in the RAS2 subspace, all possible configuration state functions (CSFs) of a particular spin and spatial symmetry are included in the expression of the multiconfigurational wave functions. The orbitals included in the RAS1 and RAS3 subspaces allow the generation of some aditional CSFs that are subject to restrictions: the maximum number of holes allowed in RAS1 and the maximum number of particles allowed in RAS3 are defined by the user. The π and π* orbitals and the oxygen lone pair can be distributed in the different subspaces in various ways. The RASSCF(2,11,2), RASSCF(4,7,4), RASSCF (6,3,6) reference wave functions were used, where the numbers in parenthesis refer to the number of orbitals in the RAS1, RAS2, and RAS3 subspaces, respectively. A maximum of 2 holes is allowed in RAS1 and a maximum of 2 particles in RAS3. Furthermore, the transition energies were computed with time-dependent DFT with the PBE0 exchange-correlation functional (TD-PBE0) and equation of motion coupled cluster singles and doubles (EOM-CCSD) methods with the def2-TZVP and cc-pVDZ basis sets, respectively. The TD-DFT calculations have been performed with NWChem 36 (release 6.1.1), while the CCSD ones have been performed with Dalton (release 2011). [START_REF]a molecular electronic structure program[END_REF] Four singlet and four triplet states were averaged in separate singlet and triplet CASSCF(12/11) calculations, respectively. When the ground state DFT geometry is used, the averaging scheme over four triplet states did not lead to convergence on a similar active space as the one obtained at the CASSCF optimized geometry. Therefore, the triplet CASSCF refer-ence wave functions in the CASPT2//DFT calculations were built with a different state-averaging scheme than the CASPT2//CASSCF ones, i.e five triplet states were considered. Regarding the CASPT2(16/15)/ANO-L-VDZP//CASSCF(12/11)/ANO-L-VDZP computations with the C 2 symmetry constraint, four singlet and four triplet states were separately averaged in each irreducible representation, leading to four sets of orbitals and states.

Results and discussions

The presentation and discussion of the results is split in two parts. First, a comparative analysis of the relevant geometries for decay mechanisms of interest is provided. Then energies and natures of the lowest-lying singlet and triplet states at the ground state geometry are discussed. The main decay channels of benzophenone are finally described in a second part.

Ground state geometry, Franck-Condon Region and Lowest-Lying Excited State Geometries

The relevant PBE0/def2-TZVP and the CASSCF(12/11)/ANO-L-VDZP optimized geometry parameters of S 0 , S 1 and T 1 are compiled in Table S1, and the geometries are displayed in Figure 1. These results can be compared with the computational data of Sett et al. [START_REF] Sett | [END_REF] A planar geometry for benzophenone could be envisaged assuming no steric repulsion between the phenyl rings. However, a steric repulsion exists between the adjacent hydrogen atoms that belong to two different phenyl rings, preventing the coplanarity of the phenyl groups. The PBE0 hybrid functional predicts a rotation of each phenyl ring with 28 • out of the plane that includes the CO bond and bissects the two phenyl planes. A similar rotation angle ( 31 • ) is predicted at the CASSCF(12/11)/ANO-L-VDZP level. The values are in a good agreement with previously reported theoretical ones. [START_REF] Sett | [END_REF]39 Note that our results are also in line with the crystal structure reported by Fleischer et al. in 1968, 40 indicating only a moderate effect of the environment on the ground state geometry of benzophenone.

Significant geometrical changes are expected for the excited states, which is confirmed by the present calculations, as can be seen in Table S1 and Figure 1. The CO bond distance in the S 0 geometry is computed to be 1.21 Å with CASSCF(12/11), which is in agreement with the DFT results. It is elongated in the excited S 1 and T 1 states for which the CASSCF values are 1.37 and 1.35 Å, respectively. These results are consistent with the fact that the electronic structure of the S 1 and T 1 states is characterized by an electron promotion from a non-bonding n orbital to an orbital with CO π* character. Also, the angle between the phenyl planes is reduced from 58 • in the S 0 optimized geometry to 40 and 44 • in the S 1 and T 1 geometries, respectively, at the CASSCF(12/11) level. This suggests electron delocalization over the two phenyl groups in the S 1 and T 1 excited states.

As mentioned above, the optimal active space for this system would correspond to 15 active orbitals, including 6 π and π* orbitals located on each phenyl ring, the π and π* on the CO double bond, and the lone pair (n O ) orbital containing the two non-bonding electrons of the oxygen atom (see Figure 2). As the number of CSFs increases with the number of active orbitals comprising the active space in the CASSCF formalism, calculations using the total amount of 16 electrons and 15 orbitals become computationally too demanding for performing CASSCF geometry optimizations. In order to find an affordable approach with enough accuracy for the aim of the present work, high-level CASPT2(16/15) computations were performed at the FC geometry and the results were compared with the values obtained with less demanding approaches including the CASPT2(12/11) one and various RASPT2 ones. Taking into account the symmetry properties of the S 0 equilibrium structure and in order to both reduce the computational cost and get insight on more excited states, C 2 (orbital) symmetry was imposed in the CASPT2(16/15) calculations. Averaging the CASSCF orbitals between several states while imposing C 2 symmetry could lead to different results than a state-average performed in the C 1 symmetry point group. Also, note that MOLCAS only averages orbitals between states of the same spin multiplicity and spatial symmetry. In order to find a good combination of active space and averaging scheme, both C 1 and C 2 point groups are considered and the obtained results compared. The highlevel CASPT2(16/15) calculations are taken as a reference to assess the validity of the CASPT2(12/11) ones.

Table 1 compiles the CASPT2(16/15)//CASSCF(12/11) vertical excitation energies ∆E V and oscillator strengths f . The lowest-energy electronic state leading to the most intense absorption is located at 5.39 eV (∼230 nm) and corresponds to the S 0 → S 4 (ππ*) transition. The S 1 excited state lies at 3.66 eV (∼338 nm). The nπ* character of this state corresponds mainly to an excitation from the oxygen lone pair (n O in Figure 2) to the anti-bonding π* orbital of the carbonyl (Π * The knowledge of the nature of these states is crucial to understand the relaxation mechanisms of photoexcited benzophenone. On the basis of the present calculations, the lowest-lying triplet state T 1 has a mixed nπ*/ππ* character and it is followed closely (0.08 eV) by a state of pure ππ* character, namely T 2 . Note that these characters have been obtained by looking at the dominant configurations in the corresponding multiconfigurational wave functions, and that the orbitals have been labeled according to their dominant characters, as can be seen in Figure 2. Note that such labeling is somehow arbitrary, since no symmetry element strictly separates the n, π and π * type orbitals in benzophenone. Taking into account these results, the population of the lowest-lying T 1 state from the S 1 (nπ*) state through ISC would be allowed according to El-Sayed rules, since T 1 is found to have a mixed nπ*/ππ* character.

Table 2 shows a comparison between the ∆E V values of several states of interest and their main corresponding excitation nature, computed at various levels of theory. Regarding the different CASPT2//CASSCF calculations, the largest discrepancies appear in the ππ* states, which can be attributed to a deficiency of CASPT2(12/11), in comparison to CASPT2(16/15), to accurately describe the excited states of ππ* nature. It is worth reminding at this point that the less relevant correlating pairs of π and π* orbitals were excluded from the active space in the CASPT2(12/11) calculations. Hence, the S 2 state (of ππ* nature) may not be perfectly described, and actually appears at higher energies than when the large active space is used. On the other hand, the S 1 (nπ*) excited state is accurately described with all the CASPT2//CASSCF approaches. A comparison with the large active space calculation shows a small difference of 0.01 eV when the C 2 symmetry is imposed and 0.02 eV when the symmetry is not considered. Regarding the ∆E V excitation energies of the lowest-lying triplet, the close-lying T 2 and T 3 states appear in different irreducible representations at the CASPT2(12/11) and CASPT2(16/15) levels when the C 2 symmetry is imposed. On the other hand, the CASPT2(12/11) approach with the C 1 point group symmetry performs more accurately as compared to the reference CASPT2(16/15) results. The differences obtained in this case are not larger than 0.2 eV for T 1 , T 2 and T 3 .

The results obtained with TD-DFT and the def2-TZVP basis set do not show large discrepancies with the reference CASPT2(16/15) values. Note that the abovementioned CASPT2 and TD-DFT values for the S 0 → S 1 excitation energy are also close to those measured in solution with various solvents 44 or in crystals 45 (around 3.6 eV in both the cases). The CCSD/cc-pVDZ level, however, overestimates this transition energy. This may be due to two factors, (i) the double-ζ Dunning basis set is too small, and (ii) excitations that are not introduced at the CCSD level of theory are important for the excitation energies. Test calculations with a larger (triple zeta) basis set showed similar results for the vertical excitation energies as when the double zeta quality basis set was used, thus issue (ii) is most probably the main one.

Further calculations were carried out in order to test the basis set convergence. The valence triple-ζ ANO-L-VTZP basis set results are thus compared with previously obtained ANO-L-VDZP ones (see Table 3). From the comparison of both sets of values, it can be seen that the triple-ζ quality basis set does not lead to significant changes as compared to the corresponding double-ζ basis set, i.e., basis set convergence is nearly achieved, and a double-ζ basis set can be safely used for more extensive explorations of potential energy surfaces.

It is known that CASSCF optimized geometries may suffer from lack of dynamic electron correlation. Therefore, we also computed the ∆E V with the same method From the analysis of the results obtained at different CASPT2//CASSCF levels, we have concluded above that a CAS space comprising only 11 orbitals is not enough to correctly describe the S 2 (ππ*) state. An accurate characterization of the potential energy surface (PES) related to S 2 would be important to interpret those experimental works in which the evolution after photoexcitation to this excited state (S 2 ) is studied instead or in addition to the analysis of the decay signals after direct population of the S 1 state. 2,46 In an attempt to improve the description, we followed the possibility of employing the RASSCF/RASPT2 method with the RAS(2,11,2), RAS (4,7,4), and RAS (6,3,6) subspaces (see Table 5). The RASPT2(2,11,2) level yields results in a very good agreement with the CASPT2(16/15) method. The RAS space is built in such a way that the two π orbitals with the occupation numbers closer to two from the CASSCF(16/15) computations are moved to the RAS1 subspace and their correlating counterparts, which are almost unoccupied, are placed in RAS3. The main disadvantage of this method is that it does not significantly reduce the computational cost. Thus, one can try to reduce the RAS2 subspace to only seven orbitals. The resulting RASPT2(4,7,4) calculations yield results which follow the same trends as the CASPT2(16/15) ones. A further reduction of the RAS2 subspace to only three orbitals leads to very poor results. This issue cannot be fixed by increasing the number of holes in the RAS1 and electrons in the RAS3 space from two to four, due to convergence problems. As already discussed, it is clear that using the CAS(12/11) space within the CASPT2//CASSCF formalism does not lead to a satisfactory description of the S 2 state. The (4,7,4) active space within the RASPT2//RASSCF formalism yields some significant improvements, but at a higher computational cost. Hence, the CASPT2(12/11)/ANO-L-VDZP//CASSCF(12/11)/ANO-L-VDZP approach will be employed in the characterization of the decay paths mainly focusing on the relaxation from S 1 , as most of the experimental works, while an approximate description will be provided for the decay route after photoexcitation to the S 2 state.

Energy decay mechanisms

The main decay paths after irradiation of benzophenone are presented and discussed here in two sub-sections. First, the evolution from the FC region along the lowestlying singlet ππ* and nπ* excited states is described. Next, the subsequent decay of the triplet states is analyzed.

4.2.1 Singlet manifold. After irradiation with high-energy UV light, the S 4 (ππ*) state, which is the third 1 (ππ*) excited state placed at 5.39 eV at the CASPT2(16/15) level (see Table 1), has the largest probability to be populated among the excited states computed in the present work. It is worth noting that a total amount of 7 triplet states of both nπ* and ππ* nature exist at lower energies than that of S 4 . Therefore, many singlet-triplet crossings (STCs) can be expected along the decay of the brighest state S 4 , which will increase the probability for population transfer to the triplet manifold. Nevertheless, the interesting electronic states of benzophenone are those appearing at lower energies (see, for example, ref 47 and references therein). Experimental works to compare with use photoexcitation wavelengths to populate S 2 (ππ*) and especially the lowestlying S 1 (nπ*) state. At the CASPT2(12/11)/ANO-L-VDZP//CASSCF(12/11)/ANO-L-VDZP level, the ∆E V transition energies to S 1 and S 2 are 3.64 and 4.67 eV, respectively, and the associated oscillator strengths f values are 0.001 and 0.008, respectively. Thus, the f related to the 1 (ππ)* state is one order of magnitude higher than that related to the 1 (nπ)* state. This behavior can be understood more clearly in a planar C 2v molecule. In such a case, the S 0 → 1 (ππ)* transition is generally allowed, while the S 0 → 1 (nπ)* transition is not. Since, in the non-planar case, S 1 has a mixed nπ*/ππ* character, this transition is allowed, and the transition probability is related to the nπ*/ππ* mixing, such that:

f (S 0 → nπ * /ππ * ) = nπ * |H|ππ * E ππ * -E nπ * 2 f (S 0 → ππ * ) (1)
where the notations nπ* and ππ* correspond, according to perturbation theory, to the zeroth-order states and the Hamiltonian H includes the (electronic) interactions that mix these two zeroth-order states. The first term in the right-hand side of Eq. 1 is defined as the mixing coefficient, λ. It follows that the magnitude of f for the S 0 → S 1 (nπ*) transition depends on the magnitude of the electronic matrix element between the zeroth-order (nπ*) and (ππ*) states, their energy gap, and the zerothorder f for the S 0 → ππ* transition. Note that since the nπ*/ππ* mixing of S 1 is weak (it is almost pure nπ* in the FC region), we will refer to this state in the remainder of the text as S 1 (nπ*).

The initial evolution of the system after light irradiation and population of the low-energy S 2 (ππ*) state is shown in Figure 3. S 2 (ππ*) evolves directly toward its relaxed non-planar minimum, S 2 (ππ*) min , placed adiabatically (i.e., with respect to the energy of the S 0 state at its optimized geometry) at 4.36 eV. The geometry of the S 2 (ππ*) minimum (Figure 4) is characterized by a slight elongation of the CO bond (1.23 Å), coplanarity of one phenyl ring with the carbonyl group, and an angle of 51 • between the phenyl rings. In Figure 3, it can be seen that the S 2 MEP does not lead to a S 2 (ππ*)/S 1 (nπ*) conical intersection (CI). The energy splitting, at the end of the MEP, between these two states is still large, 0.69 eV. Taking into account that the CASPT2(12/11)//CASSCF(12/11) level used in the present study leads to a too high S 2 relative energy (as concluded in section 4.1), a barrierless or almost barrierless decay path towards a S 2 /S 1 CI cannot be discarded. Recent experimental works studying the decay mechanisms after photoexcitation to S 2 supported a fast IC process for population transfer to the lowest-lying S 1 excited state. 46 Next, we focus on the relaxation from the S 1 (nπ * ) state that can be for instance reached after photoexcitation to S 2 or direct photoexcitation. Considering the latter case, the evolution on the PES of the S 1 (nπ*) excited state, starting at the FC region is shown in Figure 5. Note that the MEP on the S 1 surface evolves directly to the equilibrium structure of the state, S 1 (nπ * ) min , located adia- batically at 3.15 eV above S 0 (the experimental 0-0 energy difference being 3.25 eV). 3 Along the MEP reported for S 1 (nπ * ), four different decay mechanisms, which are anticipated here and described in detail below, may be active:

• S 1 (nπ)*/T 3 (ππ*) ISC followed by IC from T 3 to T 2 , IC from T 2 to T 1 and finally phosphorescence from T 1 to S 0 .

• S 1 (nπ)*/T 2 (ππ*) ISC followed by IC and then phosphorescence from T 1 to S 0 .

• direct S 1 (nπ)*/T 1 ISC, and phosphorescence from T 1 to S 0 .

• fluorescence from S 1 to S 0 .

In order to estimate the probability for singlet-triplet population transfer from S 1 to T 2 and T 3 , the SOC matrix elements between the spin components of S 1 and T 1 , S 1 and T 2 , and S 1 and T 3 are determined in the FC region, using the spin-orbit RASSI (SO-RASSI) method. 26,27 SOC magnitudes of 24 and 22 cm -1 are obtained between the S 1 (nπ*) and T 2 (ππ*) states and between the S 1 (nπ*) and T 1 (nπ*/ππ*) states, respectively, whereas no significant SOC was found between S 1 (nπ*) and T 3 (ππ*). At the CASPT2 level, the singlet-triplet splittings (∆E ST ) between S 1 and T 1 and S 1 and T 2 are 0.53 eV and 0.07 eV, respectively. Considering that an efficient population transfer from singlet to triplet states requires small energy gaps, the S 1 (nπ*)/T 2 (ππ*) ISC mentioned above is expected to be the most viable mechanism for populating the triplet state in the molecule, out of the three proposed mechanisms. To gain more insight on the relative importance of S 1 in those three possible mechanisms, we pursue our analysis along the MEP. As can be seen in Figure 5, the energy difference between S 1 and T 1 can decrease down to ∼0.25 eV, which could allow a population transfer from S 1 to T 1 . However, the SOC with the T 2 state, which also runs close in energy to S 1 , is much larger than the one with T 1 . In particular, the computed S 1 (nπ*)/T 2 (ππ*) and S 1 (nπ*)/T 1 (nπ*/ππ*) SOC magnitudes for the MEP points that follows the FC region are around 59 and 22 cm -1 , respectively. These findings point to the indirect mechanism in the gas-phase, in which an intermediate T 2 state is initially populated, instead of a direct population of the lowest-lying T 1 state. In order to estimate the solvent effects and, particularly, the plausability of the direct T 1 population in solu-tion, an analysis of the dipole moment (µ) values of the S 0 , S 1 , T 1 , and T 2 along the MEP is considered. Figure

On the basis of the present findings, it can be predicted for the gas phase and solution, in general, that after irradiation to the S 1 state, the S 1 and T 2 cross in the FC region and both states evolve very close in terms of energy along the S 1 MEP (see figure 5) with a large probability of population transfer due to the high SOC. On the other hand, T 1 is shown to have a relatively larger energy separation and lower SOC interaction.

The relaxation pathway involving fluorescence from the S 1 (nπ*) state is not likely to be efficient since, along the MEP of the S 1 excited state, the conditions for a fast singlet to triplet population transfer are fulfilled, i.e., (i) small energy gap with the lowest-lying T 2 state and (ii) large SOC magnitudes. This is in agreement with the experimental observation that the benzophenone molecule exhibits mainly phosphorescence (with an associated quantum yield of 84%), 48,49 and not fluorescence. Note that the computed CASPT2 vertical emission energy from the S 1 (nπ*) min state is 2.34 eV.

Triplet manifold.

The MEP on the PES of S 1 (nπ * ) from the FC region toward its minimum structure indicates that the T 2 (ππ*) state of benzophenone can be efficiently populated via the S 1 (nπ*)/T 2 (ππ*) ISC process described above. Figure 7 shows the subsequest evolution of the triplet manifold by the MEP computation of the T 2 state from the crossing region with S 1 . It can be seen that the MEP on the T 2 PES leads initially to a re-crossing point with the S 1 (nπ*) excited state and finally to the T 2 (ππ*)/T 1 (nπ*/ππ*) CI. Hence, two different relaxation pathways can be assumed at this point:

• via a T 2 (ππ*)/S 1 (nπ*) re-crossing favouring the repopulation of the S 1 (nπ*) state

• via a T 2 (ππ*)/T 1 (nπ*/ππ*) CI leading to the phosphorescent T 1 state

We start by discussing the ISC re-crossing channel. The computed SOC magnitude between the T 2 and S 1 states at the re-crossing point is relatively large: 47 cm -1 . This suggests that part of the population may be transferred back to the S 1 state via an efficient ISC process. From the S-T re-crossing point, a MEP, computed on the PES of the S 1 state with CASPT2//CASSCF (see Figure 8 (around 59 cm -1 along the S 1 MEP), the population lost with the re-crossing may be transferred back and forth to the T 2 state via several re-crossing processes, and finally the wave-packet may be driven toward the T 2 /T 1 CI, which appears at lower energies compared to the PES of S 1 and other points on the T 2 surface. It is worth mentioning here that the presence of the T 2 /S 1 re-crossings (Figure 7) may be related to the E-type delayed fluorescence reported experimentally by Brown and Singer. 50 Briefly, after the low fluorescence has ended, the spectrum does not show only a phosporescence spectral distribution but also some residual intensity (fluorescence) that decay at the same rate as the phosphorescence.

We finish the discussion by commenting on the second proposed relaxation mechanism from the T 2 state. The T 2 (ππ*)/T 1 (nπ*/ππ*) CI is computed to be adiabatically at 3.22 eV above the S 0 state at the CASPT2 level, favouring the population switch toward the T 1 state. At the same level of theory, the energy splitting between the two states (∆E CI ) is computed to be 0.004 eV. The geometry of T 2 at the CI shows a coplanarity of one of the phenyl rings with the carbonyl group, a CO bond distance of 1.35 Å, and a dihedral angle of 45 • between the two phenyl planes. From the T 2 (ππ*)/T 1 CI, the MEP on the T 1 (nπ*/ππ*) PES leads directly to the T 1 state minimum, T 1 (nπ*/ππ*) min (see Figure 9), which is placed adiabatically at 2.85 eV in agreement with the band origin observed experimentally at 3.00 eV. 3 The computed vertical emission energy (phosphorescence) from the T 1 (nπ*/ππ*) min equilibrium structure is 2.19 eV. Therefore, we conclude that the T 2 (ππ*) state plays a crucial role in the fast ISC in benzophenone, as in other related conjugated ketones and aldehydes. 51,52 5 Conclusion

In this work, the CASPT2//CASSCF approach with the double-ζ plus polarization ANO-L basis set was used to revisit the photophysics of benzophenone, by means of computations of MEPs and the determination of relevant energy minima and CIs. The obtained results are in good agreement with the experimental findings that suggest a two-step kinetic model (S 1 → IS → T 1 ) relaxation mechanism of the S 1 excited state 2 , and we attribute the intermediate state to T 2 (ππ*). Moreover, the results allow rationalizing the low quantum yield of fluorescence of benzophenone and suggest an explanation for the Etype delayed fluorescent emission signal observed experimentally. 50 The main relaxation mechanisms from the S 1 (nπ*) excited state are shown schematically in Figure 10. After populating the S 1 (nπ*) state in the FC region, the S 1 (nπ*) min point, which lies 3.15 eV above S 0 , can be reached. However, this path is not expected to be predominant since a strong coupling between the S 1 (nπ*) and T 2 (ππ*) states is computed already in the FC region, which may result in an efficient population transfer to the T 2 state in between the FC region and S 1 (nπ*) min . Favorable intersystem re-crossings leading to the S 1 state are found which may bring back population to the S 1 state and that explains the observable delayed fluorescence signal. 50 Eventually, the system may further de- cay along the T 2 (ππ*) state toward the T 2 /T 1 CI, which may funnel the energy to the lowest-lying triplet state T 1 (nπ * /ππ*). Finally, the molecule may arrive in a barrierless manner to the equilibrium structure of the T 1 state, T 1 (nπ*/ππ*) min , at 2.85 eV from where it efficiently phosphoresces. We note that a computational chemistry perspective of this work could consist in computing with quantum dynamics the ISC rate in benzophenone, which would be a highly demanding task owing the complexity of the involved electronic states.

2

 2 closely analyzed in 2008 the S 1 (nπ*) → T 1 (nπ*) ISC by means of ultrafast absorption spectroscopy and multivariate curve resolution, and proposed a two-step kinetic model S 1 → IS → T 1 (where IS denotes an intermediate state). With this model, they obtained ∼6.5 and ∼10 ps characteristic times, respectively.

Fig. 1

 1 Fig. 1 CASSCF(12/11)/ANO-L-VDZP optimized geometries of the S0 (top left), S1 (top right), and T1 (bottom) states. Bond lengths are given in ångström. Atom labels are displayed.

Fig. 3

 3 Fig. 3 CASPT2(12/11)//CASSCF(12/11) energies of the ground and lowest-lying singlet and triplet excited states of benzophenone along the MEP on the PES of the S2(ππ*) state from the FC region toward S2(ππ * )min.

Fig. 5

 5 Fig. 5 CASPT2(12/11)//CASSCF(12/11) energies of the ground and lowest-lying singlet and triplet excited states of benzophenone along the MEP on the PES of the S1(nπ*) state from the FC region toward S1(nπ * )min.

1- 14 | 9

 149

Fig. 4 Fig. 6

 46 Fig. 4 Lowest-energy structure of the S2(ππ*) excited state obtained at the end of the MEP computed at the CASSCF(12/11)/ANO-L-VDZP level of theory.

Fig. 7

 7 Fig. 7 CASPT2(12/11)//CASSCF(12/11) energies of the ground and lowest-lying singlet and triplet excited states of benzophenone along the MEP on the PES of the T2(ππ*) state from the S1(nπ*)/T2(ππ*) STC at the FC region toward the T2(ππ*)/T1(nπ*/ππ*) CI.

Fig. 8

 8 Fig. 8 CASPT2(12/11)//CASSCF(12/11) energies of the ground and lowest-lying singlet and triplet excited states of benzophenone along the MEP on the PES of the S1(nπ*) state from the T2(ππ*)/S1(nπ*) re-crossing point toward S1(nπ*)min.

Fig. 9

 9 Fig. 9 CASPT2(12/11)//CASSCF(12/11) energies of the ground and lowest-lying singlet and triplet excited states of benzophenone along the MEP on the PES of the T1(nπ*/ππ*) state from the T2(ππ*)/T1(nπ*/ππ*) CI toward T1(nπ*/ππ*)min.

Fig. 10

 10 Fig. 10 Schematic representation of the main potential decay mechanisms of benzophenone from the lowest-lying singlet nπ* excited state.

Table 1

 1 Vertical excitation energies computed at the CASPT2(16/15)/ANO-L-VDZP//CASSCF(12/11)/ANO-L-VDZP level of theory, and oscillator strengths and dipole moments of the corresponding reference states (C2 point group symmetry).

		A symmetry		B symmetry
	State	∆EV(eV)	f	µ(D) State	∆EV(eV) f	µ(D)
	S0	0		2.76
	S1(nπ*)	3.66	0.001 0.96
	S2(ππ*)	4.33	0.003 2.94
				S3(ππ*)	4.43	0.001 2.92
				S4(ππ*)	5.39	0.150 3.52
	T1(nπ*/ππ*) 3.33	0.000 2.47
	T2(ππ*)	3.41	0.000 1.09
				T3(ππ*)	3.69	0.000 2.95
				T4(ππ*)	4.18	0.000 2.95
	T5(ππ*)	4.18	0.000 2.70
	T6(ππ*)	4.22	0.000 2.90
				T7(ππ*)	4.26	0.000 2.92
				T8(ππ*)	6.47	0.000 2.77

Fig.

2

SA-CASSCF active space natural orbitals of benzophenone. The natural orbitals that are similar to the highest-occupied molecular orbital (HOMO) and lowest-unoccupied molecular orbital (LUMO) of a ground-state, single-configuration SCF calculation are specifically labeled.

  The dipole moments µ computed with CASSCF(16/15) are 2.76, 0.96, and 2.47 D for S 0 , S 1 , and T 1 , respectively, which is qualitatively in line with the experimental values of 2.98, 1.23, and 1.72 D, respectively, measured in benzophenone crystals.41,42 Dipole moments allow to roughly analyze the influence of different solvents. It is estimated that a polar solvent may induce a blueshift of the S 1 (nπ*) band and a redshift of the other singlet ππ* bands. Concerning triplets, T 1 is expected to be stabilized in the same manner as the ground state, while T 2 may suffer a slight blueshift. These estimations are in agreement with previous experimental results and other theoretical works on the absorption spectrum.2,43

Table 2

 2 Excitation energies computed with several methodologies at the CASSCF(12/11)/ANO-L-VDZP S0 equilibrium structure. The CASPT2(16/15) excitation energies are reproduced for convenience from Table1. The ANO-L-VDZP basis set is used for the CASPT2 calculations.

			CASPT2(12/11)		CASPT2(16/15)
		A symmetry	B symmetry	No symmetry	A or B symmetry
	State	∆E V (eV)	∆E V (eV)	∆E V (eV)	∆E V (eV)
	S 0	0.00		0.00	0.00
	S 1 (nπ * )	3.65		3.64	3.66(nπ*)
	S 2 (ππ * )		4.71	4.67	4.33(ππ*)
	T 1 (nπ*/ππ*)	3.25		3.11	3.33(nπ*/ππ*)
	T 2 (ππ*)		3.43	3.57	3.41(ππ*)
	T 3 (ππ*)	3.61		3.60	3.69(ππ*)
		TD-PBE0/def2-TZVP	EOM-CCSD/cc-pVDZ
	S 1 (nπ*)	3.66	4.00
	S 2 (ππ*)	4.74	5.06
	T 1 (nπ*/ππ*)	2.98	3.49
	T 2 (ππ*)	3.41	3.88
	T 3 (ππ*)	3.42	3.91

Table 3

 3 Vertical excitation energies computed with the CASPT2(12/11) method and two ANO-L-type basis sets at the S0 equilibrium structure optimized at the CASSCF(12/11)/ANO-L-VDZP level.

		ANO-L-VDZP ANO-L-VTZP
	State ∆E V (eV)	∆E V (eV)
	S 1	3.64	3.54
	S 2	4.67	4.63
	S 3	4.67	4.66
	T 1	3.10	3.05
	T 2	3.56	3.51
	T 3	3.59	3.56
	but using the ground state equilibrium geometry ob-
	tained at the DFT(PBE0)/def2-TZVP level of theory (see

Table 4 )

 4

. Results in good agreement with the ones obtained with the CASSCF ground state geometry (see Table

2

) are found. One should also note that a mixed nπ*/ππ* character is also obtained for T 1 , in agreement with the CASPT2//CASSCF results discussed above.

Table 4

 4

	CASPT2(12/11)/ANO-L-VDZP vertical excitation
	energies computed at the S0 PBE0/def2-TZVP optimized
	structure.
	State	A symmetry B symmetry
		∆E V (eV)	∆E V (eV)
	S 1 (nπ*)	3.58
	S 2 (ππ*)	4.59
	T 1 (nπ*/ππ*) 3.20
	T 2 (ππ*)		3.31
	T 3 (ππ*)	3.54

1-14 | 7

Table 5

 5 RASPT2 excitation energies (in eV) computed at the S0 CASSCF(12/11)/ANO-L-VDZP equilibrium structure.

	Method	S 1	S 2	T 1	T 2	T 3
	CASPT2(16/15) 3.66 4.33 3.33 3.41 3.69
	RASPT2(2,11,2) 3.66 4.32 3.32 3.42 3.68
	RASPT2(4,7,4)

3.59 4.48 3.40 3.52 3.75 RASPT2(6,3,6) 3.71 5.43 3.80 4.36 4.44

in Figure

2). There are three triplet states identified in the proximity of S 1 , located vertically at 3.33 eV (∼372 nm), 3.41 eV (∼362 nm), and 3.69 eV (∼335 nm) above the ground state. 4 | 1-14

compiles the µ values computed with CASSCF. At the FC region, a more important blueshift is estimated for S 1 than for T 1 and T 2 . It implies that the S 1 /T 2 CI may appear slightly displaced from the FC and also that the S 1 -T 1 energy separation may be larger, hence unfavouring the direct population of T 1 . Along the MEP, the µ values of S 1 and T 1 are similar to each other and lower than those of T 2 . Thus, a less important blueshift can be expected for T 2 , thus approaching the S 1 and T 2 PESs, whereas the energy separation between the PESs of S 1 and T 1 is estimated to remain as in the gas phase. Nevertheless, these µ differences and the estimated blueshifts are small, which may explain the fact that experimentalists do not observe major differences when solvents with different polarities are used.2

Acknowledgments

Helpful discussions with Prof. Manuela Merchán are acknowledged. The research was supported by the EM-TCCM master program and by the Project CTQ2010-14892 of the Spanish MEC/FEDER. R.W.A.H. acknowledges the Zernike Institute for Advanced Materials ("Dieptestrategie program") for financial support. D.R.-S. thanks the "Juan de la Cierva" program of the Spanish MINECO (Ref. JCI-2012-13431).

Table of contents:

Artwork:

Synopsis:

The dominant triplet population mechanism in benzophenone is determined from wave function based calculations, and involves the T 2 (ππ*) state as an intermediate between S 1 and T 1 .

|

1-14


Andrew E Clark
email: andrew.clark@ens.fr.

Sarah Flèche
email: s.fleche@lse.ac.uk.

Claudia Senik
email: senik@pse.ens.fr.

Dan Benjamin

François Bourguignon

Daniel Cohen

Jan-Emmanuel De Neve

Richard Easterlin

Yarine Fawaz

Francisco Ferreira

Ada Ferrer-I-Carbonell

Paul Frijters

Sergei Guriev

Ori Heffetz

Tim Kasser

John Knight

Richard Layard

Sandra Mcnally

Andrew Oswald

Nick Powdthavee

Bernard Van Praag

Eugenio Proto

Martin Ravallion

Paul Seabright

Conal Smith

Economic Growth Evens-Out Happiness: Evidence from Six Surveys

Keywords: Happiness, inequality, economic growth, development, Easterlin paradox Happiness, inequality, economic growth, development, Easterlin paradox. JEL codes: D31, D6, I3, O15

In spite of the great U-turn that saw income inequality rise in Western countries in the 1980s, happiness inequality has fallen in countries that have experienced income growth (but not in those that did not). Modern growth has reduced the share of both the "very unhappy" and the "perfectly happy". Lower happiness inequality is found both between and within countries, and between and within individuals. Our cross-country regression results argue that the extension of various public goods helps to explain this greater happiness homogeneity. This new stylised fact arguably comes as a bonus to the Easterlin paradox, offering a somewhat brighter perspective for developing countries.

I. Introduction

"Will raising the incomes of all increase the happiness of all?" Richard Easterlin asked somewhat ironically in 1995 [START_REF] Easterlin | Will Raising the Incomes of All Increase the Happiness of All?[END_REF], having shown that average self-declared happiness generally does not increase over the long run, even during episodes of sustained economic growth [START_REF] Easterlin | Does Economic Growth Improve the Human Lot?[END_REF]. This finding has helped inspire a vast empirical and theoretical literature on social comparisons and adaptation (as surveyed in [START_REF] Clark | Relative Income, Happiness and Utility: An Explanation for the Easterlin Paradox and Other Puzzles[END_REF]). Easterlin's original finding has more recently been called into question, with it being suggested that in some countries there is a positive time-series correlation between per capita GDP and average levels of subjective well-being (a well-known contribution to this extent is Stevenson and Wolfers, 2008a). At the same time as this ongoing debate about the relationship between the average happiness level and GDP growth, a striking new stylised fact has recently emerged regarding the distribution of happiness or "happiness inequality". As documented in [START_REF] Clark | The Great Happiness Moderation[END_REF], there is strong evidence across a wide variety of datasets that GDP growth is associated with systematically lower levels of happiness inequality (where this latter is picked up by the coefficient of variation). It is this happinessinequality literature to which we wish to contribute here. We provide evidence that economic growth is systematically correlated with a more even distribution of subjective well-being. This correlation for the most part holds despite the associated rise in income inequality, does not seem to be the result of any statistical artefact, is found in almost all domains of satisfaction, but is not found in placebo tests on other subjective variables. We suggest the extension of the provision of public goods as one likely candidate explanation for the lowering of happiness inequality.

None of our analyses of countries over time reveal a significant relationship between GDP growth and average happiness. It is this finding that has attracted great attention, as it suggests that economic growth may serve no (happiness) purpose. Outside of a utilitarian world, however, we may be more Rawlsian and give a certain weight to the avoidance of misery: here higher GDP does seem to chalk up points. The value we attach to GDP growth will then depend on the particular social-welfare function that we have in mind.

The remainder of the paper is organised as follows. Section 2 shows that higher income is associated with a tighter distribution of subjective well-being across countries, within countries, across individuals, and within individuals. Section 3 then presents a regression analysis emphasising the key roles of income inequality and public goods in determining happiness inequality. Section 4 considers a number of additional tests regarding the measurement of dispersion, the issue of time in panel (for our panel data results), and some placebo tests. Last, Section 5 concludes.

Income Growth and Happiness Inequality

There is a great deal of work using country-level data attempting to show a relationship, or the lack of one, between GDP growth and average levels of satisfaction or happiness over time. However, until very recently, little attention was paid to inequality in subjective well-being as economies grew. Knowing that average happiness scores remained broadly unchanged does not tell us anything about the distribution of well-being: flat happiness time profiles can be associated with a stable distribution of happiness, rising inequality, or lower inequality.

Two papers, Stevenson and Wolfers (2008b) and [START_REF] Dutta | Inequality of Happiness in the US: 1972-2010[END_REF], have explicitly addressed this issue. Both papers consider American General Social Survey (GSS) data from the early 1970s onwards (although their analytical approach differs); both underline a general fall in the inequality of happiness in the US up to around 2000, followed by an inversion of the trend (i.e. rising happiness inequality). [START_REF] Veenhoven | Inequality of Happiness in Nations[END_REF] found falling happiness inequality in EU countries (surveyed in the EuroBarometer) over the years 1973-2001, in spite of rising income inequality. He also notes a tighter distribution of happiness in "modern nations" rather than more traditional countries. [START_REF] Clark | The Great Happiness Moderation[END_REF] then looked at this issue systematically, using a wide variety of different datasets and a long time period . The crux of their argument is that countries with growing GDP per capita have also experienced falling happiness inequality. The data used there come from the World Values Survey (WVS), the German socio-economic panel (SOEP), the British Household Panel Survey (BHPS), the American GSS and the Household, Income and Labour Dynamics in Australia survey (HILDA). Happiness inequality was picked up by the coefficient of variation (the standard deviation of happiness divided by the sample happiness mean).

It is difficult to know what the correct measure of the distribution of subjective well-being is. One worry is that higher happiness levels will mechanically lead to lower dispersion as reflected in the coefficient of variation. If higher income produces greater happiness, on average, then the correlation between GDP per capita and the coefficient of variation of happiness will mechanically turn out to be negative. We here avoid the possibility of any artificial relationship by using the simple standard deviation of happiness as our measure of distribution. As we will show, this makes no difference to the main result that economic growth evens out the distribution of happiness (as we find no evidence in country panels that higher GDP per capita raises average happiness). We will also test a number of other measures of the spread of happiness.

Most countries with growing GDP per capita exhibit a tighter distribution of happiness, even though the US, with its U-shape of happiness inequality in GSS data, is an outlier in this respect. We provide a unified explanation of the experience of all countries, via a regression framework that encompasses the effect of GDP per capita, income inequality, and the role of public goods.

The following sub-sections consider the relationship between income and happiness inequality, first in country cross-sections and panels, and then in individual cross-sections and panels. It finishes by asking whether the correlation is a statistical artefact, and considering the relationship with domain satisfaction variables, rather than a single measure of overall well-being.

WVS Country Cross-Section

Our first piece of evidence follows on from [START_REF] Clark | The Great Happiness Moderation[END_REF] by looking at country cross-sections from the last available wave of the WVS (in the 2000s). Figure 1 shows the results. The left-hand panel reveals that the standard deviation of happiness is lower in richer countries. This relationship is significant at the one per cent level, as shown by the regression equation at the foot of the figure.

The right-hand side panel (taken from [START_REF] Clark | The Great Happiness Moderation[END_REF] shows the equivalent relationship using the coefficient of variation. The slope may look flatter, but of course the dependent variable is not on the same scale. In both cases richer countries have tighter happiness distributions.

WVS Country Panel

The next piece of evidence comes from changes over time within countries. We first consider WVS countries. These are a priori interviewed five years apart, although the gap may be somewhat shorter or somewhat longer. In Figure 2A we include WVS countries thatare observed at least twice, with the gap between observations being at least five years. We also only include countries that recorded positive real GDP per capita growth (where these figures come from the Penn World Tables) in all of the intervening years between the two consecutive WVS observations.1 What happened to happiness inequality in these growth periods? Figure 2A answers this question using data from a selection of Western countries. Although there is some sample variability here, happiness inequality falls over time as countries grow richer. The average relationship over time in these countries is given by the grey line in the Figure . Again, it makes little difference whether we consider the standard deviation or the coefficient of variation.

The trend line slopes downwards in both panels of Figure 2A: as countries become richer, their inequality in happiness falls. This is formalised in the regression equation at the foot of the table ,   where "year" attracts a negative and significant coefficient. 2 While suggestive, neither Figure 1 nor Figure 2A arguably provides evidence of a clean causal relationship. In the cross-section analysis, there could be some innate country characteristic that both makes a country rich and reduces the spread of its happiness distribution; in Figure 2A there has perhaps been a generic move towards reduced happiness inequality across all countries, which has nothing to do with GDP growth per se.

One obvious rejoinder to the latter is to consider countries that did not experience such periods of continuous GDP growth, and show that the relationship is different for them. This is what we do in Figure 2B, where we select countries that experienced some years with falling real GDP per capita between the two WVS observations (although there are fewer of these). Figure 2B reveals, on the contrary, a substantial rise in happiness inequality in these countries. Hence, there is no evidence of a general trend towards a tighter distribution of subjective well-being over time in the WVS: this tightening is only found for countries which have systematically become richer. It is worth noting that this finding is not particular to the WVS. We can reproduce Figure 2A using 40 years of annual data from the Eurobarometer: these results are depicted in Appendix A Figure 1.

BHPS, SOEP, HILDA and GSS: Country Panels

One drawback of the WVS is its small time-series dimension. 3 We therefore now turn to singlecountry datasets, which contain many more waves of data. We use four popular long-running single-country datasets, covering the United Kingdom (BHPS), Germany (SOEP), Australia (HILDA), and the United States (GSS). We summarise the way in which happiness and GDP comove over time in the regressions in Table 1. The first three columns refer to regressions for our three dependent variables: the log of real GDP per capita, average life satisfaction, and the standard deviation of life satisfaction. We carry out separate regressions for each of the four countries. The control variables in each regression include standard demographics (as listed in the note at the foot of the table), the year (divided by 10) and (if significant) year-squared divided by 100. We include the latter to detect non-linear movements over time. We might in particular suspect that inequality in life satisfaction in the US is U-shaped over time, following the findings of Stevenson and Wolfers (2008b) and [START_REF] Dutta | Inequality of Happiness in the US: 1972-2010[END_REF].

The results in column 1 show that real GDP per capita has trended upwards in all four countries, with no evidence of any non-linearities. Column 2 suggests that average happiness is flat in all four countries.

We are most interested in the results in column 3, which show the trend in happiness inequality. In the UK and Australia, this trend is negative and linear. However, in both the US (as previously found) and Germany there is evidence of a U-shaped relationship. Column 4 calculates the turning point in happiness inequality in these two countries. These turn out to be 1993 for the US,4 and rather later in 2007 for Germany. We will below suggest that the U-shapes in these two countries are at least partly driven by developments in income inequality.5

BHPS, SOEP, HILDA and GSS: Individual Cross-Sections

At the individual level, within each country, happiness dispersion is also smaller amongst the rich than amongst the less well-off (Figure 3): the richer seem to be more insulated against various kinds of shocks. In particular, higher income allows consumption to be protected from movements in income. 6 And it is in general likely that higher income allows the hedonic impact of various life shocks (job loss, divorce, etc.) to be smoothed.

BHPS, SOEP and HILDA: Individual Panels

Figure 3 referred to pooled data. We can also use our three panel datasets (we thus drop the GSS here) to look at the same correlation within-individual. The within-individual volatility of life satisfaction over time is lower for those in higher income deciles, as illustrated in Figure 4 (where the income decile we assign to the individual in the panel data is defined as the average rounded income decile per individual over the period during which they are observed in the panel).

A Statistical Artefact?

All of our results so far suggest that higher income goes hand in hand with less happiness inequality. However, we may worry that this is a statistical artefact. In particular, if higher incomes make people happier, and happiness is measured using a bounded scale, then inequality will fall as more people become right-censored on the top rung of the happiness ladder. At the extreme, as everyone reports the top subjective well-being score, inequality will be zero.

A first response to this worry is that there is actually no evidence in Table 1 that higher income over time does go hand-in-hand with higher happiness. We can nonetheless directly address the changing distribution of subjective well-being via histograms of happiness at the beginning and end of the periods under consideration. This is what [START_REF] Clark | The Great Happiness Moderation[END_REF] do. Their striking finding is that, as GDP grows, fewer people report the lowest happiness scores. But far from becoming more heavily populated, the top category is becoming increasingly deserted in all of the BHPS, GSS, HILDA and SOEP. As a consequence, the middle happiness categories are increasingly popular. The fall in happiness inequality then actually seems to have come from a mean-preserving contraction, as the percentage reporting the lowest happiness levels is also falling. 7A further formal test, if required, of the role of censoring comes from dropping the top and bottom scores in the panel datasets. 8 The test here is to look at happiness answers in year t+1 of those who reported happiness scores of between 2 and 6 (on a 1 to 7 scale) in the BHPS data (say) at year t.

All of these individuals can report either higher or lower happiness when re-interviewed in year t+1. As in Table 1, the results in Figure 3 in Appendix A show that there continues to be a trend to lower happiness inequality as GDP grows in the three panels.

Well-being Domains

Is there something strange about the specific distribution of overall satisfaction scores? Figure5 suggests mostly not. We here look at the trends in inequality in various domain satisfactions.

Satisfaction with health, income and job all exhibit falling inequality in the UK, Germany and Australia. As was the case for life satisfaction, the US is an outlier. Job satisfaction inequality is (slightly) falling, but not that for health and income. We suspect that income inequality may play a crucial role here: this is the topic of the next section.

Regression Analysis: Income Inequality and Public Goods

The results above come from analyses without any controls other than basic demographics. Can we now identify variables that help to explain the fall in happiness inequality over time? We here consider the role of income inequality and public goods. Recent years have seen rising incomes accompanied by rising income inequality 9 in many countries (which we would suppose to increase happiness inequality). 10 We do indeed find rising income inequality over time in the BHPS, GSS, HILDA and SOEP datasets.

At the same time, we may also reasonably expect income growth in general to produce public goods (like education, health, public infrastructure and social protection). Modern growth also comes with non-material public goods, such as lower violence and crime, greater freedom of choice in private life, political freedom, transparency and pluralism, better governance, and so on [START_REF] Inglehart | Modernization and Post-Modernization: Cultural, Economic and Political Change in 43 Societies[END_REF][START_REF] Inglehart | Development, Freedom, and Rising Happiness: A Global Perspective (1981-2007)[END_REF]. These public goods are, by definition, available to everyone (although, operate both in the cross-section and in time series, and operate in something of the same way across countries. We arguably know only little about reporting functions, and how they may change over time. 8 We cannot do this in the GSS, of course, as it is not panel. 9 See [START_REF] Atkinson | Top Incomes in the Long Run of History[END_REF].

The first column of Table 2 confirms, as expected from Figure 1, that happiness inequality does indeed fall with the log of GDP per capita. 12 Column 2 then adds income inequality, as measured by the Mean Log Deviation in Income, and our measure of government expenditure. These attract insignificant coefficients, and their addition only mildly affects the estimated coefficient on log GDP per capita. 13 Column 3 adds the two health variables, control of corruption and civil liberties.

Their inclusion turns the coefficient on GDP per capita insignificant, while the estimated coefficients on these "public-good" variables are of the expected sign (better outcomes reduce happiness inequality). Economic growth thus seems to reduce happiness inequality because it provides public goods that tighten the distribution of subjective well-being: the correlation between GDP per capita and life expectancy, mortality and the control of corruption are all around 0.8 in absolute value, while that with civil liberties is 0.67. Last, column 4 adds measures of trust and religious fractionalisation to the regression. Neither of these is significant 14 (although religious fractionalisation varies only little within countries over time, and we here include country fixed effects). Income inequality is positively correlated with happiness inequality in column (4).

Income growth would therefore seem to reduce happiness inequality because it allows for the greater provision of public goods. However, if this growth is accompanied by too large a rise in income inequality, then it is entirely possible that happiness inequality will actually rise as a result:

this is arguably what we have observed in the US data above, and perhaps to a lesser extent in Germany. 12 The errors are clustered at the country-year level here (which is the aggregation level of GDP per capita), to avoid under-estimating the standard errors: see [START_REF] Moulton | An Illustration of a Pitfall in Estimating the Effects of Aggregate Variables on Micro Units[END_REF]. 13 We might worry about reverse causality, with subjective well-being affecting productivity at work, and thus income (See Ostroff, 1992, for survey analysis and[START_REF] Oswald | Happiness and Productivity[END_REF], for an experimental approach). Our argument throughout this paper is a second-moment one, relating the level of income to the spread of the subjective well-being distribution. We actually find no relationship between average income and average well-being in our individual-country panels (see column 2 of Table 1). The only way in which movements in life satisfaction could then drive the level of income would be if the rising life satisfaction of the previously dissatisfied increases productivity more than the falling life satisfaction of the previously satisfied reduces productivity. We know of no evidence for such non-linear effects.

Additional Tests

All of our analysis above uses the standard deviation in respondents' self-declared satisfaction as our key dispersion measure. Such a measure assumes that happiness is continuous and cardinal, with equal distances between the steps. Although such an assumption is common in the field, we have no way of knowing whether it is actually true. To be on the safe side, we have reproduced our results using the Index of Ordinal Variation [START_REF] Berry | Assessment of variation in ordinal data[END_REF], a measure of variation specifically designed for ordinal measures. 15 As shown in [START_REF] Clark | The Great Happiness Moderation[END_REF], the movement over time in the IOV is practically identical to that in the standard deviation of happiness, so it is perhaps unsurprising that they produce such similar results.

One common cardinal measure of dispersion is the Gini coefficient. The movements over time in the Gini coefficient of subjective well-being are depicted in Appendix Figure 5. As was the case for the standard deviation, this falls over time, with a sharp upturn towards the end of the period in the US, and a smaller upturn for West Germany. A regression such as that in column 1 of Table 2 with the Gini of well-being as the dependent variable produces a negative significant coefficient; adding the controls in column 4 renders this coefficient insignificant.

Our empirical results in Section 3 come from both repeated cross-section and panel data. One potential problem in panel data is that of panel conditioning. [START_REF] Landua | An Attempt to Classify Satisfaction Changes: Methodological and content Aspects of a Longitudinal Problem[END_REF] underlined that individuals seem to become used to satisfaction scales after a while in SOEP data. He, in particular, noted a tendency away from the end values. This finding was reproduced on more recent SOEP data by [START_REF] Frick | Using Analysis of Gini (ANoGi) for Detecting Whether Two Sub-Samples Represent the Same Universe: The SOEP Experience[END_REF]. [START_REF] Wooden | Panel Conditioning and Subjective Well-being[END_REF] analyse HILDA data and find a clear narrowing in the dispersion of satisfaction scores over the first few waves in which the individual responds. We have checked what happens to the trend in life satisfaction when we drop the first two participation waves per individual. The results appear in Appendix Figure 6 for our three panel data sets, and continue to show a downward trend in dispersion, with again the exception of the final few years in West Germany. 16Last, it might be suspected that there is something inherent about self-reported variables, particularly in a panel-data context, which produces more homogenous responses over time. We here run a placebo test by looking at the changing standard deviation over time in other variables available in our four single-country datasets. For the BHPS, SOEP and GSS, we use self-reported interest in politics, which is measured on a one-to-four ordinal scale; in HILDA we consider the self-reported number of hours per week spent volunteering.

The results appear in Appendix A Figure 7. There is little evidence of any particular trend in the dispersion of these measures over time, and certainly no evidence of a downward trend. The shrinking dispersion of subjective well-being over time in our datasets appears to be specific to this question, rather than a general feature of self-reports.

Conclusions

In spite of the great U-turn that saw income inequality rise in Western countries in the 1980s, happiness inequality has notably fallen in countries that have experienced consistent income growth (but not in those that did not). Modern growth has reduced the share of both the "very unhappy" and the "perfectly happy". The extension of public amenities may have contributed to this greater happiness homogeneity by reducing the insecurity faced by the worst-off groups in the population.

We believe that higher national income goes hand-in-hand with a reduced spread in happiness. It can always be countered that there are potential common factors that drive both income/GDP and the inequality of happiness. One example, at the country level, might be the quality of institutions.

Higher quality institutions (including educational institutions) might lead to both higher GDP and a smaller spread in happiness. Any omitted variable which explains all of our empirical findings would however have to vary across countries and within countries over time, and across individuals and within individuals over time, in the same way that income does: no obvious candidates come to mind.

The question of why the top of the happiness is increasingly deserted is perhaps a more difficult one to answer. One salient point is that the extension of public goods and government expenditure is costly, and the happier may have borne the brunt of the necessary taxation. A second explanation, which is far more difficult to test, is that growth has enlarged the world of possibilities of those previously at the top of the well-being distribution, raising their aspirations and reducing their satisfaction. Comparisons may be made not only within country, but also across countries (as suggested by Becchettiet al., 2013), and the worldwide integration of the elite is likely an integral part of globalization [START_REF] Goldthorpe | The economic basis of social class", LSE, Centre for Analysis of Social Exclusion, Centre for Analysis of Social Exclusion[END_REF]. It is also possible that some of the byproducts of economic growth have rendered comparisons to others easier to make and more salient. This is very likely the case with the internet. [START_REF] Clark | Who compares to whom? The anatomy of income comparisons in Europe[END_REF] show that those who have internet access at home are also more likely to report that they compare their income to others (although they do not have data on exogenous changes in internet access, for which see [START_REF] Lohmann | Information Technologies and Subjective Well-being: Does the Internet Raise Material Aspirations?[END_REF]. Over the course of economic growth, this dampening effect will be concentrated amongst the richer, and perhaps also amongst those who were previously the happier.

As in much of this literature, we find no evidence that economic growth has been accompanied by higher mean levels of subjective well-being. But we certainly believe that it has come along with

S.D. of Life Satisfaction S.D. of Life Satisfaction / Mean

Note: The lines show the levels of life-satisfaction inequality for countries with some periods of negative growth, over periods of at least 5 years. Note: Happiness and life satisfaction questions are administered consistently over time within countries for all of these surveys, although the surveys use different scales: 1-3 in the GSS, 0-10 in the SOEP and HILDA, and 1-7 in the BHPS. We select people aged between 18 and 65 years old; we also drop observations corresponding to a declared income of below 500$ per year. GDP per capita figures are taken from Heston, Summers and Aten the Penn World Tables. The other controls include gender, age, age-squared, marital status, labour-force status and education. Robust standard errors are in parentheses.

Netherlands Italy Belgium Ireland

Notes: Life satisfaction is administered on a 4-point scale. We select people aged between 18 and 65. GDP per capita figures are taken from Heston, Summers and Atenthe Penn World Tables. The values plotted are SD residuals by year from a regression of life satisfaction on gender, age, age-squared, marital status, employment status and education.

Appendix.

FiguresFigure 1 .

 1 Figures

Figure 2A .

 2A Figure 2A. Happiness Inequality within Countries with Uninterrupted GDP Growth. WVS Panel. Selected Western countries

Figure 2B .

 2B Figure 2B. Happiness Inequality within Country in Countries with Falling GDP. WVS Panel.

Figure 3 .

 3 Figure 3. Income as a Buffer Stock: The Standard Deviation of Happiness is Lower in Higher Income Deciles

Figure 4 .

 4 Figure 4. Income as a Buffer Stock: The Standard Deviation of Happiness within Individual over Time is Lower for Individuals in Higher Income Deciles

Figure 3 .

 3 The Trend in the Standard Deviation of Happiness Excluding People Who Report the Highest or Lowest Levels of Happiness United Kingdom (BHPS) West Germany (SOEP) Australia (HILDA) Note: We drop observations corresponding to people who reported being in the top or bottom life satisfaction categories one year previously. drop observations corresponding to the first two years in which individuals appear in the panels in question.

Table 1 . Time Trends in GDP per capita, Happiness and Happiness Inequality. Single-country Panels.

 1

	Tables
		Ln Real GDP	Life	S.D. of Life	Estimated
		per Capita	Satisfaction	Satisfaction	Minimum Year
	United Kingdom (BHPS)
	Year/10	0.226*** (0.0140)	0.0005 (0.0165)	-0.0493*** (0.0107)	--
	Year 2 / 100	--	--	--
	West Germany (SOEP)
	Year/10	0.402*** (0.0113)	0.0007 (0.0220)	-28.90*** (4.650)	2007
	Year 2 / 100	--	--	0.0720*** (0.0116)
	Australia (HILDA)
	Year/10	0.153*** (0.0188)	-0.00135 (0.0214)	-0.181*** (0.0372)	--
	Year 2 / 100	--	--	--
	United States (GSS)
	Year/10	0.195*** (0.00617)	0.00389 (0.00586)	-3.779*** (0.499)	1993
	Year 2 / 100	--	--	0.00948*** (0.00125)

Table 2 . OLS Estimates of the Determinants of SD Life Satisfaction: WVS-EVS (1981 -2008)

 2 The other controls include country fixed effects, time trends, age-category dummies, sex, labour-force status, and marital status. The errors are clustered at the country-year level. See Appendix B for the detailed variable descriptions. * significant at the 10% level, ** significant at the 5% level and *** significant at the 1% level. Weighted estimates.

	(1)	(2)	(5)	(6)

Notes:

This Figure differs from that in[START_REF] Clark | The Great Happiness Moderation[END_REF], which only considered the coefficient of variation and which included countries with both positive growth in all years and positive growth in some years.

We restrict the sample to Western countries here only to avoid cluttering the figure excessively. The same shape appears if we include all WVS countries: the regression for the left-hand panel of Figure2Athen becomes SD(Satisfaction)=0.008***year+19.33.

The WVS started in 1981 and has been repeated every five years since 1990-1991. The sixth wave is currently in the field.

This turning point is somewhat earlier than that inStevenson and Wolfers (2008b) and[START_REF] Dutta | Inequality of Happiness in the US: 1972-2010[END_REF], which undoubtedly comes from our use of control variables in Table1.

Analogous relations between GDP and happiness inequality are found in Eurobarometer data. We show a number of the relevant single-country graphs in Appendix A Figure2. Happiness inequality trends downwards in eight out of nine countries as GDP trends upwards. The exception is Germany.

Which is why consumption varies less over time than does income: see[START_REF] Krueger | Does Income Inequality lead to Consumption Inequality?[END_REF].

In general, objective happiness is converted into reported happiness via a "reporting function". It is possible that reporting functions change as income grows. For this to explain our results, such reporting functions would have to

[START_REF] Becchetti | The drivers of happiness inequality: suggestions for promoting social cohesion[END_REF] find that education reduces and unemployment increases happiness inequality. They however do not find any role for income inequality. Their measure of the latter is the share of the poor, which is different from ours.

We find the same results in ordered probit analysis.

It is not always easy to have good measures of social capital at the cross-country level. Although trust is insignificant in this regression, some other measure of social capital may do better. The difficulty is in finding consistent measures across countries and over time.

[START_REF] Dutta | Inequality of Happiness in the US: 1972-2010[END_REF] also propose an ordinal measure of happiness inequality.

We have also taken the opposite tack, and kept only the first five responses in the panel. The same result pertains.

See Wagner et al. (2007).

We are very grateful to two anonymous referees for pertinent and helpful suggestions. We thank seminar participants at Ca'Foscari University (Venice), the Conference on "Happiness and Economic Growth" (Paris), the Fifth ECINEQ conference (Bari), the Franco-The data were originally collected by the ESRC Research Centre on Micro-social Change at the University of Essex. The German data used in this paper were made available by the German Socio-Economic Panel Study (SOEP) at the German Institute for Economic Research (DIW), Berlin. The Household, Income and Labour Dynamics in Australia (HILDA) Survey was initiated and is funded by the Australian Government Department of Families, Housing, Community Services and Indigenous Affairs (FaHCSIA), and is managed by the Melbourne Institute of Applied Economic and Social Research (Melbourne Institute). Neither the original collectors of the data nor the Archives bear any responsibility for the analyses or interpretations presented here. We thank CEPREMAP and the French National Research Agency, through the program Investissements d'Avenir, ANR-10-LABX_93-01, for financial support; this work was also supported by the US National Institute on Aging (Grant R01AG040640) and the Economic & Social Research Council.

Figure 5. Trends in Satisfaction Inequality by Domain United Kingdom (BHPS) Health (1-7) Income (1-7) Job (1-7) West Germany (SOEP) Health (0-10) Income (0-10) Job (0-10) Australia (HILDA) Health (0-10) Income (0-10) Job (0-10) United States (GSS)

Health (1-7) Income (1-3) Job(1-4)

Notes: We select people aged between 18 and 65 years old; we also drop observations corresponding to a declared income of below 500$ per year. GDP per capita figures are taken from Heston, Summers and Atenthe Penn World Tables.

of course, their marginal benefit may differ across individuals). We then suspect that their provision will reduce happiness inequality.

The role of income inequality and public goods in determining the standard deviation of life satisfaction is explored in Table 2. The data here is the WVS and these are OLS regressions. 11 The "public-good"-type variables we consider are Government expenditure, Life expectancy, the underfive Mortality rate, the Control of Corruption, Civil Liberties, Trust and Religious Fractionalisation.

While we group these together under the heading "Public Goods", they are not all 100% publiclyprovided. Amongst the variables in Table 2, health is probably a publicly-provided public good, whereas something like trust is at least partly privately-produced.

We also control separately for the level of GDP per capita and income inequality by country-year.

All of the explanatory variables are described in Appendix B. The four separate regressions in Table 2 are balanced, in that they are estimated only on the sample of observations with no missing values for the most complete specification (that in column 4).

The changes over time in our seven public-goods variables over our broad analysis period appear in Appendix Table 4. Government expenditure as a percentage of GDP has been on an upward trend over the past 20 years, and life expectancy has risen sharply since the early 1980s, at the same time as the under-five mortality rate has been falling. On the contrary the upward trend in trust (from the WVS) is only weak, and there is no discernible trend in the control of corruption and civil liberties.

The last figure here shows the rise in income inequality since 1970, using data from the Luxembourg Income Study (LIS). reduced happiness inequality. This new "augmented" Easterlin paradox therefore offers a somewhat brighter perspective: economic growth is a useful tool for policy-makers who care not only about average subjective well-being, but also its distribution. Notes: Interest in politics ranges from 1 to 4, where 1 means "very interested" and 4 "not at all interested". Participation in volunteer work is defined as the number of hours per week.

Appendix

Appendix B.

This paper uses the five currently available waves of the World Values Survey (WVS, 1981(WVS, -2008)), covering 105 countries. These include high-income, low-income and transition countries, as well as data from the ISSP and the 2002 Latinobarómetro. We also analyze individual-country surveys, such as the British Household Panel Survey (BHPS, 1996(BHPS, -2008)), the German Socio-Economic Panel (SOEP, 1 1984-2009), the American General Social Survey (GSS, 1972(GSS, -2010) ) and the Household, Income and Labour Dynamics in Australia survey (HILDA, 2001(HILDA, -2009)).

The Happiness and Life Satisfaction questions were administered in the same format across all these surveys, but with different response scales: 1-3 in the GSS, 1-10 in the WVS, 0-10 in the SOEP and the Australian HILDA, 1-7 in the BHPS. The wording of the life satisfaction question in the WVS was "All things considered, how satisfied are you with your life as a whole these days?: 1 (dissatisfied)….10(very satisfied)". In the SOEP, it was "How satisfied are you with your life, all things considered?": 0 (totally unsatisfied) … 10 (totally satisfied). The BHPS survey asked "How dissatisfied or satisfied are you with your life overall?": 1 (not satisfied at all) … 7 (completely satisfied)". The wording of the happiness question in the GSS was "Taken all together, how would you say things are these days -would you say that you are very happy, pretty happy, or not too happy?". We do not need to harmonize these scales, as we consider the evolution of the variance of happiness over time within countries. The surveys cover representative samples of the population in participating countries, with an average sample size of ten to fifteen thousand respondents in each wave. We select people aged between 18 and 65 years old; we also drop observations corresponding to a declared income of below 500$ per year.

We use the American General Social Survey because it is the only long-run survey containing a happiness or life satisfaction question in the United States. However, this data is not really suited to our purpose, as only three responses are possible (very happy, pretty happy, and not too happy), making the calculation of the variance problematic. However, as the evidence initially used to suggest the Easterlin paradox partly relied on American data, and because we would like to include data from the United States, we do report the results based on this data, although they may need to be considered with some caution.

Description of the Country-level Control Variables used in Table 2

GDP per capita. GDP per capita in constant 2000 dollars (Penn World Tables)

Mean Log

Deviation in Income is a measure of income inequality derived from WVS.

World Bank Database (http://databank.worldbank.org/data/home.aspx)

Expenditure (% of GDP). This is cash payments for operating activities of the government in providing goods and services. It includes compensation of employees (such as wages and salaries), interest and subsidies, grants, social benefits, and other expenses such as rent and dividends. When there is no information for some years, the first year available is used as a replacement.

Life expectancy at birth, total (years). This is given by the number of years a newborn infant would live if prevailing patterns of mortality at the time of its birth were to stay the same throughout its life.

Mortality rate, under-5 (per 1,000 live births). This is the probability per 1,000 that a newborn baby will die before reaching age five, if subject to current age-specific mortality rates.

Governance Indicators

Control of corruption (Transparency International). This index measures the perceived levels of public sector corruption on a scale of 0-10, where 0 means that a country is perceived as highly corrupt and a 10 means that a country is perceived as very clean. When there is no information for some years, the first year available is used as a replacement.

Limits on Civil liberties (Freedom House). Civil Liberties contain numerical ratings between 1 and 7 for each country or territory, with 1 representing the most free and 7 the least free. Civil liberties contains information on freedom of expression and belief, associational and organizational rights, rule of law, personal autonomy and individual rights.

World Values Survey

Trust is the percentage of individuals who answer "most people can be trusted" to the question "Generally speaking, would you say that most people can be trusted or that you can't be too careful in dealing with people?".

Fractionalisation Data

Religious Fractionalisation comes from the fractionalisation dataset compiled by Alberto Alesina and associates and measures the degree of religious heterogeneity in various countries. When there is no information for some years, the last year available is used as a replacement.

Electronic Appendix

The