Guillaume Daudin
email: guillaume.daudin@dauphine.fr

CROISSANCE EUROPÉENNE AU XVIII e SIÈCLE

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

revenu réel par tête a été entre 1700 et 1800 de 0,4 % en Grande-Bretagne, 0,2 % en Hollande, 0,1 % par an en Espagne et en Europe occidentale et -0,1 % en Italie du Nord 7 . J'ai moi-même proposé 0,6 % de croissance pour la France jusqu'à la Révolution 8 . Tout cela est compatible avec l'hypothèse d'un commerce moteur de la croissance : les pays à la plus forte croissance sont aussi ceux qui on vu leur commerce le plus augmenter. L'examen de l'évolution des salaires réels entre les différents pays européens entre 1300 et 1800 montre aussi que le croissance du commerce intercontinental -forcément maritime -était associé à une croissance plus forte 9 .

Sous-ensemble du commerce, le commerce atlantique a eu un effet particulièrement positif sur la croissance, d'après certains sous condition de bonnes institutions 10 . Cela est moins vrai pour le XVIII e siècle que pour le XVII e siècle. En utilisant une version modifiée de la base de données de Bairoch, Batou et Chèvre 11 , on trouve que le taux de croissance des ports pratiquant le commerce atlantique (selon la liste restreinte d'Acemoglu, Johnson et Robinson) n'était que de 0,36% par an XVIII e (à comparer avec 0,32% pour l'ensemble des villes) alors qu'il était de 0,45% par an au XVII e (à comparer avec 0,19% pour l'ensemble des villes). La population des villes maritimes en général a augmenté 7 BOLT J. et ZANDEN J.L., « The Maddison Project: collaborative research on historical national accounts », in The Economic History Review, vol. 67

Conséquences nettes de l'existence des relations intercontinentales

De +1,5 % à + 2 % De + 6,5 % à + 8 % De -3 % à -2.5 % De + 0 % à 0.5 %

La croissance smithienne et la mer

  tendance plutôt à priver l'économie intérieure de talents et de capitaux attirés par ses promesses de gains. Cela se voit dans le fait que les ports français attiraient beaucoup de marchands de l'intérieur en ascension sociale. Par contre, puisqu'il fallait atteindre un certain niveau de prospérité avant de pouvoir participer à cette activité maritime si attirante -notamment pour jouer le jeu de la diversification qui était le moyen le plus efficace pour lutter contre les fortunes de mer -il encourageait les marchands de l'intérieur à étendre leur activité, à ne pas recycler leurs fortunes trop tôt dans la terre et les honneurs, bref à faire leur travail de marchands au mieux, ce qui rendait possible la croissance smithienne.ConclusionRésumons-nous. Les taux de croissance nationaux et les taux de croissance des populations urbaines indiquent bien qu'il y avait un lien entre la croissance et le commerce maritime au XVIII e siècle. Pour comprendre ce lien, il ne faut pas être obnubilé par la Révolution industrielle dont les effets décisifs sur les taux de croissance datent plutôt du XIX e siècle. Le progrès technique et l'accumulation de capital productif ne dépendaient pas de manière cruciale du commerce maritime. Par contre, la croissance smithienne lui était beaucoup plus lié par l'augmentation de la demande, la baisse des coûts de l'échange, les nouveaux biens qu'il offrait, le renforcement de la place des marchands et l'encouragement de leur activité. L'état de nos connaissance n'est malheureusement pas encore assez avancé pour donner une idée, même approximative, de la part de la croissance européenne au XVIII e siècle qui peut être expliquée par chacun de ces phénomènes.

  du XVIII e siècle, il ne faut pas le faire en ayant les yeux fixés sur le futur : cette méthode fait courir le risque le l'anachronisme. dans les motivations du mouvement abolitionniste britannique de la fin du XVIII e siècle la fin du rôle central de l'esclavage rendue possible par la Révolution Industrielle 24 . Le débat sur le sujet porte à la fois sur le niveau des profits censés permettre l'accumulation et leur masse. Les profits du commerce atlantique, pour autant qu'on puisse les mesurer, étaient bien plus élevés que ceux qui pouvaient être espérés dans l'économie intérieure européenne. Cela est vrai même si on prend en compte le risque lié au expéditions maritimes : en effet, à partir d'un certain niveau d'investissement, celui-ci était résorbé par une importante diversification grâce à la prise de participations par chaque investisseur dans de nombreux navires25 . Certes, les différents secteurs du commerce maritime attiraient de nombreux acteurs qui réduisaient progressivement les surprofits. Cependant, au cours du XVIII e , les investisseurs français ont exploité toute une série de frontières qui chacune et pendant un temps offrait des occasions d'investissement exceptionnelles. Cela a d'abord été le commerce avec les Antilles, le commerce d'esclaves, celui avec l'Inde puis avec la Chine.Même si l'on accepte l'argument sur le niveau des profits, il reste la difficulté liée à leur masse. Les activités liées au commerce colonial ou même maritime en général ne mobilisaient que peu de travailleurs, de capitaux et de terres par rapport à la taille de l'économie européenne dans son ensemble. En conséquence même si le niveau des profits était important, la masse des profits n'a pu représenter qu'une petite partie de l'accumulation du capital totale en Europe.Comme l'a écrit de manière mémorable Patrick O'Brien, les apports en capital de la périphérie n'ont été que périphériques pour le développement de l'Europe 26 . La taille du commerce maritime est d'ailleurs trompeuse à cet

	de 0.42% par an entre 1700 et 1800 comparé à 0.27% par an pour les villes non joué un rôle très important 14 . Ainsi, ni les marchés intérieurs ni les marchés Même si on prend cet aspect en compte, en supposant par exemple que
	maritimes 12 . extérieurs français n'étaient moins étendus que les marchés britanniques, et l'investissement dans le secteur intercontinental n'aurait pas été possible durant
	Le lien entre commerce maritime au sens large et croissance en Europe est donc établi par un faisceau d'indices. La manière dont on peut l'expliquer pourtant la Grande Bretagne a connu une mutation beaucoup plus rapide que la l'ensemble du siècle, la croissance aurait été plus faible de 6 à 7,5 %, pour un Le commerce maritime, le progrès technique et France 15 . Les explications seraient plutôt à trouver du côté de l'évolution des l'accumulation du capital perte cumulée à la fin des années 1790 de 2 à 3 % du revenu national. Ces
	dépend du type de croissance auquel on s'intéresse : les prémices de la sciences et des techniques et, d'après Joel Mokyr, des lumières industrielles chiffres seraient plus importants pour des économies plus ouvertes, comme la
	Révolution Industrielle, le progrès technique et l'accumulation du capital ou la centrées sur la Grande-Bretagne 16 . L'activité maritime y a joué un rôle : les Revenons donc à l'économie pré-moderne des pays européens. Quelles étaient Grande-Bretagne ou les Provinces-Unies, mais ils resteraient de toutes les
	croissance smithienne. Dans ces trois cas, l'état actuel des connaissance nous navires de l'époque moderne étaient de grands utilisateurs des avancées les sources de sa croissance ? Encore une fois, il ne s'agissait pas de l'économie manières modestes.
	contraint à concentrer notre enquête sur les conséquences du commerce colonial techniques. Ceux utilisés pour le commerce avec l'Asie, comme les vaisseaux de de la Révolution industrielle : le progrès technique industriel, bien qu'il existait, Tableau 1 : Le rôle statique du secteur intercontinental (cas de la
	au sens large 13 . guerre, étaient particulièrement impressionnants de ce point de vue. Tous était modeste et incrémentiel. Ce n'est pas dans ses ressorts qu'il nous faut France, fin des années 1780) 27
	nécessitaient le développement d'instruments de navigation de plus en plus regarder. Quant au progrès technique agricole, d'une part il semble avoir été Revenu du Revenu de Revenu Situation Revenu total capital la terre du travail britannique
	avancés. L'histoire du développement d'un instrument de mesure du temps qui reste précis malgré les remous des voyages maritimes est bien connue 17 . 4 % -4.75 % du PIB 7 % -8 % du PIB Taille relative du 9,5 % -15 % de la 13 % -15,5 % 0 % 2.5 % -3 % 23 % -26 % de la secteur intercontinental production industrielle production industrielle
	L'activité maritime a donc joué un rôle important dans l'épanouissement d'un
	secteur de production d'instruments de précision. Celui-ci a à la fois fournis des
	outils utiles pour la « Recherche & Développement » qui s'est systématisée
	après la Révolution industrielle, et développé des savoir-faire qui ont pu être
	réutilisés dans les secteurs de la Révolution Industrielle. De plus, l'ouverture sur
	l'extérieur et la volonté d'émuler les produits asiatiques ont encouragés la
	recherche, comme le montre le développement de centre de production
	européens de porcelaine et de toiles imprimées 18 . Tout cela n'est pas négligeable,
	, n mais doit comparé à d'autres sources de transformation techniques et o 3, 2014, p. 627-651.
	vol. 56, n intellectuelles comme la mathématisation des sciences et la systématisation de la o 3, 2003, p. 403-443. Economic History Review, XXXV, n o 1, 1982, p. 1-18. Voir aussi SHERIDAN R.B., « The Wealth De nombreux chercheurs, dont le plus célèbre est Marx, ont présenté les of Jamaica in the Eighteenth Century », in Economic History Review, vol. 18, n o 2, 1965, p.
	entreprises coloniales de l'Europe moderne comme centrale pour l'accumulation 293-311 ; COELHO P.R., « The profitability of imperialism: The British experience in the West
	Institutional Change and Economic Growth », in American Economic Review, vol. 95, n 2005, p. 546-579 ; DAUDIN G., « Dynamique du commerce atlantique et dynamique de o 3, 14 MOKYR J., « Demand vs. supply in the industrial revolution », in The Journal of Economic History, vol. 37, n o 04, 1977, p. 981-1008 est un des premiers articles à presenter cet argument. indies 1768-1772 », in Explorations in Economic History, vol. 10, n o 3, 1973, p. 253-280 ; primitive de capital ayant précédé la Révolution industrielle 23 . Cette vision DEVINE T.M., « The Colonial Trades and Industrial Investment in Scotland, c. 1700-1815 », in
	l'intérieur : le cas des ports de Brest à la Corogne », in Les ports du Golfe de Gascogne : De 15 DAUDIN G., « Domestic trade and market size in late-eighteenth-century France », in Journal traditionnelle a été reprise par l'école marxiste de l'après-guerre, notamment par Economic History Review, vol. 29, n o 1, 1976, p. 1-13SOLOW B., « Caribbean Slavery and
	Concarneau à la Corogne (XVe-XXIe siècle). FERNANDEZ A. et MARNOT B. (dir.), Paris, PUPS, 2013, p. 33-46 ; DAUDIN G., « Cities of Commerce: how can we test the hypothesis », in of Economic History, vol. 70, n o 3, 2010, p. 716-743. 16 MOKYR J., The Enlightened Economy: Britain and the Industrial Revolution, 1700-1850. British Growth : the Eric Williams Hypothesis », in Journal of Development Economics, vol. 17, Eric Williams qui, avant de devenir premier ministre de Trinidad et Tobago, a n o 1, 1985, p. 99-115 ; ELTIS D. et ENGERMAN S.L., « The Importance of Slavery and the Slave
	Penguin, 2011.
	17 SOBEL D., Longitude. The True Story Of A Lone Genius Who Solved The Greatest Scientific
	Problem Of His Time. Fourth Estate, 1996.
	Berghahn Books, 2006.	16, n	o 1, 1998.

8 DAUDIN G., Commerce et prospérité, op. cit., p. 28. 9 ALLEN R.C., « Progress and Poverty in Early Modern Europe », in Economic History Review, 10 ACEMOGLU D., JOHNSON S. et ROBINSON J.A., « The Rise of Europe: Atlantic Trade, Low Countries Journal of Social and Economic History, vol. 11, n o 4, 2014, p. 103-118 11 BAIROCH P., BATOU J. et CHÈVRE P., La Population des villes européennes : Banque de données et analyse sommaire des résultats 800-1850 / The Population of European Cities: Data Bank and Short Summary of Results. Genève, Libraire Droz, 1988. Pour plus de détails sur la base, voir DAUDIN G., « Cities of Commerce », op. cit. La Révolution Industrielle et la mer La Révolution Industrielle britannique a marqué l'avènement de la croissance soutenue, sans diminution ni stagnation de longue durée du produit, dont nous bénéficions toujours. Elle est d'habitude datée de la deuxième moitié du XVIII e siècle, période où une vague d'innovations techniques ont mécanisé la production et permis d'utiliser de manière intensive le charbon, transformant ainsi durablement l'économie britannique avant de s'étendre au reste de l'Europe. Cet événement fondateur de l'économie contemporaine est l'un des plus étudiés de l'histoire économique du monde. De très nombreuses explications en ont été données : il n'est pas possible de résumer ces débats ici. La doxa actuelle est que la demande de produits, et notamment la demande extérieure, n'a pas 12 Les villes sont classes comme maritimes si elles avaient un port pouvant accueillir des navires de mer (y compris de pêche) au XIX e siècle ou avant. Les ports d'estuaire comme Rouen sont inclus. 13 À ce propos, une synthèse récente présente des discussions pays par pays intéressantes : ROITMAN J.V., PÉTRÉ-GRENOUILLEAU O. et EMMER P.C. (dir.), A deus ex machina revisited: Atlantic colonial trade and European economic development. Brill, 2006. Voir aussi FINDLAY R. et O'ROURKE K.H., Power and Plenty: Trade, War, and the World Economy in the Second Millennium. Princeton University Press, 2007 et O'BRIEN P. et DE LA ESCOSURA L.P., « The costs and benefits of European imperialism from the conquest of Ceuta, 1415, to the Treaty of Lusaka, 1974 », in La Revista de Historia Económica, vol. 18 BERG M., « From imitation to invention: creating commodities in eighteenth-century Britain », in The Economic History Review, vol. 55, n o 1, 2002, p. 1-30. procédure expérimentale à partir du XVII e siècle. Il est difficile de croire que les lumières industrielles n'auraient pas eu lieu sans le commerce maritime. Cependant, la Révolution industrielle est intimement liée au développement de l'industrie du coton. Ne dépendait-elle donc pas crucialement d'une plante qui n'était disponible pour l'industrie européene que grâce au commerce maritime avec le Proche Orient et, surtout, les Amériques ? De plus, la subsistance de la population en croissance rapide en Grande-Bretagne n'a peutêtre été rendue possible que par les calories apportées par la culture du sucre ? Kenneth Pomeranz, qui a renouvelé l'étude comparative de l'Europe et de la Chine au à l'orée de la Révolution Industrielle, voit dans l'exploitation des ressources agricoles américaines, grâce à la main d'oeuvre des esclaves africains, la principale source de l'avantage de l'Europe sur la Chine 19 . Cette exploitation n'aurait pas été possible sans commerce maritime. Si on se place du point de vue de la Grande Bretagne des années 1850, le commerce maritime était indispensable pour fournir son industrie en matières premières et sa population en nourriture 20 . Cependant, 1850 est au-delà de notre période d'intérêt. Les structures de l'économie britannique du XVIII e siècle n'étaient pas celles du XIX e . La croissance britannique au XVIII e siècle était d'ailleurs beaucoup moins rapide que ne le suggèrent les transformations du secteur du coton, de la métallurgie et du charbon 21 . Dans son ensemble, elle était moins dépendante du commerce maritime que dans les années 1850. Donc, même si on admet, par hypothèse, que la Révolution industrielle n'aurait pas été possible sans le commerce maritime, la croissance liée à celle-ci se lit plutôt dans les chiffres du XIX e siècle que dans ceux du XVIII e siècle. Si l'on veut 19 POMERANZ K., The great divergence : Europe, China, and the making of the modern world economy. Princeton, Princeton University Press, 2000. 20 CLARK, O'ROURKE et TAYLOR, « The growing dependence of Britain on trade during the Industrial Revolution », in Scandinavian Economic History Review, vol. 62, n o 2, 2014, p. 109-136 21 CRAFTS N.F.R. et HARLEY K.C., « Output Growth and the British Industrial Revolution: A Restatement of the Crafts-Harley View », in Economic History Review, vol. 45, n o 4, 1992, p. 703-730 comprendre la croissance beaucoup plus modeste qu'on ne l'a longtemps crû et d'autre part il n'a pas été influencé de manière importante par l'introduction de nouvelles plantes : la pomme de terre et le maïs n'ont pas été décisifs à court terme. Les nouvelles méthodes de culture qui ont permis l'augmentation de la productivité de la terre ne dépendaient pas des cultures du Nouveau Monde ou du commerce maritime. Utiliser plus d'animaux, plus d'engrais, plus de semence, plus de travail dans le cadre du « système de Norfolk » (assolement à quatre cultures : navet, orge, trèfle et blé) était une méthode déjà connue dans certaines régions d'Angleterre et des Pays-Bas à la fin du Moyen-Âge 22 . L'augmentation de la production par tête venait donc d'ailleurs : l'accumulation des facteurs de production, l'intensification de leur utilisation ou leur meilleure allocation. 22 ALLEN R.C., Enclosure and the yeoman: the agricultural development of the South Midlands 1450-1850. Oxford, Clarendon Press, 1992. 23 MARX K., Le Capital : Critique de l'économie politique, Livre premier. Paris, Presses Universitaires de France, 1993. inclus égard : il faut effet prendre en compte seulement les revenus en excédent de ce qui aurait été produit par les mêmes facteurs de production utilisés dans l'économie nationale. Mes calculs dans le cas français sont rapportés dans le Tableau 1. Ils indiquent qu'à la fin des années 1780, le revenu total de l'économie française aurait été plus faible de 1.5 à 2 % si le capital et le travail qui étaient investis dans le secteur intercontinental -c'est à dire la partie la mieux connue de l'économie maritime -avaient été rapatriés dans l'économie domestique. Le capital aurait plus souffert de ce rapatriement que le travail. Peut-être est-ce que l'épargne aurait été affectée de manière disproportionnée. 26 O'BRIEN P., « European Economic Development : The Contribution of the Periphery », in Trade in Industrializing Britain », in Journal of Economic History, vol. 60, n o 1, 2000, p. 123-144. POMERANZ K., The great divergence : Europe, China, and the making of the modern world economy, op. cit. ; Morgan K., Slavery, Atlantic trade and the British economy, 1660-1800. Cambridge University Press, 2000 ; EMMER P.C., The Dutch slave trade, 1500-1850.

  région isolée devait produire tout ce dont elle avait besoin pour subvenir à ses besoins, quitte à éparpiller ses capacités productives de manière inefficace. Amérique Espagnole 31 . La croissance industrielle de Nantes dépendait des biens nécessaires au commerce d'esclaves 32 . L'Angleterre connaissait les mêmes phénomènes33 . Cependant, le jeu des nombres implique que, rapportée à 29 HOFFMAN P.T., Growth in a traditional society: the French countryside, 1450-1815. 'ensemble de l'économie, la taille des secteurs répondant à la demande coloniale n'était pas considérable.Sous certaines hypothèses sur le goût pour la diversité des consommateurs, ce jeu des nombres est plus favorable à l'effet des produits offerts par le commerce maritime et le Nouveau Monde. Le sucre, le tabac, le café, le thé, le cacao, les toiles de coton imprimées, les porcelaines… ont acquis au XVIII e siècle une place de choix dans la consommation des Européens. Boire son café au lait était devenu à la fin du XVIII e siècle une manière banale de commencer sa journée pour les ouvriers parisiens. Le café devient aussi important en Allemagne. C'est à cette époque que les Britannique ont commencé à boire du thé. Ces produits, souvent addictifs, avaient aussi pénétré les campagnes les plus reculées, comme celles des Alpes 34 . Ils n'étaient pas très importants en termes de calories, mais ils apportaient des goûts et des sensations nouveaux aux consommateurs européens. Hans-Joachim Voth et Jonathan Hersh ont calculé que l'Anglais moyen aurait été disposé à voir son revenu abaissé de 15 % pour conserver un accès au thé et au sucre en 1850 35 . La consommation de biens la croissance. Cela était en partie dû à la nouvelle donne politique issue de la Glorieuse Révolution et de la puissance que les profits du commerce maritime offraient aux marchands. Grâce à eux, ils auraient été capables d'imposer des bornes étroites à l'arbitraire de l'exécutif qui est vu par cette école comme le principal obstacle institutionnel à la sûreté des droits de propriété, condition nécessaire au développement. Les Provinces-Unies l'auraient précédé sur le même chemin institutionnel. Celui-ci était par contre fermé pour le Portugal et l'Espagne à cause de leurs institutions. L'exécutif y aurait été capable d'y accaparer même les profits du commerce maritime. la France était dans une position particulière. Je fais partie de ceux qui ont des doutes sur cette thèse, qui sous-estime par exemple à quel point les marchands dans l'Angleterre du XVII 41 . Je pense moi aussi que les marchands, du négociant au colporteur, jouaient un rôle central dans la croissance smithienne au XVIII e siècle. Par contre, il me semble qu'il faut s'intéresser, au delà des questions institutionnelles, au moyens dont ils disposaient pour assurer la circulations des biens dans l'économie et payer les coûts de transaction : collecter les informations nécessaires à leur activité, se prémunir contre le méfait, et disposer du capital nécessaire. Je ne crois pas que le commerce maritime fournissait ces moyens : en tant que secteur à hauts profits, il avait

	au moins comparable, et peut-être un peu supérieur, à l'accumulation du
	capital.
	Une région ouverte au commerce -y compris maritime -pouvait L'effet sur le bien-être des consommateurs européen est important car,
	s'approvisionner grâce à celui-ci pour certains biens, par exemple les céréales, et d'après la thèse de Jan De Vries, l'un des moteurs de la croissance avant la
	développer sa production, par exemple viticole, au delà de ses besoins Révolution industrielle était la « révolution industrieuse », c'est à dire
	immédiats. Cette spécialisation avait deux effets positifs. D'une part, elle l'intensification de l'effort de travail -concomitant d'une baisse du temps
	permettait d'allouer les moyens de production vers des secteurs qui étaient réservé aux loisirs -dans les ménages européens 36 . Les biens fournis par le
	relativement favorisés dans chaque région par les circonstances géographiques ou commerce maritime s'inscrivaient en cela dans un ensemble de nouveaux biens
	autre. D'autre part, même en l'absence d'avantage a priori l'augmentation de la qui ont contribué à profondément changer les habitudes de consommation au
	spécialisation des travailleurs permettait d'augmenter leur productivité comme XVIII e siècle 37 .
	Un autre socle de la croissance smithienne était l'extension des capacités
	d'échange dans l'économie. Pour que les consommateurs travaillent plus pour
	accéder à de nouveaux produits offerts par les producteurs ou les commerçants,
	il fallait que les coûts de l'échange ne soient pas trop élevés. Les nouvelles
	routes britanniques à péage ne peuvent pas être attribuées au commerce
	maritime. Mais Daron Acemoglu et ses co-auteurs ont affirmé que cela pouvait
	Américains en Europe au XVIII e siècle était inférieure à celle de l'Angleterre de être le cas des institutions qui encadraient l'échange 38 . Ils se placent dans le
	1850. Cependant, le sucre et le thé n'étaient pas les seuls biens de cadre de l'économie de coûts de transaction dont l'importance pour l'histoire
	consommation rendus disponibles par le commerce maritime. Il faudrait économique a été soulignée par Douglass North 39 . Comme l'avaient souligné les
	d'ailleurs aussi inclure les matières premières industrielles, par exemple les observateurs contemporains -dont Voltaire dans la dixième Lettre
	teintures. Bref, les nouveaux biens avaient eu sur l'économie européenne un effet Philosophique -et comme l'affirme l'historiographie whig britannique, le statut
	social et politique exceptionnellement élevé des marchands en Angleterre y a
	1975, p. 309-330 ; INIKORI J.E., Africans and the Industrial Revolution in England: A study in 36 DE VRIES J., « The Industrial Revolution and the Industrious Revolution », in Journal of
	international trade and economic development. Cambridge University Press, 2002 ; O'BRIEN Economic History, vol. 54, n o 2, 1994, p. 249-270 ; DE VRIES J., The Industrious Revolution:
	27 DAUDIN G., « Do Frontiers Give or do Frontiers take? The Case of Intercontinental Trade in Princeton University Press, 2000. P.K. et ENGERMAN S.L., « Exports and the Growth of the British Economy from the Glorious Consumer Behavior and the Household Economy, 1650 to the Present. Cambridge, Cambridge
	France at the End of the Ancien Régime », in A Deus Ex Machina Revisited. Atlantic Colonial 30 BUTEL P., Les Négociants bordelais, l'Europe et les îles au XVIIIe siècle. Paris, Aubier, 1974. Revolution to the Peace of Amiens », in Slavery and the Rise of the Atlantic System, SOLOW University Press, 2008.
	Activities and European Economic Development. PÉTRÉ-GRENOUILLEAU O., EMMER P. et 31 TANGUY J., Quand la toile va: l'industrie toilière bretonne du xvie au xviiie siècle. Éditions B. (dir.), 1991, p. 177-209 37 ROCHE D., Histoire des choses banales : Naissance de la consommation, XVIIe-XVIIIe siècle.
	Apogée, 1994 34 RADEFF A., Du Café dans le chaudron : économie globale d'Ancien Régime, Suisse Paris, Fayard, 1997.
	Occidentale, Franche-Comté et Savoie. Lausanne, Société d'Histoire de la Suisse Romande,
	1996.
	35 VOTH H.-J. et HERSH J., « Sweet Diversity: Colonial Goods and the Rise of European Living
	Standards after 1492 », in CEPR Working Paper, 2009.

Si on se borne donc aux moteurs principaux de la croissance après la Révolution Industrielle, le progrès technique et l'accumulation directe de capital, le commerce maritime ne peut pas avoir joué un rôle décisif. Cependant la croissance avant la Révolution Industrielle était due plutôt à l'exploitation des gains de l'échange telle que décrite par Adam Smith 28 . La croissance «smithienne», lorsque le contexte institutionnel et les conditions de la demande la permettaient, conduisait à deux phénomènes : d'une part une intensification de l'utilisation des moyens de production et d'autre part leur meilleure allocation entre les différentes activités de manière à maximiser la production.

ROITMAN J. (dir.)

,

Leiden, Brill, 2006, p. 199-224

; DAUDIN G., « Profits du commerce international et croissance dans la France du XVIIIe siècle », in Revue Économique, vol. 57, n o 3, 2006, p. 605-613. Une dans le célèbre exemple de la fabrique d'épingle de Smith. Ainsi Phillip Hoffman a montré que dans la France de l'époque moderne, l'efficacité de la production agricole dépendait de manière décisive de sa proximité avec les centres urbains de demande 29 . Le commerce maritime jouait un rôle dans ces phénomènes par la baisse des coûts de l'échange qu'il permettait. Grâce à lui, des marchés lointains pouvaient offrir des débouchés importants à l'agriculture et l'industrie européenne, encourageant donc la spécialisation. L'arrière-pays de Bordeaux fournissait farines minots, pruneaux, etc. les Antilles françaises, encourageant en cela leur production 30 . L'industrie de toiles de Bretagne dépendait en partie des marchés de l'32 BOULLE P., « Slave Trade, Commercial Organization and the Industrial Growth in 18th-Century Nantes. », in Revue française d'histoire d'Outre-Mer, vol. 59, n o 214, 1972, p. 70-112. 33 BOULLE P.H., « Marchandises de traite et développement industriel dans la France et l'Angleterre du XVIIIe siècle », in Revue française d'histoire d'outre-mer, vol. 62, n o 226-227, l38 ACEMOGLU D., JOHNSON S. et ROBINSON J.A., « Rise of Europe », op. cit. encouragé e siècle était un groupe de pression comme les autres qui ont utilisé la Glorieuse Révolution pour renforcer leurs propres privilèges 40 . Toutefois, l'idée que les marchands jouaient un rôle particulier dans la croissance à l'époque moderne et qu'ils ont été renforcés de manière décisive par les profits du commerce maritime me semble raisonnable. Elle a même un parfum marxiste qui indique de manière rassurante qu'elle n'appartient pas à une chapelle idéologique particulière. J'ai gardé la thèse que j'ai défendue dans mes propres travaux pour la fin, au risque d'en exagérer l'importance 40 ZAHEDIEH N., The Capital and the Colonies: London and the Atlantic Economy 1660-1700. Cambridge University Press, 2010. 41 DAUDIN G., Commerce et prospérité, op. cit.

SMITH A., An Inquiry Into the Nature and Causes of the Wealth of Nations. London, Strahan,

COASE R.H., « The Nature of the Firm », in Economica, , n o 4, 1937, p. 386-405 ; NORTH D.C., Institutions, institutional change and economic performance. Cambridge university press,

24 WILLIAMS E.E., Capitalism & slavery. UNC Press Books, 1944 ; AMIN S., Accumulation on a World Scale. New-York, Monthly Review Press, 1974 ; FRANK A.G., World Accumulation, 1492-1789. New-York and London, Monthly Review Press, 1978 ; WALLERSTEIN I., The modern World system II : Mercantilism and the Consolidation of the European World Economy 1600-1750. New-York, Academic Press, 1980. CROUZET F. (dir.), Capital Formation in the Industrial Revolution. London, Methuen, 1972, p. 8 donne une liste des prédécesseurs d'Eric Williams. 25 Thomas R.P. et Bean R.N., « The Fishers of Men : The Profits of the Slave Trade », in Journal of Economic History, vol. 34, no 4, 1974, p. 885-914 ; Anstey R., « The volume and profitability of the British slave trade, 1761-1807 », in Genovese (eds.), Race and Slavery in the Western Hemisphere: Quantitative Studies, 1975, p. 3-31 ; Richardson D., « Profitability in the Bristol-Liverpool Slave Trade », in Revue Française d'histoire d'Outre-Mer, vol. 62, no 226-227, 1975, p. 301-308 ; Richardson D., « Profits in the Liverpool Slave Trade: The Accounts of William Davenport, 1757-1784 », in Liverpool, the African Slave Trade, and Abolition : essays to illustrate current knowledge and research. Anstey R. et Hair P.E.H. (dir.), Liverpool, Historic Society of Lancashire and Cheshire, 1976, p. 60-90 ; INIKORI J.E., « Market Structures and the Profits of the British African Trade in the Late Eighteenth Century », in Journal of Economic History, vol. 41, n o 4, 1981, p. 745-776 ; DARITY W.J., « The Number Game and the Profitability of the British Trade in Slaves », in Journal of Economic History, vol. 45 February 98, 25, p. 693-703 ; DAUDIN G., « Comment calculer les profits de la traite ? », in Outre-Mer : Revue d'Histoire, vol. 89, n o 336-337, 2002, p. 43-62 ; DAUDIN G., « Profitability of Slave and Long Distance Trading in Context: The Case of 18th Century France », in Journal of Economic History, vol. 64, n o 1, 2004, p. 144-171.


France (Dublin: The Lilliput Press, 2000. Pp. x + 244. 1 map. 16 ills. no price given)

The mutation of alcoholic beverage consumption was an important aspect of the rise of new patterns of consumption in eighteenth century Europe. Starting in the late seventeenth century, Western France supplied London, Paris, Ireland and northern Europe with brandy as a substitute to grain alcohols. During the eighteenth century, the region around Cognac specialized in producing that quality aged champagne brandy that is still known as cognac. Irish and English traders were some of the actors of this specialization.

In The Brandy Trade, Cullen provides a thorough description of the rise and the history of the brandy trade in the region between Bordeaux and La Rochelle from the late seventeenth century to very end of the eighteenth century. In The Irish Brandy Houses, he completes this study with a vivid record of the life and activity of the Irish traders who migrated to the region during the eighteenth century. Both books were made possible by the remarkable survival of the business records of many actors of the brandy trade.

Cullen starts The Brandy Trade by describing the emergence of the international trade in brandy spirit. Cullen contradicts Labrousse and affirms that distillation did not reflect conditions of surplus and low prices. He underlines that distillation was not linked to agricultural crisis: on the contrary, the rise of distillation was linked to the most dynamic aspects of the French To some extent, this last part covers the same material as The Irish Brand Houses. Yet, this second book is more thorough. After presenting the history of brandy trade, the mechanisms of brandy smuggling to Ireland, and the evolution of the trading community in the first part of the eighteenth century, Cullen gives a record of the story of the Irish families -especially Hennessy and Saule, but also Jennings, Galwey, Delamain, and others -who played a leading role in brandy trade starting in the 1760s. In the remaining five chapters, he describes the life of these traders in a chronological way. This study is all the more welcome as, if the négociants from the main French ports are well known, there is a lack of information on smaller-scale traders. The quality of the insight that Cullen gives on the life and activity of these smaller-scale traders is remarkable. Even if, as it is often the case, we suspect that trading difficulties take a much larger part in business correspondence than their successes, the precariousness of their situation is striking. Family issues appear deeply intermingled with business relationships. The book also explores the existing trading networks with Ireland and England and the effects of the French Revolution on trade.

Because of the nature of the evidence and the existing sources, the life and experience of the individual traders is much better documented than the actual evolution of regional trade and production. The agricultural and industrial producers are largely absent from Cullen's story. To some extent, the subtitle of his 1998 book, Regional Specialisation in the Charente, is misleading:

Cullen does not study the economy of the region as a whole, nor the role that brandy played in shaping the way all inhabitants were -or were not -integrated into a larger commercial economy. That could have been an alternative objective for the last part of the book, rather than going through material that Cullen explored again in The Irish Brandy Houses.

Yet, my complaints are those of a spoiled reader. The quality of both books is high. The Brandy Trade will remain for a long time the authoritative work on early modern brandy. The

  economy. It was encouraged by the development of domestic markets. It provided high quality bills to the Paris foreign exchange market. Cullen then describes the distilling process and the evolution of brandy toward the product we are now used to. He presents quickly the different actors of the industry, before describing the complex evolution of different markets and different producing regions. The importance of the Paris market was the most stable feature of the trade: Parisian demand repeatedly pre-empted the supply of other markets. By contrast, trade with Ireland was very volatile. In the sixth chapter, Cullen prepares a chronological study of the trading families in Cognac and Bordeaux that constitutes the last three chapters of The Brandy Trade.

Irish Brand Houses should be useful to anyone interested in diasporas and the life of traders in the eighteenth century.

Guillaume Daudin

Stanford University


Letterbook of

 Greg & Cunningham, 1756-57, Merchants of New York and Belfast.Edited by Thomas M. Truxes. Oxford: Oxford University Press for The British Academy, 2001. Pp. xxxi, 430. £50.00.

Researchers have been giving Irish trading networks more attention recently (cf. for example Louis M. Cullen. The Irish Brandy Houses of Eighteenth-Century France. Dublin: The Lilliput Press, 2001). Thomas M. Truxes has already written a pioneer book on the subject (Irish-American Trade, 1660-1783. New-York: Cambridge University Press, 1989). Ireland could trade directly with British America since a modification in English navigation laws in 1731. In his new book, Truxes gives us an in-depth view of the most successful Irish-American partnership of the time based on the letters sent by its New-York partner during nine months in 1756 and 1757. These were interesting times: the Seven Year's War had opened new opportunities -like (British) privateering and troop supply -and caused new difficulties -like the renewed fight against smuggling by the English authorities and (French) privateering -without disrupting fundamentally the traditional flows of trade between Ireland and British America: Ireland mainly exported linens, salted beef, pork, butter and passengers and imported flaxseeds, rum, lumber, staves and some pig iron and semi-manufactured goods.

The book is divided in three parts. A 100-page introduction covers a lot of ground. It first presents the context of the activity of the partnership: the towns of New-York and Belfast alongside with traditional Irish-American trade and the Seven Years' War. Then, it tells us whatt is known of the history of the partnership and the partners. Finally it sums up what can be found in the letterbook about the one hundred and thirty correspondents of the partnership, the goods in which they traded, their use of bills of exchange and maritime insurance, the way they managed their ships and their smuggling activity. The 338 letters sent by Wadell Cunningham between May 1756 and January 1757, presented in chronological order, form the core of the book. The Appendix are: two "deeds of copartnership" between Thomas Gregg and Wadell Cunningham in 1761 and 1768; the partners' newspaper advertisements between 1752 and 1757; the wills of the two partners; and a glossary of maritime and mercantile terms built from contemporary sources. 50 pages of index conclude the book. Truxes has also gathered eight plates of iconographic documentation.

The amount of research and the amount of erudition put into this book are impressive. Thanks to his knowledge of contemporary sources and modern works, Truxes has collected extensive information on "individual persons, business enterprises, places, ships, commodities, business practices and other features of Greg & Cunningham's economic, political and social environment" (p. 93). For example, as Cunningham is glad not to receive gunpowder in June 1756, we are given the price of gunpowder on the Amsterdam Exchange from 1 May 1754 to the end of 1758 (p. 133). In the same way, because one letter was sent to Robert Wells (or Wills) of Belfast, we are given the information that can be found about him in The Belfast News-Letter from 1754 to 1771 (p. 308).

However, after reading the whole book, the "So what?" question comes to mind. It is actually unfair. This is not an essay on Irish-American trade, or even some part of it. Even the long introduction is mainly factual, and does not try to relate the letterbook with current issues on the links between social and economic relations or the importance of trading networks to economic development. This book should be used like a tool. The non-specialist will not be interested. Yet, anyone studying Irish-American trade or more generally merchant networks and practices in the eighteenth-century will find in it very useful material -much more useful than the raw source. For its small public, this book will probably prove invaluable.

Guillaume Daudin OFCE, Paris


domain: chim phys

text:
Iciar Uriarte

Aran Insausti

Emilio J Cocinero

Atef Jabri

Isabelle Kleiner

Halima Mouhib

Ibon Alkorta

Competing Dispersive Interactions: From Small Energy Diff erences to Large Structural Eff ects in Methyl Jasmonate and Zingerone

ABSTRACT: Modern structural studies of biologically relevant molecules require an exhaustive interplay between experiment and theory. In this work, we present two examples where a poor choice of the theoretical method led to a misinterpretation of experimental results. We do that by performing a rotational spectroscopy study on two large and flexible biomolecules: methyl jasmonate and zingerone. The results show the enormous potential of rotational spectroscopy as a benchmark to evaluate the performance of theoretical methods. n the past decade, structural studies on large, flexible organic molecules of biological interest moved into the focus of interest. The three-dimensional conformation that a biomolecule adopts is intimately related to its function in the biological medium. Hence, a lot of eff ort is devoted to the structural determination of biomolecules of increasing size.

Structural studies carried out in condensed media can provide an environment comparable to biological surroundings, but the solvent or crystal packing forces may mask some key interactions. High-resolution structural studies in a supersonic jet benefit from an interaction-free environment where biomolecules can be probed without any kind of interference.

strategy in these studies employs an interplay between experimental results and theoretical calculations (molecular mechanics, molecular dynamics, and quantum chemistry). This symbiosis between theory and experiment is nowadays a cornerstone in structural studies. Because of the increasing size and degrees of freedom of the molecular systems targeted, there is a need for precise theoretical models that support the interpretation of experimental results. On the other hand, theoretical methods still require a strong and reliable experimental counterpart to safeguard the results of any implemented model. Especially the combination of rotational spectroscopy with highly accurate quantum chemistry is a

The intrinsic properties of the molecule under study can be analyzed and separated from the eff ects arising from solvent or crystal packing. Regardless of the high-resolution spectroscopic technique (laser spectroscopy, microwave, etc..), the general Scheme 1. Chemical Structures of Methyl Jasmonate and Zingerone powerful tool to study the molecular structures of large and flexible organic molecules. The efficiency of this strategy was previously shown in biologically relevant cases such as amino acids, 1,2 sugars, 3-6 nucleobases, 7-9 odorants, [START_REF] Mouhib | Cassis Odor through Microwave Eyes: Olfactory Properties and Gas-Phase Structures of All the Cassyrane Stereoisomers and Its Dihydro Derivatives[END_REF][START_REF] Nguyen | A Touch of Lavender: Gas-Phase Structure and Dynamics of the Monoterpene Linalool Validated by Microwave Spectroscopy[END_REF][12] and some drugs. [START_REF] Leon | Water Encapsulation by Nanomicelles[END_REF][START_REF] Leon | Shaping Micelles: The Interplay between Hydrogen Bonds and Dispersive Interactions[END_REF][START_REF] Lesarri | The Conformational Landscape of the Volatile Anesthetic Sevoflurane[END_REF] Recently, the unprecedented speed of rotational spectra acquisition obtained with broadband chirped-pulse Fourier transform microwave (CP-FTMW) spectrometers has opened new perspectives in structural chemistry. [START_REF] Brown | A Broadband Fourier Transform Microwave Spectrometer Based on Chirped Pulse Excitation[END_REF][START_REF] Shipman | New Techniques in Microwave Spectroscopy[END_REF][START_REF] Neill | Rotational Spectroscopy of Iodobenzene and Iodobenzene-Neon with a Direct Digital 2-8 GHz Chirped-Pulse Fourier Transform Microwave Spectrometer[END_REF][START_REF] Per Ez | Structures of Cage, Prism, and Book Isomers of Water Hexamer from Broadband Rotational Spectroscopy[END_REF][START_REF] Per Ez | Broadband Fourier Transform Rotational Spectroscopy for Structure Determination: The Water Heptamer[END_REF] Hence, it is possible to study larger molecular systems (15-25 heavy atoms) such as biomolecules [START_REF] Uriarte | Structural Studies of Nicotinoids: Cotinine versus Nicotine[END_REF] or large organic compounds, [START_REF] Fokin | Intramolecular London Dispersion Interaction Effects on Gas-Phase and Solid-State Structures of Diamondoid Dimers[END_REF][START_REF] Domingos | Cold Snapshot of a Molecular Rotary Motor Captured by High-Resolution Rotational Spectroscopy[END_REF] and their weakly bound molecular complexes. [START_REF] Seifert | Probing the C-H•••π Weak Hydrogen Bond in Anesthetic Binding: The Sevoflurane-Benzene Cluster[END_REF][START_REF] Steber | Capturing the Elusive Water Trimer from the Stepwise Growth of Water on the Surface of the Polycyclic Aromatic Hydrocarbon Acenaphthene[END_REF][START_REF] Ghosh | Rotational Spectra of Two Hydrogen-Bonded Methyl Salicylate Monohydrates: Relative Stability and Tunneling Motions[END_REF] In addition, the implementation of laser vaporization sources allows one to study thermolabile systems, avoiding the undesirable thermal decomposition processes that occur when using conventional heating techniques. [START_REF] Lesarri | The Structure of Neutral Proline[END_REF][START_REF] Cocinero | Ribose Found in the Gas Phase[END_REF] The experimental progress brought by CP-FTMW spectroscopy now demands the development of more accurate, faster and efficient theoretical methods to analyze and fit the experimental data.

One of the crucial points in the predictive value of the key parameters (structure, energy, dipole moments, quadrupolar tensor, ...) of larger biomolecular systems is the choice of the theoretical method. There is a large array of quantum chemical methods for that purpose. For instance, B3LYP [START_REF] Becke | Density-Functional Thermochemistry. III. The Role of Exact Exchange[END_REF][START_REF] Lee | Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density[END_REF] was historically very important, its cost-eff ectiveness making it very commonly used. However, the need to characterize weakly bound systems forced the inclusion of empirical parametrizations such as Grimme's approach [START_REF] Grimme | Semiempirical GGA-Type Density Functional Constructed with a Long-Range Dispersion Correction[END_REF][START_REF] Grimme | Accurate Description of van Der Waals Complexes by Density Functional Theory Including Empirical Corrections[END_REF][START_REF] Grimme | Effect of the Damping Function in Dispersion Corrected Density Functional Theory[END_REF][START_REF] Caldeweyher | Extension of the D3 Dispersion Coefficient Model[END_REF] (D, D2, D3, D4 and its subvariants) in order to account for dispersive interactions. Minnesota functionals [START_REF] Zhao | The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Function[END_REF] are more modern, more sophisticated, and highly parametrized, and Møller-Plesset perturbation theory [START_REF] Møller | Note on an Approximation Treatment for Many-Electron Systems[END_REF] is very reasonable for small-to-medium systems but not without limitations. Hence, the choice of method is usually subject to a trade-off between accuracy and computational cost. Systems with only covalent interactions or relatively strong hydrogen bonds (e.g., O-H

•••O, O-H•••N, N-H•••O,
15-25 kJ mol -1 ) can be described at almost any reasonable level of theory. However, systems stabilized by O-H van der Waals, or London dispersive interactions require specific approaches to be modeled properly. In many cases, the choice of the theoretical method is done routinely without taking into account the key interactions of the system. In addition, it is wellknown that some methods are able to describe very well some weak interactions but fail at characterizing others. [START_REF] Christensen | Semiempirical Quantum Mechanical Methods for Noncovalent Interactions for Chemical and Biochemical Applications[END_REF][START_REF] Brandenburg | Low-Cost Quantum Chemical Methods for Noncovalent Interactions[END_REF] In the present study, we demonstrate that the choice of the appropriate quantum chemical method can be vital in the structural determination of biomolecules. In order to do so, we use the case studies of methyl jasmonate and zingerone, two of the main flavor components of jasmine and ginger, respectively (see Scheme 1 for chemical formulas). [START_REF] Acree | Odor Thresholds of the Stereoisomers of Methyl Jasmonate[END_REF][START_REF] Asamitsu | Synthesis and Odor Description of Both Enantiomers of Methyl 4,5-Didehydrojasmonate, a Component of Jasmin Absolute[END_REF] The results reported in this work clearly demonstrate the importance of carefully choosing the appropriate theoretical method for the molecular system under study. We can anticipate that a poor choice of the theoretical method may compromise the interpretation of the experimental results.

•••halogen, C-H•••O, C- H•••N, C-H•••π, halogen-halogen, π•••π, lone-pair•••π,
Both substances are widely present in plants. Methyl jasmonate is a vital lipidic phytohormone produced by many plants in response to biotic stress (e.g., pathogens or pests) and abiotic stress (e.g., heat, cold, wind, precipitation). It is an important signaling molecule and plays a crucial role in the communication of plants. [START_REF] Cheong | Methyl Jasmonate as a Vital Substance in Plants[END_REF][START_REF] Avanci | Jasmonates Are Phytohormones with Multiple Functions, Including Plant Defense and Reproduction[END_REF] Zingerone is the main component of ginger and it is responsible for its pungent scent. It has been shown to exhibit diuretic and neuroprotective functions, which are still not fully understood at a molecular scale. [START_REF] Fay | A Highly Effective and Selective Male Lure for Bactrocera Jarvisi (Tryon) (Diptera: Tephritidae)[END_REF] Besides their interesting biological eff ects, these molecules represent limiting cases with respect to their size and molecular weights, for both the theoretical and the experimental studies presented in this work.

Case Study 1: Methyl Jasmonate. Methyl jasmonate (MJ) consists of a cyclopentanone ring with two large flexible side chains (see Scheme 1). There are two chiral centers in the molecule, meaning that there are two diastereoisomers to consider (RR and RS). The corresponding enantiomers (SS and SR) would be equivalent by symmetry and therefore, they would give rise to the same rotational spectra. Furthermore, the double bond in one of the side chains may adopt Z or E configurations. Therefore, there are four diff erent structural isomers to consider (RR-Z, RR-E, RS-Z, and RS-E). Each one of them may adopt diff erent conformations arising from rotations along single bonds. Moreover, there could be attractive intramolecular interactions (such as hydrogen bonds) between the two chains. We can infer from these structural and conformational degrees of freedom that the conformational casuistry of methyl jasmonate is very broad and complex.

The first step in this study was a fast conformational search with molecular mechanics (MM), using Merck's molecular force field (MMFF). About 100 plausible structures for each system were found in an energy window of 20 kJ mol -1 . Hence, 4 × ∼100 ≈ 400 structures had to be reoptimized with more accurate theoretical calculations to predict the relevant rotational parameters (structures, energies, dipole moments, ...). At that stage, we chose the B3LYP method, a fast and reasonably accurate method, to guide the experiment and reoptimized all the structures.

In the meantime, the rotational spectrum of methyl jasmonate was recorded in the frequency region of 6-18 GHz using the CP-FTMW spectrometer available at the University of the Basque Country (UPV/EHU), see Figure S1 in the Supporting Information. The sample was heated up to 200 °C in a customized heating nozzle in order to build up enough concentration of methyl jasmonate in the gas phase. Two set

) c ΔK (DK) kHz dJ (d1) kHz dK (d2) kHz V d cm -1 σ e kHz N tot NA/NE g a
The microwave data set was fitted using the program XIAM. of rotational transitions were spotted in the spectrum. They were first fitted separately to two semirigid asymmetric rotor Hamiltonians using Watson's A reduction. [START_REF] Watson | Vibrational Spectra and Structure[END_REF] The rotational transitions in the spectrum displayed splittings due to tunneling eff ects from internal rotation of a methyl top. Hamiltonian, allowing for the estimation of the barrier for internal rotation of the methyl top (V3). We used two diff erent publicly available software programs (XIAM [START_REF] Hartwig | The Microwave Spectrum of Trans-2,3-Dimethyloxirane in Torsional Excited States[END_REF][START_REF] Hansen | A Molecular Beam Fourier Transform Microwave Study of O-Tolunitrile: 14 N Nuclear Quadrupole Coupling and Methyl Internal Rotation Effects[END_REF] and BELGI [START_REF] Kleiner | Rho-Axis-Method Hamiltonian for Molecules Having One Methyl Rotor and C1 Point-Group Symmetry at Equilibrium[END_REF] ) for fitting the experimental results. A number of previous studies have shown that the XIAM code and the BELGI code are complementary. [START_REF] Nguyen | Methyl Internal Rotation in the Microwave Spectrum of Vinyl Acetate[END_REF][START_REF] Eibl | Low Barrier Methyl Rotation in 3-Pentyn-1-ol as Observed by Microwave Spectroscopy[END_REF][START_REF] Ferres | The Effect of Internal Rotation in P-Methyl Anisole Studied by Microwave Spectroscopy[END_REF] As described in the Supporting Information (eff ective Hamiltonian section), the XIAM code takes into account one torsional state at a time, and has a limited number of high order Hamiltonian terms implemented, whereas the BELGI-C1 code treats all torsional states together in one matrix and has many higher order terms. While XIAM benefits from the advantages of its ease of use and speed, in cases with rather low barriers, E species transitions are not always satisfactorily predicted because the interactions between torsional states can be large. Considering the importance of carefully deriving experimental rotational constants to compare with ab initio methods, we decided to use both methods in this work. In the present case, the results were similar (see Tables S1-S3), and in the following, we will only refer to the XIAM fits, which are summarized in Table 1. The experimental values of rotational and torsional parameters for the two conformers of MJ are presented in full in Table S1 of the Supporting Information.

The common strategy used in the microwave spectroscopy community to assign a molecular conformation to a given set of experimental rotational constants is to compare them to a predicted spectrum generated with the values arising from quantum chemical calculations. However, all attempts to assign the spectrum based on diff erent structural models of methyl jasmonate failed. This spectrum did not correspond to any of the predicted conformations for methyl jasmonate using the B3LYP method. Finally, we considered the possibility of sample fragmentation resulting in decomposition products or internal reactions, but the considered species did not match the spectrum either. By this point, we had carried out more than 1000 calculations but none of them reproduced the experiment. Obviously, the B3LYP method failed to provide accurate structures of the most abundant conformers of methyl jasmonate. Therefore, we decided to redo the optimizations of methyl jasmonate using the MP2 ab initio method, which finally led to structures that were in agreement with our experimental results. Its lowest lying energy conformers (Conf. MJ1 and Conf. MJ2) had not been detected by the B3LYP method, and these two were the structures that reproduced the experimental data. Although optimizations at the B3LYP level may be largely sufficient for average sized systems, there is a limit, when intramolecular interactions become too important and cannot be described properly by this method. Indeed, for methyl jasmonate the B3LYP method was insufficient to provide accurate structures that are consistent with the highly accurate experimental results.

Finally, we carried out numerous calculations with diff erent methods to compare their performance in reproducing experimental values. We always used the same basis set (6-311++G(d,p)). Theoretical methods yield the rotational constants at the bottom of the potential well (commonly known as Be equilibrium structure) while the experimental rotational constants refer to the structure at the vibrational ground state v = 0 (B0). We estimated ΔBvib = B0 -Be with anharmonic calculations at the B97D3/def2-SVP level of theory and subtracted it from the experimental rotational constants in order to obtain the semiempirical values of Be. The latter are directly comparable to the rotational constants obtained from quantum chemical calculations. [START_REF] Fokin | Intramolecular London Dispersion Interaction Effects on Gas-Phase and Solid-State Structures of Diamondoid Dimers[END_REF][START_REF] Vogt | The Equilibrium Molecular Structures of 2-Deoxyribose and Fructose by the Semiexperimental Mixed Estimation Method and Coupled-Cluster Computations[END_REF] The results are summarized in Table 2 for the two observed conformers of methyl jasmonate. It is noteworthy that the B3LYP method, which does not account for dispersion eff ects, fails at reproducing the experimental structure (relative errors in A around 15%). This is also the case for other methods such as PBE, TPSS or B2PLYP (although B2PLYP provides significantly better results, with ∼6% error in the A rotational constant). Only ab initio and DFT methods that considered dispersion eff ects (either intrinsically or by adding a correction) were successful at reproducing experimental values. In this case, the Minnesota functional M06-2X gives worse results (relative errors in A around 4-5%) than DFT methods corrected by Grimme's corrections (relative errors 0-2%). Figure 1 displays the steps of the optimization of Conf. MJ2 with the B3LYP and B3LYP-D3BJ methods starting from the same geometry. We can clearly see that with B3LYP, as the optimization progresses, the two arms move away from each other, the final distance CH•••O being 4.1 Å (conformer Conf. MJ2′). Hence, the CH•••O interaction is not accounted for by the B3LYP method. On the contrary, using B3LYP-D3BJ, the two arms remain close to each other, held together by the CH••• O weak interaction (final distance 2.8 Å, Conf. MJ2). We infer that the dispersion correction D3BJ has a dramatic eff ect on the rotational constants. In particular, the A rotational constant is strongly influenced by dispersion in the case of methyl jasmonate because the principal axis a lies along the two side chains. Hence, the moment of inertia Ia (to which A is inversely proportional) is very sensitive to the coordinates of the atoms in the side chains (see Figure 2 for the principal axes of inertia). The eff ect of dispersion is so pronounced in this molecule that the assignment to the molecular conformations was not possible with the B3LYP methodology.

Letter

To summarize, the two experimental conformations are assigned to Conf. MJ1 and Conf. MJ2: both of them are RR diasteroisomers (or SS), the configuration of the double bond in the side chain is Z and both conformations are stabilized by a C-H•••O weak hydrogen bond (WHB). The corresponding 3D structures are depicted in Figure 2, along with the structures of Conf. MJ1′ and Conf. MJ2′, i.e., the conformations predicted from calculations where no dispersion is included.

Case Study 2: Zingerone. Zingerone (see Scheme 1) consists of an o-methoxyphenol moiety with a flexible chain in para position with respect to the hydroxy group. In order to guide the spectral search, we predicted the most favorable structures employing the same strategy as described above and commonly used in the literature: we performed MM to have an initial screening of the conformational scope (14 structures were found in 15 kJ mol -1 ) and then we reoptimized all of them with the MP2 robust method.

Later, the rotational spectrum of zingerone in the 6-18 GHz frequency region was recorded using the FTMW and CP-FTMW spectrometers at the University of the Basque Country (UPV/EHU) see Figure S2 of the Supporting Information. The sample was heated to 200 °C in a customized heating nozzle in order to bring it into the gas phase. A set of rotational transitions corresponding to the A species of one conformation of the molecule was assigned and fitted to a semirigid asymmetric rotor Hamiltonian in Watson's S-reduction. Splittings between A and E species due to the internal rotation were accounted for in the fit by appropriate operators in the Hamiltonian. The results of the fit are summarized in Table 1.

Surprisingly, there was no correspondence between experi-ment and theory. The discrepancy in the rotational constants was remarkable. Diff erences of ∼17% indicated that the predicted structure was not the experimentally observed structure, despite having used a method as reliable as MP2 theory.

In this case, we decided to reoptimize all the structures using DFT theory, in particular the B3LYP-D3BJ functional. Interestingly, the results generated with this functional were not comparable to those obtained with the MP2 method. With both methods, all the lowest lying conformations were stabilized by an OH•••O interaction between the hydroxy and the methoxy group. However, the global minimum was diff erent for the MP2 and B3LYP-D3BJ calculations. Additionally, the lowest lying structure predicted by B3LYP-D3BJ was not predicted using the MP2 method. The two conformations predicted as global minima in zingerone are shown in Figure 3. The first one (Conf. Z1, MP2 structure exhibits an attractive CH•••π interaction between the terminal CH3 in the flexible chain and the π cloud in the aromatic ring. This interaction forces the side chain to tilt toward the ring. It should be noted that this structure is not predicted by B3LYP (or TPSS, PBE, and B2PLYP, i.e., methods that do not account for dispersive interactions). In the second conformation (Conf. Z2, B3LYP-D3BJ structure), no such interaction occurs and the side chain is perpendicular to the ring. This conformer exhibits a C O•••H-C attractive interaction. This conformation was not predicted by the MP2 method. To summarize, we have two possible global minima, one of which was not predicted at the MP2/6-311++G(d,p) level of theory.

In addition, we extended the predictions to other theoretical models. Table 3 compiles the results of those calculations. Clearly, the result of the optimizations highly depend on the applied theoretical method. The balance between the CH•••π interaction (Conf. Z1) and the CH•••O interaction (Conf. Z2) is very subtle. It is a challenging task for theoretical models to weigh the strength of the two interactions, since both are weak hydrogen bonds (WHB) with small stabilization energies below 4 kcal mol -1 . [START_REF] Desiraju | The Weak Hydrogen Bond in Structural Chemistry and Biology[END_REF] Regarding the theoretical methods, three diff erent trends are observed. They are summarized in Figure 4 with the three common methods MP2, B3LYP, and B3LYP-D3BJ. We can observe that B3LYP-D3BJ is capable of reproducing both conformations if the correct starting geometry is chosen. B3LYP always yields Conf. Z2 stabilized by CH•••O interaction, that is, with the lateral chain perpendicular to the plane of the aromatic ring. Finally, MP2 calculations result always in Conf. Z1 with a CH•••π interaction, where the chain is bent toward the benzene ring. Table 3 shows the diff erences in the rotational constants. Again, the choice of the theoretical method is crucial to success. The rotational constants are very sensitive to both conformers and we will only be able to discern unambiguously the structure of zingerone through microwave spectroscopy.

By comparing the experimental values of the rotational constants with the theoretical predictions, we can conclude that the experimental conformation of zingerone is Conf. Z2, which is stabilized by a CH•••O interaction. This means that MP2/6 311++G(d,p), which is a very commonly used and most trusted method in the rotational spectroscopy community, fails to predict the experimentally observed conformation of zingerone as minimum energy structure. In fact, the observed conformer is not even predicted by this method. MP2 overestimates the CH•••π interaction and, as the balance between WHBs is so subtle, the real structure is not predicted by the MP2 method. In order to visualize the subtlety of the competition between the two WHBs, we calculate the barrier between the two structures by performing a relaxed scan of the 1-2-3-4 dihedral angle (see Scheme 1 for atom numbering). Moreover, we perform this scan using two diff erent methods (M06-2X and B3LYP-D3BJ) to evaluate the response of both of them in the WHBs competition. Figure 5 displays the two scans. We can observe the local minima corresponding to Conf. Z1 and Conf. Z2 in both scans. It is worth noting that these minima are almost isoenergetic. The energy diff erence between both of them is less than 0.5 kJ mol -1 (this value is probably within the error of the theoretical calculation). Moreover, M06-2X and B3LYP-D3BJ predict that the barrier between the two structures is very low (∼1 kJ mol -1 ), meaning that it can be easily overcome in the supersonic expansion and therefore, the system would relax to the most stable conformation. This behavior has been previously observed in rotational spectra and is well reported in the literature. [START_REF] Ruoff | Relaxation of Conformers and Isomers in Seeded Supersonic Jets of Inert Gases[END_REF][START_REF] Godfrey | The Missing Conformers of Glycine and Alanine: Relaxation in Seeded Supersonic Jets[END_REF] The most relevant diff erence between the two This study demonstrates that rotational spectroscopy combined with theoretical methods is a powerful strategy, which can be broadly applied to flexible organic compounds. However, one must be cautious in the use of diff erent theoretical methods. There is currently no strategy to avoid brute force optimizations in this kind of molecular systems (for example, force fields are poorly parametrized for the accurate description of averaged sized biologically active ligands). Therefore, benchmark studies such as the ones presented here are extremely necessary. We can infer from the studies on methyl jasmonate and zingerone that using several distinct levels of theory allows one to get a good overview of the conformational landscape. This work demonstrates that microwave spectroscopy can benchmark new theoretical models very much in need to guide the experiments.

■ ASSOCIATED CONTENT

* S Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.8b02339.

Experimental and computational details, rotational spectra of methyl jasmonate and zingerone, experimental spectroscopic parameters of all observed species, and a list of experimental rotational transitions (PDF)

■ ACKNOWLEDGMENTS A.J., H.M., and I.K. would like to acknowledge the CNRS InFinity program and the OSU/Efluve for funding; H.M. thanks the European COST initiative MOLIM for a STSM fellowship; E.J.C. and I.U. thank the MINECO (CTQ2017-89150-R) and the UPV/EHU (UFI11/23 and PPG17/10) for funds. Computational resources and laser facilities of the UPV-EHU were used in this work (SGIker). Computational resources of the MSME laboratory of the University Paris-Est were used in this work.

  40,41 A, B, C: rotational constants. ΔJ (DJ), ΔJK (DJK), ΔK (DK), δJ (d1), δK (d2): centrifugal distortion constants for Watson's A reduction for methyl jasmonate and (S) reduction for zingerone, I r representation. Errors of the experimental data given in parentheses are in units of the last digit. b Watson's A reduction ΔJ, ΔJK, ΔK, δJ, and δK are used for methyl jasmonate conformer 1 (Ray's kappa = -0.43) and for conformer 2 (Ray's kappa = -0.19). c Watson's S reduction DJ, DJK, DK, d1 and d2 are used for zingerone (Ray's kappa = -0.75). d Barrier to internal rotation of the methoxy methyl group and carbonyl methyl group in methyl jasmonate and zingerone, respectively. e Standard deviation of the fit. f Total number of fitted lines. g Number of fitted A and E species; h Parameters set to zero. Table 2. Comparison of the Experimental Values of Ae, Be, and Ce with the Theoretical Values Calculated with Diff erent Theoretical Methods (Basis Set: 6-311++G(d,p)) for Conf. MJ1 (a) and Conf. MJ2 (b) of Methyl Jasmonate a a Note that the experimental values of Be are calculated such that Be = B0 -ΔBvib, and the anharmonic correction ΔBvib is estimated for each conformer separately at the B97D3/def2-SVP level of theory. Values shaded in orange correspond to Conf. MJ1' (part a) and Conf. MJ2' (part b).

Figure 1 .

 1 Figure 1. Steps of the optimization of Conf. MJ2′ and Conf. MJ2 of methyl jasmonate (RR, Z conformation of the double bond) using the B3LYP and B3LYP-D3BJ DFT methods, respectively, from the same starting geometry (basis set: 6-311++G(d,p)).

Figure 2 .

 2 Figure 2. Two experimentally observed conformations of methyl jasmonate (RR, Z conformation of the double bond) (Conf. MJ1 and Conf. MJ2) as optimized at the B3LYP-D3BJ/6-311++G(d,p) level of theory, as well as the corresponding conformations Conf. MJ1′ and Conf. MJ2′ calculated at the B3LYP/6-311++G(d,p) level of theory. For Conf. MJ1, we also show the principal axes of inertia a, b, and c.

Figure 3 .

 3 Figure 3. Conformations of zingerone predicted as global minima with the MP2 (Conf. Z1) and B3LYP-D3BJ methods (Conf. Z2), using the basis set 6-311++G(d,p).

Table 3 .

 3 Comparison of the Experimental Values of Ae, Be, and Ce with the Theoretical Values Calculated with Diff erent Theoretical Methods (Basis Set: 6-311++G(d,p)) for Conf. Z2 (a) and Conf. Z1 (b) of Zingerone a a Note that the experimental values of Be are calculated such that Be = B0 -ΔBvib, and the anharmonic correction ΔBvib is estimated for each conformer separately at the B97D3/def2-SVP level of theory. NA = Not Applicable, the optimization leads either to Conf. Z1 (part a) or Conf Z2 (part b).

Figure 5 .

 5 Figure 5. Potential energy scans of the 1-2-3-4 dihedral angle of zingerone to evaluate the energy barrier between Conf. Z1 (stabilized by a C-H•••π interaction) and Conf. Z2 (stabilized by a C-H•••O interaction). Two diff erent theoretical methods have been used: M06-2X (a) and B3LYP-D3BJ (b), in order to evaluate their behavior with respect to the zingerone molecular system. The basis set is 6-311+ +G(d,p).

Figure 4 .

 4 Figure 4. Schematic representation of the prototype behavior of theoretical methods in the conformational casuistry of zingerone.

Table 1 .

 1 Microwave Spectroscopic Results of the Most Abundant Conformers of Methyl Jasmonate (RR, Z conformation of the double bond) and Zingerone Observed in the Molecular Jet

	Methyl Jasmonate	Zingerone


Franc ¸ois Sarrazin Ariston Reis

Member, IEEE Philippe Besnier

Senior Member, IEEE Philippe Pouliguen

Member, IEEE Elodie Richalot

Contactless Antenna Gain Pattern Estimation From Backscattering Coefficient Measurement Performed Within Reverberation Chambers

Keywords: Reverberation chamber, radar cross-section, antenna gain, antenna backscattering coefficient, structural mode, radiation mode

This paper presents the antenna gain pattern estimation from backscattering coefficient measurement performed within a reverberation chamber (RC). The antenna backscattering coefficient is measured for two load conditions, namely an open circuit and a 50 Ω load, enabling to distinguish the radiation and structural modes from the total backscattered field. This approach is applied to the measurement of the gain pattern of a Vivaldi antenna at 6.5 GHz and compared to a conventional transmission-type measurement within an anechoic chamber. The main lobe of the radiation pattern is in good agreement between both measurements. The difference between maximal gains, retrieved after a reference measurement using a metallic plate, is equal to 0.04 dB. A second validation is performed using a log-periodic antenna at 3.5 GHz, showing less than 0.76 dB difference over the measured angular range. To the author's knowledge, this is the first time that a contactless gain pattern measurement is performed and validated within an RC.

I. INTRODUCTION

The gain as well as the radiation pattern are crucial antenna properties usually evaluated through transmission-type measurements. This measurement implies an invasive setup with excitation cables connected to the antenna under test (AUT). In the case of electrically small antennas, the presence of cables in the antenna near-field zone can disturb both the impedance and radiation properties and lead to inaccurate results. The literature reports two main paths of investigation to overcome this issue: the first one consists in either limiting the cables effect by adding ferrites, quarter-wavelength sleeves or balun [START_REF] Icheln | Reducing the influence of feed cables on small antenna measurements[END_REF], or compensating it thanks to prior electromagnetic simulations [START_REF] Huitema | Impedance and radiation measurement methodology for ultra miniature antennas[END_REF], whereas the second one aims at avoiding the use of these excitation cables connected to the AUT. Inspired from radar cross-section (RCS) measurement, such method relies on the antenna backscattering characterization to retrieve its radiation properties in a contactless or non-invasive way.

The antenna scattering properties have been first investigated by King in 1949 [START_REF] King | The measurement and interpretation of antenna scattering[END_REF] and methodologies to retrieve Submitted February 24, 2021; revised August XX,2021. This work was supported in part by the Direction Générale de l'Armement (DGA) of the French Ministry of the Armed Forces.

A. Reis, F. Sarrazin and E. Richalot are with the ESYCOM lab, Univ Gustave Eiffel, UMR CNRS 9007, F-77454 Marne-la-Vallée, France (e-mail: francois.sarrazin@univ-eiffel.fr). P. Besnier is with Univ Rennes, INSA Rennes, CNRS, IETR-UMR 6164, F-35000 Rennes, France (e-mail: Philippe.Besnier@insa-rennes.fr).

P. Pouliguen is with the Defence Innovation Agency, French Ministry of the Armed Forces. antenna parameters have been continuously of interest since then [START_REF] Garbacz | Determination of antenna parameters by scattering crosssection measurements[END_REF]- [START_REF] Wiesbeck | Wide-band multiport antenna characterization by polarimetric RCS measurements[END_REF]. The field backscattered by an antenna consists of two components: the structural mode, due to the antenna seen as a simple object, and the radiation mode, also called the antenna mode, due to the received field transmitted towards the port of the antenna and then reradiated according to impedance mismatches. Solely the radiation mode needs to be accounted for to retrieve the antenna radiation properties. Methods to distinguish between the two modes are based on a load variation approach, either using length-varying transmission line [START_REF] Happel-Hansen | Accurate determination of gain and radiation patterns by radar cross-section measurements[END_REF] or discrete load impedances [START_REF] Wiesbeck | Wide-band multiport antenna characterization by polarimetric RCS measurements[END_REF].

RCS measurements are traditionally performed within an anechoic chamber (AC) that mimics a free-space environment. However, reverberation chambers (RCs) recently became a reliable alternative to ACs for RCS measurements. Two approaches have been investigated: one relies on the line-ofsight path contribution selection through a time-gating postprocess [START_REF] Soltane | Monostatic Radar cross-section estimation of canonical targets in reverberating room using time-gating technique[END_REF], whereas the other one takes benefit from the diffuse-field properties within the RC. The original idea has been introduced in [START_REF] Besnier | Estimating radar cross-section of canonical targets in reverberation chamber[END_REF] for a fixed stirrer position whereas the RCS accuracy improvement through the stirring process has been highlighted in [START_REF] Reis | Mode-stirring impact in radar cross section evaluation in reverberation chamber[END_REF]- [START_REF] Reis | Radar cross section measurement within reverberation chamber: stirrer position issues[END_REF]. However, up to now, all RCS measurements performed within RCs have only considered simple metallic targets such as a plate or a dihedral. In this paper, we consider the specific case of an antenna whose RCS is composed of the two aforementioned modes. More specifically, the objective of this paper is to demonstrate for the first time the possibility to estimate the antenna gain and radiation pattern from antenna backscattering coefficient measurements performed within RCs. This paper is organized as follows. Section II presents the theory to extract the antenna gain from the backscattering coefficient measured within an RC. The experimental setup as well as the measurement of a Vivaldi antenna are presented in Section III where the gain pattern results are compared to the one measured in an AC. A second validation using a log-periodic antenna is performed in Section IV. Finally, a conclusion ends this paper.

II. THEORY

A. Antenna Backscattering Coefficient Within an RC

The monostatic measurement setup within an RC is presented in Fig. 1. A measurement antenna, connected to a VNA, is oriented towards the AUT located on a mast rotating over the azimuthal plane according to the θ angle. The two antennas are separated by a distance R. The scattering parameter S(f, α, θ, Γ L ) measured by the measurement antenna at the frequency f and for the stirrer angular position α, when the AUT is terminated with a load impedance leading to the reflection coefficient Γ L , can be expressed as:

S(f, α, θ, Γ L ) = S FS (f )+ 1 -|S FS (f )| 2 η × [H(f, α, θ, Γ L ) + h s (f, α)] + C(f ) × σ(f, θ, Γ L ) (1)
where S FS (f ) and η are the free-space reflection coefficient and the radiation efficiency of the measurement antenna, respectively, H(f, α, θ, Γ L ) is the complex-valued RC transfer function describing the backscattered signal towards the antenna associated to the diffuse field (related to multiple reflections and diffractions within the cavity), h s (f, α) accounts for line-of-sight or specular reflections from the RC walls and mechanical stirrer towards the measurement antenna. The term C(f ) × σ(f, θ, Γ L ) corresponds to the ballistic wave backscattered by the AUT, with σ(f, θ, Γ L ) the AUT backscattering coefficient in which the phase information is retained [START_REF] Knott | Radar Cross Section[END_REF], and the factor C(f ) can be evaluated from the radar equation as:

C(f ) = G(f )λ 1 -|S FS (f )| 2 (4π) 3/2 R 2 exp -j2πf 2R c (2)
where G(f ) is the measurement antenna gain, λ is the wavelength, and c is the speed of light. According to the antenna scattering theory, the antenna backscattering coefficient σ(f, θ, Γ L ) can be split into two parts. The first one σ s (f, θ) is called the structural mode and is related to the wave backscattered by the AUT as an object, independently of its radiation properties. The second one σ r (f, θ, Γ L ) is called the radiation mode and is due to the wave received by the antenna at its receiving port and reradiated according to the reflection coefficient Γ L . The antenna backscattering coefficient can thus be written as:

σ(f, θ, Γ L ) = σ s (f, θ) + σ r (f, θ, Γ L ). (3)
The structural mode does not depend on Γ L whereas σ r (f, θ, Γ L ) can be expressed as:

σ r (f, θ, Γ L ) = σ max r (f, θ) × Γ L (f ) (4)
where σ max r (f, θ) is the antenna backscattering coefficient due to the radiation mode for Γ L = 1.

B. Antenna Gain Extraction

Solely the radiation mode should be considered in order to retrieve antenna radiation characteristics. In order to separate it from the structural component, we consider two measurements performed for two different load conditions Γ L1 (f ) and Γ L2 (f ). The difference between the two measured Sparameters can be expressed as:

S(f, α, θ, Γ L1 ) -S(f, α, θ, Γ L2 ) = C(f ) × σ(f, θ, Γ L1 ) -σ(f, θ, Γ L2 ) + n(f, α, θ) (5)
with

n(f, α, θ) = 1 -|S FS (f )| 2 η × [H(f, α, θ, Γ L1 ) -H(f, α, θ, Γ L2 )] . (6)
Putting ( 3) and ( 4) into (5), it comes:

S(f, α , θ, Γ L1 ) -S(f, α, θ, Γ L2 ) = C(f )× σ max r (f, θ) × (Γ L1 (f ) -Γ L2 (f )) + n(f, α, θ). ( 7
)
This difference is composed of two terms: the first one is proportional to the AUT backscattering coefficient related to its radiation mode. Indeed, the structural mode being the same for both load conditions, it vanishes. We notice from (2) that this term behaves as a sine wave signal as a function of the frequency with a c/2R period. The second term n(f, α, θ) is proportional to the difference of two random variables whose real and imaginary parts are distributed according to a centered Gaussian probability density function with equal variance. This second term can be seen as an interfering signal that is summed to the wave backscattered by the AUT. As already shown in our previous studies of target RCS retrieval from measurement in an RC [START_REF] Reis | Mode-stirring impact in radar cross section evaluation in reverberation chamber[END_REF]- [START_REF] Reis | Radar cross section measurement within reverberation chamber: stirrer position issues[END_REF], the stirring process permits to drastically reduce the amplitude of this interfering signal and thus improve the extraction accuracy of the signal of interest. To this end, an average of ( 7) is performed over a stirrer rotation. Besides, to compensate the variation over the frequency range of the measurement antenna properties along with the wavelength and the load reflection coefficients that impacts the sine wave amplitude according to ( 2) and ( 7), the term σ max r (f, θ) is extracted using a regression process from the corrected difference ∆ S in (8) (see [START_REF] Reis | Radar cross-section pattern measurements in a modestirred reverberation chamber: theory and experiments[END_REF] for full details).

∆ S (f, θ, Γ L1 , Γ L2 ) = S(f, α , θ, Γ L1 ) -S(f, α, θ, Γ L2 ) α G(f )λ 1 -|S FS (f )| 2 × (Γ L1 (f ) -Γ L2 (f )) (8)
where . α indicates an average over all stirrer positions α. Once σ max r (f, θ) is known, the AUT gain G AUT (f, θ) can be computed as [START_REF] Knott | Radar Cross Section[END_REF]:

G AUT (f, θ) = √ 4π λ × |σ max r (f, θ)|. (9)

III. VIVALDI ANTENNA MEASUREMENT

A. Experiment

All experiments have been conducted in the ESYCOM RC whose dimensions are 2.95×2.75×2.55 m 3 . Its lowest usable frequency is estimated about 400 MHz and it is equipped with a metallic mechanical mode stirrer. The setup is presented in Fig. 2. The measurement antenna is a directive horn antenna whose S-parameters are measured in the 6 to 7 GHz frequency range using 20001 frequency points and 72 equally spaced stirrer positions. The AUT is a Vivaldi antenna that rotates in the azimuthal plane from θ = -90 • to θ = 90 • with a 1 • degree step, θ = 0 • corresponding to the boresight direction. The two antennas are separated by a distance R = 2.29 m. In order to maximize the reflection coefficient difference and thus enhance the measurement sensitivity, we chose an open circuit (OC) and a 50 Ω load for the two considered load impedances. A 30-cm coaxial cable connects the AUT to a mechanical switch terminated by the two loads. The reflection coefficient Γ L at the antenna access due to the mismatch between the AUT and the load impedance is deduced from the measurement of the reflection coefficient between the AUT and 50 Ω (Γ a ) and the one between the load and 50 Ω (Γ l ) from

Γ L (f ) = 1 -Γ a (f ) 1 + Γ † a (f ) × Γ l (f ) -Γ † a (f ) 1 -Γ a (f )Γ l (f ) . (10)
where the superscript † stands for the complex conjugate. The Fig. 3 presents |Γ L | for the OC and 50 Ω loads computed using [START_REF] Reis | Radar cross-section pattern measurements in a modestirred reverberation chamber: theory and experiments[END_REF]. The OC case exhibits losses of about 1.5 dB on average over the frequency band; these are mainly due to the switch insertion losses. For the 50 Ω load, |Γ L | remains lower than -13 dB on the entire frequency range. The Fig. 4 presents the real part of ∆ S for the AUT azimuthal angle θ = 0 • averaged over all stirrer positions as a function of frequency (dashed black curve). The expected sine wave behavior can be observed as a function of frequency as well as the presence of additive noise due to the RC reflections. The amplitude of the sine function for all AUT orientations θ is extracted through a regression process. The retrieved sine wave for θ = 0 • is plotted in Fig. 4 (solid red curve). The frequency period is about 52 MHz that is equivalent to a distance of 2.89 m. This is consistent with the actual distance R between antennas once the length from the antenna aperture to the calibration reference plane for both antennas has been added. Calibration is performed thanks to the RCS measurement of a 148×151 mm 2 metallic plate in the broadside incidence, using the exact same experimental setup within the RC. The AUT gain G AUT (f, θ) can then be extracted from [START_REF] Reis | Mode-stirring impact in radar cross section evaluation in reverberation chamber[END_REF]. To provide a comparison, the antenna gain pattern is measured within an AC by the conventional transmission approach whose setup is presented in Fig. 5, for the same distance R between antennas and in the same azimuthal plane. A double ridged horn antenna (Schwarzbeck BBHA 9120 B) of known gain has been used for calibration in order to extract the gain absolute value from this measurement.

B. Results

The gain pattern measured at 6.5 GHz by the contactless method in the RC is presented in Fig. 6 and compared with the one measured using the conventional transmission setup in AC. The boresight gain is very similar: 11.57 dB with the RC contactless approach versus 11.61 dB with the AC setup, as well as the half power beamwidth: 37.8 • with the RC contactless approach versus 37.9 • with the AC setup. The main lobe is good agreement between both measurements, especially when |θ| < 30 • (less than 0.70 dB difference) whereas the side lobes after |θ| = 50 • are not well retrieved using the contactless RC setup due to the very low signal to interference ratio. This ratio could be enhanced by using other types of mode stirring.

IV. LOG-PERIODIC ANTENNA MEASUREMENT

In this section, we validate the introduced gain pattern measurement method on a second antenna, namely a logperiodic antenna (Schwarzbeck VUSLP 9111-400), in the 3 to 4 GHz frequency range using the same setup than the previous measurement. Due to practical limitations regarding this antenna support, the angular range is limited to θ = -32 • to θ = 32 • with a 1 • degree step. As for the previous measurement, a 1 GHz frequency window is selected around the central frequency to perform the sinusoidal regression. The log-periodic gain pattern is presented in Fig. 7, compared to the transmission-type AC measurement at 3.5 GHz. Both results are in good agreement and exhibit, for |θ| < 30 • , a maximum difference of 0.76 dB and an average absolute difference of 0.31 dB. Measurements have been normalized by the maximum gain given by the datasheet as no calibration measurement has been performed for this study.

V. CONCLUSION

In this paper, the antenna gain pattern has been evaluated for the first time from the backscattering coefficient measured within an RC. The radiation mode of the AUT backscattering coefficient has been extracted thanks to measurements performed for two load conditions, namely an OC and a 50 Ω load. It has been validated in a 19 m 3 RC using a Vivaldi antenna in the 6 to 7 GHz frequency range and a log-periodic antenna in the 3 to 4 GHz frequency range. The main lobe for all measurements is in good agreement with traditional transmission-type measurement performed within an AC (maximum difference of 0.70 dB for the Vivaldi antenna and 0.76 dB for the log-periodic antenna). Also, the maximum gain is very well retrieved (0.04 dB difference for the Vivaldi antenna) thanks to a calibration using a metallic plate.

Fig. 1 .

 1 Fig. 1. Measurement setup within the RC. The AUT is located on a mast rotating over the azimuthal angle and is connected to a mechanical switch.

Fig. 2 .

 2 Fig. 2. Contactless RC measurement setup from the top view. Zoom on the switch connection to the AUT and the loads.

Fig. 3 .

 3 Fig. 3. Reflection coefficient |Γ L | in dB as a function of frequency for the OC and 50 Ω loads.

Fig. 4 .

 4 Fig. 4. Real part of ∆ S for AUT azimutal angle θ = 0 • averaged over all stirrer positions as a function of frequency.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Transmission-type measurement setup within anechoic chamber. The distance between antennas is identical to that of RC.

Fig. 7 .

 7 Fig. 7. Measured gain pattern of the log-periodic antenna at 3.5 GHz as a function of the angle θ.