Sezin Topçu
email: sezin.topcu@ehess.fr

Patrick Brown
email: p.r.brown@uva.nl

The impact of technology on pregnancy and childbirth: Creating and managing obstetrical risk in different cultural and socio-economic contexts

Keywords: childbirth, pregnancy, technology, medicalization, North-South

This special issue brings together empirical studies conducted in eight different countries: Brasil, Turkey, China, Japan, Italy, Senegal, Jordan, and Switzerland. Two papers focus on pregnancy surveillance and related technologies while four others tackle the birthing processes in high-technology as well as low-technology settings. The contribution of the collected papers to the existing literature on pregnancy and childbirth risks is threefold. First, the special issue puts the materiality of hospital birth environments at the centre of the analysis, by paying systematic attention to the ways different technologies and products have shaped the biomedical model(s) of birth that become diffused around the world. Second, the collected papers provide empirical findings not only from the 'developed world' but also from the 'emerging' or 'developing' economies, with the aim of analyzing the various forms of attachment to technologyand to its capacity to remediate risks-in distinct birth cultures and geographies. Finally, this special issue is composed of studies that pay special attention to the heterogeneous and even sometimes discriminatory forms of care targeting distinct categories of users, such as low SES or working class women, immigrant women or women of advanced maternal age. This special issue thus aims to address what has increasingly been criticized as a western-world-centered or privileged-women-centered literature on childbirth risks. The contributions also provide original evidence on the necessity for considering the role played by technological settings in not only the mitigation of childbirth risks but also their conceptualization, recognition, prioritization and, in some cases, their creation.

Introduction

This special issue is the outcome of a collective reflection on the impact of technology on pregnancy and childbirth which began during an international conference organized in Paris in October 2016 1 , and which we have then pursued within a research program financed by the French National Research Agency 2 . The call for papers invited the authors to reflect on the various ways medical technologies and biomedical products have shaped, or at least influenced, birth processes and the risks associated to them. Our aims were two-fold. On one hand, we sought to analyze the processes of technomedicalization -of "technology colonization" even-in pregnancy and childbirth in contrasting national and socio-economic contexts ('emerging', 'developing' versus 'developed' countries; high SES women versus low-SES or immigrant women, and so on). On the other hand, we wished to examine how these processes shape the way obstetrical risk is theorized, framed, managed, eventually contested, but also produced or avoided. We thus wanted to highlight the contemporary ways of conceiving and managing childbirth risks (including pregnancy) both from a 'materialistic' point of view (as promoted by science and technology studies) and in a transnational perspective. In particular, we have attempted to respond to the following questions: To what extent has risk become a relevant category or tool for managing childbirth in very different settings? What are the different forms of risk that are perceived, taken into account, ignored or contested in medicalized versus demedicalized environments of birth? Do the ways of considering the (positive) role played by technology in making childbirth safer vary depending on whether the patients and the health professionals already have access to that given technology? How does the presence of a technology, an instrument, or a pharmaceutical product shape the medical action on, as well as the imaginaries related to, childbirth and its risks? Does an increased focus on risk necessarily mean an increased recourse to technology, and vice versa? To what extent does access to technology reinforce stratified reproduction (Ginzburg & Rapp, 1995)? How do economic concerns shape different medical and risk rationalities?

In this editorial, we will open a discussion in these directions with the aim of highlighting the contribution of the special issue to the existing and already abundant literature on pregnancy and childbirth risks [START_REF] Armstrong | Conceiving Risk, Bearing Responsibility: Fetal Alcohol Syndrome and the Diagnosis of Moral Disorder[END_REF][START_REF] Scamell | Fateful moments and the categorisation of risk: Midwifery practice and the ever-narrowing window of normality during childbirth[END_REF][START_REF] Rothman | Pregnancy, birth and risk: An introduction[END_REF][START_REF] Coxon | Risk in pregnancy and birth: Are we talking to ourselves?[END_REF][START_REF] Löwy | Tangled diagnosis. Prenatal testing, women and risk[END_REF]. First, we will briefly review the dynamics of technologization/medicalization of childbirth in different national settings, in order to establish the context in which this renewed enquiry has its roots and justification. We will investigate the collected papers in terms of the not-always-obvious or the sometimes-paradoxical relations between technology, risk and childbirth in terms of what they reveal for the studied national contexts. We will consider for instance that the most high-tech or the most industrialized countries are not necessarily those where 1 The international conference « Medicalized Childbirth as a Public Problem : Risk Cultures, Gender Politics, Technoreflexivities » was organized by Sezin Topçu and Ilana Löwy at Ehess-Paris on 27 th -28 th of September 2016. This conference was held thanks to a grant from Labex-Tepsis (Ehess) and a financial support from CEMS (EHESS/CNRS FRE2023/INSERM U1276), Cermes3 (EHESS/CNRS UMR 8211/INSERM U988) and Centre A. Koyré (EHESS/CNRS/MNHN UMR 8560). 2 ANR Hypmedpro (Overmedicalization of childbirth as a public problem : material trajectories, public controversies, institutional changes), coordinated by Sezin Topçu (CEMS-Ehess) (2016)(2017)(2018)(2019)(2020).

techno-medical interventions in childbirth are (or previously were) the most important. We will then examine the ways by which access to technology or availability (or absence) of high-tech services conditions (or not) risk conceptualizations of women and health professionals. Finally we will tackle the relation between risk colonization and technology commodification, as well as between risk government and economic evaluation and regulation of techno-medical interventions. This is not the first time Health Risk & Society offers space for theoretical and empirical discussion on the construction and management of pregnancy and childbirth risks. In 2014 in particular, the topic was tackled in depth (vol. 16,Issue 1;vol. 16,Issue 6). This special issue enriches these debates in two ways. On one hand, the issue puts technology and technical systems (surgery-oriented maternity wards in western Turkey, noninvasive prenatal diagnosis that diffuses at an accelerated rhythm in China, surveillance centred birth settings pacified with epidural anaesthesia in Switzerland) or their absence (in for instance public hospitals in Senegal or part of Brazil and Jordan, in alternative birth hospitals in Europe, during a natural disaster in Japan…) at the centre of the analysis. The aim here is to provide more sophisticated accounts of the 'biomedical' models which have recently been emerging. On the other hand, we aim to address what has increasingly been criticized as western-world-centered or privilegedwomen-centered analyses [START_REF] Dillaway | Intersectionality and childbirth: how women from different social locations discuss epidural use[END_REF][START_REF] Coxon | Risk in pregnancy and birth: Are we talking to ourselves?[END_REF]. The six papers brought together in this issue tackle both high-income and middle/low-income economies. Several papers also place at their core the different and even sometimes discriminatory forms of care addressed to distinct categories of users/patients (low SES women or working class women, immigrant women, women at advanced maternal age, for instance). Both "Norths" and "Souths" as well as the contrasting contexts within each of these categories -where women are subject to unequal opportunities and conditions in their reproductive experiences -are thus considered in this publication.

(Over)medicalization of pregnancy and childbirth and related public debates and controversies

The growth of medical technology in pregnancy and childbirth is a well-established phenomenon. During the 20th century, the locus of childbirth shifted from women's own homes to institutional settings managed by professionals, first in the western-industrial countries and especially the US [START_REF] Leavitt | Brought to bed: Childbearing in America, 1750 to 1950[END_REF], and then gradually these technologies were diffused elsewhere [START_REF] Al-Galiani | Hospital Birth[END_REF]. In parallel to the vast movement of hospitalization of birth, obstetrical knowledge, technologies and practices transformed the maternal experience in a radical manner, while at the same time establishing the 'biomedical' or the 'techno-scientific' management of birth. During recent decades, the medical uptake of pregnancy and childbirth has been accelerated via the generalization and even the routinization of various technologies and pharmaceutical products.

Medical interventionism has nevertheless taken diverse forms from one national context to another. In some countries such as Brasil, China, Mexico, Turkey, Italy or the US, Csections have been more or less normalized as a 'no-risk and no-pain' alternative to 'natural' birth. In others, like France, Canada and UK, epidural anaesthesia is much more generalized. In many contexts, labor induction, episiotomy or ocytocine are routinized.

The 'biomedical model' which became dominant in many contexts, refers therefore not to one but to many different realities and practices [START_REF] Akrich | Multiplying obstetrics: techniques of surveillance and forms of coordination[END_REF], although activists of natural or humanized birth may often mention it in singular terms, thereby contributing to 'blackbox' it, albeit unintentionally. One way of opening the blackbox of techno-medical birth, we argue, is to look at the different technological paths or trends in different contexts.

In the past years, the 'medicalization' of childbirth, and to a lesser extent of pregnancy, has generated a new wave of public criticism and concern in many geographies. Regarding childbirth for instance, the denounciations of 'obstetrical violence' -which have recently gained a transnational dimension as they diffused from Latin America to Asia by passing through several European countries such as France or Italy -constitute one of the most visible examples. Generally speaking, the critiques of 'medicalized birth' or the biomedical birth models, which were present already in the 1950s [START_REF] Michaels | Lamaze: an international history[END_REF], currently range from : (i) a radical refusal of technology, medicine and hospital considered to be globally harmful to the birth process (e.g. home-birth movements) ; (ii) to a critique of political-organizational issues, such as the fordisation of the maternity wards or the bureaucratisation of midwifery as a profession ; to (iii) refusal not of medical techniques and instruments per se, but rather of their unjustified, routinized, or excessive use.

It should be noted in this frame that obstetrical technologies may be subject to contrasting meanings and framings by women or consumers (Rapp, 2000). For instance, epidural anaesthesia and even caesarean birth may be considered by some consumer or activist groups as empowering for women, from a feminist or a care perspective, especially in countries where high inequalities regarding access to technologies and services have transformed the medical suppression of labour pain into a class struggle [START_REF] Roberts | Scars of Nation: Surgical penetration and the Ecuadorian State[END_REF]. In others, like in the USA, some users or feminist groups may refuse them as oppressing, authoritarian or medically dangerous [START_REF] Dillaway | Intersectionality and childbirth: how women from different social locations discuss epidural use[END_REF]. Such contrasting framings can also come from medical or the regulatory bodies, which may suddenly start or stop promoting a technology or a product when for instance new international scientific evidence or controversies on their risk-benefit balance occur, or when new economic constraints or medico-legal norms are introduced. As a result, very different meanings are attributed today to what is considered as 'medical' or 'too medical' or, on the contrary, 'natural' or 'normal' ; to what is judged as empowering or on the contrary technocratic or colonising ; as safe instead of risky. The goal of the current special issue seeks precisely to highlight such heterogeneous framings and problematisations of the relations between technology, risk and childbirth in different national settings.

Our departing observations are three-fold:

-just as with techno-bio-medicalization, risk is no longer a central tool for governing childbirth in western countries only; -the recourse to technologies and medication can be greater in 'emerging' economies or in the economically favourable regions of 'developing' ones, in comparison to the 'developed' ones. The case of Turkey usefully illustrates this point, with its high C-section rates (54 per cent at a national level), the systematization of 3D or 4D ultrasounds on a monthly basis in its private clinics, or the loose character of its regulatory framework on IVF, which allowed the country to become one of the biggest markets of IVF in the world in only a few years. -more (or less) technology does not necessarily mean more (or less) 'risk colonization', as other concerns (for population control or growth, or optimization of public costs…) can, at least periodically, render these two tendencies incompatible with each-other.

Hightech vs. low-tech management of the maternal body in eight different settings

The articles in the special issue draw on empirical material collected in eight different countries: Brazil, Jordan, Switzerland, Turkey, China, Japan, Italy and Senegal. Two of them tackle risk surveillance during pregnancy while the others focus on the medical management of childbirth. Technologies or products such as non-invasive prenatal testing (NIPTs), amniocentesis, ultrasound, electronic foetal monitoring, epidural anaesthesia, C-section or ocytocin, are objects of the different analyses. Globally, the ethnographic studies presented in the articles were designed and conducted independently from one another. All except one rely on ethnographic work conducted in hospital or medical settings. The authors, either medical anthropologists, or childbirth or risk sociologists or STS scholars, tackle risk in multiple ways, following the framings and problematisations that their field studies concretely revealed or pushed into the frontline. Accordingly, risk perception (including extrasomatic risks, or those associated to invisible powers), risk categorization or risk hierarchisation (between NIPTs and amniocentesis for instance, or between vaginal vs. caesarean birth) constitute a central analytical line for certain authors.

The analytical aim of these papers is far from realising a boundary work [START_REF] Gieryn | Boundary work and the demarcation of science from non-science: Strains and interests in professional ideologies of scientists[END_REF] on real vs. perceived risks, in order for instance to determine the irrationality or the rationality of the social players. Inspired by sociocultural or socioanthropological theories of risk (Douglas & Wildawsky, 1982 ;[START_REF] Lupton | Risk and sociocultural theory: New directions and perspectives[END_REF], it is rather to situate biomedical definitions of risk and their reception by paying attention to local notions, grammars, material cultures or new policies that shape the behaviours and symbolic representations of both women and health professionnals . The techno-political and the socio-cultural construction of medical risk in relation to other competing risksorganisational risks, economic risks, social risks such as those related to loss of privileges in access to health care or 'external' risks such as the seismic one -is also the focus of several studies. [START_REF] Rothstein | Editorial. The institutional origins of risk : A new agenda for risk research[END_REF]'s theorisation of 'risk colonisation' is effectively mobilized in this frame.

Other studies meanwhile privilege the analysis of strategic reappropriations of risk. Drawing on Foucault and governmentality studies, they tackle risk as a technology of regulation of women's reproductive experiences, that both the expectant mothers and the healthcare providers make use of for their own interests. Regarding the types of actors analyzed too, the collected papers usefully complement one another : some focus on women's perception or lived experiences of risk and of 'situated meaning-making' over it [START_REF] Chadwick | Negotiating risky bodies: Childbirth and constructions of risk[END_REF][START_REF] Lane | Pluralist risk cultures: The sociology of childbirth in Vanuatu[END_REF] ; some look through the ways doctors, midwives, policy-makers or the promoters of a given technology categorize or mobilize risk ; while others tackle both the women's and the healthcare professionals' ways of evaluating and coping up with risks. The special issue offers data on both public and private health sector dynamics of the national contexts being studied, although these are not tackled in a systematic or comparative manner in each study. But in all articles socio-economic inequalities, constraints or economic interests occupy an important place.

Last but not least, the question of the organisation of birth spaces, which play a central role in the establishment of authoritative knowledge and practices [START_REF] Jordan | Authoritative Knowledge and its Construction[END_REF], is analysed in an original manner by certain contributions of the issue. Two papers in particular focus on unconventional birth spaces. One of them (Quagliariello, 2019) tackles an alternative birth hospital which set out to position itself as a low-tech alternative to mainstream obstetrics through the specific configuration of its labour rooms. The other study [START_REF] Ivry | What disasters can reveal about technomedical birth: Japanese women's stories of childbirth during the 11 March, 2011 earthquake[END_REF] poses the question of what happens when suddenly it may no longer be possible to give birth in a hospital and even not indoors, in the case for instance of a big earthquake.

Thanks to its international focus, this special issue thus provides the opportunity to simultaneously investigate, for instance, what is arguably the world's most hightechnology country -Japan-together with some leading 'emerging' economies such as China, Brasil or Turkey, in terms of their obstetrical care dynamics. This exercise reveals striking paradoxes. In Japan, the levels of recourse to obstetrical technologies such as epidural anaesthesia, C-sections or induction of labour are surprisingly low (6%, 19%, less than 10% respectively) [START_REF] Ivry | What disasters can reveal about technomedical birth: Japanese women's stories of childbirth during the 11 March, 2011 earthquake[END_REF]. In comparison, in 'emerging' economies like Turkey and Brazil, recourse to technology, especially to C-sections, have been routinized, thanks to a medical discourse that reframed vaginal birth as both very risky and too painful (McCallum, 2000;[START_REF] Topçu | Caesarean or vaginarean epidemics? Techno-birth, risk and obstetric practice in Turkey[END_REF]. In Jordan too, where hospital birth is the norm and where obstetricians play an important role in childbirth, such a risk discourse could have been influential and could have made childbirth as « technocratic » (Davis-Floyd & Sargent, 1997) as in Turkey or China, but this does not seem to be the case, at least until the last decade [START_REF] Maffi | Conceiving of risk in childbirth: obstetric discourses, medical management and cultural expectations in Switzerland and Jordan[END_REF].

These findings imply that 'risk colonisation' [START_REF] Rothstein | Editorial. The institutional origins of risk : A new agenda for risk research[END_REF] is not a process that necessarily diffuses (or is exported) from the '(post)industrial' countries towards the 'developing' ones in a homogeneous way. Far from a purely western concept, risk can and does diffuse beyond the 'global North' and, moreover, this occurs in heterogenous ways. Therefore, as Maffi & Gouilhers (2019) put, instead of assuming a global diffusion of the risk paradigm, it is necessary to pay attention to how specific technological, medical, economic and cultural settings facilitate or impede risk colonisation.

Access to technology and the ambivalence of risk

After having become an important political issue in the industrialized countries in the 1970s, risk governance in pregnancy and childbirth gradually allowed the elaboration of a whole set of regulatory tools (safety norms, benchmarking techniques, tools such as partograms, and so on) that started to diffuse beyond western-industrial countries.

International regulatory agencies such as the WHO or the development programs or goals such as the MDGs (Millennium Development Goals) played an important role in this frame. However, the adoption, reappropriation or adaptation of the risk philosophy and risk governance tools in different national-cultural settings, as a technology for managing childbirth, seem to depend on many parameters that go beyond the simple consideration according to which medically and technologically unattended chilbirth is risky. The comparative analysis of the Swiss and the Jordanian cases conducted by [START_REF] Maffi | Conceiving of risk in childbirth: obstetric discourses, medical management and cultural expectations in Switzerland and Jordan[END_REF] provides interesting insights into these processes.

While the maternal mortality and morbidity rates in childbirth are much higher in Jordan than in Switzerland (19/100000 against 5/100000), risk remained a relatively marginal concern for women and health professionals in the Jordanian case while having become a central element of concern, discussion and action in the Swiss context. In Switzerland, both the midwives, OGs and the material environment of the highlyequipped public maternity hospitals are there to remind women of uncertainty and risk during pregnancy and labour, and the corresponding necessity to follow them in a systematic way. In comparison, in Jordanian public hospitals, neither surveillance technologies nor pain management techniques are omnipresent. Only acceleration of labour appears to be a generalised tendency. Of relevance in this latter case, perhaps, is that fear of birth pain or risk of suffering seem to have deprioritised the fear or the recognition of birth accident risks. It thus appears that biomedical risk is not necessarily the main focus or tool of childbirth management in contexts where, objectively speaking, levels of maternal and newborn mortality and morbidity rates may still be significant.

Differences in access to technology or high-technology services does seem to condition such risk dynamics to a certain extent : in Jordan and Senegal prenatal tests are not available for everybody (due to their cost), and as a result, risk measurement and relevant risk categorisations and concerns are not widely diffused. Meanwhile access to or availability of a technology does not necessarily imply that it will de facto be utilized or desired by women or the health professionals, because of safety or other concerns. As Ivry and colleagues (2019) show in this issue, in Japan for instance, suffering childbirth pain is considered a necessary effort for becoming a 'good' mother, and women's perseverance is socially praised within this cultural frame. As a result, epidural anaesthesia is not automatically seen as a key to making childbirth safer or more desirable, although it can easily be made accessible. Furthermore, giving birth without epidural anaesthesia seems to be considered as a 'natural', and not less safe, way of birthing in most Japanese settings, regardless of whether the women in labour are attached to a gynecological table and to a monitoring device or are under other medications.

Ivry and colleagues (2019) add to these findings, in that they show how, in case of a forced demedicalization, such as during the 11.03.2011 Great Japan earthquake, both women and health professionals effectively discover what non-interventionist or physiological birth is, and more importantly what birthing womens' bodies are capable of doing on their own, even for accelerating or slowing down the birth process naturally when necessary. The exceptional conditions in which women find themselves seem to push them to reconsider what really counts, between access to technology and midwives' attentive care and support. That's why the authors propose to consider disaster contexts also as opportunities that may allow social players to question and even collectively criticize the established technological-medical systems, their utility, their potential negative externalities.

Even in countries with material/technological precariousness, unlike what some medical organisations, pharmaceutical companies or more fervent feminist critics assume, access to technology is often not considered nor applauded by women or patients as a homogeneous set of opportunities, but rather in a selective way. In this sense no unconditional uptake of technology exists [START_REF] Callon | Some elements of a sociology of translation : Domestication of the scallops and the fishermen of St. Brieuc Bay[END_REF]. Lay reflexivities visà-vis technical innovations do not only emerge in situations of controversy or in those in which one has become directly concerned by their potential harm [START_REF] Epstein | The construction of lay expertise: AIDS activism and the forging of credibility in the reform of clinical trials[END_REF][START_REF] Wynne | May the sheep safely graze ? A reflexive view of the expert/lay knowledge divide[END_REF], but are omnipresent in 'lifeworlds' or in everyday handing of uncertainty and risk [START_REF] Brown | From rationalities to lifeworlds: analysing the everyday handling of uncertainty and risk in terms of culture, society and identity[END_REF], as well.

In none of the contexts studied by the special issue authors were techniques or products such as labour induction, active management of labour, or forceps seen by women and their relatives as desirable tools. Some of these have even become a distinct focus of resistance for women. [START_REF] Maffi | Conceiving of risk in childbirth: obstetric discourses, medical management and cultural expectations in Switzerland and Jordan[END_REF] explain how in Jordan, obstetriciangynaecologists and midwives were gradually made more accountable for the pelvic examinations that they practiced, since in this cultural context they were considered to be harmful both to the women and the foetus. Jordanian women of low or middle SES also appeared to be quite critical about acceleration of labour and episiotomy as they feared their iatrogenic risks, whereas they saw the access to ultrasound or monitoring devices as an entry to modern and state-funded services, and even as a way to be treated as real citizens.

Quagliariello ( 2019) similarly shows in this issue, drawing on the case of the Senegalese women who migrated to Italy, that they expected and hoped to access to pregnancy and birth technologies such as ultrasound, electronic monitoring devices or epidural anaesthesia. Because these material opportunities can only be afforded by upper class women in their home country, they seemed to consider access to these as luxury health care, but also as a demonstration of the fact that they are being treated normally -that is to say without racist discrimination -in their host country. In contrast, some of the Italian-born (upper-)middle-class women in the same clinic feared the iatrogenic risks of these technologies -particularly that one intervention could lead to another (the boomerang effect), or that technology would render their birth experience too artificial, thus potentially provoking physical as well as psychological trauma. This latter group of women represent however a minority within the alternative birth hospital in Tuscany which itself is a minority case within the Italian obstetrical landscape. Yet the Senegalese women in Quagliariello's (2019) study, most of whom were of low or middle SES, also displayed a critical reflexivity despite their fascination for some of the techniques. They considered amniocentesis in a negative way, as they saw it as risky, either because they had heard about risks of miscarriage associated to it, or because they believed more generally that it can provoke extra-biological risks by attracting the 'evil agents' interested in 'eating' the amniotic liquid.

C-sections were also resisted but for other reasons. Within their socio-cultural environments both in Senegal and within their migrant community in Italy, opting for an elective caesarean was generally seen as a sign of inferiority, powerlessness or loss of prestige for these women because they are expected to prove that their bodies are capable of giving birth normally. In comparison, in Brasil -where analysts often mention a 'perinatal paradox' [START_REF] Diniz | Gênero, saúde maternal e o paradoxo perinatal[END_REF] in that in rich-urban regions there are too many technological interventions in the birth process whereas in poor-rural ones too little technology is available; with both contexts posing risks for morbidity and mortality -access to caesarean section has for long time been seen as a way of being privileged, modern, secure, cared for, and even as a means of belonging to white-(upper) middleclass groups rather than the low-SES black groups [START_REF] Mccallum | Explaining caesarean section in Salvador da Bahia, Brazil[END_REF].

Risk colonization and the commodification of technical innovations

Women's uptake of technology, then, is not always or merely catalyzed by concerns for mitigating risks associated to pregnancy and childbirth thanks to techno-scientific innovations [START_REF] Rapp | Refusing Prenatal Diagnosis: The Meanings of Bioscience in a Multicultural World[END_REF], although in some cases, this may be the dominant reason [START_REF] Maffi | Conceiving of risk in childbirth: obstetric discourses, medical management and cultural expectations in Switzerland and Jordan[END_REF]. They also have to do with the way the imaginaries of technology are socially constructed, as synonym of modernity, or as a sign of belonging to a privileged class, for instance. Beyond, women may adopt technologies not because they are fascinated by them, but because they do not have many other alternatives in front of them. [START_REF] Topçu | Caesarean or vaginarean epidemics? Techno-birth, risk and obstetric practice in Turkey[END_REF] analysis, in this issue, of the generalization of caesarean births in western Turkey illustrates this point.

Similar to their Jordanian counterparts and unlike those in Switzerland studied by [START_REF] Maffi | Conceiving of risk in childbirth: obstetric discourses, medical management and cultural expectations in Switzerland and Jordan[END_REF], birthing women analyzed in western Turkey do not consider their birth experiences (be it vaginal or abdominal) as a beautiful, special or magical one, but mostly as a 'technical process' (painful if it is vaginal, unknown if it is abdominal-because most C-sections are realized under general anesthesia) to accomplish in order to have their babies in their arms. They opt for caesarean birth, and even pay for it in private clinics, because vaginal birth is also experienced as something technocratic, disempowering, even surgical (episiotomies are massively practiced), and additionally labour pain is not relieved in the latter case. The medical risk discourses, which categorize C-sections as entirely safe and vaginal birth as risky, also play a role in this frame, but they are not the only determinants of such a 'technology colonization', that is to say, such a strong control and even domination of childbirth by medical technologies, machines and pharmaceutical products [START_REF] Topçu | Caesarean or vaginarean epidemics? Techno-birth, risk and obstetric practice in Turkey[END_REF]. Recent government policies aiming to regulate the caesarean epidemics in the country, in particular the ban of (elective) caesareans on maternal choice in the public sector, resulted in the establishment of new risks (organizational risks, reputational risks) which now compete with the biomedical ones. At the same time, they resulted in a greater delegation of 'Csections on maternal choice' to the private sector, for which women no longer benefit from health insurance coverage. Caesarean birth, or the maternal choice for C-section, was thus transformed into a commodity to sell, thereby creating new inequalities.

Women may furthermore adopt technologies not because they are fascinated by them or because they consider them as a source of security, but because not doing so may provoke social risks, in particular regarding whether they are able to afford them. The case of the Chinese pregnant women at advanced maternal age, who opt for noninvasive prenatal tests (NIPT) is emblematic of such dynamics. [START_REF] Qiu | Birth models in and between Italy and Senegal: a cross-cultural inquiry on the risks related to childbirth and birth technologies[END_REF] shows in this issue how, with the shift to the 2-child policy and the parallel development of the domestic market for prenatal tests, expectant women older than 35 years of age are widely categorized as "high-risk" by the medical bodies. Risk categorization appears in this frame to be tightly connected to imperatives of medical consumption which, according to the author, became an important element of the maternal experience in China, just like the search for the perfect foetus and child ('suzhi'). Women are expected to act as rational and responsible players in this frame by choosing the right test, preferably the NIPTs which are not invasive like amniocentesis but are far more expensive. Risk thus serves to create new consumer categories (the AMA women) for new markets (NIPTs), while these new markets themselves create new social inequalities (only middle or high SES women can afford NIPTs) as well as new forms of self-responsibilization for the expectant women.

The case of working-class pregnant women in Brazil, analyzed by Faya-Robles (2019) in this issue, reveals other dynamics however. Different from Qiu (2019), Faya-Robles considers public pregnancy care and management where clients are mostly the working class or low-SES women living in favelas of Rio de Janeiro. Faya-Robles shows that -in a context in which the healthcare services' economic costs for the state are an important concern, but where the state is at the same time supposed to improve the safety of pregnancies and childbirth, especially among precarious women who are the most 'at risk' according to national statistics -healthisation replaced medicalization as a governmentality tool.

Following Clarke and colleagues (2000), Faya-Robles (2019) tackles healthisation as a way of governing which is less characterized by the preponderance of biomedicine than by the presence of health questions in all levels of daily life, where health has become a value in itself. The self-responsabilization of individuals, or the conduct of their conducts in the name of risk prevention and mitigation (by opting for healthier diets or lifestyles and by avoiding risky behaviours), appears to be central in this frame. The governmental aim behind such a use of risk as a technology of regulation of the reproductive process involves, according to Faya-Robles (2019), the replacing costly medical technologies of pregnancy surveillance (whose generalization, as material technologies, requires important budgets) by those of self-conduct, which are immaterial, and thus cheaper or even free. Faya-Robles (2019) also shows however that risk is not a strategic tool for states, regulatory agencies or health professionals only, but also for women and even those women facing greatest precarities. Risk here becomes transformed into an identity discourse (women can insist on their belonging to a highrisk or fragile category of women, as defined by the public authorities) in order to have access to obstetrical technologies (such as ultrasound) and to proper medical care. The Brazilian case study thus provides evidence that a greater focus on risk does not necessarily lead to a greater technologization of care. Risk can serve as a tool both for technologizing and detechnologizing pregnancy and childbirth, depending on who mobilizes it (government actors versus women) and for what purposes (cost reductions in state care or access to free care).

Conclusion

For the last decades, pregnancy and childbirth processes have been transformed to a great extent under the influence of two historical movements: pathologization of the maternal body and its techno-bio-medicalization. By focusing simultaneously on the 'material' ('technology') as well as the 'immaterial' ('risk') pillars of these processes, the collected papers sought to analyze their singularity or, on the contrary, standardization ; their complementarity or, instead, uncompatibility -at different national settings. This special issue has, however, opened more research questions and paths than it has closed. The women's uptake of obstetrical technologies, more studied for pregnancy surveillance than for childbirth until today (because the former, such as echography, have often been judged 'revolutionary' while the latter, such as labour induction, have been considered routine and thus banal) deserves particular research attention in the future. In this frame, several papers of the special issue provide evidence on the pertinence of tackling the question in an intersectionality perspective, which means being attentive to questions of race, gender and class simultaneously, without falling in the trick of culturalism. The findings of the papers gathered together in this issue also suggest that the multiple meanings of risk in pregnancy and birth management can be best understood by taking into consideration the local reappriations of risk even when it is 'imported' or when it seems to have been designed as a macro-level governmentality tool. Last but not least, the special issue highlights the analytical relevance of the economic question in the governance of reproductive processes, in that financial or market concerns seem to incresingly orientate the existing as well as the future risk theorizations, discourses and practices.


Assessing the correctness of pressure correction to solvation theories in the study of electron transfer reactions

Tzu-Yao Hsu, Guillaume Jeanmairet

Assessing the correctness of pressure correction to solvation theories in the study of electron transfer reactions Tzu-Yao Hsu 1 and Guillaume Jeanmairet 1, 2, a) 1) Sorbonne Université, CNRS, Physico-Chimie des Électrolytes et Nanosystèmes Interfaciaux, PHENIX, F-75005 Paris, France 2) Réseau sur le Stockage Électrochimique de l' Énergie (RS2E), FR CNRS 3459, 80039 Amiens Cedex, France

Liquid states theories have emerged as a numerically efficient alternative to costly molecular dynamics simulations of electron transfer reactions in solution. In a recent paper [Chem. Sci., 2019, 10, 2130], we introduced the framework to compute energy gap, free energy profile and reorganization free energy using molecular density functional theory. However, this technique, as other molecular liquid state theories, overestimates the bulk pressure of the fluid. Because of the too high pressure, the predicted free energy is dramatically exaggerated. Several attempts were made to fix this issue, either based on simple a posteriori correction or by introducing bridge terms. By studying two model half reactions in water, Cl → Cl + and Cl → Cl -, we assess the correctness of these two types of corrections to study electron transfer reactions. We found that a posteriori correction, because it violates the functional principle, leads to an inconsistency in the definition of the reorganization free energy and should not be used to study electron transfer reactions. The bridge approach, because it is theoretically well grounded, is perfectly suitable for this type of systems.

I. INTRODUCTION

The widely accepted theory to describe an electron transfer reaction in solution was proposed by Marcus in 1956 1 . Assuming the solvent can be described by a polarization field that responds linearly to the electric field generated by the solute, he derived an expression linking the activation free energy to the reaction free energy. This expression depends on an unique parameter: the reorganization energy. Reorganization energy measures the cost to distort the solvent from the configuration in equilibrium with the reactant to the one in equilibrium with the product, without transferring electron. The polarization varies continuously between the equilibrium values of reactant and product. By computing the free energy of both states for out-of equilibrium polarizations, one can plot the free energies as a function of an appropriate reaction coordinate. This gives rise to the famous Marcus two parabola picture for the free energy profiles.

From a simulation point of view, electron transfer reactions have been mostly investigated using molecular dynamics (MD). Warshel proposed to use the energy gap as the relevant microscopic reaction coordinate 2 . When the probability distribution of the energy gap is Gaussian, the fee energy profiles are parabolic with identical curvatures, as predicted by Marcus 3 . The Gaussian behavior has been verified in several studies but some systems deviate from the Marcus picture [START_REF] Hartnig | Molecular dynamics simulations of solvent reorganization in electron-transfer reactions[END_REF][START_REF] Blumberger | Cuaq+/Cuaq2+ Redox Reaction Exhibits Strong Nonlinear Solvent Response Due to Change in Coordination Number[END_REF][START_REF] Vuilleumier | Extension of Marcus Picture for Electron Transfer Reactions with Large Solvation Changes[END_REF] . Resort to numerical simulations of electron transfer reactions is essential to account for such cases, unfortunately computing the free energy profiles with MD remains costly and require the use of biased sampling techniques [START_REF] King | Investigation of the free energy functions for electron transfer reactions[END_REF][START_REF] Vandevondele | Density Functional Theory Study of Tetrathiafulvalene and Thianthrene in Acetonitrile: Structure, Dynamics, and Redox Properties[END_REF] .

We recently proposed an alternative way to compute free energy profiles and reorganization free enera) guillaume.jeanmairet@sorbonne-universite.fr gies of electron transfer reactions based on molecular Density Functional Theory (mDFT) [START_REF] Borgis | Molecular Density Functional Theory: Application to Solvation and Electron-Transfer Thermodynamics in Polar Solvents[END_REF][START_REF] Jeanmairet | A molecular density functional theory approach to electron transfer reactions[END_REF] . Because it allows to compute directly the solvation free energy through functional minimization, it is computationally more efficient than MD. Moreover, mDFT allows a fine description of solvation at the molecular level and a good agreement with MD was found for the Cl → Cl + and the Cl → Cl -electron transfer reactions. The first one is found to follow Marcus theory while the second one is not.

In previous work, the hyper-netted chain (HNC) approximation was used to describe the solute-solvent correlations [START_REF] Ding | Efficient molecular density functional theory using generalized spherical harmonics expansions[END_REF] , neglecting the so-called bridge functional. However, because this functional is a Taylor expansion truncated at second-order, it drastically overestimates the free energy per solvent molecule at gas density [START_REF] Rickayzen | Integral equations and the pressure at the liquid-solid interface[END_REF][START_REF] Borgis | Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation[END_REF][START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF] . This increases considerably the cost to create a solvent cavity and thus the predicted solvation free energy. The reference interaction site model and its 3D variant (3D-RISM), another liquid state theory that have also been extensively used to tackle electron transfer reactions [START_REF] Yamaguchi | Nonequilibrium free-energy profile of charge-transfer reaction in polarizable solvent studied using solvent-polarizable three-dimensional reference interaction-site model theory[END_REF][START_REF] Chong | The Free Energy Profiles for the Electron Transfer Reactions Calculated from the Integral Equation Method of Liquids[END_REF][START_REF] Chong | Molecular Theory for the Nonequilibrium Free Energy Profile in Electron Transfer Reaction[END_REF] , suffer from the same problem [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Gillespie | Restoring the consistency with the contact density theorem of a classical density functional theory of ions at a planar electrical double layer[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF][START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Misin | Salting-out effects by pressure-corrected 3D-RISM[END_REF] . Because it is a long-standing problem, several attempts were made to fix it. By noticing that the gas phase free energy is directly linked to the pressure, simple a-posteriori corrections were proposed [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Sohvi Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF]2319,[START_REF] Palmer | Towards a universal method for calculating hydration free energies: a 3D reference interaction site model with partial molar volume correction[END_REF][START_REF] Ratkova | An Accurate Prediction of Hydration Free Energies by Combination of Molecular Integral Equations Theory with Structural Descriptors[END_REF][START_REF] Misin | Communication: Accurate hydration free energies at a wide range of temperatures from 3D-RISM[END_REF] . Essentially, they consist in evaluating the partial molar volume (PMV) created by the solute and applying a free energy correction proportional to it.

A more rigorous approach is to fix the defect of the functional by introducing an approximate bridge term, to go beyond second order. Several models of bridge functional have been proposed, either based on hard sphere theory [START_REF] Oettel | Integral equations for simple fluids in a general reference functional approach[END_REF][START_REF] Levesque | Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation[END_REF][START_REF] Fu | Fast prediction of hydration free energies for SAMPL4 blind test from a classical density functional theory[END_REF] or weighted density approximation (WDA) [START_REF] Borgis | Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation[END_REF][START_REF] Jeanmairet | Molecular density functional theory of water describing hydrophobicity at short and long length scales[END_REF][START_REF] Jeanmairet | Molecular density functional theory for water with liquid-gas coexistence and correct pressure[END_REF] .

In this paper, we evaluate the impact of those two types of correction on the free energy profiles and reorganization free energies predicted for two model elec-tron transfer reactions: Cl → Cl + and Cl -→ Cl. For a posteriori correction we selected the widely used pressure correction (PC) [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF] while the bridge functional is the recently proposed angular independent WDA functional [START_REF] Borgis | Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation[END_REF] .

II. THEORY

We remind here some basics of mDFT which are essential for the comprehension of this paper. Thorough descriptions can be found in previous reports [START_REF] Borgis | Molecular Density Functional Theory: Application to Solvation and Electron-Transfer Thermodynamics in Polar Solvents[END_REF][START_REF] Jeanmairet | A molecular density functional theory approach to electron transfer reactions[END_REF][START_REF] Sohvi Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF] .

In mDFT, solvent and solute molecules are described by rigid models interacting through classical forcefield. The solvation free energy of the solute can be computed by minimizing the solvent functional with respect to the spatially and orientationally dependent solvent density ρ(r, Ω). We introduce the following notation x ≡ (r, Ω) for concision. The functional to be minimized is

F [ρ(x)] = F id [ρ(x)] + F ext [ρ(x)] + F exc [ρ(x)]. (1)
Its minimum is reached for the equilibrium solvent density, ρ eq . In equation 1, F id corresponds to the entropy of the non-interacting liquid

F id [ρ] = k B T ˆ ρ (x) ln ρ (x) ρ b -ρ (x) + ρ b dx (2)
where k B is the Boltzmann constant, ρ b is the homogeneous bulk solvent density and T is the temperature.

The second term of equation 1 is due to solutesolvent interaction and can be expressed as

F ext [ρ] = ˆρ (x) V (x)dx ( 3
)
where V is the the external potential energy exerted by the solute. Solvent-solvent interactions are collected in the last term of equation 1, that is the excess functional F exc . Expanding it around the density ρ b of the homogeneous fluid taken as a reference, and truncating the expansion at second order gives rise to the HNC functional. All the higher order correlations are then swept into the unknown bridge functional F B ,

F exc [ρ] = F HNC [ρ] + F B [ρ] (4) = - 1 2 k B T ˘[∆ρ(r 1 , Ω 1 )c(r 1 -r 2 , Ω 1 , Ω 2 ) ×∆ρ(r 2 , Ω 2 )] dr 1 dΩ 1 dr 2 dΩ 2 + F B [ρ]
where c is the direct correlation function of homogeneous fluid at density ρ b and ∆ρ = ρ -ρ b 11 . We evaluate two types of pressure correction in this paper. The PMV correction takes the following expression [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF]

F PMV B = ρ b k B T (1 - ρ b 2 ĉ(k = 0)) ˆρeq (x) -ρ b ρ b dx (5)
where ĉ denotes the Fourrier transform of the direct correlation function. Since equation 5 does not depend on the density ρ, such a correction does not modify the optimization process. The equilibrium density remains the same as the one obtained using the HNC functional.

For the bridge correction, we take the recently proposed spherical WDA functional 13

F WDA B [ρ] = k B T n 3 b ¨a ˆ∆ρ(r , Ω) ×w(|r -r |)dr ] 3 dΩdr (6)
the value of a is uniquely defined by imposing the correct pressure [START_REF] Borgis | Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation[END_REF] and w is a Gaussian weighting function

w(r) = (2πσ 2 w ) -3/2 exp(-r 2 /2σ 2 w ) (7)
with σ w = 1 Å. We now turn our attention to the study of electron transfer half-reaction of the type Red → Ox + e -, and remind how it is possible to compute free energy profiles and reorganization free energies using mDFT.

We start by introducing a set of external potentials defined as a linear combination between the external potential of the Ox (denoted by 0) and Red (denoted by 1) states

V η = V 0 + η(V 1 -V 0 ). ( 8
)
where η is a real number. Minimizing the functional in equation 1 using the external potentials in equation 8 generate a set of solvent densities ρ η . While ρ 0 and ρ 1 correspond to the equilibrium solvent densities for the Ox and the Red states respectively, any other value of η defines a solvent density ρ η that is out of equilibrium for both states. It becomes possible to compute the free energy of an electronic state α = 0 or 1, solvated in a solvent density ρ η , by evaluating the associated functional: F α [ρ η ]. The definition of the two solvent reorganization free energies comes out naturally

λ 0 = F 0 [ρ 1 ] -F 0 [ρ 0 ] and λ 1 = F 1 [ρ 0 ] -F 1 [ρ 1 ]. (9)
We then defined the energy gap as

∆E η = ˆρη (x) [V 1 (x) -V 0 (x)] dx (10)
and showed that it is an appropriate reaction coordinate since there is a one to one mapping between the energy gap and the solvent density

10 η ↔ V η ↔ ρ η ↔ ∆E η . (11)
The free energy of any state is therefore a function of ∆E η and it is possible to compute the free energy profile associated to state α as a function of this reaction coordinate

F α ( ∆E η ) ≡ F α [ρ η ]. ( 12
)
III. COMPUTATIONAL DETAILS

We study two model electron transfer reactions: Cl→ Cl + and Cl -→ Cl in water. The solute is modeled by one Lennard-Jones site, with σ=4.404 Å , and =0.4190 kJ/mol. We consider 3 oxidation states corresponding to Cl -, Cl (0) and Cl + . We use a Lennard-Jones cut-off of 10 Å with long range corrections [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF] . Water is described using the SPC/E model. We generate a series of biased external potential by varying the atomic charge of the solute between -1.2 ≤ q ≤ 2 with a step of 1/30 elementary charge. We use a 40×40×40

Å3 box with 120 3 spatial grid points and 196 possible orientations per spatial point with periodic boundary condition. We use the type C correction of Hünenberger [START_REF] Kastenholz | Computation of methodology-independent ionic solvation free energies from molecular simulations. II. The hydration free energy of the sodium cation[END_REF][START_REF] Kastenholz | Computation of methodology-independent ionic solvation free energies from molecular simulations. I. The electrostatic potential in molecular liquids[END_REF] due to interaction between the solute and its periodic replica. The temperature is fixed at 298.15 K.

IV. RESULTS AND DISCUSSIONS

A. partial molar volume correction

The free energy profiles of the two half-reactions calculated using the HNC functional with and without the PMV correction are displayed in figure 1. All minima are shifted to zero to ease the comparison with the curves computed by Hartnig et al. using MD [START_REF] Hartnig | Molecular dynamics simulations of solvent reorganization in electron-transfer reactions[END_REF] . The agreement between the free energy profiles obtained using MD and MDFT is satisfactory for the neutral solute. However, important deviations are observed in the case of the cation and the anion. This is a know defect of the HNC functional which is not able to properly render the hydrogen bond network of water that is essential for the solvation of ions. This is due to the truncation of the functional expansion at second order. The two types of corrections studied in this paper cannot resolve this issue because of their lack of angular dependency.

At first glance, including the PMV correction do not seem to deeply modify the shape of the curve. However, a closer look reveals that the positions of the minima are shifted. This is especially true for the ions where the minima are shifted toward zero. This might appear surprising since the energy gap defined in equation 10 solely depends on the density ρ η which is not modified by the PMV correction.

In fact, the shift of the minima is a consequence of a pathological defect of this correction. This is visible in figure 2 where the free energy profiles of the ions are depicted as a function of the charge q of the fictitious solute: q = η for the Cl → Cl + reaction and q = -η for the Cl -→ Cl reaction. The minima of the PMV corrected free energy profiles no longer correspond to q = 1 and q = -1. Equivalently, this means that the minimum of the free energy curve is not reached for the equilibrium solvent density of the solute. This is a violation of the DFT variational principle which is not surprising since the PMV correction is not properly integrated in the classical DFT formalism. It is simply an a posteriori correction that does not influence the optimization. This might be acceptable when the objective is to reproduce some reference solvation free energies [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF][START_REF] Misin | Predicting Solvation Free Energies Using Parameter-Free Solvent Models[END_REF][START_REF] Sohvi Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF][START_REF] Robert | Pressure correction for solvation theories[END_REF][START_REF] Misin | Communication: Accurate hydration free energies at a wide range of temperatures from 3D-RISM[END_REF] but it is problematic to study electron transfer reactions. Indeed, the two ways to compute the reorganization free energies, either from equation 9 or from the graphical definition, i.e. as the difference between the value of the free energy of one state at the abscissa corresponding to the minimum of the second state and the value at its minimum do not coincide anymore.

Reorganization free energies are given in table I. Note that there are two sets of values for the neutral chlorine since the value of λ depends on the choice of the second oxidation state. Solvent reorganization free energies of Cl and Cl + computed with the HNC functional are similar indicating that this electron transfer reaction is well described using Marcus Theory while the Cl -→ Cl reaction deviates from Marcus theory. These results are consistent with MD simulations 4 . When the PMV correction is added, both electron transfer reactions seem to deviate from Marcus theory since the reorganization free energies of ions differ from the one of the associated neutral state. More- over, the reorganization free energies computed using equation 9 differ by up to 25 kJ/mol from the one computed with the graphical definition. The evolution of the energy gap as a function of q is displayed in figure 3. We recall that values of q between 0 and 1 correspond to the Cl → Cl + while the value between 0 and -1 corresponds to Cl -→ Cl. When an electron transfer reaction is following Marcus theory, the energy gap should vary linearly. This is the case between q = 0 and q = 1, while the linearity is not respected between q = 0 and q = -1. This is consistent with the conclusions drawn examining the reorganization free energies computing using the HNC functional and with the MD results: the Cl → Cl + reaction follows Marcus theory while the Cl -→ Cl reaction does not. Since the energy gap is not modified by the PMV correction, the linear behavior observed for the Cl → Cl + is in contradiction with the different values of λ of table I. This is another proof of the inappropriateness of the PMV correction to study electron transfer reactions.

B. Weighted density bridge functional

The same electron transfer reactions were studied using the WDA functional of equation 6 and the prediction for the free energy profiles as a function of the 3. Vertical energy gap computed using the HNC (solid black) and WDA functional (solid red) as a function of the atomic charge. The dotted and dashed lines are the tangent to the HNC curve in q = -1 and q = 1 respectively.

energy gap or of the solute charge are given in figures 1 and 2 respectively. Adding WDA bridge has a limited impact on the free energy profile as compared to the HNC predictions. More importantly, the minimum of the free energy curve of the cation (resp. anion) is correctly reached for the solute with charge q = 1 (resp. q = -1). This is because the WDA correction has a functional form and the variational principle is respected. The reorganization free energies predicted with the WDA functional are slightly reduced with respect to the one predicted using HNC but the overall conclusion is recovered: Cl → Cl + follows Marcus picture while Cl → Cl -does not. This is also supported by the evolution of the energy gap as a function of the atomic charge in figure 3.

We now attempt to understand why the reorganization free energy of the ions are reduced when the WDA bridge is used. We introduce the free energy difference for a state α as

∆F α (η) = F WDA α (η) -F HNC α (η) (13) = ∆F ext α (η) + ∆F int α (η) (14)
where ∆F ext α is the difference of the external functionals defined by equation 3 and ∆F int α is the contribution of the ideal and excess term in equation 1. Since the WDA functional modifies the minimization process, the equilibrium solvent densities associated to the same η differ for both corrections in equation 13.

Using equation 9, the difference of the reorganization free energies obtained with and without the WDA functional can be expressed as

∆λ α = λ WDA α -λ HNC α = ∆F α (0) -∆F α (α) (15) ≈ ∆F elec α (0) -∆F elec α (α) (16)
where ∆F elec α is the electrostatic part of external functional difference. In equation 16, we assume that since the considered species are ions, the main contribution is due to the electrostatic, i.e ∆F int and the Lennard-Jones contributions can be neglected. As long as we are considering spherical ions, the electrostatic potential is spherically symmetric, so is the polarization of the solvent. The equilibrium dipolar polarization density of the fluid can be computed from the equilibrium solvent density as

P (r, Ω) µ 0 = ˆρeq (r, Ω)ΩdΩ ( 17
)
where µ 0 is the molecular dipole of the solvent molecule. The spherically averaged radial component of the solvent equilibrium polarization around each ion are displayed in figure 4. The WDA functional reduces the polarization of the solvent in the vicinity of the charged solutes, as evidenced by the decrease of the first peaks in figure 4. Since the polarization is reduced, the ion is destabilized by the WDA functional when immersed in its equilibrium density, ∆F elec α (α) > 0. On the contrary, the equilibrium density of the oppositely charged ion is less destabilizing ∆F elec α (-α) < 0. Finally, the WDA correction being angular independent we can expect the polarization around the neutral solute to be almost unchanged ∆F elec α (0) ≈ 0. With this simple analysis and using equation 16 we recover the decrease of the reorganization free energy observed in table I. A verification of the hypothesis made here and a more detailed examination of the evolution of the different quantities entering equations 14-15 is available in SI.

V. CONCLUSIONS

In this paper we studied the appropriateness of two corrections to the HNC functional to study aqueous electron transfer reactions with molecular density functional theory. First, we examine a simple a posteriori correction of the pressure which has been widely used with different expression in mDFT and 3D-RISM [START_REF] Volodymyr | Fast Computation of Solvation Free Energies with Molecular Density Functional Theory: Thermodynamic-Ensemble Partial Molar Volume Corrections[END_REF][START_REF] Gillespie | Restoring the consistency with the contact density theorem of a classical density functional theory of ions at a planar electrical double layer[END_REF][START_REF] Sergiievskyi | Solvation free-energy pressure corrections in the three dimensional reference interaction site model[END_REF][START_REF] Misin | Salting-out effects by pressure-corrected 3D-RISM[END_REF][START_REF] Sohvi Luukkonen | Hydration free energies and solvation structures with molecular density functional theory in the hypernetted chain approximation[END_REF][START_REF] Robert | Pressure correction for solvation theories[END_REF][START_REF] Misin | Communication: Accurate hydration free energies at a wide range of temperatures from 3D-RISM[END_REF] . Despite its success to predict solvation free energy in good agreement with reference simulations and experiments, this correction should not be used to study electron transfer reactions. The minima of the free energy profile does not correspond to the equilibrium solvent configuration when this correction is used because it modifies the free energy without affecting the functional optimization. The reorganization free energies becomes ill-defined: the graphical definition do not coincide with the mathematical expression. Moreover, we have shown that using any of the two definitions of the reorganization free energy lead to a behavior deviating from the Marcus picture for Cl → Cl + reaction in disagreement with MD simulations and HNC mDFT calculations. This deviation form Marcus theory is not recovered when examining the evolution of the energy gap, which is a final evidence for the inconsistency of the pressure correction.

We then turned to another type of correction, a so-called bridge functional which is trying to recover some of the contribution due to the terms beyond the quadratic approximation of the HNC functional. We have chosen to use the recent and simple angular independent weighted density functional [START_REF] Borgis | Simple Parameter-Free Bridge Functionals for Molecular Density Functional Theory. Application to Hydrophobic Solvation[END_REF] that was shown to properly reproduce the solvation free energies of hydrophobic solutes. Since this bridge has a functional form, the equilibrium density is modified and this approach does not suffer from the flaws of the PMV correction. There are no ambiguities in the definitions of the reorganization free energies, and we recover results consistent with MD simulations and HNC mDFT calculations: Cl → Cl + follows Marcus picture while Cl -→ Cl deviates from it. Overall, the WDA functional does not modify significantly the results obtained without correction [START_REF] Jeanmairet | A molecular density functional theory approach to electron transfer reactions[END_REF] . This might be because it is an angular independent correction, having a low impact on the polarization of the solvent which is the dominant effect for ions.

VI. ACKNOWLEDGEMENT

The authors acknowledge Mathieu Salanne for his constructive comments about the manuscript. This work has been supported by the Agence Nationale de la Recherche, projet ANR BRIDGE AAP CE29.

  FIG.1. Free energy profiles of Cl → Cl + and the Cl -→ Cl electron transfer reactions computed using the HNC functional without (full) and with (dashed) the PMV correction and using the WDA (dotted) bridge functional. The MD data obtained by Hartnig et al 4 are the circles.

FIG. 2 .

 2 FIG.2. Free energy of Cl + and Cl -computed using the HNC functional without (full) and with (dashed) the PMV correction and using the WDA (dotted) bridge functional as a function of the charge of the fictitious solute. The value of the PMV correction is also displayed as a function of the charge of the fictitious solute.

  FIG.3. Vertical energy gap computed using the HNC (solid black) and WDA functional (solid red) as a function of the atomic charge. The dotted and dashed lines are the tangent to the HNC curve in q = -1 and q = 1 respectively.

FIG. 4 .

 4 FIG. 4. Equilibrium radial polarization density aroundCl -and Cl + computed by HNC and WDA

TABLE I

 I

. Solvent reorganization free energies (in kJ/mol) calculated with the HNC functional, with the PMV correction using equation 9, with the PMV correction using the graphical definition and with the WDA functional.

  VII. DATA AVAILABILITY STATEMENTData available on request from the authors 1 R. A. Marcus. On the Theory of Oxidation-Reduction Reactions Involving Electron Transfer. I. J. Chem. Phys., 24:966-978, 1956. 2 Arieh Warshel. Dynamics of reactions in polar solvents. Semiclassical trajectory studies of electron-transfer and protontransfer reactions. J. Phys. Chem., 86:2218-2224, 1982. 3 J. Blumberger and M. Sprik. Redox Free Energies from Vertical Energy Gaps: Ab Initio Molecular Dynamics Implementation. In Mauro Ferrario, Giovanni Ciccotti, and Kurt Binder, editors, Computer Simulations in Condensed Matter Systems: From Materials to Chemical Biology Volume 2, volume 704, pages 481-506. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006.