Morgane Ballon
email: morgane.ballon@inserm.fr

Jérémie Botton

Anne Forhan

B De Lauzon-Guillain

Blandine Lauzon-Guillain

Fabienne Maria Melchior

El Khoury

Aurelie Nakamura

Marie Aline Charles

Sandrine Lioret

B Heude

I Annesi-Maesano

J Y Bernard

P Dargent-Molina

P Ducimetière

M De Agostini

B Foliguet

X Fritel

A Germa

V Goua

R Hankard

M Kaminski

B Larroque

N Lelong

J Lepeule

G Magnin

L Marchand

C Nabet

F Pierre

R Slama

M J Saurel-Cubizolles

M Schweitzer

O Thiebaugeorges

We

Jo Ann

Which modifiable prenatal factors mediate the relation between socio-economic position and a child's weight and length at birth?

Keywords: birth weight, birth length, maternal education, smoking, BMI, mediation analysis

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

CONTRIBUTORS

MB, SL, BH and JB conceived and designed the work, with advice from MAC. MB analyzed the data with advice from BH, JB and SL. MB, BH and SL drafted and revised the manuscript. All authors interpreted the data and criticized the manuscript for important intellectual content. MAC and BH designed and led the EDEN mother-child cohort. AF is responsible for the EDEN data management. All authors have read and approved the final version of the manuscript. This article is the work of the authors. MB serves as guarantor for the contents of this article. All authors had full access to all of the data (including statistical reports and tables) in the study and take the responsibility for the integrity of the data and the accuracy of the data analysis. All researchers are independent of the funding bodies. All members in the EDEN mother-child cohort study group designed the study. The study sponsors were not involved in the study design, data collection, or data analyses.

SOURCE OF FUNDING

CONFLICT OF INTEREST

None of the authors had a conflict of interest.

Abbreviations: Body Mass Index (BMI)

INTRODUCTION

The existence of social inequalities in growth and childhood overweight in high-income countries has been highlighted in recent years [START_REF] Ballon | Socioeconomic inequalities in weight, height and body mass index from birth to 5 years[END_REF][START_REF] Barriuso | Socioeconomic position and childhood-adolescent weight status in rich countries: a systematic review, 1990-2013[END_REF][START_REF] Howe | Socioeconomic differences in childhood growth trajectories: at what age do height inequalities emerge?[END_REF][START_REF] Mccrory | Socioeconomic differences in children's growth trajectories from infancy to early adulthood: evidence from four European countries[END_REF]. These inequalities are observed as early as the first day of life, since positive associations have been described between socioeconomic position and both weight and length at birth [START_REF] Ballon | Socioeconomic inequalities in weight, height and body mass index from birth to 5 years[END_REF][START_REF] Howe | Socioeconomic differences in childhood growth trajectories: at what age do height inequalities emerge?[END_REF][START_REF] Jansen | Explaining educational inequalities in birthweight: the Generation R Study[END_REF][START_REF] Mortensen | Socioeconomic differences in perinatal health and disease[END_REF]. It is also well known that a lower birth weight, a marker of suboptimal fetal growth, is associated with a range of short-and longterm health issues, such as cardiovascular disease, diabetes, and obesity (D. J. [START_REF] Barker | Trajectories of growth among children who have coronary events as adults[END_REF][START_REF] Jornayvaz | Low birth weight leads to obesity, diabetes and increased leptin levels in adults: the CoLaus study[END_REF][START_REF] Shenkin | Birth weight and cognitive ability in childhood: a systematic review[END_REF][START_REF] Zanetti | Birthweight, Type 2 Diabetes Mellitus, and Cardiovascular Disease: Addressing the Barker Hypothesis With Mendelian Randomization[END_REF]. These findings suggest intergenerational transmission of health inequalities between the mother and her offspring [START_REF] Aizer | The intergenerational transmission of inequality: maternal disadvantage and health at birth[END_REF][START_REF] Barker | Developmental biology: Support mothers to secure future public health[END_REF]. The development of interventions to prevent or reduce these social inequalities in health programming requires that the modifiable factors mediating the positive association between socioeconomic position and birth weight be identified. Some studies have shown that prepregnancy body mass index (BMI) and smoking during pregnancy mediate the association between socioeconomic position and birth weight [START_REF] Gissler | Register based monitoring shows decreasing socioeconomic differences in Finnish perinatal health[END_REF][START_REF] Jansen | Explaining educational inequalities in birthweight: the Generation R Study[END_REF][START_REF] Van Den Berg | Educational inequalities in perinatal outcomes: the mediating effect of smoking and environmental tobacco exposure[END_REF], but only one study has investigated the joint effects of both factors on this birth outcome [START_REF] Mortensen | The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index[END_REF]. To our knowledge, only one study has examined factors involved in the association between socioeconomic position and length or height [START_REF] Galobardes | Social inequalities in height: persisting differences today depend upon height of the parents[END_REF]. This study described social inequalities for both prenatal and postnatal growth within a single longitudinal model. However they did not seek to identify mediators of the association between socioeconomic position and birth length specifically. Although weight and length at birth are strongly correlated, their separate associations with socioeconomic position may differ by age and sex [START_REF] Ballon | Socioeconomic inequalities in weight, height and body mass index from birth to 5 years[END_REF] and thus involve different mediators. Moreover, other prenatal factors remain to be studied. Indeed, healthy dietary patterns during pregnancy have been shown to be positively associated with birth weight [START_REF] Chia | Maternal Dietary Patterns and Birth Outcomes: A Systematic Review and Meta-Analysis[END_REF][START_REF] Emmett | Dietary patterns in the Avon Longitudinal Study of Parents and Children[END_REF]. Conversely, unhealthy dietary patterns and maternal depression during pregnancy have been shown to be associated with lower birth weight [START_REF] Chia | Maternal Dietary Patterns and Birth Outcomes: A Systematic Review and Meta-Analysis[END_REF][START_REF] Field | Prenatal depression effects on early development: a review[END_REF].

One study also reported a negative association between maternal depression and birth length [START_REF] Field | Prenatal depression effects on early development: a review[END_REF]. Given that social inequalities have been described for these three factors [START_REF] Emmett | Dietary patterns in the Avon Longitudinal Study of Parents and Children[END_REF][START_REF] Hein | Socioeconomic status and depression during and after pregnancy in the Franconian Maternal Health Evaluation Studies (FRAMES)[END_REF], the latter were considered as potential mediators of the relation between socioeconomic position and birth size (i.e. weight and length).

Recently, advanced methods for mediation analyses have been developed. In particular, counterfactual approaches allow for causal inference based on observational data by estimating direct and indirect effects with more power than allowed by traditional approaches [START_REF] Lange | Assessing natural direct and indirect effects through multiple pathways[END_REF][START_REF] Vanderweele | Mediation Analysis with Multiple Mediators[END_REF]. Among the various existing approaches, the counterfactual method proposed by Lange et al. [START_REF] Lange | Assessing natural direct and indirect effects through multiple pathways[END_REF] enables the simultaneous assessment of the mediating effect of several factors as well as the consideration of an exposure variable in more than two categories.

The objective of this paper was to investigate whether any of maternal prepregnancy BMI, smoking, diet, and depression during pregnancy mediate the positive association between maternal education level and offspring birth size, i.e., weight and length.

KEY MESSAGES

-A positive association exists between socioeconomic position and birth weight, but less research has been done on the prenatal factors involved -Prepregnancy BMI and smoking during pregnancy were shown to be important mediators of the positive associations between maternal education and both birth weight and birth length.

-Other modifiable factors like dietary patterns and depressive symptoms were not shown to mediate these associations.

-These results suggest that promoting a healthy prepregnancy weight and preventing smoking during pregnancy are keys to addressing socioeconomic inequalities in healthy fetal growth.

METHODS

Study design and participants

The EDEN mother-child cohort was designed to assess the pre-and postnatal determinants of children's growth, health, and development. This cohort includes 2,002 pregnant women recruited in two French maternity hospitals (in Poitiers and Nancy) between 2003 and 2006.

Exclusion criteria were multiple pregnancies, known diabetes, illiteracy, and intention to give birth elsewhere than these two university hospitals or to move outside the region within 3 years. Due to the mode of recruitment and the selective acceptance of participation, urban and well-educated mothers are over-represented in the EDEN study compared to the national population [START_REF] Heude | Cohort Profile: The EDEN mother-child cohort on the prenatal and early postnatal determinants of child health and development[END_REF]. Details of the study protocol have been published elsewhere [START_REF] Heude | Cohort Profile: The EDEN mother-child cohort on the prenatal and early postnatal determinants of child health and development[END_REF]. Both parents provided written consent. The ethics committee of the Kremlin-Bicêtre Hospital approved this study, which was also submitted to the national commission for data protection and liberties (CNIL).

Measurements

Data come from obstetric and pediatric records at birth, as well as from self-reported questionnaires completed by the mothers and clinical examinations undertaken at different stages of follow-up.

Socioeconomic position

Maternal education level, commonly studied in relation to child birth size [START_REF] Jansen | Explaining educational inequalities in birthweight: the Generation R Study[END_REF][START_REF] Mccrory | Socioeconomic differences in children's growth trajectories from infancy to early adulthood: evidence from four European countries[END_REF][START_REF] Mortensen | The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index[END_REF] and less likely to be affected by childbearing than income and occupation, was used as a proxy for socioeconomic position. Mothers were asked to self-report their highest educational attainment at study inclusion. Education level was categorized as low (did not complete high school), intermediate (high school diploma to 2-year university degree, reference category) and high (3-year university degree or more).

Intermediate category was chosen as the reference category in order to better disentangle the mediation process when comparing one category of level education to its adjacent one.

Child weight and length

At birth, weight and length were measured by midwives with an electronic scale (Seca Ltd) and a somatometer (Testut). As measurement errors are common for birth length, we used predicted length at birth, obtained from growth modeling of an average of 16 measurements of length/height between birth and 5 years [START_REF] Botton | Postnatal weight and height growth modeling and prediction of body mass index as a function of time for the study of growth determinants[END_REF][START_REF] Carles | A Novel Method to Describe Early Offspring Body Mass Index (BMI) Trajectories and to Study Its Determinants[END_REF]. For children without predicted data available, measured length was used instead (7%). Birth weight and length z-scores specific for gestational age and sex were calculated according to Audipog references (Faculté de médecine RTH Laennec Lyon, 2008).

Candidate mediators

Women self-reported their prepregnancy weight at inclusion. Maternal height was measured by research midwives with a wall stadiometer (Seca 206) during clinical examinations conducted between 24 and 28 weeks of gestation. Prepregnancy BMI was calculated as weight (kg) divided by height (m) squared and categorized as underweight (<18.5 kg/m 2 ), normal (≥18.5 and <25 kg/m 2 ), overweight (≥25 and <30 kg/m 2 ) or obese (≥30 kg/m 2 ) (WHO). During the first visit with research midwives (between 24 and 28 weeks of gestation), mothers reported their daily cigarette consumption and smoking habits at the beginning of pregnancy (smoking in the first trimester) and their smoking status at the time of the visit (smoking in the second trimester). After delivery, research midwives collected similar information for smoking at the end of the third trimester of pregnancy (third trimester smoking status). All this information was combined into one variable categorized as nonsmokers, smokers only in the first trimester, and smokers throughout pregnancy. Depressive symptoms during pregnancy were assessed with the 20-item Center for Epidemiology Studies Depression Scale (CES-D) [START_REF] Radloff | The CES-D scale: a self-report depression scale for research in the general population[END_REF]. Each response was coded between 0 and 3 points, then summed into a depressive symptoms score (ranging from 0-60). Different cut-offs, ranging from 16 to 23 [START_REF] Henry | Determining the optimal clinical cutoff on the CES-D for depression in a community corrections sample[END_REF][START_REF] Radloff | The CES-D scale: a self-report depression scale for research in the general population[END_REF][START_REF] Vilagut | Screening for Depression in the General Population with the Center for Epidemiologic Studies Depression (CES-D): A Systematic Review with Meta-Analysis[END_REF], have been proposed to detect individuals with probable depression. We chose the threshold of 23 to define women with depressive symptoms, as suggested by a validation study for the French population [START_REF] Fuhrer | La version française de l'échelle CES-D (Center for Epidemiologic Studies-Depression Scale). Description et traduction de l'échelle d'auto-évaluaion[END_REF].

Maternal diet in the last trimester of pregnancy was assessed during the maternity ward stay after delivery by a validated food frequency questionnaire (FFQ) [START_REF] Deschamps | Reproducibility and relative validity of a food-frequency questionnaire among French adults and adolescents[END_REF].

The 137 standardized items from the FFQ have previously been synthesized by principal component analysis into two dietary patterns [START_REF] Yuan | Early factors related to carbohydrate and fat intake at 8 and 12 months: results from the EDEN mother-child cohort[END_REF]: the so-called Healthy dietary pattern, characterized by a high intake of fruit, vegetables, fish, and whole grains, whereas the Western dietary pattern was characterized by a high intake of processed and snacking foods. These variables were considered to be continuous scores reflecting adherence to each dietary pattern.

Other variables

Gestational age, maternal age, and parity were collected at birth from medical records.

Preterm birth (yes/no) was defined by a gestational age <37 weeks of gestation.

Population studied

Of the 1,907 children included in the EDEN cohort, 407 were excluded because of missing values for any of the variables of interest, (i.e., outcomes, exposure, mediators, and confounders). The final sample thus included 1,500 children.

Statistical methods

Participants included in the analysis were compared to those who were not included.

Characteristics of the study population were described at birth according to maternal education level. The bivariate statistical analyses used chi-square tests, correlations, and ANOVA analyses as appropriate.

The counterfactual method developed by Lange et al. [START_REF] Lange | Assessing natural direct and indirect effects through multiple pathways[END_REF], based on marginal structural models, was used to conduct the mediation analysis. This allowed us to break down the total effect of maternal education level on birth size (i.e., weight and length z-scores) into natural direct and indirect effects through the candidate mediators. Based on this method, the total effect can be considered as the change in z-score that would be observed if, for example, maternal education level could change (e.g., from intermediate to high). The natural direct effect can be considered the difference in birth z-scores for a given change in education level (e.g., from intermediate to high), keeping mediators at the value they naturally take when maternal education is unchanged (e.g., at the intermediate level). The natural indirect effect is the difference in birth z-scores when maternal education remains unmodified, but the mediators change to the value they would naturally take if maternal education were to change (e.g., from intermediate to high). The validity of this statistical method depends on whether or not it satisfies specific hypotheses. First, for a given mediator, there should be no unmeasured confounders in associations between: 1) exposure and outcome, 2) mediator and outcome, 3) exposure and mediator, 4) mediator and outcome conditional on the exposure. Second, no causal associations should exist between the mediators.

To select candidate mediators of the association between maternal education and birth outcomes, we first conducted separate mediation analyses for each of the following potential mediators, selected a priori as most likely to explain this relation: maternal prepregnancy BMI, smoking, depressive symptoms, and maternal dietary patterns during pregnancy. We checked that they were independent of each other, conditional on the exposure and the confounders in our sample, by running a multiple regression model of one mediator on the others, adjusting for maternal education and the confounders. When a statistically significant association was observed between candidate mediators, we used the residual method to obtain independent variables and verify the model's assumptions, by generating new variables as the residuals of the regression of one mediator on the others. Simple mediation analyses were then conducted with these new variables. All individually significant mediators were next included in two multivariable marginal structural models to assess how they jointly mediated the association between maternal education and each of the birth weight and length z-scores.

To obtain robust 95% confidence intervals, we used a bootstrap approach with 5,000 replications. Analyses were adjusted for center (i.e., Nancy or Poitiers), parity, maternal height, and age.

Because analyses were conducted on the database with no missing data for any of the five mediators (Population B, Figure 1), the first sensitivity analysis used the database with no missing data for the mediators selected for the final multivariable model (Population A, Figure 1). A second sensitivity analysis excluded 84 preterm infants from the sample (Population C, Figure 1). SAS v9.3 (SAS Institute, Cary, NC, US) was used for all but the mediation analyses, which were run under R v3.4.2 as proposed by Lange et al. [START_REF] Lange | Assessing natural direct and indirect effects through multiple pathways[END_REF]. Graphs were also plotted with R v3.4.2. Statistical significance was defined at P ≤ 0.05.

RESULTS

Population characteristics

Mothers excluded from the analysis (n=407) had lower education levels and were more likely to have experienced depressive symptoms or to be underweight or obese than those who were included (low education level: 42.0% vs. 25.5%; depressive symptoms: 13.8% vs. 7.7%; underweight: 11.2% vs. 8.0%; obesity: 12.3% vs. 7.9%). No statistically significant differences were observed in birth weight and length z-scores between those included and excluded.

There was a positive gradient between maternal education level and both the mother's age and height and the z-scores for the child's birth weight and length (Table 1). All candidate mediators except depressive symptoms were strongly associated with both maternal education level and birth size z-scores (Tables 1 &2).

Simple mediation analysis

Prepregnancy BMI and smoking during pregnancy were mediators of the association between maternal education and birth weight and length; the Western dietary pattern was a mediator of the association with birth length only (Table 3). Since this dietary pattern was strongly associated with prepregnancy BMI and smoking during pregnancy, we performed an additional simple mediation analysis by using the residuals of its regression on BMI, smoking during pregnancy, and confounders, as a new mediator. Because this new variable, which assessed variability in the Western dietary pattern independent of other variables, was not significant, the multiple mediation analyses used only prepregnancy BMI and smoking during pregnancy.

Multiple mediation analyses of birth weight and length

The multiple mediation analysis of birth weight showed that the total effect of maternal education level on birth weight in children of mothers with a low, compared with an intermediate, education level was negative, with a -0.14 difference in z-scores (Figure 2). The natural indirect effect through smoking was also negative, with a -0.07 difference. There was no significant natural direct or indirect effect through prepregnancy BMI. Smoking during pregnancy mediated 52% of this relation. Similar findings were observed with birth length, although the percentage of mediation was lower (37%, Figure 2).

The comparison of children from mothers with high and with intermediate education levels

showed no total or direct effect of maternal education level on birth weight or length. There were, however, significant natural indirect effects through smoking and BMI and they worked in opposite directions for both birth weight and length (0.03 difference in z-score: positive and negative, respectively); this difference explains the absence of a total effect.

Sensitivity analyses

Results were on the whole consistent when analyses were performed for the complete database (Population B) for all mediators or repeated for the database with no missing items on smoking and BMI only (Population A, results not shown but available on request). Results remained consistent after the exclusion of children born preterm.

DISCUSSION

This study, using a validated method to assess multiple mediation, provides new and comprehensive insights into modifiable prenatal mediators of the social gradient in birth size.

Mediators of birth weight and length

Dietary patterns and depressive symptoms during pregnancy did not explain the association between maternal education level and birth size in this study. To our knowledge, no others have investigated these two candidate mediators, although this apparent lack of literature may reflect publication bias. Moreover, dietary patterns and depressive symptoms may be subject to stronger measurement errors than prepregnancy BMI and smoking, which might have reduced statistical power for these two candidate mediators. Indeed, CES-D measure is a screening tool but not a diagnosis and dietary patterns were assessed through dietary intake during the last trimester of pregnancy, which could led to a recall bias. Further research is needed to confirm or infirm our findings. Perhaps stress during pregnancy, rather than depression, and energy intake, rather than diet during pregnancy, explain the association between maternal education and birth size.

Consistent with other studies [START_REF] Jansen | Explaining educational inequalities in birthweight: the Generation R Study[END_REF][START_REF] Mortensen | The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index[END_REF][START_REF] Van Den Berg | Educational inequalities in perinatal outcomes: the mediating effect of smoking and environmental tobacco exposure[END_REF], smoking during pregnancy mediated about 52% of the association between maternal education and birth weight in the comparison of children of mothers with low, compared with intermediate, education levels. The pattern of mediation was different when we compared high vs. intermediate education levels: prepregnancy BMI and smoking during pregnancy mediated the association of interest, but their indirect effects were in opposite directions and thus cancelled each other out. Mortensen et al. also found that prepregnancy BMI and smoking during pregnancy mediated in opposite directions across the entire gradient (and did not differ whether low or high education levels were compared to the intermediate level) [START_REF] Mortensen | The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index[END_REF]. Consistent findings for mediation observed with birth length are noteworthy in our study, which adds a novel and more comprehensive perspective into the social patterning of birth size.

The method proposed by Lange et al. [START_REF] Lange | Assessing natural direct and indirect effects through multiple pathways[END_REF] enabled us to investigate maternal education in three categories and observe differential effects in these three groups. However, the lack of any mediating effect by prepregnancy BMI between the groups with low and intermediate maternal education must be confirmed in cohorts with greater social variability.

The women in these two subgroups of the Eden cohort were very similar in terms of prepregnancy BMI.

The fact that prepregnancy BMI and smoking during pregnancy cancelled each other out should not be taken lightly. It means that a baby born to an overweight mother who smokes has the same birth weight and length as another whose non-smoking mother has a normal weight. In the first case, birth size would be the result of growth restriction due to exposure to smoking, masked by excess growth due to maternal obesity (6% of the EDEN women combined these two characteristics) and misleadingly suggests that their children's growth is optimal, when it is not. For any given birth size, the distinct causes producing them are expected to affect later growth, development, and health differentially. Future studies, investigating metabolic biomarkers in cord blood, might provide evidence supporting (or contradicting) this hypothesis.

Potential explanation of mechanisms involved

Smoking during pregnancy may affect birth size through different mechanisms. It could lead to vasoconstriction and to higher maternal and fetal bloods levels of carboxyhemoglobin [START_REF] Wickstrom | Effects of nicotine during pregnancy: human and experimental evidence[END_REF], which are responsible for fetal hypoxia. Exposure to smoking during pregnancy is also suspected of modifying regulation of fetal gene expression, by altering DNA methylation and microRNA expression [START_REF] Knopik | The epigenetics of maternal cigarette smoking during pregnancy and effects on child development[END_REF]. Maternal overweight or obesity is associated with higher birth size of her offspring [START_REF] Gaudet | Maternal obesity and occurrence of fetal macrosomia: a systematic review and meta-analysis[END_REF]. Mothers with higher BMI have higher levels of circulating blood glucose [START_REF] Harmon | Continuous glucose profiles in obese and normal-weight pregnant women on a controlled diet: metabolic determinants of fetal growth[END_REF] and lipids, so that more of these nutrients are available for the fetus. Increased insulin secretion by the fetal pancreas in response to glucose in turn accelerates fetal growth [START_REF] Group | Hyperglycemia and Adverse Pregnancy Outcome (HAPO) Study: associations with neonatal anthropometrics[END_REF].

It is noteworthy that prepregnancy BMI and smoking during pregnancy are modifiable factors. As two reviews have shown, effective strategies already exist to reduce maternal BMI and smoking [START_REF] Johnson | Interventions for Reducing Body Mass Index and Other Weight-related Indicators: A Review of Systematic Reviews[END_REF][START_REF] Lancaster | Effectiveness of interventions to help people stop smoking: findings from the Cochrane Library[END_REF], and the consistency of our findings across birth weight and length suggests that promoting healthy weight or preventing smoking during pregnancy in future mothers is likely to favorably and simultaneously affect the offspring's birth weight and length.

Limitations and strengths

Study limitations include our inability to consider all potential mediators, thus we did not fully explain the relation between maternal education and birth size. For example, we did not take health care utilization into account, although it is hypothesized to improve fetal monitoring and help prevent fetal growth anomalies. Mothers from the EDEN cohort were, however, very homogeneous in this respect given the mode of inclusion at the maternity ward before 24 weeks of gestation. Maternal stress during pregnancy and passive smoking (due to paternal smoking), unmeasured in the current study, are other relevant mediators to explore.

Classification of prepregnany BMI relies on self-reported weight. Although women are likely to under-estimate their prepregnancy weight, and more importantly when overweighed or obese [START_REF] Bannon | Comparison of Self-reported and Measured Pre-pregnancy Weight: Implications for Gestational Weight Gain Counseling[END_REF], previous studies have suggested that classification of prepregnancy BMI remains usually unchanged [START_REF] Bannon | Comparison of Self-reported and Measured Pre-pregnancy Weight: Implications for Gestational Weight Gain Counseling[END_REF][START_REF] Holland | Self-reported pre-pregnancy weight versus weight measured at first prenatal visit: effects on categorization of pre-pregnancy body mass index[END_REF]. Moreover, validity of prepregnancy weight was previously validated within the EDEN study using multiple pregnancy weights collected all over pregnancy [START_REF] Diouf | Specific role of maternal weight change in the first trimester of pregnancy on birth size[END_REF]. The presence of selection bias at inclusion, as is often the case in cohort studies, has implications for the generalization of our findings. We can hypothesize that a better representation of disadvantaged families at baseline would have provided more variability and therefore more power to address the study objectives. Further research is needed to confirm our findings on more datasets with greater social variability.

The method we used to conduct mediation analyses is validated and relevant. To our knowledge, no dedicated codes to conduct multiple imputation have however been developed to address missing data. Although there are about 20% missing data in the principal analysis, sensibility analysis on Population B, which includes only 6% missing data, led to the same results. Furthermore, the Lange method relies on several assumptions. Residual confounding not taken into account cannot be ruled out; prepregnancy BMI and smoking during pregnancy might be proxies for other factors related to unhealthy lifestyles. Our data nonetheless met the principal assumption necessary for using this method: maternal smoking and prepregnancy BMI were conditionally independent of maternal education level. Examining the stability of estimates across the one-by-one mediation analysis, multiple mediation, and sensitivity analyses shows that the results appear robust. Moreover, a clear strength of our study is that data are from a prospective birth cohort and can thus identify mediators of the longitudinal association between maternal education level and birth parameters. To our knowledge, this study, as the first to investigate birth length, provides new insights into the existing literature.

Finally, unlike most mediation analyses, we investigated maternal education level in three categories to examine the social gradient in birth size in more detail.

CONCLUSION

Among the modifiable factors examined, dietary patterns and depressive symptoms did not mediate the positive association between maternal education and birth weight and length while prepregnancy BMI and maternal smoking during pregnancy did so. The latter two factors, however, act in opposite directions and thus mask the extent to which prenatal growth is socially differentiated. Although these original findings need to be replicated in more socially diverse samples, they suggest that promoting a healthy prepregnancy weight and preventing smoking during pregnancy are keys to addressing socioeconomic inequalities in healthy fetal growth and thereby attenuating the intergenerational transmission of socioeconomic health inequalities.

  Fondation pour la Recherche Médicale -French Ministry of Research -Institut Fédératif de Recherche and Cohort Program -INSERM Nutrition Research Program -French Ministry of Health Perinatal Program -French Agency for Environment Security (AFFSET) -French National Institute for Population Health Surveillance (INVS) -Paris-Sud University -French National Institute for Health Education (INPES) -Nestlé -Mutuelle Générale de l'Éducation Nationale -French Speaking Association for the Study of Diabetes and Metabolism (Alfediam) -National Agency for Research (ANR nonthematic program) -National Institute for Research in Public Health (IRESP TGIR Cohorte Santé 2008 Program)

Figure legends Figure 1 :Figure 2 :

 legends12 Figure legendsFigure 1: Flow chart of the population included in the study. EDEN mother-child cohort


Nanna Holmgaard List
email: nanna.h.list@gmail.com

Trond Saue

Patrick Norman

Rotationally averaged linear absorption spectra beyond the electric-dipole approximation

Keywords: response theory, multipole-expansion free, rotational averaging, Lebedev quadrature, X-ray absorption

Based on the recently developed implementation of the full semi-classical field-matter interaction operator [List et al. in J. Chem. Phys. 142, 244111 (2015)], we present a numerically accurate yet efficient scheme to perform rotational averaging of linear absorption spectra beyond the electric-dipole approximation. This allows for a gauge-origin independent determination of UV/vis and X-ray absorption spectra for randomly oriented systems such as multilayers, liquids, and gas phase samples. The approach is illustrated by the determination of spectral intensities of electric-dipole allowed π → π * transitions and electric-dipole forbidden n → π * transitions in the UV-vis region of the spectrum as well as electric-dipole forbidden 1s → 3d transitions in the X-ray region of the spectrum. The employed Lebedev quadrature scheme shows very fast convergence with respect to the number of symmetry-independent quadrature points-in all considered cases, the oscillator strengths for the randomly oriented systems are fully converged with use of only seven quadrature points.

Introduction

Light absorption and emission as due to the interaction between electromagnetic field radiation and molecular compounds are fundamental and essential ingredients in Nature as well as every-day life. Notable examples are given in terms of the conversion of solar energy by means of photosynthesis in green plants or, more broadly, the entire scientific field of photochemistry; [START_REF] Kagan | Organic Photochemistry: Principles and Applications[END_REF] chromphores and dyes that give rise to a spectrum of colors in liquids, pigments, and molecular materials; the production of light in chemiluminescence and bioluminescence; [START_REF] Zomer | Chemiluminescence and Bioluminescence[END_REF] artificial light harvesting and organic solar cells or, more broadly, large parts of the scientific field of photonics; [START_REF] Nalwa | Handbook of Advanced Electronic and Photonic Materials and Devices[END_REF] and, not least, the plethora of spectroscopies that we have designed to probe and investigate the physical and chemical properties of molecules and molecular materials in various environments and phases. [START_REF]Encyclopedia of Spectroscopy and Spectrometry[END_REF] We also design photoactive probes to selectively target host systems as to get indirect information of the host rather than the probe itself, as exemplified by the development of palettes of luminescent conjugated oligothiophenes for the detection and identification of protein aggregates. [START_REF] Shirani | [END_REF] In all of the phenomena, applications, and scientific fields mentioned above, the scientific community leverages the fundamental microscopic understanding obtained by means of computational chemistry to achieve rational molecular design and material performance enhancements. It comes therefore as no surprise that theoretical chemists, and in particular quantum chemists, have dedicated large efforts to develop methods and algorithms to accurately yet efficiently include the effects of external time-dependent electromagnetic fields on the molecular charge density. Most notably, there exist today several basic formulations of response theory with algorithmic and program implementations for virtually all standard electronic structure theory methods on the market. [6] In all of these instances, a semi-classical description is employed for the external field and it is common to assume that the field is sufficiently weak for perturbation theory to be applicable. [START_REF] Mukamel | Principles of Nonlinear Optical Spectroscopy[END_REF][START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF][START_REF] Boyd | Nonlinear Optics[END_REF] Thereafter differences start to appear, however, as one introduces the coupling terms between the semi-classical fields and the quantum mechanical system into the Hamiltonian. Choices are made based on the level of accuracy needed, algorithmic ease, and computational speeds. In absorption and emission studies, the component of the magnetic field can often be ignored as its effect on the molecular charges is weaker than electric fields by a factor of 1/c. If, on the other hand, a natural dichroism in isotropic samples is studied, then it is the mere presence of the magnetic component of the radiation field that gives rise to a differential absorption by the sample.

In this context, we have recently investigated the use of an exact and complete inclusion of electromagnetic fields in the interaction Hamiltonian by means of the principle of minimal coupling. [START_REF] List | [END_REF] The assumptions made in the implementation of this strategy involved the specification of a state of linear polarization and a direction of propagation of a harmonic plane wave. Since any form of multipole expansion was avoided, response functions involving this full field-matter interaction operator are intrinsically gauge-origin invariant, and our formulation thus leads to physical observables that are independent of the gauge origin. The contributions to both absorptive and dispersive parts of response functions from all higher-order combinations of multipole moment operators are included in this treatment, and for instance natural dichroism is directly accounted for in the absorptive part of the linear response function.

For studies of oriented samples, such as the calculation of a near-edge X-ray absorption fine structure (NEXAFS) spectrum of a self-assembled monolayer at normal or grazing incidence, [START_REF] Stöhr | NEXAFS Spectroscopy[END_REF] the scheme presented by List et al. [START_REF] List | [END_REF] is well suited. However, for multilayer samples, liquids, or otherwise randomly oriented systems, there is a need to perform spectrum averaging over molecular orientations in the laboratory frame of reference, or, equivalently, averaging over light propagation and polarization directions in the molecular frame. The advantage of performing a multipole moment expansion of the interaction operator becomes obvious at this point in the calculation, since, for every given order in the expansion, the orientational average of the corresponding property tensor can be performed analytically [START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF][START_REF] Friese | [END_REF][13][14][15][16][17]]-e.g. the orientational average of the electric-dipole polarizability is referred to as the isotropic polarizability and is equal to a simple tensor trace. The same approach cannot be taken in our case, but, as will be demonstrated in the present work, it is possible to introduce the numerical quadrature integration schemes by Lebedev and co-workers [START_REF] Lebedev | Angular quadrature parameters available from[END_REF][START_REF] Lebedev | [END_REF][20][21][22][23] to achieve efficient and numerically accurate spectrum calculations for randomly oriented samples.

In the following, we will present a scheme for rotational averaging of linear absorption cross sections beyond the electric-dipole (ED) approximation associated with the interaction of a molecule with a linearly polarized electromagnetic field.

We will refer to results obtained at this level of theory by the label BED and contrast these with the corresponding results obtained in the ED approximation. The remainder of this work is organized as follows: In Section 2, we give a brief recapitulation of the theoretical background relevant for the description of linear absorption in the BED framework as well as present a numerical scheme for the determination of spectra for randomly oriented samples. Computational details are given in Section 3. Representative numerical illustrations of the convergence behavior of the proposed scheme are presented in Section 4 by example calculations of UV/vis and X-ray absorption spectra of formaldehyde and acrolein and X-ray absorption spectra of ferrocene and vinylferrocene.

Theory

We will consider a collection of randomly oriented, identical, non-interacting closed-shell molecules subjected to a weak, monochromatic, linearly polarized electromagnetic field oscillating at an angular frequency ω. The interaction between the molecular sample and the external field will be treated in the commonly used semi-classical framework, where the incident radiation is described as a classical field in terms of auxiliary scalar and vector potentials, φ(r, t) and A(r, t). Using the minimal coupling formulation of the interaction, the field-matter interaction operator can in the Coulomb gauge (∇ • A = 0, choosing φ = 0) [START_REF] Schatz | Quantum Mechanics in Chemistry[END_REF] be written as

V (t) = e m e N i=1 A(r i , t) • pi = ±ω V ω e -iωt , (1)
with Fourier components given by

V ±ω = ± N i=1 eE 0 2iωm e e ±ik•ri ( • pi ). (2)
In writing Eq. ( 1), we have neglected the term quadratic in A, which does not contribute to linear absorption. [START_REF] List | [END_REF] The linear momentum operator is given by pi = -i ∇ i , is the reduced Planck constant, -e and m e are the charge and mass of an electron, respectively, and N is the total number electrons in the molecule. E 0 is the real-valued external field amplitude, and k is the wave vector that describes the propagation direction of the light beam with norm k = |k| = ω/c 0 , where c 0 is the speed of light in vacuo. Finally, is a unit vector perpendicular to k, specifying the polarization direction of the light.

Linear absorption of linearly polarized light

In this section, we will review the theory necessary for describing linear absorption using the field-matter interaction operator in Eq. [START_REF] Kagan | Organic Photochemistry: Principles and Applications[END_REF]. Following the energy-loss approach, molecular absorption cross sections, describing the rate of energy flow from an external electromagnetic field to a molecule, can be related to resonanceconvergent complex molecular response functions. [START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF][START_REF] Mahr | Two-Photon Absorption Spectroscopy[END_REF] In particular, the linear absorption cross section, [START_REF] List | [END_REF]

σ(ω) = 4ω ε 0 c 0 E 2 0 Im[G k ˆ (-ω; ω)], (3)
is proportional to the imaginary part of the general molecular polarizability G k ˆ (-ω; ω) that represents the linear response of the molecule to a linearly polarized field characterized by polarization and propagation directions (ˆ , k). In exact-state theory, the complex polarizability can be expressed as a sum over excited states according to

G k ˆ (-ω; ω) = 1 n>0 0| V -ω |n n| V ω |0 ω n0 -ω -iγ n0 + 0| V ω |n n| V -ω |0 ω n0 + ω + iγ n0 , (4)
where ω n0 is the transition energy from the electronic ground state |0 to the electronic excited state |n . The phenomenological damping parameters γ n0 = (2τ n ) -1 are related to the lifetimes τ n of the exited states, assuming exponential decay. In approximate-state theory, the resonance-convergent polarizability can be addressed in the framework of the complex polarization propagator (CPP) approach. [START_REF] Norman | [END_REF][27][28] Eq. ( 4) is then replaced by a matrix equation in which the explicit reference to the excited states is avoided. [START_REF] List | [END_REF] In other words, the linear absorption cross section can be evaluated without having to resolve the individual eigenvectors of the electronic Hessian. This is particularly important because it makes it possible to treat core excitations without imposing restrictions on the excitation manifold.

In the alternative and traditional approach, one considers the infinite-lifetime limit (γ n0 = 0) of the linear absorption cross section in Eq. ( 4). Focusing on the absorptive part of the response function, we obtain [START_REF] Merzbacher | Quantum Mechanics[END_REF][START_REF] Bransden | Physics of Atoms and Molecules[END_REF]

lim γ→0 σ(ω) = 4π 2 α e 2 ω n>0 δ(ω -ω n0 )|TM n0 BED | 2 , (5)
where α = e 2 /(4πε 0 c 0 ) is the dimensionless fine structure constant and TM n0

Integration over frequencies of the right hand side of Eq. ( 5) leads to a sum of terms

σ n0 = π 2 e 2 (4πε 0 )m e c 0 f n0 , (7)
where appears the dimensionless oscillator strength, [START_REF] Schatz | Quantum Mechanics in Chemistry[END_REF][START_REF] Bransden | Physics of Atoms and Molecules[END_REF]

f n0 = 2m e e 2 ω n0 |TM n0 BED | 2 , (8)
which represents the relative intensity of absorption with respect to a harmonic oscillator model. Eqs. ( 4) and ( 8) are typically simplified by invoking a Taylor expansion of the exponential operator in orders of the wave vector k, i.e., e ik•r = 1+ik•r-1 2 (k•r) 2 + • • • . Retaining only the zeroth-order term defines the ED approximation, which relies on the assumption that the wavelength of the electromagnetic radiation field is much larger than the molecular dimension (kr 1). In most cases, this is a good approximation for transitions in the UV/vis region, where the field indeed appears uniform across the molecule. In X-ray spectroscopies, the short wavelength of the radiation becomes comparable to the spatial extension of the probed molecular systems, and intuitively, this would put the use of the ED approximation in doubt. Recently, we have demonstrated the validity of the ED approximation for ED allowed transitions in K-edge X-ray absorption spectroscopy. It was shown that the otherwise short wavelengths are compensated by the spatial compactness of the initial 1s-orbitals, and that the kr value remains below unity for all elements in the periodic table .[10] However, when probing excitations that are ED forbidden by symmetry reasons, or when considering more complex physical phenomena, the higher-order contributions accounting for the spatial variations of the field need to be taken into account. A prominent example is pre-edge features in metal K-edge X-ray absorption spectra of transition metal complexes, which are often governed by ED forbidden transitions from metal 1s-to d-orbitals. [START_REF] Yamamoto | [END_REF][32][33] An important aspect of going beyond the ED approximation by including additional terms in the multipole expansion of the physical observable is that the individual terms, beyond the lowest non-zero multipole order, are origindependent. [START_REF] Barron | Molecular Light Scattering and Optical Activity[END_REF]34] Accordingly, special attention has to be paid to ensure origininvariance of the physical observables when using a truncated multipole expansion formulation. [35][36][37] In particular, Bernadotte et al. showed that origin-independent oscillator strengths can be obtained by including all terms in the oscillator strength consistently through a given order in the wave vector. [37] However, the cross sections introduced in this origin-independent but truncated multipole expansion formulation (see Eq. 62, Ref. [37]) can in certain cases result in unphysical, negative oscillator strengths, as recently reported in Ref. [38].

In this work, we take as starting point for the rotational averaging of the linear absorption cross section the untruncated expressions in Eqs. ( 4) and ( 8) involving the complete field-matter interaction operator. Using the exact semi-classical representation will, by necessity, lead to rotationally averaged cross sections that are origin-independent and always positive, as they should be.

Rotational Averaging

In Eqs. ( 4) and ( 8), we assumed that the components of (k, ) and (r i , pi ) refer to a common coordinate system, say a space-fixed frame with unit vectors {e X , e Y , e Z }. For convenience, the space-fixed coordinate system is defined such that the Z-axis is parallel to the propagation direction and the X-axis to the polarization vector, i.e., k = ke Z and = e X . This corresponds to an experimental setup, where the absorbing molecules are oriented in a specific way with respect to the incoming electromagnetic radiation. In the majority of experiments, the molecules in the sample are free to rotate, and the theory as described above stands in need of further refinement before its results can be related to experimentally measured absorption cross sections. Under these circumstances, it is necessary to perform rotational averages of Eqs. ( 4) and ( 8) to account for the distribution of molecular orientations in the sample. For simplicity, we shall assume an isotropic distribution of molecules in which case the problem reduces to averaging over all possible orientations of the molecule relative to the propagation and polarization directions.

We therefore consider a molecule-fixed coordinate frame with unit vectors {e x , e y , e z }. In this frame, the coordinates r are related to those of the spacefixed frame, r, by

r = R (φ, e Z ) R (θ, e Y ) R (χ, e Z ) r, (9)
where the total rotation is expressed in terms of the three Euler angles (φ, θ, χ

)[39] R (φ, e Z ) R (θ, e Y ) R (χ, e Z ) =   cos φ -sin φ 0 sin φ cos φ 0 0 0 1     cos θ 0 sin θ 0 1 0 -sin θ 0 cos θ     cos χ -sin χ sin χ cos χ 0 0   . (10)
In the molecule-fixed frame, the wave and polarization vectors are given by k = ke r ; = cos χe θ + sin χe φ [START_REF] Stöhr | NEXAFS Spectroscopy[END_REF] in terms of unit vectors e θ = e x cos θ cos φ + e y cos θ sin φe z sin θ e φ = -e x sin φ + e y cos φ e r = e x sin θ cos φ + e y sin θ sin φ + e z cos θ.

The isotropic average of the oscillator strength, or cross section, over the three Euler angles is defined as

f n0 φ,θ,χ = 1 8π 2 2π 0 dφ π 0 sin θdθ 2π 0 f n0 dχ. ( 12
)
We note that in the space-fixed frame the dependence on the Euler angles enters parametrically in the electronic wave functions describing the ground state |0 and the excited state |n , whereas in the molecule-fixed frame all dependence is in the wave and polarization vectors in the field-matter interaction operator. We now introduce an intermediate frame whose coordinates are given by

r = R (χ, e Z ) r = R (-θ, e Y ) R (-φ, e Z ) r . (13)
In this frame, the representation of the wave and polarization vectors are given by k = (0, 0, k) ; = (cos χ, sin χ, 0) .

In the intermediate frame, the dependence on the two Euler angles (θ, φ) is in the electronic wave functions, whereas the dependence on the Euler angle χ only enters the polarization vector. The averaging over the latter angle can now be performed straightforwardly in an analytical manner. It reduces to

f n0 χ = 2m e e 2 ω n0 e m e 2 α β χ 0| N i=1 e -ikz i p iα |n n| N j=1 e +ikz j p jβ |0 , (15)
where repeated Greek indices indicate summation over Cartesian coordinates. The only non-zero elements are

x x χ = y y χ = 1 2 , (16)
such that the final expression for the χ-averaged oscillator strength reads

f n0 χ = m e e 2 ω n0 e m e 2 α=x,y | 0| N i=1 e -ikz i p iα |n | 2 . ( 17
)
In a similar manner, the χ-averaged linear absorption cross section becomes

σ(ω) χ = 2ω ε 0 c 0 E 2 0 Im[G z x (-ω; ω) + Im[G z y (-ω; ω)] . (18)
The χ-averaged quantities Eqs. ( 17) and ( 18) depend on the solid angle (θ, φ), which specifies the orientation of the molecule with respect to the propagation direction. The isotropic averaging of the linear absorption has therefore been reduced to a two-angle integration of Eqs. ( 17) and ( 18) over the solid angle. To get a flavor of how the averaging work, we consider the average of the oscillator strength within the ED approximation

f ED n0 φ,θ,χ = m e e 2 ω n0 e m e 2 α=x,y | 0| N i=1 p iα |n | 2 φ,θ,χ . ( 19
)
The average over the final two angles (θ, φ) is best done in the molecule-fixed frame. Inserting the relations

p ix = cos θ cos φ p ix + cos θ sin φ p iy -sin θ p iz (20)
p iy = -sin φ p ix + cos φ p iy , (21)
and performing the average gives the well-known expression for the isotropically averaged oscillator strength in the velocity representation

f ED n0 φ,θ,χ = 2 3m e ω n0 α=x,y,z | 0| N i=1 p iα |n | 2 . ( 22
)
Since it effectively amounts to a matrix trace, it is valid in all frames considered. In the general case, the (θ, φ)-integration will be performed numerically using the schemes of Lebedev and co-workers. [START_REF] Lebedev | Angular quadrature parameters available from[END_REF][START_REF] Lebedev | [END_REF][20][21][22][23] The distribution of quadrature points over a unit sphere for a given Lebedev grid defines the propagation directions included in the numerical integration, and the χ-average is performed in accordance with Eqs. ( 17) and ( 18) by considering two orthogonal polarization directions for each wave vector, forming a right-handed coordinate system. The Lebedev grids have been constructed under octahedral symmetry, which makes them suited for the present context. In particular, the χ-averaged transition strength is identical for quadrature points corresponding to k and -k. This can be exploited to limit the (θ, φ)-integration to a hemisphere, thereby reducing the number of quadrature points needed for a given Lebedev grid by a factor of two in the general case. Additional savings can be achieved for symmetric molecules where the calculations may be restricted to those quadrature points that are not related by symmetry operations in the given point group.

Computational Details

The lowest energy valence and carbon K-edge excitations in formaldehyde and trans-acrolein have been used to study the convergence behavior of the proposed Lebedev-based scheme for obtaining rotationally averaged linear absorption cross sections involving the full field-matter interaction operator. The molecular geometries were optimized under the constraint of C s and C 2v symmetry, respectively, at the DFT level of theory using the B3LYP [START_REF] Stephens | [END_REF][41][42][43] exchange-correlation functional and the cc-pVTZ [44] basis set. The property calculations for formaldehyde and acrolein were performed using the CAM-B3LYP 100% functional, designed for the purpose of describing core electron excitations. [28] This is a modified version of the Coulomb-attenuated method CAM-B3LYP [45] functional taking into account 100% Hartree-Fock Coulombic interactions in the long-range limit (α=0.19, β=0.81, and µ=0.33a -1 0 ). For the calculation of oscillator strengths, we have employed the restricted-channel approximation. [28,46] Specifically, the excitation manifold in the linear response calculation of formaldehyde and acrolein was restricted to excitations originating from the carbon 1s-orbitals to any of the virtual orbitals to selectively target the C K-edge. Furthermore, we have considered the Fe K-edge NEXAFS spectrum of ferrocene (iron bis-cyclopentadienyl) and vinylferrocene (ferrocene with a vinyl substituent at one of the cyclopentadienyl rings) to illustrate the applicability of the presented procedure for probing quadrupole intensities of isotropic samples. The structures of ferrocene and vinylferrocene in eclipsed conformation have been taken from Ref. [47], where they were optimized using the BP86 functional and the TZP basis set. The frequency-dependent absorption cross sections, or, in other words, the linear absorption spectra, of the ferrocene derivatives were determined at the B3LYP level within the CPP formalism using the most recent CPP algorithm [48,49] and a lifetime broadening γ=1000 cm -1 (or 0.1240 eV). It is well-known that the energy separation between the pre-edge peaks in substituted ferrocenes is highly sensitive to the choice of functional, and that it is exaggerated by B3LYP. [47,50] We primarily adopt the use of this functional for the ferrocene derivatives to be able to compare our results with those of previous work. The (θ, φ)-angular integration of the linear absorption cross section was handled by a set of highly accurate Lebedev grids. [START_REF] Lebedev | Angular quadrature parameters available from[END_REF]51] All property calculations employed the aug-cc-pVDZ [44,52,53] basis set and were performed with a locally modified version of the Dalton program. [54,[START_REF]A Molecular Electronic Structure Program[END_REF]

Results and Discussion

To investigate the convergence of the numerical scheme presented above for the calculation of rotationally averaged linear absorption cross sections using the full field-matter interaction operator, we consider a set of prototypical transitions in the UV/vis and X-ray spectral regions. Specifically, we consider the lowest n → π * excitations in formaldehyde and acrolein, the π → π * transition in acrolein, C 1s → π * transitions in formaldehyde and acrolein, and mixes of Fe 1s → 3d and 1s → 4p transitions in ferrocene and vinylferrocene (see Fig. 1 for chemical structures and atom labeling used for the 1s → π * transitions in acrolein). These include ED allowed and forbidden transitions in both spectral regions, and this allows us to study how the nature of the electronic transitions, and thus the importance of BED effects, influences the numerical convergence of the adopted quadrature scheme used to determine rotationally averaged oscillator strengths. It is instructive to start by considering the behavior of the χ-averaged linear absorption cross section, or oscillator strength, to be numerically integrated with respect to the direction of the wave vector k over a unit sphere. Note that the averaging over χ means that the information about polarization dependence is lost.

Formaldehyde and acrolein

First, we consider the lowest valence and C K-edge transitions in the two aldehydes, formaldehyde and acrolein. The excitation energies as calculated at the CAM-B3LYP 100% /aug-cc-pVDZ of theory are given in the upper part of Table 1, and the (θ, φ)-dependency of the associated χ-averaged oscillator strength f χ for a representative set of the considered transitions is illustrated in Figs. 2 and3 for acrolein and formaldehyde, respectively. To dissect the impact of going beyond the ED approximation, we include in Fig. 2 (bottom row) also the difference ∆ f χ between the oscillator strengths obtained beyond and within the ED approximation. As a general comment, we note that the χ-averaged oscillator strength is seen to be identical for wave vectors k and -k, which we exploit in the Lebedev quadrature.

As demonstrated in Ref. [START_REF] List | [END_REF], the ED approximation is useful for describing linear absorption of ED allowed excitations across the entire frequency range, and BED corrections are expected to be significant only for ED forbidden transitions. In formaldehyde, the n → π * transition is strictly ED forbidden by symmetrythe excited state spans the A 2 irreducible representation of the C 2v point groupwhereas in acrolein, the corresponding transition is ED allowed, and the extent to which it may be observable in the ED approximation depends sensitively on the local symmetry of the carbonyl group.

There is a high degree of directional dependence in the linear absorption cross section for these planar systems. The π → π * transitions are most effectively induced by an electric field directed along the conjugation axis (the z-direction in Fig. 1), and thus propagating in a direction orthogonal to the conjugation axis, whereas zero absorption is associated with fields propagating along the conjugation axis. This is illustrated in Fig. 2a in terms of the χ-averaged oscillator strength for the π → π * transition in acrolein. The nearly perfect cylindrical symmetry about the z-axis reflects the fact that the main component of the leading ED contribution is invariant to the direction of propagation orthogonal to the conjugation axis (within the xy-plane). The effects of going beyond the ED approximation are shown in Fig. 2b. Note that ∆ f χ is negative and negligible in all directions of light propagation, as expected for BED corrections to ED allowed transitions. [START_REF] List | [END_REF] Similar observations can be made for the ED allowed C K-edge transitions in both aldehydes. This is exemplified in Figs. 2e and2f for the C 1 1s → π * transition in acrolein. In this case, the directional dependence reflects the π * -orbital entirely, and thus the cylindrical symmetry is about the x-axis. The BED effects for the core transitions are two orders of magnitude larger than for the valence π → π * excitation in acrolein, but they are still negligible (0.006-0.07%) for all practical purposes.

For the strictly ED forbidden n → π * transition in formaldehyde, the χ-averaged oscillator strength displays a more complex dependence on the direction of light propagation, see Fig. 3. In this case, the leading contributions stem from the terms involving the magnetic-dipole and electric-quadrupole operators. As a consequence, with formaldehyde placed in the yz-plane and the C=O bond along the z-axis in the molecular frame, the strongest absorption intensity will be due to light propagating in the x-direction with an electric-field polarization along y, and vice versa. Inspection of the individual lowest-order contributions (data not shown) reveals that the magnetic-dipole term is two orders of magnitude larger than the electricquadrupole contribution (see further discussion in Section 4.3). Note that this division into magnetic-dipole and electric-quadrupole contributions is here unambiguous since, in the present case, the vanishing ED oscillator strength implies gauge-origin invariance for these terms. In analogy to the ED term for the π → π * transition in acrolein (Fig. 2a), the main component of the magnetic-dipole contribution (magnetic field along z) is invariant to the propagation direction within the xy-plane. The observed deviation from cylindrical symmetry can be attributed to the electric-quadrupole-magnetic-dipole cross term that, however, vanishes upon rotational averaging over the solid angle. [37] Turning to the n → π * transition in acrolein, the orientational dependence and magnitude of f χ reflects the gain in ED intensity due to the broken symmetry of the carbonyl group. The main contribution to the ED component comes from light polarization along the out-of-plane direction, and a close to cylindrical symmetry about the x-axis is seen in Fig. 2c. But in the view of Fig. 2d, it is clear that the BED corrections display the features of the corresponding ED forbidden transition in formaldehyde (see Fig. 3).

砀 ⴀ 㘀 ⸀ ⸀ 㔀 ⸀ ⸀ 㔀 ⸀ ⸀ ⸀ ⸀ 㔀 ⸀ 㔀 ⸀ 㔀 ⸀ ⸀ ⸀ ⴀ ⸀ 㔀 ⴀ ⸀ 㔀 ⴀ ⸀ 㔀 ⴀ ⸀ ⴀ ⸀ ⴀ ⸀

Ferrocene and vinylferrocene

To demonstrate the behavior of the proposed scheme for X-ray induced transitions that are ED forbidden, we consider the iron near K-edge absorption spectra of eclipsed ferrocene (D 5h symmetry) and vinylferrocene. These examples have already been studied in detail in Refs. [47] and [50], both experimentally and also theoretically using truncated multipole-expanded expressions for the calculations of oscillator strengths. In Fig. 4, we show the isotropically averaged spectra as determined with use of the CPP formalism both within and beyond the ED approximation. First, we note that our theoretical spectra for the ferrocene derivatives are in good agreement with those reported by Bernadotte et al. based on a full second-order expression for the oscillator strength, [50] and this confirms the validity of the proposed procedure to obtain isotropically averaged linear absorption cross sections beyond the ED approximation.

In ferrocene, the pre-edge peak at 6961.9 eV can be assigned to the ED forbidden transitions from the Fe 1s orbital to the degenerate pair of lowest unoccupied orbitals. These virtual orbitals are composed of the antibonding linear combination of Fe 3d-orbitals and the cyclopentadienyl π-orbitals of e 1 symmetry. Such transitions are ED forbidden due to symmetry, and the leading term in the linear absorption cross section will therefore come from the electric-quadrupole and magnetic-dipole contributions, as immediately seen from the absence of this peak in the ED spectrum (Fig. 4b).

By introducing the vinyl substituent, the symmetry, and therefore the degeneracy, is lifted, and the 3d-orbitals are mixed with the π * -orbital of the vinyl group. This leads to two distinct pre-edge peaks found at about 6961.9 and 6965.5 eV, respectively (see Fig. 4a), at the B3LYP/aug-cc-pVDZ level of theory. The final states contributing to these peaks in vinylferrocene are illustrated by means of isodensity surface plots in Fig. 5. At the B3LYP level, they contain contributions from several single-excited determinants in which the Fe 1s-orbital has been replaced by a virtual orbital, and the figure shows the linear combinations of these virtual orbitals. The lowest-energy peak, corresponding to the 1s → 3d transitions discussed above in unsubstituted ferrocene, becomes slightly ED allowed following the vinyl substitution. This can be seen in terms of a very weak feature in the ED spectrum in Fig. 4a (blue dashed curve profile). The second peak originates from a transition from the Fe 1s-orbital to a final state dominated by the π * -orbital of the vinyl substituent, shown in Fig. 5c. However, due to the locality of the Fe 1sorbital, the intensity will be governed by the Fe atomic orbital components of the final state that mainly stem from the 4p z -and 3d xz -orbitals. This 4p-3d mixing in the final state explains the relative ED and BED intensity behavior of the second pre-edge peak, where the ED intensity increases due to the 1s → 4p component of the transition and now exceeds the higher-order contributions arising from the ED forbidden 1s → 3d component, the latter given by the difference between the BED and ED spectra.

The χ-averaged oscillator strength for 1s → 3d transitions shows a stronger dependency on the propagation direction compared to those for the π → π * and n → π * transitions discussed above. This is illustrated in Figs. 6a and6b for the two transitions giving rise to the lowest-energy pre-peak in the K-edge spectrum of vinylferrocene. In these cases, the transition densities are governed primarily by a combination of the very compact Fe 1s-orbital and the valence 3d xz -and 3d yzorbitals (Figs. 5a and5b, respectively). A large transition moment and thereby oscillator strength is thus obtained for a situation with an electric-field polarization along the z-direction and propagation along either the xor y-directions in the molecular frame, or vice versa. The effect of the vinyl substituent is small for the first pre-edge peak and xand y-directions are therefore close to identical in this respect. This explains the near-cylindrical symmetry of the summed oscillator strength for the two transitions shown in Fig. 6c. In view of the more complex nodal structure of f χ for the 1s → 3d transitions as compared to the corresponding behaviors of the valence and core excitations in formaldehyde and acrolein (Figs. 2 and3), one can anticipate that the numerical integration over angles (θ,φ) will be more difficult to converge. This aspect will be discussed in the subsequent section.

Convergence rate and computational cost

Having considered the behavior of the χ-averaged linear absorption cross section, we proceed to investigate the rate of convergence in the numerical integration scheme and compare the computational cost with the corresponding conventional calculations based on truncated multipole expansions.

The isotropically averaged oscillator strengths involving the full field-matter interaction operator for the valence and core transitions in the two aldehydes are listed in Table 1 for progressively larger Lebedev grids LDn, where n = {6, 14, 26, 38, 50} denotes the number of quadrature points. For comparison, we also include the corresponding oscillator strengths calculated within the ED approximation, and these results have been obtained by means of analytical integration, or, equivalently, tensor averaging.

The number of quadrature points needed when going beyond the ED approximation to integrate the absorption cross section to a certain degree of accuracy is closely associated with the importance of higher-order terms in the wave vector for a given excitation. In general, the higher the order in the expansion that is relevant for the spectrum, the larger the number of sampled orientations required to obtain sufficiently accurate absorption cross sections. This is readily understood from the fact that the Lebedev grids are constructed to exactly integrate real spherical harmonics up to a given maximum angular momentum L max as given in Table 1. [START_REF] Lebedev | Angular quadrature parameters available from[END_REF] For the rotational averaging of the linear absorption cross section, we have to treat a product of two complex conjugate transition moments involving the full field-matter interaction operator. In general, an m'th rank transition multipole moment T

[m] λ1...λm in the multipole expansion of the transition moment is expressible as a sum of irreducible tensors of integer orders l = 0, 1, . . . m which transform under rotations as spherical harmonics with angular momenta l. The product of two transition multipole moments of rank m 1 and m 2 , as relevant for the linear absorption cross section, will produce a set of irreducible tensors with orders L that lie between |m 1 -m 2 | and m 1 + m 2 . Accordingly, the smallest LD6 grid (L max = 3) is sufficient to fully integrate the pure dipole contributions (L = 2), and the LD14 scheme (L max = 5) treats fully the linear absorption through second order in the wave vector.

In comparing results obtained with different number of quadrature points, we can thus identify the character of the contributions to transition intensities in the language of multipole expansions. At the level of LD6, we retrieve in addition to the electric-dipole-electric-dipole (µ 2 ) term also the magnetic-dipolemagnetic-dipole (m 2 ) term, which transforms under rotation in the same way. The electric-dipole-magnetic-dipole (µm) and electric-dipole-electric-quadrupole (µQ) cross terms vanish for absorption of linearly polarized light. [37] At the level of LD14, we retrieve, in addition to the contributions mentioned above, also the electric-quadrupole-electric-quadrupole (Q 2 ) term as well as the electric-dipoleelectric-octupole (µO) and electric-dipole-magnetic-quadrupole cross terms (µM), which, just like the m 2 term, is of quadratic order in the wave vector expansion and therefore expected to provide a contribution to the linear absorption cross section that is of similar size.

As seen in Table 1 for formaldehyde and acrolein, the Lebedev quadrature used to rotationally average the oscillator strength converges in general very rapidly, and already at the LD14 level errors are, in all considered cases, within the adopted convergence threshold of 10 -9 for the excitation vectors. The LD6-ED difference in averaged oscillator strengths amounts to 1.2 × 10 -6 and 2.0 × 10 -6 for the n → π * transitions in acrolein and formaldehyde, respectively. These estimates of m 2 contributions are some three orders of magnitude larger than the Q 2 contributions (including also µO and µM for acrolein) as estimated from the corresponding LD14-LD6 differences. This is in stark contrast to the 1s → 3d transitions in the ferrocene derivatives, where the m 2 and Q 2 contributions are of similar size, as illustrated in the inset of Fig. 4b.

Since the inclusion of terms through second order in the wave vector has proven here and elsewhere [START_REF] List | [END_REF]37] to provide accurate results for the calculation of linear absorption spectra in both the UV/vis and X-ray regions of the spectrum, it may be natural to question the sense in turning to a treatment of the full field-matter interaction operator from a computational cost point of view. In the CPP framework, the inclusion of the terms through second order in the wave vector to the complex polarizability in a multipole formulation means that one needs to solve 3 + 3 + 6 = 12 linear response equations for a given optical frequency, corresponding to the independent components of the multipole moment operators (µ, m and Q). In the BED framework, on the other hand, the adoption of the LD14 quadrature scheme means that one needs to solve 2 × 7 = 14 linear response equations for a given optical frequency, corresponding to the seven independent wave vector points and the two components of the linear momentum operator (p x and p y ). So the conclusion is that, not only does the LD14 scheme provide a highly accurate level of numerical integration of the BED absorption cross section (with, amongst others, its virtue of gauge-origin independence) but it is on par with a conventional multipole expansion scheme in terms of computational efficiency.

Conclusion

We have presented a scheme to perform rotational averaging of linear absorption cross sections involving the full field-matter interaction operator. Our approach is based on a combination of numerical and analytical integration over the three Euler angles. The former is performed with use of Lebedev quadratures over the solid angle and corresponds to integration over all possible directions of the wave vector of the electromagnetic field, and the latter to averaging over the associated polarization vector. The computational cost to accomplish this achievement is not significantly higher than what is found in response theory calculations based on conventional multipole expansions.

This work ensures the calculation of gauge-origin independent and positive linear absorption cross sections in isotropic samples, also beyond the ED approximation. This is particularly important in cases where the magnetic-dipole and electricquadrupole contributions to the cross section are larger than or comparable to the ED contribution, such as in n → π * transitions in organic compounds and K-edge spectra of transition metal complexes. -2.17 × 10 -5

e

  -ik•ri ( • pi )|n .

Figure 1 .

 1 Figure 1. Chemical structures of the molecules considered: (a) formaldehyde; (b) acrolein; eclipsed conformation of (c) ferrocene and (d) vinylferrocene. The orientation of the molecules within the molecule-fixed coordinate system is also shown.

Figure 2 .

 2 Figure 2. (θ, φ)-dependency of the χ-averaged oscillator strength f BED χ (top row) and the corresponding differences ∆ f χ between BED and ED (bottom row) for selected transitions in acrolein: (a,b) the lowest π → π * transition (c,d) the lowest n → π * transition and (e,f) the C 1 1s → π * transition. The orientation of acrolein in the molecule-fixed frame as well as the atom labeling is given in Fig. 1.

Figure 3 .

 3 Figure 3. (θ, φ)-dependency of the χ-averaged oscillator strength f BED χ for the lowest n → π * transition in formaldehyde. The orientation of formaldehyde in the molecule-fixed coordinate system is given in Fig. 1.

  , χ [a.u.]

Figure 4 .Figure 5 .

 45 Figure 4. The low-energy part of the isotropically averaged Fe K-edge spectra of (a) vinylferrocene and (b) ferrocene from CPP calculations at the B3LYP/aug-cc-pVDZ level of theory as obtained within and beyond the ED approximation, using the LD14 scheme to compute the isotropic average in the latter case. Inset figure shows the convergence of the pre-edge peak of ferrocene with respect to the size of the Lebedev grid (see discussion in Section 4.3). A common lifetime broadening γ = 1000 cm -1 has been used.

Figure 6 .

 6 Figure 6. (θ, φ)-dependency of the χ-averaged oscillator strength f BED χ for the two transitions underlying the lowest-energy pre-edge peak in the Fe K-edge spectrum of vinylferrocene: (a) the 1s → 3dxz transition, (b) 1s → 3dyz transition and (c) the summed contribution from (a) and (b). The orientation of vinylferrocene in the molecular coordinate system is given in Fig. 1.

Table 1 .

 1 Isotropically averaged oscillator strengths for the lowest valence and C K-edge transitions in acrolein and formaldehyde within and beyond the ED approximation as given for different Lebedev grids (LDn, n = {6, 14, 26, 38, 50}), together with the difference ∆f . Lmax indicates the highest angular momentum spherical harmonic functions that a given Lebedev grid integrates exactly. NLD denotes the number of symmetry-independent Lebedev quadrature points, specifying the propagation direction. The atom labeling used for acrolein is given in Fig.1.

	acrolein

Acknowledgements

It is our great pleasure to dedicate this paper to the 60th anniversary of Prof. Hans Jørgen Aa. Jensen. His outstanding scientific contributions in general and his long-term commitment to the Dalton and DIRAC projects in particular are both of great value to us all. We are very grateful for his sharing of knowledge and scientific guidance as well as being a dear friend.

The authors thank Prof. Christoph Jacob for kindly providing the structures of ferrocene and vinylferrocene. N. H. L. acknowledges the Carlsberg Foundation for a postdoctoral fellowship. P. N. acknowledges financial support from Knut and Alice Wallenberg Foundation (Grant No. KAW-2013.0020), and the Swedish Research Council (Grant No. 621-2014-4646). Computational resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputer Centre (NSC), Sweden, are acknowledged.