Christine Paillard
email: christine.paillard@univ-brest.fr

Yannick Gueguen
email: ygueguen@ifremer.fr

Mathias Wegner

David Bass

Alberto Pallavicini

Luigi Vezzulli

Isabelle Arzul
email: isabelle.arzul@ifremer.fr

Recent advances in bivalve-microbiota interactions for disease prevention in aquaculture

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

Introduction

Marine bivalves are ecological keystone species that fulfill essential roles in coastal ecosystems by providing habitat, mediating bentho-pelagic energy fluxes, and maintaining water quality. Their benthic filter-feeding lifestyle also makes them useful sentinels to detect pollution [1**]. Bivalves are also economically highly important aquaculture species, with a world aquaculture production of molluscs for human consumption reaching 17.5 million tons in 2018 [2]. Aquaculture production and natural populations are strongly impacted by environmental fluctuations of biotic and abiotic factors. In recent years, infectious diseases have emerged as the major limiting factors for bivalve production, but very few tools are available to prevent and treat infections [3*]. Techniques developed for the production of other ani mals cannot be directly applied to bivalve aquaculture, since the lack of antibody-mediated immune memory prevents individual vaccination approaches, and culture in the open sea prevents application of large-scale disease treatments.

In marine ecosystems, including open sea aquaculture, bivalves are exposed and colonized by a plethora of microorganisms [1**] making host-microorganisms interactions a prime target for understanding bivalve health [3* ,4-6]. Microorganisms can fulfill beneficial functions for the host and vice versa, and the holobiont concept captures these relationships by merging hosts and their associated microbiota into a single evolutionary unit of selection [7]. It suggests that microbiota composition provides benefits to the hosts for survival, but also homeostasis and development. For example, studies of resident microorganisms in the haemolymph of healthy bivalves suggest that the host-microbiome could limit settlement of pathogenic strains and play a role in preventing dysbiosis and aiding health recovery [8**,9].

Facilitated by the advances in high-throughput sequencing technologies, interactions between host immune responses and their associated microbiota are increasingly investigated within the context of disease [10]. Here, we review not only the methodologies used to characterize bivalve microbiota, but also the factors that structure microbiomes associated with bivalves under aquaculture conditions and highlight 1) their composition and host specificity, 2) interactions between environment and host-associated microbiomes, and 3) how these interactions may affect the ability of bivalves to resist pathogens. This review highlights critical gaps in our understanding of microbiome function, and identifies research priorities leading to innovative management practices of molluscan aquaculture in the future.

Current methodologies

Typically, microbial communities associated with bivalves of aquaculture interest have been taxonomically characterized and analysed by massive amplicon sequencing of genes encoding small subunit ribosomal RNA gene regions (Figure 1) [1**,11*,12**,13*]. While microbiota are often characterized by 16S sequencing of bacteria, the eukaryotic genetic diversity is rarely investigated [14,15**]. A more comprehensive approach than ribosomal amplicon sequencing is shotgun metagenomics, which offers the advantage to target the whole microbial gene pool within a sample with the possibility to run functional analysis as well as assembling genomes. Shotgun metagenome analysis applied to the study of Pacific oyster microbiome recently revealed that oyster hemolymph is a complex ecosystem containing diverse bacteria, protists and viruses [15**]. However, the bioinformatic processing of metagenome data is not yet standardized and the application ofshotgun metagenomicsfor the analysis of the bivalve microbiota is still challenging [15**].

An important limitation of metagenomic or eukaryotic metabarcoding studies is co-sequencing of host DNA that largely masks target DNA from host-associated microbial communities [16]. Ultradeep sequencing or target sequencing implying removal of non-target DNA from samples before sequencing significantly overcomes this problem. In particular, hybridization-based capture with target-specific biotinylated probes is a versatile and cost effective technique for the sequencing of both viruses and bacteria in complex samples [17]. Vezzulli et al. [18] demonstrates that by using such an approach, ultra-low abundance Vibrio cholerae genomes, including specific virulence factors, could be retrieved from raw river water approximately 2500 times more effectively than shotgun sequencing alone. By applying the same methodology, 884 phylogenetic and virulence markers of the potential microbial pathogenic community were targeted in oyster tissue allowing high taxonomic resolution analysis of the bivalve pathobiota [19*]. For studying host-associated micro-eukaryotes, metabarcoding primers are available that reduce co-amplification of metazoan host sequences, enabling more thorough sequencing of micro-eukaryotic symbionts and parasites [14,20].

DNA sequencing technologies are rapidly evolving and are expected to contribute to new breakthroughs in the study of the bivalve microbiota. Latest technologies such as long-read targeted sequencing [21] or new ultra-deep sequencing platforms (e.g. NovaSeq 6000) are certainly paving the way to a more comprehensive analysis and understanding of the bivalve holobiont. Genotype associated patterns can however vary within individuals according to developmental stage or tissues [START_REF] Pimentel | Microbiome analysis reveals diversity and function of Mollicutes associated with the eastern oyster, Crassostrea virginica[END_REF]. Interestingly, species richness has been shown to decrease as larvae develop suggesting a shift towards a more specific microbiome [25*]. This trend also continued between later stages (postlarvae and adults), with adults showing the lowest diversity [START_REF] Fernàndez | Changes in the composition and diversity of the bacterial microbiota associated with oysters (Crassostrea corteziensis, Crassostrea gigas and Crassostrea sikamea) during commercial production[END_REF]. In later developmental stages, tissue type becomes a strong determinant of bacterial community composition [23,[START_REF] King | Regional and oyster microenvironmental scale heterogeneity in the Pacific oyster bacterial community[END_REF]. In particular, microbiota vary between internal organs such as gut and those that are in closer contact to the environment (e.g. hemolymph or gills). This pattern, consistent across individuals of different species [START_REF] Musella | Tissue-scale microbiota of the Mediterranean mussel (Mytilus galloprovincialis) and its relationship with the environment[END_REF], might suggest a role of microbiota in different physiological processes specific to the investigated organ.

Host factors shaping bivalve microbiota

On all levels, the effects of host-associated factors on compositional microbiome diversity are strongly influenced by environmental variation. Environmental disturbance for example can homogenize inter-individual differences [8**] or deplete genetic signals [24**]. In many cases, for example, on the population level, environmental and host genetic effects will be confounded, therefore, it is imperative to consider the interaction of host-specific and environmental effects when describing host-microbiota interactions.

Environmental factors modifying bivalve microbiota

The influence of environmental factors on bivalve microbiota can be explained by the benthic lifestyle of bivalves and its heterogeneity at small spatial scales (ecological niches, sessile versus freeliving life stages), and by the spatial and temporal variation of abiotic and biotic factors (season, transplantation, pollution, pathogens, water quality, and so on). Microbiota-environment interactions have repeatedly been studied by monitoring bivalves in the field, either over short or long periods. Specific diversity patterns suggest that bivalves select the microorganisms from the water column or sediment, depending on their microbiota compartment. This microbiota selection process is modified with the time of exposure for intertidal bivalves [13*], but also with the seasons [11**]. The influence of stochastic processes may also be a factor for microbiota diversity [1**]. The transfer of bivalves from hatcheries to rearing sites plays an essential role in shaping the microbiome from juveniles to adults [START_REF] Trabal | Molecular analysis of bacterial microbiota associated with oysters (Crassostrea gigas and Crassostrea corteziensis) in different growth phases at two cultivation sites[END_REF].

The initial microbiome (bacteria, protist and virus) is deeply modified by transplantation (strongly for transient microbiome, and lessfor resident microbiota such asthose from digestive gland), according to the environmental conditions of the new rearing sites [12**,15**,27*,34*].

Altogether, results indicate that bivalve microbiota is at least partly originated from the marine microbiota where these organisms are living. Water pollution could also impact bivalve microbiota. Titanium dioxide nanoparticles (nTiO2) -very widely used in a diverse range of products -and PS-NH2 nanoplastic have been shown to affect the Mytilus galloprovincialis microbiota [START_REF] Auguste | Exposure to TiO 2 nanoparticles induces shifts in the microbiota composition of Mytilus galloprovincialis hemolymph[END_REF][START_REF] Auguste | Impact of nanoplastics on hemolymph immune parameters and microbiota composition in Mytilus galloprovincialis[END_REF]. Similarly, oysters exposed to atrazine, a prevalent herbicide in agricultural practice, result in a significant loss of key mutualistic microbial species and a subsequent colonization of pathogenic bacteria [START_REF] Britt | The effects of atrazine on the microbiome of the eastern oyster: Crassostrea virginica[END_REF]. Exposure to estrogenic chemicals such as 17β-estradiol, Bisphenol A and Bisphenol F is also leading to changes in the microbiota composition of first larval stages of M. galloprovincialis [START_REF] Balbi | Insight into the microbial communities associated with first larval stages of Mytilus galloprovincialis: possible interference by estrogenic compounds[END_REF].

Water temperature seems to be a major driver and strongly influences the composition of oyster microbiota both in the haemolymph [8**,33] and other tissues [19*], and only a small fraction of the microbial species persist as a 'core' community throughout the different seasons.

Accordingly, climate change was observed to alter the microbiome of the Sydney rock oyster S. glomerata [START_REF] Scanes | Climate change alters the haemolymph microbiome of oysters[END_REF]. Warmer temperatures are also favoring the proliferation of potential pathogenic bacteria (e.g. Vibrio) within bivalve microbiota, thus increasing susceptibility to microbial diseases [START_REF] De Lorgeril | Inefficient immune response is associated with microbial permissiveness in juvenile oysters affected by mass mortalities on field[END_REF][START_REF] Destoumieux-Garzon | Vibrio-bivalve interactions in health and disease[END_REF].

Microbiota and bivalve health

Microbiota are hypothesized to influence disease susceptibility [1**,7]. However, in bivalves few studies have explicitly described the role of these host-microbial interactions in buffering the host against fluctuating conditions (enantiostasis). In particular, investigations into the roles of microbiota in host health and disease have so far mostly focused on oysters (Crassostrea and Saccostrea species) from a solely bacterial perspective. However, it is clear that the bivalve symbiome also includes viruses and microeukaryotes, the identities and roles of which are only just starting to be investigated [15**].

Oyster bacterial microbiota are dynamic and were shown to be modified in response to multiple stressors, including pathogens [8**,15**,24**,27*,34*, [START_REF] Green | Bacterial diversity of the digestive gland of Sydney rock oysters, Saccostrea glomerata infected with the paramyxean parasite, Marteilia sydneyi[END_REF][START_REF] De Lorgeril | Immune-suppression by OsHV-1 viral infection causes fatal bacteraemia in Pacific oysters[END_REF][START_REF] King | Characterisation of the Pacific oyster microbiome during a summer mortality event[END_REF][START_REF] Stevick | Bacterial community dynamics in an oyster hatchery in response to probiotic treatment[END_REF]. Recently, [START_REF] De Lorgeril | Immune-suppression by OsHV-1 viral infection causes fatal bacteraemia in Pacific oysters[END_REF] demonstrated that Pacific Oyster Mortality Syndrome affecting C. gigas oyster throughout the world is associated with a disruption of the host microbiota. This study showed that infection by an Ostreid herpesvirus compromises the oyster antibacterial defence and is followed by a microbiota shift leading to bacteremia and oyster death [START_REF] De Lorgeril | Immune-suppression by OsHV-1 viral infection causes fatal bacteraemia in Pacific oysters[END_REF]. Field analysis performed on C. gigas also revealed that oysters experiencing mortality outbreaks at different life stages displayed signs of dysbiosis [19*,46]. Nevertheless, it is often unclear whether an altered microbiota is a cause or result of the impact of invading pathogen(s). Examples are known of both situations, and it is important to consider both options in each case. Experimental investigations and temporal sampling are required to better understand causal relationships [START_REF] Bass | The pathobiome in animal and plant diseases[END_REF].

Recent studies highlight the potential involvement of oyster microbiota in disease resistance. They showed that the microbiomes of disease-susceptible oyster families were significantly different from the microbiomes of disease-resistant families [48,49*]. The differences observed suggested that both the structure and the composition of the microbiota might predict oyster mortalities [49*]. Interestingly, the microbiota of oysters originating from the same hatchery picked up genotypespecific signals of their different rearing environment in the field and this was correlated with different outcomesto Ostreid herpesvirus infection [34*]. Bacterial taxa explaining genotype-specific differences offer the opportunity to identify putative microbial determinants of disease resistance. Accordingly, a row of key bacterial taxa were identified, which may provide a protective or detrimental role in disease outbreaks of oysters (Table 1) [45,48,49*].

The diversity and roles of microeukaryotic symbionts are both almost certainly much greater than currently appreciated. It is increasingly recognized that bivalves are host to a diversity of protistan parasites that have an indeterminate effect on the host. Some of these can occur at relatively high prevalence without a clearly discernible effect on the populations (e.g. Minchinia mytili in Mytilus edulis; [50], and another currently undescribed Minchinia species in Cerastoderma edule; [START_REF] Lynch | Detection of haplosporidian protistan parasites supports an increase to their known diversity, geographic range and bivalve host specificity[END_REF]). The effects of these parasites may become more apparent under conditions of environmental or other physiological stress on their hosts. Further research is now required to elucidate the dynamic reciprocity between the whole microbiota and host in the context of pathogenesis, and host susceptibility and resistance to disease.

Conclusion

Over the last few years many studies have investigated the bacterial microbiome of a wide range of bivalves, in both healthy and diseased states (Figure 1). These have shown that the structure and the composition of microbiota is influenced by the ecology and geography of the sampling site, season, host life stage, and health status among other factors (Figure 2). Studies of the microbiomes of farmed bivalves to date have focused on oysters with or without associated mortality events. It will thus be important to complement these with new studies underway on other bivalves affected by chronic diseases including microparasites (e.g. Perkinsosis and Brown R Disease in Manila clams, Juvenile Oyster Disease and Perkinsosis in C. virginica). Integrating the dynamics of the whole symbiotabacteria, viruses, and eukaryotes -and its interaction with the host and surrounding environment, should be the ultimate aim to achieve a comprehensive understanding of bivalve physiological states, including health [START_REF] Bass | The pathobiome in animal and plant diseases[END_REF].

Research on microbiome modulator drugs (prebiotics, probiotics, synbiotics) for the health of cultured bivalves should be developed as its value has been demonstrated for other invertebrates such as shrimp [START_REF] Rajeev | Healthy microbiome : a key to successful and sustainable shrimp aquaculture[END_REF]. So far, our understanding of bivalve microbiota has focused on its structural aspects, but the large diversity and variability of these microbiota have precluded the identification of probiotic taxa providing protection against disease. Alternative approaches using Bacillus pumilus RI06-95 as a probiotic in oyster hatcheries has shown some evidence of decrease of pathogenic microorganisms [START_REF] Stevick | Bacterial community dynamics in an oyster hatchery in response to probiotic treatment[END_REF]. Several studies on bivalves -and also on other invertebrates and vertebrates -suggest that the gut compartment harbors a more stable microbiota, including symbionts, and could be a good lead for the identification of a potential microbiota profile. Manipulation of the gut microbiome could also improve immunity and health of bivalves as has been done for shrimps [START_REF] Rajeev | Healthy microbiome : a key to successful and sustainable shrimp aquaculture[END_REF]. Probiotics interact with the immune system and prevent pathogen colonization in multiple ways [START_REF] Knipe | Probiotics and competitive exclusion of pathogens in shrimp aquaculture[END_REF]. They can directly compete with pathogens for the same niche (competition for nutrients or production of antimicrobial molecules) and also stimulate the host immune system. These contributions to bivalve defenses are still poorly understood and could be developed as one of the additional strategies to combat disease in bivalve aquaculture. Phage therapy using specific or cocktails of bacteriophages, could be also used to limit pathogen colonization [3*].

Complementary approaches (including various 'omics, microscopy techniques, (epi)genetics, and immunology) integrating microbiome studies with host and environmental factors will allow us to better understand the dynamics of microbiota across a diversity of environments and hosts and should help us to identify, for example through network interaction studies, taxa predictive of survival as well as mortality events [START_REF] Pita | The sponge holobiont in a changing ocean: from microbentitle>[END_REF][START_REF] Timmins-Schiffman | Coupled microbiome analyses highlights relative functional roles of bacteria in a bivalve hatchery[END_REF]. The more comprehensive understanding that is resulting from these integrated approaches will enable us to better manage bivalve aquaculture, and mitigate problems. Pathobiotic systems can be reversed within certain boundaries, and host health status is a result not just of individual pathogens but a greater diversity of influences, of which we have increasing understanding and ability to manipulate. Collaboration between producers, farm managers, and scientists from diverse disciplines (e.g. microbial ecology, pathology, environmental science, modeling, chemistry, genetics, immunology, and so on) will be key to this success. 12** Nguyen VK, King WL, Siboni N, Mahbub KR, Dove M, O'Connor W, Seymour JR, Labbate M: The Sydney rock oyster microbiota is influenced by location, season and genetics. Aquaculture 2020:735472 http://dx.doi.org/10.1016/j.aquaculture.2020.735472 Nguyen et al. demonstrated that multifactorial factors (environmental and genetic links to disease resistance) act in synergy to shape the microbiota of the benthic bivalves. This study showed the strong impact of the location and season, but also the role of the composition of the microbiota to prevent QX-disease development in Saccrostrea glomerata. syndrome in hatchery and transplanted them into a natural environment either before or during a disease outbreak. Using metabarcoding and shotgun metagenomics, they showed that oyster hemolymph is a complex ecosystem containing diverse bacteria, protists and viruses, whose composition and dynamics are primarily determined by the environment. However, all of these are also shaped by oyster genetic backgrounds, indicating they indeed interact with the oyster host and are therefore not only of transient character. Co 28* Pierce ML, Ward JE: Gut microbiomes of the eastern oyster (Crassostrea virginica) and the blue mussel (Mytilus edulis): temporal variation and the influence of marine aggregate associated microbial communities. mSphere 2019, 4:e00730-19 http://dx.doi.org/10.1128/mSphere.00730-19

The microbial communities of the aggregates influence the composition of the gut core microbiota of these two bivalves, even though the composition of the core microbiota is very different from that of the aggregates. The gut core microbiota of Crassostrea virginica and Mytilus edulis show strong similarities, linked potentially to their common suspension feeding mode.

Figure 2

Host and environmental factors acting on bivalve microbiota. Some of these factors, alone or in combination, have been shown to lead to dysbiosis, diseases and recovery.

Table 1

Bacterial microbiota taxa associated with a better resistance of Crassostrea gigas oyster during disease outbreaks or experimental challenge.

4

  Dittmann KK, Rasmussen BB, Castex M, Gram L, Bentzon-Tilia M: The aquaculture microbiome at the centre of business creation. Microb Biotechnol 2017, 10:1279-1282. 5 Apprill A: Marine animal microbiomes: toward understanding host-microbiome interactions in a changing ocean. Front Mar Sci 2017, 4 http://dx.doi.org/10.3389/fmars.2017.00222. 6 King WL, Jenkins C, Seymour JR, Labbate M: Oyster disease in a changing environment: decrypting the link between pathogen, microbiome and environment. Mar Environ Res 2019, 143:124-140. 7 Rosenberg E, Zilber-Rosenberg I: The hologenome concept of evolution after 10 years. Microbiome 2018, 6:78 http://dx.doi. org/10.1186/s40168-018-0457-9. 8** Lokmer A, Wegner MK: Hemolymph microbiome of Pacific oysters in response to temperature, temperature stress and infection. ISME J 2015, 9:670-682. Lokmer and Wegner used an experimental setup to investigate the interaction of temperature, temperature stress (shifts between temperature treatments) and infection on the composition of the haemolymph microbiome. By repeatedly sampling the same individual they could show that higher temperatures lead to a more dynamic microbiome composition with acclimated animals showing distinct personalized microbiota. Temperature stress homogenized variation between individuals. Bacterial infection mainly reduced the microbiome diversity in sick animals with diseased animals showing reduced diversity giving rise to opportunists that dominating the hemolymph communities. 'Healthy' microbiomes seem to prevent dysbiosis, but environmental factors reduce microbiome stability. 9 Desriac F, Le Chevalier P, Brillet B, Leguerinel I, Thuillier B, Paillard C, Fleury Y: Exploring the hologenome concept in marine bivalvia: haemolymph microbiota as a pertinent source of probiotics for aquaculture. FEMS Microbiol Lett 2014, 350:107-116. 10 Lozupone CA: Unraveling interactions between the microbiome and the host immune system to decipher mechanisms of disease. mSystems 2018, 3:e00183-17 http://dx. doi.org/10.1128/mSystems.00183-17. 11* Neu AT, Hughes IV, Allen EE, Roy K: Decade-scale stability and change in a marine bivalve microbiome. Mol Ecol 2021, 30:1237-1250 Neu et al. quantified the diversity and composition of Donax gouldii microbiome over a decadal timescale. Despite increases in water temperature (+1.6°C due to seasonal change) and chlorophyll a concentration (more than ninefold) the diversity of microbial taxa did not change. However this stability was underlain by substantial turnover in the composition of the microbiome, with only six amplicon sequence variants (ASVs) persisting over the sampling period.

  13* Offret C, Paulino S, Gauthier O, Château K, Bidault A, Corporeau C, Miner P, Petton B, Pernet F, Fabioux C et al.: The marine intertidal zone shapes oyster and clam digestive bacterial microbiota. FEMS Microbiol Ecol 2020, 96 http://dx.doi. org/10.1093/femsec/fiaa078 PMID: 32353873 The digestive gland (DG) microbiota of two bivalves the pacific oyster -Crassostrea gigas and the manila clam Ruditapes philippinarum, is highly influenced by the site of implantation. The structuration of DG microbiota is in accordance with their ecological niche. The intertidal imprint is only observed in the clam DG microbiota (non-depurated and depurated), suggesting the existence of autochthonous bacteria in the clam DG microbiota. 14 Clerissi C, Guillou L, Escoubas J-M, Toulza E: Unveiling protist diversity associated with the Pacific oyster Crassostrea gigas using blocking and excluding primers. BMC Microbiol 2020, 20:193 http://dx.doi.org/10.1186/s12866-020-01860-1. 15** Dupont S, Lokmer A, Corre E, Auguet J-C, Petton B, Toulza E, Montagnani C, Tanguy G, Pecqueur D, Salmeron C et al.: Oyster hemolymph is a complex and dynamic ecosystem hosting bacteria, protists and viruses. Anim Microbiome 2020, 2:12 http://dx.doi.org/10.1186/s42523-020-00032-w Dupont et al. reared five oyster families differing in susceptibility to the Pacific oyster mortality

26

  King GM, Judd C, Kuske CR, Smith C: Analysis of Stomach and gut microbiomes of the eastern oyster (Crassostrea virginica) from Coastal Louisiana, USA. PLoS One 2012, 7:e51475 http:// dx.doi.org/10.1371/journal.pone.0051475. 27* Lokmer A, Goedknegt MA, Thieltges DW, Fiorentino D, Kuenzel S, Baines JF, Wegner KM: Spatial and temporal dynamics of Pacific oyster Hemolymph microbiota across multiple scales. Front Microbiol 2016, 7:1367 Lokmer et al. disturbed oyster microbiota by antibiotics and used a transplantation experiment to show that the interaction of an established microbiome can lead to increased mortality in transplanted animals. Vibrio colonization of solid tissues was also more frequent in transplanted animals further suggesting that change of microbiome composition by transplantation reduces the ability of oysters to maintain microbiome stability and prevent disease.

Figure 1

 1 Figure 1 Articles published since 2006 reporting the microbial community in bivalve molluscs. Source Scopus May 2021, Query: 16S AND bivalve AND (microbiota OR microbiome OR community OR metabarcoding), output has been checked, removing non-relevant articles.

  2 FAO Yearbook: Fishery and Aquaculture Statistics 2018/FAO Annuaire. Statistiques des Pêches et de l'Aquaculture 2018/FAO Anuario. Rome: Estadisticas de pesca y acuicultura; 2018 http:// dx.doi.org/10.4060/cb1213t. 3* Pérez-Sànchez T, Mora-Sànchez B, Balcàzar JL: Biological approaches for disease control in aquaculture: advantages, limitations and challenges. Trends Microbiol 2018, 26:896-903 Infectious diseases have become a prominent limiting factor for aquaculture. In addition, growing global concerns about antibiotic resistance have prompted the search for environmentally friendly approaches. In this context, Perez-Sanchez et al. offer an update on the successes and challenges of biological approaches for bacterial disease prevention and control in aquaculture. Examples of biological approaches include the use of probiotics, prebiotics, synbiotics, postbiotics or quorum sensing interference.

  -occurrence network analyses suggest a disruption of the microbial network after transplantation into natural environment during both non-infectious and infectious periods. Dynamics of the Pacific oyster pathobiota during mortality episodes in Europe assessed by 16S rRNA gene profiling and a new target enrichment next-generation sequencing strategy. Environ Microbiol 2019, 21:4548-4562 In this study comparative analysis of contrasting C. gigas samples collected during recurrent mortality episodes at different European sites was conducted using 16S rRNA metabarcoding and target enrichment metagenomics. This represents, to date, the largest study investigating the structure and dynamics of farmed oysters' microbiota. Comparative 16SrDNA gene-based microbiota profiles of the pacific oyster (Crassostrea gigas) and the mediterranean mussel (Mytilus galloprovincialis) from a shellfish farm (Ligurian Sea, Italy). Microb Ecol 2018, 75:495-504 Vezulli et al., carried out a comparative analysis of haemolymph and digestive gland microbiota in C. gigas and M. galloprovincialis co-cultivated at the same aquaculture site using 16SrDNA pyrosequencing. It was found that Vibrio, including potential pathogenic species, accounted for a larger fraction of the microbiota in C. gigas compared to M. galloprovincialis. In addition, microbiome analyses performed on bivalve specimens subjected to commercial depuration highlighted the ineffectiveness of such practice in removing Vibrio species from bivalve tissues. 23 Lokmer A, Kuenzel S, Baines JF, Wegner KM: The role of tissue specific microbiota in initial establishment success of Pacific oysters. Environ Microbiol 2016, 18:970-987. 24** Wegner KM, Volkenborn N, Peter H, Eiler A: Disturbance induced decoupling between host genetics and composition of the associated microbiome. BMC Microbiol 2013, 13:252 http://dx. doi.org/10.1186/1471-2180-13-252 Wegner et al. investigated the microbiome composition of oysters from genetically differentiated beds, and could show that microbiome differentiation correlated to genetic differentiation of the host, suggesting an important role of host genetics on microbiome assembly. This genotypespecific composition was however sensitive to environmental disturbance since the association fell apart after heat shock treatment, highlighting the importance not to consider environmental and genetic effect in isolation. 25* Arfken A, Song B, Allen SK, Carnegie RB: Comparing larval microbiomes of the eastern oyster (Crassostrea virginica) raised in different hatcheries. Aquaculture 2021, 531 http://dx. doi.org/10.1016/j.aquaculture.2020.735955 Arfken et al. showed that oyster larval microbiomes were significantly different from water microbiomes with significant influence of hatchery types and spawning seasons. Obtained results highlight the importance of environmental conditions on microbiome development of oyster larvae associated with hatchery success of larval cultivation.

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Acknowledgements

This work was supported by the European Project H2020 VIVALDI and has received funding from the European Union H2020 research and innovation program under grant agreement no. 678589. It was also supported by ISBLUE project, Interdisciplinary Graduate School for the blue planet (ANR-17-EURE-0015) and co-funded by a grant from the French Government under the program 'Investissements d'Avenir'.

Conflict of interest statement

Nothing declared


Marc-Antoine De Pastre
email: marc-antoine.de_pastre@ens-paris-saclay.fr

Yann Quinsat

Claire Lartigue

Joint Special Interest Group meeting between euspen and

Keywords: Computed tomography, virtual volume correlation, modal decomposition, shape defect identification

As one of the main additive manufacturing (AM) advantages, lattice structures are being studied in many applications such as vibration attenuation, weight reduction of components or optimised heat exchangers. However, lattice structures are challenging to produce, and may present some shape defects. Although significant works have been performed in lattice structure defect observations such as overhanging features or resulting porosity, there has been relatively less research in modelling shape defects by defining a geometric description approach. In this paper, a Virtual Volume Correlation (V2C) method is proposed in order to identify metal laser powder bed fusion (LPBF) BCCz struts shape defect directly from volumetric data obtained by X-ray computed tomography (XCT). In the proposed V2C method, a correlation score is calculated between the volumetric data and a virtual volume. This virtual volume is determined according to the computer-aided-design (CAD) model and a shape defect which is defined using a linear decomposition relying on a user-defined defect basis. Shape defects of the studied part are successively, according to a Newton Raphson optimisation scheme, determined by correlation score minimisation. Vertical and inclined beams have been printed and measured with XCT and focus variation (FV). Strut geometries obtained with V2C methodology are compared with extracted ISO50% point clouds, on the one hand, and measured FV point clouds, on the other hand, by computing signed cloud-to-mesh distances. These comparisons bring out that the V2C method is efficient to identify strut shape defects directly from volumetric data, without any post-reconstruction XCT data treatment. The simplification of these data treatment steps then raises the direct and accurate CAD feedback opportunity. Conclusions are drawn regarding the suitability of the proposed V2C method and its further development to more complex LPBF structures.

Introduction

With the design freedom enabled by additive manufacturing (AM), complex geometries are not limited by subtractive or formative manufacturing constraints anymore [START_REF] Gibson | Additive manufacturing technologies: rapid prototyping to direct digital manufacturing[END_REF]. Lattice structures, which are increasingly studied in the AM field, consist of an elementary cell regularly repeated in the 3D directions to form a network [START_REF] Hadi | Design Configurations and Creation of Lattice Structures for Metallic Additive Manufacturing[END_REF][START_REF] Beyer | Design and analysis of lattice structures for additive manufacturing[END_REF][START_REF] Echeta | Review of defects in lattice structures manufactured by powder bed fusion[END_REF]. Lattice structures are not free from defects, which have been reviewed by Echeta et al. [START_REF] Echeta | Review of defects in lattice structures manufactured by powder bed fusion[END_REF] for powder bed fusion (PBF). Measurement of these defects are mainly performed using computed tomography (CT) for its ability to assess either internal or external dimensional deviations [START_REF] Echeta | Review of defects in lattice structures manufactured by powder bed fusion[END_REF][START_REF] Chiffre | Industrial applications of computed tomography[END_REF]. However, CT dimensional measurements rely on the material boundary determination introducing a threshold uncertainty, prior to any further data treatments [START_REF] Dewulf | Uncertainty determination and quantification for dimensional measurements with industrial computed tomography[END_REF][START_REF] Rathore | Influence of resolution on the x-ray ct-based measurements of metallic am lattice structures[END_REF][START_REF] Lifton | On the uncertainty of surface determination in x-ray computed tomography for dimensional metrology[END_REF]. There have been significant efforts in reducing this uncertainty with sub-voxel studies [START_REF] Lifton | On the uncertainty of surface determination in x-ray computed tomography for dimensional metrology[END_REF][START_REF] Yague-Fabra | A 3d edge detection technique for surface extraction in computed tomography for dimensional metrology applications[END_REF] or more recently by conformance approach [START_REF] Petrò | Conformance and nonconformance in segmentation-free x-ray computed tomography geometric inspection[END_REF].

In this paper, a virtual volume correlation (V2C) method is proposed in order to find form defect of body-centered cubic with vertical struts geometry (BCCz cells), directly from volumetric data, without boundary thresholding nor additional data treatment. Indeed, recent works have highlighted the interest of virtual correlation techniques for contour or envelope identification in different fields. In the medical field, virtual correlation methods are applied for modelling pelvic organs from magnetic resonance imaging (MRI) volumetric data [START_REF] Jiang | Virtual image correlation of magnetic resonance images for 3d geometric modelling of pelvic organs[END_REF]. In the mechanical field, virtual correlation techniques have proven their interest for shape boundary identification of curved and elongated structures [START_REF] François | Metrology of contours by the virtual image correlation technique[END_REF][START_REF] Semin | Accurate measurement of curvilinear shapes by virtual image correlation[END_REF].

The aim of this paper is to assess the proposed V2C method applied to lattice beams relying on a defect basis description. V2C estimated form defect will be compared to extracted ISO50% and measured focus variation (FV) [START_REF]Geometrical product specification (GPS). Surface texture: Areal. Part 606: Nominal characteristics of non-contact (focus variation) instruments[END_REF] point clouds. In section 2, the V2C method is presented and assessed using vertical and inclined struts, which is representative of a BCCz lattice structure. Results are presented in section 3 and discussed in section 4.

Methodology

Virtual volume correlation

Let f be a physical volume, containing a closed envelope whose displacement field u should be identified in comparison to an original regular shape. Let {𝑿 𝑓 } be the voxels defining that envelope. Similarly, let g be a virtual volume and {𝑿 𝑔 } the voxels defining the virtual envelope. Each point of the physical envelope can be written as: 𝑿 𝑓 = 𝑿 𝑔 + 𝒖 V2C consists in iteratively minimising the grey level differences between physical and virtual volumes using the least square criterion. A correlation score Φ should be introduced as:

Φ(𝒖) = ∭[𝑓(𝑿) -𝑔(𝑿 + 𝒖)] 2 𝑑Ω 𝑅𝑂𝐼
where ROI refers to the region of interest in terms of considered voxels, and {𝑿} refers to considered voxels in the physical volume f. The displacement field 𝒖 only applies to the virtual contour points, and can be described as a sum of modes i.e. elementary displacement fields. Whereas previous works often modelled strut shape defects as first order ellipse approximations [START_REF] Rathore | A methodology for computed tomography-based non-destructive geometrical evaluations of lattice structures by holistic strut measurement approach[END_REF][START_REF] Murchio | Additively manufactured ti-6al-4v thin struts via laser powder bed fusion: Effect of building orientation on geometrical accuracy and mechanical properties[END_REF], modal decomposition methods [START_REF] Samper | Form defects tolerancing by natural modes analysis[END_REF][START_REF] Etievant | A modal approach for shape defect measurement based on global stereocorrelation[END_REF][START_REF] Thiebaut | Non-rigid registration for form defect identification of thin parts[END_REF] consist in expressing the form defect as a linear combination of elementary displacements:

𝒖 = ∑ 𝜆 𝑘 𝒖 𝑘 𝑘
where 𝜆 𝑘 are the components of {𝝀} and refer to the modal amplitude i.e. the amplitude of the elementary displacement descriptor 𝒖 𝑘 and are defined by:

{𝝀 * } = 𝑎𝑟𝑔𝑚𝑖𝑛 Φ(𝒖)
𝝀 𝜆 𝑘 * are found by combining equations 2 and 3 using a Newton-Raphson optimisation. An illustration of V2C application is shown in Figure 1 for a 2D example of a strut cross-section. From volumetric data (Figure 1a), the shape defect is initially taken as a nominal circle (Figure 1b), and is iteratively identified (Figure 1c).

Shape defect basis

This section particularly focuses on the elementary displacement descriptors 𝒖 𝑘 introduced in equation 3. In previous virtual correlation works, Semin et al. [START_REF] Semin | Accurate measurement of curvilinear shapes by virtual image correlation[END_REF] used segmentation 2D descriptors to identify elongated curvilinear shapes whereas Jiang et al. [START_REF] Jiang | B-spline based multi-organ detection in magnetic resonance imaging: B-spline based multiorgan detection in mri[END_REF] and Rhétoré et al. [START_REF] Réthoré | Curve and boundaries measurement using b-splines and virtual images[END_REF] used Bspline curves. Jiang et al. [START_REF] Jiang | Virtual image correlation of magnetic resonance images for 3d geometric modelling of pelvic organs[END_REF] extended the methodology to the 3D modelling of pelvic organs relying on NURBS geometric descriptors. In this work, as lattice beams are studied, displacement descriptors 𝒖 𝑘 are introduced considering cylinder defects. Indeed, for this geometry, Homri et al. [START_REF] Homri | Tolerance analysis: Form defects modeling and simulation by modal decomposition and optimization[END_REF] noted the modal decomposition usefulness for cylinder shape defect identification.

Therefore, the displacement field is assumed to be four-fold:

• Rigid transformations • Dilatation • Vertical defects:

-Vertical section modification descriptors such as taper or barrel modes -Vertical rippled modes

• Plane defects defined by sinusoidal descriptors Figure 2 illustrates some of the considered modes for the chosen shape defect basis.

Sample manufacturing and measurement

Sample manufacturing

Samples consist in vertical and inclined beams in order to be representative of vertical and inclined beams defining a BCCz strut-based lattice structure (see Figure 3). Beam radii have been set to 0.6 mm and have a 5 mm length. Samples were produced by laser PBF on an Addup FormUp 350 using Inconel 718 powder and the printing parameters displayed in Table 1. Hatch space 55 µm

Sample measurements

Each beam was measured using X-ray CT and FV instruments. Measurement setups are provided hereafter:

• XCT: geometric magnification of 33 % leading to a voxel size of 9 µm. Volumetric reconstruction was performed from 900 projections, tube voltage 150 kV, tube current 40 µA. Projections were saved in a .raw file format.

V2C assessment pipeline

Using the chosen shape defect basis, volumetric data from XCT are directly used by V2C to compute the correlated virtual envelopes for each strut, saved as a point cloud which is thus meshed. For each strut, the correlated mesh is compared to respectively extracted ISO50% and measured FV point clouds, by computing cloud-to-mesh distances. Figure 4 summarizes the different steps of the process.

Results

Figure 5 displays, for both 45° and 90° struts, cloud-to-mesh distances between correlated envelopes and point clouds stemming from ISO50% and FV sets. Results show that mean discrepancies between correlated mesh and point clouds are relatively the same for both ISO50% and FV comparative sets, in both strut cases. In addition, standard deviation estimations are consistent whatever the comparative set, with values around 26 µm.

Discussion

For the 45° strut, standard deviations show more discrepancies between ISO50% and FV sets when compared to the correlated mesh obtained using V2C. This observation may be explained by the surface topography of the 45° strut in comparison to the 90° strut. As a matter of fact, during the manufacturing process, 45° struts down-facing surfaces have no supporting structures, resulting in a poor strut surface quality. When measured using XCT, these rough surfaces alter V2C identification. Moreover, these surfaces have an increased FV measurement uncertainty, impacting the resulting standard deviation when compared to correlated mesh. Indeed, as shown in Figure 5 for the 45° strut, standard deviations are 4 µm higher in the correlated-FV set than in the correlated-ISO50% set. Conversely, the 90° strut is not impacted by the absence of supporting structures, resulting in a regular surface topography all around the strut. That is why, in Figure 5, for the 90° strut, standard deviations are about the same value.

For both 45° and 90° struts, estimated standard deviations are about the same value which is between 2.5 and 3 times the XCT resolution.

Correlated vs ISO50%

Correlated vs FV 45° strut These values are relatively low regarding the objective of studying shape defect. Thus, V2C provides a good estimation of lattice strut shape defect, performing the strut roughness separation from shape defect.

Conclusion

This paper showed the suitability of the proposed V2C method to identify strut shape defect directly from volumetric measurements, without any additional XCT data treatment. Moreover, the noteworthiness of V2C is to extend shape defect determination relying on modal decomposition relatively to a user-defined defect basis. However, modal decomposition raises the number of considered modes question. Future works will focus on more precise sensitivity studies in order to better understand how the number of considered modes, as well as the chosen defect basis, will impact the correlated shape defect. Future works will also focus on a meshed version on V2C adapted to entire lattice structures in order to broaden CAD feedback opportunities.

Figure 1 .

 1 Illustration of V2C on a 2D strut cross-section: (a) XCT physical measurement; (b) initial virtual shape; (c) computed virtual shape defect

Figure 2 .

 2 Figure 2. Illustration of some of the considered modes in the chosen basis description (amplitudes have been enhanced for more visibility)

Figure 3 .

 3 Figure 3. Vertical and inclined BCCZ representative printed struts. (One strut from each set has been considered in the following)

•

  FV: 10× objective lens with long working distance, numerical aperture 0.3, field of view (2.05 × 2.05) mm, pixel lateral resolution 2.07 µm, optical lateral resolution 0.91 µm, contrast lateral resolution 0.53 µm, rotation step of 50°, fusion of multiple field of views performed in the manufacturer's software, measured volume after data fusion (1.3 × 1.3 x 5.6) mm

Figure 4 .

 4 Figure 4. Data processing pipeline. Inputs are filled in green and outputs are filled in blue.

Figure 5 .

 5 Computed cloud-to-mesh distances for both struts: (a) and (c) correlated vs ISO50% point clouds; (b) and (d) correlated vs FV point cloud

Table 1 : Printing parameters

 1

	Powder	Inconel 718
	Layer thickness	40 µm
	Laser power	220 W
	Scan speed	2100 mm.s -1
	Contour scan power	210 W
	Contour scan speed	1800 mm.s -1

Acknowledgments This work was performed using HPC resources from the "Mésocentre" computing center of CentraleSupélec and École Normale Supérieure Paris-Saclay supported by CNRS and Région Île-de-France (http://mesocentre.centralesupelec.fr/). CT measurements at ENS Paris-Saclay has been financially supported by the French "Agence Nationale de la Recherche," through the "Investissements d'avenir" program (ANR-10-EQPX-37 MATMECA Grant)


L H Coudert

R A Motiyenko

L Margulès

F Kwabia Tchana

The rotation-torsion spectrum of CD 2 HOH

Keywords: CD 2 HOH, internal rotation, rotation-torsion, asymmetrical methyl group, hindering potential, line-position analysis, submillimeter wave, terahertz, far-infrared, 4-dimensional model

New transitions are reported in the submillimeter wave, terahertz, and far-infrared rotation-torsion spectrum of doubly-deuterated methanol CD 2 HOH. The newly assigned transitions allow us to spectroscopically characterize torsional states with 0 ≤ K ≤ 12 and 0 ≤ v t ≤ 2. Three line position analyses are carried out. In the first one, restricted to rotation-torsion lines involving torsional states with 3 ≤ K ≤ 12 and v t ≤ 2, rotational energies are evaluated with a J(J + 1) Taylor-type expansion for each torsional state. 4853 transitions were accounted for with a unitless standard deviation of 11.4. In the second analysis, 126 torsional subband centers are fitted to obtain refined torsional parameters including the hindering potential. In the third analysis, 5911 rotation-torsion transitions involving torsional states with v t ≤ 2, and J ≤ 26 are reproduced with a unitless standard deviation of 3.2 using a four-dimensional rotation-torsion fitting Hamiltonian.

Introduction

Internal rotation of an asymmetrical monodeuterated CH 2 D methyl group has been spectroscopically investigated in monodeuterated methyl formate [1,2], methanol [3][4][5][6][7][8][9][10][11][12], propene [13], and acetaldehyde [14][15][16]. In the case of monodeuterated methanol, a global analysis based on a fourdimensional rotation-torsion fitting Hamiltonian could be carried out [17]. Although internal rotation of a monodeuterated CH 2 D methyl group is only quantitatively different from that of a doublydeuterated CD 2 H methyl group, the latter has been studied in fewer molecules. The microwave spectrum of doubly-deuterated methyl formate was analyzed [18] accounting for the internal rotation with an effective approach. Doubly-deuterated methanol CD 2 HOH, first studied a long time ago, was more difficult to model. It was first spectroscopically investigated by Quade and coworkers [5][6][7]19] using microwave spectroscopy and transitions involving torsional levels up to K = 2 and v t = 6 were assigned. Later, new transitions were recorded in the microwave and far-infrared (FIR) domains [20][21][22][23][24] allowing us to spectroscopically characterize all torsional levels with 3 ≤ K ≤ 9 and v t ≤ 2. Rotation-torsion transitions were analyzed approximating the rotational energy of each torsional state with a J(J +1) Taylor-type expansion. Various torsional parameters, including the hindering potential, were also determined thanks to an analysis of the torsional subband centers [20]. However, unlike in the monodeuterated species CH 2 DOH, no global analysis of the rotation-torsion spectrum with a rotation-torsion fitting Hamiltonian [17] could be carried out.

In this paper, new assignments are reported in the microwave and FIR spectra of CD 2 HOH for rotation-torsion transitions involving torsional states with 0 ≤ K ≤ 12. A satisfactory line position analysis restricted to the 4853 lines with K ≥ 3 and v t ≤ 2 is carried out. Improved torsional parameters are retrieved fitting the 126 available torsional subband centers. More importantly, the four-dimensional rotation-torsion fitting Hamiltonian developed for the monodeuterated species [17] is slightly modified and applied to the fitting of rotation-torsion transitions up to the e 1 torsional state and to J = 26, K = 12. This rotationtorsion fitting Hamiltonian was key to the assignment of rotation-torsion transitions involving torsional states with K < 3 as their rotational energy cannot be correctly evaluated with a Taylor-type expansion.

The paper has four remaining sections. In Section 2, the new assignments are reported and the results of the line position analysis for lines with K ≥ 3 are given. Section 3 deals with the analysis of the torsional subbands centers and Section 4 with the global analysis of the rotation-torsion transitions. Section 5 is the discussion.

Assignments and analysis for K ≥ 3

The spectra used in the present work have been described in our previous investigation [20]. They consist of millimeter wave (MM) and submillimeter wave (SMM) spectra recorded from 80 to 600 GHz and of terahertz (THZ) spectra spanning the range 0.7-1.52 THz. FIR Fourier transform spectra measured in the 19-670 cm -1 region were also used. The frequencies measured in the MM and SMM domains are characterized by an experimental uncertainty of 30 kHz. In the THZ domain, the uncertainty is 30 kHz below 1 THz and 50 kHz above. For FIR spectra, an experimental uncertainty of 0.3 × 10 -3 cm -1 was adopted.

Torsional levels are labeled using either the scheme of Su and Quade [4] or that of El Hilali et al. [11]. The former makes use of the rotational quantum number K, with K ≥ 0, and of the torsional label L. This label, denoted gn, consists of the letter g, either e or o, and of the positive integer n which is a counter. g = e (o) indicates that the torsional function correlates with an even (odd) torsional function for K = 0. The scheme of El Hilali et al. [11] requires the same rotational quantum number K and the torsional quantum number v t . A cross reference table between both schemes is given in Table 1 of our previous investigation [20]. In this investigation, rotation-torsion energy levels are assigned using four quantum numbers J, K, p, v t , where p is an integer equal to 1 or 2, related to the C s symmetry species of the level, allowing us to distinguish the two members of an asymmetry doublet. For the |J, K, v t , ± rotation-torsion basis set wavefunctions defined in Eqs. ( 26) and ( 27) of Lauvergnat et al. [10], the quantum number p is defined as follows. For K > 0, p is 1 and 2 when ±(-1) K is +1 and -1, respectively; for K = 0, p = 1 (2) when the torsional function is an even (odd) function of the torsional angle. This result is consistent with a p = 1 (p = 2) rotation-torsion energy level belonging to the C s symmetry species A (A ) for J even and A (A ) for J odd.

The quantum number p is used instead of the usual asymmetric-top quantum numbers JK a K c because there is no straightforward way to assign in terms of K c the rotation-torsion levels corresponding to the |J, K a , v t , + and |J, K a , v t , -rotationtorsion wavefunctions. This is due to the slight asymmetry of CD 2 HOH, to the large amplitude torsional motion, and to the dependence of the inertia tensor on the angle of internal rotation. Symmetry considerations [10] allow us to derive the electric dipole selection rules for the quantum number p. For ∆J = 0 we have 1 ↔ 2 and for ∆J = 1 1 ↔ 1 and 2 ↔ 2.

The Taylor-type expansion used to compute the rotational energy for a given K, v t torsional state, based on that of Fisher et al. [25], is expressed with the help of the rotational quantum number p as

E(J, K, p, v t ) = m≥0 a m (K, v t )[J(J + 1)] m ± 1 2 W (J, K, v t ) (1)
where a m (K, v t ) are expansion parameters; the upper (lower) sign is for p = 1 (2); and W (J, K, v t ) is the asymmetry splitting written

W (J, K, v t ) = (J + K)! (J -K)! {S + T J(J + 1) + U [J(J + 1)] 2 + V [J(J + 1)] 3 } (2)
where S, T, U, and V are expansion parameters depending on K, v t . As already stressed in our previous investigation [20], in the case of CD 2 H-OH, the Taylor-type expansion of Eq. ( 1) allows us to satisfactorily reproduce rotational energies of torsional states with K ≥ 3. It fails for K < 3 because for such low K-values the levels are close together and the effects of Hamiltonian operators with ∆K = 1 matrix elements become important. Equation (1) of Ref. [17], where the zeroth-order rotation-torsion Hamiltonian is expressed in terms of the inverse generalized inertia tensor [26,27], emphasizes that coupling terms with ∆K = 1 matrix elements include Coriolis coupling terms of the form {P α , µ(α) α,x }P x and {P α , µ(α) α,y }P y and asymmetry terms in {P x , P z }µ(α) x,z and {P y , P z }µ(α) y,z .

Perpendicular K , v t ← K , v t torsional subbands with K , K ≥ 3 and v t , v t ≤ 2 were reported in our previous investigation [20]. For some of them, additional transitions could be assigned extrapolating to higher-J values with the Taylor-type expansion. New subbands were also observed making use of the four-dimensional rotation-torsion approach. Figure 1, where the assigned torsional subbands are shown schematically, should be compared to Fig. 4 of our previous investigation [20]. New assignments were carried out for the K = 10 ← 9, 11 ← 10, and 12 ← 11 subbands with v t , v t ≤ 2. Parallel transitions with 11 ≤ K ≤ 12 and v t ≤ 2 were also observed for the first time. 72 torsional subbands could be assigned and the number of available parallel and perpendicular transitions, including those assigned in this work and those reported in Refs. [21,23,24], is 4853. These transitions were fitted simultaneously using the Taylor-type expansions of Eqs. ( 1) and (2). This analysis provides us with a means to verify the assignments since, as confirmed by Fig. 1, all torsional states are connected. Each transition was given a weight equal to the inverse of its experimental uncertainty squared. A total of 4853 transitions were analyzed with a unitless standard deviation of 11.4. Table 1 summarizes the results of the line position analysis for all subbands and for perpendicular transitions only. Most subbands display a satisfactory χ 2 close to 1. It is mainly for subbands characterized by low K-values equal of 3 or 4 that unsatisfactory large values of χ 2 tend to arise. Such torsional states, although much less perturbed than those with K < 3, are still coupled to neighboring states through interactions becoming important when J increases. Assignments, observed frequencies or wavenumbers, and observed minus calculated residuals are listed in Table S1 available in the supplemental material. The a m (K, v t ) expansion parameters of Eq. ( 1) and the S, T, U, and V parameters of Eq. ( 2) determined in the analysis are reported in Table 2 for 3 ≤ K ≤ 4 and in Table S2 available in the supplemental material for all K-values. As emphasized by Table 2, the parameter a 0 (K = 3, v t = 0) was set to zero as the lowest lying K = 3, v t = 0 was used as reference state. The sign of the asymmetry parameter S is not constant. For K = 3, it is negative for v t = 0 and 1, and positive for v t = 2. For v t = 1 it is negative for K = 3 and positive for K = 4. This further confirms that the rotational level assignment in terms of K c is not straightforward.

Several subbands reported in our previous investigation [20] were reported almost at the same time by Mukhopadhyay [21][22][23]. The rotational assignments reported by this author in Tables 1 and2 of Ref. [21] for the K = 5, o 1 ← 4, e 1 subband, for parallel transitions within the e 0 , o 1 , and e 1 torsional states, and for a few perpendicular transitions are only partially consistent with those in our previous investigation [20]. In Table 2 of Ref. [21], at least four lines are missassigned. The parallel transition at 333335.082 MHz does not belong to the K = 3, e 1 torsional state, but to K = 3, o 1 ; the parallel transition at 333163.863 MHz does not belong to K = 7, o 1 but to K = 4, o 1 ; the perpendicular transition at 353983.299 MHz is not a J = 6 ← 7 line, but a J = 7 ← 8; and the perpendicular transition at 312334.867 MHz does not belong to the K = 6, e 1 ← 5, e 1 torsional subband. The Q-branch origins given in Table 1 of Ref. [22], retrieved from the FIR spectrum recorded in this reference, are within ±0.01 cm -1 from those listed in Table 6 of our previous investigation [20] for K + 1, e 0 ← K, e 0 torsional subbands with 3 ≤ K ≤ 8. The wavenumbers given in Table 2 of Ref. [22] for the K + 1, e 0 ← K, e 0 torsional subbands with 6 ≤ K ≤ 8 are within ±0.002 cm -1 from those listed in the supplementary data associated with our previous investigation [20]. The two wavenumbers given in Table 2 of Mukhopadhyay [23] for the two lowest J-value lines of perpendicular subbands originating from K, e 0 , with K ranging from 3 to 11, agree with the assignments reported in our previous investigation [20] and the present one. The K = 4, o 1 ← 3, e 1 torsional subband reported in Table 3 of Mukhopadhyay and Billinghurst [24] was already reported in our previous investigation [20].

Torsional spectrum analysis

The fitting torsional Hamiltonian used in the present investigation is the same as the one used in our previous investigation [20], except that additional distortion terms were used. The fitting torsional Hamiltonian, denoted H t , depending on the torsional angle α and on the rotational quantum The line position analysis of rotation-torsion transitions belonging to torsional subbands involving e 0 , o 1 , and e torsional levels with K ≥ 3 is described in Section 2. b K and L for the upper and lower torsional levels, σ the subband center in cm -1 , N the number of assigned perpendicular transitions, J Max the maximum J-value, and the unitless χ 2 are given for each torsional subband. c Reported in our previous investigation [20]. d Reported in our previous investigation [20], additional transitions assigned in this work. a Parameters determined in the torsional subband analysis described in Section 2 are listed. Each column corresponds to a torsional level labeled with K and L. Only parameters for 3 ≤ K ≤ 4 are reported. b Parameters, defined in Eqs. ( 1) and ( 2), are given in cm -1 and their uncertainty is given in parentheses in the same units as the last digit. A blank entry means that the parameter was set to 0. number K, is expressed as

K L K L σ N J Max χ 2 K L K L σ N J Max χ 2 4 e 0 3 e 2
H t = 1 2 P α µ(α) α,α P α + 1 2 {P α , µ(α) α,z }K + 1 2 {µ(α) z,z -1 2 [µ(α) x,x + µ(α) y,y ]}K 2 + 6 i=1 V i (1 -cos iα)/2 -D KK K 4 + k 1 P α K 3 + k 2 P 2 α K 2 + k 3 P 3 α K + k 4 P 4 α + 6 i=1 V iK (1 -cos iα)K 2 + 6 i=1 V iKα {P α , cos iα}K + 6 i=1 V iαα {P 2 α , cos iα} (3)
where µ(α) α,α , µ(α) α,z , µ(α) x,x , µ(α) y,y , and µ(α) z,z are α-dependent components of µ(α) the generalized inverse inertia tensor [26,27]; P α is the momentum conjugated to α; {, } is the anticommutator; V i , with 1 ≤ i ≤ 6, are six zerothorder potential energy parameters; and

D KK , k 1 , k 2 , k 3 , k 4 , V iK , V iKα , V iαα , with 1 ≤ i ≤ 6, are 23
distortion parameters. The eight kinetic energy parameters involved in the generalized inverse inertia tensor are defined in Table I of Ref. [11]. These kinetic energy parameters and the parameters defined in Eq. (3) were fitted to the available torsional subband centers. The integers N and p Max introduced in Ref. [10] for the calculation of the matrix elements of the generalized inverse inertia tensor components were respectively set to 11 and 9. The integer n t , also defined in this reference, which is the maximum value of |n| for the free internal rotation basis set functions |n , was set to 20. The data set includes the 6 torsional subbands involving torsional levels up to K = 2 and e 2 already considered [20]; the 72 subband centers retrieved in Section 2; and 46 subbands involving torsional levels with v t > 2. This last data set is almost identical to the one reported in Section 4.2 of our previous paper [20] except that the subbands 4, o 5 ← 3, e 5 and 4, o 5 ← 3, e 2 at respectively 232.28 and 654.69 cm -1 were added based on the rotational assignment reported by Mukhopadhyay [24]. The 7, e 3 ← 6, o 1 subband was rotationally reassigned using the results of this reference and its band center decreased from 366.90 [20] to 366.74 cm -1 leading to a better agreement with its calculated value. The 1, o 5 ← 2, e 1 and 3, o 5 ← 4, e 1 subbands at respectively 715.60 and 773.06 cm -1 were also added using the wavenumbers reported by Mukhopadhyay and Billinghurst [24]. 4 subbands with no analysis of their rotational structure were removed from this last data set. Table S3 available in the supplemental material gives the observed minus calculated tables for the torsional subbands belonging to this last data set. Line position analyses were carried out for each subband using the J(J +1) Taylor-type expansion of Section 2.

In the torsional spectrum analysis, weights equal to the inverse of the subband center uncertainty squared were used. For the 6 torsional subbands involving torsional levels up to K = 2 and e 2 , an uncertainty of 0.02 cm -1 was assumed since there is no satisfactory analysis of their rotational structure. For the 72 subbands centers retrieved in Section 2, a smaller accuracy of 0.01 cm -1 was taken since their rotational structure was successfully analyzed. For the remaining subbands, a satisfactory analysis could not always be carried out and a larger experimental uncertainty of 0.1 cm -1 was assumed. Altogether, 126 torsional subbands were fitted. The kinetic energy parameters, involved in the generalized inverse inertia tensor, which were not varied were set to the value in Table 3 of our previous investigations [20]. Similarly, the zeroth-order potential energy parameters of Eq. (3) which were not varied were set to the value in Table 4 of Ref. [10]. Table 3 of the present paper lists the torsional subbands and the observed minus calculated residuals. The root mean square (RMS) value of the residuals is 0.048 cm -1 . The 0, e 2 ← 1, e 2 subband, with a band center of 2.3 cm -1 , as retrieved from Ref. [6], could not be included in the data set because it displayed a too large observed minus cal-culated residual of -0.17 cm -1 . For such low Kand v t -values, such a large discrepancy may mean that this subband was missassigned in Ref. [6]. Table 4 reports the values of the fitted parameters. A comparison with the values in Table 5 of our previous paper [20] emphasizes that most parameters display small changes except the zeroth-order parameters I 0

x , V 1 , and V 2 . Concerning theses last two constants, the largest changes are displayed by the difference V 1 -V 2 while the sum V 1 + V 2 barely changes. The sum and difference are respectively -11.217 and -5.3415 cm -1 , with Table 5 of Ref. [20], and -11.317 and -4.573 cm -1 with Table 4 of the present papers.

Assignments for K < 3 and global analysis

The rotation-torsion transitions considered in this third analysis include those assigned in Section 2 and those involving at least one torsional state with K < 3. As stressed in Section 2, a J(J + 1) Taylor-type expansion is not appropriate to compute their rotational energy and assignment for these states were performed with the help of the 4-dimensional rotation-torsion approach. A preliminary fit of the transitions assigned in Section 2 and of those reported by Quade and coworkers [5][6][7]19] and by Mukhopadhyay [21] was performed. From this fit parallel and perpendicular transitions characterized by a higher J-value were predicted and assigned. This bootstrap procedure was repeated until no more transitions could be found. Table 5 gives the number of parallel and perpendicular transitions thus assigned. Although the analysis carried out in Section 2 allowed us to check the assignments of transitions with K ≥ 3, the assignment of the rotation-torsion transitions considered in this third analysis was further checked using 4-sided combination loops. All loops could be closed within experimental uncertainty when only unblended lines were involved. It should be emphasized that for all torsional states with K < 3, the maximum J-value and the transitions number are much higher than in the previous investigations [5-7, 19, 21].

The rotation-torsion Hamiltonian used in the present investigation differs slightly from the one introduced in Section III of Ref. [17]. Distortion effects were taken into account for the eight structural parameters, I 0

x , I 0 y , I 0 z , I 0 xz , I 1 x , I α , I x , and I 0 α , involved in the rotation-torsion Hamiltonian H 0 in Eq. ( 1) of Ref. [17]. These parameters are replaced by a J(J + 1) expansion so that that P → P + P JJ J(J + 1) + P JJJJ [J(J + 1)] 2 , where P is any of the eight structural parameters. Two rotationtorsion distortion operators were added, P α P x P 2 and {sin α, P α }P y , with corresponding constants ρ xJ and ρ yα .

In the analysis, energy levels were computed using the results of Section III C of Ref. [17]. The integers N , p Max , and n t , described in the previous section of this paper, were respectively set to 11, 9, and 22. The integers v tMax and k Max , describing respectively the number of torsional energies selected and the maximum value of |k| in the rotation-torsion energy level calculation, were set to 18 and 15, respectively.

The fitted data set consists of 4335 microwave transitions and 1459 FIR transitions. 312 microwave and 10 FIR transitions were treated as unresolved doublets. The torsional subband centers fitted in Section 3 were also considered excluding those with K < 3 and v t ≤ 3. This amounts to 117 subband centers. A total of 5911 transitions were least squares fitted with a weight equal to the inverse of their uncertainty squared. For torsional subband centers, the uncertainty values are the same as in Section 3. The RMS of the observed minus calculated residuals is 0.135 MHz for microwave transitions, 0.21 × 10 -3 cm -1 for FIR transitions, and 0.372 cm -1 for the torsional subband centers. The unitless standard deviation of the fit is 3.2. Tables S4, available in the supplementary material, gives the observed minus calculated table for the microwave and FIR rotation-torsion transitions.

The 95 parameters varied in the analysis are listed in Table 6 and include the eight structural parameters, their distortion coefficients, the six potential energy parameters, and the constants of the rotation-torsion distortion operators introduced Section III of Ref. [17] and in the present section. This table shows that the values obtained for the 6 kinetic energy parameters I 0

x , I 0 y , I 0 z , I α , I x , and I 0 α are within a few percent from those calculated in Table 3 of our previous investigation [20]. For the two smaller parameters I 0 xz and I 1 x , larger discrepancies arise. For the potential energy parameters V 1 , V 2 , V 3 , and V 4 , there is a satisfactory agreement with the values reported in Table 4 of Ref. [10], in Table 5 of our previous investigation [20] and in Table 4 of the present work. Values for the distortion parameters k 1 , k 2 , k 3 , k 4 , and D K are also in some- a Torsional subbands are assigned using K and v t for the upper and lower torsional levels.

K v t K v t Obs O-C K v t K v t Obs O-C K v t K v t Obs O-C 2 0 1 1 0.87(2) -2 6
b Subband centers are given in cm -1 in the column headed Obs. Their uncertainty is given in parentheses in 10 -2 cm -1 . c Observed minus calculated residuals are given in 10 -2 cm -1 in the column headed O-C and correspond to the spectroscopic constants in Table 4. a Torsional parameters are defined in Eq. ( 3).

b Kinetic energy parameters involved in the generalized inertia tensor are given in amu• Å2 . The remaining parameters are in cm -1 . Uncertainties are given in parentheses in the same units as the last digit. c Constrained to the value in Table 3 of Ref. [20]. d Constrained to the value in Table 4 of Ref. [10].

what good agreement with those reported in these last two tables.

Discussion

New assignments were carried out in the spectrum of CD 2 HOH. Compared to previous spectroscopic investigations [5-7, 19-21, 23, 24], assignments were extended to torsional states characterized by a higher K-values and to rotation-torsion transitions characterized by a higher J-values. A first analysis, reported in Section 2, was performed with the help of a J(J + 1) Taylor-type expansion for the limited set of transitions involving torsional states with K ≥ 3. As the fitted data set includes a large number of parallel and perpendicular transitions allowing us to connect all observed torsional states, as emphasized by Fig. 1, this first analysis provides us with a means to verify the new assignments. It yielded 72 torsional subband centers which, along with those used in our previous investigation [20], were analyzed to retrieve various torsional parameters such as those describing the hindering potential. This second analysis is reported in Section 3 and allowed us to reproduce 126 subband centers with an RMS of 0.048 cm -1 . It should be a For parallel transitions, K and L are given for each torsional state. For perpendicular transitions, K and L are given for the upper and lower torsional levels. b N and J Max are respectively the number of assigned transitions and the maximum J-value for each torsional subband.

emphasized that in this second analysis, the 72 torsional subbands with v t ≤ 2 reported in Section 2 only involve torsional levels with 3 ≤ K. Torsional levels with K < 3 are found in the 46 torsional subbands involving torsional levels with v t > 2, reported in our previous paper [20], as well as in the 6 microwave torsional subbands, also reported in our previous paper, involving torsional levels with K ≤ 2.

The main result of this investigation is the global analysis with the four-dimensional rotation-torsion fitting Hamiltonian developed for the monodeuterated species CH 2 DOH [17]. This global analysis, reported in Section 4, was carried out up to J = 26 and v t = 2 and observed frequencies and wavenumbers were reproduced with a unitless standard deviation of 3.2. Although, the rotation-torsion fitting Hamiltonian approach was key to the assignment of rotation-torsion transitions, especially those with K < 3, the value obtained for the unitless standard deviation is slightly too large. While somewhat satisfactory RMS values were obtained for the microwave and FIR data, it is for the torsional subband centers that an unsatisfactory RMS arises. Its value of 0.372 cm -1 is almost 10 times that retrieved in the analysis of Section 3. This stems from the large values of the torsional subband centers uncertainties compared to those of the microwave and FIR transitions. Inspection of the observed minus calculated table reveals that the largest residuals can be seen for torsional subbands involving high lying torsional levels with v t > 2, not involved in the microwave and FIR data sets. It should be stressed that a global analysis of the microwave and FIR transitions only, without the torsional subband centers, was attempted but was not satisfactory.

The reason why the rotation-torsion fitting Hamiltonian approach does not allow us to reproduce the observed line positions satisfactorily is not understood yet. It should be pointed out that even though the inverse inertia tensor µ(α) displays larger variations with the torsional angle α in CH 2 DOH than in CD 2 HOH, the results of the global analysis for the former species were more satisfactory [17].

In the spirit of the reduced Hamiltonian derived by Watson for asymmetric top molecules [28][29][30], one may wonder if a similar reduction could be carried out for the rotation-torsion fitting Hamiltonian used in this work so as to achieve a more satisfactory analysis. In the case of molecules displaying internal rotation of a symmetrical methyl group, such a reduction is available [31,32]. Unfortunately a similar reduction cannot be carried out for the present rotation-torsion fitting Hamiltonian. The first step of the reduction process [28][29][30][31][32] requires expanding the unreduced Hamiltonian as a series. In the case of the rotation-torsion used in this work, this is not possible since even the zerothorder Hamiltonian, H 0 in Eq. (1) of Ref. [17], cannot be written as a sum of rotation-torsion operators times a spectroscopic parameter. a Parameter constants and operators are described in Ref. [17] and in Section 4.

b Kinetic energy parameters involved in the exact Hamiltonian H 0 are given in amu• Å2 ; the remaining parameters are in cm -1 . Uncertainties are given in parentheses in the same units as the last digit. c Constrained to the value in Table 3 of Ref. [20]. d Constrained to the value in Table 4 of Ref. [10].

Figure 1 :

 1 Figure 1: Transitions assigned in the MM, SMM, THZ, and FIR spectra for torsional levels with 3 ≤ K ≤ 12 and vt ≤ 2 are shown schematically. Torsional levels are indicated by a full circle with a diameter proportional to the number of parallel transitions. Solid lines indicate perpendicular transitions connecting different levels. Their width is proportional to the number of assigned transitions.

Table 1 :

 1 Results summary a of the torsional subbands b analysis

Table 2 :

 2 Taylor-type expansion parameters determined in the torsional subbands a analysis

	Parameter b	K = 3, e 0	K = 3, o 1	K = 3, e 1
	a 0		3.849112(13)	12.7302658(123)
	a 1	0.69479101(57)	0.69467349(40)	0.69340586(16)
	a 2 × 10 6	3.850(9)	2.139(6)	-0.6357(11)
	a 3 × 10 9 -4.06(7)	-0.34(4)	0.013(3)
	a 4 × 10 12	18.46(21)	1.78(7)	0.3250(27)
	a 5 × 10 15 -29.249(243)
	S × 10 9 -1.58(3)	-1.337(10)	0.4698(16)
	T × 10 12 -19.63(48)	-0.32(12)	0.10(1)
	U × 10 15	160.80(326)	-3.4(4)	-0.074(29)
	V × 10 18 -495.85(941)	4.2(5)	0.044(22)
	W × 10 21	568.00(981)
	Parameter b	K = 4, e 0	K = 4, o 1	K = 4, e 1
	a 0	15.0663474(116) 18.4045900(113) 27.6349712(118)
	a 1	0.69480449(19)	0.69453481(21)	0.69340189(11)
	a 2 × 10 6	1.299(2)	0.3030(22)	-0.86596(72)
	a 3 × 10 9 -0.159(8)	-0.114(10)	0.0698(17)
	a 4 × 10 12	0.480(14)	0.044(21)	0.0932(14)
	a 5 × 10 15 -0.338(10)	0.065(15)
	S × 10 12 -0.9969(85)	0.319(20)	0.0676(34)
	T × 10 15	0.32(7)	0.02(21)	0.010(21)
	U × 10 18 -0.75(22)	-0.5(8)	0.04(4)
	V × 10 21	1.9(3)	0.6(13)	-0.02(3)
	W × 10 24 -1.1(1)	-0.5(8)

Table 3 :

 3 Assignments, a observed subband centers, b and residuals c in the torsional spectrum of CD 2 HOH

Table 4 :

 4 Parameters determined in the torsional subband center analysis Parameter a Value b Parameter a Value b

	I 0 x I 0 y	25.074(71) V 3 24.0877 c V 4	370.687(83) -0.7770 d
	I 0 z I 0 xz I 1 x I α	6.0487(21) V 5 -0.0982 c V 6 0.0157 c k 1 -1.1002 c k 2	0.0350 d -0.7460 d 0.03441(54) -0.05300(78)
	I x	-1.0787 c	k 3	0.03645(54)
	I 0 α	5.2949(21) k 4	-0.00948(15)
	D KK	0.00841(15) V 3K	0.0141(16)
	V 1	-7.947(46)	V 1αα	0.000682(60)
	V 2	-3.369(50)

Table 5 :

 5 Transitions assigned for K < 3 subbands a

	K L N b J Max	b K L N b J Max	b
	0 e 0 20	24	1 e 1 31	22
	0 o 1 20	24	2 e 0 28	21
	0 e 1 19	23	2 o 1 33	24
	1 e 0 37	24	2 e 1 29	23
	1 o 1 36	23
	K L	K	L	N b J Max	b
	1 e 0	0	e 0	28	25
	1 e 0	0	o 1	31	20
	1 e 0	0	e 1	29	22
	1 o 1	0	e 0	36	22
	1 o 1	0	o 1	15	23
	1 o 1	0	e 1	6	23
	1 o 1	1	e 1	11	19
	1 e 1	0	e 0	27	21
	1 e 1	0	o 1	19	20
	1 e 1	0	e 1	25	20
	2 e 0	1	e 0	44	22
	2 e 0	1	o 1	51	22
	2 e 0	1	e 1	60	19
	2 o 1	1	e 0	43	22
	2 o 1	1	o 1	47	23
	2 o 1	1	e 1	23	22
	2 e 1	1	o 1	47	24
	2 e 1	1	e 1	62	19

Table 6 :

 6 Rotation-torsion spectroscopic parameters obtained in the global line position analysis

	Parameter a	Value b	Parameter a		Value b
	I 0 x I 0 y I 0 z I 0 xz I 1 x Iα	H H H H H H	24.475(2) 24.216(2) 6.00876(1) -9.8188 c 9.25(13) -1.2330(3)	×10 -2 ×10 -3	c 4 D 1xy D 1yz ρyα λv k 2j	Pα{Pz, Px 2 -Py 2 } sin α{Px, Py} sin α{Py, Pz} {sin α, Pα}Py PαPz 3 P 2 Pα 2 Pz 2 P 2	5.85(8) 6.11(5) 8.08(4) -8.80(7) -1.30(3) 4.20(5)	×10 -4 ×10 -4 ×10 -2 ×10 -2 ×10 -5 ×10 -5
	Ix	H	8.09(32)	×10 -2	k 3j	Pα 3 PzP 2	-4.46(3)	×10 -5
	I 0 α I 0 zJJ I 0 xzJJ I 1 xJJ I αJJ	H H H H H	5.2551 c -3.26(4) -1.69(160) 6.55(17) 1.27(3)	×10 -5 ×10 -6 ×10 -7 ×10 -5	Mv H K H KJ H JK H J	Pα 4 P 2 Pz 6 Pz 4 P 2 Pz 2 P 4 P 6	1.558(10) -5.38(24) -6.55(680) -2.96(11) -1.57(25)	×10 -5 ×10 -6 ×10 -8 ×10 -8 ×10 -11
	I xJJ	H	-2.63(7)	×10 -5	h K	{Pz 4 , Px 2 -Py 2 }	6.37(24)	×10 -8
	I 0 xJJJJ I 0 yJJJJ I 1 xJJJJ V 1	H H H (1 -cos α)/2	-2.23(13) 1.45(21) -2.64(51) -9.193(10)	×10 -9 ×10 -9 ×10 -11	h KJ h J G 3 G 6	P 2 {Pz 2 , Px 2 -Py 2 } 2P 4 (Px 2 -Py 2 ) (1 -cos 3α)(Px 2 -Py 2 ) (1 -cos 6α)(Px 2 -Py 2 )	-1.80(18) -2.47(22) -6.84(11) 2.98(5)	×10 -9 ×10 -11 ×10 -3 ×10 -3
	V 2	(1 -cos 2α)/2	-3.310(10)		lv	PαPzP 4	5.73(21)	×10 -8
	V 3	(1 -cos 3α)/2	370.629(8)		gv	Pα 2 P 4	-2.74(11)	×10 -8
	V 4	(1 -cos 4α)/2	1.86(2)		k 3jj	Pα 3 PzP 4	2.92(4)	×10 -10
	V 5	(1 -cos 5α)/2	3.5000 d	×10 -2	k 4jj	Pα 4 P 4	-2.24(3)	×10 -10
	V 6	(1 -cos 6α)/2	-7.4600 d	×10 -1	V 1KKJJ	(1 -cos α)Pz 2 P 2	-1.10(2)	×10 -6
	D K	-Pz 4	7.66(5)	×10 -3	V 2KKJJ	(1 -cos 2α)Pz 2 P 2	1.63(14)	×10 -7
	D KJ	-Pz 2 P 2	1.22(1)	×10 -3	V 3KKJJ	(1 -cos 3α)Pz 2 P 2	-1.935(2)	×10 -4
	D J	-P 4	1.174(6)	×10 -6	V 6KKJJ	(1 -cos 6α)Pz 2 P 2	4.498(3)	×10 -4
	d k	-{Pz 2 , Px 2 -Py 2 }	2.78(4)	×10 -4	H 1KK	(1 -cos α){Px, Pz 3 }	-4.97(51)	×10 -4
	d j	-2P 2 (Px 2 -Py 2 )	3.09(32)	×10 -8	H 1JJ	(1 -cos α){Px, Pz}P 2	-2.58(1500) ×10 -7
	k 1	PαPz 3	3.14(2)	×10 -2	D 2yz	sin 2α{Py, Pz}	1.56(3)	×10 -3
	k 2	Pα Pz 2	-4.84(3)	×10 -2	D 3yz	sin 3α{Py, Pz}	2.19(4)	×10 -2
	k 3	Pα Pz	3.31(2)	×10 -2	D 2xz	(1 -cos 2α){Px, Pz}	-6.05(29)	×10 -4
	k 4	Pα	-8.49(5)	×10 -3	D 3xz	(1 -cos 3α){Px, Pz}	-9.38(4700) ×10 -5
	V 1KK	(1 -cos α)Pz 2	5.93(9)	×10 -3	D 2xy	sin 2α{Px, Py}	1.579(10)	×10 -3
	V 3KK	(1 -cos 3α)Pz 2	1.47(2)	×10 -2	D 3xy	sin 3α{Px, Py}	-2.23(4)	×10 -3
	V 5KK	(1 -cos 5α)Pz 2	-1.85(8)	×10 -3	V 1KKKK	(1 -cos α)Pz 4	1.65(5)	×10 -5
	l k	PαPz 5	1.46(6)	×10 -5	V 2KKKK	(1 -cos 2α)Pz 4	-1.91(6)	×10 -5
	k 2k	Pα Pz 4	-1.03(4)	×10 -5	V 3KKKK	(1 -cos 3α)Pz 4	-1.95(12)	×10 -5
	k 4B	Pα	1.10(5)	×10 -6	V 2KKαα	{1 -cos 2α, Pα 2 Pz 2 }	-6.39(47)	×10 -4
	V 1JJ	(1 -cos α)P 2	-7.82(8)	×10 -4	V 1αα	{1 -cos α, Pα 4 }	-1.01(5)	×10 -3
	V 2JJ	(1 -cos 2α)P 2	1.218(8)	×10 -3	c 1j	2Pα 2 (Px 2 -Py 2 )P 2	2.02(21)	×10 -9
	V 3JJ	(1 -cos 3α)P 2	9.98(93)	×10 -5	c 3	2Pα 4 (Px 2 -Py 2 )	-3.34(30)	×10 -8
	V 4JJ	(1 -cos 4α)P 2	1.36(2)	×10 -3	G 1j	(1 -cos α)(Px 2 -Py 2 )P 2	5.30(150)	×10 -9
	V 5JJ	(1 -cos 5α)P 2	-1.05(2)	×10 -3	G 2j	(1 -cos 2α)(Px 2 -Py 2 )P 2	1.36(9)	×10 -7
	∆xz	Pα {Px, Pz}	-3.87(50)	×10 -4	G 3j	(1 -cos 3α)(Px 2 -Py 2 )P 2	1.44(3)	×10 -7
	H 1	(1 -cos α){Px, Pz}	-2.05(29)	×10 -3	L KKJ	Pz 6 P 2	1.44(21)	×10 -12
	δxz	Pα{Px, Pz 2 }	9.13(100)	×10 -4	L KJ	Pz 4 P 4	7.39(77)	×10 -12
	ρ xJ	PαPxP 2	-2.84(3000) ×10 -7	L JJK	Pz 2 P 6	-1.84(2)	×10 -13
	Lv	PαPzP 2	1.80(2)	×10 -3	L J	P 8	1.01(18)	×10 -15
	Gv	Pα P 2	-8.79(11)	×10 -4	l k	{Pz 6 , Px 2 -Py 2 }	-1.19(2)	×10 -10
	G 1	(1 -cos α)(Px 2 -Py 2 ) -5.09(5)	×10 -4	l jk	P 2 {Pz 4 , Px 2 -Py 2 }	2.67(29)	×10 -13
	c 1	2Pα 2 (Px 2 -Py 2 )	-3.08(5)	×10 -4