H Resit Akçakaya
email: resit.akcakaya@stonybrook.edu

Ana S L Rodrigues

David A Keith

E J Milner-Gulland

Eric W Sanderson

Simon Hedges

David P Mallon

Molly K Grace

Barney Long

Erik Meijaard

P J Stephenson

Assessing Ecological Function in the Context of Species Recovery

Keywords: Conservation impact, Conservation planning, Green List of Species, Species recovery, Conservation optimism

Species interactions matter to conservation. Setting an ambitious recovery target for a species requires considering the size, density and demographic structure of its populations such that they fulfill the interactions, roles and functions of the species in the ecosystems in which they are embedded. A recently proposed framework for an IUCN Green List of Species formalizes this requirement by defining a fully recovered species in terms of representation, viability and functionality. Defining and quantifying ecological function from the viewpoint of species recovery is challenging, both in concept and application, but also an opportunity to insert ecological theory into conservation practice. We propose two complementary approaches to assessing a species' ecological functions: a confirmation approach that starts with a list of the interactions of the species, identifying the ecological processes and the other species that are involved in these interactions, and quantifying the extent to which the species contributes to the identified ecological process; and an elimination approach that infers functionality by ruling out symptoms of reduced functionality, analogous to the Red List approach that focuses on symptoms of reduced viability. Despite the challenges, we believe that incorporation of functionality into species recovery planning is possible in most cases. It is also an essential element of an aspirational conservation vision that goes beyond preventing extinctions, aiming to restore a species to levels beyond what is required only for its own viability. This vision focuses on conservation and recovery at the species level, but also sees species as embedded in ecosystems, influencing and being influenced by the processes in those ecosystems. Thus, it connects and integrates conservation at the species and ecosystem levels.

Introduction

Conservation biologists have long considered a species' place in the complex web of its interactions when setting species-focused conservation goals and targets (e.g., [START_REF] Janzen | The deflowering of Central America[END_REF][START_REF] Redford | The empty forest[END_REF][START_REF] Soulé | Ecological effectiveness: Conservation goals for interacting species[END_REF][START_REF] Sanderson | How many animals do we want to save? The many ways of setting population target levels for conservation[END_REF]. This aspect of conservation was recently formalized within a framework proposed for an IUCN Green List of Species, which aims to quantify species recovery to This article is protected by copyright. All rights reserved. levels beyond those necessary to avoid extinction, and to provide metrics for measuring the success of conservation measures in terms of progress towards full recovery [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF]. The proposed Green List of Species framework defines a fully recovered species as one that is viable, and ecologically functional, in each part of its indigenous and projected range, and defines functionality of a species as "the degree to which it performs its role as an integral part of the ecosystem in which it is embedded." Different aspects of this role include the species' "influence on or contribution to ecosystem-level processes (e.g., primary production), interactions with other species (e.g., trophic relationships), structural effects (e.g., ecosystem engineering), and intra-specific processes (e.g., migration)." In many cases, ecological functionality of a population will depend on its abundance, density, and demographic structure, which determine the behaviors and interactions of the organisms.

In this essay, we explore the application of the concept of ecological functionality to species conservation. We first review the justification for considering ecological function in the context of species recovery, and explore why this idea, despite a long history of discussion in the conservation literature, has not been explicitly or systematically implemented as a criterion of species recovery.

To address this shortcoming, we discuss conceptual and practical challenges of defining and assessing functionality, recommend practical approaches that are based on ecological theory for defining a species' function and determining whether a population is functional, and suggest directions of future research. Although our essay is motivated by the proposed Green List of Species, our ideas apply to all aspects and systems of species recovery planning.

Why consider function in species recovery?

The irreversibility of extinction, and the rapid decline of many species towards extinction, have largely, and appropriately, focused past conservation efforts on ensuring first and foremost that species retain viable populations that ensure their continued existence into the future. However, conservationists have long recognized the need to go beyond this minimal requirement, and to conserve species with "ecologically functional populations" [START_REF] Conner | Wildlife populations: minimally viable or ecologically functional?[END_REF]), or at "ecologically effective densities" [START_REF] Soulé | Ecological effectiveness: Conservation goals for interacting species[END_REF].

Ecological interactions between species are at the core of the complex web of life on Earth.

Conserving species interactions, beyond the conservation of the species themselves, is thus a major conservation goal [START_REF] Redford | The half-empty forest: Sustainable use and the ecology of interactions[END_REF][START_REF] Soulé | Strongly interacting species: Conservation policy, management, and ethics[END_REF]. Species interactions, like viability of individual species, are sensitive to human impact (Tylianakis et al. 2008), and may cease to occur in any ecologically meaningful way, even if the species are still present and viable in the ecosystem [START_REF] Janzen | The deflowering of Central America[END_REF], a process captured by the terms "ecological extinction" or "functional extinction" [START_REF] Redford | The empty forest[END_REF][START_REF] Janzen | Latent extinction-the living dead[END_REF]). This article is protected by copyright. All rights reserved. Species are embedded in ecosystems, so they influence and are influenced by ecosystem processes.

A major focus of the literature on ecosystem processes and functions has been the relationship between biodiversity and ecosystem functionality (e.g., [START_REF] Cardinale | Biodiversity loss and its impact on humanity[END_REF]. Although research on this relationship has often focused on aggregated measures of ecosystem function, notably productivity, that are not directly relevant to biodiversity conservation [START_REF] Srivastava | Biodiversity-ecosystem function research: Is it relevant to conservation? Annual Review of Ecology[END_REF], it is generally recognized that conserving processes in an ecosystem requires conserving its components, including species.

A separate but related goal is to conserve species in their natural or wild state. [START_REF] Sanderson | How many animals do we want to save? The many ways of setting population target levels for conservation[END_REF] expressed this concept as "having animals acting like animals, not just persisting", and stated that demographic sustainability (i.e., long-term viability) "should be seen only as a threshold requirement, a necessary but not sufficient level" and that conservation targets should consider "interactions, including ecological functions and social dynamics, along with demographic requirements". Related to this are considerations of "wildness". [START_REF] Redford | What does it mean to successfully conserve a (vertebrate) species?[END_REF] recognized a range of states of species conservation, ranging from captive to self-sustaining; [START_REF] Sanderson | How many animals do we want to save? The many ways of setting population target levels for conservation[END_REF] proposed "ecological integrity" of animal populations to synthesize function, behavior, and demography in setting conservation targets; and [START_REF] Caro | Vanishing behaviors[END_REF] defined and reviewed disappearance of behaviors (ethodiversity loss). These considerations imply that, in addition to interactions of species with other species, there are certain patterns of intra-specific behavior and social dynamics (such as migration, aggregations, patterns of social hierarchy, etc.) that are characteristic of a species that may disappear as a species declines, even if these smaller populations are not at risk of extinction. Such characteristics may not appear to comfortably fit into a general definition of ecological function of a species (see below), but are nonetheless important indicators of the successful conservation of a species that go beyond the criterion of low risk of extinction.

An important aspect of the concept of functionality is its contribution to a comprehensive approach to conservation across levels of biological organization. Indeed, considering interactions of a species in a community and its contributions to ecosystem processes forms a bridge between conservation at the species level and conservation at the ecosystem level. Another key concept of the proposed Green List of Species framework-the representation of the species across the range of ecological settings to which it is native, as quantified by viability and functionality across areas comprising the native range-forms a bridge between conservation at the species level and conservation at the population level. Hence the proposed framework is designed to integrate conservation at population, species, and ecosystem levels. This article is protected by copyright. All rights reserved.

Challenges

In this section, we identify several challenges related to assessing species-level ecological functions in the context of species recovery.

Defining function from a species recovery viewpoint

The terminology of ecological function can become confusing, especially when functions of both species and ecosystems are considered. From the perspective of species recovery in general, and the proposed Green List of Species framework specifically, we define ecological function as specific to a species, and ecosystem process as specific to an ecosystem (see Table 1 for definitions). So, for instance, the contribution of a species to nutrient cycling in a particular ecosystem is an ecological function of that species, whereas nutrient cycling itself is an ecosystem process, which, if it benefits humans, leads to an ecosystem service.

From a species conservation point of view, we are concerned with several general types of ecological functions, all of which arise from interactions among organisms (see Table 2 for examples). Some functions arise from inter-specific interactions, others from intra-specific interactions (discussed later). Some inter-specific interactions are direct interactions of the focal species with one or few other species; others involve indirect or diffuse interactions of the focal species with many species, and thus are better considered to be at the ecosystem level (Table 1). For example, dispersal of the seeds of a particular tree species by a bat species is an ecological function of the bat species at the level of direct interaction with another species (i.e., the tree species). The contribution of the same bat species to productivity and forest regeneration (for example, through dispersal of seeds of multiple plant species it interacts directly with) is an ecological function of the bat species at the ecosystem level.

These types of inter-specific ecological function are closely related to the concept of specific effect function (SEF), which is the "per-unit capacity of a species to influence an ecosystem property or service," measuring the "difference made to a particular process at the ecosystem level by a standard 'amount' of a species" [START_REF] Díaz | Functional traits, the phylogeny of function, and ecosystem service vulnerability[END_REF]. For the purposes of the Green List of Species, SEF multiplied by population size is more relevant than SEF itself, because a species Green List assessment is based on the total contribution of the species' population to the ecosystem process in question.

Another general type of ecological function arises from intra-specific interactions and processes, and involves patterns of behavior and social dynamics that are characteristic of a species, and which may disappear as a result of human impact (Table 2). These include long-distance migration, large-scale This article is protected by copyright. All rights reserved. aggregations of individuals, and patterns of social hierarchy. Although this type of ecological function often affects, and is affected by, larger scale ecological processes (such as nutrient cycling; Doughty et al. 2016; see also discussion on "mobile links" below), it involves species-specific behaviors that are considered worthy of conservation effort [START_REF] Caro | Vanishing behaviors[END_REF], even in cases where they do not affect ecosystem processes, and even if (like the other types of functions considered here) they may not be required for a low risk of extinction.

In summary, a working definition of function for the purposes of species recovery may be summarized as totality of the species' interactions, determining its influence on, or contribution to, ecosystem processes, and the patterns of intra-specific interactions, behavior and social dynamics that are characteristic of that species. Note that "function" is a property of the species, while "functionality" (the degree to which the function is performed) is a property of a particular population of the species at a particular time, as it depends on the size, density and demographic structure of the species' population at that place and time.

Although "functionality" is continuous, for practical purposes, it can be assessed in terms of few (or even just two) categories; for instance, [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF] proposed to use it as a binary variable (functional or not functional) in the context of the Green List of Species, and [START_REF] Sanderson | The ecological future of the North American bison: Conceiving long-term, large-scale conservation of wildlife[END_REF] suggested breaking the continuum of ecological interactions by bison populations into five categories, from "no contribution" to "exceptional contribution" to local interactions. A binary concept of ecological functionality is also implicit in the concept of "ecological extinction" or "functional extinction"; a population is considered to be functionally extinct (or not functional) if its abundance is too low, or its demographic structure is not suitable, for it to fulfill its ecological roles in the community or ecosystem. Dividing a continuous variable into categories for ease of assessment and communication is common in many fields, including conservation, where the probability that a species is extinct is divided to list species as extinct, possibly extinct and extant [START_REF] Akçakaya | Inferring extinctions III: A cost-benefit framework for listing extinct species[END_REF]; and the probability that a species will go extinct in the future is divided to sort species into threat categories [START_REF] Collen | Clarifying misconceptions of extinction risk assessment with the iucn red list[END_REF]. Some uses of the term "functional extinction" and "ecological extinction" refer to termination of basic demographic processes such as reproduction (e.g., [START_REF] Fan | Ecological extinction of the critically endangered northern white-cheeked gibbon Nomascus leucogenys in China[END_REF]Roberts et al. 2017). We do not consider these processes as ecological functions in the context of the proposed Green List of Species framework, because populations that do not perform such basic demographic functions would not be viable, and viability is a separate, indispensable component of recovery in the proposed framework and a stated pre-requisite of functionality [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF].

Our approach to interpreting ecological function differs from other efforts by focusing on the species, not the function. For example, [START_REF] Brodie | Ecological function analysis: Incorporating species roles into conservation[END_REF] propose a method for conserving This article is protected by copyright. All rights reserved.

functions by focusing on strong interactions, arguing that "critical ecological functions should be another facet of biodiversity that we try to conserve ... in tandem with protecting the taxa ... and the habitats in which they occur." We agree that conservation of functions [START_REF] Brodie | Ecological function analysis: Incorporating species roles into conservation[END_REF]) and ecosystems [START_REF] Keith | Scientific foundations for an IUCN Red List of Ecosystems[END_REF]) are complementary to conservation of species. Our focus here is incorporating ecological functionality into the assessment of species recovery, which was first formalized by the Green List of Species protocol.

Function as contribution to ecosystem resilience

The ecological function of a species, in terms of its contribution to, or influence on, an ecosystem process, can be described in different ways. In some cases, the more important influence may be on the stability of the ecosystem process (e.g., its resistance or resilience to perturbations), rather than the magnitude of the contribution. A simplistic way of explaining this is in terms of mean vs. variance: some species may contribute in quantity (make a large difference in mean), others may influence the quality, e.g., by helping reduce variability, thus increasing stability or consistency of an ecosystem process. For example, perennial plants may be critical to maintaining soil stability and forage for herbivores during drought, but may contribute only a small portion of these functions in wetter times when short-lived palatable plants are more abundant (Westoby et al. 1989). So, the ecological function of a species may be in terms of its contribution to the stability of ecosystems, e.g., their capacity to resist regime shifts between alternative states, or to recover following a disturbance [START_REF] Nimmo | Vive la résistance: reviving resistance for 21st century conservation[END_REF]. However, the above dichotomy is complicated by the fact that many ecosystem processes are not "stable" in the simple sense of being constant or having a small variability. They are often naturally variable in space and time, especially in small spatial and temporal scales (Oliver et al. 2016), or have natural regime shifts between alternative states. So, the contribution of a species to an ecosystem process may not necessarily stabilize the process when considered at the scale of the species' population(s) in a particular area (a "spatial unit", in the proposed Green List of Species terminology). The functional contribution of some species may even promote variability, as appears to be the case in fire-promoting plants [START_REF] Bond | Kill thy neighbour: An individualistic argument for the evolution of flammability[END_REF]. Nevertheless, a species may contribute to the resilience and recovery of ecosystems at broader spatial and temporal scales, through its contributions to ecological processes at critical times and places as, for example, certain fire regimes maintain diversity at landscape scales [START_REF] Keith | Functional traits: their roles in understanding and predicting biotic responses to fire regimes[END_REF]. Thus, contributions of a species to an ecosystem process may appear disruptive at one scale but promote resilience at another.

Understanding the scale-dependence of the relationship between measures of resilience of ecosystem processes (such as variability) and the properties of the populations of species (such as This article is protected by copyright. All rights reserved. size, density and demographic structure) contributing to those processes remains a challenge. Using resilience as a predictor of functionality would require a deeper understanding this relationship.

Functional and non-functional species in the same ecosystem

Because function is often thought of at the ecosystem level, and because a general goal of conservation is preserving well-functioning ecosystems, it may appear that if one species in an ecosystem is not fully functional, then the ecosystem is not fully functional; and, therefore, no species in the ecosystem can be considered functional. However, this would not be a useful conclusion from a species conservation perspective, because it would not let us know which species need further recovery to make them fully functional. In the context of species Green List assessments, ecological function is species-specific; therefore, assessment of functionality should focus on the species being assessed.

For instance, a prey species could be deemed functional even if its predator populations are not functional or even present. Indeed, what is relevant to assessing functionality at the level of the prey species is whether there are conservation or management actions focused on it that are necessary to its recovery. If the only recovery necessary in this case is that of the predator, regarding the prey species as non-functional would not be useful. However, if the lack of predators (or the fact that the predators are below their functional densities) causes populations of a prey species to become overabundant, and thereby cause disruption of ecological processes or threaten native species, then management actions at the level of the prey may be appropriate and the prey population should be considered non-functional (e.g., seagrass-turtle-shark system; [START_REF] Heithaus | Seagrasses in the age of sea turtle conservation and shark overfishing[END_REF]. Thus, relationships between functionality and population density or size can be non-linear; a population's density may be too low or too high for the species to be functional. In other words, some species (in this example, predators) being absent or non-functional may cause other species (in this case, prey species) to become non-functional. But, the non-functionality of the prey species in such a case would be based on their influence on ecosystem processes, not on their lack of interaction with the predators.

Functional redundancy

Functional redundancy occurs when there is overlap between the functions performed by a number of species, such that changes in one species' contribution to ecosystem processes can be compensated for by contributions from other species [START_REF] Lawton | Redundancy in ecosystems[END_REF]. Functional redundancy is an active topic of research and is documented in many ecosystems. It is important to note that redundancy is not a property of the species, but a property of the particular function of a particular species (or set of species). According to ecological niche theory, there are no redundant This article is protected by copyright. All rights reserved. species, because each species has a unique niche. However, some functions of a species may be redundant, if those functions are also performed by other species in the ecosystem.

Even those functions that appear to be redundant may be performed by different species in different contexts, for example in different times (of day or year) or in different places (microhabitats, strata, ecoregions), making each species' influence or contribution unique and therefore not redundant. In other words, redundancy, like function, is dynamic and spatially heterogeneous, and varies depending on the functions considered.

From the point of view of species recovery assessments, the challenge functional redundancy brings is mostly practical, involving the need to identify contributions of a species and the conditions that allow those contributions. We further discuss these needs in the context of approaches to determining whether a population is functional (see Assessing Functionality below). In addition, functional redundancy may appear to reduce the importance of a species' functionality, and hence the priority for its recovery. However, as we discuss in the next section, the point of considering function in species recovery is not to assign importance to species, but to establish a higher threshold for their recovery compared to what is required for their viability. Thus, functional redundancy does not invalidate the need to consider a species ecological function in setting targets for its recovery.

Species "importance" and ecosystem services

A discussion of species functionality in the context of species conservation may be misinterpreted as justifying the conservation of a species based on its functional importance. It is critical to emphasize that neither the proposed Green List of Species framework, nor the IUCN Red List (into which the Green List metrics will likely be integrated) assume any judgment about how "worthwhile" it is to conserve a species. The point of including functionality in species recovery assessments is about setting more ambitious conservation goals than extinction avoidance, and aiming to restore a species to levels beyond what is required only for its own viability, reflecting its overall role in the ecosystem(s) it inhabits. Thus, assessing functionality is a way of expecting more from our conservation efforts, not from the species. Although some species do make larger contributions to ecosystem processes (e.g., keystone species), our purpose is not to use functionality to compare species to each other. Rather, it is to evaluate, for each species, contributions of populations in different parts of its range, relative to the reference level of full functionality across the species range. Hence, being unable to identify "important" functions, or a lack of "strong interactions" (as defined by [START_REF] Brodie | Ecological function analysis: Incorporating species roles into conservation[END_REF], does not preclude the establishment of ambitious recovery targets: they can be based on proxies of functionality, to determine population levels at which to consider a This article is protected by copyright. All rights reserved.

species "fully recovered" (as defined by [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF]) in a particular area (see Assessing Functionality below). Some ecological functions of a species considered in recovery planning may also provide ecosystem services. In other words, the species may contribute to ecological processes that produce benefits for people [START_REF] Gascon | The importance and benefits of species[END_REF]. These may include, for example, supporting services (such as nutrient recycling and soil formation), regulating services (such as decomposition and detoxification) and cultural services (such as recreation and artistic inspiration). In some cases, the most visible (or the only quantified) function of a species may be related to an ecosystem service. For example, there may be a lot more known about the contribution of an insect to pollination of crops than to pollination of rare native plants, even if the latter risk going extinct if the population of the insect species in question is no longer functional.

Even in these cases, we argue that the assessors should consider function from the perspective of the native non-human species that the focal species interacts with, and the natural ecosystems the focal species is native to, rather than benefits to humans. This focus on natural interactions reflects our emphasis on conservation of species for their own sake, rather than an anthropocentric perspective. However, the considerations of functionality (especially during a formal Green List of Species assessment) may also be a good opportunity to catalogue the benefits of biodiversity for human well-being in a more systematic way.

Spatial and temporal variability in functionality

A species may occur in multiple ecosystems, and perform different roles in each, for instance by contributing to different ecological processes, or by interacting with a different assemblage of species. Therefore, its functions can vary across its range, changing in type or magnitude. In order to incorporate this spatial variability, functionality must be assessed separately for each ecosystem or unique ecological setting the species exists in, i.e., within each of the "spatial units" considered in a Green List assessment.

In addition, a species may perform its ecological function across several ecosystems and transfer resources, biota and matter, providing 'mobile links' between them [START_REF] Lundberg | Mobile link organisms and ecosystem functioning: Implications for ecosystem resilience and management[END_REF]Doughty et al. 2016). In order to incorporate these types of spatial structure and dynamics, functionality for such species needs to be assessed not only in terms of population size and structure, but also in terms of movement dynamics, at the relevant spatial scale.

A species' function can vary across time, as its environment and cohabiting biota changes. Even when the function remains the same, functionality of populations may change, e.g., because of fluctuations in abundance. Some of this can be due to natural cycles or fluctuations in the This article is protected by copyright. All rights reserved. environment. In these cases, the species' function, and the functionality of its populations, should be assessed over whole natural cycles, at the temporal scales relevant to the ecosystem processes and their resilience (see discussion above on scale and resilience).

Temporal changes in a species' function can also be a result of human impacts, either locally and regionally (e.g., because of habitat alteration) or globally (e.g., because of climate change). In some cases, human impacts may cause functional species to become deleterious and thus nonfunctional [START_REF] Carey | Native invaders -challenges for science, management, policy, and society[END_REF]). These types of temporal shifts or trends in species function can be incorporated into successive Green List assessments, similar to how the Red List status of a species is periodically updated.

Assessing Functionality

Despite the challenges reviewed, we believe that incorporation of functionality into species recovery planning is possible in most cases; it is also an essential element of an aspirational conservation vision. In this section, we present ideas and preliminary guidance for identifying the ecological functions of a species, and determining if a population is ecologically functional.

We propose two approaches, a confirmation approach and an elimination approach. The former aims to identify specific functions of a species, and based on these, determine the functionality of a particular population in a specific time and place. The latter approach infers functionality by ruling out each of a set of symptoms of reduced functionality. For a given species either or both approaches may be applicable, depending on the type of information and expert knowledge available.

The confirmation approach is based on the fact that all species interact with other species, and uses a list of the interactions of the focal species with others as a starting point for identifying its ecological functions. Then, we envision a process of identifying the ecological processes (such as predation, dispersal, facilitation, etc.) that are involved in these interactions (see Table 2 for examples). This information, together with a knowledge of the functional traits that are often associated with such interactions, would allow assessors to identify the ecological conditions that determine the extent to which a particular population of the species (at a specific time and place) contributes to these identified ecological process (i.e., the determinants of functionality; Table 2). The next step is identifying the variable to assess functionality. As the examples in Table 2 imply, the variable to be measured depends on the function; it could be the total number of seeds dispersed by a mammal, the number of plant species pollinated by an insect, or the contribution (in units of mass or volume per unit time) to distribution of a particular nutrient. This article is protected by copyright. All rights reserved.

Finally, the relationship between this response variable and population density (or other characteristic, such as demographic structure) of the focal species is established. In some cases, this relationship is non-linear (e.g., a step function, or sharp peak, and even hysteresis), naturally leading to a categorical assessment of functionality (e.g., as functional vs. not functional), based on threshold values of population size, density and structure. Many of the studies cited in Table 2 If the function-density relationship is gradual (i.e., close to linear), functionality can be assessed either as a continuous rather than binary variable, or with a subjective threshold (e.g., >50%) that is consistent with the level of ambition of the species recovery objectives. Note that species recovery can be quantified as a percentage of fully recovered even with a binary definition of functionality (see Fig 1 in [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF]. If more than one function can be identified for a species, functionality can be assessed based on the function that is better studied, the function that is unique among the species in the same ecosystem, the function that allows a better approximation of the species' role and population characteristics prior to major human impacts, the function that requires the highest density, or the function that represents a strong interaction [START_REF] Brodie | Ecological function analysis: Incorporating species roles into conservation[END_REF].

The elimination approach considers the same types of information discussed above for the confirmation approach, but focuses on the end-result rather than the mechanism. It looks for symptoms of reduced functionality, analogous to the Red List approach of identifying symptoms of reduced viability. We propose a list of questions and considerations to guide the assessors in this process (Table 3), aimed at allowing a systematic consideration of the criteria and evidence for determining whether the size, density and the demographic structure of the species' populations are appropriate for its ecological function(s).

These proposed approaches for determining functionality may not be applicable in some cases, or they may give results that are too uncertain for practical application. In these cases, the Green List framework [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF]) recommends a number of proxies, such as population density in This article is protected by copyright. All rights reserved.

areas not impacted by human activities (Table 3), which can be used even if no function can be identified.

Future Directions

There is a need for further refinement of concepts and methods to identify and quantify ecological functions of species. Research on functional traits, functional rarity, and the relationships of these concepts to the goals of species recovery may contribute to addressing this need. Functional traits are characteristics of an organism that are relevant to its response to the environment or its effects on ecosystem processes or properties (Violle et al. 2007). Traits that determine a species' response to the environment (called 'Response traits'; [START_REF] Lavorel | Predicting changes in community composition and ecosystem functioning from plant traits: Revisiting the holy grail[END_REF] are more relevant to its viability than to its functionality, whereas traits that determine a species' effects on ecosystem processes or properties (called 'Effect traits') are more relevant to its functionality than to its viability. The same traits can be important both for a species' response to the environment and for its effect on ecosystem processes. But the reverse is also possible: characteristics or traits of a species that determine its response to the environment (thus its viability) may differ from those that determine its effect on ecosystem processes (thus its functionality). Thus, a trait may have no or minor effect on individual fitness but a strong effect on ecosystem properties [START_REF] Shipley | Reinforcing loose foundation stones in trait-based plant ecology[END_REF].

Different species may contribute to the same ecosystem process through different combinations of traits and their values. Thus, a unique mapping between traits and functions may not exist.

Nevertheless, across species in a particular taxonomic group, there may be a pattern of dependence between traits and functions. Such patterns may be uncovered by appropriate statistical methods.

For example, [START_REF] Díaz | Functional traits, the phylogeny of function, and ecosystem service vulnerability[END_REF] used a phylogenetic comparative method (Freckleton et al. 2002) to model how species functionality depends on species trait values.

Although the relationship between traits and functions (e.g., the predictability of ecosystem-level processes from traits) may not be directly relevant to the practical aspects of recovery planning and Green List assessments, its improved understanding may help to identify ecological function(s) of a species through its traits. This would require an analysis involving a group of species whose traits and functions are already known. If such an analysis uncovers strong patterns of dependence between traits and functions, the results may help identify functions of species that share traits with those analyzed and thereby help in situations where the functions of a particular, related, species are not well known.

A related concept is functional rarity of a species, which combines the rarity of the species with the rarity of its traits (Violle et al., 2017). Species rarity is often considered in terms of combinations of geographic range (restricted vs. widespread) and local abundance (scarce vs. abundant). These two This article is protected by copyright. All rights reserved. forms of species rarity are combined with two parallel forms of trait rarity, measuring the extent to which species traits are "more or less distinct or redundant within local communities or larger-scale species assemblages" (Violle et al. 2017). The functional rarity framework suggests that the types of rarity relevant to viability (range restriction and local scarcity) may be different from those relevant to functionality (trait distinctiveness and uniqueness), and that many combinations of species rarity and trait rarity are possible. The most distinct combinations of functional traits seem to be supported predominantly by rare species [START_REF] Mouillot | Rare species support vulnerable functions in high-diversity ecosystems[END_REF]). Thus, even in diverse ecosystems where functional redundancy is expected, rare species disproportionately increase the diversity of ecosystem processes, and they may potentially insure against future uncertainty arising from climate change and other human impacts. More importantly, methods developed to quantify functional rarity could bring insights into the challenging questions of identifying the function of a species, and determining if its populations are functional.

Conclusion

A basic tenet of ecology is that species are not isolated entities. The interactions of a species with other species and other components of biota are an important aspect of its essence, its intrinsic value, and its fundamental connection to Earth's evolutionary heritage. Thus, "conserving nature" requires conserving the interactions among species, as well as the species themselves. This can be achieved in different ways, for example by conserving particular types of functions and interactions [START_REF] Brodie | Ecological function analysis: Incorporating species roles into conservation[END_REF] or by conservation at the ecosystem level [START_REF] Keith | Scientific foundations for an IUCN Red List of Ecosystems[END_REF]. The proposed Green List of Species [START_REF] Akçakaya | Quantifying species recovery and conservation success to develop an IUCN Green List of Species[END_REF]) is a third approach, which identifies species' current functionality across its range relative to its potential functionality, and so incentivizes the conservation of this functionality. This framework focuses on conservation and recovery at the species level, but also sees species as embedded in ecosystems, and influencing and being influenced by the processes in those ecosystems. Thus, it connects and integrates conservation at the species and ecosystem levels. We recognize, and are working to address, the many challenges to our goal of developing this framework into a practical tool for species assessments and recovery planning that goes beyond the minimal requirement of maintained presence through extinction avoidance.

This article is protected by copyright. All rights reserved. Ecological function of a species: The totality of the species' interactions, determining its influence on, or contribution to, ecosystem processes, and the patterns of intra-specific interactions, behavior and social dynamics that are characteristic of that species.

Ecological functionality of a population:

The extent to which the population fulfils the ecological function(s) of the species in a particular place and time, as determined by its size, density and demographic structure. Ecological functionality can be assessed either as continuous (e.g., a percentage) or as categorical (e.g., as functional vs. not functional).

Direct interactions:

A category of the ecological function of a species as determined by its interactions with one or few other species, including pollination, seed dispersal, herbivory and predation (i.e., effects of the species on "ecological processes" as defined by Martinez 1996 and[START_REF] Pettorelli | Satellite remote sensing of ecosystem functions: Opportunities, challenges and way forward[END_REF].

Indirect interactions (structural functions): A category of the ecological function of a species as determined by its effects on other species through creation of habitat structures, features and conditions that affect the dynamics of those species.

Diffuse interactions:

A category of the ecological function of a species as determined by its effects on other species through contributions to ecosystem processes, such as decomposition, nutrient cycling and redistribution, and maintenance of fire regimes (i.e., effects of the species on "ecosystem processes" as defined by Lovett et al. 2006 and[START_REF] Pettorelli | Satellite remote sensing of ecosystem functions: Opportunities, challenges and way forward[END_REF].

Intra-specific interactions:

A category of the ecological function of a species as determined by within-species processes and patterns of behavior that are characteristic of the species, such as colony formation and other aggregations, and spatial patterns of movement and dispersion. This article is protected by copyright. All rights reserved.

e. a change in the typical patterns of behavior (e.g., social interactions, patterns of aggregation, movement) among individuals of the species being assessed or other species.

2. Comparing areas or subpopulations with different densities or abundances of the species, consider any evidence which suggests that the reduced population size or density of the species, or a change in its demographic (e.g., age) structure has caused or may cause any of the outcomes a-e listed above. It is important to consider that ecological function of a species and its natural density or carrying capacity may be different in different ecological settings. So, this comparison is more relevant between areas or subpopulations with similar ecological characteristics.

3. Comparing time periods when the species was at different densities or abundances, consider any evidence which suggests that the reduced population size or density of the species, or a change in its (e.g., age) structure has caused or may cause any of the outcomes a-e listed above.

4. Based on information on the functional traits of the species, and an analysis of relationships between trait and function in similar species, consider the potential that reduced population size or density of the species, or a change in its demographic (e.g., age) structure may cause any of the outcomes a-e listed above.

5. If no function can be identified for a species, consider the following proxies. a. Pre-impact: Use the natural or pre-disturbance population size or carrying capacity of a species as a proxy for functional density, assuming that at pre-impact densities the species did fulfill its ecological roles and functions. It is important to consider that carrying capacities vary naturally across the range and over time for many species. b. Non-impact: If impacts change over the range of the species, use the population size, density or carrying capacity in apparently non-impacted (or least impacted) areas as a proxy. It is important to consider that carrying capacities vary naturally across the range and over time for many species. c. Similar species: Information from similar species can be useful in determining either the principal ecological functions of the species, and densities that allow these functions; or the non-impact densities that can be used as proxy for functional density. If data allow, information from a number of similar species can be integrated to find relationships between functionality and density.

  have identified such threshold values in specific times and places. For example, Estes et al. (2010) estimated a threshold density of 6.3 Sea Otters (Enhydra lutris) per km surveyed, below which kelp forests are not maintained along the coast of California. McConkey and Drake (2006) estimated a Flying Fox (Pteropus tonganus) abundance index of about 0.8 required for effective long-distance seed dispersal in Tonga. Outcalt et al. (1999) recommend that a minimum Wiregrass (Aristida stricta) density of 1 m -2 to maintain cover and allow fires to spread in the Atlantic coastal plain. Age structure for pine trees (Pinus spp.) must include 80 years old individuals to allow Red-Cockaded Woodpeckers (Leuconotopicus borealis) to excavate nest cavities in southeast US (U.S. Fish and Wildlife Service 2003).

  Tylianakis JM, Didham RK, Bascompte J, Wardle DA. 2008. Global change and species interactions in terrestrial ecosystems. Ecology Letters 11:1351-1363. U.S. Fish and Wildlife Service. 2003. Recovery plan for the red-cockaded woodpecker (Picoides borealis): second revision. U.S. Fish and Wildlife Service, Atlanta, GA. 296 pp. https://www.fws.gov/rcwrecovery/files/RecoveryPlan/finalrecoveryplan.pdf (accessed 23 Oct 2018) Violle C, et al. 2007. Let the concept of trait be functional! Oikos 116:882-892. Violle C, et al. 2017. Functional rarity: The ecology of outliers. Trends in Ecology & Evolution 32:356-367.

TABLE 1 : Definitions related to ecological function in the context of species recovery

 1

TABLE 3 : Types of information to consider in inferring functionality of populations 1

 3 . Based on available information on the interactions of the species being assessed with other species, and its ecology in general, consider whether a reduction in population size or density of the species being assessed, or a change in its demographic (e.g., age) structure has the potential to cause non-trivial changes of any of the following types.a. a reduction in the abundance of another native species; b. an increase in the abundance of a non-native species or over-abundance of another species; c. a reduction in a demographic rate in any life stage of another native species (e.g., germination, seed production, nest success, natal dispersal, etc.) that has the potential to decrease its abundance or otherwise reduce its viability; d. a change in any ecosystem process or structural feature (see examples in Table2);

Acknowledgments:

We thank the participants of the workshops and meetings of the Task Force on Assessing Conservation Success, convened under the auspices of the IUCN SSC Red List Committee; WWF-US for funding several of the meetings. Stony Brook University OVPR Seed Grant Program supported H.R.A. NERC Knowledge Exchange Fellowship supported M.K.G. This article is protected by copyright. All rights reserved.


Eduardo Machado-Charry

Cesar Gonzalez Pascual

Yannick J Dappe

Laurence Magaud

Normand Mousseau

Pascal Pochet

Order and disorder at the C-face of SiC: an hybrid surface reconstruction

In this letter, we explore the potential energy surface (PES) of the 3×3 C-face of SiC by means of density functional theory. Following an extensive and intuitive exploration, we propose a model for this surface reconstruction based on an all-silicon over-layer forming an ordered honeycomb-kagome network. This model is compared to the available experimental Scanning Tunneling Microscope (STM) topographies and conductance maps. Our STM simulations reproduce the three main characteristics observed in the measurements, revealing the underlying complex and hybrid passivation scheme. Indeed, below the ordered over-layer, the competition between two incompatible properties of silicon induces a strong disorder in the charge transfer between unpassivated dangling bonds of different chemistry. This effect, in conjunction with the glassy-like character of the PES, explain why it took decades to explain this structure.

Semiconductor surfaces reconstruct mainly through ad-atoms either self-reorganization. 1 In the former, the dangling bonds are passivated by forming new bonds with the additional atoms, while in the latter, the dangling bonds are partially or completely eliminated via some charge transfer. Although reconstructions usually follow one or the other scheme, both mechanisms are sometimes mixed. In most cases, a hybrid mechanism takes place when some unpassivated surface atoms (known as rest atoms 1 ) undergo a charge transfer with the ad-atoms. As a consequence, the surface can exhibit a commensurate and ordered super-structure. A deep understanding of these reconstruction mechanisms is central in applied physics and, more particularly, for growth science as well as for tuning new devices based on the use of interfacial physics such as 2D electron gas or surface superconductivity. 2 Though being of relative importance for the growth of graphene from hexagonal SiC, a reconstruction at the carbon face (000 1) surface (known as 3×3) has remained unresolved for more than two decades. We show in the following that it corresponds to a hybrid reconstruction, in which an ordered network of ad-atoms and a partial intermixing at the level of the rest-atoms lead to a hybrid incommensurate and ordered structure.

The 3×3 was first imaged by means of Scanning Tunneling Microscopy (STM) in 1997. 3 This system shows a clear distinction between STM features at occupied states (stm o ) and unoccupied states (stm u ). Several attempts [3][4][5][6][7] to explain these observations with an atomic model of the surface were unsuccessful since the exact composition of the surface as well as the affected number of layers cannot be attained from the experimental data. The difficulty of determining the composition of the bare 3×3 is increased as this surface undergoes an evolution by increasing the temperature, with the subsequent formation of graphene on it. 8 In 2012, cutting-edge STM data 5,[START_REF] Hiebel | Investigation of the graphene -SiC(000-1) (carbon face) interface using scanning tunneling microscopy and ab initio numerical simulations[END_REF] revealed new features (stm g ) at an energy range close to zero bias. The three distinct STM features of the 3×3 are reproduced in Fig. 1(a-c). Following the notation in Ref. 3, these features are explained in terms of three symmetry points of the surface cell, namely A, B and C; at

( 1 3 , 1 3 ), ( 2 3 , 2
3 ) and (1, 1) respectively. The stm o shows a simple kagome-like pattern where sites A and B, at the center of each triangle, have the same intensity and are equivalent by rotation. This equivalence is broken at stm u where only at sites A a clear wide spot is visible in the surface cell. On the other hand, sites C

show low intensity values in stm o and stm u . However, the picture is quite different at low bias, i.e. stm g [Fig. 1(c)], where only signal from some C sites is observed when analyzing conductance images of several adjacent surface cells. Bright and dark signals at C sites (labeled C + and C -) appear without any clear order. The analysis of different images shows that the C + sites represent between 60% and 70% of the total number of C sites. 5 This puzzling situation motivates us to explore the potential energy surface (PES) for this system in order to identify the structure. An exhaustive search for the PES, with methods such as ART 10 is, however, not possible today for two main reasons. First, with both silicon and carbon easily adopting complex structures, the PES of the SiC surface is complex, almost glassy-like in character. 11 This suggests that a complete exploration of the landscape would require sampling a large number of states, many accessible only through highly non-equilibrium pathways. Second, and more important, the PES exploration must take place in the grand canonical ensemble and current systematic exploration tools are not able to treat this ensemble efficiently. In view of these limitations, we select to pursue an extensive manual search mainly driven by user intuition and comparison spanning over a large range of surface concentration as depicted in Fig. 2(a). Energy optimization for all the configurations are determined using BigDFT with surface conditions 12 in a 3×3 orthorhombic cell containing four SiC bilayers stacked in 4H, the silicon atoms of the bottom bilayer being hydrogen passivated. The LDA functional and 2 k-points in the zigzag direction are used in addition to standard parameters. [START_REF] Machado-Charry | [END_REF] Based on the surface formation energy (FE) plot, selected configurations (see below) are further analyzed with FIREBALL package 13 for extracting the hamiltonian partial DOS for the STM simulation with a home-made code, using a realistic W-tip. 14 In 2015, while analyzing new STM topographies for ultra-thin silicon dioxide layer grown on 2×2 Ru oxide, 15 we identified some similarities between both systems. Figures 1(a) and 1(d) show that there is a clear correspondence between the stm o of the 3×3 and the STM image of the silica ad-layer on Ru oxide. Both show a kagome network, the only difference being that the C site is always bright in the latter. Although both images were taken at different polarity, the known configuration of the silica (Si 2 O 3 ) on Ru oxide [Figs. 1(e-f)] is used as a template for the 3×3. Thus, we simply transmute each bridging oxygen of the silica into a bridging silicon atom. This results in an almost flat over-layer made of five silicon ad-atoms [Figs. 1(g-h)]. Due to its two dimensional character this model is called 2DL 1 in our exploration nomenclature [Fig. 2(b)]. As can be seen in Figs. 1(i-j) the simulated STM images based on the 2DL 1 model nicely reproduce both stm o and stm u . Through careful scrutiny of these images, sites A and B in Figs. 1(a-b) can be identified with the two non-bridging silicon atoms, i.e. with the two silicon atoms of the silica model forming a honeycomb network. On the other hand, the three bridging silicon atoms are responsible for the bright STM features in the kagome pattern, as the bridging O atoms do for silica [Fig. 1(d)]. We will refer to these bridging sites as O sites in the following.

The 2DL 1 over-layer, despite the previous results, fails to reproduce the third STM condition, i.e. stm g . This discrepancy is a weak but is yet a clear signal that something is still missing in the 2DL 1 model. Indeed, in addition to the three previously discussed STM conditions, the low formation energy FE condition should also be fulfilled for a model to be a reasonable candidate to explain this reconstruction. As demonstrated by previously published models, [3][4][5][6][7] that also have failed to reproduce one or the other condition, meeting all the constraints in a model is a challenge.

To raise to this challenge, it is necessary to explore a wide range of possible configurations.

Figure 2(c) depicts the grand potential energy for the more than 700 models considered in our study, demonstrating the potential richness of this surface. Colorized lines in Fig. 2(c) highlight the models with the lower formation energy in a given chemical potential window. These lower bound models fulfill by definition the FE condition. Indeed, the 2DL 1 model is the one with lower

FE at µ C -E bulk C = 0,
where it is also close in energy with the (2×2) C phase (model DA 1 in the plot). 16 It has been observed 17 that the latter is in coexistence with the 3×3 when they are formed in ultrahigh vacuum around 1075 • C. We also colorize the three significant models of literature that claim to reproduce the 3×3 reconstruction but failed either for thermodynamics reasons 3,7 or STM incompatibility. 6 The model proposed by Nemec et al. 6 (3DL 1 in Fig. 2(c)) is interesting as it was the first to be a pure silicon structure that turns out to have a large stability range compatible with that of the bulk SiC. This team has pursued its search and independently proposed the same ad-layer 2DL 1 as a possible candidate for the 3×3. 18 In order to understand why the 2DL 1 silicon over-layer does not reproduce the stm g condition it is necessary to analyze the interaction of this layer with the last carbon layer. Among the nine dangling bonds in the 3×3 cell, two are passivated by the silicon ad-atoms in top position (A and B sites), six are passivated by the three bridging Si connecting (O sites), the remaining dangling bond (C site) being not passivated. These atoms present a large peak in the DOS at the Fermi level (not shown) associated to their half-filled dangling-bond.

Pursuing with our analogy with the silica over-layer for which we recently got more insight, 19 we build a supercell of the 2DL 1 where some of the carbon atoms at the C site are substituted by silicon atoms. The calculations are done in a 9×6 orthorhombic supercell containing 12 C sites that allow a substitution ranging from 25 to 66%. The formation energy for different ratios of silicon substitution is depicted in the inset of Fig. 2(c): from a thermodynamic point of view, these substitutions at the C site are less stable than the parent 2DL 1 and can thus be viewed as defective.

In order to observe the effect of these substitutions on the STM images at energies close to zero bias, we make use of a smaller supercell, because we find that the contrast in the simulated stm u image is quite sensitive to the thickness of the slab. Therefore, in a 3×3 orthorhombic supercell with twice the thickness of our original 4H-SiC slab, we recalculate a model with 50% of substitution. Thus, the model contains only two inequivalent C sites [Fig. 3(d)]. The corresponding STM simulations are depicted in Fig. 3(a-c) together with the partial DOS for the surface atoms lying at the four characteristic sites A, B, C and O in panel (e).

In the partial DOS, we observe a change in the charge transfer on one hand between the carbon atom and the silicon atom at both C sites, and on the other hand between the both atoms at C sites and the silicon over-layer. Interestingly, a new state appears in the gap due to the silicon atom substituting one of the carbon atom at the C site. The energy range of this silicon state is broad and centered around -0.6V. More precisely, this -0.6V state corresponds to p x and p y orbitals hybridized with the three neighboring silicon atoms. The state at the Fermi level is of s and p z character. The simulated STM image at -0.6V [Fig. 3

(c)] clearly allows to assign this Si defect at

C site as the C + defect observed by Hiebel et al.. 5,[START_REF] Hiebel | Investigation of the graphene -SiC(000-1) (carbon face) interface using scanning tunneling microscopy and ab initio numerical simulations[END_REF] Subsequently, the C -sites are ascribed to the last carbon atom presenting a dangling bond in the parent 2DL 1 model. Interestingly, the carbon atoms at the C -sites are drawn towards the surface whereas silicon atoms at the C + sites relax outward it [Fig. 3d] with a height difference of 0.5Å between both sites. This height difference is consistent with differences observed in the profile lines from STM images (≈ 0.4 and 0.5Å in the Fig. 1(c) in Ref. 5) further confirming the present assignment of C + and C -sites. This small difference, moreover, has an impact in the low intensity values of the these sites in the simulated STM image at -2.5V [Fig. 3(a)]. This is also confirmed by the experimental topographies (Fig.

1(b) of Ref. 5).

Finally, the disorder seems to derive from the complex charge transfer between C + and C - sites and the over-layer. In order to introduce the idea of disorder, we have to acknowledge that the charge transfer is incomplete as some electrons stay on C -sites, leading to a peak at the Fermi level. We infer that this comes from a complex balance of the electron in the over-layer on one hand and in the substitutional silicon at C + site on the other hand. This is an important point for graphene growth as this over-layer is interfaced with graphene. The occurrence of such a disorder at C sites in the experimental STM [Fig. 1(c)] is an additional proof that a perfect passivation coming from complete charge transfer can not be found in a finite size super-cell.

To get more insight on the physics of the charge transfer between C sites, we substitute in the 3×3 super-cell one of the two carbon atoms at C sites by a Ga atom and the second one by an As atom. This choice being driven by the reminiscence of the topology of the C + and C -sites with these of a <110> GaAs surface 20 . In this well documented surface reconstruction case, the difference of electronegativity between Ga and As atoms drives a charge transfer from Ga to As sites leading to a Ga atom with a quasi sp2 configuration (with a empty p z orbital) and an As atom with a quasi sp3 configuration that displays a lone pair. Such a charge transfer leading to a selfreconstruction of all the dangling bonds. As a matter of fact, we found that such a complete charge transfer also occurs between the Ga-substituted C site and the As-substituted one (as calculated by a Bader analysis 21 ). The corresponding topologies match those of C + and C -for As and Ga atoms respectively. In the case of silicon substituted C sites, the charge transfer occurs but only partially as already discussed. We suspect that this partial charge transfer comes from a competition between the two following properties: a larger bond length for Si-Si with respect to C-Si and a lower electronegativity for silicon with respect to C. Indeed, the first property drive Sisubstituted C site to a C + topology while the second property drive them to the C -one thus leading the whole system to an incommensurate charge transfer in our two C sites supercell. This analysis is consistent with the experimental ratio of C + sites being ≈ 65%, as indeed a 50% ratio would correspond to a full and commensurate charge transfer as in the case of Ga and As substitution.

In summary, based on a extensive exploration of the surface PES, we propose an energeticallyfavorable atomic model that reproduces for the first time the three main STM conditions reported for the 3x3 reconstruction of the C-face of SiC. 5 This model is close to that of the silica overlayer grown on Ru oxide. 15 It forms an all-silicon honeycomb-kagome network at the surface of a 3×3 supercell that passivate 8 over the 9 carbon dangling bonds in the supercell. The last carbon dangling bond is passivated through a charge transfer between Si-substituted sites and un-substituted one. This charge transfer is, however, not commensurate with the lattice due to a competition between the bond length and the electronegativity of silicon with respect to C. This competition explains the disordered character in the two different types of C sites, thus leading to a strong disorder below the highly ordered 3×3 overlayer. eV. Some previous models are: 3DL 1 (blue), 6 3DC 2 (lime green) 7 and 3DC 1 (dark red). 3 12

FIGURE CAPTIONS FIG. 1 . 3 FIG. 3 .

 CAPTIONS133 FIGURE CAPTIONS

FIGURESFIG. 1 . 11 FIG. 2 .

 1112 FIGURES

FIG. 3 .

 3 FIG. 3. STM images of a 2DL 1 supercell with 50% silicon substituted C sites at different voltages: (a) -2.5V, (b) 2.5V and (c) -0.6V. The five different site types are marked by colored circles: A (orange), B (green), C + (light blue), C -(dark blue) and O sites (red). The surface cell of the 3×3 is represented by a cyan rhombus. (d) 3D view of the surface with the color marks used in a-c panels. (e) The partial DOS for the five type of atoms highlighted in (d). Color lines follow the same color code as in (d).

ACKNOWLEDGMENTS E.MC, LM, NM and PP acknowledge financial support from the NanoScience Fundation. NM also acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada. CG acknowledges financial support by Spanish government through project MAT2017-88258-R and the "María de Maeztu" Program for Units of Excellence in R&D (MDM-2014-0377). The BigDFT calculations were done using French supercomputers (GENCI) through Project 6194 and the supercomputers of Calcul Québec/Compute Canada. E.MC and PP acknowledge stimulating discussions with Dr. Fanny Hiebel, Dr. Jean-Yves Veuillen and Dr. Pierre Mallet and their valuable collaboration.