Anis Bkakria

Frédéric Cuppens
email: nora.cuppens@telecom-bretagne.eujose.fernandez@polymtl.ca

Nora Cuppens-Boulahia

José M Fernandez

Confidentiality-Preserving Query Execution of Fragmented Outsourced Data

Keywords: Data confidentiality, Privacy-preserving, Data fragmentation, Data outsourcing

Ensuring confidentiality of outsourced data continues to be an area of active research in the field of privacy protection. Almost all existing privacy-preserving approaches to address this problem rely on heavyweight cryptographic techniques with a large computational overhead that makes inefficient on large databases. In this paper, we address this problem by improving on an existing approach based on a combination of fragmentation and encryption. We present a method for optimizing and executing queries over distributed fragments stored in different Cloud storage service providers. We then extend this approach by presenting a Private Information Retrieval (PIR) based query technique to enforce data confidentiality under a collaborative Cloud storage service providers model.

Introduction

In the last few years, database outsourcing has become an important tool in IT management, as it offers several advantages to the client companies, especially for small ones with limited IT budget. In most models of database outsourcing, data storage and management (e.g. data backup and recovery) are completely operated by external service providers that take advantages of economies of scale to reduce the cost of maintaining computing infrastructure and data-rich applications in comparison with the cost of in-house data management.

Nonetheless, outsourcing gives rise to major security issues due to the usually sensitive nature of information saved in databases. Storing such sensitive information with external service providers requires them to be trusted and that they should protect data against external threats. Even if these service providers protect data adequately and process queries accurately, they may be curious about the stored data and may attempt to profit by disclosing this sensitive information to third parties. This raises the question of whether it is possible to protect confidentiality and privacy of the outsourced data. A recent study [START_REF]Fourth annual us cost of data breach study[END_REF] by the Ponemon Institute indicates that since 2005 more than 250 million customer records containing sensitive information have been lost or stolen. The same study reports the average organizational cost of data leakage between 2005 and 2009, to be about $6.6 million. Hence, there seems to be a clear financial motivation to protect data privacy and confidentiality even when records are leaked.

Related Work. One approach to protect confidentiality and privacy of the outsourced data is based on encrypting all tuples in outsourced databases [START_REF] Hacigümüs | Providing database as a service[END_REF][START_REF] Hacigümüs | Executing sql over encrypted data in the database-service-provider model[END_REF]. With this approach, one crucial question is that of how to efficiently execute queries. Clearly, when using deterministic encryption techniques equality-match queries are simple to evaluate. However, range queries and aggregations become more difficult to perform as we must decrypt all records to evaluate this kind of queries, which makes query execution on the outsourced encrypted data much more difficult. Therefore, our main focus in this paper is about preserving both privacy and confidentiality of outsourced database while ensuring a secure way for querying outsourced data. One promising method to meet this requirement is to use data fragmentation. Basically, data fragmentation procedures are not particularly designed for preserving data security, they are aimed to improve data manipulation process, optimize storage, and facilitate data distribution. Nevertheless, in the last few years two significant alternatives have been proposed. The first one [START_REF] Hudic | Data confidentiality using fragmentation in cloud computing[END_REF][START_REF] Biskup | On the inference-proofness of database fragmentation satisfying confidentiality constraints[END_REF] relies on data fragmentation alone to protect confidentiality. The distribution model used in this approach is composed mainly of two domains: a trusted local domain from which the data originates, and a honest but curious domain in which the data will be distributed. Because of its trustworthiness, the local domain is used to store fragments that contain highly sensitive data without the need to encrypt them. This is not so efficient in that it forces data owners to always protect and manage fragments containing highly sensitive data. A more promising alternative [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF][START_REF] Ciriani | Enforcing confidentiality and data visibility constraints: An obdd approach[END_REF] attempts to combine data fragmentation together with encryption. In this proposal, the main idea is to partition the data to be externalized across two or more independent service providers, and furthermore to encrypt all information which can not be secured by fragmentation (e.g. employees' bank account numbers of a company). While they do constitute an interesting way to ensure confidentiality of outsourced database, these approaches have the major limitation that it assumes that data to be outsourced is represented within a single relation schema (or table). Clearly, this assumption is too strong and seldom satisfied in the real environments, as generally, relational databases are normalized to minimize redundancy and dependency by dividing large tables into smaller (and less redundant) tables and defining relationships between them.

In this paper, we strive to protect the confidentiality and the privacy of sensitive outsourced database using both encryption and fragmentation, combining the best features of both approaches. Furthermore, we aim to overcome the previously mentioned limitations of [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF][START_REF] Ciriani | Enforcing confidentiality and data visibility constraints: An obdd approach[END_REF] by presenting an approach which is able to deal efficiently with multi-relation normalized databases. In a distributed environment, the problems encountered in one-relation 1 databases take on additional complexity when working with multi-relation normalized databases as it gives rise to new problems such as protecting the relatioships between relational schemas and defining a secure and efficient technique allowing authorized users to query these sensitive relationships. We will show how to protect data unlinkability of different fragments of the original database by protecting user query privacy using a practical Private Information Retrieval (PIR) technique. Unlinkability of two items of interest (e.g., records stored into different fragments) means that within the system, from an adversary point of view, these items of interest are no more and no less related. In our approach, a relation containing sensitive information will be fragmented into two or more fragments. Unlinkability of fragments means that despite the fact that an adversary has knowledge about the fragments of a relation, it remains unable to link records from different fragments.

The remainder of the paper is organized as follows. Section 2 illustrates through an example the problem and need for an approach like ours. In Section 3, we detail the threat model, security model, and assumptions. Section 4 describes our approach to enforce privacy and confidentiality of outsourced data. Section 5 presents the query optimization and execution model. In Section 6, we present a PIR-based technique to achieve query privacy and enforce data confidentiality under a collaborative Cloud storage service providers model. Finally, we conclude the paper in section 7.

Motivating Example

Consider a relational hospital database D with relations: Patient(Id, Name,ZIP, Illness, Id Doctor *) and Doctor(Id, Name, Specialty) where Id Doctor is a foreign key referencing the column Id of the relation Doctor. Let us assume that the hospital is going to outsource the database to a third party. Sensitive information contained in D should be protected. If we look carefully, we can consider that the list of patients and their attributes (Id, N ame, Zip) are not sensitive, and also that the list of illnesses could be made public. Nevertheless, the relationship between these two lists (list of patients and list of illnesses) should be protected. Therefore if we can find a way (e.g. vertical fragmentation [START_REF] Ceri | Vertical partitioning of algorithms for database design[END_REF]) to break relationships between patients and their respective illnesses, there is no need to encrypt all records of the P atient relation. On the other hand, the relationship between a patient and his doctor should be protected. Since the list of doctors itself is not sensitive, the simplest way to protect the relationship between the two relations P atient and Doctor consists in encrypting the foreign key Id Doctor. Actually, we can either encrypt the foreign key Id Doctor of the relation P atient or the primary key Id of the relation Doctor, because in the two cases, relationship between relations P atient and Doctor will be protected. However, encrypting the foreign key seems to be more beneficial as a foreign key references only one relation (only the relationship between the two relations is protected) while a primary key can be referenced by many relations. Therefore, if we encrypt the primary key, we will protect all relationships between the relation containing the primary key and other relations referencing the encrypted primary key. Thus, when the security requirement specifies that only the a relationship between data is sensitive, our apporach is more appropriate than the one based on full encryption.

Threat Model and Security Assumptions

Our approach is based on a typical client-server architecture, where servers are managed by different service providers. These service providers are considered "honest-but-curious", in agreement with most related work [START_REF] Biskup | On the inference-proofness of database fragmentation satisfying confidentiality constraints[END_REF][START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF][START_REF] Hacigümüs | Executing sql over encrypted data in the database-service-provider model[END_REF][START_REF] Hudic | Data confidentiality using fragmentation in cloud computing[END_REF]. Service providers are assumed to be "honest" in that they do not manipulate outsourced data in order to respond incorrectly to user queries. In other words, we suppose that responses to user queries received from these service providers are always accurate. In the first part of this paper, we will assume that service providers are "curious" in that they will try to infer and analyze outsourced data, and will also actively monitor all received user queries and try to derive as much information as possible from these queries. In the second part of the paper, we further assume that service providers can collude and cooperate together to link outsourced data. The client part of this architecture is assumed to be trustworthy and all interactions between the user and the client are secured. Protecting the client part against external attacks is beyond the scope of this article.

Confidentiality using fragmentation and encryption

Our approach extends in several ways the vertical fragmentation-based approach described in [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF][START_REF] Ciriani | Enforcing confidentiality and data visibility constraints: An obdd approach[END_REF]. This approach considers that all data is stored in a single relation, while in our approach data can be stored in several relations, which is the case in typical database environments. In our approach, we consider that databases to be externalized are normalized so that two relations can be only associated together through a primary key/foreign key relationship. For this purpose, we introduce a new type of confidentiality constraint for fragmentation, the inter-table fragmentation constraint. The aim of this new fragmentation constraint is to protect the relationship between relations. This section first presents the different kinds of confidentiality constraints used to achieve our goals of protecting the confidentiality by encryption and fragmentation, and second formalises the concept of fragmentation in our approach which extends ideas presented in [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF][START_REF] Ciriani | Fragmentation design for efficient query execution over sensitive distributed databases[END_REF][START_REF] Ciriani | Enforcing confidentiality and data visibility constraints: An obdd approach[END_REF].

Definition 1 (Confidentiality Constraint). Consider that data to be secured are represented with a relational database D, which is composed of a list of relational schemas R = (R 1 , . . . , R n ), with each of these relational schemas R i containing a list of attributes A Ri = (a 1,i , a 2,i , . . .). A confidentiality constraint over D can be one of the following: Singleton Constraint (SC). It is represented as a singleton set SC Ri = {a j,i } over the relation R i . This kind of confidentiality constraint means that the attribute a j,i of the relational schema R i is sensitive and must be protected, typically by applying encryption. Association Constraint (AC). This kind of confidentiality constraint is represented as a subset of attributes AC Ri = {a 1,i , . . . , a j,i } over the relational schema R i . Semantically, it means that the relationship between attributes of the subset AC Ri is sensitive and must be protected.

Inter-table Constraint (IC).

It is represented as a couple of relational schemas IC = {R i , R j } of the relational database D. Relations R i and R j should be associated through a primary key/foreign key relationship. The use of this kind of confidentiality constraint ensures protection of the primary key/foreign key relationship between the two relational schemas concerned with the inter-table constraint IC.

Note that protecting the relationship between two tables relies on protecting the primary key/foreign key relationship and storing involved relations separately. The association constraint can also be addressed through encryption (encrypt at least one of attributes involved in the constraint), but clearly this will increase the number of encrypted attributes and make interrogation of the database more complicated. A more adapted way to resolve this kind of confidentiality constraint was proposed in [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF], which is based on splitting involved attributes in a manner that their relationships cannot be reconstructed.

In the case of an inter-table confidentiality constraint, protecting the foreign key using encryption is the simplest way to secure the relationship between the two relational schemas. However encrypting only the foreign key is not enough to keep the relationship between relational schemas secure, as service provider may be able to link records in two relational schemas by observing and analyzing user queries over these relational schemas. To overcome this problem, the two relational schemas in question should be split into different fragments, and each of these fragments should be distributed to a different Cloud storage provider. An interesting approach for modeling constraints and resolving the data fragmentation problem was proposed in [START_REF] Ciriani | Enforcing confidentiality and data visibility constraints: An obdd approach[END_REF], that efficiently computes data fragments that satisfy the confidentiality constraints. It is based on Boolean formulas and Ordered Binary Decision Diagrams (OBDD) and uses only attribute-based confidentiality constraint (Singleton Constraints and Association Constraints). However, it cannot deal as-is with Inter-table Constraints. In order to use this approach, we define a way to reformulate Inter-table Constraint as a set of Singleton Constraints and Association Constraints. We explain this transformation in the definition and theorem below.

Definition 2 (Inter-table Constraint transformation). Consider a relational database with two relations R 1 (a 1 , . . . , a n ) and R 2 (b 1 , . . . , b m * ). Let us assume that R 1 and R 2 are related through a foreign key/primary key relationship in which the foreign key b m of the relation R 2 references the primary key a 1 of relation R 1 . We assume that R 1 and R 2 contain respectively p and q records, with p > 1 and q > 1. An Inter-table Constraint c = {R 1 , R 2 } over relations R 1 and R 2 states that the relationship between these two relations must be protected by encrypting the foreign key b m and by storing R 1 and R 2 in two different fragments. Therefore, the constraint c can be written as follows:

i A singleton constraint SC = {b m } to state that the value of b m should be protected. ii A list of (m × n) association constraints AC = {(a i , b j )|i ∈ [1, n], j ∈ [1, m]}.
We propose the notion of a correct transformation of Inter-table constraints. A transformation of an Inter-table constraint c to a set of confidentiality constraints C is correct if the satisfaction of C enforce the protection of the unlinkability between records of the two relations involved in c. The following Theorem formalizes this concept.

Theorem 1 (Transformation correctness). Given a relational database D composed from two relational schemas R 1 (a 1 , . . . , a n ) and R 2 (b 1 , . . . , b m * ) related through relationship between the foreign key b m of R 2 and the primary key a 1 of R 1 . Let c = {R 1 , R 2 } be an Inter-table constraint, the set of constraints C be the result of the transformation of c, and F = {F 1 , . . . F q } be a fragmentation of D that satisfies C. The Inter-table constraint c is correctly transformed into a set of constraints C if all the following conditions hold :

1.

b m does not appear in clear in any fragment of F.

2.

∀ AC

i,j = {a i , b j } ∈ C, i ∈ [1, n], j ∈ [1, m], if a i ∈ F k and b j ∈ F l then k = l
The main advantage of the Inter-table constraint is that it allows treatment of multi-table relational databases. In addition, it gives a simple way to formulate confidentiality constraints between relations. As we have seen in Item (i) of Definition 2, the attribute b m (foreign key of the relation R 2 ) should be encrypted. However, to be able to query data and construct relationship between relations, the chosen encryption algorithm must be deterministic [START_REF] Bellare | Deterministic encryption: Definitional equivalences and constructions without random oracles[END_REF] in order to preserve uniqueness and allow the construction of relationship between relations (e.g. through JOIN queries). As is known, in normalized multi-relation databases, three types of relationship between relations exist: (1) one-to-one, (2) one-to-many and (3) many-to-many relationships. Inter-table constraints over relations associated using (1) or (2) can be simply transformed as shown in Definition 2, while others associated using (3) need a pre-transformation step before applying the transformation of Definition 2, as they are normally linked through a third relation known as a linking table. The pre-transformation steps is described in the example below.

Definition 3 : Fragmentation [5]

Let us consider a relational database D with relations R 1 , . . . , R n and A the list of all attributes contained in different relations. Given A f the list of attributes to be fragmented, the result of the fragmentation is a list of fragments F = {F 1 , . . . , F m } where each of these fragments satisfies:

i ∀F i ∈ F, i ∈ [1..m], F i ⊆ A f . ii ∀a ∈ A f , ∃F i ∈ F : a ∈ F i . iii ∀F i , F j ∈ F, i = j : F i ∩ F j = ∅.
Note that the list of attributes to be fragmented A f contains all attributes in A, except those concerned with Singleton Constraints (attributes to be encrypted). Condition (i) guarantees that only attributes in A f are concerned by the fragmentation, condition (ii) ensures that any attribute in A f appears in clear at least in one fragment and condition (iii) guarantees unlinkability between different fragments.

Logically, to be able to get information about the original database, we should be able to reconstruct original database from fragments. So after defining the fragmentation process, we shall define a mechanism to combine fragmentation and encryption. More clearly, a mechanism to integrate attributes involved in the Singleton Constraints (attributes to be encrypted) in the suitable fragment. These encrypted attributes allow only authorized users (users who know the encryption key) to construct the sensitive relationships. Based on the definition of P hysicalf ragment proposed in [START_REF] Ciriani | Fragmentation and encryption to enforce privacy in data storage[END_REF], we define our mechanism called Securef ragment to combine fragmentation and encryption. Definition 4 : Secure Fragment : Let D be a relational database with a list of relations R = {R 1 (a 1,1 , . . . , a j,1 ), . . . , R n (a 1,n , . . . , a k,n )}, F = {F 1 , . . . , F m } a fragmentation of D and A f be the list of fragmented attributes. Each fragment F i ∈ F is composed of a subset of attributes A i ⊆ A f . Each A i is composed of a subset of attributes of one or more relations R j ∈ R. We denote by R Fi the list of relations in R which a subset of their attributes belongs to the fragment F i ∈ F . The secure fragment of F i is represented by a set of relations schema R e Fi in which each relation is represented as follows R e j (salt, enc, a 1 , . . . , a k ) where {a 1 , . . . , a k } ⊂ A i ∩ R j and enc is the encryption of all attributes of R j that do not belong to {a 1 , . . . , a k } (all attributes of R j involved in a singleton constraint except those concerned by a singleton constraint over the foreign key), combined before encryption in a binary XOR with the salt. All foreign key attributes which are involved in singleton constraints are encrypted using a deterministic encryption algorithm (e.g., AES) to ensure their indistinguishability. The Algorithm 1 shows the construction of secure fragments.The main reason for reporting all original attributes (except foreign keys involved in the Singleton constraints) in an encrypted form for each relation in a fragment, is to guarantee that a query Q over the original relation R j can be executed by querying a single fragment (which contains R e j ) while preserving confidentiality of sensitive relationships, so we do not need to reconstruct the original relation R j to perform the query Q. Furthermore, encrypting foreign keys ensure the protection of sensitive relationships between relations involved into Inter-table constraints. However, using deterministic encryption algorithm has two issues. First, a major advantage is to enforce indistinguishability of records which allows only authorized users who know the encryption key to execute queries associating these relations. Second, a minor drawback is that it allows an adversary to infer information about repeatedly occurring values of the encrypted foreign keys, but this information does not allow the adversary to break the unlinkability between relations. The attribute salt which is used as a primary key of different relations in the secure fragments protects encrypted data against frequential attacks. In addition, there is no need to secure the salt attribute because knowledge of the value of this attribute will not give any advantage when attacking encrypted data.

Example 4.2. Assume that we have a relational database D of a medical insurance company that contains two relations P atient and Doctor represented respectively in Table 1 and Table 2. The insurance company has defined the a set of confidentiality constraints CC = {C 1 = {SSN }, C 2 = {N ame pat, Illness}, C 3 = {P atient, Doctor}}. As shown before, the first step in the fragmentation pro- ). Relations P atient and Doctor are linked through the foreign key Id doc in the relation P atient, therefore C 3 will be replaced by C 4 = {Id doc} and all possible Association constraints composed of an attribute of the relation Doctor and an attribute of the relation P atient (Guarantee that the relation P atient will not be in the same fragment as the relation Doctor). In our example, attributes SSN and Id doc of the relation P atient are involved in singleton constraints C 1 and C 4 respectively. So they will not be concerned by the fragmentation. As a result C 3 will be replaced by :

• C4 = {Id doc} • C8 = {Dob, N ame doc} • C5 = {N ame pat, Id doctor} • C9 = {Illness, Id doctor} • C6 = {N ame pat, N ame doc} • C10 = {Illness, N ame doc} • C7 = {Dob, Id doctor}
A possible fragmentation of D that satisfies all confidentiality constraints is the set of fragments {F 1 , F 2 , F 3 } with: F 1 = {P atient(N ame pat, Dob)}, F 2 = {P atient(Illness)} and F 3 = {Doctor(Id doctor, N ame doc)}. Next step is the Securef ragmentation transformation (Definition 3). We assume that encryption of the protected attributes uses the deterministic encryption algorithm E with the encryption key K. The result of applying the SecureF ragmentation over different fragments is represented as follows.

• Note that F 3 has not been changed because there is no singleton constraints over the Doctor attributes. Lastly data fragments F 1 , F 2 and F 3 are distributed to different Cloud storage providers.

Query Execution Model

Before discussing techniques for processing query over distributed fragments, we will first present the architecture of our proposed approach. It includes three principal entities : (1) a U ser which attempts to execute queries over the original database, (2) a Client which rewrites user queries by splitting them to create an optimized distributed Query Execution Plan QEP; QEP is a set of sub-queries and other operations (e.g., decryption, join...), it is created by the Query T ransf ormative based on the M etaData which contains information (relations, clear attributes, encrypted attributes, selectivity of attributes) about data distribution in different fragments. Furthermore, the Query Executor executes each of these sub-queries over the appropriate fragments and sends back the results to the client. ( 3) Server represented by different Cloud storage providers in which fragments are distributed.

Query Transformation and Optimization: In our querying model, query transformation is performed by the Query Transformative (QT ) entity on the client side. When receiving a user query, the query is analyzed syntactically and semantically so that incorrect queries are rejected as earlier as possible. Next, based on the M etadata stored on the client side, the QT will attempt to find a fragment on which the user query can be executed, i.e. a fragment in which QT can find all attributes and relations involved in the user query. If such a fragment does not exist, QT will decompose the user query into queries expressed in relational algebra, find out which fragments are involved in the query, and finally transform the user query into a set of fragments queries. Using this set of fragment queries and other operations such as encryption, decryption, join and aggregation, the QT creates a QEP and sends it to the Query Executor. A query can have more that one QEP. Logically, each QEP may have a different execution cost. Thus, the QT should have the capability to pick out the best QEP in terms of execution cost. This capability is explained later in the Query Optimization section.

For multi-fragment query2 , QT will use local join operations as it should combine results of execution of subqueries over fragments. There are two different ways to perform local join operation : (1) Execute all sub-queries in a parallel manner, then join the result on the client side. (2) Execute sub-queries in a sequential manner to have the ability to perform semi-joins using the result of previous sub-queries. While (1) can be cheaper than (2) in terms of sub-query execution, it is much more costly in the join operation because in (1), sub-queries results might contain a lot of records that will not be part of the final results.

In addition to traditional query optimization methods such as selecting conditions as earlier as possible, the QT attempts to minimize the execution cost of the created QEP by applying the selection condition with the most selective attribute, i.e the selection condition which is satisfied by the smallest number of tuples. To give this ability to the QT , we assign a selectivity 3 to each attribute contained in the original database to the M etadata stored in the Client. We apply the optimization method to the example below.

Preserving Data Unlinkability

Ensuring data confidentiality is achieved by preserving unlinkability between different data fragments and by encrypting all sensitive information that cannot be protected using only fragmentation. However, we have seen in the previous section that evaluation of some queries may use semi join in order to join data from different fragments. This will not be a concern in the case of non-colluding Cloud storage providers, but it becomes a serious concern when Cloud Storage Providers (CSP) can collude. In this section, we present our solution to overcome this privacy concern when we assume that CSP can collude to link data stored in different fragments.

To overcome this problem, the Client should have the ability to execute semi join queries and retrieve data from a fragment without the CSP (which stores the fragment) learning any information about the semi join condition values. To meet this requirement, we will use a Private Information Retrieval keyword-based technique. PIR keyword-based was presented in [START_REF] Gilboa | Private information retrieval by keywords[END_REF] to retrieve data with PIR using keywords search over many data structures such as binary trees and perfect hashing. Later, [START_REF] Olumofin | Privacy-preserving queries over relational databases[END_REF] investigated the use of SQL for PIR, based on the use of B+ tree. In the next part of this paper, we will explain how we can use technique presented in [START_REF] Olumofin | Privacy-preserving queries over relational databases[END_REF] to ensure our semi join queries privacy requirement.

Theorem 2. Let D be a multi-relation normalized database, F = {F 1 , F 2 } be a fragmentation of D, and Q be a multi-fragment query that joins records from both fragments F 1 and F 2 . Consider that SCPs in which the fragments F 1 and F 2 are stored can collude to link data stored in these fragments, and that Q is evaluated using semi join operations. Sensitive relationships between F 1 records and F 2 records remain protected if and only if the privacy of the semi join subqueries is guaranteed.

PIR System design. In the Client of our architecture, we give to Query Executor the ability to communicate with different Cloud storage providers through the PIR keyword-based protocol. In the Server, we add on each CSP a P IR Server as a front-end entity to answer Query Executor's PIR queries. An adversary (a Cloud storage provider administrator) who can observe Query Executor's PIR-encoded queries is unable to find out the clear content of the queries. Enforcing integrity on the P IR server side is straightforward since we assume that PIR servers will not attempt to wrongly answer Query Executor's PIR queries.

Keyword Index Structures. The main purpose for using PIR keyword-based is to ensure the privacy of semi join queries. In our approach, this kind of queries is mainly executed over primary or foreign key attributes. Therefore each P IR Server will create a B+ Tree over each indexed attribute (Primary or foreign key). The advantage of such an index structure is that data appears only in the leaves while other nodes are used to control the search. On the leaves of B+ tree and for each key in the tree, we store the tuple corresponding to the key as the data part linked to the key. We consider the act of retrieving a node's data as a PIR operation over all nodes in the B+ tree. In all existing PIR schemes, a common assumption is that the client should know the address of the block or the item to be retrieved. To satisfy this assumption in our approach, the P IR server after creating the index structure, sends an index helper containing the B+ tree's root node to the client.

Semi-Join PIR keyword-based query. Using the PIR keyword-based query requires a setup phase in which the Query Executor and the P IR server exchange information. This setup phase is divided into two steps:

1. The Query Executor sends the Relation schema name and the attribute name over which the semi join query is to be performed to the corresponding P IR server. 2. When receiving the Relation schema name and the attribute name, the P IR server selects the corresponding B+ tree and sends its root node to the Query Executor.

After receiving the root node sent by the P IR server, the Query Executor will compare the list of keys contained in the root node with values used in the condition of the Semi join query in order to find out the indexes of the next nodes to be retrieved. The Query Executor will subsequently perform PIR queries over chosen indexes to retrieve corresponding nodes. Once all items have been retrieved, The Query Executor combines them to build the result of the original Semi join query. Refer to Appendix D for a description of the PIR keyword-based protocol algorithms used in the Client and the Server parts.

Conclusion

Existing approaches based on fragmentation and encryption have focused on single-relation schema database which is a strong and rarely encountered assumption in real environment. In this paper, we have presented a new approach based on fragmentation, encryption and query privacy techniques which ensures confidentiality of outsourced multi-relation databases. We have presented also a way to optimize and execute queries over distributed data fragments.

Our future work will include the implementation of our approach. It will also include enhanced query optimization and execution techniques to overcome some limitations of our approach, such as processing nested queries.

A f key = {a ∈ Ci, Ci ∈ C : |Ci| = 1 and isForeignKey(a) = T rue} /* A f key : The set of foreign keys to be encrypted*/

F := Fragment(D, C f ) for all Fi = {ai 1 , ai 2 , • • • , ai n } in F do R f = classifyAttributes(Fi) /* Classify attributes*/ for all R f i in R f do for all r in R f i do r s [salt] := GenerateSalt(R f i , r) /* r : record */ r s [enc] := E k (t[aj 1 , • • • , aj q ] ⊕ r s [salt]) /* aj 1 , • • • , aj q = Ri -R f i */ for all a in R f i do r s [a] := r[a] /

B Proof of Theorem 1

Proof. According to Item (ii) of Definition 2, the Inter-table constraint will be replaced by all possible associations constraint composed from an attribute of relation R 1 and another from relation R 2 . Due to the fact that an association constraint between two attributes means that the relationship between these attributes will be protected using fragmentation (each attribute will be stored in different fragments), Item (ii) guarantees that relations R 1 and R 2 will be stored in different fragments which hold condition [START_REF] Gilboa | Private information retrieval by keywords[END_REF].

Item (i) of Definition 2 creates a singleton constraint over the foreign key b m of the relation R 2 . Thus b m will be considered as a sensitive attribute and will be protected using encryption, which means that the foreign key b m will not appear in clear in any fragment. As a result, if an adversary succeeds in having access to the fragments in which R 1 and R 2 have been stored, she is unable to link data stored in these relations.

C Proof of Theorem 2

Proof. To prove the Theorem 2, we will use the following two sketches. The fist sketch proves that without ensuring semi join sub-queries privacy, collaborative CSPs can, in some cases, break data unlinkability, while the second sketch proves that, under a collaborative Cloud storage service providers model, protecting data unlinkability can only be guaranteed with the protection of the privacy of the semi join sub-queries.

SKETCH without using the PIR keyword-based protocol: Suppose that the Cient wants to execute a query which joins records from two fragments F 1 and F 2 . Let us consider that the sub-query Q 1 executed over the fragment F 1 has returned n tuples. And the semi-join query Q 2 executed over F 2 has returned m tuples. Therefore, if CSPs that store F 1 and F 2 collude together to link tuples from Q 1 and Q 2 results, the probability to guess correctly the relationship between tuples is:

P ROB[Result(Q 1 ) ↔ Result(Q 2 )] = 1 m×n
Clearly, if m and n are small, CSPs will have a great chance to break data unlinkability.

SKETCH using the PIR keyword-based protocol: Let us consider that the Client attempts to perform a query which joins records from two fragments F 1 and F 2 . According to our defined PIR keyword-based protocol, the Client will execute Q 1 over the fragment F 1 without using the keyword-based protocol. Next, the Client will send the table name T and the attribute name a on which the semi-join will be performed, the Server replies with the root node of the corresponding B+ tree. It is clear from the previous step that the CSP which stores F 2 can only know the attribute name and the table name on which the semi-join will be performed. After receiving the root node, the Client will use the PIR protocol to retrieve internal corresponding nodes until the leaves of the B+ tree are reached. The PIR protocol will ensure that the server will not know which nodes were retrieved by the Client. Moreover, all tuples are stored in the leaf level of the B+ tree. Therefore, in order to retrieve each record, the Client shall execute the same number of PIR queries. Rightfully, the only revealed information when using the PIR keyword-based protocol is the table name and the attribute name on which the semi-join has been performed. Therefore, if CSPs storing F 1 and F 2 collude together to break data unlinkability, they will be able only to infer that the relation T 1 in F 1 over which Q 1 has been executed is linked to the relation T through the attribute a. Due to the fact that the foreign key in T 1 referencing the attribute a in T is encrypted, linking records is not possible.

D SemiJoin PIR keywordbased query algorithms

F 1 :

 1 P atient(salt, enc, N ame pat, Dob, E k (Id doc)) with enc = E K ( SSN, Illness salt) • F 2 : P atient(salt, enc, Illness, E k (Id doc)) with enc = E K ( SSN, N ame pat, Dob salt) • F 3 : Doctor(Id doctor, N ame doc)

Table 1 .

 1 Patient relation

	SSN	Name pat Dob	Illness Id doc
	865746129 A. Barrett 20-08-1976 Illness1 doc 3
	591674603 C. Beat	18-01-1981 Illness2 doc 3
	880951264 N. Baines 14-09-1986 Illness1 doc 2
	357951648 S. Brandt 18-01-1981 Illness3 doc 1

Table 2 .

 2 Doctor relation

	Id doctor Name doc
	doc 1	C. Amalia
	doc 2	D. Annli
	doc 3	P. Amadeus

cess consists in transforming Inter-table constraint (C 3

i.e. a query that cannot be executed over only one fragment

A Secure fragmentation algorithm


Christel Gervais
email: christel.gervais_stary@upmc.fr

Cameron Jones

Christian Bonhomme

Danielle Laurencin
email: danielle.laurencin@umontpellier.fr

Insight into the local environment of magnesium and calcium in low-coordination number organocomplexes using 25 Mg and 43 Ca solid state NMR: a DFT study

Keywords: NMR, 25 Mg, 43 Ca, organocomplexes, GIPAW

come

Insight into the local environment of magnesium and calcium in low-coordination-number organo-complexes using 25 Mg and 43 Ca solid-state NMR: a DFT study

Introduction

Calcium and magnesium are the two most abundant alkaline earth metals. They are widespread in nature, being among the 10 most abundant elements in the Earth's crust and among the 12 most abundant elements in the human body. They belong to the so-called family of "rock-forming elements", and are present in natural minerals in association with carbonates (e.g. calcite, magnesite, dolomite), phosphates (e.g. francolite, struvite), silicates (e.g. akermanite), sulfates (e.g. gypsum, epsomite), and chlorides (e.g. carnallite, antarcticite)… In the body, they can be found in the mineral phase of bones and teeth, in the active sites of enzymes (e.g. calmodulin), and in the physiological fluids, and they play a key role in signal transduction.

In addition to their high abundance, their low toxicity and light weight makes them interesting elements for the fabrication of functional materials. For example, cement and plaster are two key construction materials which contain calcium, while Mg-based alloys are essential parts of modern vehicles. Mg-based electrodes for Mg-batteries and calcium-pyrophosphate based bioglasses are other examples of the many advanced functional materials currently under study in research laboratories (Murgia et al., 2015;Soulié et al., 2016).

In the field of molecular chemistry, organomagnesium and organocalcium complexes are of great interest for organic synthesis. While Grignard reagents have been known and used for over 100 years, an increasing number of purposely designed Mg and Ca complexes find applications in catalysis.

Within these complexes, the Lewis acidity of the metal cation and the nucleophilicity of the organic ligands both play an important role in chemical conversion [START_REF] Harder | Alkaline-earth metal compounds: oddities and applications[END_REF]Torvisco et al., 2011;Kobayashi et al., 2011;Sarrazin et al., 2016). The reactivity of the metal ion strongly depends on its local environment and oxidation state, and hence it is important to determine them precisely. While Xray diffraction is commonly used to characterize the metal ion environments in crystalline forms of these molecules, to the best of our knowledge, 43 Ca and 25 Mg solid state NMR have not been used yet. 25 Mg and 43 Ca are both difficult nuclei to study by NMR (Freitas et al., 2012;Laurencin et al., 2013;[START_REF] Schurko | Solid-State NMR of the Light Main Group Metals. In "The Lightest Metals[END_REF]. Indeed, magnesium-25 is a quadrupolar nucleus (I = 5/2) with a moderate quadrupole moment (Q = 199.4 mbarn), a low natural abundance (10%) and a very low Larmor frequency (ν0 = 52.0 MHz at B0 = 20 T), while calcium-43 is a quadrupolar nucleus (I = 7/2) with a small quadrupole moment (Q = -44.4 mbarn), a very low natural abundance (0.14%) and also a very low Larmor frequency (ν0 = 57.2 MHz at B0 = 20 T). Despite these "unfavourable" nuclear spin properties, the development of (ultra-)high magnetic field magnets and of pulse sequences specific to quadrupolar nuclei has resulted in 25 Mg and 43 Ca NMR becoming more accessible (Perras et al., 2013;Schurko, 2013), and it has been shown that these spectroscopies can be very informative on the local structure around calcium and magnesium in a variety of materials (Cahill et al., 2009;Pallister et al., 2009;Freitas et al., 2012;Laurencin et al., 2013), including crystalline molecular complexes [START_REF] Sene | [END_REF]Sene et al., 2015;Burgess et al., 2013;Burgess et al., 2014). In a number of cases, computational methods involving density-functional theory (DFT) calculations of NMR parameters have also been used to assist in the interpretation of the NMR spectra (Bonhomme et al., 2012a), because the calculation of 25 Mg and 43 Ca NMR parameters can be performed with high accuracy (Laurencin et al., 2012;Gervais et al., 2008). Moreover, such computational methods can also serve as a guideline to determine the acquisition conditions that need to be used for recording the NMR spectra, which can be particularly useful when starting studies on new families of compounds. Indeed, the calculated quadrupolar parameter CQ can help determine at what field and under what conditions (static or magic angle spinning) the measurements need to be performed.

In this article, we report DFT calculations of the 25 Mg and 43 Ca NMR parameters for a series of crystal structures involving organomagnesium and organocalcium complexes, in which the alkaline earth metal ions have low coordination numbers and are mainly coordinated to N-bearing ligands. The compounds studied were selected as representative examples of the types of systems currently developed in molecular chemistry (Torvisco et al., 2011). The values of the calculated NMR parameters are discussed in view of the local environment of the magnesium and calcium, and it is shown in what way they can be used for "NMR crystallography" types of applications [START_REF] Harris | NMR Crystallography[END_REF], to help refine crystal structures. Moreover, acquisition conditions for recording the spectra of such complexes are also proposed, opening the way to future experiments on these systems.

Computational details

DFT calculations were performed on a total of 20 different structures. These were selected on the basis of (i) the quality of the X-ray diffraction data, (ii) the size of the unit cell, (iii) the presence of hydrogen atoms in the published structures (these H positions having been either determined from the F-maps and refined with isotropic thermal parameters, or placed geometrically and refined using a riding model). The reference CCDC .cif file numbers of the structures which were studied are the following (together with the experimental temperatures at which the XRD measurements were performed): S1).

NMR chemical shift calculations were performed within the DFT formalism using the QUANTUM ESPRESSO (QE) software (Giannozzi et al., 2009). The PBE generalized gradient approximation (Perdew et al., 1996) was used and the valence electrons were described by norm conserving pseudopotentials (Troullier et al., 1991) in the Kleinman Bylander form (Kleinman et al., 1982). The Mg pseudopotential was generated by D. Ceresoli and exported from his web site (Ceresoli, 2016). The Ca pseudopotential used here is the same as in some of our previous work (Gras et al., 2016). Spinorbit relativistic effects were neglected in the present account; their inclusion in the DFT calculations would possibly lead to small changes for light-weight elements like Ca and Mg (Komorovsky et al., 2015). Dispersion corrections were not included, as they were found to have no strong effect on the calculated NMR parameters for the systems studied here (see Table S2). Because of the large cell sizes, the Γ point was used for the Brillouin zone integration in most calculations to reduce computational cost, because no strong variation in the calculated NMR parameters was found for higher k-point meshes (see supporting information Tables S1 andS3; for structures 198654 and 247152). A plane wave energy cut-off of 80 Ry was used for Ca-based systems, and 60 or 80 Ry for Mg systems.

Comparisons of calculations with cut-off values of 60 and 80 Ry on the Mg structure 198354 are shown in the supporting information (Table S1).

The shielding tensor was computed using the Gauge Including Projector Augmented Wave (GIPAW) approach (Pickard et al., 2001), which enables the reproduction of the results of a fully converged all electron calculation (Lejaeghere et al., 2016). Absolute shielding tensors were obtained. To set the 43 Ca chemical shift scale, as previously described (Gervais et al., 2008), the calculated δiso for a series of reference compounds were compared to experimental values so that the average sum of experimental and calculated shifts coincide (equations can be found in supporting information, Table S4 and Figure S1). The same procedure was applied for the 25 Mg chemical shift scale, and the comparison between calculated and experimental data is presented in supporting information (Table S5 andFigure S2); the slope of the linear correlation was fitted freely to allow extraction of σref.

Diagonalization of the symmetric part of the calculated tensor then provides its principal components σ11, σ22, σ33 from which the chemical shift components δ11, δ22, δ33 can be calculated. δ11, δ22 and δ33 are defined such as |δ33 -δiso| ≥ |δ11 -δiso| ≥ |δ22 -δiso|, and δiso = 1/3(δ11 + δ22 + δ33). The chemical shift anisotropy (CSA) parameters are defined by ΔCSA = δ33 -δiso and ηCSA = |(δ22 -δ11)/ΔCSA| (Haeberlen convention) [START_REF] Haeberlen | Advances in Magnetic Resonance[END_REF]. The principal components Vxx, Vyy, and Vzz of the electric field gradient (EFG) tensor (defined as |Vzz| ≥ |Vxx| ≥ |Vyy|) are obtained by diagonalization of the traceless EFG tensor.

The quadrupolar interaction can then be characterized by the quadrupolar coupling constant CQ and the asymmetry parameter ηQ, which are defined as: CQ = eQVzz/h and ηQ = (Vyy-Vxx)/Vzz (e is the proton charge, h Planck's constant and Q the quadrupole moment of the considered nucleus). The experimental values of the quadrupole moment of 25 Mg (Q = 19.94 × 10 -30 m 2 ) and 43 Ca (Q = 4.44 × 10 -30 m 2 ) were used to calculate CQ (Pyykkö 2008;Burgess et al, 2014). All calculated values are at 0 K. Simulations of spectra were performed using the DMFit (Massiot et al., 2002) and QUEST (Perras et al., 2012) programs. The Euler angles reported describe the relative orientation of the EFG tensor with respect to the shielding tensor; they are provided in the relevant Tables in Supporting Information.

Results and discussion

Calcium complexes

The coordination environments around calcium in six of the complexes studied here are shown in Figure 1. The cation is linked to 3-to 4-ligands, two of which correspond to bulky silylated amines. For the 3coordinate systems (Figures 1a-b), a nearly planar arrangement of the atoms around the metal cation is observed, the X-Ca-X' angles ranging between ~105 and 135°. In the 4-coordinate systems (Figures 1c-f), the ligands are positioned in a tetrahedral configuration around the metal, but these tetrahedra are distorted, as shown by the unequal Ca…X distances (see supporting information, Tables S6 to S11), and the range of X-Ca-X' angles (between ~80° and 140° depending on the complex). These environments are very different from the 5 to 8-coordination spheres of most Ca(II) compounds (Sowrey et al., 2004;Wong et al., 2008;Burgess et al., 2014;Gambuzzi et al., 2015). The calculated 43 Ca NMR parameters for the 3-coordinate structures are reported in Table 1, and those of the 4coordinate structures in Table 2. (Tang et al., 2005a), b) 704097 (Barrett et al., 2008), c) 255195 (Tang et al., 2005a), d) 255196 (Tang et al., 2005a), e) 198356 (Vargas et al., 2002), f) 247152 (Gillett-Kunnath et al., 2005). or CNHC, where NHC refers to the N-heterocyclic carbene ligand). The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (Figure S1).) 8

Overall, the calculated isotropic chemical shifts (δiso > 80 ppm), quadrupolar coupling constants (|CQ| > 5 MHz) and CSA spans (|ΔCSA| > 90 ppm) were found to be larger than those reported in previous experimental 43 Ca NMR studies on organic and inorganic materials, in which the metal is mainly surrounded by O-bearing ligands and has higher coordination numbers (Laurencin et al., 2013;Burgess et al., 2014;Widdifield et al., 2014). The high isotropic chemical shifts can be explained by the short average Ca…X distances (Laurencin et al., 2013), while the large CQ values are due to the anisotropy of the electron distribution around the calcium, which depends on the way in which the 3 (or 4) ligands are positioned around the metal. For the complexes presented in Figure 1, smaller quadrupolar parameters were found for the 4-coordinate systems than the 3-coordinate ones, in agreement with the more symmetric electron distribution in the former case. Interestingly, the calculated CSA span |ΔCSA| was found to globally increase as a function of |CQ| (Figure 2a), and the CSA and EFG sensors appeared to have similar directions (Principal Axes Sytems, PAS) for several compounds (Figures 2b-d). In the case of the 3-coordinate complex involving an N-heterocyclic carbene ligand (Figure 1b), the eigenvectors corresponding to δ22 and Vyy eigenvalues were actually along the Ca-CNHC direction (Figure 2c). S12). The calculated values provide important guidelines for determining how to record 43 Ca NMR spectra for such compounds. For measurements under MAS (Magic Angle Spinning) conditions, from which δiso and the quadrupolar parameters can be extracted, ultra-high magnetic fields (B0 ≥ 20 T) with rotors spinning at 10 kHz (at least) should to be used, in order to avoid any overlap between the central transition and the associated spinning sidebands (see supporting information, Figure S3). Measurements under static conditions may also be interesting to carry out at high field, in order to extract the CSA parameters (Figure S4). The 1 H… 43 Ca dipolar couplings being very weak (~400 Hz to the most), 1 H decoupling is not expected to significantly improve the measurement under MAS conditions, while it will be needed for static measurements. Due to the very low natural abundance of calcium-43, isotopic enrichment will most probably be necessary for both static and MAS experiments (Laurencin et al., 2013), which implies that the synthetic procedures of these compounds will need to be adapted, because the commercially available 43 Ca-enriched precursor is *CaCO3.

The comparison of calculated 43 Ca parameters after H-atom optimization and full geometry optimization is also worth discussing. Indeed, it shows the very strong sensitivity of δiso( 43 Ca) to the average Ca…X distance, which depends on the nature of the atoms directly linked to the Ca and the bulkiness of the ligands involved. For changes of more than 0.03 Å in this average distance, the calculated δiso( 43 Ca) varies by more than 5 ppm (see Tables 1 and2). In contrast, for the two structures for which there are the least variations in the local geometry around calcium (CCDC 704097 and 255195 -see Tables S7 andS8), δiso( 43 Ca) varies by less than 2 ppm. Overall, this shows that 43 Ca NMR experiments, in combination with DFT calculations, can be used to contribute to the determination of crystal structures in cases when no single crystal data is available, and/or to their refinement. A typical system for which this would be useful is that of a diketiminato complex described by Avent and coworkers (Figure S5, supporting information), for which the crystal structure published shows disorder in some of the carbon positions (Avent et al., 2005). DFT calculations were performed on a model of this structure (after proposing atomic positions for the partially occupied crystallographic sites). A comparison of the NMR parameters derived from H-relaxation and full relaxation is reported in the supporting information (Table S13). The value of δiso( 43 Ca) changes by nearly 15 ppm between Hrelaxed and fully relaxed structures, as a reflection of the significant modifications in the Ca local environment upon full geometry optimization (Table S14). Recording the experimental 43 Ca NMR spectrum for this compound, and comparing experimental and calculated 43 Ca NMR parameters would therefore be useful in trying to refine better this structure.

Magnesium complexes 10

The local environments around Mg in the 12 complexes which we studied here are shown in Figure 3.

In most cases, the oxidation state of Mg is +II, except for compounds CCDC 1477718 (Figure 3d), 745088 (Figure 3h) and 661565 (Figure 3i), which correspond to diamagnetic Mg(I) dimers. The ligands which are coordinated to the metal cation through N-atoms are bulky silylated amines, guanidinates, diketiminates and pyridine (or its derivative), while the O-bearing ligands are ethers (diethylether and tetrahydrofuran). The coordination numbers for these compounds range from 2 to 4, which is low compared to what is most often observed for Mg complexes (Freitas et al., 2009). For each complex, the distances between the Mg and its nearest neighbors are reported in Tables S15-S26. In the 2coordinate Mg(II) complexes, the N-Mg-N angles vary from ~140° (structure 266469), to ~160° (structure 1207191) and 180° (structure 198354). Concerning the 3-coordinate Mg(II) systems, the coordinating atoms are almost in the same plane as the Mg, and the X-Mg-X' angles vary from ~90 to ~145°. The 4-coordinate complexes display a tetrahedral arrangement of the coordinating atoms around the magnesium, although, just like in the case of calcium, the tetrahedra are distorted due to distributions in the Mg…X distances (see Tables S24 to S26) and the X-Mg-X' angles (from ~95 to ~135°). The calculated 25 Mg NMR parameters for each of these structures are reported in Tables 3 to 5. The CCDC references of these complexes are: a) 198354 (Vargas et al., 2002), b) 1207191 (Bartlett et al., 1994) ,c) 266469 (Tang et al., 2005b), d) 1477718 (Boutland et al., 2016), e) 175074 [START_REF] Kennedy | [END_REF], f) 266470 (Tang et al., 2005b), g) 266468 (Tang et al., 2005b), h) 745088 (Bonyhady et al., 2010), i) 661565 (Green et al., 2007), j) 247151/266465 (Gillett-Kunnath et al., 2005;Tang et al., 2005b), k) 266466 (Tang et al., 2005b) and l) 266464 (Tang et al., 2005b). Values in parenthesis (in italic) were obtained after optimization of all atomic positions, while the other values correspond to H-atom optimization only. d(Mg…N) corresponds to the average Mg…N distance. The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (Figure S2). Values in parenthesis (in italic) were obtained after optimization of all atomic positions, while the other values correspond to H-atom optimization only. d(Mg…X) corresponds to the average Mg…X distance (X = O, N). The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (Figure S2). Values in parenthesis (in italic) were obtained after optimization of all atomic positions, while the other values correspond to H-atom optimization only. d(Mg…X) corresponds to the average Mg…X distance (X = O, N). The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (Figure S2).

As in the case of calcium, the calculated isotropic chemical shifts (δiso > 30 ppm) and spans (|ΔCSA| > 40 ppm) were found to be generally higher than those reported in previous experimental 25 Mg NMR studies. The high δiso values can be explained by the very short Mg…X distances (Freitas et al., 2009;Freitas et al., 2012;Burgess et al., 2013;Laurencin et al., 2012;Wong et al., 2006). The calculated quadrupolar coupling constants (|CQ| > 13 MHz) were also found to be very high, and to the best of our knowledge |CQ| values exceeding 20 MHz had never been reported to date (considering both experimental and computational studies) (Freitas et al., 2012). Such high values can be explained by the very small coordination numbers of Mg within the complexes, which lead to highly unevenly distributed electron densities around the metal ion. It is worth noting that just like in the case of calcium, the CSA span |ΔCSA| increases globally as a function of |CQ| (Figure 4a). Moreover, in several cases, the shielding and EFG tensors were found to be co-axial (Figures 4b,d-f). A noticeable exception concerns the Mg-Mg dimer (CCDC 1477718); in this case, the Vzz and δ33 components are nearly aligned with the Mg-Mg bond direction (Figure 4c). Based on these NMR parameters and the nuclear spin properties of magnesium-25, it appears that experiments performed under MAS conditions would be very difficult to interpret even when working at ultra-high magnetic fields (B0 ≥ 20 T), due to the overlap of the sidebands with the central transition resonance. However, under static conditions, using the latest wide-line acquisition methods (Schurko, 2013), it should be possible to record the static spectra of these compounds. Some of the possible methods include the VOCS-WURST-QCPMG sequence (Variable-Offset Cumulative Spectroscopy -Wideband, Uniform Rate, Smooth Truncation -Quadrupolar Carr Purcell Meiboom Gill), with additional 1 H decoupling to increase the transverse relaxation rates. Such experiments will be challenging, as shown by the simulations of the expected spectra at 20 T (see supporting information, Figure S6), the full breadth of the central transition signal exceeding 1 MHz in some cases. This implies that isotopic enrichment in 25 Mg will most probably be needed to obtain high quality spectra. Due to the very large |CQ| values, it appears that it will not be possible to determine the CSA parameters from the static spectra (see supporting information, Figure S6), but it should be possible to extract |CQ| and ηQ values. Nevertheless, it is worth noting that in cases when there are two different Mg sites with similar local environments (compounds 198354 (Figure 3a) and 1477718 (Figure 3d)), the calculated NMR parameters are very close, with for example |CQ| values differing by less than 1 MHz. This implies that resolving the 2 crystallographic sites using 25 Mg NMR may be very difficult experimentally. An alternate way to measure the quadrupolar parameters could be to perform NQR (nuclear quadrupolar resonance) characterizations at different temperatures.

When comparing the different structures studied here, it appears that |CQ| globally increases as the coordination number decreases from 4 to 2 (Figure 5). Calculations on more species would of course be needed to further confirm these trends, and to include other Mg local environments such as 4coordinate Mg compounds in which the metal ion lies in the same plane as the 4 atoms to which is it bound. Nevertheless, from the data presented here, as a first approximation, it could be tempting to suggest that the measurement of |CQ| can be used as a guideline for determining the Mg coordination number in complex low-coordinate structures, and notably those in which only powder X-ray diffraction data of poor quality is available. However, previous studies on 5-and 6-coordinate Mg(II) complexes show that it may not always be so simple. Indeed, while |CQ| values are generally much lower than 12 MHz in 5-and 6-coordinate complexes (Freitas et al., 2009), i.e. below those observed here for the 4-coordinate compounds, noticeable exceptions exist for porphyrin or phtalocyanine Mgcomplexes, in which the metal ion sits in the cavity of the macrocycle, with distorted octahedral (CN = 6) or square pyramidal (CN = 5) environments. Indeed, |CQ| values were observed in the ~13-15 MHz range for these systems (Wu et al., 2003;Wong et al., 2006), which is close to what we have found for 4 coordinate complexes. Therefore, the |CQ| parameter should be considered carefully, when it comes to predicting Mg coordination numbers. 3l). Indeed, while only small differences in calculated δiso and |CQ| values are observed for compounds 3j and 3k, more significant changes were found for compound 3l, which reflects the fact that the local geometry around the Mg in this complex is more different than in the other two, in terms of N-Mg-X bond angles. Overall, these observations underscore the very high sensitivity of 25 Mg NMR to changes in local environments of the metal, and hence its potential utility for assisting in the refinement of crystallographic data, or even to understand the structure of non-crystalline/disordered structures.

Conclusion

In this manuscript, we have described the first 25 Mg and 43 Ca NMR computational study of a series of low-coordination Mg and Ca complexes (CN = 2 to 4), which are of interest in organometallic chemistry and catalysis. By focusing on the calculated NMR parameters of the metal ions, it was demonstrated how the combination of solid state NMR and GIPAW calculations of NMR parameters can be useful for assisting in the resolution and/or refinement of complex crystal structures. The 43 Ca chemical shifts and 25 Mg quadrupolar parameters were indeed found to be very sensitive to small changes in the coordination environment of the metal ions. The NMR parameters calculated in this study expand by far the ranges of values reported so far for these nuclei. Particularly large quadrupolar coupling constants |CQ| and chemical shift anisotropies |ΔCSA| were notably calculated for both 43 Ca and 25 Mg, and the consequences on the measurement conditions to use for the NMR experiments were discussed. In particular, considering the current instruments and pulse sequences currently available, isotopic enrichment in 43 Ca or 25 Mg will be needed to ensure the acquisition of high quality spectra. Considering the cost of the isotopically-labeled 25 Mg and 43 Ca precursors, it will unfortunately not be possible to perform such experiments on a routine basis.

Attempts were made to correlate the calculated NMR parameters of 43 Ca and 25 Mg to a structural feature around the metal ion, for example its coordination number. The general observation was that |CQ| increased as the coordination number decreased from 4 to 3 (in the case of Ca), or 4 to 2 (in the case of Mg). However, it was shown on the basis of previously published data that this trend needs to be taken very carefully, because in some cases relatively large |CQ| values can also be found for higher coordination numbers if the coordination polyhedron is distorted. It is worth noting that for other s-block metal cations like Sr 2+ and Na + , it has also been shown that it can be very difficult to relate the |CQ| values to a given coordination number (Bonhomme et al., 2012b;Wong et al., 2000). Overall, this indicates that it is only through the combination of NMR experiments and DFT calculations that it will be possible to establish the coordination environment of Ca and Mg, and subsequently use this information as a support to X-ray structure determination.

Table S3 DFT calculations on calcium complex CCDC 247152: influence of the k-point mesh.

Figure S1

Plots of σiso(calc) as a function of δiso(exp) (left), and PQ(calc) as a function of PQ(exp) (right), for the calcium compounds shown in Table S4 (PQ = CQ(1 + ηQ 2 /3) 0.5 )). S5. 1 and2 for the H-relaxed structures.

Simulations were performed assuming an "infinite" spinning speed; only central transitions are represented here. The relative orientation between the two tensors using three Euler angles (,,) was extracted from the firstprinciples calculations, which allows the simulation of the static spectra taking all these parameters into account.

The QUEST software was used to simulate these spectra (Figure S4). The Euler angles reported therefore describe the relative orientation of the EFG tensor with respect to the shielding tensor along the following conventions : The relative orientation between the two tensors using three Euler angles (,,) was extracted from the firstprinciples calculations which allows the calculation of the static spectra taking all these parameters into account.

Ar-=

The QUEST software was used to simulate spectra (Figure S6). The Euler angles reported therefore describe the relative orientation of the EFG tensor with respect to the shielding tensor along the following conventions: 345for the H-relaxed structures were used in both cases. Euler angles are given in Table S27 (above), while the relative orientation of the shielding and quadrupolar tensors is shown in Figure 4.

  255198 (T = 213 K), 704097 (T = 150 K), 255195 (T = 100 K), 255196 (T = 100 K), 198356 (T = 90 K), 247152 (T = 96 K) and 244505 (T = 173 K) for Ca, and 198354 (T = 96 K), 1207191 (T = 130 K), 266469 (T = 100 K), 1477718 (T = 123 K), 175074 (T = 123 K), 266470 (T = 100 K), 266468 (T = 100 K), 745088 (T = 123 K), 661565 (T = 123 K), 247151 (T = 95 K), 266465 (T = 100 K), 266466 (T = 100 K), 266464 (T = 100 K) for Mg. For each structure, calculations of NMR parameters were performed after geometry optimization of either the H positions or all atom positions. The cell parameters were kept fixed to the experimental values during the optimization. In the case of the structure CCDC 198354 (in which the highest quadrupolar coupling constants were found), no strong 4 changes in calculated NMR parameters were observed when optimizing the cell parameters (see Supporting information, Table

Figure 1

 1 Figure 1 Representation of the calcium local environment in the complexes studied by DFT. The CCDC references of the corresponding crystal structures are: a) 255198 (Tang et al., 2005a), b) 704097

Figure 2

 2 Figure 2 a) Correlation between |ΔCSA| and |CQ| for the organocomplexes shown in Figure 1, depending on the coordination number (CN) of the Ca (CN = 3 in green, CN = 4 in red); b-d) Relative orientations of the shielding and EFG tensors in structures CCDC 255198, 704097 and 247152 (for clarity, only the non-H atoms within a 5 Å radius around the Ca are shown here; Euler angles can be found in Supporting Information -TableS12).

Figure 3

 3 Figure 3 Representation of the magnesium local environment in the complexes studied here by DFT.

Figure 4

 4 Figure 4 a) Correlation between |ΔCSA| and |CQ| for the organocomplexes shown in Figure 3, depending on the coordination number (CN) of the Mg (CN = 2 in blue, CN = 3 in green, CN = 4 in red). b-f) Relative orientations of the shielding and quadrupolar tensors in structures CCDC 198354, 1477718, 266470, 266465 and 266464 (for clarity, only the non-H atoms within a 5 Å radius around the Mg are shown, except for 1477718 as the figure would have been too complex; Euler angles can be found in Supporting Information -TableS27).

Figure 5

 5 Figure 5 Representation of the variations in calculated δiso and |CQ| parameters as a function of the coordination number, for the Mg complexes shown in Figure 3.

  ) vs Pq(exp)(Q = 4.44)

Figure S2

 S2 Figure S2 Plots of δiso(exp) as a function of σiso(calc) (left), and |CQ(calc)| as a function of |CQ(exp)| (right), for the magnesium compounds shown in TableS5.

  environments of calcium in the organocomplexes studied by DFT

  δ11 ≥ δ22 ≥ δ33 and |V33| ≥ |V22| ≥ |V11|). Span δ11 -δ33 and skew δ22 -δiso)/

  Figure S5 Representation of the Ca local environment in the crystal structure CCDC 244505.

  d(Ca…X) corresponds to the average Ca…X distance (X = N, O). The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (FigureS1).

Figure S6

 S6 Figure S6Simulation of the static25 Mg NMR spectra at 20 T, for structuresCCDC 198354, 1477718, 266470, 266465, 266464, using the QUEST program. Simulations are performed considering a) both the EFG and shielding tensors, or b) only the EFG tensor, showing that for these compounds, the lineshape is dominated by quadrupolar effects. The values calculated in Tables3-5for the H-relaxed

b) CCDC 704097 a) CCDC 255198 c) CCDC 255195 d) CCDC 255196 e) CCDC 198356 f) CCDC 247152

	Table 1 DFT-calculated 43 Ca NMR parameters for the 3-coordinate Ca-organocomplexes shown in
	Figure 1.
	Ca
	C
	H
	N
	O
	Si
	Values in parentheses (in italic) were obtained after geometry optimization of all atomic positions, while all other
	values correspond to H-atom optimization only. d(Ca…X) corresponds to the average Ca…X distance (X = N, O

Table 2

 2

	CCDC	255195	255196	198356		247152
	Site	Ca(1)	Ca(1)	Ca(1)	Ca(2)	Ca(1)
	δiso (ppm)	103.3 (101.3)	106.0 (99.4)	96.3 (82.9)	94.2 (81.1)	102.1 (89.6)
	CQ (MHz)	-6.11 (-6.02)	-6.68 (-6.32)	5.08 (5.25)	5.17 (5.35)	7.22 (6.28)
	ηQ	0.85 (0.86)	0.61 (0.72)	0.87 (0.84)	0.97 (0.93)	0.15 (0.41)
	ΔCSA (ppm)	121.7 (122.8)	113.4 (109.2)	-99.8 (-102.5) -95.7 (-99.7) -141.6 (-128.6)
	ηCSA	0.95 (0.94)	0.88 (0.97)	0.81 (0.81)	0.82 (0.81)	0.17 (0.28)
	d(Ca…X) (Å)	2.44 (2.45)	2.43 (2.46)	2.34 (2.38)	2.34 (2.38)	2.32 (2.36)

DFT-calculated 43

Ca NMR parameters for the 4-coordinate Ca-organocomplexes shown in Figure

1

.

Values in parentheses in italic were obtained after geometry optimization of all atomic positions, while all other values correspond to H-atom optimization only. d(Ca…X) corresponds to the average Ca…X distance (X = N, O). The equation used to derive calculated shifts from calculated shieldings can be found in supporting information (Figure

S1

).)

Table 3

 3 DFT-calculated 25 Mg NMR parameters for the "2-coordinate" Mg-organocomplexes shown in Figure3.

	CCDC	198354		1207191	266469	1477718
	Site	Mg(1)	Mg(2)	Mg(1)	Mg(1)	Mg(1)	Mg(2)
	δiso (ppm)	46.3 (43.3)	49.0 (45.5)	41.6 (40.7)	40.0 (37.7)	53.5 (50.0)	62.1 (59.1)
	CQ (MHz)	-44.80 (-43.36) -45.69 (-44.15) -36.68 (-35.94) -29.78 (-29.26)	-30.97 (-30.02) -32.01 (-31.27)
	ηQ	0.23 (0.24)	0.23 (0.23)	0.06 (0.06)	0.16 (0.14)	0.20 (0.22)	0.11 (0.11)
	ΔCSA (ppm)	-144.4 (-140.6) -143.7 (-139.4) -120.8 (-119.5) -89.3 (-90.6)	-148.1 (-142.5) -158.5 (-154.2)
	ηCSA	0.25 (0.26)	0.36 (0.38)	0.06 (0.07)	0.09 (0.09)	0.15 (0.20)	0.05 (0.07)
	d(Mg…N) (Å)	1.92 (1.94)	1.91 (1.93)	1.97 (1.97)	1.97 (1.97)	1.99 (2.00)	1.98 (2.00)
	d(Mg…Mg) (Å) -	-	-	-	2.85 (2.86)	2.85 (2.86)

Table 4

 4 DFT-calculated 25 Mg NMR parameters for the "3-coordinate" Mg-organocomplexes shown in Figure3.

	CCDC	175074	266470	266468	745088	661565
	δiso (ppm)	43.4 (38.0)	54.7 (53.4)	54.8 (52.7)	86.8 (84.1)	70.6 (69.1)
	CQ (MHz)	-27.36 (-26.73) 27.34 (26.87)	26.81 (26.12)	23.70 (23.13)	-24.88 (-24.53)
	ηQ	0.70 (0.66)	0.96 (0.98)	0.86 (0.85)	0.87 (0.90)	0.76 (0.76)
	ΔCSA (ppm)	-82.3 (-81.5)	-91.4 (-91.6) 51.6 (51.0)	-88.4 (-87.9)	-106.5 (-105.8)
	ηCSA	0.27 (0.28)	0.35 (0.34)	0.99 (0.97)	0.59 (0.59)	0.20 (0.20)
	d(Mg…X) (Å)	2.00 (2.02)	2.04 (2.05)	2.07 (2.08)	2.04 (2.05)	2.07 (2.09)
	d(Mg…Mg) (Å)	-	-	-	2.81 (2.80)	2.85 (2.85)

Table 5

 5 DFT-calculated 25 Mg NMR parameters for the "4-coordinate" Mg-organocomplexes shown in Figure3.

	CCDC	247151	266465	266466		266464
	Site	Mg(1)	Mg(1)	Mg(1)	Mg(1')	Mg(1)
	δiso (ppm)	34.8 (29.6)	34.8 (28.9)	34.5 (30.5)	36.5 (31.9)	54.1 (50.5)
	CQ (MHz)	-16.86 (-16.98) -16.83 (-16.80) -15.51 (-14.88) -16.21 (-15.48) -13.93 (-13.89)
	ηQ	0.34 (0.34)	0.34 (0.35)	0.60 (0.57)	0.52 (0.51)	0.51 (0.51)
	ΔCSA (ppm)	-54.3 (-55.4)	-54.3 (-54.7)	-42.6 (-41.8)	-45.5 (-44.4)	-46.5 (-46.9)
	ηCSA	0.17 (0.17)	0.17 (0.18)	0.85 (0.87)	0.70 (0.75)	0.57 (0.56)
	d(Mg…X) (Å)	2.04 (2.07)	2.04 (2.07)	2.10 (2.12)	2.10 (2.12)	2.08 (2.10)

Table S5

 S5 DFT-calculated 25 Mg NMR shieldings and |CQ| values of a selection of crystalline phases, used for the calibration of 25 Mg NMR calculations, together with the previously reported experimental25 Mg isotropic chemical shifts and |CQ| values (experimental errors are given in parenthesis).

	Structure	ICSD	σ(iso) (calc) / ppm	δ(iso) (exp) / ppm |CQ| (calc) / MHz	|CQ| (exp) / MHz
	MgO	52026	541.3	26.26 (5) a
	Mg(OH)2	203212	553.4	11.8 (2) a	2.95	3.09(1)
	MgCO3	73731	566.2	-3.8 (2) a	1.29	2.24(2)
	α-Mg2P2O7 15326	577.5	-11 (1) b	3.96	4.05 (4)
			566.5	-6 (3) b	11.76	12.45 (4)
	Mg(PO3)2	4280	580.9	-10 (1) b	3.42	3.09 (4)
			571.5	-2 (2) b	4.38	4.62 (6)
	Mg3(PO4)2	31005	554.7	17 (2) b	7.08	7.12 (2)
			578.0	-4 (2) b	6.95	5.77 (6)

Experimental values taken from [a] Pallister, P. J. et al (2009), Phys. Chem. Chem. Phys. 11, 11487-11500, and [b] Laurencin, D. et al (2012) J.

Phys. Chem. C, 116, 19984-19995.

Table S6

 S6 Closest neighbors around the 2 calcium sites (within a 3.0 Å radius) in the organocalcium structure CCDC 255198.

		H-optimization	Full -optimization
	Ca Neighbor	Distance (Å)	Distance (Å)
	N	2.32	2.34
	N	2.33	2.34
	O	2.38	2.43
	H	2.67	2.68
	C	2.86	2.86
	H	2.86	2.90
	Table S7 Closest neighbors around the 2 calcium sites (within a 3.0 Å radius) in the organocalcium
	structure CCDC 704097.
		H-optimization	Full -optimization
	Ca Neighbor	Distance (Å)	Distance (Å)
	N	2.29	2.29
	N	2.29	2.29
	C	2.60	2.61
	H	2.76	2.78
	H	2.76	2.78

Table S8

 S8 Closest neighbors around the calcium (within a 3.0 Å radius) in the organocalcium structure CCDC 255195.

		H-optimization	Full optimization
	Ca Neighbor	Distance (Å)	Distance (Å)
	N	2.35	2.33
	N	2.36	2.37
	N	2.52	2.53
	N	2.55	2.57
	H	2.73	2.74
	H	2.76	2.77
	H	2.83	2.84
	C	2.94	2.95
	Table S9 Closest neighbors around the calcium (within a 3.0 Å radius) in the organocalcium
	structure CCDC 255196.
		H-optimization	Full optimization
	Ca Neighbor	Distance (Å)	Distance (Å)
	N	2.36	2.37
	N	2.37	2.39
	N	2.47	2.50
	N	2.52	2.57
	H	2.66	2.67
	H	2.87	2.90
	H	2.89	2.97
	C	2.98	2.99

Table S10

 S10 Closest neighbors around the 2 calcium sites (within a 3.0 Å radius) in the organocalcium structure CCDC 198356.

	S4.
		H-optimization	Full -optimization
	Ca(1) Neighbor	Distance (Å)	Distance (Å)
	N	2.33	2.35
	N	2.33	2.35
	O	2.35	2.39
	O	2.37	2.44
	C	2.95	2.98
	C	2.98	2.99
	Ca(2) Neighbor	Distance (Å)	Distance (Å)
	N	2.32	2.34
	N	2.33	2.36
	O	2.35	2.40
	O	2.36	2.41
	C	2.84	2.87
	Table S11 Closest neighbors around the calcium (within a 3.0 Å radius) in the organocalcium
	structure CCDC 247152.
		H-optimization	Full optimization
	Ca Neighbor	Distance (Å)	Distance (Å)
	N	2.30	2.31
	N	2.31	2.32
	O	2.34	2.39
	O	2.35	2.41
	H	2.97	2.87

Simulation of 43 Ca NMR spectra based on the GIPAW calculated NMR parameters Figure S3 Simulation

  of the43 Ca MAS NMR spectra at 20 T, for structures CCDC 198356 (Ca (1) site) and 704097, based on the values calculated in Tables

Table S12

 S12 Calculated 43 Ca NMR parameters after H-optimization of structures CCDC 255198,

	704097 and 247152.

  Figure S4Simulation of the static43 Ca NMR spectra at 20 T, for structures CCDC 255198, 704097 and 247152, using the QUEST program. The NMR parameters calculated in Tables1 and 2for the Hrelaxed structures were used. Only central transitions are represented here. Euler angles are given in TableS12(above). The relative orientation of shielding and EFG tensors is shown in Figure2. The simulations were performed with (a) and without (b) the shielding tensor, showing that for 43 Ca, it is necessary to take shielding tensors into account to properly interpret the static lineshape at high field.

		198356
		704097
		255198	704097	247152
	δiso (ppm)	99.5	135.4	102.1
	|CQ| (MHz)	8.0	9.9	7.2
	ηQ	0.8	0.4	0.2
	Ω (ppm)	286.5	310.2	224.6
		0.3	-0.7	0.8
	α	-100	90	115
	β	175	-90	175
	γ	0	0	5

Table S13

 S13 Calculated 43 Ca NMR parameters after H-optimization or full optimization of structure CCDC 244505.

Table S14

 S14 Closest neighbors around the calcium (within a 3.0 Å radius) in the organocalcium structure CCDC 244505.

	S6.
	H-optimization		Full optimization
	Ca Neighbor	Distance (Å)	Ca Neighbor	Distance (Å)
	N	2.37	N	2.40
	N	2.40	N	2.44
	N	2.41	N	2.45
	O	2.43	N	2.46
	N	2.43	O	2.49
	H	2.91	H	2.94
	H	2.93	H	2.99
				31

Local environments of magnesium in the organocomplexes studied by DFT.

Table S15

 S15 Closest neighbors around the 2 Mg sites (within a 3.0 Å radius) in the organomagnesium structure CCDC 198354 (H-optimized structure).

	Rel H		Rel tot		Rel H		Rel tot
	Mg(1) site				Mg(2) site
	Neighbor	Dist. (Å)	Neighbor	Dist. (Å)	Neighbor	Dist. (Å)	Neighbor	Dist. (Å)
	N	1.92	N	1.94	N	1.91	N	1.93
	N	1.92	N	1.94	N	1.91	N	1.93
	H	2.82	H	2.84	H	2.65	H	2.67
	H	2.82	H	2.84	H	2.65	H	2.67
	H	2.85	H	2.85	H	2.69	H	2.69
	H	2.85	H	2.85	H	2.69	H	2.69
	H	2.88	H	2.90	C	2.86	C	2.86
	H	2.88	H	2.90	C	2.86	C	2.86
	C	2.94	C	2.94	C	2.97	C	2.98
	C	2.94	C	2.94	C	2.97	C	2.98
	H	2.97	H	2.98
	H	2.97	H	2.98
	C	2.98	C	2.99
	C	2.98	C	2.99

Table S16

 S16 Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium structure CCDC 1207191.

		H-optimization	Full optimization
	Mg Neighbor	Distance (Å)		Distance (Å)
	N	1.96		1.97
	N	1.97		1.98
	C	2.65		2.66
	H	2.69		2.75
	C	2.82		2.83
	C	2.92		2.91
	Si	2.93		2.93
	C	2.96		2.96
	Table S17 Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium
	structure CCDC 266469.
	H-optimization		Full optimization
	Mg Neighbor	Distance (Å)	Mg Neighbor	Distance (Å)
	N	1.98	N		1.99
	N	1.99	N		2.00
	H	2.53	H		2.48
	C	2.56	C		2.58
	C	2.65	C		2.65
	C	2.79	C		2.86
	Si	2.89	Si		2.91
	C	2.94	C		2.93
	H	2.95
	Si	2.98

Table S18

 S18 Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium structure CCDC 266469.

	H-optimization		Full optimization
	Mg(1) site
	Mg(1) Neighbor	Distance (Å)	Mg (1) Neighbor	Distance (Å)
	N	1.99	N	2.00
	H	2.74	C	2.73
	C	2.77	H	2.75
	Mg	2.85	Mg	2.86
	H	2.92	H	2.89
	C	2.93	C	2.93
	C	2.96	H	2.99
	Mg(2) site
	Mg (2) Neighbor	Distance (Å)	Mg (2) Neighbor	Distance (Å)
	N	1.98	N	2.00
	H	2.75	H	2.69
	Mg	2.85	Mg	2.86
	C	2.97	H	2.98
			C	2.99

Table S19

 S19 Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium structure CCDC 175074.

	H-optimization		Full optimization
	Mg Neighbor	Distance (Å)	Mg Neighbor	Distance (Å)
	N	1.98	N	2.00
	N	1.99	N	2.00
	O	2.53	O	2.07
	H	2.56	C	2.81
	C	2.65	H	2.84
	H	2.79	H	2.87
	H	2.89	C	2.91
	H	2.94	H	2.91
	C	2.95	H	2.98
	H	2.98

Table S20

 S20

	Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium
	structure CCDC 266470.
	H-optimization		Full optimization
	Mg Neighbor	Distance (Å)	Mg Neighbor	Distance (Å)
	N	1.99	N	2.00
	N	1.99	N	2.00
	N	2.13	N	2.15
	H	2.73	H	2.75
	H	2.73	H	2.75
	H	2.93	H	2.99
	H	2.93	H	2.99
	C	2.97
	C	2.97

Table S21

 S21 Closest neighbors around the magnesium (within a 3.0 Å radius) in the organomagnesium structure CCDC 266468.

	H optimization		Full optimization
	Neighbor	Dist. (Å)	Neighbor	Dist. (Å)
	N	1.97	N	1.99
	N	2.10	N	2.11
	N	2.13	N	2.15
	H	2.41	H	2.43
	H	2.46	H	2.46
	H	2.56	H	2.57
	H	2.73	H	2.75
	H	2.78	C	2.82
	C	2.80	H	2.82
	C	2.85	C	2.88
	C	2.86	C	2.88
	Mg	2.94	Mg	2.96
	C	2.96	C	2.98
	C	2.99

Table S22

 S22 Closest neighbors around the magnesium (within a 3.0 Å radius) in the organomagnesium structure CCDC 745088.

	H optimization		Full optimization
	Neighbor	Dist. (Å)	Neighbor	Dist. (Å)
	N	2.04	N	2.05
	N	2.04	N	2.06
	Mg	2.81	Mg	2.80
	C	2.96	C	2.95
	C	2.96	C	2.96

Table S23

 S23 Closest neighbors around the magnesium (within a 3.0 Å radius) in the organomagnesium structure CCDC 661565.

		H optimization	Full optimization
	Mg Neighbor	Distance (Å)	Distance (Å)
	N	2.07	2.09
	N	2.07	2.09
	C	2.49	2.50
	H	2.73	2.75
	H	2.73	2.75
	Mg	2.85	2.85

Table S24

 S24 Closest neighbors around the magnesium (within a 3.0 Å radius) in the organomagnesium structures CCDC 247151 and 266465.

		CCDC 247151		CCDC 266465
		H optimization	Full optimization H optimization	Full optimization
	Mg Neighbor	Distance (Å)	Distance (Å)	Distance (Å)	Distance (Å)
	N	2.02	2.04	2.02	2.04
	N	2.03	2.05	2.03	2.05
	O	2.05	2.09	2.05	2.09
	O	2.06	2.10	2.06	2.10
	C	2.90	2.92	2.90	2.92
	C	2.91	2.93	2.91	2.94

Table S25

 S25 Closest neighbors around the calcium (within a 3.0 Å radius) in the organomagnesium structure CCDC 266466.

	H-optimization		Full optimization
	Mg(1) site
	Mg(1) Neighbor	Distance (Å)	Mg (1) Neighbor	Distance (Å)
	N	2.03	N	2.05
	N	2.03	N	2.05
	N	2.17	N	2.18
	N	2.19	N	2.22
	H	2.83	H	2.83
	C	2.91	C	2.93
	C	2.93	C	2.95
	H	2.94	H	2.98
	Mg(1') site
	Mg (1') Neighbor	Distance (Å)	Mg (1') Neighbor	Distance (Å)
	N	2.03	N	2.05
	N	2.03	N	2.05
	N	2.16	N	2.18
	N	2.18	N	2.20
	H	2.94	C	2.95
	C	2.96	C	2.98
	C	2.97	H	2.99

Table S26

 S26 Closest neighbors around the magnesium (within a 3.0 Å radius) in the organomagnesium structure CCDC 266464.

	S7.
		H optimization	Full optimization
	Mg Neighbor	Distance (Å)	Distance (Å)
	N	2.01	2.03
	N	2.01	2.03
	N	2.14	2.17
	N	2.14	2.17
	H	2.97
	H	2.97
			39

Simulation of 25 Mg NMR spectra based on the GIPAW calculated NMR parameters Table S27

  Calculated 25 Mg NMR parameters after H-optimization of structuresCCDC 198354, 1477718, 266470, 266465 and 266464.

Acknowledgements NMR spectroscopic calculations were performed using HPC resources from GENCI-IDRIS (Grant 2016-097535). The CNRS and the Embassy of France in Australia are acknowledged for financial support.

Supporting information S1. Additional DFT calculations on organocalcium and organomagnesium complexes

Table S1 DFT calculations on magnesium complex CCDC 198354: influence of the k-point mesh, the energy cut-off, and the optimization of unit cell parameters.

Optimization