Alda Mari

DOES THE DIFFERENCE IN THE HUMAN PROCESSING OF "chaque" AND "tout" MATCH THE DIFFERENCE BETWEEN THE PROOF RULES FOR CONJUNCTION AND QUANTIFICATION?

"Chaque vin a sa lie." versus "Toute nuit a un jour."does the difference in the human processing of " chaque" and " tout" match the difference between the proof rules for conjunction and quantification?

Alda Mari, Christian Retoré

To cite this version:

Alda Mari, Christian Retoré. "Chaque vin a sa lie." versus "Toute nuit a un jour."

Semantic and discursive properties of tout and chaque In spite of the abundant literature on quantification in French, little if not any attention has been paid to the types of discourses in which quantifiers are used. Likewise, relatively few studies have investigated the differences between tout and chaque, which are both universal quantifiers, ranging over singular entities.

Our starting observation with [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF], is that tout is naturally used in generic sentences (see also [START_REF] Jayez | tout" as a genuine free choice item[END_REF]), whereas chaque is blocked. Not to create confusion, we do not provide English translations for tout and chaque and use the metalinguistic TOUT/CHAQUE.

(1) a. Tout lion a une crinière "TOUT lion has a mane." b. * Chaque lion a une crinière. (no generic reading)

"CHAQUE lion has a mane."

1

(2) a. Tout homme est mortel. "TOUT man is mortal." b. *Each man is mortal. (no generic reading) "CHAQUE man is mortal."

The general tout, comparison with n'importe quoi Tout has been argued to be a Free Choice Item (FCI) (see [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF]) and to have an intrinsic modal semantics (although there exist a variety of proposal, their common core is that FCI are modals). A comparison between n'importe quoi and tout that enriches the already noted differences, can help us spelling out in greater details the semantics of tout. Firstly, FCI are not as natural as tout in generic sentences, or, at least, they do not lead to the same interpretation.

(3) # N'importe quel homme est mortel.

"Any man is mortal."

Clearly tout can sustain a case in which an infinite set is used. N'importe quoi does not. To interpret (3), we would need to fix a set of relevant men, and pick any one of those. This leads us to conclude that tout is the absolute general universal quantifier in language.

Another piece of data leading to this conclusion is the contrast between tout and n'importe quoi/qui with respect to sub-trigging. Sub-trigging, is the term coined by [START_REF] Ehrenkranz | OR and ANY: the syntax of two logical operators[END_REF] to describe the fact that episodic sentences can be rescued when the NP head noun is modified by an adjective or a post-nominal modifier. For English any [START_REF] Dayal | ANY as inheritently modal[END_REF] proposes that the sub-trigger introduces a spatio-temporal restriction that prevents the any-quantifier to range over the totality of possible worlds or situations.

(4) a. * Mary read any book.

b. Mary read any book that she bought.

We observe an opposite behavior with respect to sub-trigging. With imperatives, FCI do not require sub-trigging, tout does (note that tout, despite [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF] can be used in imperatives, granted that sub-trigging is used).

(5) a. Prend n'importe quelle carte ! "Take FCI card!" b. Prend *toute carte/toute carte qui puisse te faire gagner ! "Take *TOUT card / TOUT card that allows you to win!"

Why do we have to accommodate here a restriction ? Because the context (a card game) presupposes the existence of a limited set of cards, and this restriction clashes with the default information of absolute generality of tout. A restriction for the domain of quantification of tout is thus needed for the sentence to be felicitous.

As we mentioned, current analyses of FCI and tout in particular, rely on a modal semantics. It is tempting to extend this line of analysis and to use a modal semantics also for capturing the absolute generality of tout (even if our description in fact departs in some ways from [START_REF] Jayez | tout" as a genuine free choice item[END_REF]).

While modal analyses are enlightening in many respects, they raise the question of how we can construct or compute the set of all possible worlds. Restricting via ordering sources the set of relevant worlds in a natural way out of the problem. However, here we explore an alternative route asking when can one assert a given sentence and how can one refute an asserted sentence. Studying the condition of asserting and refuting a statement is a different but worth studying semantics. This will be our choice. However, we want to first consider the types of sentences in which tout and chaque are used, further justifying the use of a semantics that can capture the conditions for assertion and refutation, rather than a purely truth conditional approach.

Chaque vs. tout: an analytical and a synthetic quantifier The starting point of our description of the types of statements in which tout and chaque are used, will consist in acknowledging that tout and chaque are employed, respectively, in prescriptive and descriptive statements. We substantiate this labels by spelling out the ingredients of prescriptivity and descriptivity.

Prescriptive statements are grounded in rules of the form P(x) → Q(x) [START_REF] Jayez | tout" as a genuine free choice item[END_REF]. The rule must pre-exist, and it is meant to reveal a non-accidental association between the P property and the Q property. We will observe that the statements in which tout is used are analytical generic ones, akin to indefinite generic statements. [START_REF] Cohen | On the generic use of indefinite singulars[END_REF][START_REF] Mari | Another look at italian generic sentences[END_REF][START_REF] Corblin | The roots of genericity: indefinite singulars vs definite plurals[END_REF][START_REF] Krifka | Definitional generics[END_REF][START_REF] Mari | Covert and overt modality in generic sentences[END_REF] We will argue that observation of each of the entities is not needed, as the ability of the domain of quantification of being infinite reveals. Tout-statements, being universals, hardly tolerate exceptions but they nevertheless do: if one of the entities is not conformed to the rule, one might even discuss whether it really belongs to the class one quantify over.

Chaque is used in descriptive statements. It requires the domain of quantification to be finite; moreover, the content in the scope of the quantifier can be accidental to the entities in the restriction (unlike what happens with tout, for which only intrinsic properties of the class are targeted). The notional category 'universal quantification', with chaque, we show, takes the form of a closure over a domain, each of the entities of which has been inspected. Typically, chaque, cannot be used as a generic [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF].

We refer to tout-universal quantifier as analytical universal quantifiers and to chaque-type of quantifier as synthetic universal quantifier in order to disentangle the type of the statements in which they are used.

The two proof-theoretical views of universal quantification The model theoretic view of universal quantification is completely naive: ∀xP(x) is true whenever P(x) holds for all x in the domain. Most authors consider ∀xP(x) is nothing than a short hand for & x∈D P(x) which is not necessarily a first order formula e.g. when D is infinite (or worse, uncountable, like instants or places). This conjunctive view presupposes that the domain is clear, and this is rather rare in natural language.

In order to model meaning, we think that sense (Sinn) is more faithful than reference or denotation (Bedeutung) and a natural candidate for the sense of a sentence is the set of its proofs -and this differs from the usual interpretation of sentences as sets of situations in which the sentence happens to be true. [START_REF] Frege | Über sinn und bedeutung[END_REF] From this proof-theoretical view, there are two natural ways to assert a universal statement.

One is the standard proof rule (∀ i ): for a variable x about which nothing is assumed you are able to infer P(x), hence you can conclude ∀x.P(x) (example: simply assuming that n is integer you show that there exists four squares whose sum is n, so you can conclude that every integer is the sum of four squares, or for more linguistic examples, with "tout" see [START_REF] Jayez | tout" as a genuine free choice item[END_REF]). Gentzen deductive systems NK or LK [START_REF] Gentzen | Untersuchungen über das logische Schließen I[END_REF] give a clear account of this rule, and Hilbert generic element τx. P(x) introduced in [START_REF] Hilbert | Die logischen Grundlagen der Mathematik[END_REF]. This is an ideal element that, with respect to P has nothing particular, so when it enjoys P so does every other element: ∀x.P(x) ≡ P(τx.P(x)). This τ that is an in situ quantifiers is also the dual of the better known ε operator that has been used for modelling definite and indefinite noun phrases-see e.g. [START_REF] Retoré | Typed Hilbert epsilon operators and the semantics of determiner phrases[END_REF] The other natural rule, known as the ω-rule is quite different. It was introduced by Gentzen in [START_REF] Gentzen | Die Widerspruchsfreiheit der reine Zahlentheorie[END_REF] to establish the consistency of arithmetics: assume you have a proof of P(n) for each n, then you can conclude that ∀nP(n). This rule is closer to the model theoretic view, and it presupposes that the domain D is known, here D = N. Observe that the ω-rule requires an infinite number of premisses, so a proof with an ω-rule has an infinite width, although any of its branches is finite.

The difference between the two rules ∀ i and ω can be made intuitive as follows: the usual proof rule ∀ i yields to statements that are true in any model, while the later rule ω only derives statements that are true in the intended model with domain D. Observe that there do exist statements that are true in one model and not in others: completeness theorem says that the formulae of first order logic that are true in any model are exactly the ones that are provable in first order classical logic; a non provable formulae can be true in one model and false in another model. There is also a structural difference: formulae and proofs cannot refer to entities in the model; although the logical language may include constants, those constants cannot properly for elements of models: elements of models vary from a model to another one, and furthermore an interpretation may map different constants onto a single element in the model. Hence these two views of quantification are quite different although they may coincide for a particular language and theory, in particular on a well-defined finite domain.

"Chaque" as a conjunction and "tout" as a generic In common French, as far as our intuition and data are correct (see below), it seems that chaque needs a precise the domain which on the other hand can be totally contingent. Exceptions are less welcome with chaque than they are with tout -the collective universal quantifier tous les is the one that better tolerates exceptions. [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF] This is absolutely consistent with the interpretation of chaque as & x∈D . On the other hand, "tout" especially in "tout X" may be applied to a possibly vague class.

As opposed to "chaque", "tout" requires the assertion to be perennial in some sense, which prevents "tout" from applying to very particular classes that are not perennial as suggested in [START_REF] Jayez | tout" as a genuine free choice item[END_REF][START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF]. This makes "tout" close to the proof theoretical ∀x or better to the τx.P(x). Observe that it is also close to the generic "un" in this respect.

Next, how do we refute an assertion with "chaque"? There is only one possibility which consists in finding one element that does not enjoy the property. This is consistent with "chaque" being a conjunction. Now how does one refute a "tout" assertion? One way is to exhibit an element a not satisfying the property, in the (often imprecise) domain D under consideration: this is a switch from a real quantification to a conjunction over the domain, and the asserter may object that a is not in the imprecise domain D he was thinking of. Another way is to object a "tout" assertion is to remain in the conceptual level, and to say that a subdomain of the domain of the "tout" does not enjoy the property. Here as well the asserter may object that they are not part of the intended domain, but it is going to be more difficult.

We explored a bit the proof theoretical interpretation of natural language quantification in [START_REF] Vito | Quantification in ordinary language and proof theory[END_REF].

Verifying our intuitions: ongoing experiments We have attempted to use corpora to substantiate our hypothesis, by looking for data in a dialog corpus of rather spontaneous speech (CID). Universal quantifiers were too rare to draw any conclusion though, and it seems necessary to deploy some specific experiments. First, we are testing the sensitivity of these quantifiers to different types of domains and most notably vague and precise ones. We expect a complementary distribution between tout and chaque. Second, we are testing how tout and chaque sentences are refuted. We test for (i) individual exceptions and (ii) type exceptions.

Our tests take the form of judgments elicitations in the first place. We are planning to create more elaborate experiments, where fillers and control sentences are used thus completing our questionnaire. This type of experiments will be finalized, once we will have sharpened the hypothesis with standard introspective methods. The design of the data base, the web programming of the questionnaires and the storage of the answers for statistics will be the project for two groups of four third year students, from January to May.

Future work There are further questions completing this study that we would like to develop.

Firstly we would like to compare the processing of quantification by standard subjects and dyslexic subjects, and children in particular. Indeed, a recent study on the difference between the understanding of negation (and to a lesser extend of Aristotle quantified sentences A E I O) by standard and dyslexic children has greatly helped to understand the human processing of such sentences. [START_REF] Delfitto | Towards a pragmatics of negation: the interpretation of negative sentences in developmental dyslexia[END_REF] Secondly we would like to extend the study to "tous_les", a (the?) third wording of universal quantification in French [START_REF] Kleiber | La quantification universelle en trio: tous les, chaque et tout[END_REF], which in contrast with tout and chaque insofar as it refers to the domain as a whole. How does "tous_les" compare with "tout" and "chaque"? The same question may apply to "les" (although les N is considered to be a referential expression, rather than a quantificational one [START_REF] Corblin | The roots of genericity: indefinite singulars vs definite plurals[END_REF]), and to the generic "un" (which we can foresee close to "tout")

Thirdly we would like to also use experiments whereby subjects, after listening to a sentence, match pictures that are presented to them with the relevant sentence. Pictures are presented on the screen of an eye-tracker which records their eye glaze. This will allows us to gain some insights into how subjects understand quantifiers. We already used such experiments in a study showing that "chaque" in object position takes scope more easily than "tous_les" does in the same object position [START_REF] Capelier-Mourguy | Quantifier scope: a formal and experimental study[END_REF] Concluding remark The difference between the "chaque" and "tout" proverbs in the title possibly comes from the actual existence of the domain of quantification, "vins" possibly being understood here as barrels -this is comforted by the dependency of the "lie" on the "vin" via "sa" -while the "tout" proverb seems to be speaking of the eternal essence of "nuit" -observe that "jour" also is an essence, introduced by the generic "un", hence not dependent on the "nuit"which do not precisely determine the domain of the quantifier.


S Poncé

G Antonius

P Boulanger

E Cannuccia

A Marini

M Côté

X Gonze

Verification of first-principles codes: comparison of total energies, phonon frequencies, electron-phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo

Keywords: Density functional perturbation theory, Electron-phonon coupling, Temperature dependence, Verification, Allen-Heine-Cardona theory, Zero-point motion renormalization, Diamond

Verification of first-principles codes: Comparison of total energies, phonon frequencies, electron-phonon coupling and zero-point motion correction to the gap between ABINIT and QE/Yambo

Introduction

First-principles electronic-structure codes develop and evolve : they adapt to increasing computational capabilities and also include new formalisms, approximations, and numerical methods. In addition to the validation of new formalisms and approximations, the verification of implementations is of utmost importance if one wants to deliver reliable new results or compare them to existing ones. This concern has been the subject of increased attention in the recent years, as witnessed by the set up of ESTEST, a framework for the validation and verification of electronic structure codes [1], and the organization of several related activities under the auspices of the "Centre Européen de Calcul Atomique et Moléculaire" (CECAM) [2].

Email address: samuel.ponce@uclouvain.be (S. Poncé) In particular, the first-principle computation of electronic properties, quasiparticles band structures and optical spectra of crystalline solids has reached an unprecedented level of sophistication. Many-body GW calculations [START_REF] Aulbur | Quasiparticle Calculations in Solids[END_REF], dynamical-mean-field theory [START_REF] Georges | [END_REF] and Bethe-Salpeter [5] calculations, that includes excitonic effects, sometimes claim to agree with experimental data at the level of 0.1-0.2 eV. However, the influence of lattice vibrations on electronic properties is usually neglected because it is assumed to lead only to minor corrections, on the order of a few tens of meV. Actually, as reviewed in Ref. [6], for materials that contain light atoms like diamond, the inclusion of the influence of lattice vibration is non-negligible, since the renormalization is larger than the claimed accuracy of quasiparticle methods.

For the case of diamond, the closing of the electronic gap has been measured experimentally at different temperatures, and an Einstein oscillator fit has been used to extrapolate the data at zero Kelvin [7], giving a value of 0.37 eV for the renormalization of the indirect band-gap due to the zero-point motion renormalization (ZPR) of atoms. The temperature dependence of the direct bandgap of diamond was also studied experimentally [8].

The direct band-gap renormalization has been studied from first-principles approaches. Ramírez obtained 0.7 eV using path-integral Monte Carlo simulations [9] and more recently Giustino et al. [10] obtained a value of 0.615 eV using the Allen-Heine-Cardona (AHC) [11,12] theory, with the Local Density Approximation (LDA) [13,14] of Density Functional Theory (DFT) [START_REF] Martin | Electronic Structure. Basic Theory and Practical Methods[END_REF], a basis of plane waves, and normconserving pseudopotentials [START_REF] Martin | Electronic Structure. Basic Theory and Practical Methods[END_REF].

However, the first-principle computation of such quantity is particularly delicate, because it is the outcome of several layers of consecutive first-principle calculations : computation of the total energy (and associated relaxation of cell geometry), computation of the phonon frequencies and eigenvectors, computation of the electron-phonon (EP) coupling, and finally, computation of the zero-point motion effect. Not only the choice of a mathematical formalism, with associated approximations (like the above-mentioned Monte Carlo versus DFT possibility), might deliver different values, but the implementation of one well-defined mathematical formalism, with given approximations, needs to be carefully verified.

At variance with the 0.615 eV result of Giustino et al. [10], calculations made by us lead to a smaller value, on the order of 0.4 eV, on the basis of the implementation partly described in Ref. [START_REF] Gonze | [END_REF]. However, the mathematical formalism and numerical approximations were, to our understanding, equivalent to that of Ref. [10]. This raised the question on whether the accumulation of layers of calculations could yield numerical errors that are as large as 0.2 eV, or whether there might be a problem in the implementations.

In this work, we present a rigorous and careful study of all the quantities that enters into the calculation of the ZPR of the direct band gap of diamond due to EP coupling, on the basis of two different implementations, and provide the values of the numerical discrepancies. We work within the AHC formalism with exactly the same numerical approximations, as implemented in ABINIT [17], on one side, and in Yambo [18] on top of Quantum Espresso (QE) [19], on the other side. These implementations have been done completely independently by two different groups. The ABINIT implementation has been used earlier to study zero-point motion effects on the electronic structure in the abovementioned Ref. [START_REF] Gonze | [END_REF], while the YAMBO+QE implementation has been used, independently, in Ref. [20][21][22][23]. Unfortunately, we did not have access to the code used by Giustino et al. [10].

We found only small numerical discrepancies between the ABINIT and QE+YAMBO results: less than 10 -5 Ha/at on the total energy, 0.07 cm -1 on the phonon frequencies, 0.005 on the electron-phonon matrix elements squared (relative difference), and less than 4 meV on the ZPR. Given our choice of formalism, and associated approximations, the numerically converged value for the renormalization of the direct band gap in diamond due to electron-phonon coupling in the AHC formalism is -0.409 eV (reduction of the band gap), from both implementations. Changing the pseudopotential can lead to larger differences, in any case not larger than 50 meV.

The structure of the article is as follows. In section 2, we discuss the mathematical theory used in this work. In section 3, we give details about the material studied as well as computational parameters and approximations. In section 4, we review the results and discuss their impact. We draw the conclusions in section 5.

Theory and methods

Ground-state and phonons

The decomposition of the total energy differs between ABINIT and QE, such that a comparison of energy components needs to be done with care. The expression for the total ground-state energy per unit cell of a periodic insulator at zero Kelvin, within DFT is [START_REF] Martin | Electronic Structure. Basic Theory and Practical Methods[END_REF]24]:

E Total = 1 N k k occ n nk T + Vpsp nk + E Hxc + E Ew + E psp-core , ( 1
)
where n is the band number, k the wavevector, |nk represents a Kohn-Sham orbital, T the kinetic energy operator, Vpsp the operator corresponding to the external potential of the electronic system (composed by a local and a non-local part when the implementation is based on the pseudopotential concept), the n-summation is over the occupied bands and the k-summation over a discretization of the Brillouin zone. E Hxc is the Hartree and exchange-correlation energy functional of the electronic density (expressed per unit cell), E Ew is the Ewald energy per unit cell (periodic positively charged particles placed in a negatively charged homogeneous background), and finally E psp-core is the pseudo-core energy per unit cell. It is also possible to define a one-electron contribution per unit cell as:

E One-el = 1 N k k occ n nk T + Vpsp nk + E psp-core . (2)
See the appendix for more details concerning the Ewald energy, the pseudocore energy and the oneelectron contribution.

The phonon frequencies and eigenvectors can be obtained from Density Functional Perturbation Theory (DFPT) following Refs. [24][25][26][27][28]. With Csα,s ′ β (q) being the interatomic force constant matrix in reciprocal space, the phonon frequencies ω qλ and eigendisplacements ξ α (qλ|s) are linked by the dynamical equation

s ′ β Csα,s ′ β (q)ξ β (qλ|s ′ ) = M s ω 2 qλ ξ α (qλ|s), (3)
where s labels the atom in the cell (at position τ s and with atomic mass M s ) and α is a Cartesian coordinate.

Using the orthonormalisation relation

δ λ ′ λ = sα M s ξ * α (qλ ′ |s)ξ α (qλ|s), (4)
the eigenfrequencies can also be expressed as

ω 2 qλ = sα s ′ β ξ * α (qλ|s) Csα,s ′ β (q)ξ β (qλ|s ′ ). ( 5
)

Electron-phonon coupling and zero-point motion renormalisation

The computation of the ab initio temperature dependence implies the calculation of the electron-phonon interaction. Following Ref. [23] the first-order electronphonon matrix elements can be computed thanks to DFPT as

g qλ nn ′ k = sα (2M s ω qλ ) -1/2 e iq • τ s × nk ∂ Vscf ∂R sα n ′ k -q ξ α (qλ|s), ( 6
)
where Vscf is the self-consistent mean potential felt by the electrons (which depends on the atomic positions):

Vscf = Vpsp + VHxc . (7)
The first-order electron-phonon matrix element, g qλ nn ′ k , that will be referred to as the "GKK" matrix element, describes the probability amplitude for an electron to be scattered from k to kq, with the emission or the absorption of a phonon with crystalline momentum q belonging to the phonon branch λ.

The second-order electron-phonon matrix element is:

Λ qλq ′ λ ′ nn ′ k = 1 2 s,α,β ξ * α (qλ|s)ξ β (q ′ λ ′ |s) 2M s (ω qλ ω q ′ λ ′ ) 1/2 × nk ∂ 2 V(s) scf ∂R sα ∂R sβ n ′ k -q -q ′ . ( 8
)
The AHC theory [12] allows one to calculate the temperature-dependent change in the electronic eigenenergies, as well as their zero-point renormalization, as the sum of a Fan [29,30] and a Debye-Waller (DW) self-energy term. These two terms can be deduced from the more general many-body formalism [23] as:

Σ FAN nk (iω, T ) = 1 N q n ′ qλ g qλ nn ′ k 2 × 2n qλ (T ) + 1 (iω -ε n ′ k-q -i0 + ) (iω -ε n ′ k-q -i0 + ) 2 -ω 2 qλ + ω qλ (1 -2 f n ′ k-q (T )) (iω -ε n ′ k-q -i0 + ) 2 -ω 2 qλ , (9)
and

Σ DW nk (T ) = 1 N q qλ Λ qλ-qλ nnk (2n qλ (T ) + 1), (10)
where n qλ (T ) is the Bose-Einstein distribution function for the phonon mode (q, λ) at temperature T , and f n ′ k-q (T ) is the electronic occupation.

The ZPR of the traditional AHC theory [12] is recovered by using the following approximations for the Fan term: ω ≈ ε nk (the on-the-mass-shell (OMS) limit), |ε nk-ε n ′ k-q | ≫ ω qλ (the adiabatic limit) and by considering only the real part of the self energy:

∆ε AHC nk (T ) = Σ DW nk (T ) + 1 N q n ′ qλ g qλ nn ′ k 2 (2n qλ (T ) + 1) ε nk -ε n ′ k-q .
(11) From a practical point of view, the DW term is very difficult to calculate, as one needs access to the second derivative of the self-consistent potential (that is not provided by a DFPT calculation of phonons). Making use of the translational invariance (if all atoms are displaced by the same amount in the same direction, all physical quantities should be conserved) [12], one can rewrite the DW term as a sum of a diagonal contribution and a non-diagonal one. The diagonal Debye-Waller (DDW) contribution is the product of first-order electron-phonon matrix that is easy to calculate [START_REF] Gonze | [END_REF]:

Σ DDW nk (T ) = - 1 N q qλ s,s ′ ,α,β ξ * α (qλ|s)ξ β (-qλ|s) 4M s ω qs × n ′ n 1 ε nk -ε n ′ k nk ∂ Vscf ∂R s ′ α n ′ k n ′ k ∂ Vscf ∂R sβ nk + nk ∂ Vscf ∂R sβ n ′ k n ′ k ∂ Vscf ∂R s ′ α nk . ( 12
)
The non-diagonal contribution comes from the modification of the screening due to atomic motion. By opposition with the case of small molecules [START_REF] Gonze | [END_REF], the effect of the non-diagonal Debye-Waller term is expected to be small in extended system, thanks to the screening of the periodic lattice. Neglecting it corresponds to the rigid-ion approximation.

From a numerical point of view, the term with an energy denominator in Eq. ( 11) is omitted when the difference of eigenenergies is smaller than 10 -6 or is smoothed by introducing a small imaginary component.

Finally, following Sternheimer [31], one can largely speed up the calculation of the sum over states appearing in the Fan and DDW terms. In that case, they are rewritten in terms of a sum limited to an active space (spanning the occupied state with a few extra bands over the valence band maximum) :

-

n ′ n |n ′ k n ′ k| ∂ V(s) scf (r) ∂R sα |nk ε nk -ε n ′ k = P a ⊥ ∂nk ∂R sα - n ′ ≤a ⊥ n ′ n |n ′ k n ′ k| ∂ V(s) scf (r) ∂R sα |nk ε nk -ε n ′ k , (13)
with P a ⊥ the projector over the states whose eigenenergies is above the active space threshold and therefore orthogonal to the active space. The result of such a projection is an outcome of a phonon DFPT calculation, and, as such it is available at no additional cost. More informations about this last derivation can be found in Ref. [START_REF] Gonze | [END_REF].

Material and calculation

The ABINIT, QE and Yambo software applications are described in Refs. [17][18][19], respectively.

Ground-state and phonons

The calculation of structural properties in this work is based on DFT [START_REF] Martin | Electronic Structure. Basic Theory and Practical Methods[END_REF]32,33] using the LDA [13,14]. A norm-conserving pseudopotential [34] accounts for the core-valence interaction and a plane-wave basis set is then used to expand the electronic wavefunctions. The pseudopotential was generated using the fhi98PP code [35] with a 1.5 atomic unit cut-off radius for pseudization. The valence electrons of Carbon, treated explicitly in the ab initio calculations, are generated for the 2s 2 2p 2 3d 0 configuration. Quite importantly for the comparison between codes, the same pseudopotential file was used by ABINIT and QE. Moreover this pseudopotential is the same as the one used in Ref. [10]. We refer to this pseudopotential as our "reference" pseudopotential.

Careful convergence checks (error below 0.5 mHa per atom on the total energy) leads to the use of a 6x6x6 Γ centered Monkhorst-Pack k-point sampling [36] of the Brillouin zone and an energy cut-off of 30 Hartree for the truncation of the plane wave basis set. The lattice parameter of 6.652 Bohr was obtained by structural relaxation of the diamond system.

Additional tests were performed to assess the influence of the pseudopotential choice.

In addition to our "reference" pseudopotential, we considered five other ones. We will refer to the first one as 06-C.LDA.fhi also generated using the fhi98PP code. It is a Troullier-Martins pseudopotential with the Perdew/Wang [37] parametrization of LDA, an atomic cut-off radius of 1.0247 atomic unit and a maximum angular channel of l = 3. The second one is the 6c.pspnc Troullier-Martin [34] pseudopotential with a 1.4851 atomic unit cut-off radius and a maximum angular channel of l = 1. The third one is the 06-C.GGA.fhi Troullier-Martin pseudopotential with the GGA Perdew/Burke/Ernzerhof [38] parametrization and a 1.0247 atomic unit cut-off radius. The maximum angular channel used is l = 3 for this pseudopotential. The required cut-off energy for the truncation of the basis set for those three pseudopotential was also 30 Hartree. The fourth one is the 6c.4.hgh Hartwigsen-Goedecker-Hutter pseudopotential [39] with a 1.2284 atomic unit cut-off radius and a maximum angular channel of l = 1. An energy cut-off of 60 Ha was required for this pseudopotential. The last one is the C.pz-vbc.UPF VonBarth-Car pseudopotential with a maximum angular channel of l=1 and an energy cut-off of 45 Ha and 1.5 atomic unit cut-off radius. The lattice parameter of the five additional pseudopotential after structural relaxation were 6.648, 6.694, 6.729, 6.675 and 6.663 Bohr, respectively. All the calculations with these pseudopotential were also done with the 6x6x6 unshifted Monkhorst-Pack k-point grid.

Electron-phonon coupling and zero-point motion renormalisation

In order to converge the ZPR below 1 meV, in the original AHC formulation, around 300 unoccupied bands needs to be explicitly included in the summation present in the Fan and DDW terms for diamond. In contrast, only 12 bands were needed to describe the active space when the Sternheimer re-writing is used.

To avoid high symmetry points that might slow down the convergence study (some EP matrix elements might be zero by symmetry and are not representative of the discretization of an integral) we computed the ZPR correction on a random q-wavevector grid, as described in Ref. [23]. The rate of convergence of homogeneous wavevector grid will also be discussed. The statistical analysis to converge the results is explained in the next section. The Sternheimer implementation, which speeds up significantly the calculation, is only present in the ABINIT software. Therefore we did the statistical analysis only with ABINIT.

Results and Discussion

Ground-state and phonons

We started by comparing DFT ground-state total energies between ABINIT and QE using the same "reference" norm-conserving pseudopotential and the same numerical parameters (plane wave kinetic energy cut-off and wavevector sampling). The total energy in ABINIT and QE is decomposed in different terms detailed in Eqs. ( 1) and (2). The comparison between the terms and the total energy is given in the upper panel of Table 1. The agreement is excellent: one gets a discrepancy on the order of 10 -5 Ha/atom between the total energies computed using the two codes. The disagreement is even smaller for selected contributions : on the order of 10 -7 Ha/atom for the exchange-correlation and Hartree contributions, and about 10 -9 Ha/atom for the Ewald energy. We did not try to track down the origin of the total energy discrepancy, the agreement being beyond practical needs.

Table 1 also shows the agreement between the two codes on the phonon frequencies at some high symmetry points obtained from the DFPT Eq. ( 5). The agreement is also rather good, with less than 0.07 cm -1 differences after imposition of the acoustic sum rule (ASR) at Γ. The imposition of the ASR is discussed e.g. in Ref. [40]. Without the imposition of the ASR, the frequency of acoustic modes at Γ are small, but nonnegligible : 3.335 cm -1 for ABINIT, and 8.832 cm -1 for QE. Such a variation between codes is however sufficient to lead to significant differences in the absolute value of the Fan and DDW terms computed separately, as we shall see later. Concerning the electronic properties, the nine lower eigenenergies, relative to the top of the valence band at Γ are compared for the two codes in Table 2. One can see that there is less than 0.0003 eV differences between the two codes.

Electron-phonon coupling and zero-point motion renormalisation

We now move forward and compare the GKK electron-phonon matrix elements given in Eq. ( 6). This quantity is actually subject to an arbitrary dependence on the phase factors of the wavefunctions, and cannot be compared directly between codes. We have therefore compared the square norm of the GKK (the GKK times its complex conjugate). Such a quantity, termed "GKK2" is relevant in the present context, since the square of GKK is used to build the ZPR, see Eq. ( 9). When wavefunctions are degenerate, we also sum them inside the degenerate space, to remove any arbitrariness. Moreover, to decrease the number of handled data, we sum the GKK2 over the six phonon modes, giving SGKK2.

A measure of the relative difference between the two codes for SGKK2, for different high symmetry qwavevectors is displayed in Fig. 1 and Fig. 2. We plot, for each pair of electronic state (or degenerate state) the difference of the SGKK2 divided by their sum :

∆ = S GKK2(ABINIT ) -S GKK2(QE) S GKK2(ABINIT ) + S GKK2(QE) . ( 14
)
The absolute values of SGKK2 are reported in Table 3 One can see that the relative differences are in all three cases lower than 0.005 for all matrix elements on the 300x300 matrix bands.

Finally, we have compared the ZPR computed with ABINIT and Yambo using the GKKs of QE in the AHC framework of Eq. ( 11). The energy denominator was smoothened by introducing a small imaginary component of 100 meV, following Ref. [10,41].

We have first compared the two codes without the Sternheimer rewriting and then, in the case of ABINIT, we have used the Sternheimer rewriting of Eq. 13 and we have summed over 300 bands in the case of Yambo. In Table 4 we show a comparison between ABINIT and YAMBO for different number of q-wavevectors. The 47 q-wavevector case corresponds to a homogeneous, nonshifted 10x10x10 grid, folded in the irreducible part of the Brillouin zone. For the 1000 and 2000 q-wavevector cases, the wavevectors are randomly generated once and then used in both codes. In the last two columns of Table 4, we can see that the disparity between the two codes on the ZPR is lower than 4 meV.

We have imposed in both codes the phonon frequencies to be 0 for the acoustic modes at q = Γ (ASR).

One can nevertheless see that the absolute value of the Fan (last term in Eq. ( 11)) and DDW (Eq. ( 12)) terms display more variation between both codes than the total ZPR (which is the sum of both terms). The reason for this is that the acoustic modes tends to have a larger relative difference than the optical ones between the two codes. Their separate contributions in the Fan and DDW tends to the same value, with opposite sign, when the limiting behaviour for vanishing wavevector is considered. There is thus a cancellation of error between the Fan and DDW terms, that allows one to obtain a much better accuracy on the sum of these terms. Indeed, due to the presence of the phonon frequency in the denominator of Eqs. ( 6) and ( 12), the acoustic modes will be the one that contributes mostly to the Fan and DDW terms. It can be shown that, due to translational invariance, the eigendisplacement vectors of Fan and DDW will tends to cancel out for acoustic modes (especially those close to Γ). As a result, mostly the optical modes will contributes to the ZPR. This explains why the discrepancy is larger on the absolute value of Fan and DDW terms separately, than on the total ZPR between the two codes. Note that the Fan and DDW terms are not observable quantities separately. They come from a perturbation series, whose sum is an observable.

Analysis of the convergence with respect to the number of q-wavevectors

We have just provided an analysis of the level of agreement that one can expect from two different codes that implement the same physics. We now turn ourselves to a careful convergence study of the ZPR within the AHC formalism. Since the calculations are heavy in YAMBO due to the band summations we decided to make that convergence study in ABINIT only, with the Sternheimer rewriting.

We have performed DFPT calculations on 20,000 randomly generated q-wavevectors in the full Brillouin Zone. We have then performed a statistical analysis of these results. We have computed the ZPR over N (N=250, 500, 750, 1000, 2000, 3000, 4000, 5000, 6000, 10000) q-wavevectors taken randomly between the 20,000 set and we have done such calculation 100 4: Comparison of the ZPR for different electronic states at Γ, for a 6x6x6 unshifted k-point grid with an energy cutoff of 30 Hartree for the plane wave basis set, using the same norm-conserving LDA pseudopotential. In the case of YAMBO, 300 bands were explicitly included into the calculation. In the case of ABINIT, the Sternheimer equation (SEq) was used to limit the computational effort (12 active bands were needed). Moreover, for the set of 47 q-wavevectors, the value obtained without the Sternheimer equation and with a summation over 300 bands, is also displayed. The energies are in meV.

times for each N. This gave us, for each N, a set of 100 different ZPR values whose statistical characteristics are given in figure 3. We can see that the ZPR converges smoothly towards 409 meV, the mean of the ZPR for the 20,000 set. Since the rate of convergence of the variance of a normal distribution goes as 1/N q with N q the number of random q-wavevectors, the rate of convergence of the associated standard deviation goes as 1/ N q . We can see on Fig. 3 that the 1/ N q of the continuous line follow neatly the lower 25% and upper 75%.

The drawback of the random q-points methods is that one is forced to test a sufficiently large set of random q-wavevectors. The homogeneous grid approaches might be more appealing. The red dots on Fig. 3 corresponds to non shifted homogeneous Monkhorst-Pack grids closest to the random points number we have chosen to analyse (e.g. the last grid is a 70x70x70 unshifted q-point grid that lead to 8112 q-wavevectors in the irreducible Brillouin-Zone.). As we can see, the red dots are always well inside the 50% windows.

One can set an upper limit on the convergence rate if one does not use an imaginary component to smooth the function.

In this case, when the difference of eigenenergies in the denominator of Eq.( 11) vanishes, the integrand to be considered over the whole Brillouin zone diverges. This happens around Γ, with a divergence that behaves like 1 q 2 . Treating separately a small volume around Γ, set Figure 3: (Color online) Convergence with respect to the number of random q-wavevetors included in the ZPR calculation using the software ABINIT in the static AHC formulation. 100 ZPR calculations have been performed for each subset of q-wavevectors taking N q q-wavevectors among 20,000 (the total number of computed qwavevectors). The upper and lower bars are the maximal and minimal values in each set. The top and bottom of the boxes represent 25% and 75% of all the data in the set. The middle line is the median and the blue diamonds are outliers. The red dots comes from non shifted homogeneous Monkhorst-Pack grids, for which the number of q-wavevectors corresponds to those in the irreducible Brillouin zone aside of the regular discretization, one can estimate its contribution by replacing it by the integral over a sphere with a cut-off radius q c whose length is inversely proportional to the linear density of q-wavevectors. The contribution of this small sphere is ∝ q c 0 1 q 2 q 2 dq so that the rate of convergence of these integrals goes as q c ∝ N -1/3 q , slightly worse than in the case of the random sampling.

Moreover, there are also regions distant from Γ, but where the eigenenergies of the q-wavevectors are very close to the one at Γ (diamond is indeed an indirect gap semiconductor). On the surface S (ǫ Γ ) where the eigenenergy is exacly equal to the Γ eigenenergy, the denominator also vanishes. In the neighbourhood of the surface, the divergence is inversely proportional to the linear difference between the energy at Γ and the actual eigenenergy in its neighbourhood. To estimate the rate of convergence with respect to the number of qwavevectors of the discretized integral, we have to consider the disctretization of an integral in the Brillouin zone, in a zone of width q c around S (ǫ Γ ), in which the distance with respect to the surface is denoted as q ⊥ giving a behaviour ∝ S (ǫ Γ ) × +q c -q c 1 q ⊥ dq ⊥ . Although the principal value of this integral vanishes identically, fluctuations due to the discretization will not be small, and hence the convergence is non-monotonic.

Nonetheless, in practice, the small imaginary component at the denominator is present. One can observe, in Fig. 3, that the fluctuations, in the case of the homogeneous grid, are quite acceptable. The error with respect to the q-wavevector sampling might be estimated at 5 meV, for the set of 20,000 q-wavevectors.

After this careful comparison between codes, and this convergence analysis, we obtain that the ZPR converges smoothly towards 409 meV. This value disagrees with the one (0.615 eV) provided by Ref. [10]. The latter was actually first confirmed using QE+YAMBO, see e.g. Ref. [22]. However, while performing the cross verification between ABINIT and QE+YAMBO for the present study, we found a misuse of the symmetries at Γ in the interfacing between QE and YAMBO, affecting only the DDW term. After correction, we obtain the results provided in this work, with the numerical uncertainty being much smaller than 0.2 eV. Documentation describing how to generate data at Γ with the same standard meaning as data at other k-points appeared in QE version later than 4.0.5 (input variable nogg). Work relying on such data might have been affected by this ambiguity.

Pseudopotential choices

We will now assess the influence of the pseudopotential choices. Such a study would not be mandatory in the present context of comparison between codes for the same pseudopotential (the reference pseudopotential has indeed been used with ABINIT, YAMBO, and also in the study of Ref. [10]). This comparison will be performed only using the Abinit software. We have tested all the norm-conserving pseudopotentials available on the Abinit website as well as two UPF pseudopotentials, one of which is the reference pseudopotential. In the Table 5, we give a comparison of the ZPR using different pseudopotentials for Carbon. The calculations are made on an homogeneous 10x10x10 q-point grid (47 q-points in the IBZ), for a 6x6x6 unshifted k-point grid with the energy cut-off reported in Section 3.1 for the plane wave basis set, and 12 bands were used (with the Sternheimer equation). The low density q-wavevector grid used in this study allows for a fair comparison between pseudopotentials, but does not yield converged final results.

Pseudo

Band Fan DDW Fan+DDW ZPR reference Table 5: Comparison inside Abinit of the ZPR (and its Fan + DDW decomposition) for different pseudopotentials at Γ. Homogeneous 10x10x10 q-point grid (47 q-points in the IBZ), for a 6x6x6 unshifted k-point grid with the adapted energy cut-off for the plane wave basis set, 12 bands with were used (with the Sternheimer equation). The energies are in meV. Due to the low sampling on the q-wavevector, these value are not converged one, although the comparison between different pseudopotentials is meaningful.

One can see that although the pseudopotentials are very different (various exchange-correlation functional, different angular momentum channel include and different atomic cut-off radius) the spread on the ZPR is only around 50 meV. Fluctuations for the Fan and DDW terms, treated separately, are much larger. As emphasized earlier, the decomposition is indeed non-physical, and prone to large numerical uncertainties.

Conclusions

In this work, we have carefully compared all the quantities entering into the calculation of the ZPR in the AHC formalism in two different softwares: ABINIT and Yambo on top of QE. We show that one can get less than 10 -5 Ha/atom discrepancy on the total energy, 0.07 cm -1 on the phonon frequencies, 0.005 on the electronphonon matrix elements squared (relative discrepancy) and less than 2 meV on the zero-point motion renormalization. We also discuss the absolute value of the Fan and DDW terms taken separately. We have also presented the converged result of the band-gap reduction due to electron-phonon renormalization, that is 409 meV at 0 Kelvin and discussed its discrepancy with previously published result. We have also performed an analysis of the convergence rate of q-wavevector samplings.

Finally we have discussed the impact of the pseudopotential choices and shown that it was relatively small (around 10% of the total ZPR) thus increasing our confidence in the results and methodology.

Acknowledgements

The authors acknowledge many interesting discussions with Giustino and for sharing with us the reference pseudopotential. The authors are also thankful for the scientific and technical help from J.-M. Beuken, A. Jacques, Y. Pouillon, and G.-M. Rignanese. This work was supported by the FRS-FNRS through a FRIA grant (S.P.). A. M. acknowledges funding by MIUR FIRB Grant No. RBFR12SW0J. Computational ressources have been provided by the supercomputing facilities of the Université catholique de Louvain (CISM/UCL) as well as from the Consortium des equipements de Calcul Intensif en Fédération Wallonie Bruxelles (CECI) that is funded by the Fonds de la Recherche Scientifique de Belgique (FRS-FNRS).

Appendix

In this appendix, we detail several terms of the decomposition of the total energy, as provided by ABINIT and QE. As concern ABINIT, a decomposition of the total energy can be inferred from Refs. [24,40], but the Ewald and psp-core terms are actually mixed in these references, which is misleading.

We define first the psp-core energy.

The external potential originates from sum of atomic pseudopotentials : v psp (r, r ′ ) = ls v s (rτ s -R l , r ′ -τ s -R l ) [START_REF] Martin | Electronic Structure. Basic Theory and Practical Methods[END_REF] Each atom contribution to this external potential is made of a local and a nonlocal part:

v s (r, r ′ ) = v loc s (r)δ(r -r ′ ) + v non-loc s (r, r ′ ) ( 16
)
For each atom the local part is long ranged, with an asymptotic behaviour -Z s /r. Such behaviour implies a divergence at G = 0 in reciprocal space. Divergencies at G = 0 also happen in the Hartree energy and the Ewald energy.

A careful treatment of the divergencies lead to their mutual cancellation, albeit with some finite residual. The residual specifically linked to the long-range behaviour of the local pseudopotential is denoted as the psp-core energy:

E psp-core = 1 2Ω 0        s Z s        s ′ v loc s ′ (r) + Z s ′ r dr. ( 17
)
The Ewald energy is the energy of an infinite number of periodic positively charged particle placed in a negative homogeneous background:

E Ew = 1 2 s,s ′ Z s Z s ′ G 0 4π Ω 0 G 2 e iG • (τ s -τ s ′ ) e -G 2 4Λ 2 - l e iq • R l erfc(Λ |R l + τ s ′ -τ s |) |R l + τ s ′ -τ s | - 2 √ π Λδ ss ′ - π Ω 0 Λ 2 (18)
with Z s the charge of ion s, Ω 0 the unit cell volume, Λ a parameter that can assume any value and is adjusted to obtain the fastest convergence.

Figure 1 :Figure 2 :

 12 Figure 1: (Color online) Relative differences ∆ of the SGKK2 between ABINIT and QE, at k = Γ and q = Γ, for 20x20 and 300x300 pairs of bands.

Table 1 :

 1 Comparison of selected quantities related to the ground state and to phonon calculations, for diamond, with a 6x6x6 unshifted kpoint grid and a kinetic energy cutoff of 30 Hartree for the plane wave basis set. The same norm-conserving LDA pseudopotential is used. The lattice parameter is 6.652 Bohr. All the energies are in Hartree, are expressed per cell (two atoms per cell) and the phonon frequencies are in cm -1 .

				ABINIT 7.3.2	QE 4.0.5
		Kinetic energy	8.450310501	-
	One-electron energy	-	4.135925595
		Hartree energy	0.943336981	0.943337120
		XC energy	-3.567609861	-3.567609935
		Ewald energy	-12.955782342	-12.955782345
	Psp-core energy	0.581222385	-
	Loc. psp. energy	-5.093200787	-
		NL psp. energy	0.197606844	-
		Total energy	-11.444116277	-11.444129565
	Phonon freq. at q = Γ 1	3.335 (x3)	8.832 (x3)
	q = Γ 25 ′ with ASR imposed	1330.408 (x3) 0.000 (x3)	1330.428 (x3) 0.000 (x3)
				1330.403 (x3)	1330.400 (x3)
		q = L 3	555.305 (x2)	555.319 (x2)
		q = L 2 ′ q = L 1	1076.250 1235.429 (x2)	1076.268 1235.440 (x2)
		q = L 3 ′ q = X 3	1273.840 795.900 (x2)	1273.860 795.964 (x2)
		q = X 1	1098.461 (x2)	1098.489 (x2)
		q = X 4	1224.570 (x2)	1224.590 (x2)
	Sym.	Band	eigenergies in Abinit	eigenergies in QE
	Γ 1	1	-21.7959	-21.7957
	Γ 25 ′ Γ 15	2-3-4 5-6-7	0.0000 5.6698	0.0000 5.6699
	Γ 2 ′ Γ 1	8 9	14.3020 19.4714	14.3023 19.4716

Table 2 :

 2 Comparison between Abinit and QE of the nine lower eigenenergies in eV, relative to the top of the valence band at Γ.

Table 3 :

 3 for the two codes.

	Band	Soft.	1	2-3-4	5-6-7	8	9
	1	AB QE	0 0
	2-3-4	AB QE	1.530449 1.530401	5.803074 5.803088
	5-6-7	AB QE	0.493950 0.493932	0.292491 0.292472	4.495984 4.496296
	8	AB QE	0 0	4.635255 4.635284	2.430641 2.430665	0 0
	9	AB QE	0 0	1.565383 1.565460	4.002821 4.002843	0 0	0 0

Comparison between Abinit (AB) and Quantum Espresso (QE) of absolute value of the SGKK2, at k = Γ and q = Γ in 10 -6 a.u. (1 a.u. = 4.78599 • 10 12 J/kg). Matrix elements with values lower than 10 -11 Ha have been put to 0.