François Marmier
email: francois.marmier@mines-albi.fr

Maria Gonzales-Blanch

Naoufel Cheikhrouhou
email: naoufel.cheikhrouhou@epfl.ch

A new structured adjustment approach for demand forecasting

Keywords: Forecasting, Demand planning, Judgmental forecasting, Judgmental factors, Time series

Demand forecasting consists of using data of the past demand to obtain an approximation of the future demand. Mathematical approaches can lead to reliable forecasts in deterministic context, extrapolating regular patterns in series. However unpredictable events that do not appear in the historical data can make the forecasts obsolete. As forecasters have a partial knowledge of the context and probable future events (such as strikes, promotions), this work investigates structuring the implicit as well as the explicit knowledge in order to be easily and fully integrated into final forecasts. This paper presents a judgmental-based approach in forecasting where mathematical forecasts are considered as a basis and the structured knowledge of the experts is provided to adjust the initial forecasts. This is achieved using the identification of four factors characterizing specific events that could not have been considered in the initial forecasts. The validation of this approach has been conducted on 2 industrial case studies, a plastic bag manufacturer and a distributor on the food market. The results show that structuring the expert knowledge could lead not only to high improvements of forecasts accuracy but also to a better initial data cleaning and outlier identifications.

INTRODUCTION

Demand planning and forecasting using time series consists of using historical data of the past demand to obtain an approximation of the future demand. Many different ways of developing forecasts have been studied or proposed, like the mathematical and statistical methods or probabilistic approaches to model random variables [START_REF] Makridakis | Forecasting, methods and applications[END_REF]. Although mathematical approaches can lead to reliable forecasts in some contexts, extrapolating regular patterns in series to unpredictable events can render the forecasts obsolete. On the other hand, forecasters have for asset their partial knowledge of the context and of potential external stunner. This knowledge is not available within the statistical methods as it corresponds to particular events, such as personnel strike or limited promotions. Nowadays, it has been recognized that judgment is an indispensable component of forecasting [START_REF] Lawrence | Judgmental forecasting: A review of progress over the last 25years[END_REF] and that there still is a need for studies which could help companies to improve their forecast quality using contextual information [START_REF] Martins | Strategic analysis of products related to the integration of human judgment into demand forecasting[END_REF], [START_REF] Sanders | Integrating judgmental and quantitative forecasts: Methodologies for pooling marketing and operations information[END_REF]. The objective of this work is the development of a methodological approach for the improvement of forecasting accuracy by structuring and using expert knowledge. A judgmental adjustment based approach, combining mathematical forecast and judgmental factors, is discussed and its performances are compared to classical mathematical forecasting techniques. The comparison of the forecasting accuracy of the considered approaches is done using error measures. Furthermore, the increasing accuracy of the proposed methods in forecasting is evaluated on the basis of two industrial applications within a plastic bag manufacturer in Spain and a food distributor in France. In Section 2, an overview of the features of the forecasting techniques over the literature including the judgmental forecasting methods is presented and discussed. Section 3 outlines the details of the proposed judgmental adjustment-based approach and its main features. The performances of the proposed approach are compared in Section 4. Two industrial case studies using the developed approach are presented in section 5. Finally, Section 6 contains the conclusions of this work and future research directions.

COMBINED JUDGMENTAL-MATHEMATICAL FORECASTING SYSTEMS

In the literature review [START_REF] Webby | Judgemental and statistical time series forecasting: a review of the literature[END_REF], four main types of approach integrating human judgment into forecasting methods are defined: the model building approach is a methodology which uses judgment in the selection and development of the quantitative forecast [START_REF] Bunn | Interaction of judgmental and statistical forecasting methods, issues and analysis[END_REF]. This approach requires high technical knowledge. The combination of judgmental and mathematical forecasts is a pragmatic approach for integrating the personal analysis of contextual information that may influence the forecasts [START_REF] Fildes | Good and bad judgement in forecasting: Lessons from four companies[END_REF]. It has been shown that it gives good results for time series which have a low variation coefficient [START_REF] Sanders | Bringing judgment into combination forecasts[END_REF]. Moreover, both (mathematical and contextual) forecasts have to present similar accuracies to lead to reliable global forecast. The judgmental decomposition consists of identifying and analysing the effects of past contextual information in time series before establishing mathematical forecasts. Once the mathematical forecasts are established, future-oriented factors are taken into account and the forecasts are adjusted judgmentally [START_REF] Edmundson | Decomposition; a strategy for judgemental forecasting[END_REF]. From the planner point of view, it is possible that decomposition reduces the cognitive workload, but it may be risky and ineffective in some circumstances. In the judgemental adjustment process, the mathematical forecasts are reviewed by a forecaster and then adjusted on the basis of his/her knowledge and experience. It is a common practice in companies and it is the major competing alternative to the combination process for integrating judgment with mathematical forecasts. However, the practice of judgmental adjustment has been criticized because of its informal nature [START_REF] Bunn | Interaction of judgmental and statistical forecasting methods, issues and analysis[END_REF]. Using a procedure or a decision support system to structure information is helpful in both decision-making and forecasting. Judgmental adjustments lead to greater improvements in accuracy when the process is structured. The forecasting process is performed sequentially with the quantitative forecasts generated first, and then adjusted based on judgment. One of the advantages of this approach is time reduction, allowing the judgment to rapidly incorporate the latest updated information [START_REF] Sanders | Integrating judgmental and quantitative forecasts: Methodologies for pooling marketing and operations information[END_REF]. Flores et al. [START_REF] Flores | Judgmental adjustment of forecasts: A comparison of methods[END_REF] compare different approaches of judgemental adjustment; The forecaster ranks the impacts due to different factors to obtain an average adjustment of an ARIMA forecast.

In order to integrate human judgment factors in mathematical models in a structured way, this work proposes a factor-based approach to assist the forecaster to focus selectively on different events and to structure its judgment when adjusting forecasts and enhance the accuracy of forecasts. The approach proposes a classification of event-based factors used in structuring the information. Similar factors are proposed by Lee et al. [START_REF] Lee | UNIK-FCST: Knowledge-Assisted Adjustment of Statistical Forecasts[END_REF] to determine the reaction of the demand faced to different situations; These factors are used in a casebased reasoning software playing the role of forecasting expert. In our case, the forecaster will be able to structure and communicate his implicit knowledge concerning markets evolution (total volume, number of competitors…), customers (numbers, needs, potential demand…) and contractors (development status, special offers…) by representative factors.

PROPOSED APPROACH

The global approach consists of integrating the expert judgment in a structured manner as complementary information to mathematical forecasts, established by means of classical mathematical models. Figure 1 describes this idea composed of three steps: first, data are filtered and initial mathematical forecasts are developed. Second, different probable events are identified by experts and the related factor variables are weighted and finally, the mathematical forecasts are adjusted.

Figure 1. Conceptual scheme of the general method developed

First step: data filtering and mathematic forecasting

This step consists of identifying the outliers and removing the effect of non-periodic events on the data series. The time series is decomposed, taking into account past contextual information. In other words, the data Y(t-k) of the period t-k of the time-series is cleaned (noted Yo(t-k)). Historical judgmental events can usually be identified by examining the corresponding historical records. In addition, in order to prevent from the risk of missing or ignoring some historical judgmental events, outliers may be examined and analyzed in detail. Outliers are considered as unusual or abnormal observations in data and are replaced by representative values (f.i. the average of the neighboring values). If the time series presents a strong trend or a seasonal pattern, a confidence interval is evaluated for each year and each time period. Thus, the mathematical forecasts for the given filtered or "cleaned" time-series for period t are

0 0 0 ^( ) ( ( 1), ( ( 2),...) Y t f Y t f Y t = - - (1)

Second step: Identification and formalization of influencing factors

A judgmental factor is defined as a "factor that cannot be fully incorporated into the time series models, and thus cannot be effectively identified by the extrapolation of past patterns in the data set" [START_REF] Lee | Judgmental adjustment in time series forecasting using neural networks[END_REF]. The attributes or causal variables are the variables that cause changes in the factors. The attributes related to the a factor are: duration, intensity, type, etc. The realization of judgmental factor is considered as a judgmental event. Thus, a judgmental event is accompanied by its impact and may affect the demand during more than one time

Step 1

Step 2

Step 3

Then : i = w. max Δ (2)

Transient factors

Transient factors influence time series data only during the period in which a particular event occurs (such as a strike) (cf. figure 2.a). When the event is over, the effect does not last any longer. The total impact i covers the effective number of impacted periods. So, the time can be decomposed into periodical instances tp.

Adjustments that encompass n consecutive periods p are detailed in equation 3.

( , ) ( , )

n p p p F i t f i t = = ∑ (3)
where f are the periodical adjustment functions, i p and t p corresponds to the impacted period p with the intensity i p . Transient factors render the data be upper or lower than the value of the main trend ^0 Y . The expert establishes first, the sign of the adjustment (positive or negative) then, the maximal value of the adjustment max Δ and finally, following his knowledge of the industrial and commercial context, the expert deduces the weight w in order to determine the impact.

Transferred impact factors

The global impact i of this kind of event is transferred from one set of periods S p1 to another one S p2 , without changing the global forecasts of the related consecutive n periods. Figure 2.b presents this factor. This phenomenon happens when the price changes are announced beforehand at time t a . In this case, the temporary change in demand due to the expected price change is transferred and compensated in the following time period at time t e . Then, S p1 precedes S p2 . But if t a = t e , S p2 precedes S p1. Periodical instances that encompass n consecutive periods is computed using equation ( 4):

1 ( , ) ( , ) n p p p F i t f i t = = ∑ (4)
Where i p is the proportional impact (positive or negative) on the period p. As for the other factors, the expert determine the maximal value max Δ subject to be transferred then uses the scale of impact to rate the event and to obtain i.

Quantum jump factors

The quantum jump factor occurs when the effect of a non-repetitive new event is permanent. Figure 3.a presents this factor. The adjustment created by a quantum factor is evaluated and separated in different periodically instances in order to be adjusted to the original mathematical forecast. This adjustment is detailed in the equation 5 where t e is the effective date (the date of the event generating the demand jump)

( , ) . F i t i Qj =

(5) Where Qj is a binary variable which means that the Y-intercept, which define the elevation of the demand trend, is impacted with the intensity i at t e : If t < t e then Qj =0 Else Qj =1. The weighting process of the jump factor variables is similar to the weightings of the other factors. After determining the adjustment sign, an expert determines the maximal value of the adjustment max Δ and express his /her opinion about the percentage that weights that variable. ( , ) . . F i t i t Tc = (6) Where Tc is a binary variable which means that the slope of a linear regression of the demand trend is impacted with the intensity i at t e : If t < t e then Tc =0 Else Tc =1.

Third step: the adjustment process

The final adjusted forecast is defined as: ^a

Y (t) = ^o Y (t) + Ŷ Δ (t) (7)
Where Ŷ Δ (t) is the adjustment computed with the judgmental established by experts. It corresponds to the total impact on demand of the different event occurring at time t. Some factors influenced the demand over a restricted period or a set of periods but others are definitely modifying the demand behaviour. When multiple judgmental factors impact simultaneously the demand, it is necessary to decompose forecast adjustments according to the effects of each individual judgmental event. Then, the global adjustment at time t, defined in equation 8, corresponds to the total K of the different adjustments related to the K factors impacting the demand at time t :

Ŷ Δ (t) = ( ) 1 , K k k k F i t = ∑ (8)

EVALUATION OF ADJUSTED FORECAST QUALITY

Accuracy measures how well the results of the forecasting model correspond to the actual data.

Accuracy plays an important role in evaluating forecasting methods and demand planning. Forecast accuracy is evaluated using different error measures. In general, the error measure used is the one that most closely related to the decision being made. )

n t t t t Y F MAPE n Y = - = ∑ (10)

INDUSTRIAL APPLICATION

First case study

Consider the case of the Company X, a plastic bag manufacturer in the south of Spain. The time series used are composed by the aggregated monthly demand collected over three years (2004)(2005)(2006) as it can be observed in figure 7. The aim of this scenario is to plan demand for the year 2007. Company X has three main customers in this area which are all supermarket chains. An expert analyzes all the coming events and the factors influencing in the plastic bag demand. The model that gives the smallest MAE and MAPE is ARIMA(5, 0, 4). Figure 4 presents the time series and the mathematical forecasts obtained. The forecasting expert of company X identifies and classifies different factors for the forecasted year: Plastic price: The supply and demand model determines price and quantity sold in the market. This means that if plastic prices increase the clients will ask for lower quantities. The expert knows that the trend of the demand is going to change because the raw material price is going to increase from January 2007 due to new framework contracts. It corresponds to a trend change factor. Special offer: The company X wants to reduce stocks in the beginning of the year 2007. A special offer to their clients is then planned to be launched consisting of delivering the double quantity with lower price. The consequence of this factor is that a big part of the demand of January will be transferred to February and March 2007. In this case, this special offer is classified as a transferring factor. Technical improvements: Nowadays, big companies care about the fulfillment of the environmental demanded requirements. In this frame, the expert predicts that using ecological non-polluting plastic bags is an interesting additional purchase decision criterion for supermarkets. This factor is classified as a jump factor because there would be new clients trying this new concept of plastic bags.

Client situations: Company X has a client that asks for a fixed quantity of plastic bags per month. This client has announced that its facilities will close during a month after the summer holidays. The consequence of this event will be a decrease in the total monthly demand. However, this client will order new plastic bags for November. This factor is classified as a transient factor. Table 1 presents the total impact of the different factors on demand per month using the information on the different factors. Figure 5 compares the real 2007 data to the mathematical forecast and to the judgmentally adjusted forecast. The adjusted forecast presents better results in the forecast and percentage error over nine months. Table 2 presents the forecast performance of the proposed approach. The adjusted forecast gives the best results since the MAE has been decreased by 14% and the MAPE by 18%.

Second case study

In this second application, we are interested in a set of 4 different products provided by a fresh food distributor with historical sales depicted on figure 6. For these different series, the expert identifies that mathematical forecast would not be sufficient to achieve reliable forecast. Different factors due to the customers' demand behavior are considered. The proposed approach is applied and the mathematical forecasts are then adjusted. The deviation between mathematical forecasts and real sales shows the limit of the past data extrapolation to obtain reliable forecast (cf. Fig. 6). By using the structured expert judgement (jump factor for the case on 3 shows the improvement of the forecast accuracies relying on the error measures of the equations 9 and 10.

CONCLUSION

We propose a factor-based approach to assist the forecaster, structure his judgment when adjusting forecasts and to enhance the accuracy of global forecasts. Therefore, the forecaster is able to structure and communicate his knowledge regarding future market modification and non-periodic events. It allows forecasters to identify different factors, in order to integrate these events as well as their impacts with mathematical models into global and reliable forecasts. For both presented industrial case studies, the proposed approach has a substantial impact on the accuracy of the resulting forecasts. In fact, by applying our approach, the MAE and the MAPE have been both decreased. Forecasts are logically improved since the achieved adjustments are based on the expert knowledge. However, if the adjustments are done globally, without structuring or without real analysis of their impacts, their durations and their locations in time, the deviation and the error might increase. The developed approach is simple and complete, thus allowing the expert to transcribe each possible situation. It presents interesting perspectives concerning the analysis of experts' perceptions, in particular the cognitive process of impact estimation for each considered factor, and their integration into the forecasting process; different experts could have different perceptions of factors, leading to a range of different evaluations for a same factor. In addition, it is interesting to investigate the evolution of forecast accuracy if experts get different feed-backs based on the comparison between adjusted forecasts and past real demand and occurred events.

Figure 2

 2 Figure 2.a Transient factor

Figure 3

 3 Figure 3.a Quantum jump factor

  Some factors like price variation can modify the demand by a percentage. These factors are considered as Trend change factors as shown in figure3.b. The expert establishes the sign of the possible trend change and its limit max Δ . Then he weighs the impact using the impact scale. The trend adjustment is detailed in equation 6 where t e is the effective date of the event generating the demand jump.

Figure 4 .

 4 Figure 4. Company X's demand (2004-2007)

  fig 6.a, transient factor on several time periods on fig 6.b and two transient factors on fig 6.c and 6.d), substantial improvements of the forecast accuracy are achieved. Table

Figure 5 .

 5 Figure 5. Mathematical forecast, forecast judgmentally adjusted and reality

  to its unreliability in explaining the results. Like MSE, RMSE penalizes errors according to their magnitude. Both MSE and RMSE are not unit-free and therefore comparisons across series are difficult. With MAD there is no way to know if MAD error is large or small with respect to the actual data. With the ME and MPE approach, the real significance of the error can be difficult to evaluate because errors can be annulated by summing negative and positive components. Contrary to ME, the use of MAE makes it possible to take into account both positive and negative errors. With MAE, the total magnitude of error is provided but not the real bias or direction of the error. MAPE measures the deviation as a percentage of actual data with respect to the forecast. Thus, positive and negative errors do not cancel out. In this paper, MAE and MAPE are used to calculate the forecast errors as they are considered as being the most performing approaches to measure errors.

	Mean Absolute Error	(	MAE	)	1 n = ∑ n	t e	(9)
	Mean Absolute Percentage Error
	Some
	commonly used error measures are Mean Absolute
	Deviation (MAD), Mean Absolute Error (MAE) (cf.
	equation 9), Mean Percentage Error (MPE), Mean
	Square Error (MSE) and Root Mean Square Error
	(RMSE), R2 (or R-squared), Mean Absolute Percentage
	Error (MAPE) (cf. equation 10).

[START_REF] Armstrong | Judgmental Bootstrapping: Inferring Experts' Rules for Forecasting[END_REF]

alleged that R2 (which assesses the pattern of the forecasts relative to that of the actual data), is not particularly useful to forecasters and its use is probably more harmful when using time-series data. Moreover, Armstrong explains why MSE should not be used due

Table 2 .

 2 Comparison of the error measures

	Measure	MAE	MAPE
	Math. forecast	112.2086	0.081931
	Judg. forecast	97.54472	0.066673

Table 3 .

 3 Improvement of the forecast accuracy

	Acurracy increase	a	b	c	d
	MAE	67%	38%	77%	72%
	MAPE	64%	25%	20%	31%

Table 1 .

 1 Final monthly adjustments

		Trend change	Transferring factor	Jump factor	Transient factor
	Month	adjustment	adjustment	adjustment	adjustment	Total adjustment
	January	-23	400			377
	February	-46	-200	200		-46
	March	-69	-200	200		-69
	April	-92		200		108
	May	-115		200		85
	June	-138		200		62
	July	-161		200		39
	August	-184		200		16
	September	-207			-250	-457
	October	-230				-230
	November	-253				-253
	December	-276				-276

Figure 6. Mathematical forecast, mathematical forecast judgmentally adjusted and reality


Mauro Causá

Andreas Savin

Bernard Silvi

Richard F W Bader

Atoms and bonds in molecules and chemical explanations *

The concepts of atoms and bonds in molecules which appeared in chemistry during the XIX th century are unavoidable to explain the structure and the reactivity of the matter at a chemical level of understanding. Although they can be criticized from a strict reductionist point of view, because neither atoms nor bonds are observable in the sense of quantum mechanics, the topological and statistical interpretative approaches of quantum chemistry (QTAIM, ELF and MPD) provide consistent definitions which accommodate chemistry and quantum mechanics.

1 About the explanation of the bonding in chemical systems.

One of the aims of chemistry, as a science, is to study and explain the structure of the matter and its transformations. By explain it is meant that science has to tell how and why a given phenomenon occurs in given circumstances. Scientific explanation has been discussed by philosophers who proposed different accounts (for a review see [START_REF] Strevens | Scientific explanation[END_REF][START_REF] Woodward | Scientific explanation[END_REF]), for example, Hempel's deductive-nomological, or DN, account [START_REF] Hempel | Studies in the logic of explanation[END_REF], the unification account [START_REF] Friedman | Explanation and scientific understanding[END_REF][START_REF] Kitcher | Explanatory unification[END_REF], the causal account [START_REF] Salmon | Explanation and the causal structure of the world[END_REF] and the kairetic account [START_REF] Strevens | The causal and unification approaches to explanation unified-causally[END_REF] which attempts to catch the advantages of the unification and causal accounts. In 1 brief, the DN account provides a scheme for any deterministic explanation of a particular event and consists in a deductive derivation of the occurrence of the event from a set of true propositions involving at least a scientific law or principle. The unification approach intends to derive the occurrence of the event using a theory that unifies many phenomena or the theory that unifies the phenomena better than any other. In the causal model the explanation will trace the causal processes and interactions leading to the event (i.e. the causal history) , or at least a subset of these, as well as describing the processes and interactions that make up the event itself. Except for a dogmatic acceptation of the unification account, these different approaches leave a rather great freedom provided the explanation contains no false representation of reality. They do not invalidate the recourse to empirical laws, idealization or conjectures and enable to consider different levels of explanation. As pointed out by E. Scerri [START_REF] Scerri | Philosophy of chemistrys-a new interdisciplinary field?[END_REF] If one believes only in fundamental explanations, this form of activity appears to be seriously mistaken. However, as chemists we are also aware of the need to operate on many levels and the fact that explanations can be genuinely levelspecific. Such approaches must be used very carefully. They should not degenerate into the introduction of ad hoc explanations that are invoked in the explanation of particular chemical facts but cannot be generalized to other situations.

In principle, explanations are not essentially communicative and exist independently of the intention of explaining anything to anyone. However, the availability of different levels of explanation is essential for the transmission of knowledge and therefore for the development of scientific ideas.

In the special issue of the Annals of the New York Academy of Sciences entitled "CHEMICAL EXPLANATION: Characteristics, Development, Autonomy" published in May 2003, G. del Re exemplified the difference between a chemical explanation and other explanations of facts. He considered the expected answers to the following question [START_REF] Del Re | Reaction mechanisms and chemical explanation[END_REF]:

Why are most glass bottles green? A technologist would respond by saying that their production cost is lower; a physicist would answer that their glass absorbs light selectively, so that the transmitted light has a different spectrum from that of sunlight; but a chemist would answer that it is because ordinary glass contains ferrous ions.

This example shows that several causal processes may be invoked to explain a fact. In the present case all answers are relevant and complementary since they partially address the question. This example insists on the cultural differences between scientific communities.

Chemical explanations of the structure of the matter and its transformations mostly rely upon a reductionist approach which describes the matter as an assembly of molecules or crystals themselves composed of elemental atoms. The mutual interaction of these elemental atoms, in other words the bonding, is ruled by the location of the involved elements in the periodic table which determines their "atom in molecules" properties such as electronegativity, valence, ionic and covalent radii. This description can be viewed as a synthesis of the early mechanistic and substantialist conception of the matter. The electronic origin of the bonding is accounted for by very simple and efficient phenomenological models: the ionic and covalent bond models and the mesomery. Moreover, the Valence Shell Electron Pair Repulsion (VSEPR) and Ligand Close Packing (LCP) models [START_REF] Gillespie | Inorganic stereochemistry[END_REF][START_REF] Gillespie | Molecular Geometry[END_REF][START_REF] Gillespie | Electron domains and the vsepr model of molecular geometry[END_REF][START_REF] Gillespie | Improving our understanding of molecular geometry and the vsepr model through the ligand close-packing model and the analysis of electron density distributions[END_REF][START_REF] Gillespie | Chemical Bonding and Molecular Geometry[END_REF] state rules which explain the molecular geometry. Many chemical explanations belong to either the DN account or the the causal approach. For example the formation of an ionic bond can be deduced from the electronegativities of the involved elemental atoms and from the electronegativity equalization principle which implies the exchange of an extensive quantity (here electrons) yielding opposite charge ions attracted by the Coulomb electrostatic laws. Reaction mechanisms depicted by curly arrows can be viewed as causal stories.

Rather than atoms, the reduction of chemistry to physics is an example of the unification account in action. It considers interacting particles ruled by the laws of quantum mechanics which enable an almost complete description of the interaction between electron and nuclei and therefore the quantitative exact predictions of observables. It corresponds to a mechanistic working program anticipated by [START_REF] Dirac | Quantum mechanics of many-electron systems[END_REF][START_REF] Dirac | Quantum mechanics of many-electron systems[END_REF]:

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

Although the predictive power of quantum mechanics is indisputable, its ability to provide explanations has been questioned by René Thom in Prédire n'est pas expliquer [START_REF] Thom | Prédire n'est pas expliquer[END_REF]:

La mécanique quantique est incontestablement le scandale intellectuel du siècle! Comment entendez vous scandale? C'est que la science a renoncé à l'intelligibilité du monde; elle y a réellement renoncé! C'est quelque chose qui s'impose et qui n'est pas intelligible. 1Thom's provocative remark echoes Niels Bohr's attitude reported by Karl R. Popper in the introductory comments of Quantum Theory and the Schism in Physics [START_REF] Popper | Quantum Theory and the Schism in Physics: From The Postscript to the Logic of Scientific Discovery[END_REF] When he accepted quantum mechanics as the end of the road, it was partly in despair: only classical physics was understandable, was a description of reality. Quantum mechanics was not a description of reality. Such a description was impossible to achieve in the atomic region; apparently because no such reality existed: the understandable reality ended where classical physics ended. and Popper's assessment from the 1982 preface of the same book: Today, physics is in a crisis. Physical theory is unbelievably successful; it constantly produces new problems, and it solves the old ones as well as the new ones. And part of the present crisisthe almost permanent revolution of its fundamental theories -is, in my opinion, a normal state of any mature science. But there is also another aspect of the present crisis: it is also a crisis of understanding.

The reservation of René Thom concerning the lack of intelligibility of quantum mechanics as well as Popper's crisis of understanding can be explained by the status of quantum systems as discussed by Heelan [START_REF] Heelan | Paradoxes of measurement[END_REF]:

An individual quantum object is intuitively given to an observer only by the actual isolated footprints it leaves in the perceptual world of the laboratory, the record of individual measurements. It is not just a conceptual object, nor is it an "embodied object in its own right. It is, however, physical and material because of the footprints it makes in the world. It seems then to exist and function ontologically prior to and i some way independently of the phenomenological constitution of classically scientific laboratory space-time.

The interpretation of approximate wavefunctions is the most popular approach used by quantum chemist (and others) to build up explanations of the structure and reactivity of chemical species. It had been encouraged by Dirac who wrote in the last sentence of his 1929 paper "Quantum mechanics of Many-Electron Systems" [START_REF] Dirac | Quantum mechanics of many-electron systems[END_REF]:

It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. This prescription has been anticipated two years before Dirac's paper by the calculations of the dihydrogen molecule of Heitler and London [START_REF] Heitler | Wechselwirkung neutraler atome und homoöpolare bindung nach quantenmechanik[END_REF] on the one hand and of Condon [START_REF] Condon | Wave mechanics and the normal state of the hydrogen molecule[END_REF] on the other hand.2 Heitler and London's paper was immediately recognized as a milestone in the history of chemistry. Here was found the mathematical dynamic formulation of Lewis' covalent bond, the energy of the electron pair bond being given as a resonance energy due to the interchange of two electrons. As pointed out by Pauling [START_REF] Pauling | Modern structural chemistry[END_REF]):

Condon's treatment is the prototype of the molecular-orbital treatment that has been extensively applied in the discussion of aromatic and conjugated molecules, and Heitler and London's treatment is the prototype of the valence-bond method.

These two methods constitute the backbone of the theory of chemical bonding [START_REF] Hoffmann | A conversation on vb vs mo theory: A never-ending rivalry?[END_REF]:

Quantum mechanics has provided chemistry with two general theories, valence bond (VB) and molecular orbital (MO) theory. The two theories were developed at about the same time, but quickly diverged into rival schools that have competed, sometimes fervently, on charting the mental map and epistemology of chemistry.

Though these methods have proved their efficiency in both predictive and interpretative purposes, they have a constitutive epistemological weakness because they mostly rely on the interpretation of the approximate wave function in terms of its atom related components rather than upon observable quantities. As pointed out by [START_REF] Coulson | Valence[END_REF]:

This epistemological difficulty is mostly due to the weakness of interpretative methods that give a physical significance to quantities, such as molecular orbitals or valence bond structures, appearing as intermediates during the course of solution of the many-body Schrödinger equation.

In a second approach, no more than interpretative postulates is required and no approximation is considered in the principles. The key is then to import a mathematical method of analysis to perform the partition. Theories derived accordingly should [START_REF] Aslangul | Aspects of the localizability of electrons and molecules: Loge theory and related methods[END_REF]) "provide the mathematical bridge between the chemical intuition and wave mechanics, which may be considered as a theoretical justification of the main chemical ideas." Several interpretative methods have been developed in this spirit: the loge theory [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF][START_REF] Daudel | Théorie de la localisabilité des corpuscules .1. la notion de loge et la signification géometrique de la notion de couche dans le cortège électronique des atomes[END_REF][START_REF] Daudel | Localizability of electrons in atoms and molecules-application to the study of the notion of shell and of the nature of chemical bonds[END_REF][START_REF] Aslangul | Aspects of the localizability of electrons and molecules: Loge theory and related methods[END_REF][START_REF] Aslangul | The loge theory as a starting point for variational calculations. i. general formalism[END_REF], the quantum Theory of Atoms in Molecules (QTAIM) [START_REF] Bader | Binding regions in polyatomic molecules and electron density distributions[END_REF][START_REF] Bader | Molecular charge distributions and chemical binding[END_REF][START_REF] Bader | Partitioning and characterization of molecular charge distributions[END_REF][START_REF] Bader | Molecular fragments or chemical bonds?[END_REF][START_REF] Bader | Quantum topology of molecular charge distributions. ii. molecular structure and its change[END_REF][START_REF] Bader | Quantum theory of atoms in molecules -dalton revisited[END_REF][START_REF] Bader | Atoms in molecules[END_REF][START_REF] Bader | Atoms in Molecules: A Quantum Theory[END_REF][START_REF] Bader | A quantum theory of molecular structure and its applications[END_REF]Bader et al., 1996a;[START_REF] Bader | The quantum mechanical basis of conceptual chemistry[END_REF][START_REF] Bader | Everyman's derivation of the theory of atoms in molecules[END_REF], the topological analysis of the electron localization function (Häussermann et al., 1994b;[START_REF] Silvi | Classification of chemical bonds based on topological analysis of electron localization function[END_REF][START_REF] Savin | Elf: The electron localization function[END_REF][START_REF] Savin | On the significance of elf basins[END_REF][START_REF] Silvi | The topological analysis of the electron localization function : a key for a position space representation of chemical bonds[END_REF] and of the electron localizability indicator [START_REF] Kohout | Electron localizability indicator for correlared wavefunctions. i. parallel spin pairs[END_REF][START_REF] Kohout | Electron localizability indicator for correlared wavefunctions. i. antiparallel spin pairs[END_REF][START_REF] Wagner | Charge decomposition analysis of the electron localizability indicator: A bridge between the orbital and direct space representation of the chemical bond[END_REF] as well as the determination of maximum probability domains [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF][START_REF] Gallegos | Maximal probability domains in linear molecules[END_REF][START_REF] Lopes | Understanding Maximum Probability Domains with Simple Models[END_REF] and the determination of the electron number probability distribution functions [START_REF] Francisco | Electron number probability distributions for correlated wave functions[END_REF]Martín Pendás et al., 2007;Martín Pendás et al., 2007;[START_REF] Martin Pendas | An electron number distribution view of chemical bonds in real space[END_REF]. This route has been recently reexamined by Richard Bader [START_REF] Bader | On the non-existence of parallel universes in chemistry[END_REF] and by [START_REF] Bader | Atoms in molecules as nonoverlapping, bounded, space-filling open quantum systems[END_REF] to justify the universality of the QTAIM approach. Richard Bader clearly rejects the chemical approach claiming [START_REF] Bader | On the non-existence of parallel universes in chemistry[END_REF] Readers of this paper must be prepared to encounter the view that chemistry has been reduced to physics and that the existing language of chemistry stated in terms of models based on valence bond and molecular orbital theories and related notions such as electronegativity, resonance, Coulson's overlap integral, non-bonded and steric interactions as well as 'Pauli repulsions', is to be replaced in its entirety by physics.

A third approach, the conceptual DFT, [START_REF] Parr | Electronegativity: The density functional viewpoint[END_REF][START_REF] Parr | Absolute hardness: Companion parameter to absolute electronegativity[END_REF][START_REF] De Proft | Conceptual and computational dft in the study of aromaticity[END_REF][START_REF] Geerlings | Conceptual density functional theory[END_REF] provides definitions of global (chemical potential/electronegativity, hardness, softness, electrophilicity) and local (Fukui functions, local hardness) reactivity related properties in terms of energy derivatives. This approach rationalizes chemical laws such as the electronegativity equalization or the hard soft acid base principle often used to explain bonding properties.

The autonomy of chemistry

Chemistry has both a scientific side (study of the matter and of its evolution) and a technological side (synthesis, applied analytical chemistry) which are intricate even in quantum chemistry. In the hierarchy of sciences, chemistry, is generally ranked after physics. As pointed out by [START_REF] Van Brakel | The ignis fatuusa of reduction and unification[END_REF] This prejudice goes back to Kant's view that science is "proper" to the extent that mathematics is applied within it. According to Kant, chemistry is rational (because it uses logical reasoning), though not a proper science, because it misses the basis of mathematics and the synthetic a priori chemistry is an uneigentliche Wissenschaft -an "improper" science.

Kant followed the evolution of chemistry considering Stahl's phlogistic theory as a scientific revolution before accepting Lavoisiers's views [START_REF] Vasconi | Sistema delle scienze naturali e unitsà della conoscenza nell'ultimo Kant[END_REF][START_REF] Van Brakel | Kant's legacy for the philosophy of chemistry[END_REF]. Nevertheless, Kant writing at the end of the XVIII th century considered the mathematics and chemistry of those days. During the XIX th century mathematical models have been developed in chemistry independently of any reductionist attempt, either implicitly such as in Dalton's law of multiple proportions or explicitly such as in the law of mass action or in kinetics.

Chemistry has known a considerable development along the XIX th during which a large number of the natural elements have been discovered as well as many fundamental chemical concepts have been introduced. The periodic law of Mendeleev remains a culminating achievement of this area. At the turn of the XIX th , the law of the definite proportions and the law of multiple proportions enabled the introduction of a consistent description of the matter through the definition of stoichiometry coefficients. These latter are integers for a large majority of chemical compounds (exceptions are the consequence of a random substitution in crystals and glasses yielding formulas such as Zn 1-x Cd x Cl 2 ) which implies a discontinuous description of the matter in terms of atoms. In the model of Dalton, atoms are small indivisible bodies surrounded by an envelope of caloric. [START_REF] Dalton | New System of Chemical Philosophy[END_REF] The model was further improved by Laming who was aware of Faraday's electrochemical equivalent. Laming's hypotheses [START_REF] Laming | Observation on a paper by prof. faraday concerning electric conduction and the nature of matter[END_REF] replaces caloric by electricity:

1. different sorts of atoms are naturally associated with unequal quantities of electricity 2.

a mass of electrical matter, or electricity, may be regarded as composed of electrical atoms, just as a mass of ordinary matter contains ordinary atoms; and thus the sphere of electricity which surrounds an ordinary atom will consist of a number of electrical atoms arranged in concentric strata. The number of electrical atoms belonging to a given ordinary atom may be assumed to be such as to complete its external spherical stratum,or, on the contrary, it may be such as to leave that external spherical stratum more or less imperfect.

At the beginning of the XX th century, a few years after Thomson's discovery of the electron, G. N. Lewis proposed in a memorandum dated March 28, 1902[START_REF] Lewis | The atom and the molecule[END_REF], his cubic atomic model in which the vertices are occupied or not by electrons according to the element's column in the periodic table.

In this way, he established a direct link between electrons and the concept of valence which provides a foundation to Abegg's valence and countervalence law [START_REF] Abegg | Die valenz und das periodische system. versuch einer theorie der molekularverbindungen[END_REF]. It is worth noting that these different atomic models have been conceived on the only basis of chemical arguments. Moreover, Lewis atom is closer to nowadays representations than Thomson's 1904 plumpudding model [START_REF] Thomson | On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure[END_REF].

The derivation of a model explaining the bonding was made by the same Lewis in the 1916 paper "The Atom and the Molecule" published by the American Chemical Society [START_REF] Lewis | The atom and the molecule[END_REF]). Lewis's approach is based on six postulates 1. In every atom is an essential kernel which remains unaltered in all ordinary chemical changes and which possesses an excess of positive charges corresponding in number to the ordinal number of the group in the periodic table to which the element belongs.

2. The atom is composed of the kernel and an outer atom or shell, which, in the case of the neutral atom, contains negative electrons equal in number to the excess of positive charges of the kernel, but the number of electrons in the shell may vary during chemical change between 0 and 8.

3. The atom tends to hold an even number of electrons in the shell, and especially to hold eight electrons which are normally arranged symmetrically at the eight corners of a cube.

4. Two atomic shells are mutually interpenetrable. 5. Electrons may ordinarily pass with readiness from one position in the outer shell to another. Nevertheless they are held in position by more or less rigid constraints, and these positions and the magnitude of the constraints are determined by the nature of the atom and of such other atoms as are combined with it. 6. Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances.

The third postulate is the consequence of a chemical fact: the stoichiometry of almost all chemical species implies that they have an even number of electrons, this fact is the chemical footprint of the fermionic nature of the electron. The importance of the electron pair was recognized by Lewis 6 years before the Stern and Gerlach experiment and 9 years before the theoretical introduction of the electron spin3 [START_REF] Uhlenbeck | Ersetzung der hypothese vom unmechanischen zwang durch eine forderung bezglich des inneren verhaltens jedes einzelnen elektrons[END_REF][START_REF] Pauli | über den einfluss der geschwindigkeitsabhängigkeit der elektronenmasse auf den zeemaneffekt[END_REF][START_REF] Uhlenbeck | Spinning electrons and the structure of spectra[END_REF]. As a consequence of the pair formation, electron occupying the vertices of a same edge are drawn together towards the middle of the edge. Therefore the group of 8 electrons is reorganized in 4 pairs arranged tetrahedrally. This model accounts for the characteristics of the carbon atoms as well as for the formation of triple bonds and for the free rotation around a single bond.

Lewis emphasized the concept of electron pair as the cornerstone of molecular structure and proposed to write the formulas of chemical compounds by using atomic symbols surrounded by a number of dots corresponding to the number of electrons in the atomic shell. In spite of its simplicity, Lewis's approach is remarkably efficient and therefore remains fundamental for basic chemical education. Although Lewis's model explains the structure of a majority of molecular species, it fails, for example, to account for the hexagonal structure of benzene or for the paramagnetism of dioxygen. The attempts of Huggins to understand benzene by a single Lewis structure yielded chimerical representations [START_REF] Huggins | The structure of benzene[END_REF] whereas the concept of mesomery, pioneered by Ingold, [START_REF] Ingold | The structure of the benzene nucleus, part i: Intranuclear tautomerism[END_REF][START_REF] Ingold | Significance of tautomerism and of the reactions of aromatic compounds in the electronic theory of organic reactions[END_REF] which considers a weighted superposition of structures has been very successful with this respect and therefore constitutes an important complement to Lewis's model. In order to be able to treat dioxygen, Linnett modified the original Lewis model by splitting the initial octet into two sets of four electrons, one having one spin quantum number and the other the opposite spin [START_REF] Linnett | A modification of the lewis-langmuir octet rule[END_REF][START_REF] Linnett | The Electronic Structure of Molecules. A new approach[END_REF]. [START_REF] Sidgwick | Bakerian lecture. stereochemical types and valency groups[END_REF] remarked that the spatial arrangement of ligands around a central atom conforms to a quite limited number of types. They related this grouping to the size of the shared valency groups, the number of involved shared electrons together with that of the unshared groups. This description implicitly introduces a spatial extension of the bonding and non bonding pairs which is more consistent with the physical reality than the fixed pairs of Lewis. In the Sidgwick and Powell's model both shared and unshared groups have the same size which is uniquely determined by the type of spatial arrangement considered. This model was considerably improved by [START_REF] Gillespie | Inorganic stereochemistry[END_REF] who assumed that the arrangement of the pairs around a given center aroused from their mutual repulsion. The repulsion depends upon the type of pairs involved: a lone pair repels other electron pairs more than a bonding pair, a double bond repels other bonds and the repulsion depends on the electronegativity. In revised account of the Valence Shell Electron Pair Repulsion (VSEPR), electron pair domains are considered rather than the point on a sphere representation of the earlier version. Electronic pair domains are defined as a charge cloud which occupies a given region of space and excludes other pairs from this region as a consequence of the Pauli exclusion principles. This electron pair domain version of VSEPR emphasizes the shape and size of the domains rather than the magnitude of their mutual repulsion. In addition to bond and lone pair domains, Gillespie considers single electron domains which are expected to be smaller than an electron pair domain [START_REF] Gillespie | Electron domains and the vsepr model of molecular geometry[END_REF]. The VSEPR model is very successful in predicting qualitatively the shape of molecules. It enables to understand many features of the molecular geometry in a qualitative fashion. Implicitly, the motion of nuclei is not considered. Both Lewis's and VSEPR models provide rather simple complementary explanations of molecular structure, a reason which explains their wide use in chemical education. The formation of individualized electron pairs of different types is the keystone of these models. It is not an experimental fact and it is questionable from a strict theoretical point of view. Most of the justification rely on the Pauli's principle and implicitly on the orbital approximation, for example Salem considers [START_REF] Salem | Faithful couple: The electron pair[END_REF]:

Two electrons-hut no more than two-can "form" or "enter" or "occupy," as it were, the same wave. For this to occur, there must be a special matching. Electrons have an intrinsic, intimate, characteristic property. For two electrons to enter the same wave, they must have opposite characteristics. Hence, without pushing the analogy too far, it is legitimate to speak of an electron couple.

The properties associated to pairs, such as bonding or not bonding character, location, size of their domain, are even more hypothetical and it appears necessary to revisit their definitions.

Linking chemical concepts and quantum mechanics

The concepts of chemistry often lack of scientificity or have not enough precise or inconsistent definitions. For example, although the concept of chemical bond is at the heart of chemistry Pauling's definition [START_REF] Pauling | The Nature of the Chemical Bond[END_REF] adopted in the IUPAC gold book [START_REF] Mcnaught | Compendium of Chemical Terminology The Gold Book[END_REF] There is a chemical bond between two atoms or groups of atoms in the case that the forces acting between them are such as to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent 'molecular species'.

introduces forces acting on nuclei (rather than atoms) which implicitly depend on the choice of a force field model (valence force field, Urey-Bradley force field, . . . ), and should be therefore determined by an arbitrary decomposition of the resultant of the force vectors on each centre which is expected to vanish at the equilibrium geometries. Moreover, it recourses to the decision of an ideal chemist. In spite of a precise definition, the concept of chemical bond is supported by a multiplicity of criteria derived from experimental structural and thermodynamical measures as well as theoretical ones.

As pointed out by Alvarez et al [START_REF] Alvarez | A bonding quandary-or-a demonstration of the fact that scientists are not born with logic[END_REF] If one allows oneself to use a multiplicity of criteria, bonds may exist by one measure, not by another. This is not a reason to wring our hands, nor complain how unscientific chemistry is (or how obstinate chemists are). chemistry has done more than well in creating a universe of structure and function on the molecular level with just this "imperfectly defined" concept of a chemical bond. Or maybe it has done so well precisely because the concept is flexible and fuzzy.

This makes difficult the construction of bridges between chemistry and quantum mechanics and leads to ask the following question: are chemical and physical approaches mutually exclusive or complementary? It seems therefore interesting to consider the hypothetical features of the Lewis's and VSEPR models in the context of exact quantum mechanics. By exact quantum mechanics we do not mean exact wave functions but that we will not consider mathematical intermediates due to the approximation introduced in the calculus of the wavefunctions. We adopt here the point of view of the statistical interpretation of quantum mechanics which emphasizes the importance of electron densities further put forward by the DFT formulation.

Quantum mechanical background

This technical subsection presents the derivation of the density of probability functions and related objects used to build links between phenomenological chemical models and quantum mechanics. Assuming the Born-Oppenheimer separation, the electronic wavefunction is a function of 4N variables: each electron, labelled by i, is described by the three components of its position vectors r i and by its spin coordinates σ i . In order to simplify the notation r i and σ i are gathered in a four component (space+spin) vector ξ i . According to the statistical interpretation of quantum mechanics the product

P N (ξ 1 , ξ 2 , . . . , ξ N ) = Ψ(ξ, ξ 2 , . . . , ξ N )Ψ * (ξ, ξ 2 , . . . , ξ N ) (1)
represents the probability of finding the electrons labelled 1, 2, . . . , N in the volume elements dτ 1 , dτ 2 , . . . , dτ N located at points r 1 , r 2 , . . . , r N with the spin coordinates σ 1 , σ 2 , . . . , σ N . The indiscernability of the electrons implies that the probability of finding any N electrons at these positions is:

Γ (N ) (ξ 1 , ξ 2 , . . . , ξ N ) = N !P N (ξ 1 , ξ 2 , . . . , ξ N ) (2)
The probability of finding k electrons with the ordered space and spin coordinates specified by the list of variables ξ 1 , ξ 2 , . . . , ξ k ), the remaining N -k being anywhere is given by:

Γ (k) (ξ 1 , ξ 2 , . . . , ξ k ) = N ! (N -k)! dτ k+1 dτ k+2 . . . dτ N P N (ξ 1 , ξ 2 , . . . , ξ k , (3) ξ k+1 , . . . , ξ N )
Particularly important are the first and second order distribution functions Γ (1) (ξ) and Γ (2) (ξ, ξ ). Integration of Γ (1) (ξ) over the spin coordinate σ yields the electron density ρ(r), since the spin coordinate is discrete this integration is actually the sum of two contributions ρ α (r) and ρ β (r) which correspond to the two values ± 1 2 of σ:

ρ(r) = ρ α (r) + ρ β (r) (4)
The procedure described above to get the electron density and its spin components is applied to Γ (2) (ξ, ξ ) in order to obtain the pair function and its four spin components:

Π(r, r ) = Γ (2) (ξ, ξ )dσdσ = Π αα (r, r ) + Π αβ (r, r ) + Π βα (r, r ) + Π ββ (r, r ) (5)
Though the pair function appears to be the suitable quantity enabling the discussion of the pairing, it is a six-dimensional function and therefore it should be helpful to grasp its properties in a 3-dimensions mathematical object. It is convenient to define the conditional probability as

Π σσ (r, r ) ρ σ (r ) = ρ σ (r)[1 + f σσ (r, r )] (6)
where σ and σ stand for the spin labels. The function f σσ (r, r ) is the correlation factor and the product ρ σ (r)f σσ (r, r ) describes the correlation hole around the reference electron. According to the values of σ and σ there are two kinds of correlation holes: the Coulomb hole (σ = σ ) accounts for the Coulombic interaction whereas the antisymmetry of the wavefunction is responsible for the Fermi hole (σ = σ ).

The representation of the conditional probability, also called the Lennard-Jones function, [START_REF] Lennard-Jones | The spatial correlation of electrons in molecules[END_REF][START_REF] Gillespie | The lennard-jones function: A quantitative description of the spatial correlation of electrons as determined by the exclusion principle[END_REF] or of the Fermi hole requires to fix the position of the reference electron. The contour maps [START_REF] Luken | Properties of the fermi hole in molecules[END_REF][START_REF] Luken | Localized orbitals based on the fermi hole[END_REF]Bader et al., 1996b) clearly reveal the electron pairing and yield patterns anticipated by the VSEPR model. The difficulty raised by the choice of the reference electron location led to develop different methods such as the Fermi hole mobility [START_REF] Luken | Mobility of the fermi hole in a single-determinant wavefunction[END_REF], which represents the rate of change of the shape of the Fermi hole with respect to motion of the probe electron. Another related method is the domain averaged Fermi hole developed by R. Ponec and co-workers [START_REF] Ponec | Electron pairing and chemical bonds. chemical structure, valences and structural similarities from the analysis of the fermi holes[END_REF][START_REF] Ponec | Electron pairing and chemical bonds. molecular structure from the analysis of pair densities and related quantities[END_REF][START_REF] Ponec | Electron pairing and chemical bonds: Bonding in hypervalent molecules from analysis of fermi holes[END_REF][START_REF] Ponec | Domain-averaged fermi holes-a new means of visualization of chemical bonds. bonding in hypervalent molecules[END_REF]Cooper & Ponec, 2008;[START_REF] Ponec | Analytic models of domainaveraged fermi holes: A new tool for the study of the nature of chemical bonds[END_REF] defined as:

g Ω (r) = ρ(r) Ω ρ(r )dr - Ω Π(r, r )dr (7)
which corresponds to a reference electron localized in the domain Ω. Strictly speaking eq. 7 describes a "domain averaged exchange-correlation hole" rather than a "domain averaged Fermi hole" since this expression involves spinless densities.

The Electron Localization Function (ELF) was originally designed by Becke and Edgecombe to identify "localized electronic groups in atomic and molecular systems" [START_REF] Becke | A simple mesure of electron localization in atomic and molecular systems[END_REF].

ELF = η(r) = 1 1 + χ(r) 2 (8)
It relies, through its kernel, χ(r) on the laplacian of the conditional same spin pair probability scaled by the homogeneous electron gas kinetic energy density:

χ σ (r) = D σ (r) D 0 σ (r) (9)
in which

D σ (r) = t σ (r) - 1 4 (ρ σ (r)) 2 ρ σ (r) = t σ (r) -t W σ (r) (10)
where τ σ (r) is the σ spin contribution to the positive definite kinetic energy density. For a closed shell singlet D σ (r) is the difference between the total positive definite kinetic energy density T s (r) and the von Weizsäcker kinetic energy density functional T vW (r), [START_REF] Von Weizsäcker | Zur theorie der kermassen[END_REF]:

D σ (r) = T s (r) -T vW (r) (11) whereas D 0 σ (r) = 3 5 (6π 2 ) 2/3 ρ 5/3 σ (r) (12)
is the kinetic energy density of the homogeneous electron gas. This formulation led to an interpretation of ELF in terms of the local excess kinetic energy because the Pauli repulsion enabled its calculation from Kohn-Sham orbitals. [START_REF] Savin | A new look at electron localization[END_REF][START_REF] Savin | Electron localization in the solid-state structures of the elements: the diamond structure[END_REF][START_REF] Savin | Elf: The electron localization function[END_REF] Orbital-based interpretations of ELF have been proposed by Burdett [START_REF] Burdett | Electron localization in molecules and solids: The meaning of elf[END_REF] and more recently by Nalewajski et al. [START_REF] Nalewajski | Electron localization function as information measure[END_REF] who considered the non additive interorbital Fisher information. Another route pioneered by Dobson [START_REF] Dobson | Interpretation of the fermi hole curvature[END_REF] explicitly considers the pair functions. It has been independently developed by Kohout et al. with the Electron Localization Indicator (ELI) [START_REF] Kohout | Electron localizability indicator for correlared wavefunctions. i. parallel spin pairs[END_REF][START_REF] Kohout | Electron localizability indicator for correlared wavefunctions. i. antiparallel spin pairs[END_REF][START_REF] Wagner | Charge decomposition analysis of the electron localizability indicator: A bridge between the orbital and direct space representation of the chemical bond[END_REF] and by one of us, [START_REF] Silvi | The spin pair compositions as local indicators of the nature of the bonding[END_REF] with the spin pair composition c π (r) enabling to generalize ELF to correlated wave functions. [START_REF] Matito | Electron localization function at the correlated level[END_REF] From a simple statistical viewpoint, the concept of electron density localization at a given position r relies on the definition of a local standard deviation of the electron density. This can be achieved either by a procedure due to P. Ayers who introduced the local covariance measure [START_REF] Ayers | Electron localization functions and local measures of the covariance[END_REF] ξ h σ (ρ; r) = ρ σ h σσ (r, r )dr ( 13) the evaluation of which requires a model for the correlation hole or by the finite sampling procedure described below which considers the standard deviation of the electron density integrated over a sampling volume V (r) encompassing the reference point and containing a given quantity of matter, in other words a given charge q. The smaller the standard deviation is, the higher the localization is. Instead of the standard deviation, it is advantageous to use its square, the variance σ 2 , which can be expressed as the expectation value of the variance operator [START_REF] Diner | Statistical and stochastic aspects of the delocalization problem in quantum mechanics[END_REF][START_REF] Silvi | How topological partitions of the electron distributions reveal delocalization[END_REF]:

σ2 ( N [V (r)]) = Π(V (r), V (r)) -N (V (r)) N (V (r)) -1 = Π(V (r), V (r)) -q 2 + q (14)
in which N (V (r)) = q and Π(V (r), V (r)) are respectively the one particle and two particle densities integrated over the sample V (r). In the expression of the variance given above, only Π(V (r), V (r)) is function of the position and therefore -q 2 + q can be regarded as a constant and deleted. The integrated pair density is the sum of an opposite spin contribution, 2 Παβ (V (r), V (r)) almost proportional to q 2 and of a same spin contribution Παα (V (r), V (r)) + Πββ (V (r). The expectation value of the variance of the integrated opposite spin pair density,

σ2 ( Παβ (V (r), V (r))) = Παα (V (r), V (r)) Πββ (V (r), V (r)) + N α (V (r)) Πββ (V (r), V (r)) + N β (V (r)) Παα (V (r), V (r)) + N α (V (r)) N β (V (r)) -( Παβ (V (r))) 2 (15)
also depends upon the integrated same spin pair densities. The integrated same spin pair density has numerically been shown proportional to a function of the reference point position, say c π (r), times q 5/3 . In the limit q = 0, the ratio Π(V (r), V (r))

q 5/3
tends to the spin pair composition, c π (r), a local function independent from the size of the sample [START_REF] Silvi | The spin pair compositions as local indicators of the nature of the bonding[END_REF]. The ability of this function to localize "electronic groups" can be illustrated by a very simple example in which two α and two β spin electrons are confined in a box of volume Ω as represented in figure 1.

figure 1 near here

For the sake of simplicity we assume the electron density probability to be uniform, i. e. ρ(r) = 4/Ω without spin polarization (ρ α (r) = ρ β (r) = 2/Ω, such as the opposite spin pair functions, Π αβ (r 1 , r 2 ) = Π βα (r 1 , r 2 ) = 4/Ω 2 are constant. This model enables to consider two localization cases. On the one hand, the opposite spin pairs are delocalized over the box and the same spin pair functions are constant: Π αα (r 1 , r 2 ) = Π ββ (r 1 , r 2 ) = 2/Ω 2 and therefore c π (r) is also constant. On the other hand, each opposite spin pair occupies one half of the box such as:

Π αα (r 1 , r 2 ) = Π ββ (r 1 , r 2 ) = 0 r 1 , r 2 ∈ same half box 4/Ω 2 r 1 , r 2 ∈ different half boxes (16)
It follows that

c π (r) = 0 r / ∈ the boundary ∝ 4/Ω 2 r ∈ the boundary (17)
which enables to locate the boundary between the two opposite spin pair regions. For Hartree-Fock wavefunction, it can be easily demonstrated [START_REF] Silvi | The spin pair compositions as local indicators of the nature of the bonding[END_REF] that:

c π (r) ≈ χ σ (r) (18)
The ELF itself is further obtained through the transformation of χ σ (r) into a lorentzian function

η(r) = 1 1 + χ 2 σ (r) (19)
so as it tends to 1 in those regions where the localization is high and to small values at the boundaries between such regions. There exist other localization functions based on other criteria such as the "localized orbital locator" (LOL) [START_REF] Schmider | Chemical content of the kinetic energy density[END_REF] which relies of the positive definite kinetic energy density and the "electron pair localization function" (EPLF) [START_REF] Scemama | Electron pair localization function: A practical tool to visualize electron localization in molecules from quantum monte carlo data[END_REF] which considers the averaged distance between a reference electron and the closest spin-like or spin unlike electron of the system.It is worth noting that several proposals have been made in order to evaluate ELF from experimental densities [START_REF] Tsirelson | Determination of the electron localization function from electron density[END_REF][START_REF] Jayatilaka | Electron localization functions obtained from x-ray constrained hartree-fock wavefunctions for molecular crystals of ammonia, urea and alloxan[END_REF].

Revisiting some chemical concepts

The purpose is not to justify or invalidate the Lewis's and VSEPR models but rather to establish correspondences between the concepts they assume and quantum mechanics.

Spatial arrangement of valence electrons

The spatial arrangement of electrons and electron pairs is a key concept of the two models since it enables to distinguish bonding from non bonding pairs. As electron are not at fixed positions, electron positions are not helpful. However, the distribution function Γ (N ) (ξ 1 , ξ 2 , . . . , ξ N ) (eq. 2) describes the probability of finding the systems at the configuration given by the list of the arguments. Therefore, an equivalence can made between the Γ (N ) maxima electron positions and the arrangement of the N electrons in the models. This can be graphically represented by plotting N points at the values of the arguments of Γ (N ) at a maximum, the spin being indicated by the color of the point. The N/2 opposite spin pairs are further selected on a distance criterion. It has been initially proposed to consider the electron coordinates of the absolute maximum of Γ (N ) [START_REF] Artmann | Zur quantentheorie der gewinkelten valenz, i. mitteilung: Eigenfunktion und valenzbetätigung des zentralatoms[END_REF][START_REF] Howard | Directed valence as a property of determinant wave functions[END_REF] but distribution functions may have either several maxima close in magnitude if the point group of the geometry of the nuclei is finite or continua of maxima otherwise. Moreover, equivalent maxima are generated by spin inversion. The most probable valence electron arrangements in the water molecule has been recently investigated [START_REF] Scemama | Maximum probability domains from quantum monte carlo calculations[END_REF] for both Hartree-Fock and correlated (VQMC) wave functions. The single determinant wave function yields a tetrahedral arrangement of opposite spin electron pairs (figure 2(a)). It corresponds to the expectation made by Lewis in order to explain the tetrahedral carbon and the formation of triple bonds and also to the VSEPR AX 2 E 2 type. The account for Coulomb correlation separates the two electrons of each pair and gives rise, among others, to the configurations of figure 2(b) and (c). Configuration (d) recovers both Lewis's cubic model and Linnett double quartet. It is worth noting that Lewis anticipated the multiplicity of spatial configurations with the concept of tautomerism [START_REF] Lewis | The atom and the molecule[END_REF]:

In such cases we have the phenomenon of tautomerism, where two or more forms of the molecule pass readily into one another and exist together in a condition of mobile equilibrium. . . . In the simplest case where we deal with a single tautomeric change we speak of the two tautomers and sometimes write definite formulae to express the two. But we must not assume that all of the molecules of the substance possess either one structure or the other, but rather that these forms represent the two limiting types, and that the individual molecules range all the way from one limit to the other. figure 2 near here The spatial organization of the couples of opposite spin electron positions which maximizes Γ (N ) enables to identify which pairs belong to the valence shells of two atoms and which others to the valence shell of just one atom and gives sense to the concepts of bonding and non-bonding pairs in agreement with Lewis's definition of the bond [START_REF] Lewis | Valence and the Structure of Atoms and Molecules[END_REF]:

Two electrons thus coupled together, when lying between two atomic centers, and held jointly in the shells of two atoms, I have considered to be the chemical bond.

Electron pair domains

Instead of considering the maxima of Γ (N ) , it should interesting consider the spatial extension of opposite spin electron pairs and therefore to investigate the regions in which the probability of finding them are large. A first kind of technique enabling a quantitative study, the loge theory [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF][START_REF] Daudel | Théorie de la localisabilité des corpuscules .1. la notion de loge et la signification géometrique de la notion de couche dans le cortège électronique des atomes[END_REF][START_REF] Daudel | Localizability of electrons in atoms and molecules-application to the study of the notion of shell and of the nature of chemical bonds[END_REF], has been developed for this purpose in the fifties by R. Daudel and co-workers. The loge theory divides the space in p connected non overlapping volumes, V n , within which one has to evaluate the probability P n of finding n and only n electrons of given spins. The difficulty of finding the best decomposition has limited applications of the loge theory to very small systems. In the seventies, the introduction of concepts from Shannon's information theory [START_REF] Shannon | The Mathematical Theory of Communication[END_REF] brought some noticeable improvements [START_REF] Aslangul | Introduction of information theory in study of electron localizability in atoms and molecules[END_REF][START_REF] Aslangul | Aspects of the localizability of electrons and molecules: Loge theory and related methods[END_REF][START_REF] Aslangul | The loge theory as a starting point for variational calculations. i. general formalism[END_REF]. The performances of the computers of the seventies forbid the applications of this method to chemically interesting molecular systems.

Recently, the evaluation of P n has been revisited by Cancès et al [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF] who derived a series of efficient recurrence formulas in the case of a single determinant representation of the wave functions. An algorithm optimizing the shape of the maximum probability domains (MPDs), Ω n , has been designed accordingly and applied to linear molecules [START_REF] Gallegos | Maximal probability domains in linear molecules[END_REF] and other simple molecules [START_REF] Scemama | Maximum probability domains from quantum monte carlo calculations[END_REF]. This approach which applies to any multiplet in any spin configuration and therefore is not limited to the determination of opposite spin pair MPDs. For example, it had been recently used to characterize ions in rock-salt structures [START_REF] Causà | Maximum probability domains in crystals: The rock-salt structure[END_REF]. For a given value of n < N there are several solutions. In the case of closed shell N -electron systems N/2 opposite spin pair MPDs are expected which correspond to the core, valence bonding and valence non bonding pairs. It often happens that more MPDs can be determined. This is the case of the FHF -complex where two symmetry related overlapping domains containing the proton correspond to the two F-H bonds. In this particular case, the overlap of the domain is interpreted as a manifestation of the resonances between the [F-H + F -] and [F -+HF] structures.

The single electron densities (SEDs) of Lüchow and Petz [START_REF] Lüchow | Single electron densities: A new tool to analyze molecular wavefunctions[END_REF] provide a partition of the total density in terms of single electron contributions containing the many-body information from the wave function. The method is technically based on an assignment of the electrons which corresponds to a selection by a distance criterion of the permutations which leaves constant the number of electrons assigned to core, bonds, lone pairs, etc.

A partition of the space in adjacent non overlapping regions can be achieved by applying the dynamical system theory [START_REF] Abraham | Foundations of Mechanics[END_REF] to the ELF gradient field. [START_REF] Silvi | Classification of chemical bonds based on topological analysis of electron localization function[END_REF]Häussermann et al., 1994a) This yields basins of attractors which can be thought as corresponding to atomic cores, bonds, and lone pairs and therefore recovering the Lewis picture of bonding and the electronic domains of the VSEPR approach. In a limit model built with perfectly localized orbitals, the ELF basins and MPDs are identical [START_REF] Savin | On the significance of elf basins[END_REF]. However, in the cases where the MPD approach yields overlapping domains the ELF basins provide an averaged picture which resemble the Ingold's mesomeric state [START_REF] Ingold | Mesomerism and tautomerism[END_REF]. Integration of the one electron density over an ELF basin yields the basin population which often noticeably deviates from 2, the population of an electron pair MPD. The core basins surround nuclei with atomic number Z > 2 and are labeled C(A) where A is the atomic symbol of the element. The union of the valence basins encompassing a given core C(A) constitutes the valence shell of atom A. A valence basin may be shared by several valence shells, this is a generalization of Lewis's fourth postulate "Two atomic shells are mutually interpenetrable" [START_REF] Lewis | The atom and the molecule[END_REF]. The valence basins are characterized by the number of atomic valence shells to which they participate, or in other words by the number of core basins with which they share a boundary. This number is called the synaptic order. Thus, there are monosynaptic, disynaptic, trisynaptic basins, and so on. Monosynaptic basins, labeled V(A), correspond to the lone pairs of the Lewis model and polysynaptic basins, V(A,B,C,. . . ), to the shared pairs of the Lewis model. In particular, disynaptic basins correspond to two-centre bonds and trisynaptic basins to three-centre bonds, and so on [START_REF] Silvi | The synaptic order: a key concept to understand multicenter bonding[END_REF]. In the example of the furan molecule displayed figure 3 the oxygen valence shell delineated by a red dotted line results of the union of the two V(C, O) basins corresponding to the C-O bonds and of the V(O) of the oxygen lone pair. In a carbon valence shell there are two V(C,C) and one V(C, H), this latter constituting the hydrogen valence shell. Three examples of multi-centre bonds are given in figure 4. The ELF picture of diborane recovers the protonated double bond picture of Pitzer [START_REF] Pitzer | Electron deficient molecules. i. the principles of hydroboron structures[END_REF] in which the two boron atoms are linked by two trisynaptic basins embedding a proton, in C 2 H 4 (AlCH 3 ) 2 the two V(C, Al, C) basins account for the addition of the AlCH 3 on the C 2 H 4 molecule given for comparison. Finally, in the lithium crystal the ELF valence basin occupy the octahedral interstitial site.

ELF basins and VSEPR domain.

The topology of the laplacian of the charge density has been invoked as a physical basis of the VSEPR model [START_REF] Bader | A physical basis for the vsepr model of molecular geometry[END_REF]. Qualitatively, the valence shell charge concentrations (VSCCs) of the central atom correspond to the electronic domains of the model [START_REF] Gillespie | The vsepr model revisited[END_REF][START_REF] Gillespie | Electron domains and the vsepr model of molecular geometry[END_REF] and they have also been used to explain the geometries of non-VSEPR molecules (Gillespie et al., 1996). From a quantitative point of view, Malcolm and Popelier attempted to justify the assumptions on the domain sizes considering the full topology of the laplacian of the charge density [START_REF] Malcolm | The full topology of the laplacian of the electron density: scrutinising a physical basis for the vsepr model[END_REF]. They conclude that non bonding domains are larger than bonding domains of the same valence shell and that multiple bond domains are larger than single bond ones, but the full topology of the laplacian fails to account for the the electronegativity effects.

Although the ELF basins do not exactly fulfill the requirements of the VSEPR electronic domain definition, Gillespie and Robinson wrote [START_REF] Gillespie | Models of molecular geometry[END_REF]:

These basins correspond to the qualitative electron pair domains of the VSEPR model and have the same geometry as the VSEPR domains.

The VSEPR model relies on the interaction between the electronic domains around a given atomic center which is ruled by either electrostatic forces in the earliest version and on the relative size of the domain in the last version. The electrostatic repulsion between ELF valence basins, Ω and Ω has been investigated by Martín Pendás et al [START_REF] Martín Pendás | Electron-electron interactions between elf basins[END_REF] who considered the integral

V Ω,Ω ee = Ω dr 1 Ω π(r 1 , r 2 ) |r 2 -r 1 dr 2
They found a very fair agreement with the VSEPR assumptions: monosynaptic basins (lone pairs) are more repulsive than disynaptic ones, multiple bonds are more repulsive than single bonds, the repulsion decreases when the electronegativity of the substituent implied in the disynaptic basin increases.

A thorough analysis of the ELF basin volumes has been carried out on a sample of about 150 molecules [START_REF] Silvi | The elf topological analysis contribution to conceptual chemistry and phenomenological models[END_REF]. Since valence basins may in principle extend to infinity the molecular space has been limited by the density isosurface ρ(r) = 10 -4 which ensures less than 0.2 e are not considered in each molecule. The volumes of the V(X), V(X, H) and V(X, X) basins are nicely anti correlated with the electronegativity χ(X) of the X atom, they decrease as χ(X) increases and are moreover independent of the period of the element. Around the X centre the V(A, X) basins are always smaller than the V(X) ones whereas the V(X, H) are smaller for X belonging to the third and fourth periods and almost equal for the second period elements. The case of the hydrogen ligand is however particular because the proton is embedded in the basin which is not the case of the nucleus and core basin of other ligands. For elements of the same group and a given ligand the volume of the V(A, X) basin increases with the period (and in some extent with the decrease of the electronegativity of the central atom), for example the V(Si, F) basin volume is twice that of V(C, F). Finally, the multiple bonds yield basin larger than single bonds mostly for geometrical reason: in the case of a A=B double bond the basins are located on both sides of the plane containing the ligands of A and B. Therefore the repulsion from the ligands is weaker than for a single A-B bond, for example the volume of each V(C, O) basin of formaldehyde is calculated to be ten times larger than that of methanol. The analysis of the core basin volumes and of the shapes of the outermost core shells shows that these properties cannot be always neglected in bonding models which provides indications for further improvements of such models.

Bonding classification of chemical interactions

In the description provided by ELF, an atom X in a molecule is composed of its nucleus and of the electron density of its core and valence basins which form its valence shell. As already mentioned a valence basin may belong to the valence of different atoms and therefore two atoms bonded by a disynaptic basin do overlap. This description is consistent with the Lewis's atom and differs from the QTAIM atoms which are non overlapping bodies. In the QTAIM framework the interactions have been initially divided in "shared electron " and "closed shell (or unshared electron)" interaction classes according to the sign of the laplacian of the electron density at the bond critical point [START_REF] Bader | The characterization of atomic interactions[END_REF][START_REF] Bader | Atoms in Molecules: A Quantum Theory[END_REF]. Other indicators such as the energy density at the BCP [START_REF] Cremer | Chemical bonds without bonding electron density -does the difference electron-density analysis suffice for a description of the chemical bond?[END_REF], 1983;[START_REF] Kraka | Description of chemical reactions in terms of the properties of the electron density[END_REF] and the delocalization indexes [START_REF] Fradera | The lewis model and beyond[END_REF] have been further introduced in order to refine the classification of interactions and to approach the current chemical terminology [START_REF] Bianchi | Experimental electron density analysis of mn 2 (co) 10 : Metal-metal and metal-ligand bond characterization[END_REF][START_REF] Macchi | Experimental electron density in a transition metal dimer: Metal-metal and metal-ligand bonds[END_REF][START_REF] Mori-Sánchez | A classification of covalent, ionic, and metallic solids based on the electron density[END_REF].

The dichotomous classification into "shared electron" and "unshared electron" interaction classes naturally emerges from the organisation of the ELF basins. The "shared electron" interaction implies the presence of a disynaptic or polysynaptic basin and gathers the covalent, dative and metallic types of bond which can be further discriminated on the basis of other criteria such as the dissociation products or the flatness of the function at it valence maxima. The "unshared electron" class consists on the one hand of interactions governed by electrostatic or dispersion forces (ionic bonding, Hydrogen bonding, van der Waals complexes) and on the other hand of interactions in which electron delocalization plays a dominant role (two-center three-electron bonding, charge-shift bonding). The calculation of the basin populations and basin electrostatic moments [START_REF] Pilme | Advancing beyond charge analysis using the electronic localization function: Chemically intuitive distribution of electrostatic moments[END_REF] sheds light on the electrostatic aspects of the bonding whereas the basin population variances and covariances [START_REF] Savin | Topological analysis of the electron localization function applied to delocalized bonds[END_REF][START_REF] Silvi | How topological partitions of the electron distributions reveal delocalization[END_REF] provide a measure of the delocalization. Examples of application to metallic bond [START_REF] Silvi | Direct space representation of the metallic bond[END_REF], two-center three-electron bonding [START_REF] Fourré | What can we learn from two-center threeelectron bonding with the topological analysis of elf?[END_REF] and charge-shift bonding [START_REF] Shaik | Charge-shift bonding-a class of electron pair bonds emerges from valence bond theory and supported by electron localization function approach[END_REF] can be found in the literature.

Conclusions

The statistical approaches provide, at least formally, mathematical bridges between quantum mechanics and chemistry which enable to critically think about the content and the definition of many chemical concepts. The improvement of these latter can be achieved in order to remove any assumption in contradiction with the laws of quantum mechanics and take into account important features which had been nelected. The substitution of a classical statement by its statistical formulation is often sufficient to make it consistent. The chemical objects used in the bonding models (electron pairs, electronic domains, valence shells) are not physical observables but correspondences can be established with objects defined in the statistical frameworks (loges, MPDs, basins). These approaches are on the way to solve the historical problems of communication between chemical phenomenological systems and quantum mechanics.

Chemistry and physics belong to the same universe but often use their proper objects and their proper laws because they have to explain different phenomena. The laws of chemistry account for the complexity and the diversity of the matter in a rather qualitative fashion which embodies the physical laws. In fact, there is not a unique level of understanding and nobody is obliged to strictly follow the unification account. Although quantum electrodynamics is the physical theory which treats the interaction of light with matter it is of little help to explain how works a telescope for which classical geometric optics provides a satisfactory answer. In the same way, the explanations of chemical phenomena based on chemical concepts are indisputably useful since they grasp the salient feature. As an example, we can mention a paper of C. [START_REF] Gourlaouen | Is an electronic shield at the molecular origin of saturnism? a computational modelling experiment[END_REF] in which the authors addressed the electronic origin of lead toxicity. In brief, the calmodulin protein forms complexes with Ca ++ which are involved in important biological processes such as mitosis neurotransmission, and regulation of the calcium pump. In these complexes the calcium dication is heptacoordinated. Substitution of Ca ++ by Pb ++ induces strong spatial distorsions which modify the properties of the complex. The striking chemical difference in the ELF pictures of the Ca ++ and Pb ++ bioinspired model complexes is the presence of a lone pair basin on the lead centre which then provides a VSEPR type explanation of the distorsions of the Pb ++ complex with respect to the Ca ++ one. Figure 1: Each half space contains two of opposite spin electrons and the density is assumed to be constant. The population N (r) of the sampling volume being denoted by q, the antiparallel pair population of the sample is also constant and equal to q 2 2 . The spin pair composition is identically zero when the sample is entirely in the same half space, it reaches its maximum when the sample crosses the boundary.

  Figure 2: Γ (N ) maxima electron positions in the water molecule. The spin coordinate is represented by two colors, yellow and blue. The oxygen atom kernel is represented by a red sphere, the hydrogen nucleus by a light blue sphere. (a) single determinant wave function, opposite spin electron, in green, positions coincide at the maximum, the white lines show the tetrahedral arrangement of the electron pairs, (b) Correlated VQMC wavefunction at a first maximum of Γ (N ) , the white lines link opposite spin points corresponding to pairs. (c) Correlated VQMC wavefunction at a second maximum. (d) same as (c) and the white lines link same spin positions showing the Linett's double quartet arrangement. From ref.[START_REF] Scemama | Maximum probability domains from quantum monte carlo calculations[END_REF].

Figure 3 :

 3 Figure 3: ELF basins of the furan molecule. The red and blue dotted lines delineate the oxygen and carbon valence shells

Figure 4 :

 4 Figure 4: ELF=0.8 isosurface for B 2 H 6 , C 2 H 4 (AlCH 3 ) 2 and C 2 H 4 , ELF=0.5 of the body centered cubic structure of lithium.

Quantum mechanics is undeniably the intellectual scandal of the century! What do you mean by scandal? I mean Science has renounced intelligibility, It has really renounced! That's something which asserts itself and which is not understandable.

Condon's calculation uses the H + 2 analytical result of Burrau[START_REF] Burrau | Berechnung des energiewertes des wasserstoffmolekel-ions (h + 2 ) im normalzustand[END_REF] to build the H 2 approximate wavefunction. The MO method has been independently developed by Hund[START_REF] Hund | Zur deutung einiger erscheinungen in den molekelspektren[END_REF][START_REF] Hund | Zur deutung der molekelspektren[END_REF][START_REF] Hund | Zur frage der chemischen bindung[END_REF] Mulliken, 1928a,b;[START_REF] Lennard-Jones | The electronic structure of some diatomic molecules[END_REF].

Lewis's sixth postulate which seems in contradiction with quantum mechanics can be considered as an anticipation of Pauli's exclusion principle[START_REF] Pauli | über den einfluss der geschwindigkeitsabhängigkeit der elektronenmasse auf den zeemaneffekt[END_REF]


Jean-Charles Manceau

Dimitrios G Hatzignatiou

Louis De Lary

Niels Bo Jensen

Kristin Flornes

Thomas Le Guénan

Arnaud Réveillère

For

  Request from the IEAGHG For the risk scenario undesired CO 2 migration in the subsurface The state of knowledge of novel and standard mitigation and remediation practices and associated costs Review of mitigation plans in place on current/past/future CO 2 geological projects Context Less information in the public domain on this topic compared to other ones, but increasing interest Europe, Directive on GCS: the permit application shall include a corrective measures plan USA, U.S. Federal Requirements under the UIC Program for CGS Wells: the owner or operator must provide an emergency and remedial response plan as part of the permit application Australia, Offshore Petroleum and Greenhouse Gas Storage Act 2006: well operations management plan with explanations on how the risks can be dealt with Approach, three main questions Which are the current mitigation and remediation techniques contemplated in the literature? From an operational perspective which measure is achievable? how to distinguish/prioritize among various feasible measures? How is this measure incorporated/integrated in the intervention plan or set-up in case of a detected CO In the study, comprehensive state of knowledge of mitigation and remediation practices, standard and technically feasible or innovative and under development Considered measure types: 1. Interventions on wells 2. Fluid management techniques 3. Breakthrough technologies 4. Remediation measures on potential impacts Interventions on wells 6 After Randhol et al. 2007 Operations on: Wellhead (repair) Packers (replacement) Tubing (repair) Casing (squeeze cementing, casing patch, swaging) Well (well killing, relief well) Similar operations for managing abandoned wells Requirements for wells P&A to be followed for injection wells and leaking abandoned wells 7 th Trondheim CCS Conference, June 5 -6, 2013 Fluid management techniques Akervoll et al., 2009. as an alternative option for well cementing 2-Foams and gels for controlling fluids migration 3-Biofilms notably to decrease formation and pathways permeability Operational perspective: Costs Site-specificity, degree of maturity: difficulties in estimating the costs of mitigation and remediation In this study: Literature review on elements given to estimate the intervention costs Qualitative elements given for the described measures Quantitative elements only possible for mature techniques with different scenarios (onshore/offshore, for implementing mitigation and remediation plans Two public plans: Goldeneye project (UK), Gorgon project (Australia) In this study, survey sent to CO 2 storage projects (integrated to the risk assessment plan and to the monitoring plan (site specific) Designed by experts, reviewed by stakeholders Very diverse format depending on the regulations Which measure? Distinction man-made/natural pathways Mostly imply actions on wells Pressure management -considered as preventive rather than corrective measure Immature technologies are not contemplated Remediation measures based essentially on environmental clean-up experience Flexible plan, seen as a support for decisionmaking at the time of an irregularity detection 14 comprehensive portfolio of the remediation and mitigation technologies mentioned in literature 2-qualitative or quantitative criteria to enable a knowledgeable choice 3-methodologies for decision-making 4-best practices in terms of intervention plan set-up and implementation To go forward, some avenues for research: Migration processes understanding and consequences characterization Technologies development and adaptation to the CGS conditions Development of tools for the intervention optimization

th Trondheim CCS Conference, June 5 -6, 2013

7 th Trondheim CCS Conference, June

-6, 2013

7 th Trondheim CCS Conference,June 5 -6, 2013

Mitchel et al., 2010 7 th Trondheim CCS Conference, June 5 -6, 2013

7 th Trondheim CCS Conference,June 5 -6, 2013

th Trondheim CCS Conference, June 5 -6, 2013 7 th Trondheim CCS Conference, June 5 -6, 2013 16 Thank you.

In the study, techniques for the remediation of : (Benson and Hepple, 2005;Rohmer et al., 2010;Zhang et al., 2004;and Sweatman et al., 2010) pH adjustment (spreading of alkaline supplements, irrigation and drainage) Highly site-specific and situation-dependent

In the study, generic elements:

On each measure, on the maturity and intervention specifications The possible methods and approaches for decisionmaking (CBA, CEA, MCA)