Anthony Scemama
email: scemama@irsamc.ups-tlse.fr

Andreas Savin
email: andreas.savin@lct.jussieu.fr

The effect of uncertainty on building blocks in molecules

Keywords: atoms, chemical bond, effective number of electrons, probabilities

Probabilities to find a chosen number of electrons in flexible domains of space are calculated for highly correlated wave functions. Quantum mechanics can produce higher probabilities for chemically relevant arrangements of electrons in these regions.

However, the probability to have a given arrangement, e.g., that corresponding to chemical formulas (bonds or atoms), is low in spite of often being maximal. Like in valence bond theory, it is useful to consider alternative distributions of the electrons. Exchanges of electrons should be considered not only between atoms, but also between other types of regions, like those attributed to lone pairs. It is useful to have definitions flexible enough to allow the user to find the reference representations he considers most relevant. We tentatively suggest a tool (the effective number of parties) to help one make the choice.

Building blocks are often used to understand, and even solve problems. For two centuries, the atom has been considered the most important building block of molecules. Associating the words "atomic" and "hypothesis" (as was done in the 19th century to deny their existence) is -in popular view -ridiculous: we know that atoms exist. However, in quantum chemistry we perform our calculations using nuclei and electrons: the Coulomb forces exerted by the nuclei, and the requirement of antisymmetry shape the wave function. Due to these factors, the electrons do not behave independently from one another, they do not roam freely through space. This leaves room for searching quantities that correspond to building blocks, in spite of difficulties that arise when doing it:

1. Quantum calculations cover a wider field. There is room for aspects not covered by traditional concepts. For example, Lewis wrote that, for him, the three-electron bond was unthinkable. 16 2. The association between quantities derived from quantum calculations and traditional building blocks is not unique. Even a rigorous choice of a definition of an atom in a molecule remains nothing but one of possible choices.

3. Although the attractive forces of the nuclei confine electrons to the vicinity of nuclei, and the Pauli exclusion principle drives electrons to divide space among themselves, the separation into building blocks (atoms or electron pairs) is not perfect. Despite the constraints with confining effect, electrons can escape the regions of space we associate with the building blocks.

4. The choice of the blocks we look at is not obvious. For example, is it not better to look for ions in an ionic crystal than for atoms?

In this paper we focus on the last two points above. We can quantify the uncertainty of electrons to be within a given region of space by calculating probabilities. We can go even further, and ask about the effect a given number of electrons in one region of space has on the number of electrons in another region of space. Furthermore, we choose to give sufficient flexibility to the definitions to allow comparisons between different choices of the building blocks, for example having a number of electrons in a region corresponding to the neutral atom, or to one of its ions.

We would like to avoid -as much as possible -effects due to artefacts due to the level of computation. Although our conclusions could be reached with less elaborate wave functions, we base our numerical results on quite accurate wave functions (linear combinations of up to millions of determinants, see details in appendix A).

PROBABILISTIC MEASURES

The probability to have n electrons present in a spatial region

The number of electrons in a spatial region is a random variable. For a given wave function, Ψ, Monte Carlo sampling according to |Ψ| 2 produces a large set of M configurations (sets of positions for the N electrons of the system, {r 1 , r 2 , . . . r N }; note that we use in this paper the term "configuration" in the sense of Monte Carlo, and not in the meaning of a linear combination of Slater determinants). We estimate from the set of configurations the probability of a chosen number of electrons, n, 0 ≤ n ≤ N to be in a selected region of space, Ω.

P Ω (n) = m Ω (n) M , (1)
where m Ω (n) is the number of configurations {r 1 , r 2 , . . . r N } for which n of the r i are within Ω. Another way to describe P Ω (n) is to use the indicator function, 1 Ω (n). It is equal to one, if n of the r i are within Ω, and equal to zero otherwise. Its expectation value gives the probability

P Ω (n) = E(1 Ω (n)) (2)
Note that, as 1 2 = 1, the variance is easily obtained as

σ Ω (n) 2 = P Ω (n) -P Ω (n) 2 (3)
so that we can estimate the error of P Ω (n) from the variance of the mean, σ Ω (n)/ √ M , using that P Ω (n) -P Ω (n) 2 is between 0 and 1/2. This shows that using M of the order of 10 4

already gives reasonable estimates of the probability, which in our case is not required with more than two decimals. The results shown in is paper are produced using sets of the order of 10 6 Monte Carlo configurations. When not impeding clarity, we omit to indicate explicitly n or Ω; for example we will write P Ω ≡ P Ω (n), or P (n) ≡ P Ω (n).

We can also obtain the population of the region Ω,

Q Ω = N n=0 n P Ω (n) (4)
It gives the average number of electrons in Ω. The same number would be obtained by integrating the electron density over the region Ω.

Other than in earlier papers (see, e.g., reference 24), we are interested in considering more than one spatial region. Let us use for the spatial regions the symbols A, B, . . . (instead of Ω A , Ω B ,. . . ). The joint probability of having n A electrons in A and n B electrons in B is

P A∩B (n A , n B ) = m A∩B (n A , n B ) M (5)
where m A∩B (n A , n B ) is the number of configurations for which n A electrons are in A and n B electrons in B. We can also write

P A∩B (n A , n B ) = E(1 A (n A ) 1 B (n B )) (6)
Note that P A (n A ) can be seen as a marginal,

P A (n A ) = N n B =0 P A∩B (n A , n B ). (7)
Of course, P A∩B can be extended to more than two regions, to the probability to have n A electrons in A, n B electrons in B, n C electrons in C, etc.: P A∩B∩C... . If we have a partitioning of space, we will denote it by n A |n B |n C . . .

One might naively expect that the probability of n A electrons to be in A and n B electrons to be in B, P A∩B , is given by P A P B . In general, this is not true, because our random variables are not necessarily independent. A trivial example for this shows up when B is the part of space not covered by A, R 3 \A, and n B = N -n. (One can see it as the situation n A |N -n A .)

In this case, P B = P A and P A∩B = P A = P A P B = P 2 A . A way to detect the dependence of random variables is given by conditional probabilities, the probability of n A electrons to be in A given that n B electrons are in B,

P A|B = P A∩B P B . (8)
When P A|B = P A the variables are independent. This reminds us that, in general, P A|B = P B|A , although in daily life we easily mix them up. Note that P A|B can be larger or smaller than P A . For example, for

A = R 3 \B, P A|B (n A = N -n B , n B ) = 1, P A|B (n A = N -n B , n B ) = 0, no matter what P A ∈ (0, 1) is.

CHOICE OF THE SPATIAL REGION

Sharp boundaries

In this paper we choose regions having sharp boundaries: an electron either is, or is not in Ω. This is close to what is done, for example, in loge theory is due to considering atomic orbitals). However, using fuzzy boundaries introduces a supplementary arbitrariness. Furthermore, contrary to intuition, fuzzy boundaries can have an equalizing effect on the probabilities (see, appendix B). This is opposed to our aim to have one of the probabilities to be significantly larger than the other ones.

In principle, the regions A and B can overlap. This appears, for example, when producing maximum probability domains 24 , or single electron densities 17 . Furthermore, the spatial regions may not cover all of R 3 . The latter case reminds of the population analysis first promoted by Davidson. 7 He argues that population analysis has a meaning only in a minimal basis. A part of the density is not recovered this way, thus Roby extended it 22 to take into account the missing contributions by defining two-, and multi-center contributions. Such extensions are also possible for our type of discussion, but we leave it for further studies.

Partial optimization of the spatial domain

For the present study, defining the spatial region by maximizing the probability to have a given number of electrons in it 23 seems appropriate. However, in the present paper we perform only partial optimizations, for three reasons. First, the points we try to make with our paper can be made also with a partial optimization (or just considering symmetry).

of the regions considered may contain a small number of configurations, and thus shape optimization needs a more careful treatment. Finally, we would like our results to be easily reproducible without any important programming effort (as the optimization of the spatial domain requires). For the spatial regions Ω, we consider simple geometrical shapes, that we optimize. More details are given in the appendix C.

Ambiguity in the choice of the spatial region

The choice for a region depends on the object of study. For example, in an ionic crystal, we may be more interested in finding ions than in finding atoms. For some methods, one can explicitly formulate the requirement. For finding maximum probability domains Let us start with a simple analytic example. We consider two non-interacting particles in a one-dimensional box: 0 ≤ x ≤ 1. The eigenfunctions are

φ k (x) = √ 2 sin(k x), with k = 1, 2, . . . . (9)
We occupy φ 1 with one electron and put the other electron on φ k . A different spin is important only when both electrons are in φ 1 . Otherwise, it does not matter: there is no interaction in our model. We ask about the probability to have one electron in one half of the box, and the other electron in the other half of the box. For all odd k it is equal to 1/2.

For even k, it is given by

1 2 + 8 π 2 k k 2 -1 2
It is no surprise that for k = 1 we obtain the urn result: the particles are independent.

The probability is maximal (0.86) when k = 2: the two electrons are now separated to a higher degree. If the spins of the electrons were parallel, we would say this is due to the Pauli principle. However, this holds also for antiparallel spins; we can attribute it to the orthogonality of the orbitals. For odd k = 1, this principle works, too, but in each of the half-boxes: it does not help separating the electrons each into a half-box.

This can be seen by orbital localization, by considering the equivalent orbitals 9 (φ 1 ± φ k )/ √ 2 instead of φ 1 and φ k . For odd k, the orbitals do not localize into one of the halves of the boxes as the linear combination does not change the symmetry. For even k, the degree of localization decreases with increasing k, see figure 1.

The observation that for odd k there is no orthogonalization (Pauli principle) effect on the probability of having one electron in the left part of the box, and one in the right part of the box -and there is no orbital localization in the half-boxes -does not mean that, with a proper choice of the region, the probability to have one and only one electron cannot be made larger than for Ehrenfest urns. For k = 3, the probability to have one electron in the interval is 0.89 in the maximum probability domain, x ∈ (0.32, 0.68). The price to pay for it, is that the other electron is -with the same probability -in spatially disjoint regions. This quantum effect has no analogy in classical descriptions, but could be relevant, for example, in the description of time-dependent processes, such as chemical reactions. In order to get our bearings, let us first consider a simple example, that of the CH 4 molecule. We define the core region, by choosing Ω to be a sphere maximizing the probability of n = 2 electrons to be inside it, see figure 2. For a radius of 0.56 bohr, P core (n) = 0.81,

while P core (n ± 1) = 0.1.
We define the valence space by excluding the sphere attributed to the core. We divide the valence space into four Voronoi cells. A Voronoi cell is the collection of points closest to one of the H nuclei. We associate such a regions to a CH bond, and find the probability of having two electrons in it to be P CH (n = 2) = 0.54. This value is lower than that obtained for two electrons in the core. At the same time, the probabilities to have one more electron in this region (or to have one less) is around 0.2, and thus higher than for the core. P CH (n = 2), it is significantly higher than given by the multiple urn model (0.31 for eight balls in four Other examples of electron pairs (bonds or lone pairs) are the F-F bond or a lone pair in F 2 , see figure 5, or one of the banana bonds in HCCH (figure 4). In these cases

P Ω (n = 2) = 0.4 that is lower than seen for the CH bond, but still larger than in the multiple urn model. Of course, the probability to have n = 2 in this region is increased when compared to CH 4 . P (2 ± 1) is still around 0.2. Now, the probability to have 2 electrons in the region is comparble to that of having 2 ± 1 electrons in it.

Let us now look at the triple bond in HCCH, and the group of three lone pairs in F 2 .

After a shift of the central value (2 for one pair, 6 for three pairs), we find roughly the same distribution of probabilities as for the pairs of electrons just presented (see figs. electrons in the CC region is slightly higher than that of having n = 6 electrons in it.

Atoms: F, Xe

We can define also spatial regions for atoms. In the F 2 molecule there is not much to discuss: we divide the molecule by a plane perpendicular to the line passing through the two F nuclei, at half distance from each of these. As we discuss atoms, let us now also include the cores. The probability to have 9 electrons on each F atom is around 0.6, higher than in the two urn model (see figure 6).

Let us now consider another molecule, XeF 2 . We now move two planes symmetrically around the Xe nucleus, separating the Xe atom from the F atoms. The probability of having n = 54 electrons within these planes is maximal (P Xe = 0.35) for z Xe = 2.72 bohr, see figure 7. 20 Even more pronounced than in HCCH, we have P Ω (n) < P Ω (n-1)+P Ω (n+1):

it is less likely to find a number of electrons equal to that expected for the Xe atom (54) than to find a different one. Of course, we used just planes, and did not fully optimize the spatial regions to maximize P (n = 54), but we do not expect a qualitative change. Further below, we argue that this might be related to the difficulty of recovering the historical chemical concepts.

Populations

In all cases, we find that the regions Ω maximizing the probability to have n electrons in it also have a population Q Ω , eq. ( 4), close to n -to an accuracy comparable to that we used in obtaining the probabilities. This can be explained by the symmetry of the distributions observed.

One may wonder if it would not be better to optimize the regions to have the desired average number of electrons, and forget about fluctuations. Finding a region that satisfies the Q Ω = n does not seem advisable. First, this would not detect the presence of resonant ionic structures having on average the same number of electrons as the covalent structure.

Second, there are infinitely many partitions of space that give exactly the same average. atoms, and still obtain the same average number of electrons in it.

The population, Q Ω , eq. ( 4), is only the average number of electrons in a spatial domain.

We have to associate some uncertainty to it. We could try the standard deviation, σ Ω , eq. ( 3). When discussing Gaussian distributions, one often indicates as error margin 2σ.

One can instead state that the probability to get a result between theses error margins is higher than ≈ 0.95. The latter characterization can be used also for other probabilistic distributions. So. let us return to our probability distributions. From the expression of σ, eq. ( 3), we see that it reaches its maximum, 0.5, only for P Ω = 0.5. So, in general, σ < 0.5, and 2σ < 1. As the population Q Ω is very often close to the number of electrons for which we have considered the probability, n Ω -that is always an integer -there is only one integer in the interval (Q Ω -2σ, Q Ω + 2σ). Thus, in general, this extension of the allowed range of values does not increase the probability of having electrons beyond P (n). As we have seen, even for the clear-cut core/valence separation P (n) reaches only 0.8. So, in order to be on the safe side (of having a probability larger than 0.95), to accept in our picture at least the transfer of at least one electron between spatial regions.

Joint probabilities

We can extend the previous discussion to multiple regions. Let us come back to the CH 4 molecule. The numbers given for CH 4 correspond to the probability to have 2 electrons in one CH bond region, no matter how the electrons are distributed in the other regions. The joint probability to have two electrons in each of the CH regions is lower (0.2). This distribution, 2|2|2|2 is the most probable distribution of electrons over the four regions. However, exchange of one electron between two regions can occur in many ways (1|3|2|2, 3|1|2|2, 1|2|3|2, . . . ). Summing up all the probabilities related to all these distribution yields a value more than twice as large (0.5). However, the value obtained for 2|2|2|2 it is five times larger than the probability of obtaining 2 balls in each of 4 urns. This low joint probability of the most probable distribution of electrons is no exception.

For example, in HCCH, the partition 2|6|2 is the most probable (P = 0.2). However, the partitions where one electron is transferred between the two regions (like 1|7|2, 3|5|2) have a probability of 0.3. For the F 2 molecule, the valence partition 6|2|6 yields a joint probability of only 0.1, while those with one electron transferred (like 7|1|6, 5|3|6, or 5|2|7 produce 0.3.

In this case the transfer between lone pair regions on different atoms that is more important here than, e.g., between the CH regions in HCCH: the lone pair regions have a common border in our partition. In fact the probability of 5|2|7 and that of 7|2|5 is smaller than that of 6|2|6, but also around 0.1.

The probability to have 2 electrons in each of the valence electron pairs (for example, 2 in each of the CH bond regions, and 2 in each of the banana bond regions in HCCH) would be even lower. This lowers the probability of having a Lewis-type structure. Furthermore, were we to take into account the probability of electrons to move between the core and valence region, the probabilities would further drop. For example, that of having one of the six electrons of the region attributed to the CC triple bond being in one of the two C core regions, 2|5|2, is 0.06.

Conditional probabilities

One way to understand the low values for joint probabilities is by starting to assume that the probability to have n A electrons in A and n B in B are independent of each other, 0 < P A (n A )P B (n B ) < P A (n A ) or P B (n B ) < 1 (10) . However, in general, independence cannot be assumed. For example, when an electronegative element, say, F takes a charge, it takes it from somewhere. We have a probability that is was being taken from some other spatial domain. So we ask: What is the probability to have n A in region A (say, 54-1 electrons in the Xe region of XeF 2 ), given that there are n B in region B (say, 9 in one of the F regions)? If the fluctuations were due to independent events, it would be easy to answer, because P A|B = P A .

Let us start with the simple case when regions A and B correspond to CH bond regions in CH 4 , see figure 8. Let us consider n A = 2 and n B = 2, the chemically assumed values.

The probability to have the chemically assumed value in B increases that of having the chemically assumed value in A (compared to independent events). Furthermore, having n B + 1 also increases the probability to have n A -1 in the neighbouring region. Evidently, this holds for n B -1 and n A + 1. However, a simultaneous increase or decrease of the number of electrons in neighbouring regions is less favorable than for independent particles. P A|B (n A + 1, n B + 1) < P A (n A + 1) and P A|B (n A -1, n B -1) < P A (n A -1). A similar pattern is seen, e.g., in HCCH when A is the CH region, and B the CC region (see figure 9, top panel), or in F 2 when when A is the region of the three lone pairs, and B that of the bonding region. We suspect these trends to be quite common.

A further pattern is present when looking at distant pairs, like the two CH bonds in HCCH (figure 9, middle panel). We see practically no reciprocal influence.

An intermediate pattern shows up, when looking how one banana bond influences the other (figure 9, lower panel), or how lone pairs on the same atom interact (in F 2 , figure 10, lower panel): 3). This increased electron transfer between two electron regions only reminds of ionic structures, but it is not: it happens on the same atom (in F 2 ), or within the triple bond (in HCCH).

P A|B (n A = 2, n B = 2) ≈ P A (n A = 2). However, P A|B (n A = 1, n B = 3) > P A (n A = 1) and P A|B (n A = 3, n B = 1) > P A (n A =
Finally, let us look at XeF 2 (see figure 11, upper panel). We consider the atomic regions, Xe and F and notice also a significant increase in cases that do not fit at all in the "standard" cases discussed above. For example, for n B -1 = 53 we have an increase on n A + 1, but also on n A . Furthermore, we notice an significant increase for n B -2 on n A + 1.

As the transfer of electrons occurs between atoms, it is more easy to describe it by "ionic structures" that get more weight than what one would expect by simple "urn-like" fluctuation. This leads us to ask whether it would be not convenient to use ions as a reference, and not atoms.

CHOOSING THE SIGNIFICANT REGION

Alternatives

Figure 12 shows how probabilities evolve when changing the positions of the planes perpendicular to the z-axis, and symmetrically placed with respect to the origin (the position of the Xe nucleus, the F nuclei being at z = ±3.74 bohr). We see that there is indeed a maximum in the probability of having n = 54 electrons in a region around the Xe nucleus, but the probability to have n = 53 and even n = 52 is larger. The effect is even more pronounced if we look at the region around one of the F nuclei. The probability to have n = 10 is clearly more pronounced than that of having n = 9.

A similar feature is present when looking at the F 2 molecule, figure 13. The probability to have one electron in the region described by cones between the two nuclei can be made larger than that of having two. Correspondingly, the probability of having seven electrons for the region of three lone pairs. The dashed lines correspond to specific partitions.

. in the region attributed to three lone pairs on an atom is also getting larger than the best we can get for having six. 21 This is in contrast to what we have seen for the core-valence separation (shown for CH 4 , figure 2) where the expected values were also the best ones.

As we have the signature of ionic structures, let us compare the evolution of the probabilities of having different numbers of electrons in a region, as we change the region is changed.

For this let us choose a set of integers, I, and look at

w Ω (n) = P Ω (n) k∈I P Ω (k) , n ∈ I (11)
Of course, w Ω changes with Ω. Let us consider XeF 2 , and displace the planes that separate the region around the Xe nucleus (placed at the origin) from the F nuclei (placed at z = Instead of looking at the ratios w for P (n), we can look at the ratio of the joint probabilities corresponding to partitions, e.g., n 1 |n Xe |n 2 that of having n 1 electrons in one F region, n Xe in the region around Xe, and n 2 around the other F nucleus. We see (in figure 14) that the two types of ratios do not differ much in spite of P A∩B being lower than P (A).

We find, that for a certain value of z our 49, 406 determinant wave function produces ratios of probabilities (w Ω ) that are pretty close to the weights obtained in valence bond calculations [START_REF] Braïda | The essential role of charge-shift bonding in hypervalent prototype xn-XeF-t00a[END_REF] . The value of z where this happens is slightly larger than that that we would have attributed to Xe + . This indicates that what we call a Xe atom, is, in fact, closer to a Xe + ion. So let us better consider the probabilities (see figure 15) for the domain that maximizes the probability of having 53 electrons around the Xe nucleus (as in figure 12). At the same time, this means that we should consider having in the region around the F nucleus either 9 or 10 electrons, F ↔ F -.

Effective number of parties

We are now confronted with the existence of alternative descriptions. In fact, already

Lewis saw the need for it, and used the term tautomerism. 15 Choosing one of the spatial region is in detriment for another. For the example above, the optimal region Xe is not the best for Xe + . The P (53) and P (54) are different in the two regions. There is an arbitrariness in the choice of the region. Of course, we can optimize regions by weighting their importance.

The choice of these weights has also a certain degree of arbitrariness. In some cases, chemical intuition can decide about having equal weights, e.g., F and F -, as equivalent to the choice of Xe + , see fig. 15.

Another possibility is to choose the region that minimizes the number of alternative structures. In the study of political systems, the concept of "effective number of parties" exists: if a party is weakly represented in a parliament, it weighs less. Laakso and Taagepera 14 use for it

N = 1 n P 2 n ( 12
)
where P n is the fractional number of seats occupied by a party, and N is the total number of parties present in the parliament. N → for the CH bonds in CH 4 ). A possible limitation of such a procedure is seen when we consider perturbations of the system (chemical reactions): equivalence disappears, grouping is abandoned, and abrupt changes can appear.

For our example, XeF 2 , the distinction between the two variants above is irrelevant; with, or without grouping, the same partitions are selected.

CONCLUSIONS

In 1933, Lewis called his approach analytical 16 : deducing from a large body of experimental material some simple laws consistent with the known phenomena. He contrasted it with the approach of the mathematical physicist who uses postulates to synthesize a molecule. He recognized its power of producing quantitative results (in contrast to the analytical method that produces only qualitative results). However, he feared that the inaccuracy of a single postulate might completely invalidate the results while the "results of the analytical method can never be far wrong, resting as they do upon so numerous experimental supports". After one century, the postulates of quantum mechanics not only remained valid, they also allowed to go beyond what had been experimentally known. However, our minds still aspire for simple laws, in spite of computations becoming more convenient. This paper tries to extract some simple information from quite accurate computations without trying to explore the origins of the effects observed: we now that they lie in the Schrödinger equation and the Pauli principle.

As the core of quantum mechanics is probabilistic, we try to provide probabilistic measures. We define the space into regions, and compute the probability to have a given number of electrons in it.

We note that the probabilities are low: already the probability to have core-valence separation in CH 4 is only of 0.8. Considering an arrangement as in a Lewis structure (for example, having 2 electron pairs in each of the CH bonds) is so low (0.2) that it is more likely to have electrons not distributed according the Lewis structure. A simple explanation for it is that there are many ways for an electron to quit its pair. We would like to stress that this is not contradicting the Pauli principle, because we consider electrons in a region of space, not in an infinitesimal volume element. Not always can we see it as a contribution of ionic structures: such effects show up also between lone pairs on the same atom, or multiple bonds.

Quantum mechanical effects enhance some of the probabilities, and reduce others. However, such changes are normally around 0.1-0.2. Such enhancements can be quantified by comparing with models (such as the Ehrenfest urn model), or by considering conditional probabilities.

We try to defend the idea that there is a need to giving the user some freedom in the definitions allowing to adapt the tool to the purpose of the analysis (for example to choose if it is better to refer to atoms or to ions). Not all interpretative methods give it. The probabilistic nature provides several interpretations, and depending on what we ask this may be simple or complicated. We tentatively suggest -without exploring it systematically -an Occam's razor approach inspired from political analyses: the effective number of parties, the parties corresponding to the different distributions of electrons over the spatial regions.

We choose among the partitions that are having the minimal number of representative distributions.

This paper could have had the title: "Concepts in search for observations". It is not clear to us how relevant our remarks are for chemists. However, these effects are present, and it might be worth looking for them. To conclude, let us mention an aspect that we could treat with our approach (but have only mentioned in passing in this paper). One may have regions of space where electrons are localized, producing spatially disconnected regions between which electrons are delocalized. yielding for the two particle wave function:

Φ(x 1 , x 2 ) = φ 1 (x 1 )φ 2 (x 2 ) -φ 1 (x 2 )φ 2 (x 1 )
We can compute analytically the probability to have one and only one electron in the left (or right) half of the box:

P left (1) = P right (1) = 1 2 + 32 9π
S k (x) = x k+1 k j=0 (k + j)! k! j! (2k + 1)! (k -j)! (k + j + 1)! (-x) j (B3)
In the limit k → ∞, it becomes the step function at x = 1/2, while S 0 = x. The expression for the probability to have one and only one particle in one of these "fuzzy half-boxes": for the sharp separation. This value increases as the indicator function is smoothed out, giving more weight to the minority distribution as the k increases.

P lef t (1) = P right (1) = Φ|S k (x 1 ) (1 -S k (x 2 )) + (1 -S k (x 1 )) S k (x 2 )|Φ (

Appendix C: Geometrical shapes

As a first example, let us consider the atomic cores. Although they can be polarized by the molecular environment, we find that they are close to spheres centered on the nuclei.

We adjust the radius to maximize the probability of n core electrons to be in the sphere (n core is the number of electrons commonly assumed to be in the core, e.g., 2 for C and for F).

As the common image of atoms is spherical, let us first try to construct spheres around nuclei such that the probability of finding n electrons is maximal. Let us take, as an example, the XeF 2 molecule (see, figure 17, panel bottom right). The dots show 10 4 configurations (each of 72 electron positions). The sphere in the center maximizes the probability to find 54 electrons around the Xe nucleus. It has a radius of ≈ 3.1 bohr. The other sphere, with a radius of ≈ 2.0 bohr, maximizes the probability to find 9 electrons around one of the F nuclei. It is clear that the spheres significantly overlap (the Xe-F distance is ≈ 1.4 bohr smaller than the sum of the radii of the spheres). Furthermore, as one can see in figure 17, a non-negligible number of electron positions around a given atom are not taken in consideration for computing the probability. This can be explained in the following way. As the radius of the sphere around nucleus A increases from 0, the probability to have n > 0 electrons also increases. Beyond a certain value of the radius, P A (n) continues to increase by including electrons that by there positions would normally be attributed to another atom, although -for a flexible shape -it would be possible to extend without doing it by deforming the sphere. For n < N , after a certain radius, the P A (n) starts decreasing, as P A (n + 1), P A (n + 2), . . . P A (N ) increase. In particular, P A (N ) = 1 for an infinite radius.

As the probabilities sum to one for a certain radius, the sphere stops increasing leaving out a number of electron positions that would have been included allowing deformations of the sphere. In the most accurate calculations of the domains maximizing the probability to find a pair of electrons describing bonds [START_REF] Braida | Shape and topology optimization for maximum probability domains in quantum chemistry[END_REF] , it has been often found that they have shapes of cones with the apex on nuclei. Of course, the core regions must be excluded. A schematic drawing is shown in figure 17, top left panel. We construct such regions by varying the aperture (2α) to maximize the probability to find the chosen numbers of electrons in it. When the aperture is π, we have planes perpendicular to the rotation axis. We found it convenient to displace the position of such planes, in particular when trying to define "atoms", e.g. in the XeF 2 molecule. In fact, this corresponds to using weighted Voronoi cells. For example, in XeF 2 , oriented along the z-axis with the Xe atom at the origin, we find the value of z for which the probability of having n = 54 electrons between the two planes ±z is maximal (see figure 17, middle panels). Note that by using cones or planes, we have space-filling regions, and the problem of leaving out configurations (present for spheres) does not show up (see figure 17, panels top and middle right).

1 .4

 1 The probability to have n electrons present in a spatial region 3. Choice of the spatial region 3.1. Sharp boundaries 3.2. Partial optimization of the spatial domain 3.3. Ambiguity in the choice of the spatial region 4. Fluctuations 4.1. Ehrenfest urn model 4.2. Orbital nodal structure and localization: two particles in a box 4.3. Electron pairs: CH 4 , HCCH, F 2 , XeF 2

FIG. 1 .

 1 FIG. 1. Localized (equivalent 9 ) orbitals for two independent particles in a one-dimensional box, in a state where the canonical orbitals φ 1 and φ k are occupied; for k = 2 (top panel), k = 3 (middle panel), and k = 4 (bottom panel).

FIG. 2 .

 2 FIG.2. The probabilities of n = 0, 1, . . . , 4 electrons to be present in a sphere of radius r around the C nucleus in CH 4 .

1

 1

4. 3 .

 3 Electron pairs: CH 4 , HCCH, F 2 , XeF 2

FIG. 3 .

 3 FIG. 3. Probability to find n electrons in the region attributed to the CH bond in CH 4 . The region is shown in the top right corner of the plot, the (excluded) C core region being also shown, in red.

1 ΩFIG. 4 .

 14 FIG. 4. Probability to find n electrons in the region attributed to the CH bond (top panel), the triple bond (middle panel), and the banana bond in HCCH.

  FIG. 5. Probability to find n electrons in the region attributed to the F-F bond (top panel), the three lone pairs on one of the atoms (middle panel), and a single lone pair in F 2 .

FIG. 7 .

 7 FIG. 7. Probability to have n electrons in the region attributed to the Xe atom (top panel), and the F atom (bottom panel) in XeF 2 .

FIG. 8 .

 8 FIG. 8. The conditional probability P A|B of having n A electrons in region A given that there are n B electrons in region B, for CH 4 ; A and B correspond to two different CH bond regions. The horizontal semi-transparent blue planes show P A (n A ).

FIG. 9 .

 9 FIG. 9. The conditional probability P A|B of having n A electrons in region A given that there are n B electrons in region B, for HCCH (top: A=CH, B=CC, middle A, B are two different CH regions, bottom: A, B are two different banana bonds. The horizontal semi-transparent blue planes show P A (n A ).

FIG. 10 .

 10 FIG. 10. The conditional probability P A|B of having n A electrons in region A given that there are n B electrons in region B, for F 2 (top: A is the group of 3 lone pairs, and B is the bond, bottom: A, B are two different lone pairs on the same atom. The horizontal semi-transparent blue planes show P A (n A ).

FIG. 11 .

 11 FIG. 11. Conditional probability in XeF 2 , for finding n A = 7, 8, . . . , 11 electrons in the F atom regions, given that there are n B = 51, 52, . . . , 55 electrons in the Xe atom region (top panel), or Xe + ion region (bottom panel).

FIG. 12 .

 12 FIG. 12. Probability to have n electrons between two planes at ±z, in XeF 2 . The Xe nucleus is at z = 0, the F nucleus at z ≈ 3.74. The curves marked -|n|-(top panel) show the probability to have n electrons in the region containing the Xe nucleus, the curves marked n| -|-(bottom panel) show the probability to have n electrons in a region containing one of the F nuclei.

  FIG.13. F 2 molecule, probability for the regions corresponding to the FF bond (top), and the three lone pairs (bottom); full lines are for having 1, 2, or 3 electrons in the bond region, or6, 7, 8

F

  FIG.14. Ratio of probabilities as changing with the position of the planes defining the regions in the XeF 2 molecule; for Ω between -z and z, around the Xe atom. P Ω (n)/ i P Ω (n i ), n i = 52 (blue), 53 (green), 54 (red), full curves. The dashed curves, show the relative contributions of the probabilities for the partitions 10|52|10 (blue), 9|53|10 together with 10|53|9 (green), and 9|54|9 (red). The vertical dot-dashed lines indicate the values of z for which P (n = 53) and P (n = 54) are maximal. The horizontal dot-dashed lines show weights obtained in valence bond calculations.[START_REF] Braïda | The essential role of charge-shift bonding in hypervalent prototype xn-XeF-t00a[END_REF]

  FIG. 15. Probability to find n electrons in the region attributed to the Xe + ion (top panel), and to F - F (bottom panel) in XeF 2 .

4 ≈

 4 Figure16shows the change of the probability with the change of the smoothness parameter.(Note that the points in figure16show P left as a function of 1/k: the values at the left of the graph correspond to a strong cutoff.) The highest value of the probability is reached for

  The Ehrenfest model can be extended to more than two urns, 27 leading to a multinomial distribution under assumptions similar to those made for two urns (see, e.g., ref.13). Let us consider, for example, the case of having 6 balls in three urns. The probability to have exactly two balls in one of the urns is 80/243 ≈ 0.33. That of having 2 balls in each of the urns is lower, 30/243 ≈ 0.12.

	23 the
	user specifies the number of electrons attributed to it. 19 This is the path we choose in this
	paper. For other methods, it shows up in the result. For example, in ionic crystals, the
	QTAIM "atoms" are closer to ions, see, e.g., refs. 5 and 18.)
	4. FLUCTUATIONS
	4.1. Ehrenfest urn model
	In order to have a reference case, let us consider the Ehrenfest urn model (constructed
	to describe the fluctuations at thermal equilibrium). 10 N balls (electrons for us) are in
	distributed in two urns (spatial domains for us), A and B. One ball is randomly picked from
	any of the urns (naturally, more likely with the urn with more balls) and put in the other
	urn. As time evolves, the number of balls in the urns stabilizes to a distribution 12 , that -
	for even N -has maximum at N/2 equal to 2 -N N !/[(N/2)!] 2 ; the number of balls in an urn
	follows a binomial distribution. The ratio between the probability to have N/2 balls in the
	urn and that to have N/2 ± 1 is 1 + 2/N . For example, for N = 4 (two electron pairs), we
	would have P Ω (n = 2) = 0.375 and P Ω (n = 1) = P Ω (n = 3) = 0.25. As N increases, not
	only P (N/2) decreases, but P (N/2 + 1) and P (N/2 -1) get closer to it. For example, if we
	consider N = 14 (two F atoms, valence shells), P (7) ≈ 0.21 and P (6) ≈ 0.19.

4.2. Orbital nodal structure and localization: two particles in a box

  1 if one party dominates, and N = N if all parties have an equal number of seats. Of course, this definition can be used for other types, by replacing P n by probabilities. Let us try to use this definition in the case of XeF 2 . 1. Let us use the probabilities of n 1 |n Xe |n 2 (to have n Xe electrons in the Xe atom region,n 1 in that of one of the F atoms, and n 2 in that of the other F atom). If we use the partition to maximize to probability to have a Xe atom, we get N = 8, dominated by F|Xe|F, F -|Xe + |F and F|Xe + |F -, F + |Xe -|F and F|Xe -|F + , and F -|Xe 2+ |F -, F -|Xe|F + and F + |Xe|F -. If we use the partition to maximize to probability to have a Xe + ion, we get N = 6, dominated by F -|Xe 2+ |F -, F -|Xe + |F and F|Xe + |F -, F|Xe|F, F -|Xe|F + and F + |Xe|F -. This eliminated the partitions requiring Xe -: and thus reduced the number of descriptors needed.2. It may be more reasonable to group together the chemically equivalent descriptions, e.g., F -|Xe + |F and F|Xe + |F -. This way, N is reduced, because we count such pairs only once. Also, the importance is modified, for example, for n 1 = n 2 instead of having P (n 1 , n Xe , n 2 ) = P (n 2 , n Xe , n 1 ) we consider both possibilities together, and have P (n 1 , n Xe , n 2 ) + P (n 2 , n Xe , n 1 ). In general, grouping equivalent partitions favors

	cases where many equivalent distributions exist over cases where the partitions are
	unique unique ones (for example, e.g. twelve partitions 1|2|2|3, 1|2|3|2, . . . over 2|2|2|2

TABLE I .

 I Energies (a.u) of the Hartree-Fock and CI wave functions used to generate the configurations. N det is the number of determinants, and exFCI is an estimate of the full CI energy obtained by extrapolation of the CIPSI energies. The percentage of correlation energy are computed with respect to the exFCI energy. Probability of having one electron in dominantly in the left half of the box (and another electron dominantly in the right half of the box) as a function of the smoothing (using the smoothstep function S k , eq. B3). The dotted line is connecting the points to guide the eye.

		0.85
		0.80
		0.75
	P
		0.70
										1|1
		FHF -0.65		KrF 2		XeF 2		F 2		HCCH	CH 4
	N det	1083789 0.60		554150		49406		260758	38038	1245216
	Frozen MOs	1-2 0.0	0.2	1-11	0.4	1-20 0.6	0.8	1-2	1.0	1-2	1
						1/m
	Active MOs	3-89		12-137		21-321		3-80		3-91	2-73
	HF FIG. 16.	-199.586	-2950.704	-7430.097		-198.765	-76.851	-39.561
	exFCI	-200.161	-2951.587	-7431.468		-199.311	-77.186	-39.913
	CI	-200.138	-2951.486	-7431.322		-199.289	-77.148	-39.908
	% Correlation	96%		89%		89%		96%		87%	98%

Appendix A: Technical details

For all the systems presented, we generated a correlated wave function with a frozencore configuration interaction using a perturbative selection made iteratively (CIPSI) in the full valence CI space. For CIPSI calculations, we used natural orbitals of a preliminary CIPSI calculation made with the Hartree-Fock orbitals to generate a more compact CI expansion. For XeF 2 , the Universal Gaussian Basis Set (UGBS) was used, [START_REF] De Castro | Accurate universal Gaussian basis set for all atoms of the Periodic Table[END_REF] and for all other compounds we used the def2-TZVPD basis set. 26 The CIPSI calculations were performed with the Quantum Package software [START_REF] Garniron | Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs[END_REF] and the QMC calculations were made without any modification of the wave function with QMC=Chem. 25 The energies of the generated wave functions are given in table I.

Appendix B: Smooth boundaries for two particles in a box

Let us consider two same-spin fermions in a box of length L = 1. The eigenfunctions corresponding to the lowest energies are