# Population analysis from variational chemical partition of molecular position space

Bernard Silvi, esmail alikhani

► To cite this version:

Bernard Silvi, esmail alikhani. Population analysis from variational chemical partition of molecular position space. Journal of Computational Chemistry, 2025, 46, pp.e70184. 10.1002/jcc.70184. hal-05158931v2

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Submitted on 1 Aug 2025

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RESEARCH ARTICLE

OPEN ACCESS

# Population Analysis From Variational Chemical Partition of Molecular Position Space

Bernard Silvi¹ | M. Esmaïl Alikhani²

¹Sorbonne Universités, UPMC, Univ Paris 06, UMR 7616, Laboratoire de Chimie Théorique, Paris, France | ²Sorbonne Universités, UPMC, Univ Paris 06, UMR 8233, MONARIS, Paris, France

**Correspondence:** M. Esmaïl Alikhani (esmail.alikhani@sorbonn-e-universite.fr)

**Received:** 15 May 2025 | **Revised:** 20 June 2025 | **Accepted:** 26 June 2025

**Keywords:** Chemical bonding | Electron localization | Partitioning | Population analysis | Statistical analysis

## ABSTRACT

Two years ago, a partition was proposed in the position space of the charge distribution of atoms and molecules, relying on the optimization of the boundaries of space-filling non-overlapping localization regions. This is achieved with the help of global objective functions providing either a measure of the dispersion of numbers of electrons in each localization region or of the dependence of these numbers between them. The output of the method reproduces very well the expected chemical structures in terms of atomic shells, bonds, and lone pairs domains. This method, inspired by the data processing of experimental results, is by construction free of any approximation since it only requires as input the one and two-particle densities of probability. It is based on a strict definition of the concepts of chemical entity, i, e, a set of nuclei and of a given number of localized electrons, for which an electron count followed by a multivariate analysis is carried out. In its first version, the software was extremely inefficient, and its use was therefore limited to atoms and small molecular systems to keep the computational effort within reasonable limits. A significant improvement has been recently achieved, enabling calculations of molecules as large as polyaromatic hydrocarbons. In the present paper, we first discuss the philosophy of previous population schemes, then we describe the features of our new family of population analysis, and then report the results obtained on a rather large sample of molecules and complexes, including polyaromatic hydrocarbons, propellanes, transition metal carbonyl 1:1 complexes and hydrogen-bonded systems as well as reaction mechanisms.

## 1 | Forewords (Bernard Silvi)

There exists an impression, I would say an emotion, that one can experience in the presence of things charged with history. When a physical contact is established between you and such an object, you may feel that the time gap between the present and the past is shrinking. Thanks to Shridar Gadre, I had the opportunity to discover this on a Tuesday afternoon in June 2002. On that day, I had accompanied Shridar to the library of the French Academy of Sciences, where he wished to consult original manuscripts of Coulomb and Poisson. By the end of our visit, the librarian put several notebooks in my hands. I remember that they had black cardboard covers and turned out to be Lavoisier's laboratory

notebooks. I opened one of them and read the report of his observations, written by him and by his wife. More than twenty years later, I still have a memory of these exceptional moments for which I would like to renew my thanks to Shridar.

Shridar Gadre is a very influential scientist in the field of computational molecular physics. I have always been impressed by the intrinsic quality of Shridar Gadre's scientific texts. The molecular electrostatic potential (MESP) is undoubtedly his choice topic as illustrated by the publication up to now of more than 80 articles [1–84] and of a book [85]. Most of his studies are based on a topographical approach exploiting the dynamical structure of the gradient field of the MESP, here the electric field, enabling

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2025 The Author(s). *Journal of Computational Chemistry* published by Wiley Periodicals LLC.
to highlight bonding properties, intermolecular interactions
in complexes as reactivity and reaction mechanisms. Many of
his other publications concern the study of atomic and molec-
ular electron densities as well as electron momentum densities
[86–111], I share with him a common, genuine interest in the
elaboration of interpretative methods, from their physical or
chemical underlying to the realization of computational tools. It
is therefore an honour and a pleasure for us to contribute to this
special issue of the Journal of Computational Chemistry.

2 | From Electron Densities to Population Analysis

The one-electron density of probability function relies on the
statistical interpretation of Quantum Mechanics. It is one of
the most important observables and it is involved in important
theorems such as the electrostatic expression of the Hellmann-
Feynman theorem [112–115] and Hohenberg–Kohn theorem
[116] which is at the root of the Density Functional Theory
(DFT). It is the glue that packs nuclei into molecules and solids,
ensuring the stability of the matter. It is a physical observable,
which can be expressed as the expectation value of a density op-
erator, and, in principle, determined experimentally. However,
experimental densities are not directly measured but derived
from the footprints of the interaction of crystalline materials
with high-energy photons or light particles. The calculation
of electron density from molecular orbitals is straightforward
and the Z-vector method of Handy and Schaeffer [117] enables
its evaluation for post-Hartree-Fock methods like MP3, MP4,
and CCSD. Two-electron densities (or pair functions) provide
complementary pieces of information. They are also quantum
mechanical observables that can be rigorously evaluated for
Hartree-Fock and CI wave functions.

The analysis of one-electron density of probability provides
information on the bonding in molecules and solids, enabling
a step-by-step recognition of chemical objects. In this way
“Chemistry emerges from Quantum Mechanics” [118]. The
study of deformation density maps is an intuitive qualitative ap-
proach initially proposed by R. Daudel [119] which use a promo-
lecular density built as the superposition of the densities of the
constitutive ground state atoms.

A more quantitative strategy distributes the total charge density
and, if possible, the pair functions over parts of the investigated
system related to chemical entities and then condenses the infor-
mation into finite sets of numbers. Population analysis methods
belong to this strategic choice. They usually consider a partition
into atoms and yield atomic populations and interatomic bond
orders. They rely on the concepts of atom in molecule and chem-
ical bond, which have been developed by chemists for almost
two centuries. They are not explicitly associated with quantum
mechanical operators, and the definitions of atoms in molecules
and bonds are not precise enough to provide unique answers.
This has led to a methodological entanglement entailing a large
variety of partitioning strategies which can be distributed in
(i) orbital-based, (ii) physical space partitioning methods, and
(iii) procedures based on the fit of a property depending on the
charge distribution. A thorough description of the different pop-
ulation analysis schemes and their performances can be found
in a chapter of the Encyclopedia of Physical Organic Chemistry

[120]. The data in the output are on the one hand one-electron
properties such as atomic populations and distributed multipole
moments [121–127] and on the other hand two-electron ones
like bond orders and delocalization indices [128–141] which are
related to the pair density. The principal weakness of population
analysis methods is that the hypotheses underlying the parti-
tion schemes are not clearly related to chemical concepts. For
example, in the ELF population analysis, the chemical meaning
of the basin population belongs to a grey zone because it is not
possible to establish that the electron density located there has
a specific chemical role. This role in the bonding is supposed
and assumed, not demonstrated. This is due in part to the lack
of precision in the definition of concepts related to bonding,
and particularly the concept of the bond itself. QTAIM does not
suffer from such defects insofar as we are interested in physi-
cal properties. In recent article [142], we proposed a loge the-
ory [143, 144] inspired strategy to overcome this difficulty. The
initial code being rather inefficient, the applications have been
limited to atoms and small molecules, for which encouraging
results have been obtained. It has been significantly improved
and now calculations of molecules as large as phenalene are fea-
sible at a reasonable CPU cost. In this article, we first recall the
principles of our method and explore its possibilities on a series
of test cases.

3 | Toward a Partitioning in Terms of Chemical Entities

Before describing our method, it is worth explaining the ambi-
tions which have led us to propose this alternative to the ELF
population analysis, which remains, however, our choice of in-
terpretative tool. Chemistry offers very efficient predictive tools
based on a reductionist representation of the matter in terms of
functional groups, ligands, atoms in molecules, bonds, electron
pairs ... completed by coordination and electron count rules.
Chemical explanations make use of these chemical entities and
rules to account for the stoichiometry, structure and the reactiv-
ity of molecules. The Lewis's valence theory [145, 146], Ingold's
mesomery [147–149] and the VSEPR model [150–152] are suc-
cessful examples of mainly intuitive models in use in Chemistry.
The purpose of the loge theory [143, 144] is to “provide a math-
ematical bridge between chemical intuition and wave mechan-
ics, which may be considered as a theoretical justification of the
main chemical ideas” [144]. Despite this noble ambition, the
theory has remained in draft form due to difficulties in imple-
mentation. One of the reasons might be that its promoter have
chosen to go from Quantum Mechanics to Chemistry rather
than from Chemistry to Quantum Mechanics. The definition of
the concept of loge proposed by Raymond Daudel in the seminal
article [143] is rather unclear and cumbersome, in brief “a family
of normed functions fᵢ of one or several points of a space is lo-
calized with an accuracy ≥ ε if it is possible to find a decomposi-
tion of this space into non-overlapping domains (or loges) Vᵢ such
as ∫Vᵢ |fᵢ|² dτ ≥ 1 - ε”. An alternative definition has been given
twenty years later: Loges are space-filling non-overlapping vol-
umes “in which there is a high probability of finding a given
number n of electrons (but not always the same) with a certain
organization of their spins.” [144]. We opted to go in the opposite
direction by giving up the ambition to find a theoretical justifi-
cation for the chemical ideas.
The concepts mentioned earlier all imply the presence of specific groups of electrons in specific regions of the molecule or of the crystal. This conjecture is generally accepted, although no experimental evidence has been provided, whereas it is strongly supported by theoretical tools such as Maximum Probability Domains (MPDs) [153–157]. In contrast to Loge theory and MPDs, the approach presented in our 2022 article [142] does not a priori set the size of the groups. Instead, starting from guessed volumes assigned to the groups, an objective target function is minimized with respect to the region boundaries to find optimal group localization volumes. The nature of the groups, their number, and the guess of are determined by the underlying chemical explanation and therefore depend on the user's decision. For example, if the explanation invokes Lewis's or VSEPR models, core, bonding, and non-bonding group localization volumes have to be found.

## 4 | Overview of the Method

The partitioning method described here has been developed for isolated molecular systems. Its generalization to periodic systems, although formally possible, seems very difficult to implement due to limitations imposed by the present state of the art of periodic ab initio programs. The objective of the method is to obtain a partition of the molecular space of a N electron system in terms of M space-filling non-overlapping volumes $V_A$ localizing electron groups. By localizing, we mean that the fluctuations of the number of electrons in each volume should be as small as possible.

Each volume is characterized by its limiting surfaces and by its electron population, $\overline{N}_A$, given by the integral of the one-electron density over the volume $V_A$:

$$ \overline{N}_A = \int_{V_A} \rho(\mathbf{r})d\mathbf{r} \qquad (1) $$

and interpreted as of the averaged number of electrons visiting $V_A$ obtained after an infinite number of electron counts. It is accordingly rewritten in terms of $p_i(V_A)$, the probability functions of finding i and only i electrons in $V_A$:

$$ \overline{N}_A = \sum_{0}^{N} ip_i(V_A) \qquad (2) $$

This multidimensional count distribution is adequately accounted for by a multivariate multinomial distribution since the sums of the numbers of electrons and the populations are equal to N:

$$ \sum_{A=1}^{M_A} N_A = \sum_{A=1}^{M_A} \overline{N}_A = N \qquad (3) $$

whereas the sum of the probabilities $p_i(V_A)$ satisfies:

$$ \sum_{i=0}^{N} p_i(V_A) = 1 \qquad (4) $$

To achieve our objective, we must be able to compare the spread of the data of two M-variate distributions. The comparison is done with the help of a global indicator, hereafter referred to as the objective target function (OTF). For univariate distributions, the ratio of variance to the mean, or Fisher dispersion index [158], is a choice indicator [159]. It is defined as:

$$ DI(X) = \frac{\mathrm{Var}X}{EX} \qquad (5) $$

where $EX$ and $VarX$ denote the mean and the variance of the random variable. It is the ratio of two variances since $EX$ is the variance of the Poisson distribution $P(EX)$. $DI(X)$ is a positive real number, $DI(X) = 0, DI(X) < 1, DI(X) = 1$, and $DI(X) > 1$ correspond to not-, under-, equi-, and over-dispersed distributions. Generalization to bivariate [160] and multivariate distributions [161] have been proposed during the last decade. In our 2022 article [142] we proposed to use as OTFs the sum of the dispersion indexes $\Sigma DI$ and the mutual information [162] (MI)

$$ MI = \sum_{A} \sum_{B} p_{AB}(1, 2) \log \frac{p_{AB}(1, 2)}{p_{A}(1)p_{B}(2)} \qquad (6) $$

where $p_A(1)$ is the probability of finding a given electron (1) in $V_A$, $p_B(2)$, the probability of finding a given electron (2) in $V_B$ and $p_{AB}(1, 2)$ is the probability of the joint event, a given electron (1) in $V_A$ and another given electron (2) in $V_B$. These probabilities are expressed in terms of the populations $\overline{N}_A$ and $\overline{N}_{AB}$:

$$ p_A(1) = \frac{\overline{N}_A}{N} \qquad (7) $$

$$ p_{AB}(1, 2) = \frac{\overline{N}_{AB}}{N(N-1)} \qquad (8) $$

in which $\overline{N}_{AB}$ is the integral over $V_A$ and $V_B$ of the two-electron density $\Pi(\mathbf{r}_1, \mathbf{r}_2)$. The results obtained on atoms and small molecular systems with these two OTFs are close to one another and in good agreement with the expectations of Lewis's model.

In practice, the calculation of dispersion indices and mutual information only requires the electron and electron pair populations of the volumes of the partition. These quantities, as well as the variances and covariances, are quantum mechanical observables, for which operators have been proposed (see Appendix A). The optimization of the partition is carried out by minimizing the OTF with respect to the volume boundaries as described in the next section. Starting from an initial guess, the volumes are varied until convergence of the OTF to a local or global minimum value. Different OTFs are expected to yield different but nevertheless close optimized partitions, and therefore, the choice of the OTF is an open subject.

## 5 | Implementation

The computer program handles Hartree-Fock and Kohn-Sham molecular orbitals calculations carried out with Cartesian Gaussian-type basis functions. The evaluation of the OTFs considered, $\Sigma DI$, $MI$, $GDI$ and $MDI$, requires the calculation of the
volumes populations $\overline{N}_A$ and pair populations $\overline{N}_{AB}$ which are expressed in terms of their spin components:

$$ \overline{N}_A = \overline{N}_A^\alpha + \overline{N}_A^\beta \quad (9) $$

$$ \overline{N}_{AB} = \overline{N}_{AB}^{aa} + \overline{N}_{AB}^{a\beta} + \overline{N}_{AB}^{\beta\alpha} + \overline{N}_{AB}^{\beta\beta} \quad (10) $$

with

$$ \overline{N}_A^\sigma = \sum_{i \in \sigma} \sum_{j \in \sigma} S_A(i, j) \quad (11) $$

$$ \overline{N}_{AB}^{\sigma\sigma} = \sum_{i \in \sigma} \sum_{j \in \sigma} \sum_{k \in \sigma} \sum_{l \in \sigma} S_A(i, k) S_B(j, l) - S_A(i, l) S_B(j, k) \quad (12) $$

$$ \overline{N}_{AB}^{\sigma\sigma'} = \sum_{i \in \sigma} \sum_{j \in \sigma'} \sum_{k \in \sigma} \sum_{l \in \sigma'} S_A(i, k) S_B(j, l) \quad (13) $$

where $S_A(i, j)$ is the overlap integral of the orbitals labeled i and j over the volume $V_A$.

The molecular volume enclosing at least 99.5% of the charge density is a rectangular prism divided in $N_x N_y N_z$ cubic cells of edge $\Delta l$ centred at $x = (i_x + 0.5)\Delta l, y = (i_y + 0.5)\Delta l, z = (i_z + 0.5)\Delta l$ with $i_x \in [0, N_x - 1], i_y \in [0, N_y - 1], i_z \in [0, N_z - 1]$. These indexes are condensed in a single index $ic = 1 * i_z + i_y N_z + i_x N_y N_z$.

The program requires as input the wave function stored in a .wfn file generated by the ab initio software and a file containing a guessed assignment of each cell to a volume of the partition. The optimization of the volumes $V_A$ is achieved by an iterative trial-and-error algorithm. The overlap integrals of the molecular orbitals over each cell volume, $s_{ij}(i_c)$, are calculated and stored in RAM. The size of the $s_{ij}$ array is of the order of $3.7 N_x N_y N_z N_{mol} (N_{mol} - 1) \times 10^{-9}$ Gb. For example, 64 Gb is necessary to run a 90-electron closed-shell system with a 200 × 200 × 200 grid. The memory required is, in fact, smaller because only cells for which the integrated density is larger than a given threshold are considered, and a further reduction is achieved by taking the symmetry of the system into account. This step is rather expensive in CPU time because the number of operations is proportional to the $N_x N_y N_z \times N_{mol}^2 \times N_{basis}^2$, where $N_{mol}$ and $N_{basis}$ are the numbers of molecular orbitals and basis functions, respectively. The preconditioning avoids its replication for each step of the optimization at the cost of a large random access memory requirement. The overall structure of the program is given below.

In addition to the populations of the optimized partition, the output displays the covariance matrix and the electron number probability distributions [163–165]. The Fortran 90/95 program VARPOP can be freely downloaded at the URL: https://www.lct.jussieu.fr/pagesperso/silvi/ together with the input manual and a tutorial with examples of application.

**Data:** wave function, guessed volume assignment

**Result:** optimized volumes, population analysis

/* initialization section

*/

Input wave function;

Find molecular space group and build symmetry equivalence tables;

Input guessed assignment: the assignment of each grid point is stored in the vector

label;

Build the $s_{ic}(i, j)$ array;

Calculate the $S_A(i, j)$ array: $S_A(i, j) = \sum_{i_c} \delta_{A,\text{label}(i_c)} s_{ic}(i, j)$;

Calculate $f_0$, the OTF value for the guess partition;

$f \leftarrow f_0 + 2\epsilon$; /* optimization section

*/

while $f - f_0 > \epsilon$ do

$f \leftarrow f_0$;

find border cells;

foreach border cell do

$i_1, i_a$: index and label of the border cell, $i_2, i_b$ index and label of the neighbor cell;

$S_A(i, j) = S_A(i, ) - s_{i_1} i, j + s_{i_2} (i, j)$;

$S_B(i, j) = S_B(i, ) - s_{i_2} i, j + s_{i_1} (i, j)$;

Calculate $f_n$ the OTF value for the new partition;

if $f_n < f$ then

$f \leftarrow f_n$;
label($i_1$) $\leftarrow i_b$;
label($i_2$) $\leftarrow i_a$;

else

$S_A(i, j) = S_A(i, ) + s_{i_1} i, j - s_{i_2} (i, j)$;
$S_B(i, j) = S_B(i, ) + s_{i_2} i, j - s_{i_1} (i, j)$;

end

end

end

Carry out population analysis for the converged partition;
6 | On the Choice of the Objective Target Function

The ability of several OTF candidates to account for the shell structure of atoms has been investigated in our 2022 article [142]. The most convincing results are obtained by the sum of the dispersion indexes and the mutual information, whereas the Frobenius norm of the covariance matrix and the sum of the coefficients of variation fails to find the correct shell for atoms beyond the first series of transition. The status of Shannon's mutual information (MI) [162, 166] is particular because it has the following properties:

1. For independent distributions $MI = 0.0$

2. For any partition $MI \ge 0$.

3. For any N-electron system, the upper bound of $MI$ is provided by the continuous distribution limit, that is,

$$MI_{\max} = \int_{\mathbf{r} \in \mathbb{R}^3} \int_{\mathbf{r}' \in \mathbb{R}^3} p_{12}(\mathbf{r}, \mathbf{r}') \ln \frac{p_{12}(\mathbf{r}, \mathbf{r}')}{p(\mathbf{r})p(\mathbf{r}')} d\mathbf{r}d\mathbf{r}' \quad (14)$$

4. For a partition into $M$ localization regions, the best boundaries are those which yield the maximum value of $MI$, $MI_{\max}(M)$.

The last property suggests that starting from $M = 1$, the optimal partition is reached when $MI_{\max}(M) = MI_{\max}$ and therefore, no further significant partitioning can be achieved. Particularly, $MI_{\max} = 0$ indicates that there is only one group that of $N$ electrons occupying the full position space. Except for model cases for which the integration is rather trivial, an accurate enough evaluation of $MI_{\max}$ requires a huge computational effort. For this reason, this calculation is not done in our program, where— $MI$ is used as OTF instead of $DI$. The mutual information appears to be a measure of the quality of the group localization enables the comparison between two optimized partitions. Moreover, for a given number of electrons, it is possible to build very simple zero-dispersion model distributions, such as idealized Lewis systems consisting in $N/2$ electron pairs localized in $N/2$ group localization volumes, $V_A$. The expression of the pair density $\Pi(\mathbf{r}, \mathbf{r}')$ and of $p_{12}(\mathbf{r}, \mathbf{r}')$ are:

$$\Pi(\mathbf{r}, \mathbf{r}') = \frac{1}{1 + \delta_{AB}} \rho(\mathbf{r}) \rho(\mathbf{r}') \qquad \mathbf{r} \in V_A, \mathbf{r}' \in V_B \quad (15)$$

$$p(\mathbf{r}) = \frac{1}{N} \rho(\mathbf{r}) \qquad (16)$$

$$\begin{aligned} p_{12}(\mathbf{r}, \mathbf{r}') &= \frac{1}{N(N-1)} \Pi(\mathbf{r}, \mathbf{r}') \\ &= \frac{N}{(1 + \delta_{AB})(N-1)} p(\mathbf{r}) p(\mathbf{r}') \end{aligned} \qquad (17)$$

$MI_{\max}$ can be easily calculated as a sum of the contributions:

$$\begin{aligned} MI_{\max} &= \sum_{A}^{N/2} \sum_{B}^{N/2} \int_{V_A} \int_{V_B} p_{12}(\mathbf{r}, \mathbf{r}') \ln \frac{p_{12}(\mathbf{r}, \mathbf{r}')}{p(\mathbf{r})p(\mathbf{r}')} \\ &= \sum_{A}^{N/2} \sum_{B}^{N/2} \frac{4}{(1 + \delta_{AB})N(N-1)} \ln \frac{N}{(1 + \delta_{AB})(N-1)} \quad (18) \\ &= \frac{1}{N-1} \ln \frac{N}{2(N-1)} + \frac{N-2}{N-1} \ln \frac{N}{N-1} \end{aligned}$$

The $MI$ values calculated from the populations and pair populations, that is, $\overline{N}_A = 2$ and pair populations, $\overline{N}_{AB} = \overline{N}_A \overline{N}_B / (1 + \delta_{AB})$ are equal to $MI_{\max}$ obtained with Equation (18). The zero-dispersion mutual information is a true upper bound that can be used to estimate the localization/delocalization in a given system.

An important argument in favor of OTFs based on the Fisher dispersion indexes is that they are almost directly related to a measure quantifying whether the electron counts in the volumes are clustered or scattered with respect to a statistical model. Recently, we found the article by Kokonendji and Puig [161] in which the generalization of the dispersion index is discussed. We found it interesting to question the proposals of these authors. For this purpose, the results obtained on the partition into two subspaces in very simple test cases by the bivariate Fisher Index (FI2) [160]:

$$FI2 = \frac{\{DI(A) + DI(B) - 2\rho^2 \sqrt{DI(A)DI(B)}\}}{(1 - \rho^2)} \quad (19)$$

with $\rho = \text{cov}(A, B) / \sqrt{\text{cov}(A, A)\text{cov}(B, B)}$ the global dispersion index (GDI):

$$GDI = \frac{\sqrt{\overline{N}^{\dagger}} (\text{cov}N) \sqrt{\overline{N}}}{\overline{N}^{\dagger} \overline{N}} \qquad (20)$$

and the multiple marginal dispersion index (MDI):

$$MDI = \frac{\sqrt{\overline{N}^{\dagger}} (\text{diagvar}N) \sqrt{\overline{N}}}{\overline{N}^{\dagger} \overline{N}} \qquad (21)$$

are compared with the output of $\Sigma$DI and $MI$.

The closure relation of the covariance matrix elements (Equation A15) implies:

$$\text{cov} = \begin{pmatrix} \text{cov}(A, A) & -\text{cov}(A, A) \\ -\text{cov}(A, A) & \text{cov}(A, A) \end{pmatrix} \qquad (22)$$

for any partition into two volumes. The structure of the covariance matrix implies that $\rho = -1$ and therefore, the denominator in the expression of FI2 is equal to zero, making FI2 useless for any minimization procedure. Consider now the partition in two subspace $V_A$ and $V_B$ for two simple test cases, namely those of a single particle and of two particles of opposite spins. For both cases, all the quantities necessary to calculate the OTFs can be expressed in terms of $\overline{N}_A$:

<table><thead><tr><th></th><th>Single particle</th><th>2 particles<br>α and β</th></tr></thead><tbody><tr><td>\overline{N}_B</td><td>1 - \overline{N}_A</td><td>2 - \overline{N}_A</td></tr><tr><td>\overline{N}_{AA}</td><td>0</td><td>\overline{N}_A<sup>2</sup> / 2</td></tr><tr><td>\overline{N}_{AB}</td><td>0</td><td>\overline{N}_A(2 - \overline{N}_A) / 2</td></tr><tr><td>\overline{N}_{BB}</td><td>0</td><td>(2 - \overline{N}_A)<sup>2</sup> / 2</td></tr><tr><td>cov(A, A)</td><td>\overline{N}_A(1 - \overline{N}_A)</td><td>\overline{N}_A(2 - \overline{N}_A) / 2</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>**FIGURE 1** | ΣDI (full line), MI (dashed line), GDI (dotted line), and DMI (dashed-dotted line) versus $\overline{N}_A$. Left: Single particle; Right: Two particles with opposite spins.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**FIGURE 2** | ELF = 0.8 localization domains of CH<sub>3</sub>COOH. Color code: Magenta = core; Green = disynaptic valence.</figcaption></figure>

In the first case *MI* is not defined because *p*<sub>12</sub> = 0, whereas *MI*<sub>max</sub> = 0 in the second because *p*<sub>12</sub>(*r*, *r*') = *p*(*r*)*p*(*r*'). In both cases, the optimal number of regions is 1, and any OTF presenting a minimum for $\overline{N}_A \neq 0$ and $\overline{N}_A \neq N$ has to be discarded. Figure 1 displays the OTF values as a function of the population of subspace $V_A$. GDI, which presents a minimum at $\overline{N}_A = N/2$, is not tractable. We have further checked this OTF on molecular systems with different partition schemes and obtained partitions with equal populations, that is, $\overline{N}_A = N/M$.

## 7 | Examples of Application

The nomenclature used for ELF basins has been adopted to designate the electron group localization volumes: On the one hand are core volumes surrounding nuclei with $Z > 2$ and labeled C(A) where A is the atomic symbol of the element, and on the other hand, are the valence volumes characterized by the valence shells to which they participate and denoted $V(A, B, ...)$). To illustrate the possibilities of this new method, several chemically significant cases have been selected. The calculation of the wave functions has been performed with the Gaussian 09 quantum chemical package [167]. The DFT calculations have been carried out with the Hartree-Fock density functional B3LYP [168–171] with

different basis sets: 6-31G** [172], 6-311G(2d) [173–175], cc-pVTZ [176].

### 7.1 | A Tutorial Example CH<sub>3</sub>COOH

This molecule has been chosen because it contains in the Lewis representation, various bonds and valence shells as well as lone pairs. It enables, therefore, the comparison of different partitioning guesses. Figure 2 displays the ELF = 0.8-localization domains with the basin labels.

#### 7.1.1 | Lewis Guess

The Lewis guess is built starting from the ELF basin assignment by halving the V(C,O) and V(O) basins symmetrically with respect to the $\sigma_h$ plane. Table 1 reports the populations and variance of the ELF, $\Sigma DI$ and *MI* partitions for B3LYP/6-31G** calculations. In addition, the program yields the spin component covariance matrices, the total covariance matrix, and the electron number probabilities in each group localization volume.

The noticeable deviations from the Lewis model, which occur in ELF populations for V(C2, O1) and V<sub>2</sub>(O1) are significantly reduced with both $\Sigma DI$ and *MI* partitioning. These two latter partition schemes also yield valence populations closer to 8 than ELF. As shown in Figure 3, the electron number probability distributions in the V(C2, O1) and V(O2) volumes are, as expected, more peaked with $\Sigma DI$ and *MI* than with ELF.

Table 2 reports the $\Sigma DI$ group localization volume populations calculated at the HF level with different basis sets ranging from STO-3G to cc-pVTZ for the B3LYP/6-31G(dp) optimized geometry. The choice of the basis set has little impact on the results. The largest discrepancy occurs for V(C1, C2) for which the polarized basis sets yield a value lower by 0.1 e⁻-than those calculated with STO-3G and STO-6G. The differences between the HF/6-31G(dp) and B3LYP/6-31G(dp) of Table 1 have almost the same order of magnitude.
<table><caption>**TABLE 1** | Group localization volume populations, $\overline{N}_A$, covariance, $\sigma^2(\overline{N}_A)$, mutual information, *MI*, and valence shell populations for *ELF*, $\Sigma DI$, and *MI* partitions.</caption><thead><tr><th rowspan="2"></th><th colspan="2">ELF</th><th colspan="2">Σ DI</th><th colspan="2">MI</th></tr><tr><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th><th>N&#773;<sub>A</sub><br>σ²(N&#773;<sub>A</sub>)</th></tr></thead><tbody><tr><td>K(C1)</td><td>2.09</td><td>0.26</td><td>2.05</td><td>0.26</td><td>2.04</td><td>0.25</td></tr><tr><td>K(C2)</td><td>2.09</td><td>0.25</td><td>2.04</td><td>0.24</td><td>2.02</td><td>0.24</td></tr><tr><td>K(O1)</td><td>2.14</td><td>0.34</td><td>2.05</td><td>0.33</td><td>2.04</td><td>0.33</td></tr><tr><td>K(O2)</td><td>2.15</td><td>0.35</td><td>2.08</td><td>0.33</td><td>2.06</td><td>0.33</td></tr><tr><td>V(C1, H1)</td><td>2.01</td><td>0.64</td><td>1.90</td><td>0.62</td><td>1.94</td><td>0.63</td></tr><tr><td>V(C1, H2)<sup>a</sup></td><td>1.97</td><td>0.65</td><td>1.94</td><td>0.64</td><td>1.93</td><td>0.63</td></tr><tr><td>V(C1, C2)</td><td>2.09</td><td>1.03</td><td>2.21</td><td>1.07</td><td>2.25</td><td>1.09</td></tr><tr><td>V(C2, O1)<sup>a</sup></td><td>1.21</td><td>0.82</td><td>1.83</td><td>1.07</td><td>1.79</td><td>1.06</td></tr><tr><td>V(C2, O2)</td><td>1.73</td><td>1.02</td><td>2.01</td><td>1.12</td><td>2.0'</td><td>1.14</td></tr><tr><td>V<sub>1</sub>(O1)</td><td>2.48</td><td>1.15</td><td>2.02</td><td>1.00</td><td>2.06</td><td>1.01</td></tr><tr><td>V<sub>2</sub>(O1)</td><td>2.84</td><td>1.25</td><td>2.17</td><td>1.05</td><td>2.21</td><td>1.06</td></tr><tr><td>V(O2)<sup>a</sup></td><td>2.12</td><td>1.06</td><td>1.98</td><td>1.01</td><td>1.97</td><td>1.00</td></tr><tr><td>V(O2, H4)</td><td>1.77</td><td>0.80</td><td>1.90</td><td>0.84</td><td>1.93</td><td>0.8</td></tr><tr><td>MI × 10<sup>3</sup></td><td colspan="2">3.427</td><td colspan="2">3.488</td><td colspan="2">3.491</td></tr><tr><td>C1<br>valence<br>shell</td><td colspan="2">8.04</td><td colspan="2">8.02</td><td colspan="2">8.05</td></tr><tr><td>C2<br>valence<br>shell</td><td colspan="2">7.03</td><td colspan="2">7.93</td><td colspan="2">7.93</td></tr><tr><td>O1<br>valence<br>shell</td><td colspan="2">7.74</td><td colspan="2">7.85</td><td colspan="2">7.87</td></tr><tr><td>O2<br>valence<br>shell</td><td colspan="2">7.74</td><td colspan="2">7.88</td><td colspan="2">7.90</td></tr></tbody></table>

### 7.1.2 | Merging Electron Group Localization Volumes

Usually, only a single part of a molecule will be examined in depth, so it's best to treat the other components in a more comprehensive manner for clarity's sake. In the case of acetic acid, the methyl will be considered as a whole. This can be done by merging the K(C1), V(C1, H1), V(C1, H2), and V(C1, H3) volumes after the optimization process, providing references, at the guess level or by the removal of the basis functions centred on the methyl atoms. Table 3 compares the populations calculated by these three approaches. A very good agreement with the reference is obtained when the volumes are merged at the guess construction level. More surprising and encouraging

is the success of the basis function removal approach, which saves a noticeable computational effort, giving, therefore, access to the study of a small group of atoms within a large system.

### 7.1.3 | Atom in Molecules Guess

An alternative possibility is to start from a QTAIM partition to associate an electron group with each atom. Except for ionic bonding, this guess would be rather far from the optimized group localization volumes. It is expected that the population of an atom will consist of the core electrons, the lone pair electrons, and the bonding electrons if the considered atom is the most electronegative one involved in the bond. This expectation is verified in the case of CH₃COOH except for the hydrogen atoms, which retain two electrons in their localization volumes because these latter belong to the C and O2 valence shells. $\Sigma DI$ and *MI* yield both consistent results. The electron localization-delocalization (LDMs) matrices have been introduced by C. F. Matta [177–179] in the context of the Quantum Theory of Atoms in Molecules as a mathematical object capturing electron distributions. They are originally expressed in terms of Bader's localization indexes [137]. However, they appear to be the difference between the atomic populations' diagonal matrix and the covariance matrix. Table 4 compares the LDMs obtained from the standard QTAIM partition and the *MI* optimized one. The largest differences occur for the C1 and hydrogen atoms diagonal elements and the methyl H – H'off-diagonal ones.

## 7.2 | Selected Examples

### 7.2.1 | Polycyclic Aromatic Hydrocarbons

The ability of the program to handle rather large systems is illustrated by a series of B3LYP/6-31G** calculations carried out on four polycyclic aromatic hydrocarbons: naphthalene (C₁₀H₈), phenalene (C₁₂H₁₀), phenanthrene (C₁₄H₁₀), and anthracene (C₁₄H₁₀). The $\Sigma DI$ and *MI* group localization volumes have been determined from an *ELF* initial guess, and the C-C bond populations are presented in Figure 4. The small increase of these latter populations obtained with both $\Sigma DI$ and *MI* with respect to *ELF* is correlated to a decrease in those of the C-H bonds. The calculations have been performed on rack servers with Intel Xeon 4110 8-core 2.1 GHz processors. The computer resources necessary for each PAH are given in Table 5. The use of the molecular symmetry leads to significant savings: Although anthracene and phenanthrene have the same stoichiometry this latter requires three times more memory and CPU time than the former.

### 7.2.2 | Propellanes

Propellanes have been investigated in an *ELF* study [180] of the bridging bonds, which were shown to be correlated with the ring strain. The B3LYP/cc-pVTZ calculations on [1.1.1], [2.1.1], [2.2.1], [3.1.1] and [3.2.1] propellanes have been repeated with our new program to support or invalidate the conclusions of the *ELF* study. The populations of the bridging bond V(C, C')

<footnote><sup>a</sup>Volume above the $\sigma_h$ plane.</footnote>
<figure><img src="image_5.png" /><figcaption>**FIGURE 3** | Number probability distributions in V(C2, O1) and V(O2) group localization volume. Blue: ELF; Red: $\Sigma DI$; and Yellow: MI.</figcaption></figure>

<table><caption>**TABLE 2** | Group localization volume populations calculated with different basis sets.</caption><thead><tr><th></th><th>STO-3G</th><th>STO-6G</th><th>3-21G</th><th>6-31G(dp)</th><th>cc-pVTZ</th></tr></thead><tbody><tr><td>K(C1)</td><td>2.02</td><td>2.05</td><td>2.06</td><td>2.06</td><td>2.06</td></tr><tr><td>K(C2)</td><td>2.02</td><td>2.03</td><td>2.05</td><td>2.04</td><td>2.04</td></tr><tr><td>K(O1)</td><td>2.02</td><td>2.04</td><td>2.06</td><td>2.05</td><td>2.05</td></tr><tr><td>K(O2)</td><td>2.02</td><td>2.04</td><td>2.07</td><td>2.08</td><td>2.08</td></tr><tr><td>V(C1, H1)</td><td>1.95</td><td>1.94</td><td>1.95</td><td>1.96</td><td>1.95</td></tr><tr><td>V(C1, H2)<sup>a</sup></td><td>1.94</td><td>1.94</td><td>1.94</td><td>1.95</td><td>1.94</td></tr><tr><td>V(C1, C2)</td><td>2.26</td><td>2.26</td><td>2.22</td><td>2.16</td><td>2.16</td></tr><tr><td>V(C2, O1)<sup>a</sup></td><td>1.76</td><td>1.76</td><td>1.78</td><td>1.83</td><td>1.83</td></tr><tr><td>V(C2, O2)</td><td>1.95</td><td>1.96</td><td>1.92</td><td>1.97</td><td>1.98</td></tr><tr><td>V<sub>1</sub>(O1)</td><td>2.11</td><td>2.10</td><td>2.04</td><td>2.03</td><td>2.02</td></tr><tr><td>V<sub>2</sub>(O1)</td><td>2.22</td><td>2.21</td><td>2.22</td><td>2.18</td><td>2.18</td></tr><tr><td>V(O2)<sup>a</sup></td><td>2.10</td><td>2.06</td><td>2.96</td><td>2.01</td><td>2.00</td></tr><tr><td>V(O2, H4)</td><td>1.79</td><td>1.78</td><td>1.84</td><td>1.91</td><td>1.91</td></tr></tbody></table>

<table><caption>**TABLE 3** | $\Sigma DI$ and MI group localization volume populations, calculated with three merging schemes. I: Sum of Table 1 values, II: Merging at the guess level, and III: Removal of basis functions.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Σ DI</th><th colspan="3">MI</th></tr><tr><th>I</th><th>II</th><th>III</th><th>I</th><th>II</th><th>III</th></tr></thead><tbody><tr><td>CH<sub>3</sub></td><td>7.88</td><td>7.87</td><td></td><td>7.85</td><td>7.91</td><td></td></tr><tr><td>V(C1, C2)</td><td>2.20</td><td>2.22</td><td></td><td>2.24</td><td>2.18</td><td></td></tr><tr><td>V(C2, O1)<sup>a</sup></td><td>1.82</td><td>1.82</td><td>1.84</td><td>1.78</td><td>1.78</td><td>1.84</td></tr><tr><td>V(C2, O2)</td><td>2.09</td><td>2.10</td><td>2.14</td><td>2.13</td><td>2.13</td><td>2.20</td></tr><tr><td>V<sub>1</sub>(O1)</td><td>2.04</td><td>2.05</td><td>2.08</td><td>2.10</td><td>2.10</td><td>2.14</td></tr><tr><td>V<sub>2</sub>(O1)</td><td>2.17</td><td>2.15</td><td>2.16</td><td>2.21</td><td>2.20</td><td>2.18</td></tr><tr><td>V(O2)<sup>a</sup></td><td>1.95</td><td>1.94</td><td>1.95</td><td>1.94</td><td>1.94</td><td>1.95</td></tr><tr><td>V(O2, H4)</td><td>1.89</td><td>1.89</td><td>1.90</td><td>1.89</td><td>1.90</td><td>1.91</td></tr></tbody></table>

localization volume calculated with the $\Sigma DI$ and MI partitions listed in Table 6 are significantly larger than those calculated with the ELF partition for the three molecules which have V(C) volumes. There is, nevertheless, an overall qualitative agreement that support the conclusions of Polo et al. [180], cc-pVTZ, aug-cc-pVTZ [176, 181].

### 7.2.3 | M···C ≡ O Complexes

Carbon monoxide is a molecule for which a large difference between the ELF populations and the $\Sigma DI$ and MI ones are large enough to modify the interpretation of the bonding [142]. The ELF populations of the V(C,O) and V(O) basins are respectively 3.01 and 4.22 whereas one finds 4.06 and 3.19 with the $\Sigma DI$ partition, 4.08 and 3.19 with the MI one. This makes questionable the interpretation of the bonding in complexes involving carbonyl drawn from the ELF analysis and more particularly on transition metal 1:1 complexes [182]. This latter article proposed occupation schemes in terms of the C(M), V(M), and V(C, M)

<footnote><sup>a</sup>Volume above the $\sigma_h$ plane.</footnote>

<footnote><sup>a</sup>Volume above the $\sigma_h$ plane.</footnote>
<table><caption>TABLE 4 | QTAIM and *MI* partition electron localization-delocalization matrices of CH₃COOH.</caption><thead><tr><th>QTAIM</th><th>H<sub>1</sub></th><th>C<sub>1</sub></th><th>C<sub>2</sub></th><th>O<sub>1</sub></th><th>H<sub>2</sub></th><th>H<sub>3</sub></th><th>O<sub>2</sub></th><th>H<sub>4</sub></th><th>N&#773;</th></tr></thead><tbody><tr><td>H<sub>1</sub></td><td>0.41</td><td>0.48</td><td>0.02</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.00</td><td>0.95</td></tr><tr><td>C<sub>1</sub></td><td>0.48</td><td>4.03</td><td>0.47</td><td>0.06</td><td>0.47</td><td>0.47</td><td>0.05</td><td>0.00</td><td>6.03</td></tr><tr><td>C<sub>2</sub></td><td>0.02</td><td>0.47</td><td>2.81</td><td>0.64</td><td>0.02</td><td>0.02</td><td>0.43</td><td>0.00</td><td>4.41</td></tr><tr><td>O<sub>1</sub></td><td>0.01</td><td>0.06</td><td>0.64</td><td>8.30</td><td>0.01</td><td>0.01</td><td>0.16</td><td>0.01</td><td>9.19</td></tr><tr><td>H<sub>2</sub></td><td>0.01</td><td>0.47</td><td>0.02</td><td>0.01</td><td>0.40</td><td>0.01</td><td>0.01</td><td>0.00</td><td>0.94</td></tr><tr><td>H<sub>3</sub></td><td>0.01</td><td>0.47</td><td>0.02</td><td>0.01</td><td>0.01</td><td>0.40</td><td>0.01</td><td>0.00</td><td>0.94</td></tr><tr><td>O<sub>2</sub></td><td>0.01</td><td>0.05</td><td>0.43</td><td>0.16</td><td>0.01</td><td>0.01</td><td>8.26</td><td>0.27</td><td>9.19</td></tr><tr><td>H<sub>4</sub></td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.01</td><td>0.00</td><td>0.00</td><td>0.27</td><td>0.05</td><td>0.33</td></tr><tr><td>MI</td><td>H<sub>1</sub></td><td>C<sub>1</sub></td><td>C<sub>2</sub></td><td>O<sub>1</sub></td><td>H<sub>2</sub></td><td>H<sub>3</sub></td><td>O<sub>2</sub></td><td>H<sub>4</sub></td><td>N&#773;</td></tr><tr><td>H<sub>1</sub></td><td>1.36</td><td>0.06</td><td>0.15</td><td>0.02</td><td>0.19</td><td>0.19</td><td>0.03</td><td>0.00</td><td>2.01</td></tr><tr><td>C<sub>1</sub></td><td>0.06</td><td>1.79</td><td>0.05</td><td>0.00</td><td>0.06</td><td>0.06</td><td>0.00</td><td>0.00</td><td>2.04</td></tr><tr><td>C<sub>2</sub></td><td>0.15</td><td>0.05</td><td>2.95</td><td>0.35</td><td>0.16</td><td>0.16</td><td>0.21</td><td>0.02</td><td>4.05</td></tr><tr><td>O<sub>1</sub></td><td>0.02</td><td>0.00</td><td>0.35</td><td>9.15</td><td>0.03</td><td>0.03</td><td>0.30</td><td>0.05</td><td>9.94</td></tr><tr><td>H<sub>2</sub></td><td>0.19</td><td>0.06</td><td>0.16</td><td>0.03</td><td>1.35</td><td>0.19</td><td>0.01</td><td>0.00</td><td>2.00</td></tr><tr><td>H<sub>3</sub></td><td>0.19</td><td>0.06</td><td>0.16</td><td>0.03</td><td>0.19</td><td>1.35</td><td>0.01</td><td>0.00</td><td>2.00</td></tr><tr><td>O<sub>2</sub></td><td>0.03</td><td>0.00</td><td>0.21</td><td>0.30</td><td>0.01</td><td>0.01</td><td>6.41</td><td>0.82</td><td>7.79</td></tr><tr><td>H<sub>4</sub></td><td>0.00</td><td>0.00</td><td>0.02</td><td>0.05</td><td>0.00</td><td>0.00</td><td>0.82</td><td>1.26</td><td>2.16</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>FIGURE 4 | V(C, C') populations of naphtalene, antracene, phenalene, and phenanthrene. Black: ELF; Blue: Σ DI; and Red: MI.</figcaption></figure>
<table><caption>**TABLE 5** | Number of electrons *N*, memory requirement (Gb), mesh size, and CPU cost for PAH calculations.</caption><thead><tr><th>Molecule</th><th>N</th><th>RAM</th><th>Mesh size</th><th>CPU</th></tr></thead><tbody><tr><td>Naphthalene</td><td>68</td><td>3</td><td>256 × 216 × 102</td><td>01:26</td></tr><tr><td>Anthracene</td><td>94</td><td>7</td><td>276 × 196 × 102</td><td>05:28</td></tr><tr><td>Phenanthrene</td><td>94</td><td>22</td><td>298 × 227 × 102</td><td>18:22</td></tr><tr><td>Phenalene</td><td>88</td><td>22</td><td>256 × 259 × 154</td><td>31:43</td></tr></tbody></table>

<table><caption>**TABLE 6** | V(C, C') (redbrick) and V(C) (yellow) group localization volumes and populations of propellanes.</caption><thead><tr><th></th><th>[1.1.1]</th><th>[2.1.1]</th><th>[2.2.1]</th><th>[3.1.1]</th><th>[3.2.1]</th></tr></thead><tbody><tr><td>V(C, C')</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ELF</td><td>0.13</td><td>0.25</td><td>1.17</td><td>0.89</td><td>1.83</td></tr><tr><td>Σ DI</td><td>1.06</td><td>1.07</td><td>1.58</td><td>1.35</td><td>1.75</td></tr><tr><td>MI</td><td>0.90</td><td>0.96</td><td>1.51</td><td>1.34</td><td>1.74</td></tr><tr><td>V(C)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ELF</td><td>1.21</td><td>0.99</td><td></td><td>0.51</td><td></td></tr><tr><td>Σ DI</td><td>1.17</td><td>1.07</td><td></td><td>0.90</td><td></td></tr><tr><td>MI</td><td>1.12</td><td>1.01</td><td></td><td>0.76</td><td></td></tr></tbody></table>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

populations and the net charge transfer to the ligand. The electron group populations reported in Table 7 show significant differences between the ELF partition and the Σ DI and MI ones.

For ScCO, TiCO³Δ, MnCO, CrCO⁷Σ⁺, FeCO, and CuCO complexes, there is an overall agreement which qualitatively preserves the previous interpretation. All systems except NiCO¹Σ⁺ have Σ DI and MI V(M) populations at least of the order of 1.0 and larger than the ELF ones. This effect is important for those systems where the ELF V(M) population is around 0.5. Consider, for example, the VCO⁶Σ complex where the V(M) population of the ELF guess equals 0.21 with a variance of 0.17, after minimization of the OTF, the Σ DI population is 1.09 and σ² = 0.70 therefore the V(M) contribution to Σ DI decreases by 0.17/0.21 - 0.10/1.09 = 0.16. Moreover, the electron number probabilities become more consistent with the picture of a single electron group associated with V(M): With ELF these probabilities are 0.64 and 0.31 for 0 and 1, whereas one gets 0.25, 0.49, and 0.22 for 0, 1, and 2 with Σ DI. For NiCO¹Σ⁺ the V(M) volume present in the ELF disappears during the minimization process.

### 7.2.4 | Hydrogen Bonded Systems

The study of hydrogen-bonded complexes by the ELF approach [183, 184] has revealed correlations of the complexation energy

and the proton donor stretching frequency shift with η<sub>vv'</sub>, the ELF value at the saddle point between the protonated valence basin of the Lewis acid and the valence basin of the base. It had been further shown that η<sub>vv'</sub> varies as the square of the variance of the proton donor electron population [185]. The correlation of the hydrogen bond strength with the variance of the electron populations of the proton donor and acceptor molecules support the interpretation of Weinhold and Klein [186, 187] identifying the resonance along the hydrogen bridge as a dominant contribution to the “nature” of the hydrogen bond. The partition of the position space in two volumes, V<sub>AH</sub> and V<sub>B</sub>, corresponding to the proton donor and acceptor moieties provides a very simple model. Once the optimization of the boundary has been done, we can reasonably assume that the electron density transfer towards the acid is very small and therefore:

$$ 
\begin{align} 
\overline{N}_{AH} &= (N_{AH} - 1)P_{N_{AH}-1} + N_{AH}P_{N_{AH}} \nonumber \\ 
&\quad + (N_{AH} + 1)P_{N_{AH}+1} \approx N_{AH} \nonumber \\ 
\sigma^2(\overline{N}_{AH}) &= P_{N_{AH}-1}(N_{AH} - 1 - \overline{N}_{AH})^2 \tag{23} \\ 
&\quad + P_{N_{AH}}(N_{AH} - \overline{N}_{AH})^2 + P_{N_{AH}+1}(N_{AH} + 1 - \overline{N}_{AH})^2 \nonumber \\ 
&\approx P_{N_{AH}-1} + P_{N_{AH}+1} = 1 - P_{N_{AH}} = 1 - P_{N_B} \nonumber 
\end{align} 
$$

where N<sub>AH</sub> and N<sub>B</sub> are the numbers of electron of the isolated proton donor and acceptor molecules. As P<sub>N<sub>AH</sub>-1</sub> = P<sub>N<sub>B</sub>+1</sub> and
<table><caption>TABLE 7 | B3LYP/6-311G(2d) C(C), V(C), and V(C, M) group localization volume populations.</caption><thead><tr><th rowspan="2"></th><th colspan="3">C(M)</th><th colspan="3">V(M)</th><th colspan="3">V(C,M)</th></tr><tr><th>ELF</th><th>Σ DI</th><th>MI</th><th>ELF</th><th>Σ DI</th><th>MI</th><th>ELF</th><th>Σ DI</th><th>MI</th></tr></thead><tbody><tr><td>ScCO<sup>4</sup>Σ<sup>-</sup></td><td>19.15</td><td>19.08</td><td>19.18</td><td>0.76</td><td>0.93</td><td>0.84</td><td>3.45</td><td>3.19</td><td>3.18</td></tr><tr><td>ScCO<sup>2</sup>Π</td><td>18.68</td><td>18.50</td><td>18.71</td><td>1.76</td><td>1.88</td><td>1.72</td><td>2.90</td><td>2.94</td><td>2.80</td></tr><tr><td>TiCO<sup>5</sup>Δ</td><td>20.65</td><td>20.13</td><td>20.26</td><td>0.41</td><td>1.01</td><td>0.88</td><td>3.37</td><td>3.09</td><td>3.08</td></tr><tr><td>TiCO<sup>3</sup>Δ</td><td>19.80</td><td>19.62</td><td>19.68</td><td>1.41</td><td>1.76</td><td>1.71</td><td>3.17</td><td>2.81</td><td>2.79</td></tr><tr><td>VCO<sup>6</sup>Σ</td><td>21.83</td><td>21.09</td><td>21.24</td><td>0.21</td><td>1.09</td><td>0.97</td><td>3.26</td><td>2.96</td><td>2.93</td></tr><tr><td>VCO<sup>4</sup>Δ</td><td>21.35</td><td>20.99</td><td>21.13</td><td>0.44</td><td>0.94</td><td>0.82</td><td>3.60</td><td>3.31</td><td>3.29</td></tr><tr><td>CrCO<sup>7</sup>A'</td><td>22.99</td><td>22.53</td><td>22.70</td><td>0.38</td><td>0.90</td><td>0.83</td><td>3.16</td><td>2.72</td><td>2.69</td></tr><tr><td>CrCO<sup>7</sup>Σ<sup>+</sup></td><td>22.57</td><td>22.53</td><td>22.69</td><td>0.90</td><td>0.97</td><td>0.82</td><td>2.99</td><td>2.70</td><td>2.67</td></tr><tr><td>MnCO<sup>6</sup>Π</td><td>23.22</td><td>23.24</td><td>23.40</td><td>1.09</td><td>0.98</td><td>0.84</td><td>3.13</td><td>3.04</td><td>3.01</td></tr><tr><td>MnCO<sup>4</sup>Π</td><td>22.95</td><td>22.66</td><td>22.81</td><td>1.14</td><td>1.13</td><td>1.05</td><td>3.82:</td><td>3.37</td><td>3.30</td></tr><tr><td>FeCO<sup>3</sup>Σ<sup>-</sup></td><td>23.79</td><td>23.57</td><td>23.64</td><td>1.07</td><td>1.63</td><td>1.55</td><td>3.46</td><td>3.03</td><td>2.99</td></tr><tr><td>FeCO<sup>5</sup>Σ<sup>-</sup></td><td>24.45</td><td>24.32</td><td>24.50</td><td>0.53</td><td>0.96</td><td>0.81</td><td>3.40</td><td>2.94</td><td>2.92</td></tr><tr><td>CoCO<sup>2</sup>Δ</td><td>25.53</td><td>24.52</td><td>24.76</td><td>0.41</td><td>1.74</td><td>1.61</td><td>3.41</td><td>3.00</td><td>2.94</td></tr><tr><td>CoCO<sup>4</sup>A'</td><td>25.64</td><td>25.51</td><td>25.68</td><td>0.48</td><td>0.90</td><td>0.75</td><td>3.34</td><td>2.83</td><td>2.80</td></tr><tr><td>NiCO<sup>1</sup>Σ<sup>+</sup></td><td>26.70</td><td>27.17</td><td>27.27</td><td>0.24</td><td></td><td></td><td>3.38</td><td>3.09</td><td>2.97</td></tr><tr><td>NiCO<sup>3</sup>Δ</td><td>26.69</td><td>26.39</td><td>26.54</td><td>0.68</td><td>1.16</td><td>1.04</td><td>3.04</td><td>2.72</td><td>2.65</td></tr><tr><td>CuCO<sup>2</sup>A'</td><td>27.87</td><td>27.65</td><td>27.78</td><td>0.52</td><td>0.92</td><td>0.80</td><td>3.12</td><td>2.69</td><td>2.67</td></tr><tr><td>CuCO<sup>2</sup>Σ<sup>+</sup></td><td>27.80</td><td>27.59</td><td>27.75</td><td>0.65</td><td>0.97</td><td>0.85</td><td>2.94</td><td>2.68</td><td>2.62</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>FIGURE 5 | Left: FH frequency shift versus $\sigma^2(\overline{N}_{FH})$; Right: Hydrogen bond dissociation energy versus $\sigma^2(\overline{N}_{FH})$.</figcaption></figure>

$P_{N_{AH}+1} = P_{N_B-1}$, the estimate contribution of delocalization to the interaction energy is:

$$E_{\text{deloc}} = - \frac{1 - P_{N_{AH}}}{R_e} = - \frac{\sigma^2(\overline{N}_{AH})}{R_e} \quad (24)$$

The intermolecular distance $R_e$ is taken as the internuclear separation of the atoms bridged by the hydrogen bond and their

values are in the range $R_e = 3.0 \pm 0.5$ Å. Equation (24) indicates that $E_{\text{deloc}}$ should be roughly proportional to $\sigma^2(\overline{N}_{AH})$, if $E_{\text{deloc}}$ is the dominant contribution to the bond dissociation energy; this latter would have the same behavior.

B3LYP/aug-ccpVTZ calculations have been carried out on a series of hydrogen-bonded complexes FH...B, where B = N₂, HF, HCl, CO, H₂O and NH₃ such as to cover a large range of H-bond
<table><caption>**TABLE 8** | FH population, $\overline{N}$, variance $\sigma^2$, and electron number ptobailities $P_i$, $i = 9, 10, 11$.</caption><thead><tr><th></th><th>$\overline{N}$</th><th>$\sigma^2$</th><th>$P_9$</th><th>$P_{10}$</th><th>$P_{11}$</th></tr></thead><tbody><tr><td>FH...N<sub>2</sub></td><td>10.0</td><td>0.057</td><td>0.03</td><td>0.94</td><td>0.03</td></tr><tr><td>FH...FH</td><td>10.0</td><td>0.065</td><td>0.03</td><td>0.93</td><td>0.03</td></tr><tr><td>FH...ClH</td><td>10.01</td><td>0.076</td><td>0.03</td><td>0.92</td><td>0.04</td></tr><tr><td>FH...CO</td><td>10.01</td><td>0.083</td><td>0.04</td><td>0.92</td><td>0.04</td></tr><tr><td>FH...H<sub>2</sub>O</td><td>10.01</td><td>0.108</td><td>0.04</td><td>0.90</td><td>0.06</td></tr><tr><td>FH...NH<sub>3</sub></td><td>10.03</td><td>0.139</td><td>0.05</td><td>0.87</td><td>0.08</td></tr></tbody></table>

strengths. The guess is built as the union of the atomic basins of each molecule of the complex. The optimizations of the electron group localization volumes done either with $\Sigma DI$ or $MI$ yield identical results, which corresponds to the partition minimizing the variance of the volume populations. Table 8 displays the population, the variance, and the number probabilities of the FH moiety, whereas Figure 5 shows the dependence of the hydrogen bond dissociation energy and the proton donor stretching frequency shift upon the variance of the proton donor electron population. Both methods minimize the variance and yield a partition in which the density transfer from the Lewis base is found to be almost negligible. The values listed in Table 8 validate the hypothesis made above. As expected from previous studies [183, 185], the left side of Figure 5 shows frequency shifts correlated with the square of the variance. On the right side, the bond dissociation energy appears to be a linear function of $\sigma^2(\overline{N}_{AH})$. The least squares linear regression yields a correlation coefficient $r = 0.92$ and a slope equal to $515 \text{ kJmol}^{-1}\AA^{-1}$, a value corresponding to $R_e = 2.74\AA$. These values give support to the conclusions drawn from Equation (24) and to the hypothesis that $E_{deloc}$ is a driving contribution to the bond dissociation energy.

### 7.2.5 | Reaction Mechanisms

The study of reaction mechanisms in the ELF approach has given rise to an abundant literature since the article of Krokidis et al. [188] in which the evolution of the gradient field of ELF is analyzed with the help of Thom's Catastrophe theory [189]. The bonding evolution is accounted for by the number and the nature of the ELF basins. In the variational approach, the number of electron group localization volumes depends on the user's guess choice, and therefore the ELF procedure cannot be directly transposed, it is nevertheless possible to follow the electron density flows along the reaction coordinate. The proton transfer mechanism has been the subject of several studies [188, 190–192] which suggest a cleavage of the covalent A–H bond followed by the formation of a covalent B–H bond, AH and B standing for the proton donor and acceptor moieties. The strong hydrogen bond complex FH–OH$^-$ provides an example of incomplete proton transfer characterized by a V(H) basin in the ELF picture. The proton transfer reaction has been modeled within the bonding evolution theory [188] framework showing that the mechanism is governed by two geometrical (control space) parameters, the

F–O and F–H distances [192]. For a short enough F-O separations, the energy curve along the F-H distance presents a single minimum corresponding to the detachment of the proton from the hydrogen fluoride group, whereas at larger F-O values two minima, FH–OH$^-$ and F$^-$–H$_2$O are found. In contrast with ELF, which is a scalar function defined on $\mathbb{R}^3$, neither $\Sigma DI$ nor $MI$ enables the construction of a mathematical model similar to that proposed by Krokidis [188] which provides a reaction mechanism. Instead, one can check the validity of a proposed representation along a trajectory linking two chemically distinct nuclear configurations. A series of B3LYP/6-311G(d,p) calculations has been carried out on the FH···OH$^-$ system. The F-O distance being fixed at 2.80Å, the starting and final geometries correspond to the FH–OH$^-$ and F···H$_2$O partially optimized structures, and the linking pathway is built for a series of $r(FH)$ distances. The donor-acceptor description with two groups of formally 10 electrons is expected to be valid for the starting and ending structures. $\Sigma DI$ and $MI$ give identical results with group populations ranging in the 9.90–1.10 interval all along the trajectory, the largest charge transfer being calculated for the FH–OH$^-$ minimum. Surprisingly, the variance of the group populations and the charge transfer are negatively correlated, at $r(FH) = 1.4\AA$ the calculated charge transfer is only $0.2 \ e^-$ whereas the variance is about 1.7 times that calculated for the F···H$_2$O structure. This result suggests that the populations of the valence shells of the heavy atoms are almost constants along the trajectory. This expectation is verified by the $\Sigma DI$ and $MI$ populations calculated for the Lewis initial guess. The bonding nature of two localization volumes changes at $r(FH) \sim 1.35\AA$. The $MI$ population of the V(FH)/F(F) localization volume which is $2.05 \pm 0.03 \ e^-$ below this value decreases to $1.90 \pm 0.04 \ e^-$ at larger distances whereas that of V(O)/V(OH) amounts respectively 1.87 and 1.66 $e^-$ for the two sets of geometries. Moreover, the surface separating the V(FH)/V(F) and the V(O)/V(OH) volume almost coincides with that of limiting the donor and acceptor group volumes. In complementary calculations carried out at $r(HF) = 1.30\AA$ and $r = 1.40\AA$, the V(FH)/V(F) and V(O)/V(OH) localization volumes are successively cleaved to add a new volume to the Lewis guess set. The resulting optimized localization volumes are independent of the side where the cleavage occurs, and the boundary surface remains close to that found for the Lewis guess. The population of the V(FH) volume decreases of about $0.4 \ e^-$ and a new V(F) volume with a population of $1.30 \ e^-$ is created mostly at the expense of the other V(F) volumes. These results do not support the proton transfer mechanism involving the dissociation of the FH bond followed by the formation of the OH one; instead, the electron groups appear to be hardly sensitive to the location of the proton

## 8 | Perspectives

The method described in this paper, as well as many other population analyses, aims to interpret the one- and two-electron density of probability functions in terms of traditional structural chemistry concepts. This necessarily implies a reductionist approach in which the molecular system is decomposed into interacting subsystems characterized by their number of electrons. Unlike structural chemistry models, population analysis methods have neither the ambition to be explanatory nor predictive. Instead, they have to be considered as tools enabling the
falsification of the constituent assumptions of the models. In the development of our method, we have considered as mandatory the consistency, at least formal, with Quantum Mechanics. However, many issues can be identified and have to be addressed in this last section.

As already discussed, our choice of a partition scheme in terms of non-overlapping space-filling volumes has been guided by the parsimony principle. This choice yields a formal representation of the spatial distribution of the electrons over these latter volumes in terms of multivariate multinomial distributions, which preserves the additivity of the electron count. Is it then possible to assign a chemical meaning to the electron density enclosed in these volumes? For arbitrary volumes, the answer is no. Nevertheless, we assume possible to find correspondences with chemical objects if a series of conditions are fulfilled. For example, when the population of a given volume $V_A$ is close to 2.0 and $p_2(V_A)$ is the largest electron number probability, $V_A$ can be considered as an approximation of an electron pair domain in the sense of the VSEPR model. This seems reasonable, but it is possible to define alternative conditions, such as the requirement that the volume population be exactly 2. In our work, we have achieved the global optimization of the volumes with the help of an objective target function supposed to account for the correspondence conditions before tuning the volume boundaries. The choice of the minimized target function is another difficulty, all the more problematic because our knowledge and practice of statistics and information theory are relatively basic. Moreover, the optimization process requires an initial guess, which is inferred from the chemical structure model we intend to corroborate or invalidate. There is no evidence that the result is a global minimum. If other minima exist, we should examine them to propose alternative structural models. Unlike partitioning methods based on dynamical system theory, the present approach is not able to provide any definite number of volumes. The Mutual Information upper bound is given by the continuous distribution, in other words, by a collection of infinitesimal volumes without any particular signification in the context of structural chemistry. Perfect pairing models provide an indication that volumes optimized with a Lewis initial guess yield a *MI* value close to the upper bound; however, for real systems, we can at least expect an asymptotic behavior, such as exhibited in the case of the water molecules. The Lewis partition in five electron pair volumes (oxygen core, two oxygen lone pairs, and two HO bonds) yields $MI = 0.74710^{-2}$, halving the OH bond volumes by the $\sigma_h$ plane increases *MI* by $0.0310^{-2}$ and a further division of the oxygen lone pair volumes by the $\sigma_v$ plane increases *MI* to $0.80510^{-2}$. Such variations of *MI* are not negligible and raise the question of the meaning of a single electron partition of the valence density. Therefore, it remains preferable to confine the use of the population analysis method to falsification purposes for which we can't rule out the risk that our method is self-fulfilling.

Although these remarks highlight the vanity of our ambition, they nonetheless suggest many ways to deepen and enhance this method. First, it would be advantageous to clarify the contents of many concepts used in chemistry to improve their definition and eliminate any inconsistencies with quantum mechanics. Our statistical approach may be helpful for this

purpose, but we must also improve the definition of objects that are inherent to the method, such as electron group localization volumes. A related point concerns the interpretation of the results and, more particularly, the definition of criteria allowing one to say whether a volume is indeed a group localization volume and, if so, of a group of how many electrons. The dispersion index alone does not always enable a conclusion. For example, in the case of benzene, the dispersion indexes calculated for the populations of the CH and CC bond volumes are both less than 0.5, clearly characterizing underdispersed distributions. However, the probability numbers may tell a different story. For CH, the distribution is almost symmetrical and clearly centred on $n=2$ ($p_1 = 0.21, p_2 = 0.55, p_3 = 0.19$) and thus the volume can be unambiguously assigned to a group of two electrons. For CC, the distribution is spread around $n=3$ with $p_1 = 0.08, p_2 = 0.26, p_3 = 0.34, p_4 = 0.21, p_5 = 0.08$. Although the assignment to a group of three electrons cannot be excluded, the superposition of two groups, respectively of two and four electrons, is realistic and compatible with the concept of mesomerism. Developing a quantitative approach to mesomerism in the context of our method would certainly be a significant improvement.

It is necessary to consider further technical improvements to our tool. The search for global maxima is an objective that could be achieved by the randomization of the trial-and-error algorithm coupled with the simulated annealing technique. The use of adaptive grids is likely to increase the accuracy of the determination of the surfaces bounding the group localization volumes. Finally, the calculation of the mutual information in the continuous distribution limit can provide very useful information to assess the quality of the actual partition.

The input Cartesian coordinates of the PAH and propellane molecules are provided as Supporting Information.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

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45. C. H. Suresh, N. Koga, and S. R. Gadre, "Molecular Electrostatic Potential and Electron Density Topography: Structure and Reactivity of (Substituted arene)Cr(CO)3Complexes," *Organometallics* 19 (2000): 3008-3015.

46. S. R. Gadre, K. Babu, and A. P. Rendell, "Electrostatics for Exploring Hydration Patterns of Molecules. 3. Uracil," *Journal of Physical Chemistry. A* 104 (2000): 8976-8982.

47. S. S. Pingale and S. R. Gadre, "Polarization-Corrected Molecular Electrostatic Potential for the Cation Binding Problem," *Chemical Physics Letters* 340 (2001): 604-610.

48. S. P. Gejji, S. R. Gadre, and V. J. Barge, "Theoretical Investigations on Structure, Electrostatic Potentials and Vibrational Frequencies of Diglyme and Li⁺-(Diglyme) Conformers," *Chemical Physics Letters* 344 (2001): 527-535.

49. C. H. Suresh, N. Koga, and S. R. Gadre, "Revisiting Markovnikov Addition to Alkenes via Molecular Electrostatic Potential," *Journal of Organic Chemistry* 66 (2001): 6883-6890.

50. G. Mehta, S. R. Singh, P. Balanarayan, and S. R. Gadre, "Electrophilic Additions to a 2-Methylenebicyclo[2.1.1]Hexane System: Probing π-Face Selectivity for Electrostatic and Orbital Effects," *Organic Letters* 4 (2002): 2297-2300.

51. S. R. Gadre, "Foreword," *Journal of Computational Chemistry* 24 (2003): 484-495.

52. S. R. Gadre, M. M. Deshmukh, and T. Chakraborty, "Electrostatics-Guided Ab Initio Studies on Weakly Bonded Complexes of Substituted Naphthalenes," *Chemical Physics Letters* 384 (2004): 350-356.

53. A. K. Phukan, R. P. Kalagi, E. D. Gadre, and S. Rand Jemmis, "Structure, Reactivity and Aromaticity of Acenes and Their BN Analogues: A Density Functional and Electrostatic Investigation," *Inorganic Chemistry* 43 (2004): 5824-5832.

54. A. Pramanik, R. P. Kalagi, V. J. Barge, and S. R. Gadre, "Adsorption of Water on Sodium Chloride Surfaces: Electrostatics - Guided Ab Initio Studies," *Theoretical Chemistry Accounts* 114 (2005): 129-136.

55. P. Balanarayan and S. R. Gadre, "Why Are Carborane Acids So Acidic? An Electrostatic Interpretation of Brønsted Acid Strengths," *Inorganic Chemistry* 44 (2005): 9613-9615.

56. D. S. Patel, P. Daga, P. V. Bharatam, R. K. Dongare, and S. R. Gadre, "Indian Journal of Chemistry - Section A Inorganic," *Physical, Theoretical & Analytical Chemistry* 45 (2006): 13.

57. C. H. Suresh and S. R. Gadre, "Electrostatic Potential Minimum of the Aromatic Ring as a Measure of Substituent Constant," *Journal of Physical Chemistry A* 111 (2007): 710-714.

58. P. Balanarayan, R. Kavathekar, and S. R. Gadre, "Electrostatic Potential Topography for Exploring Electronic Reorganizations in 1,3 Dipolar Cycloadditions," *Journal of Physical Chemistry A* 111 (2007): 2733-2738.

59. M. M. Deshmukh, S. R. Gadre, R. Tonner, and G. Frenking, "Molecular Electrostatic Potentials of Divalent Carbon(0) Compounds," *Physical Chemistry Chemical Physics* 10 (2008): 2298-2301.

60. C. H. Suresh, P. Alexander, K. P. Vijayalakshmi, P. K. Sajith, and S. R. Gadre, "Use of Molecular Electrostatic Potential for Quantitative Assessment of Inductive Effect," *Physical Chemistry Chemical Physics* 10 (2008): 6492-6499.

61. A. D. Kulkarni, S. R. Gadre, and S. Nagase, "Quantum Chemical and Electrostatic Studies of Anionic Water Clusters,(H2O)n," *Journal of Molecular Structure (THEOCHEM)* 851 (2008): 213-219.

62. J. K. V. Jovan and S. R. Gadre, "Electrostatic Guidelines and Molecular Tailoring for Density Functional Investigation of Structures and Energetics of (Li)n Clusters," *Journal of Chemical Physics* 129 (2008): 164314.

63. D. Roy, P. Balanarayan, and S. R. Gadre, "An Appraisal of Poincaré-Hopf Relation and Application to Topography of Molecular Electrostatic Potentials," *Journal of Chemical Physics* 129 (2008): 174103.

64. K. V. J. Jose and S. R. Gadre, "Ab Initio Study on (CO2)n Clusters via Electrostatics- and Molecular Tailoring-Based Algorithm," *International Journal of Quantum Chemistry* 109 (2009): 2238-2247.

65. D. K. Roy, P. Balanarayan, and S. R. Gadre, "Signatures of Molecular Recognition From the Topography of Electrostatic Potential," *Journal of Chemical Sciences* 121 (2009): 815-821.

66. F. B. Sayyed, C. H. Suresh, and S. R. Gadre, "Appraisal of Through-Bond and Through-Space Substituent Effects via Molecular Electrostatic Potential Topography," *Journal of Physical Chemistry. A* 114 (2010): 12330-12333.

67. S. D. Yeole and S. R. Gadre, "Molecular Cluster Building Algorithm: Electrostatic Guidelines and Molecular Tailoring Approach," *Journal of Chemical Physics* 134 (2011): 084111.

68. S. Sarmah, A. K. Guha, A. Kumar, and S. R. Gadre, "Stabilization of Si(0) and Ge(0) Compounds by Different Silylenes and Germylenes: A Density Functional and Molecular Electrostatic Study," *Dalton Transactions* 42 (2013): 13200-13209.

69. N. Mohan, C. H. Suresh, A. Kumar, and S. R. Gadre, "Molecular Electrostatics for Probing Lone Pair-π Interactions," *Physical Chemistry Chemical Physics* 15 (2013): 18401-18409.

70. A. Kumar, S. R. Gadre, N. Mohan, and C. H. Suresh, "Lone Pairs: An Electrostatic Viewpoint," *Journal of Physical Chemistry. A* 118 (2014): 526-532.

71. G. Singh and S. R. Gadre, "Indian Journal of Chemistry - Section A Inorganic," *Physical, Theoretical & Analytical Chemistry* 53 (2014): 1019.

72. A. Kumar, S. R. Gadre, C. Xiao, T. Xu, S. R. Kirk, and S. Jenkins, "Hybrid QTAIM and Electrostatic Potential-Based Quantum Topology Phase Diagrams for Water Clusters," *Physical Chemistry Chemical Physics (PCCP)* 17, no. 23 (2015): 15258-15273.

73. A. Kumar and S. R. Gadre, "On the Electrostatic Nature of Electrides," *Physical Chemistry Chemical Physics* 17 (2015): 15030-15035.

74. A. Kumar, S. D. Yeole, R. López, et al., "DAMQT 2.1.0: A New Version of the DAMQT Package Enabled With the Topographical Analysis of Electron Density and Electrostatic Potential in Molecules," *Journal of Computational Chemistry* 36 (2015): 2350-2359.

75. A. Kumar and S. R. Gadre, "Molecular Electrostatic Potential-Based Atoms in Molecules: Shielding Effects and Reactivity Patterns," *Australian Journal of Chemistry* 69 (2016): 975.

76. A. Kumar and S. R. Gadre, "Exploring the Gradient Paths and Zero Flux Surfaces of Molecular Electrostatic Potential," *Journal of Chemical Theory and Computation* 12 (2016): 1705-1713.

77. S. R. Gadre and A. Kumar, "Bonding and Reactivity Patterns From Electrostatic Landscapes of Molecules," *Journal of Chemical Sciences* 128 (2016): 1519-1526.

78. R. López, J. F. Rico, G. Ramírez, et al., "Topology of Molecular Electron Density and Electrostatic Potential With DAMQT," *Computer Physics Communications* 214 (2017): 207-215.

79. P. V. Bijina, C. H. Suresh, and S. R. Gadre, "Electrostatics for Probing Lone Pairs and Their Interactions," *Journal of Computational Chemistry* 39 (2018): 488-499.

80. S. S. Pingale, A. P. Ware, and S. R. Gadre, "Unveiling Electrostatic Portraits of Quinones in Reduction and Protonation States," *Journal of Chemical Sciences* 130 (2018): 1.
81. P. Ahuja, M. Molayem, and S. R. Gadre, "Electrostatics-Assisted Building-Up Procedure for Capturing Energy Minima of Metal Clusters: Test Case of AgnClusters," *Journal of Physical Chemistry. A* 123 (2019): 7872-7880.

82. P. K. Anjalikrishna, C. H. Suresh, and S. R. Gadre, "Electrostatic Topographical Viewpoint of π-Conjugation and Aromaticity of Hydrocarbons," *Journal of Physical Chemistry. A* 123 (2019): 10139-10151.

83. P. K. Anjalikrishna, S. Gadre, and C. Suresh, "Topology of Electrostatic Potential and Electron Density Reveals a Covalent to Non-Covalent Carbon-Carbon Bond Continuum," *Physical Chemistry Chemical Physics* 25 (2023): 25191-25204.

84. P. Anjalikrishna, S. Gadre, and C. Suresh, "Electrostatic Potential for Exploring Electron Delocalization in Infinitenes, Circulenes, and Nanobelts," *Journal of Organic Chemistry* 55 (2023): 4123-4133.

85. S. R. Gadre and R. N. Shirsat, *Electrostatics of Atoms and Molecules* (Universities Press, 2000).

86. R. K. Pathak and S. R. Gadre, "Estimation of $>p≥$ and $>p-1≥$ From Atomic Electron Densities," *Journal of Chemical Physics* 74 (1981): 5925-5926.

87. S. R. Gadre and R. K. Pathak, "Direct and Reverse Transformations Between Electron Density and Electron Momentum Density," *Physical Review A* 24 (1981): 2906-2912.

88. S. R. Gadre and R. L. Matcha, "On the Monotonicity of Atomic Momentum Densities and Inequalities Among Atomic Expectation Values," *Journal of Chemical Physics* 76 (1982): 748-749.

89. S. R. Gadre, S. P. Gejji, and N. Venkatalaxmi, "Electron Momentum Distributions and Atomic(Rn)Expectation Values," *Physical Review A* 26 (1982): 1768-1770.

90. G. Sr, S. Chakravorty, and R. K. Pathak, "On the Monotonicity of the Atomic Electron Momentum Density and Shell Structure of the Radial Momentum Density," *Journal of Chemical Physics* 78 (1983): 4581-4584.

91. S. R. Gadre, S. P. Gejjoo, and R. K. Pathak, "Direct and Reverse Transformations Between Electron Density and Electron Momentum Density: Connection With the Locally Averaged Method," *Physical Review A* 28 (1983): 462-463.

92. S. R. Gadre, S. P. Gejji, and R. K. Pathak, "Electron Density to Electron Momentum Density: The Use of an Energy Constraint," *Physical Review A* 27 (1983): 3328-3331.

93. S. R. Gadre and S. P. Gejji, "Extraction of Molecular Electron Momentum Densities From Electron Density Contour Maps," *Chemical Physics Letters* 112 (1984): 45-48.

94. S. R. Gadre and S. P. Gejji, "Refinement of Electron Momentum Densities of Ionic Solids Using an Experimental Energy Constraint," *Chemical Physics Letters* 109 (1984): 584.

95. S. R. Gadre, "Information Entropy and Thomas-Fermi Theory," *Physical Review A* 30 (1984): 620-621.

96. R. K. Pathak, S. P. Gejji, and S. R. Gadre, "From Molecular Electron Density to Electron Momentum Density," *Physical Review A* 29 (1984): 3402-3405.

97. S. R. Gadre, R. D. Bendale, and S. P. Gejji, "Analysis of Atomic Electron Momentum Densities: Use of Information Entropies in Coordinate and Momentum Space," *Chemical Physics Letters* 117 (1985): 138-142.

98. S. R. Gadre and R. D. Bendale, "Maximization of Atomic Information-Entropy Sum in Configuration and Momentum Spaces," *International Journal of Quantum Chemistry* 28 (1985): 311-314.

99. T. N. G. Row and S. R. Gadre, "A Novel Approach for the Study of Intermolecular Interactions: Molecular Deformation Densities," *Proceedings - Indian Academy of Sciences, Chemical Sciences* 95 (1985): L437-L438.

100. S. R. Gadre and S. J. Chakravorty, "Interconnections Between Atomic-Electron Density and Electron-Momentum Density: Leading and Tail Corrections," *Physical Review A* 33 (1986): 1374-1377.

101. S. R. Gadre and S. J. Chakravorty, "The Average Electron Momentum Density and Rigorous Bounds to Average Electron Densities for Atoms and Molecules," *Chemical Physics Letters* 132 (1986): 535-540.

102. S. J. Chakravorty and S. R. Gadre, "Coulomb Energy, Total X-Ray Scattering Intensities and Average Electron Densities," *Chemical Physics Letters* 142 (1987): 205-208.

103. S. R. Gadre, S. A. Kulkarni, and R. K. Pathak, "Reduced First-Order Density Matrices and "Exchange-Only" Correlation Factors for Some Closed-Shell Atomic Systems," *Physical Review A* 40 (1989): 4224-4231.

104. S. R. Gadre, S. A. Kulkarni, and I. H. Shrivastava, "Cross-Entropy Minimization for Refinement of Gaussian Basis Sets," *Chemical Physics Letters* 166 (1990): 445-451.

105. R. K. Pathak, S. A. Kulkarni, and S. R. Gadre, "Momentum Space Atomic First-Order Density Matrices and "Exchange-Only" Correlation Factors," *Physical Review A* 42 (1990): 2622-2626.

106. S. R. Gadre, A. C. Limaye, and S. A. Kulkarni, *Journal of the Chemical Society. Perkin Transactions* 1 94 (1991): 8040-8046.

107. S. A. Kulkarni, S. R. Gadre, and R. K. Pathak, "Topographical View of Molecular Electron-Momentum Densities," *Physical Review A* 45 (1992): 4399-4406.

108. S. A. Kulkarni and S. R. Gadre, "On the Topography of Electron Momentum Densities of Linear Molecules," *Zeitschrift Fur Naturforschung A: A Journal of Physical Sciences* 48 (1993): 145-150.

109. S. A. Kulkarni and S. R. Gadre, "How Reliable Are Topographical Characteristics of Hartree-Fock Level Molecular Electron Momentum Densities?," *Chemical Physics Letters* 274 (1997): 255-263.

110. P. Balanarayan and S. R. Gadre, "Topography of Molecular Scalar Fields. II. An Appraisal of the Hierarchy Principle for Electron Momentum Densities," *Journal of Chemical Physics* 122 (2005): 164108.

111. P. Balanarayan and S. R. Gadre, "Atoms-In-Molecules in Momentum Space: A Hirshfeld Partitioning of Electron Momentum Densities," *Journal of Chemical Physics* 124 (2006): 104109.

112. H. Hellmann, *Einführung in Die Quantenchemie* (Franz Deuticke, 1937).

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115. A. C. Hurley, "The Electrostatic Calculation of Molecular Energies. 2. Approximate Wave Functions and Electrostatic Methods," *Proceedings of the Royal Society A* 226 (1954): 179.

116. P. Hohenberg and W. Kohn, "Inhomogeneous Electron Gas," *Physics Review* 136 (1964): B864-B871.

117. N. C. Handy and H. F. Schaefer, III, "On the Evaluation of Analytic Energy Derivatives for Correlated Wave Functions," *Journal of Chemical Physics* 81 (1984): 5031-5033.

118. P. L. A. Popelier, "Preface," *Faraday Discussions* 135 (2007): 3-5.

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123. P. L. A. Popelier and M. Rafat, "The Electrostatic Potential Generated by Topological Atoms: A Continuous Multipole Method Leading to Larger Convergence Regions," *Chemical Physics Letters* 376 (2003): 148-153.

124. A. J. Stone, "Distributed Multipole Analysis: Stability for Large Basis Sets," *Journal of Chemical Theory and Computation* 1 (2005): 1128-1132.

125. M. Rafat and P. L. A. Popelier, "The Electrostatic Potential Generated by Topological Atoms. II. Inverse Multipole Moments," *Journal of Chemical Physics* 123 (2005): 204103.

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138. R. C. Bochicchio, R. Ponec, L. Lain, and A. Torre, "Pair Population Analysis Within AIM Theory," *Journal of Physical Chemistry A* 104 (2000): 9130-9135.

139. J. Poater, X. Fradera, M. Duran, and M. Solà, "The Delocalization Index as an Electronic Aromaticity Criterion: Application to a Series of Planar Polycyclic Aromatic Hydrocarbons," *Chemistry - A European Journal* 9 (2003): 400-406.

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142. B. Silvi and E. Alikhani, "Electron Group Localization in Atoms and Molecules," *Journal of Chemical Physics* 156 (2022): 244305.

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182. J. Pilmé, B. Silvi, and M. Alikhani, "Structure and Stability of M-CO, M = First-Transition-Row Metal: An Application of Density Functional Theory and Topological Approaches," *Journal of Physical Chemistry. A* 107 (2003): 4506-4514.

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## Supporting Information

Additional supporting information can be found online in the Supporting Information section. **Data S1.** Supporting Information.

## Appendix A

### Electron Count, Pair Population and Covariance Operators

The electron count, or population operator, was introduced in 1976 by S. Diner and P. Claverie [193]; it enables the measure of the number of particles within a given volume $V_A$:

$$ \hat{N}_A = \sum_{i}^{N} \hat{y}(\mathbf{r}_i) \quad \text{with} \quad \hat{y}(\mathbf{r}_i) = \begin{cases} \hat{y}(\mathbf{r}_i) = 1 & \mathbf{r}_i \in V_A \\ \hat{y}(\mathbf{r}_i) = 0 & \mathbf{r}_i \notin V_A \end{cases} \qquad (A1) $$

where N is the total number of particles of the system. The count operator of the union of two non-overlapping volumes $V_A$ and $V_B$ is given by:

$$ \hat{N}_{A \cup B} = \sum_{i}^{N} \hat{y}(\mathbf{r}_i) \quad \text{with} \quad \hat{y}(\mathbf{r}_i) = \begin{cases} \hat{y}(\mathbf{r}_i) = 1 & \mathbf{r}_i \in V_A \lor \mathbf{r}_i \in V_B \\ \hat{y}(\mathbf{r}_i) = 0 & \mathbf{r}_i \notin V_A \land \mathbf{r}_i \notin V_B \end{cases} \qquad (A2) $$

which is not the sum $\hat{N}(V_A) + \hat{N}(V_B)$. Considering the whole space, that is, $\bigcup_A V_A$ one gets the sum rule:
$$ \hat{N}_{\cup_A A} = N \qquad (A3) $$

The eigenvalues $N_A$ of the count operator belong to the series of integers 0, ..., N, which correspond to the accessible numbers of electrons within $V_A$. They fulfill the closure sum rule:

$$ \sum_A N_A = N \qquad (A4) $$

The expectation values $\langle N_A \rangle = \overline{N_A}$ are real numbers which are the average of an infinite number of measurements and also fulfils the closure relation:

$$ \sum_A \overline{N_A} = N \qquad (A5) $$

The pair population operator $\Pi_{AB}$ (Equation A6):

$$ \hat{\Pi}_{A,B} = \sum_{i}^{N} \sum_{j \neq i}^{N} \hat{y}(\mathbf{r}_i) \hat{y}(\mathbf{r}_j) \quad \text{with} \quad \left\{ \begin{array}{l} \hat{y}(\mathbf{r}_i) = 1 \quad \mathbf{r}_i \in V_A \\ \hat{y}(\mathbf{r}_j) = 1 \quad \mathbf{r}_j \in V_B \\ \hat{y}(\mathbf{r}_i) = 0 \quad \mathbf{r}_i \notin V_A \\ \hat{y}(\mathbf{r}_j) = 0 \quad \mathbf{r}_j \notin V_B \end{array} \right. \qquad (A6) $$

enables the count of ordered electron pairs. Its eigenvalues, $N_{AB}$ and expectation values $\overline{N}_{AB}$ satisfy the closure relations:

$$ \sum_B N_{AB} = N_A(N-1) \qquad (A7) $$

$$ \sum_B \overline{N}_{AB} = \overline{N}_A(N-1) \qquad (A8) $$

$$ \sum_A \sum_B N_{AB} = N(N-1) \qquad (A9) $$

$$ \sum_A \sum_B \overline{N}_{AB} = N(N-1) \qquad (A10) $$

The covariance operator Equation (A12) is deduced by the correspondence principle from the classical expression of the covariance matrix elements (Equation A11):

$$ \mathrm{cov}(N_A, N_B) = \langle N_A N_B \rangle - \langle N_A \rangle \langle N_B \rangle \qquad (A11) $$

$$ \mathrm{cov}(\overline{N_A}, N_B) = \overline{N_A} \overline{N_B} - \overline{N_A} \overline{N_B} \qquad (A12) $$

The product of count operators in Equation (A12) can be expressed in terms of the pair population operator $\hat{P}_{iAB}$:

$$ \mathrm{cov}(\overline{N_A}, N_B) = \hat{\Pi}_{AB} + \delta_{AB} \hat{N}_A - \overline{N}_A \overline{N}_B \qquad (A13) $$

$$ \langle \mathrm{cov}(N_A, N_B) \rangle = \overline{N}_{AB} + \delta_{AB} \overline{N}_A - \overline{N}_A \overline{N}_B \qquad (A14) $$

Combining Equations (A8) and (A5) one obtains the closure relation of the covariance matrix elements:

$$ \sum_B \langle \mathrm{cov}(N_A, N_B) \rangle = \sum_B \overline{N}_{AB} - N_A \sum_B \overline{N}_B + \overline{N}_A = 0 \qquad (A15) $$