Is the Pauli exclusion principle the origin of electron

Molecular Physics An International Journal at the Interface Between Chemistry and Physics

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Is the Pauli exclusion principle the origin of electron localisation? Luis Rincón, F. Javier Torres & Rafael Almeida To cite this article: Luis Rincón, F. Javier Torres & Rafael Almeida (2017): Is the Pauli exclusion principle the origin of electron localisation?, Molecular Physics, DOI: 10.1080/00268976.2017.1363921 To link to this article: http://dx.doi.org/10.1080/00268976.2017.1363921

Published online: 16 Aug 2017.

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Date: 16 August 2017, At: 06:19

MOLECULAR PHYSICS,  https://doi.org/./..

57TH SANIBEL SYMPOSIUM

Is the Pauli exclusion principle the origin of electron localisation? Luis Rincóna,b , F. Javier Torresa and Rafael Almeidab a

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Universidad San Francisco de Quito (USFQ), Grupo de Química Computacional y Téorica (QCT-USFQ) and Instituto de Simulación Computacional (ISC-USFQ), Departamento de Ingeniería Química, Colegio Politecnico de Ciencias e Ingeniería, Diego de Robles y Vía Interoceánica, Quito, Ecuador; b Departamento de Qímica, Facultad de Ciencias, Universidad de Los Andes, La Hechicera, Merida, Venezuela

ABSTRACT

ARTICLE HISTORY

In this work, we inquire into the origins of the electron localisation as obtained from the information content of the same-spin pair density, γ σ , σ (r2 r1 ). To this end, we consider systems of non-interacting and interacting identical Fermions contained in two simple 1D potential models: (1) an infinite potential well and (2) the Kronig–Penney periodic potential. The interparticle interaction is considered through the Hartree–Fock approximation as well as the configuration interaction expansion. Morover, the electron localisation is described through the Kullback–Leibler divergence between γ σ , σ (r2 r1 ) and its associated marginal probability. The results show that, as long as the adopted method properly includes the Pauli principle, the electronic localisation depends only modestly on the interparticle interaction. In view of the latter, one may conclude that the Pauli principle is the main responsible for the electron localisation.

Received  April  Accepted  July 

1. Introduction Electron localisation, described as the propensity of electron pairs of opposite spin to accumulate in the space, has certainly become a deeply rooted concept in the chemical intuition. Indeed, this apparently simple concept has been extensively used to rationalise the structural and reactivity properties of molecules and solids [1]. For instance, the valence electrons in non-polar molecules and many classical semi-conductors are considered to be localised between the bonded atoms; in contrast, π CONTACT Luis Rincón

[email protected]; [email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

KEYWORDS

Electron localisation; conditional pair density; information theory

electrons of aromatic molecules and valence electrons in metals are assumed to be delocalised along the entire nuclear framework. Clearly, the interplay between localisation and delocalisation play an important role for the understanding of the physical-chemical properties of multi-electron systems. Moreover, the quantification of electron localisation can provide a link between sophisticated quantum chemistry calculations and some intuitive models of electronic structure, as the Lewis electron pair model [2] or the Gillespie Valence Shell Electron Pair Repulsion theory [3,4]. However, even today, there


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L. RINCÓN ET AL.

is not an unique prescription to compute electron localisation [5–30]. The earliest attempt to obtain a measurement of electron localisation dates back to 1955 with the Daudel’s loges theory [5], which was, afterwards, rewritten in terms of the minimal missing information function [6]. Some years later, Bader showed that acummulation or depletion of electrons, a quantity closely related to electronic localisation, can be obtained from the analysis of the Laplacian of the electron density [7–9]. Beyond these initial efforts, the electron localisation function, ELF, remains as the most prominent method to examine the electron localisation. This function was proposed, in 1990, by Becke and Edgecombe from the leading term of the spherical average conditional pair density [10]. Its original interpretation was further extended by Savin in terms of the kinetic energy density [11], and afterwards, by Silvi and Savin who presented a chemical classification of the molecular space in terms of ELF basins [31,32]. More recently, a time-dependent extension of the ELF was proposed by Burnus, Marques and Gross in 2005 [19], and an ELF version for correlated levels was presented by Matito, Silvi, Duran and Sola [22]. Possibly, the ELF widespread acceptance is connected with its intuitive graphical interpretation in terms of classical chemical theories, its numerical stability with respect to the theoretical level of calculation and its easy accessibility in modern quantum chemistry softwares. Even with the success of the ELF, other methods for electron localisation have been proposed. In this vein, interpretations derived from information theory concepts and tools have been presented by Nalewajski, Koster and Escalante [21], by Astakhov and Tsirelson [28] and by Urbina, Torres and Rincon [29]. Furthermore, localisation measurements, based on the analysis of the same-spin pair density [13–16,20,27], and on the kinetic energy density [23] have been also suggested. Finally, the analysis of molecular scalar fields has been proposed as a complementary approach for the electron localisation quantification [33]. In spite of all these efforts, it is not possible to claim that an unequivocal method for the quantification of the electron localisation exists at the present time. Thus, the electronic localisation issue remains as an open problem maybe due to the lack in understanding its precise physical origin. In order to advance in this direction, the question: why do electrons localise in space? must be necessarily addressed. According to the most accepted explanation, a pair of electrons of different spins is localised due to the Pauli principle, in other words, electrons are capable to localise because they are Fermions. This idea was first suggested by Lennard-Jones in 1952 [34], who stated [35] ‘Electrons of like spin tend to avoid each other. This effect

is more powerful, much more powerful than that of electrostatic forces. It does more to determine the shapes and properties of molecules than any other single factor. It is the exclusion principle that plays the dominant role in chemistry. Its all-pervading influence does not seem hitherto to have fully realized by chemists, but it is safe to say that ultimately it will be regarded as the most important property to be learned by those concerned with molecular structure’. This argument was later extended by many authors, in particular by Richard Bader who investigated the localisation of electron pairs in the space by studying the properties of the same spin conditional pair density and the Fermi hole. The latter can be interpreted as the exclusion of one electron, due to the localisation of another electron with the same-spin at some fixed point [12,36–39]. Thus, in 1974, Bader and Stephens showed that electronic localisation is connected with the Fermi hole localisation [36,37]. Later in 1996, Bader and co-workers used the integration of the Fermi hole density inside atomic basins defined within the quantum theory of atoms in molecules (QTAIM) to obtain the so-called QTAIM localisation and delocalisation indexes [12]. As a major finding of this work, it was concluded that, due to the antisymmetry of the wave-function, the electron exclusion is a result of the Fermi hole and all measurements of localisation (or delocalisation) of electrons are associated with the localisation or delocalisation of its associated Fermi hole [38]. It is interesting to point out that independent of the theoretical method employed, the Fermi hole can be obtained with a high degree of accuracy, as long as the method reveals correctly the nodal properties of the occupied orbitals [40]. At this point, it is important to emphasise that the Fermi hole is not a product of an electron– electron interaction, and it is present even in the case of a non-interacting set particles as long as these particles are considered to be Fermions. The existence of the Fermi hole induces a decrease in the probability of finding same-spin electrons close together; nevertheless, there is nothing preventing that the electron repulsion may affect this probability. Therefore, one may wonder if the Pauli principle is the only origin of electron localisation, or if potentials, as the Coulomb repulsion between electrons or the attraction interaction with the embedding nuclear framework, affect the electronic localisation, and if it is so, to which extent. A first indication of the latter matters can be inferred considering that the electrons in atoms, molecules and solids are embedded in external potentials created by the nuclear framework or by any other external field. These potentials may have a significant effect in the localisation of the electronic density, and small changes in the potentials can produce drastic changes in the localisation of the electrons. For instance, a small distortion of


MOLECULAR PHYSICS

the benzene hexagonal arrangement results in the localisation of electrons in the regions between the carbons, in contrast to the usual delocalised character of the π electrons. In a similar fashion, imperfections created in metal crystal induce a similar localisation in the otherwise totally delocalised metallic electrons. Furthermore, in the one-electron density picture, the embedding potential determines the nodal and symmetry properties of the occupied orbitals, and in this manner, it determines the average electronic position in a molecule. However, despite these facts, it is still not clear how the influence of these potentials fields would be reflected in the properties of the pair density and of the Fermi hole. In order to contribute to gain some insights on this subject, in this work, we consider two systems constituted by identical interacting and non-interacting Fermions. In the first case, the particles are considered to interact through a soft Coulomb potential, whereas for both systems, the Fermions are subjected to two one-dimensional (1D) external potentials: a unit length box with infinite potential well, and the Kronig–Penney periodic potential [41,42]. Despite of their simplicity, these models are quite versatile, and are able to capture the essential features of more complex cases. It is important to comment that, for the non-interacting particle case, the exact solutions of the Schrödinger equation corresponding to these potential models are well known, and are taken as the starting point to build the solutions of the interacting particle system. For the first potential, we describe the interaction at two levels: a mean field, using the Hartree–Fock theory, and a correlated one, employing a configuration interaction with double excitations. In this manner, we are able to analyse how the electronic correlation affects the description of the electron localisation in this simple 1D model. For the Kronig–Penney periodic potential, the interparticle potential is only considered at the Hartree–Fock level. The electronic localisation characterisation is performed through an information theory measurement, recently proposed by us in Ref. [29]. This is based on computing the information gained by an electron by ‘knowing’ about the position of another electron with the same spin, which is the result of calculating the Kullback–Leibler divergence between the same-spin conditional pair probability density and the one electron marginal probability. By analysing the set of results rendered by all the previously described systems, we expect to be able to contribute in the understanding of the role of the electron–electron and external potentials fields interactions on the electron localisation properties. This work is organised as follows, in the next section we briefly explained the Kullback–Leibler divergence employed here. This is followed by the study

3

of the fermions localisation in an 1D box, and in a Kronig–Penney periodic potential. Finally, some summary and remarks are provided.

2. Electron localisation an the information content of the conditional pair density Following our previous work [43], here the electronic localisation will be analysed using the information content of the same spin conditional pair density, and in particular through the Kullback–Leibler divergence of the conditional pair density. The same spin conditional pair probability, γ σ , σ (r2 r1 ), is defined as the probability of finding an electron with spin σ at position r2 when a reference electron with the same spin rest, with certainty, at position r1 . Thus, γ σ , σ (r2 r1 ) is the electron probability when the position of a reference electron with the same spin is known. The same spin conditional pair probability is calculated from the same spin pair density probability, σ , σ (r1 , r2 ), and the σ spin electron density, ρ σ (r), through the following equation, γ σ,σ (r2 | r1 ) =

2 σ,σ (r1 , r2 ) . ρ σ (r1 )

(1)

For any position of the reference electron, γ σ , σ (r2 r1 ) normalises to the number of σ electrons minus one, i.e. γ σ , σ (r2 r1 )dr2 = Nσ − 1. From the latter, the conditional probability density can be defined as follows: ρ2σ,σ (r2 | r1 ) =

γ σ,σ (r2 | r1 ) . Nσ − 1

(2)

Morover, since ρ2σ,σ (r2 | r1 ) contains all the information regarding the exchange-correlation between the same-spin electrons, it properly enclose the Pauli Principle, and it can be employed for the quantification of electron localisation within the frame of the information theory [44–46]. In this context, an electron localisation spatial measure, based on the information gained by ρ2σ,σ (r2 | r1 ), was introduced recently [29]. There, the electron localisation is computed from the Kullback–Leibler divergence between ρ2σ,σ (r2 | r1 ) and the marginal one electron probability, σ (r ) [29], DKL (r) =

dr ρ2σ,σ (r

| r) log2

ρ2σ,σ (r | r) . (3) σ (r )

In this equation, the logarithm is taken in base 2 to compute DKL in units of bits. From the analysis of DKL , it is obtained that regions with high DKL values correspond to zones of the atomic and molecular space where electrons are localised. After taking into account the scaling of DKL with the number of σ -spin electron, Nσ [29], the


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L. RINCÓN ET AL.

Figure . Example of χ (Equation ) for atoms and molecule.

following function is introduced as a general descriptor of electron localisation in the space, χ (r) = (N σ − 1)DKL (r) fcut (r).

(4)

where fcut is a cut-off function that goes smoothly to zero for negligible density values (ρ cut = 1.0 × 10−4 a.u.), fcut (r) =

1 (1.0 + ERF 0.5 log10 (ρ(r)/ρcut ) ). (5) 2

Here, ERF is the error function, defined in such a way that fcut = 0.5 at distances satisfying ρ(r) → ρcut . The function χ is easily computed as long as the same-spin pair density is accessible. In previous studies, ρ2σ,σ (r | r) is obtained by assuming a mono–determinantal approximation, i.e. by employing Hartree–Fock or Kohn–Sham orbitals for closed-shell systems. For these cases, the following expression is employed: ρ2σ,σ (r2 | r1 ) | i φiσ (r1 )φiσ (r2 ))|2 1 σ ρ (r2 ) − . (6) = σ N −1 ρ σ (r1 ) Previous results have shown that large χ values correspond to regions that chemical intuition would associate

with core, bond or lone pair electrons [29]. Thus, for systems well-represented by a single Lewis picture, χ properly resembles the results rendered by other localisation tools like ELF [10,31]. However, it is important to remark that there are a number of differences between these two functions that make of χ a particularly interesting tool for studying electron localisation in chemical systems. To illustrate this, Figure 1(a) displays the dependence of χ (solid line) and ELF (dotted line) computed for the Kr atom with respect to the distance between a reference electron and the nuclei. Clearly, both functions reveal the same shell structure for Kr. However for the outer shells, the ELF values are approximately as large as the value corresponding to the core shell. In contrast, the χ value (in units of bits time electrons, bte) decreases in roughly a half, from the most inner shell (χ 0.80 bte) to the outer ones (χ 0.4 bte). This information decrease has been traced to an increase in the fluctuation, or variance, of the average number of electrons in each shell. Thus, in core regions the fluctuation in the electron number is small, typically on the order of 0.10 − 0.30 e2 , whereas for the valence shell these values are as large as 1.00 e2 . Another example is the water molecule, for which Figure 2(b) exhibits the 0.40 bte isosurface of χ. The plotting colour scheme is the same as that employed in our previous publication: red


MOLECULAR PHYSICS

5

Figure . χ for a set of six particles inside a square box.

for core electrons, green for covalent bond electrons, and blue for non-bonding electrons (lone pairs). For both, χ and ELF, five basin are obtained, each one with nearly 2.0 electrons: 2.14 e for the O core basin , 1.99 e for the O–H bond basin, and 1.85 e for the O lone pairs one. What is noticeable here is the change in the χ average value for each basin. Thus, for the O core basin an average value of 0.781 bte is obtained, which decreases to 0.418 bte for the O–H basins, and to 0.382 bte for the O lone pairs one. As in the case of atoms, this decrease is traced to an increase in the fluctuation of electron number in each basin, from 0.36 e2 in the O core basin, to 0.75 e2 for the O–H bonds and 1.10 e2 for the O lone pairs. These results show that

in comparison with the ELF, χ provides a more detailed electronic localisation measure of the different regions of the atomic and molecular spaces. In the next sections, the function χ will be employed to characterise the electronic localisation in two simple quantum mechanical models. Throughout this work, atomic units ( = e = me = (4πϵ0 )−1 = 1) are employed and spin-unpolarised systems are considered.

3. Localisation of particles in a 1D box The first considered system consists of a set of 2N noninteracting Fermions in a 1D box of unit length. For this

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L. RINCÓN ET AL.

system, the wave function is a Slater determinant with N double occupied orbitals. The orbitals are given by, √ (7) φn (x) = 2 sin(nπx). Since the particles are Fermions, they have to obey the Pauli principle. For this problem, ρ2σ,σ (r | r) can be exactly computed by using Equation (6). It is possible to compute the integral of Equation (2) very accurately by employing the SciPy libraries of Python [47,48]. Figure 2(a) shows the χ results for the six particles case (solid line), N = 3. For comparison, the ELF results are also shown (dashed line), being computed as,


ELF(x) =

1 1+

D(x) D0 (x)

,

(8)

where D(x) is the excess kinetic energy density, that is the difference between the non-interacting system and the von–Waizsacker kinetic energy densities, D(x) =

1 1 | ρ(x) |2 , | ∇φi (x) |2 − 2 i 8 ρ(x)

(9)

and ρ(x) is the Fermionic density given by ρ(x) = i φ i (x)2 where φ i (x) is the i − th occupied orbital. The term D0 (x) is the reference kinetic energy density of the homogeneous electron gas, the Thomas–Fermi kinetic energy density, D0 (x) =

3 (3π 2 )2/3 ρ(x)5/3 . 10

(10)

Figure 2(a) shows that, both χ and ELF presents three basins, each one corresponding to one of the σ spin particles. For both functions, the Fermions at the border of the box are more localised than that at the centre. However, beyond that, some differences have to be mentioned. First, for positions close to the box border, i.e. for x → 0 or x → 1, it is found that χ → 0, while ELF → 1. This can be understood by recalling that the ELF is a relative electron localisation measure, between the system excess kinetic energy density and the homegeneous electron gas kinetic energy density, taken as reference, fact that hides many of the localisation details. Thus, for the box border positions, D(x) D0 (x) and ELF → 1. However, close to the box edges, the particle densities decrease strongly and DKL (r) → 0 and consequently also χ → 0. In the region about the box centre, the particle kinetic energies become larger than their kinetic energy close to the box border, which translates in larger D(x) and smaller ELF values. On the other hand, as mentioned before, for the χ function the localisation is related to each basin population fluctuation. For the edge

basins, this fluctuation is calculated to be 0.0082 e2 , which is roughly half of that computed at the central basin, 0.0155 e2 , which explains the smaller localisation at these zones. It is remarkable that in spite of these differences, Figure 2(a) shows that even for the non-interacting case both χ and ELF recover the same Fermion localisation pattern. As a next step, the interparticle interaction is included. Since in one dimension, the Coulomb interaction is illdefined [49], an alternative potential is required. Here we have chosen the soft-Coulomb potential defined as, V1D (x, x ) =

ASC (x − x )2 + α

,

(11)

where ASC and α are parameters, whose values in the present work are taken as 1.0 and 0.01 au, respectively. For this interacting system, first the Hartree–Fock equation is solved with a basis set of orbitals of the non-interacting system, Equation (7), including all φ n (x) until n = 20. Thus, in this basis set, the one-electron core Hamiltonian in the Fock matrix is diagonal, and the two-electrons integrals are evaluated numerically using the SciPy libraries of Python [47,48]. The results for χ are shown in Figure 2(b) (dashed line), together with the solution for the noninteracting case (solid line). From there it is clear that the introduction of the Hartree-Fock (HF) mean field interaction does not induce any qualitative change in the localisation pattern, if compared with the results obtained for the non-interacting case. The main difference is the de crease of the χ value at the two minima of the inter basin region by about 12%. From this result, it is possible to state that the mean field interaction tends to decrease the Fermion localisation. Additionally, for the maxima close to the edge, only a small increase of approximately 1% is obtained, while the maximum about the centre remains nearly constant, indicating that the Fermion basin localisation remains largely unchanged by the inclusion of the mean field interaction. At this point, one may wonder if this result is only the consequence of considering a mean field interaction within the HF picture of the system. Thus, as a further step, we have included the correlation in the interacting model. This is achieved by employing a Configuration Interaction wave function with double excitations (CID),

CID = C0 0 +

nm

nm nm Cnm

nm .

(12)

nm

In CID expansion, it is assumed that n > m, and the Slater determinants are constructed directly from the solutions of the non-interacting orbitals, Equation (7), instead of employing the Hartree–Fock orbitals. Hence

MOLECULAR PHYSICS

7

in Equation (12), 0 corresponds to the non-interacting Fermions wave-function. In this equation, only excitations between the same symmetry orbitals are included. From this CID wave-function, the same spin pair density probability renders,

σ,σ

n m 1 2 n m n m C 0 + (x1 , x2 ) = C0Cnm nm 2 0 nm

+

nm


lp nm

(n m |l p ) (nm|l p)

.

(13)

lp

Here, 0 is the non-interacting pair density probabiln m ity, the pair density nm is given by,

nm nm = φn (x1 )∗ φn (x1 )φm (x2 )∗ φm (x2 )

− φn (x1 )∗ φm (x1 )φm (x2 )∗ φn (x2 ),

(14)

(n m |l p )

and the pair density (nm|l p) only is non-zero for pair densities that come from the same determinant, that is n = l, m = p, n = l and m = p , and the cases in which two determinants differ by a double excitation, for example n = l, m = p and n l , m p , or n l, m p and n = l , m = p . From this, the same-spin conditional pair density is calculated by employing Equation (1). The calculated χ are showed in Figure 2(b) (doted line). From there, it is clear that the inclusion of the electron correlation does not qualitatively change the localisation outcome obtained before. Only a small increase in the maxima with respect to those rendered by the HF calculation are observed. Thus, this set of results leads us to conclude that for this system, the Pauli principle is indeed the main responsible for the Fermion localisation.

4. Localisation of particles in the Kronig–Penney periodic potential In this section, we consider the localisation of a system constituted by 2N non-interacting and interacting Fermions per unit cell in a 1D Kronning–Penney model potential. This model potential consists of a unit cell with a square repulsive barrier of strength V0 and width b, and with a spacing between the barriers of a. Hence, the potential has a period equal to a + b. Figure 3(a) shows the 1D Kronig–Penney potential, which takes the following functional form,

Figure . χ for a set of six particles in the Kroning–Penney potential.

with the unit cell potential, VKP , given by,

V (x) =

∞ n=−∞

VKP (x − n(a + b)),

(15)

VKP (x) =

0 V0 .

for 0 < x < a for − b < x < 0

(16)

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L. RINCÓN ET AL.

For this periodic potential, the Bloch orbitals are expressed as, ψk (x) = eikx φ(x),

(17)

where φ(x) is a function with period a + b, and the wavevector takes values between −π/(a + b) and π/(a + b). Inside the unit cell, the solutions are similar to those found for a particle in a square box, namely, ψ1 (x) = Aeiαx + Be−iαx


ψ2 (x) = Ce

βx

+ De

−βx

0