Journal of Self-Assembly and Molecular Electronics (SAME)

Same-spin dynamical correlation effects on the electron localization

Author: Stefano Pittalis1, Alain Delgado1,2, and Carlo Andrea Rozzi1

Thematic Issue:  2015

Article No: 006     doi: 10.13052/jsame2245-4551.2015008

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Same-Spin Dynamical Correlation Effects on the Electron Localization

Received 1 July 2015; Accepted 18 November 2015; Publication 25 November 2015

Stefano Pittalis1, Alain Delgado1,2,3 and Carlo Andrea Rozzi1

  • 1CNR - Istituto Nanoscienze, Via Campi 213a, 41125 Modena, Italy
  • 2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
  • 3Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear, Calle 30 # 502, 11300 La Habana, Cuba


Abstract

The Electron Localization Function (ELF) – as proposed originally by Becke and Edgecombe – has been widely adopted as a descriptor of atomic shells and covalent bonds. The ELF takes into account the antisymmetry of Fermions but it neglects the multi-reference character of a truly interacting many-electron state. Electron-electron interactions induce, schematically, different kind of correlations: non-dynamical correlations mostly affect stretched molecules and strongly correlated systems; dynamical correlations dominate in weakly correlated systems. Here, within an affordable computational effort, we estimate the effects of same-spin dynamical correlations on the electron localization by means of a simple modification of the ELF.

1 Introduction

At the core of quantum chemistry, there is the concept of chemical bond. Since early times, there have been ongoing efforts to set up effective descriptors of bonding in molecules and solids. Pair of electrons of opposite spins were proposed as the basis of bonding in the Lewis theory [1]. More recently the idea emerged that bonding and other structure, such as atomic shells, should reflect the “localization” of electrons in the system.

An effective way to estimate the degree of localization is to exploit the fact that, if the position of one electron is fixed at a given point in space, it must be less likely to find a second electron around the same position. If the pair is made of electrons with parallel spin states, the localization should be related to the properties of the Fermi hole function which reflects the effect of Pauli exchange repulsion [2].

Becke and Edgecombe have followed these lines of thoughts and have analyzed a state of the form of a single Slater determinant [3]. In this case, the conditional distribution Pcondσ(r1,r2) of having a σ-spin electron at position r2 given that there is a σ-spin electron at the reference position r1 is given by

Pcondσ(r1,r2)=ρσ(r2)+hxσ(r1,r2)(1)

(we employ a simplified notation with a single spin index as the pair of electrons is restricted to parallel spin state), where the exchange-hole (x-hole) function

hxσ(r1,r2):=| ρ1σ(r1,r2) |2ρσ(r1)(2)

is expressed through the spin-dependent one-body reduced density matrix

ρ1σ(r1,r2)=iψiσ*(r1)ψiσ(r2).(3)

Here, the summation is restricted to occupied σ-state single-particle orbitals, ψ(r), and the particle density is readily obtained as ρσ(r)=ρ1σ(r,r).

Among the fundamental features of the x-hole function, there are:

  • its normalization

    d3r2hxσ(r1,r2)=1,(4)

    i.e, it accounts for −1 electron;

  • its (local) average extent

    RXσ(r):=[ hxσ(r,u)ud3u ]1(5)

    where

    hxσ(r,u):=14πdΩhxσ(r,r+u)(6)

    is the spherical average of the x-hole function determined around the reference position and u=|u|.

  • its short-range behavior for small u

    hxσ(r,u)=ρσ(r)16[ 2ρσ(r)2Dσ(r) ]u2+(7)

    where the first term is also known as the on-top x-hole and the coefficient of the second term is the curvature of the x-hole. In detail

    Dσ(r):=[ τσ(r)14(ρσ(r))2ρσ(r) ],(8)

    with

    τσ(r):=i| ψiσ(r) |2(9)

    being the (double) of the (positively defined) kinetic energy density.

    Similarly to Equation (7),

    ρσ(r,u)=ρσ(r)+162ρσ(r)u2+(10)

    and insertion of both Equations (7) and (10) in Equation (1) yields

    ρcondσ(r,u)=13Dσ(r)u2+(11)

Note, Equation (11) is a key expression that we will later modify [see Equation (19)] to take into account correlations beyond exchange effects to some extent.

For an inhomogenous system, Dσ(r) exhibits a non-trivial dependence on the reference position r. It turns out useful to make a comparison with respect to a local uniform reference with same density as the actual system. Becke and Edgecombe introduced the quantity

xσ(r)=Dσ(r)/Dσunif(r)(12)

where

Dσunif(r)=35(6π2)2/3ρσ5/3(r)(13)

from which they defined the Electron Localization Function (ELF) as

ELF(r)=11+xσ2(r)(14)

The relevant information provided by the ELF is contained in its variations. The function varies from zero to one: for ELF=0.5, the electrons are localized as in the uniform reference, otherwise they are more (less) ELF > 0.5 (ELF < 0.5) localized. The modulation of the ELF nicely reproduces the shell structure in atoms and well emphasizes covalent molecular bonds in molecules [4].

The ELF discussed above is built from a wavefunction of non-interacting electrons. Works in the literature have proposed generalizations of the ELF [5–7] at the full correlated level. Here, we find it useful to reconsider this topic by distinguishing different type of correlations. While non-dynamical correlations affect stretched molecules and strongly correlated systems, dynamical correlations dominate in most of the molecules at typical equilibrium distances and weakly correlated systems. In the language of multi-reference methods, non-dynamical correlations are related to the mixing of configurations of low energy (which may be degenerate or almost degenerate), while dynamical correlations have to do with the mixing of configurations at higher energy. Can one estimate the effects of dynamical correlations on the electron localization? Can these estimates be made within a well-affordable computational approach? As we illustrate in the next sections, these objectives can be achieved by exploiting the ability of density functional approximations to capture semilocal correlations that are of dynamical type. The approach introduced here neglects opposite-spin contributions, not because they are expected to be unimportant but because we are concerned with an estimation implying a simple and direct modification of the original ELF.

Finally, we point out that the study of electron localization has been fundamental for the understanding of how to capture non-trivial non-localities in the exchange-correlation functional by means of simple approximations. This and related topics continuously attract attention and novel insights have been discussed in recent contributions [8–11]. The analysis presented here adds useful information.

The work is organized as follows: in Section 2, we present a modified ELF; in Section 3, we analyze results of applications to atoms, molecules, and Jellium; in the Section 4, we discuss conclusions and outlooks.

2 Modification of the ELF

In an interacting system, we must account for the fact that electrons tend to avoid each other not only because of the antisymmetry requirement but also because of the electron-electron repulsion. As mentioned in the introduction, we shall focus on same-spin pairs, for which the conditional distribution is given by

Pcondσ(r1,r2)=ρσ(r2)+hxσ(r1,r2)+hcσ(r1,r2)(15)

where hcσ(r1,r2) describes many-body corrections beyond the effects included in the (Generalized) Kohn-Sham [(G)KS] state (notice that this definition differs from the coupling-constant averaged correlation hole).

For a pair of electrons at small inter-particle distance, the interaction is strong yet the relative kinetic term is not suppressed. Thus, the pair in this configuration looks like a “hydrogen atom” but with a repulsive interaction and much lighter nucleous [12]. Density functional theory (DFT) has successfully contributed to clarify that semilocal density functional approximations are very effective to capture such short-ranged correlations – that are, in fact, of dynamical type.

Our choice is to resort to the model proposed (and explained in full details) by Becke [13] for the spherical average of hcσ(r1,r2) for r2 ≈ r1. Becke’s final target was the DFT hole, whereas we are interested in the hole at full coupling strength:

hcσ(r,u)=u2[uzσ(r)]Dσ(r)6(1+zσ(r)/2)F(γσ(r)u),(16)

where

zσ(r):=2cRXσ(r),(17)

has the meaning of a “correlation length” (it sets the shortest inter-particle distance at which the correlation hole vanishes at each reference position). In the following, we shall employ c=0.88 as determined by Becke by fitting DFT correlation energies. The stability of the results with respect to relatively large variation of c will be demonstrated in the next section. F (x) is a damping function with quadratic small-x behavior (F (x) ~ 1 − ax2 + …). γσ(r) will not enter in our final expression but it is instructive to appreciate that, it is determined by enforcingproper zero-normalization condition of the correlation hole and as a result, γσ(r) ≈ 1/zσ(r). For small u, Equation (16), reads

hcσ(r,u)=zσ(r)Dσ(r)6(1+zσ(r)/2)u2+.(18)

The model in Equation (16) has several appealing features: it sets a natural correlation length; it contains zero electron as the corresponding exchange-hole accounts for –1 electrons; it gives vanishing correlation energy for one electron systems; and its short-range behavior is expected to be realistic particularly for inhomogenous systems.

Upon insertion of Equations (18), (10) and (7) into (the spherical average around r1=r of) Equation (15), we get

Pcondσ(r,u)13{ Dσ(r)[ 1zσ(r)2(1+zσ(r)/2) ] }u2+ (19)

from which we define a “modified” ELF (mELF) as follows

mELF(r):=11+xσr2(r)(20)

where

xσ'(r)=xσ(r)[ 1zσ(r)2(1+zσ(r)/2) ]=xσ(r)1+zσ(r)/2(21)

Notice that the numerator of Equation (21) includes the reference to the uniform gas as originally defined by Becke and Edgecombe.

Finally, also notice that Equation (20) – through Equations (21), (17) and (5) – entails the calculation of the Slater potential Uxσ(r)

Uxσ(r)=1RXσ(r);(22)

i.e., the Slater potential is the potential due to the exchange-hole function. Features and implications of the modified ELF are explored in Section 3.

2.1 Further Simplifications

Since an evaluation of the Slater potential, UσX(r), may not be readily available in some DFT codes, we remind that this quantity can be approximated in terms of the Becke-Russel (BR) model UσBR89, X(r) [14]. The BR potential is appealing for finite systems because it entails a model for the x-hole which is exact for finite 1-particle system and reproduces exactly the shot-range behavior of hσx(r,u) in Equation (7) of any N-electron system. One should keep in mind that, by construction, it produces a potential of a “localized” hole function. Within our scope, the BR model can be straightforwardly applied using orbitals obtained from a converged (G)KS calculation.

In the case of molecules, where the exchange-hole can delocalize over multi centers, the option to adopt the full Slater potential is available from direct post processing of the orbitals obtained from a converged (G)KS calculation.

For applications in extended metallic-like systems, we may consider the LDA expression for the Slater potential

UxLDA,σ(r)=3[ 34πρσ(r) ]1/3.(23)

This expression is exact for Jellium.

2.2 Current-Carrying and Time-Dependent States

In the expressions above, we have tacitly assumed real-valued orbitals. For current-carrying states in the case of ground state degeneracy [16–19] and time-dependent processes [15], complex-valued orbitals must be considered instead. The proper expressions are obtained with the substitution τ(r)τ(r)(jp(r)ρ(r))2 where jp(r) is the paramagnetic current. Of course, a time-dependence (r) (r,t) must also be admitted in the time-dependent case. In the reminder of this work, we shall restrict ourselves to closed-shell ground states.

3 Applications and Further Analyses

In this section, we apply Equations (14) and (20) to several systems in order to quantify their differences.

3.1 Atoms

As a first system, we consider the Argon atom. Figure 1 reports the ELF and its modified version (i.e, mELF). Qualitatively, the resulting pictures of the shell structure exhibits strong similarities. The included short-ranged correlations do not change the position of the maxima and minima. However, we can appreciate quantitative differences: minima are less shallow and maxima higher. Localization is enhanced and it increases and extents as moving outwardly from the center of the atom.

Figure 1 was obtained by employing the BR model for the Slater potential. In order to make the illustration self-contained, we report the BR and Slater potential for the Ar atom in Figure 2.

Finally, the dependence of the results on the values of the parameter c of Equation (17) is studied in Figure 3. It is apparent that variations of about ±20% do not change the results significantly. Hence, we do not find any evidence of the necessity to optimize the value of c further.

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Figure 1 (X) Electron Localization Function in its original form, Equation (14), and (X+C) in its modified version, Equation (20), computed for the Argon atom. Kohn-Sham orbitals obtained with the APE code [20], using a LDA for the exchange-correlation energy functional, are used as input to the two expressions. The Slater potential, which enters Equation (20) through the correlation length zσ [see also Equations (17) and (22)], is evaluated within the BR approximation (see also Figure 2).

3.2 Molecules

In order to assess the effect on the ELF of dynamic correlation, as introduced in this work, we have compared the usual ELF with the mELF for a few small hydrocarbons with different CC bond orders (Ethane, Ethene and Ethyne).

We have verified that the pictures obtained with the standard ELF is essentially unchanged with respect to what reported, for instance, in Ref. [4]. The corrections introduced by mELF are only minor in these cases. More noticeable differences may be found in molecules involving heavier atoms.

Let us consider the case of Iodine molecule in Figure 4. The qualitative behavior of mELF follows that of the ELF with larger values in the bonding region. We have verified that, when the two atoms are brought closer and closer, the relative minimum at the midpoint of both ELF and mELF is turned into a maximum when the bonding distance is reached (approximately 2.7 Å). At this point, the electrons pairing from the two 5p atomic orbitals becomes a π type molecular orbital. For all the molecules considered in this section, we have input the mELF with the Slater potential evaluated with converged LDA KS-DFT results.

images

Figure 2 Dots show the Slater potential for the Argon atom evaluated within the OEP-KLI method [21] as implemented in the OCTOPUS code [22] (a grid with equally spaced points is employed). The solid line shows the same quantity evaluated within the BR approximation (this was then employed in the calculation of the modified ELF reported in Figure 1). The BR potential is evaluated on quantities produced with the APE code as explained in the caption of Figure 1. It is apparent that, in the region of interest, the two potentials do not differ significantly.

3.3 Jellium

The application of Equation (20) to Jellium (an extended uniform gas of interacting electrons, whose charge is balanced by a smeared positive background) gives us the opportunity to put forward some explorative speculations.

For this system, the original ELF is a constant independent of the particle density (the values of the constant being 0.5). Instead, our mELF gives a constant that depends on the values of the particle density. Equation (17) together with Equations (22) and (23) show that the correlation length is in inverse relation with the particle density zσ(r)ρσ1/3(r).

images

Figure 3 Demonstration of the stability of the results reported in Figure 1 for the modified ELF [Equation (20)]. Here, the parameter c – entering the final expression through Equations (17) and (21) – is varied ± 20% around its prescribed valued.

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Figure 4 (X) Electron Localization Function in its original form, Equation (14), and (X+C) in its modified version, Equation (20), for the Iodine molecule. The calculation was carried out with the OCTOPUS code using a LDA for the exchange-correlation energy functional. In (X+C), we have employed the Slater potential obtained post-LDA. We employed Troullier-Martins norm-conserving pseudo potentials (as the focus here is on the state of the valence electrons).

For Jellium, the low-density limit corresponds to the strongly interacting regime. If we require the KS wave functions to be plane waves, we find that the correlation length diverges zσ(r) + ∞ and thus mELF → 1. Since it is believed that, in the same limit, the electrons in Jellium tend to localize into a crystalline structure (i.e., the Wigner’s crystal) [23], we may say that, mELF captures this instability.

In the opposite limit, electrons in Jellium get weakly interacting. Correspondingly zσ(r) 0 and thus mELF 0.5. We may say that, electrons fully delocalize in order to get packed together to develop the high-density limit.

4 Conclusions and Outlooks

We have shown how to account for the effects of same-spin dynamical correlations on the electron localization within a model which requires an affordable computational effort. We could visually asses that the effects of these correlations are somewhat unimportant in small organic molecules. They imply some noticeable effects for atomic shells and bonds involving relatively heavier atoms. However, qualitative features such as the positions of the shells and the bond character obtained at the exchange-only level are unchanged.As a novel interesting feature, the proposed modified electron localization function appears to connect the degree of localization in Jellium to different interaction regimes. For the future, it is appealing to attempt to include information on the opposite-spin channels and of non-dynamical correlation effects.

Acknowledgments

This work was financially supported by the European Community through the FP7’s MC-IIF MODENADYNA, grant agreement No. 623413.

References

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Biographies

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S. Pittalis is a researcher at the National Research Council of Italy. He obtained his Ph.D. degree in Physics at Free University Berlin (Germany) in 2008. After the first postdoc at the same institution, he moved to University of Missouri-Columbia (USA) in 2009 and then to University of California-Irvine (USA) in 2011. Returning to Europe in 2013, he was awarded a Marie Curie International Incoming Fellowship. His research directions encompass the development and application of first-principle computational methods for the equilibrium and non-equilibrium properties of materials and devices on the nanoscale.

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A. D. Gran is a visiting researcher at the Advanced Research Complex of the University of Ottawa in Canada. He got his Ph.D. degree in Physical Sciences at the Institute of Cybernetics, Mathematics and Physics (CIMAF), La Habana (Cuba) in 2006. After a postdoc at the same institution, he moved to the National Research Council of Italy (CNR) with a Marie Curie International Incoming Fellowship in 2010. During the period 2012–2015, he worked as research associate at CNR within the European Network CRONOS. His expertise expands over the wide area of many-body techniques, including wave function based methods, density functional theory, and hybrid approaches.

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C. A. Rozzi is a researcher at the National Research Council of Italy. He obtained his Ph.D. degree in Physics at the University of Modena (Italy) in 2000. He was a postdoc at Free University Berlin (Germany) between 2003 and 2006. His main research focus is in the theoretical and numerical simulation of electronic properties of nanostructured materials, with particular regard to ultrafast dynamics in nanostructures for energy photo-conversion. He is a developer of the ab initio open source scientific software “octopus”.

Abstract

1 Introduction

2 Modification of the ELF

2.1 Further Simplifications

2.2 Current-Carrying and Time-Dependent States

3 Applications and Further Analyses

3.1 Atoms

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3.2 Molecules

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3.3 Jellium

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4 Conclusions and Outlooks

Acknowledgments

References

Biographies


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