**Author:** Stefano Pittalis^{1}, Alain Delgado^{1,2}, and Carlo Andrea Rozzi^{1}

**Thematic Issue: 2015**

**Article No: **006 doi: 10.13052/jsame2245-4551.2015008

**Same-Spin Dynamical Correlation Effects on the Electron Localization**

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

Stefano Pittalis^{1}, Alain Delgado^{1,2,3 }and Carlo Andrea Rozzi^{1}

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

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.

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 ${P}_{\text{cond}}^{\sigma}({\text{r}}_{1},\text{\hspace{0.17em}}{\text{r}}_{2})$ of having a *σ*-spin electron at position **r**_{2 }given that there is a *σ*-spin electron at the reference position **r**_{1 }is given by

(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

$$\begin{array}{rr}\hfill {h}_{\text{x}}^{\sigma}({\text{r}}_{1},\text{\hspace{0.17em}}{\text{r}}_{2}):=-\frac{{\left|{\rho}_{\text{1}}^{\sigma}({\text{r}}_{1},{\text{r}}_{2})\right|}^{2}}{{\rho}_{\sigma}\left({\text{r}}_{1}\right)}& \hfill (2)\end{array}$$is expressed through the spin-dependent one-body reduced density matrix

$$\begin{array}{rr}\hfill {\rho}_{1}^{\sigma}\left({\text{r}}_{1},{\text{r}}_{2}\right)={\sum}_{i}{\psi}_{i\sigma}^{*}({\text{r}}_{1}){\psi}_{i\sigma}\left({\text{r}}_{2}\right).& \hfill (3)\end{array}$$Here, the summation is restricted to occupied *σ*-state single-particle orbitals, *ψ _{iσ}*(

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

its normalization

$$\begin{array}{rr}\hfill {\displaystyle \int {d}^{3}{r}_{2}{h}_{\text{x}}^{\sigma}\left({\text{r}}_{1,}{\text{r}}_{2}\right)}=-1,& \hfill (4)\end{array}$$i.e, it accounts for −1 electron;

its (local) average extent

$$\begin{array}{rr}\hfill {R}_{\text{X}}^{\sigma}\left(\text{r}\right):=-{\left[{\displaystyle \int \frac{{h}_{\text{x}}^{\sigma}\left(\text{r,u}\right)}{u}}{d}^{3}u\right]}^{-1}& \hfill \left(5\right)\end{array}$$where

$$\begin{array}{rr}\hfill {h}_{\text{x}}^{\sigma}\left(\text{r,u}\right):=\frac{1}{4\pi}{\displaystyle \int d\Omega {h}_{\text{x}}^{\sigma}\left(\text{r},\text{r}+u\right)}& \hfill (6)\end{array}$$is the spherical average of the x-hole function determined around the reference position and

*u*=*|***u***|*.its short-range behavior for small

$$\begin{array}{rr}\hfill {h}_{\text{x}}^{\sigma}\left(\text{r},u\right)=-{\rho}_{\sigma}\left(\text{r}\right)-\frac{1}{6}\left[{\nabla}^{2}{\rho}_{\sigma}\left(\text{r}\right)-2{D}_{\sigma}\left(\text{r}\right)\right]{u}^{2}+\cdots & \hfill \left(7\right)\end{array}$$*u*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

$$\begin{array}{rr}\hfill {D}_{\sigma}\left(\text{r}\right):=\left[{\tau}_{\sigma}\left(\text{r}\right)-\frac{1}{4}\frac{{\left(\nabla {\rho}_{\sigma}\left(r\right)\right)}^{2}}{{\rho}_{\sigma}\left(r\right)}\right],& \hfill \left(8\right)\end{array}$$with

$$\begin{array}{rr}\hfill {\tau}_{\sigma}\left(\text{r}\right):={\sum}_{i}{\left|\nabla {\psi}_{i\sigma}\left(\text{r}\right)\right|}^{2}& \hfill \left(9\right)\end{array}$$being the (double) of the (positively defined) kinetic energy density.

Similarly to Equation (7),

$$\begin{array}{rr}\hfill {\rho}_{\sigma}\left(\text{r},u\right)={\rho}_{\sigma}\left(\text{r}\right)+\frac{1}{6}{\nabla}^{2}{\rho}_{\sigma}\left(\text{r}\right){u}^{2}+\cdots & \hfill \left(10\right)\end{array}$$and insertion of both Equations (7) and (10) in Equation (1) yields

$$\begin{array}{rr}\hfill {\rho}_{\text{cond}}^{\sigma}\left(\text{r},u\right)=\frac{1}{3}{D}_{\sigma}\left(\text{r}\right){u}^{2}+\cdots & \hfill \left(11\right)\end{array}$$

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, *D _{σ}*(

where

$$\begin{array}{rr}\hfill {D}_{\sigma}^{\text{unif}}\left(\text{r}\right)=\frac{3}{5}{\left(\text{6}{\pi}^{2}\right)}^{2/3}{\rho}_{\sigma}^{5/3}\left(\text{r}\right)& \hfill \left(13\right)\end{array}$$from which they defined the Electron Localization Function (ELF) as

$$\begin{array}{rr}\hfill \text{ELF}\left(\text{r}\right)=\frac{1}{1+{x}_{\sigma}^{2}\left(\text{r}\right)}& \hfill \left(14\right)\end{array}$$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.

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

$$\begin{array}{rr}\hfill {P}_{\text{cond}}^{\sigma}\left({\text{r}}_{1},{\text{r}}_{2}\right)={\rho}_{\sigma}\left({\text{r}}_{2}\right)+{h}_{\text{x}}^{\sigma}\left({\text{r}}_{1},{\text{r}}_{2}\right)+{h}_{\text{c}}^{\sigma}\left({\text{r}}_{\text{1}},{\text{r}}_{2}\right)& \hfill \left(15\right)\end{array}$$where ${h}_{\text{c}}^{\sigma}\left({\text{r}}_{1},{\text{r}}_{2}\right)$ 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 ${h}_{\text{c}}^{\sigma}\left({\text{r}}_{1},{\text{r}}_{2}\right)$ for r_{2} ≈ r_{1}. Becke’s final target was the DFT hole, whereas we are interested in the hole at full coupling strength:

where

$$\begin{array}{rr}\hfill {z}_{\sigma}\left(\text{r}\right):=2c{R}_{\text{X}}^{\sigma}\left(\text{r}\right),& \hfill \left(17\right)\end{array}$$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 − *ax*^{2} + …). *γ _{σ}*(

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 **r**_{1=}**r** of) Equation (15), we get

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

$$\begin{array}{rr}\hfill \text{mELF}\left(\text{r}\right):=\frac{1}{1+{x}_{\sigma}^{r2}\left(\text{r}\right)}& \hfill \left(20\right)\end{array}$$where

$$\begin{array}{rr}\hfill {x}_{\sigma}^{\text{'}}\left(\text{r}\right)={x}_{\sigma}\left(\text{r}\right)\left[1-\frac{{z}_{\sigma}\left(\text{r}\right)}{2\left(1+{z}_{\sigma}\left(\text{r}\right)/2\right)}\right]=\frac{{x}_{\sigma}\left(\text{r}\right)}{1+{z}_{\sigma}\left(\text{r}\right)/2}& \hfill \left(21\right)\end{array}$$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 ${U}_{\text{x}}^{\sigma}\left(\text{r}\right)$

$$\begin{array}{rr}\hfill {U}_{\text{x}}^{\sigma}\left(\text{r}\right)=-\frac{1}{{R}_{\text{X}}^{\sigma}\left(\text{r}\right)};& \hfill (22)\end{array}$$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.

Since an evaluation of the Slater potential, *U ^{σ}*

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

$$\begin{array}{rr}\hfill {U}_{\text{x}}^{\text{LDA},\sigma}\left(\text{r}\right)=-3{\left[\frac{3}{4\pi}{\rho}_{\sigma}\left(\text{r}\right)\right]}^{1/3}.& \hfill (23)\end{array}$$This expression is exact for Jellium.

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 $\tau \left(\text{r}\right)\to \tau \left(\text{r}\right)-{\left(\frac{{\text{j}}_{p}\left(\text{r}\right)}{\rho \left(\text{r}\right)}\right)}^{2}$ where **j**_{p}(**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.

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

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.

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.

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 ${z}_{\sigma}\left(\text{r}\right)\sim {\rho}_{\sigma}^{-1/3}\left(\text{r}\right)$.

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 *z _{σ}*(

In the opposite limit, electrons in Jellium get weakly interacting. Correspondingly *z _{σ}*(

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.

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

[1] G. N. Lewis. The atom and the molecule. *J. Am. Chem. Soc.*, **38**, 762–785 (1916).

[2] R. F. W. Bader and M. E. Stephens. Spatial localization of the electronic pair and number distributions in molecules. *J. Am. Chem. Soc.*, **97**, 7391–7399 (1975); (b) R. F. W. Bader, R. J. Gillespie and P. J. MacDougall. A physical basis for the VSEPR model of molecular geometry. *ibid*, **110**, 7329 (1988).

[3] A. D. Becke and K. Edgecombe. A simple measure of electron localization in atomic and molecular systems. *J. Chem. Phys*., **92**(9), 5397–5403 (1990).

[4] A. Savin, R. Nesper, S. Wengert and T.E. Fässler. ELF: The electron localization function. *Angew. Chem. Int. Ed.*, **36**, 1808–1832 (1997).

[5] F. Feixas, E. Matito, M. Duran, M. Solá and B. Silvi. Electron localization function at the correlated level: a natural orbital formulation. *J. Chem. Theory Comput.*, **6**, 2736–2742 (2010).

[6] M. Kohout, K. Pernel, F. R. Wagner and Y. Grin. Electron localizability indicator for correlated wavefunctions. I. Parallel-spin pairs. *Theor. Chem. Acc.*, **112**(5), 453–459 (2004).

[7] M. Kohout, K. Pernel, F. R. Wagner and Y. Grin. Electron localizability indicator for correlated wavefunctions. II Antiparallel-spin pairs. *Theor. Chem. Acc.*, **113**(5), 287–293 (2005).

[8] J. Sun, B. Xiao, Y. Fang, R. Haunschild, P. Hao, A. Ruzsinszky, G. I. Csonka, G. E. Scuseria and J. P. Perdew. Density Functionals that Recognize Covalent, Metallic, and Weak Bonds. *Phys. Rev. Lett.*, **111**, 106401–5 (2013).

[9] S. Pittalis, F. Troiani, C. A. Rozzi and G. Vignale. Ab initio theory of spin entanglement in atoms and molecules. *Phys. Rev. B*, **91**(7), 075109 (2015).

[10] M. J. P. Hodgson, J. D. Ramsden, T. R. Durrant and R. W. Godby. Role of electron localization in density functionals. *Phys. Rev. B*, **90**(24), 241107(R) (2014).

[11] T. R. Durrant, M. J. P. Hodgson, J. D. Ramsden and R. W. Godby. Electron localization in static and time-dependent systems. arXiv, 1505.07687 (2015).

[12] R. T. Pack and W. B. Brown. Cusp Conditions for Molecular Wave-functions. *J. Chem. Phys.*, **45**, 556–559 (1966).

[13] A. D. Becke. Correlation energy of an inhomogeneous electron gas: a coordinatespace model. *J. Chem. Phys.*, **88**(2), 1053–1062 (1987).

[14] A. D. Becke. Exchange holes in inhomogeneous systems: a coordinate-space model. *Phys. Rev. A*, **39**(8), 3761–3767 (1988).

[15] T. Burnus, M. A. L. Marques and E. K. U. Gross. Time-dependent electron localization function. *Phys. Rev. A*, **71**(1), 010501(R) (2005).

[16] J. Dobson.Alternative expressions for the Fermi hole curvature. *J. Chem. Phys.*, **98**(11), 8870–8872 (1993).

[17] A. D. Becke. Current-density dependent exchange-correlation functionals. *Can. J. Chem.*, **74**(6), 995–997 (1996).

[18] J. Tao. Explicit inclusion of paramagnetic current density in the exchange-correlation functionals of current-density functional theory. *Phys. Rev. B*, **71**(20), 205107, 2005.

[19] S. Pittalis, E. Ränen and E. K. U. Gross. Gaussian approximations for the exchange-as energy functional of current-carrying states: Applications to two-dimensional systems. *Phys. Rev. A*, **80**(3), 032515 (2009).

[20] M. Oliveira and F. Nogueira. Generating relativistic pseudo-potentials with explicit incorporation of semi-core states using APE, the Atomic Pseudo-potentials Engine. *Comput. Phys. Commun.*, **178**(7), 524–534 (2008).

[21] T. Grabo, T. Kreibich, S. Kurth, and E. K. U. Gross. In Strong Coulomb correlations in electronic structure calculations: beyond local density approximations, V. I. Anisimov (ed.), Gordon and Breach Science, pp. 203, London (2000).

[22] A. Castro, H. Appel, M. Oliveira, C. A. Rozzi, X. Andrade, F. Lorenzen, M. A. L. Marques, E. K. U. Gross and A. Rubio. Octopus: a tool for the application of time-dependent density functional theory. *Phys. Status Solidi B*, **243**(11), 2465–2488 (2006).

[23] G. F. Giuliani and G. Vignale. *Quantum Theory of the Electron Liquid*, *Cambridge University Press*, Cambridge (2005).

**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.

**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.

**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”.

*Journal of Self-Assembly and Molecular Electronics, Vol. 3,* 1–14.

doi: 10.13052/jsame2245-4551.2015008

© 2015 *River Publishers. All rights reserved.*