Local correlation measures and atomic shell structure

Chemical Physics Letters 438 (2007) 330–335 www.elsevier.com/locate/cplett

Local correlation measures and atomic shell structure Robin P. Sagar *, Nicolais L. Guevara

1

Departamento de Quı´mica, Universidad Auto´noma Metropolitana Apartado, Postal 55-534, Iztapalapa, 09340 Me´xico D.F., Mexico Received 20 December 2006 Available online 12 March 2007

Abstract The correlation coefficient and mutual information, used to measure the interdependence between two variables, are generalized to the local level and employed to examine the radial distribution of electron correlation. We compare the behavior of the two local measures with regard to their emphasis on core and valence correlation and show that while there are differences, both are able to correctly reproduce the shell structure in atomic systems.  2007 Elsevier B.V. All rights reserved.

1. Introduction Coulomb correlation results from the presence of repulsive interactions between electrons while Fermi or exchange correlation is due to Fermi–Dirac statistics (antisymmetric wave functions). Both are important factors in electronic structure theory and responsible for physical phenomena such as superconductivity and quantum phase transitions in condensed matter systems. Atomic shell structure and chemical binding in molecules are thought to be mainly due to Fermi correlation. The usual manner of measuring Coulomb correlation in quantum chemistry is by means of the correlation energy, defined as the difference between the exact nonrelativistic energy and the Hartree–Fock (HF) value [1]. Other measures of electron correlation have also appeared in the literature such as the correlation coefficient [2], the degree of correlation [3] and the Jaynes or correlation entropy [4,5]. More recently, other measures have also appeared [6–9], among them mutual information [10]. Each measure has its own particular characteristics such as using information beyond the HF level in the form of configuration

*

Corresponding author. Fax: +52 55 5804 4666. E-mail address: [email protected] (R.P. Sagar). 1 Present address: Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma, de Me´xico, D.F., 04510 Me´xico, Mexico. 0009-2614/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.03.009

interaction expansion coefficients or the eigenvalues of the one-electron reduced density operator. The correlation coefficient from statistics and mutual information from information theory, share the characteristic that both measure the interdependence between two variables and employ differences between one- and twoelectron densities. The correlation coefficient in position space, s, is defined as s¼

hr1  r2 i  hri hr2 i  hri

2

2

;

ð1Þ

where Æræ and Ær2æ are the expectation values of the spin-free one-electron distribution, qu(r) = q(r)/N, while Ær1 Æ r2æ is the expectation value of the spin-free two-electron distribution, Cu(r1,r2) = C(r1,r2)/N(N  1) or pair density. Both qu(r) and Cu(r1,r2) are normalized to one and N is the number of electrons in the system. The marginal is connected to its parent distribution by Z ð2Þ qu ðrÞ ¼ Cu ðr; r2 Þdr2 : The mutual information in position space, Ir, is defined as  u  Z C ðr1 ; r2 Þ u I r ¼ C ðr1 ; r2 Þ ln u dr1 dr2 ¼ 2S uq  S uC P 0 q ðr1 Þqu ðr2 Þ ð3Þ

R.P. Sagar, N.L. Guevara / Chemical Physics Letters 438 (2007) 330–335

where

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where D(r), the radial distribution function is defined as

S uq ¼ 

Z

qu ðrÞ ln qu ðrÞdr

ð4Þ

is the one-electron unity normalized Shannon entropy and Z S uC ¼  Cu ðr1 ; r2 Þ ln Cu ðr1 ; r2 Þdr1 dr2 ; ð5Þ is the two-electron unity normalized Shannon entropy. These two measures have been used to study the correlation between variables in a very wide range of scientific activity. It is accepted that mutual information is able to detect more general non-linear dependencies which the correlation coefficient is incapable of finding. s and Ir have also been employed to examine electron correlation in atomic systems [2,10–12]. The correlation coefficient may be positive or negative for correlated systems, is zero for uncorrelated ones and bounded by 1 6 s 6 1. Positive values occur when the variables are correlated in the same sense or direction while negative values occur when they respond in the opposite sense. Note that the denominator in Eq. (1) is always positive thus the sign of s is governed by the numerator (covariance). On the other hand, mutual information is always positive for correlated variables and is also zero for independent variables. Larger absolute values of both the correlation coefficient and mutual information are indicative of larger correlation between variables. In electronic systems, depending on the particular form of the two-electron density, the correlation coefficient may be separated into contributions resulting from the Hartree part and from the exchange-correlation part of the two-electron density. Such a separation cannot be done for the mutual information due to the presence of the logarithm. If one is interested in how correlation is radially distributed in an atomic system, one may define local correlation measures or correlation profiles from the respective parent measures [10,13]. The interpretation of such a local measure is that it represents the correlation projected on to one of the electronic variables. The numerator of Eq. (1), or covariance, may be generalized within the HF approximation to yield a radially dependent local correlation (covariance) function,   Z 4p qðrÞ rðrÞ ¼ r3 hr1 i  4p r31 Cx ðr; r1 Þdr1 ð6Þ N ðN  1Þ N where q is the spherically averaged charge density and Cx is the spherically averaged exchange density. Likewise, Eq. (3) may be generalized to yield a local information function 



N N 1

S q ðrÞ 16p2 r2 þ N ðN  1Þ N N  Z  r21 Cðr; r1 Þ ln qðr1 Þdr1 ;

I r ðrÞ ¼ ln

DðrÞ þ

Z

r21 Cðr; r1 Þ ln Cðr; r1 Þdr1

ð7Þ

DðrÞ ¼ 4pr2 qðrÞ

ð8Þ

and S q ðrÞ ¼ 4pr2 qðrÞ ln qðrÞ

ð9Þ

is a radially dependent or local Shannon entropy. The purpose of this letter is to compare and contrast the behavior of the local correlation (covariance) function with that of the local information or information density. Although both parent measures, the correlation coefficient and mutual information, are good global measures of correlation, we wish to gain insights into their differences or similarities. We can gain such insights by examining their local behavior. We ask: How is correlation radially distributed in atomic systems and how is this dependent on the particular measure employed for the analysis? How do these measures differ in their prediction of which atomic regions (core–valence) are more correlated than others? One signature of electron correlation (Fermi and Coulomb) in atomic systems is the presence of shell structure. The Coulomb correlations serve to slightly enhance the Fermi effects [14]. Shell structure also appears in trapped bosonic systems as the strength of the interaction increases [15]. Atomic shell structure has been amply studied over the years [16–26]. It is an important structural feature of atomic electronic densities and provides a means by which to gauge the quality of an approximate density. For example, it is known that Thomas–Fermi type densities do not exhibit the correct shell structure [27]. We will explore the relationship between correlation and structure by examining the ability of r(r) and Ir(r) to correctly reproduce the shell structure of atomic systems. Atomic units are used throughout. 2. Results and discussion The radial correlation function, r(r), and the radial information density, Ir(r), were calculated for the Ar, Kr, Xe and Rn noble gas atoms utilizing HF wave functions [28]. Thus all correlation effects are due to Fermi correlation. We present these plots in Figs. 1–3. One observes that r(r) and Ir(r) have very different behaviors for all the studied atoms. First, Ir(r) P 0, while r(r) possesses both positive and negative regions. The positive regions in r(r) correspond to the core or inner shell regions while the negative areas to the outer parts including the valence shell. Thus r(r) treats the correlation in the core (positive values) differently from the correlation in the outer regions (negative values). The interpretation of this is that the variables, r and r1, are correlated in the same sense in the core, while R they move in opposite directions in the outer regions. rðrÞdr is negative in these atoms. This implies that this corresponding global correlation measure (the radial covariance) is governed by the correlation in these outer or valence regions.

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Fig. 1. Plot of r(r)(green, dashed) and Ir(r) (red, solid) for the Kr atom. The inset is the plot for the Ar atom.

Fig. 2. Plot of r(r)(green, dashed) and Ir(r) (red, solid) for the Xe atom.

Ir(r) displays a different behavior with all regions contributing in the same sense, i.e. this function does not make a distinction between core and valence regions in terms of sign but only in terms of magnitudes. The zero in r(r), which may be interpreted as a separation between the inner and outer regions, roughly coincides with a local minimum

Rr in Ir(r). Integrating 0 min I r ðrÞdr, where rmin is the position of the local minimum that corresponds to the zero in r(r), yields a result that is between 60% and 70% of the total value in the studied atoms. Thus, in contrast to the radial covariance, the mutual information is governed by contributions from the core or inner regions.

R.P. Sagar, N.L. Guevara / Chemical Physics Letters 438 (2007) 330–335

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Fig. 3. Plot of r(r)(green, dashed) and Ir(r) (red, solid) for the Rn atom.

The similarities between r(r) and Ir(r) lie in the inner regions where both are positive and display local maxima. The difference is that the innermost local maxima in r(r) appear as shoulders while they are clearly defined in Ir(r). In the Rn atom, the two innermost local maxima in r(r)

are reduced to shoulders. Thus, r(r) smooths out the shell structure performing an average in these core regions. The outer local maximum in Ir(r) coincides with a negative local minimum in r(r). Both functions display the shell structure however there are differences in the manifestations.

Fig. 4. Plot of Ir(r) (red, solid) and D(r) (green, dashed) for the Xe atom. The inset shows the broad fifth maximum in Ir(r) at larger values of r. The scale used for D(r) is D(r) Æ 5 · 105.

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R.P. Sagar, N.L. Guevara / Chemical Physics Letters 438 (2007) 330–335

Fig. 5. Plot of Ir(r) (red, solid) and D(r) (green, dashed) for the Rn atom showing the outer fifth and sixth maxima in Ir(r). The inset contains the plots of the inner regions. The scale used for D(r) is D(r) Æ 2 · 105.

In Ir(r), the shell structure is manifested as positive local maxima. In r(r), the inner shells show up as positive local maxima but the outer shells show up as negative local minima. In the case of the larger atoms (Xe and Rn), the outer and next-to-outer shells are visible as negative local minima. From Eq. (7), one can see that D(r) is a component of Ir(r), thus it should not be surprising if there is similarity in behavior between the two functions. The local maxima in the D(r) function have been studied regarding their ability to correctly capture the shell structure in atomic systems [16–18]. This function has also been used to partition the atom into core and valence regions [29]. D(r) was shown to be unable to capture all shells at most recovering five shells [16,17]. These missing shells manifested themselves as shoulders in D(r) with the use of numerical HF wave functions [18]. More recently, D(r) has been separated into inner and outer functions [30]. In Figs. 4 and 5 we compare the behavior of Ir(r) with D(r) for the Xe and Rn atoms. One notes the similarity in behavior between the two functions. In Fig. 4, the most important point is that while D(r) displays four local maxima indicative of the shells in Xe, Ir(r) is capable of capturing the fifth shell which D(r) does not. In the Rn atom of Fig. 5, the fifth shell in D(r) appears as a shoulder while the sixth shell is not even present as a local maximum. Ir(r) is able to capture all six shells as observed from the six local maxima. Note also that the fifth local maximum in Ir(r) occurs in the same region of r as the shoulder in D(r). The outer local minima in both Xe and Rn occur at values of r which correspond to inflection points in D(r).

A valid question is what are the factors that lead to the appearance of the outer two local maxima in Ir(r). These local maxima must be a result of the remaining three terms in Eq. (7). Our analysis shows that no one term dominates, with more weight coming from the third and fourth terms. Thus these local maxima are the result of the interaction between all terms of Eq. (7). Lastly, we have analyzed the differences between the correlation coefficient and mutual information by examination of their corresponding local measures. These results are based on the particular behavior of one- and two-electron densities. Thus care should be taken in generalizing these results to other situations outside of electronic structure theory where the densities are not of this particular form. 3. Conclusions The behavior of the local correlation and information functions, whose parent measures are the correlation coefficient and mutual information, are compared in a series of noble gas atoms to ascertain which radial regions are responsible for Fermi correlation and how this depends on the particular perspective. The results show that r(r) treats correlation from the core regions (positive values) differently to correlation from the valence regions (negative values). On the other hand, the local information function treats correlation from the core and valence in the same sense in a cumulative manner. r(r) stresses the correlation from the outer regions while Ir(r) emphasizes the correlation from the inner regions. The most important point is

R.P. Sagar, N.L. Guevara / Chemical Physics Letters 438 (2007) 330–335

that both r(r) and Ir(r), as local correlation functions, reveal the shell structure indicative of correlation, albeit differently. r(r) displays positive local maxima for the core regions and negative local minima for the outer regions. Ir(r) shows positive local maxima and is similar in behavior to D(r), the radial distribution function. We illustrate for the Xe and Rn atoms that in contrast to D(r), Ir(r) is capable of recovering the complete shell structure in the form of local maxima. Acknowledgements The authors thank the Consejo Nacional de Ciencias y Tecnologia (CONACyt) and the PROMEP program of the Secretario de Educacio´n Pu´blica in Me´xico for support. References [1] [2] [3] [4] [5] [6] [7] [8]

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