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Relative entropy and atomic structure
Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77 www.elsevier.com/locate/theochem
Relative entropy and atomic structure Robin P. Sagar *, Nicolais L. Guevara Departamento de Quı´mica, Universidad Auto´noma Metropolitana, Apartado Postal 55-534, Iztapalapa, 09340 Me´xico D.F., Mexico Received 12 December 2007; received in revised form 25 January 2008; accepted 11 February 2008 Available online 20 February 2008
Abstract Relative entropy, using a hydrogen-like model density as the reference, is studied in atomic systems. We find that both the position and momentum space measures display the shell effects, being similar in behavior to the expectation value, hri. We also show that the electronic momentum density is closer to the hydrogen-like reference than is the position space density. A connection between relative entropy, and the atomic radius and quantum capacitance, is also discussed. Ó 2008 Elsevier B.V. All rights reserved. PACS: 31.10.+z; 31.90.+s Keywords: Relative entropy; Atomic structure; Shell effects in momentum space; Quantum capacitance
1. Introduction Questions of spatial electronic localization are pervasive in atomic and molecular physics. The localization properties of quantum particles also forms the basis of explanations for phenomena associated with quantum systems in other realms of physics. It is thought that correlation, or the interaction between particles, drives localization, which at its extreme results in decoherence or the appearance of classical behavior. There is a limit to the extent of this localization in quantum systems due to the Heisenberg uncertainty relationship. The congregation(localization) of (different-spin)electrons in particular regions of space gives rise to the characteristic shell effects in atoms, pairing in molecules to form bonds, and superconductivity in condensed matter systems. The limit to the degree of localization that an electronic(fermionic) system may experience is due to the Pauli exclusion principle which keeps same-spin electrons apart. This is a consequence of the antisymmetric requirement of a wave function which obeys Fermi-Dirac statistics. The *
Corresponding author. Tel.: +52 55 58044675; fax: +52 55 5804 4666. E-mail address: [email protected] (R.P. Sagar).
0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.02.009
effects of this Fermi correlation and entropic uncertainty type relationships have been discussed [1,2]. The effects of coulomb correlation on opposite-spin electrons might modify the degree of localization of the system, enhancing or reducing the effects of the Fermi correlation. Recently, studies of coulomb correlation effects on localization using information measures have been presented [1–3]. The spatial electron localization has typically been measured by focusing not on the N-particle wave function, Wðx1 ; x2 ; ; xN Þ, or the N-particle density matrix, but rather on the spin-traced one-electron charge density, qðrÞ ¼
Z
W ðx1 ; x2 ; ;xN ÞWðx1 ;x2 ; ; xN Þdr1 dx2 dxN ; ð1Þ
and functions related to it. The co-ordinate, xj ¼ ðrj ; rj Þ, is a combined space-spin one. The manner in which the antisymmetric property of the electronic wave function is projected onto the charge density and the conditions which need to be obeyed is called the N-representability problem [4]. The study of the localization properties of an electronic charge density is also an examination of how the antisymmetric requirements
R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
on the wave function influence or restrict the localization characteristics of the density. We also remark that electron localization, and its consequences, may also be formulated and studied in momentum space [5–7], by examining the properties of PðpÞ, the electronic momentum density [8]. This avenue has not received the attention that the position space charge density has. Among the global measures of localization in position space, the Shannon entropy, taken from information theory [9] is defined as [10] Z S q ¼ qðrÞ ln qðrÞdr: ð2Þ qðrÞ is normalized to N, the number of electrons in the system. The momentum space Shannon entropy is given as Z S p ¼ PðpÞ ln PðpÞdp ð3Þ and PðpÞ is normalized to N. On comparing two distributions, the one with the smaller entropic value is indicative of the more localized distribution in the respective space. One important aspect of the entropies is that they are related by an uncertainty-type inequality [11] S uq
þ
S up
P 3ð1 þ ln pÞ
ð4Þ
for unity normalized entropies (S uq;p ), which are calculated from Eqs. (2) and (3) using unity normalized densities, qðrÞ and PðpÞ . The lower bound in Eq. (4) corresponds to N N the harmonic oscillator. This inequality establishes limits on the localization in one space, given the other. S q and S p have been used to study localization–delocalization phenomena in atoms and molecules [12–16], in confined systems [17] and in bosonic systems [18,19]. S q , as a function of atomic number Z, has been shown to demonstrate periodic effects [10]. The noble gas atoms are the most localized of their period. This is not the case for S p which is an increasing function of Z displaying much smaller shell effects or transitions from one period to the next [10,3]. Their behavior is shown in Fig. 1. The difference between the behavior of S q and S p can be phrased in the questions: Is atomic structure, due to the fermionic nature of electrons, different in position space as compared to momentum space, throughout the periodic table? How does the momentum density carry the information about the fermionic behavior of electrons, i.e., shell effects, periodicity? Fermi effects separate same-spin electrons in position space but how does this mechanism apply in momentum space at the one-electron level? Do these effects translate into electrons travelling with different momenta, or with the same momentum, i.e., a condensation of momenta? A natural question concerning all measures of localization is, localized as compared to what? Another measure taken from information theory is the Kullback-Leibler or relative entropy, defined in terms of unity-normalized densities,
qðrÞ qðrÞ ln dr N qo ðrÞ Z PðpÞ PðpÞ ln dp; S pK ¼ N Po ðpÞ
S rK ¼
Z
73
ð5Þ ð6Þ
where qo ðrÞ and Po ðpÞ are suitably chosen reference densities, normalized to N, in the respective spaces. The relative entropy is widely used in information theory as a measure of the distance between the densities, qðrÞ and qo ðrÞ; and PðpÞ and Po ðpÞ [20]. It is the difference between the surprisal or uncertainty of the reference density, averaged over the realistic density, and the Shannon entropy of the realistic density. It is attractive for our purposes since it is a measure which classifies the distance between two densities in terms of the uncertainty or localization properties of the densities. In these definitions, all densities are per electron and normalized to one. As such, they may be considered as intensive quantities. Such unity-normalized densities are also called shape factors in the literature [21]. The positivity of the relative entropy establishes that S rK þ S pK P 0:
ð7Þ
Using this and Eqs. (5) and (6), we obtain an upper bound for the entropy sum, Z 1 qðrÞPðpÞ ln½qo ðrÞPo ðpÞdr dp: ð8Þ Sq þ Sp 6 N Relative entropy has been used to study atomic structure [16,22–25] with different selections for the reference density. Indeed, the physical insights into the problem lies in a suitable selection of the reference density. In this work, we choose for the reference density a model which represents a hydrogenic 1s orbital occupied with N particles. Since it is hydrogenic, it has no interaction or Fermi(exchange) effects. For N > 2, the model density may be considered as being ground state boson-like. The hydrogen-like reference density in position space is given by qo ðrÞ ¼
Z 3 N 2Zr e p
ð9Þ
and is normalized to N. Note that Z = N in atomic systems, however we keep the variables apart for clarity in the discussion that follows. Furthermore, the shape factor or unity-normalized density is just the hydrogenic density (N = 1). In momentum space, the reference density can be obtained from Dirac-Fourier transformation of the hydrogenic orbital to yield, Po ðpÞ ¼
8NZ 5 p2 ðZ 2 þ p2 Þ
4
:
ð10Þ
These model densities have been previously introduced and are called cusp constrained models [23,26]. They have been tested as a model for the Shannon entropy in isoelectronic series, and provide fair results. However, their use in study-
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R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
a
b
c
d
Fig. 1. Plot of (a) S q (red) (b) S Cq (blue) (c) S p (red) (d) S Cp (blue) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
ing the atomic structure is limited since they are not capable of reproducing the shell or periodic effects present in the periodic table. We show their behavior in Fig. 1 (S Cq and S Cp ) which can be compared to S q and S p . This model corresponds to a limiting case as far as localization in position space is concerned. S Cq < S q while S Cp > S p . That is, the realistic densities with interaction and Fermi effects are more delocalized in position space, and more localized in momentum space, in comparison to the models. Thus these effects serve to localize the density in momentum space. These models also yield unity normalized entropies which satisfy the entropic inequality in Eq. (4) [26]. A linear fit of S q against S Cq values, and S p against S Cp , yielded S q ¼ 0:246S Cq þ 16:4 and S p ¼ 0:499S Cp þ 3:70 with R2 coefficients of 0.991 and 0.999, respectively. The fit is better in momentum space since the shell effects in S p are not as pronounced as in S q . The use of the hydrogen-like (HL) density as the reference density in the relative entropy would provide insights into how a realistic one-electron density, containing Fermi and interaction effects, differs from the HL or reference
density with respect to the localization(delocalization) features of the densities. The structure of the periodic table and shell effects exist due to the fermionic nature of electrons. Thus, one would expect that any measure of the difference between a realistic electron (fermion) density and the reference density should reflect the periodic nature or atomic structure. The purpose of this paper is to study the relative entropy, defined with the HL reference densities, for the ground state atoms Z = 2–54, in position and in momentum space, to ascertain its capacity in exhibiting shell effects or periodic structure. Also of interest is a comparison between the position and momentum space measures to examine the extent at which momentum space provides information about shell effects, or more generally, the fermionic nature of electrons in momentum space. Atomic units are used throughout. 2. Results and discussion Relative entropies in position and in momentum space were calculated for the atoms with Z = 2–54, using Har-
R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
75
tree-Fock (HF) wave functions [27]. In such functions, one expects the Fermi effects to dominate over the interaction effects for same-spin electrons. Spherically averaged densities were used in these calculations. The relative entropy in position space using the HL reference density is 3 2Z Sq Z N hri ln S rK ¼ ð11Þ N p N R where hri is 4p r3 qðrÞdr. Even though the shape factors or density per particle are used in the calculation of the relative entropy, the result of the leading term,hri, is that it is extensive. The first two terms in Eq. (11) measure the spread or the extent of the localization of qðrÞ. hri can also be considered as a measure of the spatial extent or volume of the atom. We present plots of S rK and hri in Fig. 2. One observes that S rK increases with Z with some exceptions. That is, the distance from the reference density is larger with increasing Z. The exceptions occur in the atoms with configurations which display d orbital filling characteristics. The essential point is that S rK behaves very similarly to hri, exhibiting the shell effects as jumps from one period to another. Thus the leading term dominates the expression in Eq. (11). Note also that these shell effects are accentuated in S rK as compared to S q . The result that S rK is dependent on hri [See Eq. (11)], and behaves similarly to it, may be interpreted as that the fermionic nature of electrons is responsible for the extensivity of the result, i.e., the result depends on a measure of the size or volume of the system. This is consistent with the argument that matter has volume due to the fermionic properties of electrons [28]. The relative entropy in momentum space is, Z 16pZ 3 Sp 8N S pK ¼ p2 PðZpÞ lnð1 þ p2 Þdp ln : N N Z 3 p2 ð12Þ
We show the behavior of S pK vs. atomic number in Fig. 3, which reveals a structure that is very similar in behavior to S rK and hri. This is significant since S pK is a momentum space measure while S rK is a position space measure. Thus the relative entropy shows the shell effects on the densities, due to the fermionic nature of electrons, in momentum space and also in position space. Note also that these effects are magnified in S pK as compared to S p in Fig. 1. The ability of momentum space measures to reproduce shell effects has also been recently commented on [29,30]. One could also ask if the analogy with position space holds in that if it is the behavior of the first term in Eq. (12) which dominates the behavior of S pK . This however is not the case. We show in Fig. 3 that this term is a decreasing function of Z. Furthermore, it is the combination of all of the terms of Eq. (12) which is responsible for the behavior similar to hri. Note also that the integral in Eq. (12) is written in a compact form which groups a factor of 8 ln Z into the third term. If one includes this with the first term the result is an increasing function of Z with no shell effects. The relative sizes of atoms have been reported to be observed in the momentum density by using information about local maxima and minima [31] which differs from the global analysis considered here. In Figs. 2 and 3, one notices that S rK > S pK . Thus the electronic density is behaving more hydrogen-like in momentum space than in position space, with respect to localization features of the density. Furthermore, the shell effects on the density, as seen from S pK , are not as clearly defined in momentum space, especially for larger Z. A linear fit of the S rK values against the S pK ones yields, S rK ¼ 25:0S pK 24:5, with a R2 coefficient of 0.912, which suggests that the relationship is not very linear. S rK and S pK are similar in behavior to the extensive quantity, hri. One can obtain intensive quantities, or per electron measures that do not grow with the size of the system, by considering S rK =N , S pK =N and hri=N . We plot
Fig. 2. Plot of S rK (red, solid) and 2 hri (blue, dash) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
Fig. 3. Plot of S pK (red, solid) and the first term in Eq. (12) (blue, dash) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
these quantities in Fig. 4. These results, based on this particular analysis, show that the shell effects are present in all measures but less defined in momentum space. It is an open question as to whether the position density carries more shell effect information than the momentum density or vice-versa. See Refs. [29,30] for another line of analysis. 2.1. Atomic radii and capacitance One might conclude from the results of the previous section [Eq. (11)] that the relative entropy is related to the atomic radius due to its dependence on hri. However, it is known [32] that hri does not provide a good measure of the atomic radius. AR better measure of the atomic radius is hria , defined as, 4p r3 qa ðrÞdr, where qa ðrÞ is the unity normalized density of the highest occupied Hartree-Fock orbital. If one is interested in constructing a bridge between relative entropy and atomic radii, one may define a relative entropy with dependence on hria , which can be obtained from Eq. (5) if one considers a realistic density corresponding to the density of the highest occupied HF orbital, and a hydrogen 1s orbital reference density. The relative entropy with this definition would be a measure of the difference between the localization properties of the highest occupied orbital density as compared to the 1s orbital density. This yields S aK ¼ 2hria S aq þ ln p; ð13Þ R where S aq ¼ 4p r2 qa ðrÞ ln qa ðrÞdr is the Shannon entropy of the highest occupied atomic orbital. The benefit of such a definition is that it provides a link between the localization features of the highest occupied orbital charge density and atomic radii. We present a plot of S aK , hria and S aq vs. Z in Fig. 5. One notes that the shell effects are very evident and that all measures have an almost identical structure. Differences
Fig. 4. Plot of S rK =N , (blue, dash) 3 S pK =N (green, dot) and hri=N (red, solid) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
between S aK and hria occur for Z = 9,10 where S aK increases while hria and S aq decrease. Within the noble gas atoms, S aq has a stronger relationship with hria since both increase, while S aK increases and decreases within the group. This behavior of S aK is not a general tendency since it increases as does hria within the alkali and alkali earth groups. Recently, the observation has been made [33] that the quantum capacitance of a neutral atom scales linearly with hria . The quantum capacitance, important in the design of nanoscale computing devices, is defined as 1=ðI AÞ where I is the ionization potential and A is the electron affinity of the system. It is also inversely proportional to the hardness, an important parameter in density functional theory [4], and is the definition of softness [34,35] which is related to the polarizability [36]. The relationship between the capacitance and hria , also suggests a relationship between the capacitance and S aK , S aq . This would provide a link between an atom’s ability to accept charge, given in terms of its detachment energies, and the localization features of its valence orbital density relative to a 1s hydrogen orbital density as given by entropic measures. A linear regression of the S aK and S aq values with those of the capacitance given in Ref. [33] yields R2 values of 0.995 and 0.993, respectively, for the alkali metals, and 0.992, 0.999 for the alkali earths. This demonstrates a linear tendency between the capacitance and S aK , S aq , in these groups. The slopes of the fits are positive; that is the increased capacitance as one moves down the group is associated with a larger S aq , or a delocalization of the valence charge density. In the case of S aK , the interpretation is that larger capacitance implies that the localization distance, or difference from the hydrogen 1s reference density, increases. 3. Conclusions Relative entropy is used to measure the distance between realistic atomic electron densities and reference densities
Fig. 5. Plot of S aK (green), hria (blue) and S aq (red) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
which are hydrogen-like. We show that relative entropy in position space is able to reproduce the shell effects present in the periodic table (Z = 2–54). Its behavior as a function of Z is very similar to one of its components, the expectation value hri. Hence this distance is extensive and proportional to the spatial extent or volume of the system. The momentum space relative entropy also displays shell effects in the periodic table and its behavior is also similar to that of hri. The results from this particular perspective suggest that the shell effects are also contained in the momentum space density, albeit not as pronounced or to the same extent as in position space. A relative entropy between the highest occupied orbital density and the hydrogen 1s orbital density, is used to obtain an expression dependent on the Shannon entropy of the highest occupied atomic orbital, and a measure of the atomic radius which in turn has been related to the capacitance of an atom. These results provide an interpretation of the capacitance in terms of the Shannon entropy and the localization(delocalization) properties of the atomic valence density. 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] R.P. Sagar, N.L. Guevara, J. Chem. Phys. 123 (2005) 044108. [2] R.P. Sagar, N.L. Guevara, J. Chem. Phys. 124 (2006) 134101. [3] N.L. Guevara, R.P. Sagar, R.O. Esquivel, Phys. Rev. A 67 (2003) 012507. [4] R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford, New York, 1989. [5] M. Kohout, F.R. Wagner, Y. Grin, Int. J. Quantum Chem. 106 (2006) 1499. [6] S.A. Kulkarni, Phys. Rev. A 50 (1994) 2202. [7] R.P. Sagar, A.C.T. Ku, V.H. Smith Jr., A.M. Simas, J. Chem. Phys. 90 (1989) 6520.
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[8] A.J. Thakkar, Adv. Chem. Phys. 128 (2003) 303. [9] C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379, reprinted in Claude Elwood Shannon: collected papers, C.E. Shannon, IEEE Press New York, 1993. [10] S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 2602. [11] I. Bialynicki-Birula, J. Mycielski, Commun. Math. Phys. 44 (1975) 129. [12] M. Hoˆ, R.P. Sagar, J.M. Perez-Jorda, V.H. Smith Jr., R.O. Esquivel, Chem. Phys. Lett. 219 (1994) 15. [13] M. Hoˆ, V.H. Smith Jr., D.F. Weaver, R.P. Sagar, R.O. Esquivel, S. Yamamoto, J. Chem. Phys. 109 (1998) 10620. [14] Z. Huang, S. Kais, Chem. Phys. Lett. 413 (2005) 1. [15] Q. Shi, S. Kais, Chem. Phys. 309 (2005) 127. [16] K.Ch. Chatzisavvas, Ch.C. Moustakidis, C.P. Panos, J. Chem. Phys. 123 (2005) 174111. [17] K.D. Sen, J. Chem. Phys. 123 (2005) 074110. [18] Ch.C. Moustakidis, K.Ch. Chatzisavvas, C.P. Panos, Int. J. Mod. Phys. E 14 (2005) 1087. [19] K.Ch. Chatzisavvas, S.E. Massen, Ch.C. Moustakidis, C.P. Panos, Int. J. Mod. Phys. B 20 (2006) 2189. [20] T.M. Cover, J.A. Thomas, Elements of Information Theory, John Wiley and Sons, 2006. [21] F. De Proft, P.W. Ayers, K.D. Sen, P. Geerlings, J. Chem. Phys. 120 (2004) 9969. [22] M. Hoˆ, R.P. Sagar, H. Schmider, D.F. Weaver, V.H. Smith Jr., Int. J. Quantum Chem. 53 (1995) 627. [23] R.P. Sagar, J.C. Ramirez, R.O. Esquivel, M. Ho, V.H. Smith Jr., J. Chem. Phys. 116 (2002) 9213. [24] A. Borgoo, M. Godefroid, K.D. Sen, F. De Proft, P. Geerlings, Chem. Phys. Lett. 399 (2004) 363. [25] K.B.K. Raju, P.S.V. Nair, K.D. Sen, Chem. Phys. Lett. 170 (1990) 89. [26] R.P. Sagar, J.C. Ramirez, R.O. Esquivel, M. Hoˆ, V.H. Smith Jr., Phys. Rev. A 63 (2001) 022509. [27] E. Clementi, C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [28] E.H. Lieb, Rev. Mod. Phys. 48 (1976) 553. [29] J.C. Angulo, J. Antolin, J. Chem. Phys. 126 (2007) 044106. [30] K.D. Sen, J. Antolin, J.C. Angulo, Phys. Rev. A 76 (2007) 032502. [31] T. Koga, J. Chem. Phys. 116 (2002) 6910. [32] R.J. Boyd, J. Phys. B 11 (1976) L655. [33] J.C. Ellenbogen, Phys. Rev. A 74 (2006) 034501. [34] P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793. [35] P.W. Ayers, Faraday Discuss. 135 (2007) 161. [36] T.K. Ghanty, S.K. Ghosh, Chem. Rev. 116 (1994) 8801.
Relative entropy and atomic structure Robin P. Sagar *, Nicolais L. Guevara Departamento de Quı´mica, Universidad Auto´noma Metropolitana, Apartado Postal 55-534, Iztapalapa, 09340 Me´xico D.F., Mexico Received 12 December 2007; received in revised form 25 January 2008; accepted 11 February 2008 Available online 20 February 2008
Abstract Relative entropy, using a hydrogen-like model density as the reference, is studied in atomic systems. We find that both the position and momentum space measures display the shell effects, being similar in behavior to the expectation value, hri. We also show that the electronic momentum density is closer to the hydrogen-like reference than is the position space density. A connection between relative entropy, and the atomic radius and quantum capacitance, is also discussed. Ó 2008 Elsevier B.V. All rights reserved. PACS: 31.10.+z; 31.90.+s Keywords: Relative entropy; Atomic structure; Shell effects in momentum space; Quantum capacitance
1. Introduction Questions of spatial electronic localization are pervasive in atomic and molecular physics. The localization properties of quantum particles also forms the basis of explanations for phenomena associated with quantum systems in other realms of physics. It is thought that correlation, or the interaction between particles, drives localization, which at its extreme results in decoherence or the appearance of classical behavior. There is a limit to the extent of this localization in quantum systems due to the Heisenberg uncertainty relationship. The congregation(localization) of (different-spin)electrons in particular regions of space gives rise to the characteristic shell effects in atoms, pairing in molecules to form bonds, and superconductivity in condensed matter systems. The limit to the degree of localization that an electronic(fermionic) system may experience is due to the Pauli exclusion principle which keeps same-spin electrons apart. This is a consequence of the antisymmetric requirement of a wave function which obeys Fermi-Dirac statistics. The *
Corresponding author. Tel.: +52 55 58044675; fax: +52 55 5804 4666. E-mail address: [email protected] (R.P. Sagar).
0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.02.009
effects of this Fermi correlation and entropic uncertainty type relationships have been discussed [1,2]. The effects of coulomb correlation on opposite-spin electrons might modify the degree of localization of the system, enhancing or reducing the effects of the Fermi correlation. Recently, studies of coulomb correlation effects on localization using information measures have been presented [1–3]. The spatial electron localization has typically been measured by focusing not on the N-particle wave function, Wðx1 ; x2 ; ; xN Þ, or the N-particle density matrix, but rather on the spin-traced one-electron charge density, qðrÞ ¼
Z
W ðx1 ; x2 ; ;xN ÞWðx1 ;x2 ; ; xN Þdr1 dx2 dxN ; ð1Þ
and functions related to it. The co-ordinate, xj ¼ ðrj ; rj Þ, is a combined space-spin one. The manner in which the antisymmetric property of the electronic wave function is projected onto the charge density and the conditions which need to be obeyed is called the N-representability problem [4]. The study of the localization properties of an electronic charge density is also an examination of how the antisymmetric requirements
R.P. Sagar, N.L. Guevara / Journal of Molecular Structure: THEOCHEM 857 (2008) 72–77
on the wave function influence or restrict the localization characteristics of the density. We also remark that electron localization, and its consequences, may also be formulated and studied in momentum space [5–7], by examining the properties of PðpÞ, the electronic momentum density [8]. This avenue has not received the attention that the position space charge density has. Among the global measures of localization in position space, the Shannon entropy, taken from information theory [9] is defined as [10] Z S q ¼ qðrÞ ln qðrÞdr: ð2Þ qðrÞ is normalized to N, the number of electrons in the system. The momentum space Shannon entropy is given as Z S p ¼ PðpÞ ln PðpÞdp ð3Þ and PðpÞ is normalized to N. On comparing two distributions, the one with the smaller entropic value is indicative of the more localized distribution in the respective space. One important aspect of the entropies is that they are related by an uncertainty-type inequality [11] S uq
þ
S up
P 3ð1 þ ln pÞ
ð4Þ
for unity normalized entropies (S uq;p ), which are calculated from Eqs. (2) and (3) using unity normalized densities, qðrÞ and PðpÞ . The lower bound in Eq. (4) corresponds to N N the harmonic oscillator. This inequality establishes limits on the localization in one space, given the other. S q and S p have been used to study localization–delocalization phenomena in atoms and molecules [12–16], in confined systems [17] and in bosonic systems [18,19]. S q , as a function of atomic number Z, has been shown to demonstrate periodic effects [10]. The noble gas atoms are the most localized of their period. This is not the case for S p which is an increasing function of Z displaying much smaller shell effects or transitions from one period to the next [10,3]. Their behavior is shown in Fig. 1. The difference between the behavior of S q and S p can be phrased in the questions: Is atomic structure, due to the fermionic nature of electrons, different in position space as compared to momentum space, throughout the periodic table? How does the momentum density carry the information about the fermionic behavior of electrons, i.e., shell effects, periodicity? Fermi effects separate same-spin electrons in position space but how does this mechanism apply in momentum space at the one-electron level? Do these effects translate into electrons travelling with different momenta, or with the same momentum, i.e., a condensation of momenta? A natural question concerning all measures of localization is, localized as compared to what? Another measure taken from information theory is the Kullback-Leibler or relative entropy, defined in terms of unity-normalized densities,
qðrÞ qðrÞ ln dr N qo ðrÞ Z PðpÞ PðpÞ ln dp; S pK ¼ N Po ðpÞ
S rK ¼
Z
73
ð5Þ ð6Þ
where qo ðrÞ and Po ðpÞ are suitably chosen reference densities, normalized to N, in the respective spaces. The relative entropy is widely used in information theory as a measure of the distance between the densities, qðrÞ and qo ðrÞ; and PðpÞ and Po ðpÞ [20]. It is the difference between the surprisal or uncertainty of the reference density, averaged over the realistic density, and the Shannon entropy of the realistic density. It is attractive for our purposes since it is a measure which classifies the distance between two densities in terms of the uncertainty or localization properties of the densities. In these definitions, all densities are per electron and normalized to one. As such, they may be considered as intensive quantities. Such unity-normalized densities are also called shape factors in the literature [21]. The positivity of the relative entropy establishes that S rK þ S pK P 0:
ð7Þ
Using this and Eqs. (5) and (6), we obtain an upper bound for the entropy sum, Z 1 qðrÞPðpÞ ln½qo ðrÞPo ðpÞdr dp: ð8Þ Sq þ Sp 6 N Relative entropy has been used to study atomic structure [16,22–25] with different selections for the reference density. Indeed, the physical insights into the problem lies in a suitable selection of the reference density. In this work, we choose for the reference density a model which represents a hydrogenic 1s orbital occupied with N particles. Since it is hydrogenic, it has no interaction or Fermi(exchange) effects. For N > 2, the model density may be considered as being ground state boson-like. The hydrogen-like reference density in position space is given by qo ðrÞ ¼
Z 3 N 2Zr e p
ð9Þ
and is normalized to N. Note that Z = N in atomic systems, however we keep the variables apart for clarity in the discussion that follows. Furthermore, the shape factor or unity-normalized density is just the hydrogenic density (N = 1). In momentum space, the reference density can be obtained from Dirac-Fourier transformation of the hydrogenic orbital to yield, Po ðpÞ ¼
8NZ 5 p2 ðZ 2 þ p2 Þ
4
:
ð10Þ
These model densities have been previously introduced and are called cusp constrained models [23,26]. They have been tested as a model for the Shannon entropy in isoelectronic series, and provide fair results. However, their use in study-
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a
b
c
d
Fig. 1. Plot of (a) S q (red) (b) S Cq (blue) (c) S p (red) (d) S Cp (blue) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
ing the atomic structure is limited since they are not capable of reproducing the shell or periodic effects present in the periodic table. We show their behavior in Fig. 1 (S Cq and S Cp ) which can be compared to S q and S p . This model corresponds to a limiting case as far as localization in position space is concerned. S Cq < S q while S Cp > S p . That is, the realistic densities with interaction and Fermi effects are more delocalized in position space, and more localized in momentum space, in comparison to the models. Thus these effects serve to localize the density in momentum space. These models also yield unity normalized entropies which satisfy the entropic inequality in Eq. (4) [26]. A linear fit of S q against S Cq values, and S p against S Cp , yielded S q ¼ 0:246S Cq þ 16:4 and S p ¼ 0:499S Cp þ 3:70 with R2 coefficients of 0.991 and 0.999, respectively. The fit is better in momentum space since the shell effects in S p are not as pronounced as in S q . The use of the hydrogen-like (HL) density as the reference density in the relative entropy would provide insights into how a realistic one-electron density, containing Fermi and interaction effects, differs from the HL or reference
density with respect to the localization(delocalization) features of the densities. The structure of the periodic table and shell effects exist due to the fermionic nature of electrons. Thus, one would expect that any measure of the difference between a realistic electron (fermion) density and the reference density should reflect the periodic nature or atomic structure. The purpose of this paper is to study the relative entropy, defined with the HL reference densities, for the ground state atoms Z = 2–54, in position and in momentum space, to ascertain its capacity in exhibiting shell effects or periodic structure. Also of interest is a comparison between the position and momentum space measures to examine the extent at which momentum space provides information about shell effects, or more generally, the fermionic nature of electrons in momentum space. Atomic units are used throughout. 2. Results and discussion Relative entropies in position and in momentum space were calculated for the atoms with Z = 2–54, using Har-
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tree-Fock (HF) wave functions [27]. In such functions, one expects the Fermi effects to dominate over the interaction effects for same-spin electrons. Spherically averaged densities were used in these calculations. The relative entropy in position space using the HL reference density is 3 2Z Sq Z N hri ln S rK ¼ ð11Þ N p N R where hri is 4p r3 qðrÞdr. Even though the shape factors or density per particle are used in the calculation of the relative entropy, the result of the leading term,hri, is that it is extensive. The first two terms in Eq. (11) measure the spread or the extent of the localization of qðrÞ. hri can also be considered as a measure of the spatial extent or volume of the atom. We present plots of S rK and hri in Fig. 2. One observes that S rK increases with Z with some exceptions. That is, the distance from the reference density is larger with increasing Z. The exceptions occur in the atoms with configurations which display d orbital filling characteristics. The essential point is that S rK behaves very similarly to hri, exhibiting the shell effects as jumps from one period to another. Thus the leading term dominates the expression in Eq. (11). Note also that these shell effects are accentuated in S rK as compared to S q . The result that S rK is dependent on hri [See Eq. (11)], and behaves similarly to it, may be interpreted as that the fermionic nature of electrons is responsible for the extensivity of the result, i.e., the result depends on a measure of the size or volume of the system. This is consistent with the argument that matter has volume due to the fermionic properties of electrons [28]. The relative entropy in momentum space is, Z 16pZ 3 Sp 8N S pK ¼ p2 PðZpÞ lnð1 þ p2 Þdp ln : N N Z 3 p2 ð12Þ
We show the behavior of S pK vs. atomic number in Fig. 3, which reveals a structure that is very similar in behavior to S rK and hri. This is significant since S pK is a momentum space measure while S rK is a position space measure. Thus the relative entropy shows the shell effects on the densities, due to the fermionic nature of electrons, in momentum space and also in position space. Note also that these effects are magnified in S pK as compared to S p in Fig. 1. The ability of momentum space measures to reproduce shell effects has also been recently commented on [29,30]. One could also ask if the analogy with position space holds in that if it is the behavior of the first term in Eq. (12) which dominates the behavior of S pK . This however is not the case. We show in Fig. 3 that this term is a decreasing function of Z. Furthermore, it is the combination of all of the terms of Eq. (12) which is responsible for the behavior similar to hri. Note also that the integral in Eq. (12) is written in a compact form which groups a factor of 8 ln Z into the third term. If one includes this with the first term the result is an increasing function of Z with no shell effects. The relative sizes of atoms have been reported to be observed in the momentum density by using information about local maxima and minima [31] which differs from the global analysis considered here. In Figs. 2 and 3, one notices that S rK > S pK . Thus the electronic density is behaving more hydrogen-like in momentum space than in position space, with respect to localization features of the density. Furthermore, the shell effects on the density, as seen from S pK , are not as clearly defined in momentum space, especially for larger Z. A linear fit of the S rK values against the S pK ones yields, S rK ¼ 25:0S pK 24:5, with a R2 coefficient of 0.912, which suggests that the relationship is not very linear. S rK and S pK are similar in behavior to the extensive quantity, hri. One can obtain intensive quantities, or per electron measures that do not grow with the size of the system, by considering S rK =N , S pK =N and hri=N . We plot
Fig. 2. Plot of S rK (red, solid) and 2 hri (blue, dash) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
Fig. 3. Plot of S pK (red, solid) and the first term in Eq. (12) (blue, dash) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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these quantities in Fig. 4. These results, based on this particular analysis, show that the shell effects are present in all measures but less defined in momentum space. It is an open question as to whether the position density carries more shell effect information than the momentum density or vice-versa. See Refs. [29,30] for another line of analysis. 2.1. Atomic radii and capacitance One might conclude from the results of the previous section [Eq. (11)] that the relative entropy is related to the atomic radius due to its dependence on hri. However, it is known [32] that hri does not provide a good measure of the atomic radius. AR better measure of the atomic radius is hria , defined as, 4p r3 qa ðrÞdr, where qa ðrÞ is the unity normalized density of the highest occupied Hartree-Fock orbital. If one is interested in constructing a bridge between relative entropy and atomic radii, one may define a relative entropy with dependence on hria , which can be obtained from Eq. (5) if one considers a realistic density corresponding to the density of the highest occupied HF orbital, and a hydrogen 1s orbital reference density. The relative entropy with this definition would be a measure of the difference between the localization properties of the highest occupied orbital density as compared to the 1s orbital density. This yields S aK ¼ 2hria S aq þ ln p; ð13Þ R where S aq ¼ 4p r2 qa ðrÞ ln qa ðrÞdr is the Shannon entropy of the highest occupied atomic orbital. The benefit of such a definition is that it provides a link between the localization features of the highest occupied orbital charge density and atomic radii. We present a plot of S aK , hria and S aq vs. Z in Fig. 5. One notes that the shell effects are very evident and that all measures have an almost identical structure. Differences
Fig. 4. Plot of S rK =N , (blue, dash) 3 S pK =N (green, dot) and hri=N (red, solid) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
between S aK and hria occur for Z = 9,10 where S aK increases while hria and S aq decrease. Within the noble gas atoms, S aq has a stronger relationship with hria since both increase, while S aK increases and decreases within the group. This behavior of S aK is not a general tendency since it increases as does hria within the alkali and alkali earth groups. Recently, the observation has been made [33] that the quantum capacitance of a neutral atom scales linearly with hria . The quantum capacitance, important in the design of nanoscale computing devices, is defined as 1=ðI AÞ where I is the ionization potential and A is the electron affinity of the system. It is also inversely proportional to the hardness, an important parameter in density functional theory [4], and is the definition of softness [34,35] which is related to the polarizability [36]. The relationship between the capacitance and hria , also suggests a relationship between the capacitance and S aK , S aq . This would provide a link between an atom’s ability to accept charge, given in terms of its detachment energies, and the localization features of its valence orbital density relative to a 1s hydrogen orbital density as given by entropic measures. A linear regression of the S aK and S aq values with those of the capacitance given in Ref. [33] yields R2 values of 0.995 and 0.993, respectively, for the alkali metals, and 0.992, 0.999 for the alkali earths. This demonstrates a linear tendency between the capacitance and S aK , S aq , in these groups. The slopes of the fits are positive; that is the increased capacitance as one moves down the group is associated with a larger S aq , or a delocalization of the valence charge density. In the case of S aK , the interpretation is that larger capacitance implies that the localization distance, or difference from the hydrogen 1s reference density, increases. 3. Conclusions Relative entropy is used to measure the distance between realistic atomic electron densities and reference densities
Fig. 5. Plot of S aK (green), hria (blue) and S aq (red) vs. atomic number. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)
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which are hydrogen-like. We show that relative entropy in position space is able to reproduce the shell effects present in the periodic table (Z = 2–54). Its behavior as a function of Z is very similar to one of its components, the expectation value hri. Hence this distance is extensive and proportional to the spatial extent or volume of the system. The momentum space relative entropy also displays shell effects in the periodic table and its behavior is also similar to that of hri. The results from this particular perspective suggest that the shell effects are also contained in the momentum space density, albeit not as pronounced or to the same extent as in position space. A relative entropy between the highest occupied orbital density and the hydrogen 1s orbital density, is used to obtain an expression dependent on the Shannon entropy of the highest occupied atomic orbital, and a measure of the atomic radius which in turn has been related to the capacitance of an atom. These results provide an interpretation of the capacitance in terms of the Shannon entropy and the localization(delocalization) properties of the atomic valence density. 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] R.P. Sagar, N.L. Guevara, J. Chem. Phys. 123 (2005) 044108. [2] R.P. Sagar, N.L. Guevara, J. Chem. Phys. 124 (2006) 134101. [3] N.L. Guevara, R.P. Sagar, R.O. Esquivel, Phys. Rev. A 67 (2003) 012507. [4] R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford, New York, 1989. [5] M. Kohout, F.R. Wagner, Y. Grin, Int. J. Quantum Chem. 106 (2006) 1499. [6] S.A. Kulkarni, Phys. Rev. A 50 (1994) 2202. [7] R.P. Sagar, A.C.T. Ku, V.H. Smith Jr., A.M. Simas, J. Chem. Phys. 90 (1989) 6520.
77
[8] A.J. Thakkar, Adv. Chem. Phys. 128 (2003) 303. [9] C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379, reprinted in Claude Elwood Shannon: collected papers, C.E. Shannon, IEEE Press New York, 1993. [10] S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 2602. [11] I. Bialynicki-Birula, J. Mycielski, Commun. Math. Phys. 44 (1975) 129. [12] M. Hoˆ, R.P. Sagar, J.M. Perez-Jorda, V.H. Smith Jr., R.O. Esquivel, Chem. Phys. Lett. 219 (1994) 15. [13] M. Hoˆ, V.H. Smith Jr., D.F. Weaver, R.P. Sagar, R.O. Esquivel, S. Yamamoto, J. Chem. Phys. 109 (1998) 10620. [14] Z. Huang, S. Kais, Chem. Phys. Lett. 413 (2005) 1. [15] Q. Shi, S. Kais, Chem. Phys. 309 (2005) 127. [16] K.Ch. Chatzisavvas, Ch.C. Moustakidis, C.P. Panos, J. Chem. Phys. 123 (2005) 174111. [17] K.D. Sen, J. Chem. Phys. 123 (2005) 074110. [18] Ch.C. Moustakidis, K.Ch. Chatzisavvas, C.P. Panos, Int. J. Mod. Phys. E 14 (2005) 1087. [19] K.Ch. Chatzisavvas, S.E. Massen, Ch.C. Moustakidis, C.P. Panos, Int. J. Mod. Phys. B 20 (2006) 2189. [20] T.M. Cover, J.A. Thomas, Elements of Information Theory, John Wiley and Sons, 2006. [21] F. De Proft, P.W. Ayers, K.D. Sen, P. Geerlings, J. Chem. Phys. 120 (2004) 9969. [22] M. Hoˆ, R.P. Sagar, H. Schmider, D.F. Weaver, V.H. Smith Jr., Int. J. Quantum Chem. 53 (1995) 627. [23] R.P. Sagar, J.C. Ramirez, R.O. Esquivel, M. Ho, V.H. Smith Jr., J. Chem. Phys. 116 (2002) 9213. [24] A. Borgoo, M. Godefroid, K.D. Sen, F. De Proft, P. Geerlings, Chem. Phys. Lett. 399 (2004) 363. [25] K.B.K. Raju, P.S.V. Nair, K.D. Sen, Chem. Phys. Lett. 170 (1990) 89. [26] R.P. Sagar, J.C. Ramirez, R.O. Esquivel, M. Hoˆ, V.H. Smith Jr., Phys. Rev. A 63 (2001) 022509. [27] E. Clementi, C. Roetti, At. Data Nucl. Data Tables 14 (1974) 177. [28] E.H. Lieb, Rev. Mod. Phys. 48 (1976) 553. [29] J.C. Angulo, J. Antolin, J. Chem. Phys. 126 (2007) 044106. [30] K.D. Sen, J. Antolin, J.C. Angulo, Phys. Rev. A 76 (2007) 032502. [31] T. Koga, J. Chem. Phys. 116 (2002) 6910. [32] R.J. Boyd, J. Phys. B 11 (1976) L655. [33] J.C. Ellenbogen, Phys. Rev. A 74 (2006) 034501. [34] P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793. [35] P.W. Ayers, Faraday Discuss. 135 (2007) 161. [36] T.K. Ghanty, S.K. Ghosh, Chem. Rev. 116 (1994) 8801.