- Email: [email protected]
The Fisher–Shannon information plane for atoms
Physics Letters A 372 (2008) 2428–2430 www.elsevier.com/locate/pla
The Fisher–Shannon information plane for atoms J.B. Szabó a , K.D. Sen b , Á. Nagy a,∗ a Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Received 27 November 2007; accepted 4 December 2007 Available online 8 December 2007 Communicated by V.M. Agranovich
Abstract The Fisher–Shannon information product and plane for atoms are presented analytically assuming Thomas–Fermi–Gáspár statistical model. A comparison with the Hartree–Fock densities reveals that the atomic shell structure is inadequately expressed information theoretically in the statistical model. The shape complexity measure of Lopez et al. is found to have a better large Z dependence than the one obtained from nonrelativistic Hartree–Fock densities. © 2007 Elsevier B.V. All rights reserved.
The concept of Shannon information [1] has been applied in various fields of physics for a long time. There have been efforts to elucidate physical and chemical properties of atomic and molecular systems from an information theoretical point of view [2–13]. Nowadays Fisher information [14] is also playing an increasing role. The importance of Fisher information in quantum mechanics was first noticed by Sears et al. [2]. The equations of non-relativistic quantum mechanics [15] were derived using the principle of minimum Fisher information [16]. The timeindependent Kohn–Sham equations and the time-dependent Euler equation of the density functional theory were also derived from the principle of minimum Fisher information [17, 18]. The Fisher information of single-particle systems with a central potential was also determined [19] and that of a twoelectron ‘entangled artificial’ atom proposed by Moshinsky was also studied [20]. In recent papers atomic Fisher information [21] was investigated and phase space Fisher information was defined [22]. In a recent paper Romera and Dehesa [23] introduced the concept of Fisher–Shannon information plane as a specific correlation measure. In this Letter the Fisher–Shannon information plane for atoms is presented. The electron density, the basic * Corresponding author.
E-mail address: [email protected] (Á. Nagy). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.12.008
variable of the density functional theory is defined as 2 (r) = N Ψ (r, r2 , . . . , rN ) dr2 · · · drN ,
(1)
where Ψ (r, r2 , . . . , rN ) is the normalized wave function of the N -electron system. In this Letter we use the shape function σ (r) = (r)/N , that is the electron density normalized to 1. The Shannon information is defined as Sσ = − σ (r) ln σ (r) dr. (2) The Shannon information power 1 exp(2/3)Sσ (3) 2πe is introduced to measure the spreading of the density [23]. The Fisher information [∇σ (r)]2 Iσ = (4) dr σ (r)
Jσ =
measures the ‘narrowness’ of the electron distribution. Iσ and Jσ provide complementary description of the electron localization, Iσ is a local, Jσ is a global measure of the smoothness. Romera and Dehesa [23] introduced the concept of the information product 1 P = J σ Iσ . 3
(5)
J.B. Szabó et al. / Physics Letters A 372 (2008) 2428–2430
2429
They showed that a uniform scaling of coordinates leads to the scaled density σλ (r) = λ3 σ (λr),
(6)
the scaled Fisher information I σλ = λ 2 Iσ
(7)
and the scaled Shannon information power Jσλ = λ−2 Iσ .
(8)
Consequently, the information product P is invariant under uniform scaling. To derive an analytic Z-dependence of the information product P of atoms consider first the universal electron density of Gáspár [24] e−3ax , σG (r) = B (1 + Ax)3
Fig. 1.
(9)
where the quantity x is defined as x=
r . μ
μ = ν/Z 1/3 , ν = 1/2(3π/4)2/3 = 0.8853 and B = C 3 Z. The constants have the following values: C = 3.1, a = 0.04 and A = 9. In a recent paper [25] we showed that for neutral atoms (N = Z) the Shannon information coming from this density has the form (11)
where c1 = 4.321345 and c2 = 0.636126. It is interesting to compare SσG with the Thomas–Fermi–Shannon information [3]: SσTF = c3 − ln Z,
(12)
where c3 = 5.59. Then, the Shannon entropy power can be written as JσG = AG Z −c
G
(13)
and JσTF = ATF Z −2/3 ,
(14)
where cG = 2c2 /3, AG =
1 exp(2c1 /3) 2πe
(15)
and 1 exp(2c3 /3). (16) 2πe In a recent paper [21] the Z-dependence of the Fisher information has been studied. Accurate numerical calculations showed an almost linear dependence on the atomic number, while Gáspár’s–Fisher information was found to be proportional to the two-third power of the atomic number: ATF =
IσG = c4 Z 2/3 ,
2 a πνC 3 1 1 121 2 23 3 + − η− η + η ν A 3 3 8 2 15 + 3η2 2 + 12η + η2 Ei(−3η) , 2
c4 = 9 (10)
SσG = c1 − c2 ln Z,
where
(17)
(18)
where η = a/A and x Ei(x) = −∞
et dt. t
(19)
c4 can be evaluated: c4 = 27.463. The Fisher information calculated by Romera [26] using Koga–Roothaan–Hartree–Fock atomic wave functions [27] has been fitted as IσHF = KZ k ,
(20)
where k = 1.0801 and K = 5.8325. Now, we turn to the information product P (Eq. (5)): P G = B G Z 2(1−c2 )/3 ,
(21)
where B G = c4 AG /3. It is interesting to note that if we combine Gáspár’s–Fisher information and the Thomas–Fermi Shannon information power we obtain a constant for the information product 1 P TFG = B TFG = c4 ATF . 3
(22)
To compare these results with the more realistic Z-dependence of the atomic information product, we used the Koga– Roothaan–Hartree–Fock functions [27] for neutral atoms with atomic numbers Z = 2–102 to calculate the information product PHF , which is the product of exponential Shannon entropy and the Fisher information measure in the position space. A plot of PHF as a function of Z is displayed in Fig. 1. The shell structure is clearly displayed in this plot which is the departure from the linear behavior predicted by the Thomas–Fermi– Gáspár product P TFG .
2430
J.B. Szabó et al. / Physics Letters A 372 (2008) 2428–2430
Finally, we would like to consider the shape complexity measure [28] defined by CLMC = H . D,
(23)
where H denotes a measure of information and D represents the so-called disequilibrium or the distance from equilibrium (most probable state). The form of CLMC is designed such that it vanishes for the two extreme probability distributions corresponding to perfect order (H = 0) and maximum disorder (D = 0), respectively. In particular, H = eSr , D = ρ 2 (r) dr.
(24) (25)
ical Innovation Foundation and the Ministry of Science and Technology. K.D.S. is grateful to Professor S. Suhai, DKFZ, Heidelberg for warm hospitality and encouragement. This work was also supported by the grant OTKA No. T 67923. References [1] [2] [3] [4] [5] [6] [7]
Using the Gáspár density in Eq. (9) and the Shannon entropy given by Eq. (11) we combine the following result for the disequilibrium D = 0.3403901Z 3 ,
(26)
to obtain G CLMC = 25.5641Z 2.36 .
(27)
s derived from the Hartree–Fock densities [29] A fit to the CLMC leads to HF = 5.6351Z 0.5603 , CLMC
(28)
with R 2 value of 0.8663. It was found in [29] that CLMC obtained from relativistic densities increases sharply at large Z. G The statistical complexity measure CLMC obtained here shows this behaviour leading to, in this sense, a more reasonable result than the non-relativistic Hartree–Fock method, as Z approaches a large value. In summary, based on the Thomas–Fermi–Gáspár statistical model, we have obtained the analytic results for Fisher– Shannon information product. As compared to the Hartree– Fock densities, the atomic shell structure is inadequately expressed in the statistical model from the viewpoint of inforHF , the shape complexity mation theory. As compared to CLMC G measure, CLMC , is found to exhibit a better dependence when Z becomes large, suggesting that atoms grow in complexity as Z becomes large, a conclusion supported by the relativistic Dirac–Fock calculations [29]. Acknowledgements This Letter was written in the frame of the Bilateral Intergovernmental Scientific and Technological Cooperation between Hungary and India sponsored by the Research and Technolog-
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
[28] [29]
C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. S.B. Sears, R.G. Parr, U. Dinur, Israel J. Chem. 19 (1980) 165. S.R. Gadre, Phys. Rev. A 30 (1984) 620. S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 2602. R.J. Yánez, W. Van Assche, J.S. Dehesa, Phys. Rev. A 32 (1994) 3065. M. Hó, R.P. Sagar, V.H. Smith Jr., R.O. Esquivel, J. Phys. B 27 (1994) 5149. M. Hó, R.P. Sagar, J.M. Pérez-Jordá, V.H. Smith Jr., R.O. Esquivel, Chem. Phys. Lett. 219 (1994) 15. Á. Nagy, R.G. Parr, Int. J. Quantum Chem. 58 (1996) 323. N.L. Guevara, R.P. Sagar, R.O. Esquivel, Phys. Rev. A 67 (2003) 012507. N.L. Guevara, R.P. Sagar, R.O. Esquivel, J. Chem. Phys. 119 (2003) 7030. N.L. Guevara, R.P. Sagar, R.O. Esquivel, J. Chem. Phys. 122 (2005) 084101. Ch.C. Moustakidis, S.E. Massen, Phys. Rev. B 71 (2005) 045102. K.D. Sen, J. Chem. Phys. 123 (2005) 074110. R.A. Fisher, Proc. Cambridge Philos. Soc. 22 (1925) 700. M. Reginatto, Phys. Rev. A 58 (1998) 1775. B.R. Frieden, Physics from Fisher Information. A Unification, Cambridge Univ. Press, 1998. R. Nalewajski, Chem. Phys. Lett. 372 (2003) 28. Á. Nagy, J. Chem. Phys. 119 (2003) 9401. E. Romera, P. Sánchez-Morena, J.S. Dehesa, Chem. Phys. Lett. 414 (2005) 468. Á. Nagy, Chem. Phys. Lett. 425 (2006) 157. Á. Nagy, K.D. Sen, Phys. Lett. A 360 (2006) 291. I. Hornyák, Á. Nagy, Chem. Phys. Lett. 437 (2007) 132. E. Romera, J.S. Dehesa, J. Chem. Phys. 120 (2004) 8906. R. Gáspár, Acta Phys. Hung. 3 (1954) 263 (in German), J. Mol. Struct. Theochem. 501–502 (2000) 1 (English translation T. Gál). J.B. Szabó, Á. Nagy, Acta Phys. Chim. Debr. 40 (2006) 131. E. Romera, Mol. Phys. 100 (2002) 3325. T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Int. J. Quantum Chem. 71 (1999) 491; T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Theor. Chem. Accounts 104 (2000) 411. R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321; R.G. Catalan, J. Garay, R. Lopez-Ruiz, Phys. Rev. E 66 (2002) 011102. A. Borgoo, F. De Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett. 44 (2007) 186.
The Fisher–Shannon information plane for atoms J.B. Szabó a , K.D. Sen b , Á. Nagy a,∗ a Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Received 27 November 2007; accepted 4 December 2007 Available online 8 December 2007 Communicated by V.M. Agranovich
Abstract The Fisher–Shannon information product and plane for atoms are presented analytically assuming Thomas–Fermi–Gáspár statistical model. A comparison with the Hartree–Fock densities reveals that the atomic shell structure is inadequately expressed information theoretically in the statistical model. The shape complexity measure of Lopez et al. is found to have a better large Z dependence than the one obtained from nonrelativistic Hartree–Fock densities. © 2007 Elsevier B.V. All rights reserved.
The concept of Shannon information [1] has been applied in various fields of physics for a long time. There have been efforts to elucidate physical and chemical properties of atomic and molecular systems from an information theoretical point of view [2–13]. Nowadays Fisher information [14] is also playing an increasing role. The importance of Fisher information in quantum mechanics was first noticed by Sears et al. [2]. The equations of non-relativistic quantum mechanics [15] were derived using the principle of minimum Fisher information [16]. The timeindependent Kohn–Sham equations and the time-dependent Euler equation of the density functional theory were also derived from the principle of minimum Fisher information [17, 18]. The Fisher information of single-particle systems with a central potential was also determined [19] and that of a twoelectron ‘entangled artificial’ atom proposed by Moshinsky was also studied [20]. In recent papers atomic Fisher information [21] was investigated and phase space Fisher information was defined [22]. In a recent paper Romera and Dehesa [23] introduced the concept of Fisher–Shannon information plane as a specific correlation measure. In this Letter the Fisher–Shannon information plane for atoms is presented. The electron density, the basic * Corresponding author.
E-mail address: [email protected] (Á. Nagy). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.12.008
variable of the density functional theory is defined as 2 (r) = N Ψ (r, r2 , . . . , rN ) dr2 · · · drN ,
(1)
where Ψ (r, r2 , . . . , rN ) is the normalized wave function of the N -electron system. In this Letter we use the shape function σ (r) = (r)/N , that is the electron density normalized to 1. The Shannon information is defined as Sσ = − σ (r) ln σ (r) dr. (2) The Shannon information power 1 exp(2/3)Sσ (3) 2πe is introduced to measure the spreading of the density [23]. The Fisher information [∇σ (r)]2 Iσ = (4) dr σ (r)
Jσ =
measures the ‘narrowness’ of the electron distribution. Iσ and Jσ provide complementary description of the electron localization, Iσ is a local, Jσ is a global measure of the smoothness. Romera and Dehesa [23] introduced the concept of the information product 1 P = J σ Iσ . 3
(5)
J.B. Szabó et al. / Physics Letters A 372 (2008) 2428–2430
2429
They showed that a uniform scaling of coordinates leads to the scaled density σλ (r) = λ3 σ (λr),
(6)
the scaled Fisher information I σλ = λ 2 Iσ
(7)
and the scaled Shannon information power Jσλ = λ−2 Iσ .
(8)
Consequently, the information product P is invariant under uniform scaling. To derive an analytic Z-dependence of the information product P of atoms consider first the universal electron density of Gáspár [24] e−3ax , σG (r) = B (1 + Ax)3
Fig. 1.
(9)
where the quantity x is defined as x=
r . μ
μ = ν/Z 1/3 , ν = 1/2(3π/4)2/3 = 0.8853 and B = C 3 Z. The constants have the following values: C = 3.1, a = 0.04 and A = 9. In a recent paper [25] we showed that for neutral atoms (N = Z) the Shannon information coming from this density has the form (11)
where c1 = 4.321345 and c2 = 0.636126. It is interesting to compare SσG with the Thomas–Fermi–Shannon information [3]: SσTF = c3 − ln Z,
(12)
where c3 = 5.59. Then, the Shannon entropy power can be written as JσG = AG Z −c
G
(13)
and JσTF = ATF Z −2/3 ,
(14)
where cG = 2c2 /3, AG =
1 exp(2c1 /3) 2πe
(15)
and 1 exp(2c3 /3). (16) 2πe In a recent paper [21] the Z-dependence of the Fisher information has been studied. Accurate numerical calculations showed an almost linear dependence on the atomic number, while Gáspár’s–Fisher information was found to be proportional to the two-third power of the atomic number: ATF =
IσG = c4 Z 2/3 ,
2 a πνC 3 1 1 121 2 23 3 + − η− η + η ν A 3 3 8 2 15 + 3η2 2 + 12η + η2 Ei(−3η) , 2
c4 = 9 (10)
SσG = c1 − c2 ln Z,
where
(17)
(18)
where η = a/A and x Ei(x) = −∞
et dt. t
(19)
c4 can be evaluated: c4 = 27.463. The Fisher information calculated by Romera [26] using Koga–Roothaan–Hartree–Fock atomic wave functions [27] has been fitted as IσHF = KZ k ,
(20)
where k = 1.0801 and K = 5.8325. Now, we turn to the information product P (Eq. (5)): P G = B G Z 2(1−c2 )/3 ,
(21)
where B G = c4 AG /3. It is interesting to note that if we combine Gáspár’s–Fisher information and the Thomas–Fermi Shannon information power we obtain a constant for the information product 1 P TFG = B TFG = c4 ATF . 3
(22)
To compare these results with the more realistic Z-dependence of the atomic information product, we used the Koga– Roothaan–Hartree–Fock functions [27] for neutral atoms with atomic numbers Z = 2–102 to calculate the information product PHF , which is the product of exponential Shannon entropy and the Fisher information measure in the position space. A plot of PHF as a function of Z is displayed in Fig. 1. The shell structure is clearly displayed in this plot which is the departure from the linear behavior predicted by the Thomas–Fermi– Gáspár product P TFG .
2430
J.B. Szabó et al. / Physics Letters A 372 (2008) 2428–2430
Finally, we would like to consider the shape complexity measure [28] defined by CLMC = H . D,
(23)
where H denotes a measure of information and D represents the so-called disequilibrium or the distance from equilibrium (most probable state). The form of CLMC is designed such that it vanishes for the two extreme probability distributions corresponding to perfect order (H = 0) and maximum disorder (D = 0), respectively. In particular, H = eSr , D = ρ 2 (r) dr.
(24) (25)
ical Innovation Foundation and the Ministry of Science and Technology. K.D.S. is grateful to Professor S. Suhai, DKFZ, Heidelberg for warm hospitality and encouragement. This work was also supported by the grant OTKA No. T 67923. References [1] [2] [3] [4] [5] [6] [7]
Using the Gáspár density in Eq. (9) and the Shannon entropy given by Eq. (11) we combine the following result for the disequilibrium D = 0.3403901Z 3 ,
(26)
to obtain G CLMC = 25.5641Z 2.36 .
(27)
s derived from the Hartree–Fock densities [29] A fit to the CLMC leads to HF = 5.6351Z 0.5603 , CLMC
(28)
with R 2 value of 0.8663. It was found in [29] that CLMC obtained from relativistic densities increases sharply at large Z. G The statistical complexity measure CLMC obtained here shows this behaviour leading to, in this sense, a more reasonable result than the non-relativistic Hartree–Fock method, as Z approaches a large value. In summary, based on the Thomas–Fermi–Gáspár statistical model, we have obtained the analytic results for Fisher– Shannon information product. As compared to the Hartree– Fock densities, the atomic shell structure is inadequately expressed in the statistical model from the viewpoint of inforHF , the shape complexity mation theory. As compared to CLMC G measure, CLMC , is found to exhibit a better dependence when Z becomes large, suggesting that atoms grow in complexity as Z becomes large, a conclusion supported by the relativistic Dirac–Fock calculations [29]. Acknowledgements This Letter was written in the frame of the Bilateral Intergovernmental Scientific and Technological Cooperation between Hungary and India sponsored by the Research and Technolog-
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
[28] [29]
C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. S.B. Sears, R.G. Parr, U. Dinur, Israel J. Chem. 19 (1980) 165. S.R. Gadre, Phys. Rev. A 30 (1984) 620. S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 2602. R.J. Yánez, W. Van Assche, J.S. Dehesa, Phys. Rev. A 32 (1994) 3065. M. Hó, R.P. Sagar, V.H. Smith Jr., R.O. Esquivel, J. Phys. B 27 (1994) 5149. M. Hó, R.P. Sagar, J.M. Pérez-Jordá, V.H. Smith Jr., R.O. Esquivel, Chem. Phys. Lett. 219 (1994) 15. Á. Nagy, R.G. Parr, Int. J. Quantum Chem. 58 (1996) 323. N.L. Guevara, R.P. Sagar, R.O. Esquivel, Phys. Rev. A 67 (2003) 012507. N.L. Guevara, R.P. Sagar, R.O. Esquivel, J. Chem. Phys. 119 (2003) 7030. N.L. Guevara, R.P. Sagar, R.O. Esquivel, J. Chem. Phys. 122 (2005) 084101. Ch.C. Moustakidis, S.E. Massen, Phys. Rev. B 71 (2005) 045102. K.D. Sen, J. Chem. Phys. 123 (2005) 074110. R.A. Fisher, Proc. Cambridge Philos. Soc. 22 (1925) 700. M. Reginatto, Phys. Rev. A 58 (1998) 1775. B.R. Frieden, Physics from Fisher Information. A Unification, Cambridge Univ. Press, 1998. R. Nalewajski, Chem. Phys. Lett. 372 (2003) 28. Á. Nagy, J. Chem. Phys. 119 (2003) 9401. E. Romera, P. Sánchez-Morena, J.S. Dehesa, Chem. Phys. Lett. 414 (2005) 468. Á. Nagy, Chem. Phys. Lett. 425 (2006) 157. Á. Nagy, K.D. Sen, Phys. Lett. A 360 (2006) 291. I. Hornyák, Á. Nagy, Chem. Phys. Lett. 437 (2007) 132. E. Romera, J.S. Dehesa, J. Chem. Phys. 120 (2004) 8906. R. Gáspár, Acta Phys. Hung. 3 (1954) 263 (in German), J. Mol. Struct. Theochem. 501–502 (2000) 1 (English translation T. Gál). J.B. Szabó, Á. Nagy, Acta Phys. Chim. Debr. 40 (2006) 131. E. Romera, Mol. Phys. 100 (2002) 3325. T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Int. J. Quantum Chem. 71 (1999) 491; T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Theor. Chem. Accounts 104 (2000) 411. R. López-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209 (1995) 321; R.G. Catalan, J. Garay, R. Lopez-Ruiz, Phys. Rev. E 66 (2002) 011102. A. Borgoo, F. De Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett. 44 (2007) 186.