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Maximum Rényi entropy principle and the generalized Thomas–Fermi model
Physics Letters A 373 (2009) 844–846
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Maximum Rényi entropy principle and the generalized Thomas–Fermi model Á. Nagy a , E. Romera b,∗ a b
Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 10 December 2008 Received in revised form 27 December 2008 Accepted 2 January 2009 Available online 8 January 2009 Communicated by V.M. Agranovich
The Thomas–Fermi model is generalized using the maximum Rényi entropy principle. A simple relation between the dimension and the Rényi parameter is emphasized. The Rényi entropy of atoms is approximately expressed with the logarithm of the kinetic energy. An approximate relation between the value of the density at the nucleus and the kinetic energy is found. © 2009 Elsevier B.V. All rights reserved.
Recently, there has been a growing interest in using information concepts in several fields of science. Although Rényi entropy has been introduced by Rényi [1] as early as 1961, this kind of entropy has only recently obtained a wide range of application e.g. in the analysis of quantum entanglement [2], quantum communication protocols [3], quantum correlations [4] or localization properties [5]. The present authors have recently studied the Rényi information of atoms both in position and momentum spaces [6]. In this Letter we explore how the maximum Rényi entropy principle can be applied to generalize the Thomas–Fermi model. Earlier the principle of extreme physical information [7] was used to derive the Euler and the Kohn–Sham equations of the density functional theory [8,9]. There are several ways to derive the Euler equation of the density functional theory. Consider now a system of N electrons moving in a local external potential v. Imagine a system of noninteracting electrons having the same density (r) as that of the original interacting electrons. The kinetic energy T s [] of the noninteracting system is defined by the Levy–Lieb [10] constraint search as
T s [] = min Φ| Tˆ |Φ = Φ[] Tˆ Φ[] , Φ→
(1)
where Tˆ is kinetic energy operator and the minimum is searched over the non-interacting wave functions Φ having the density . Then the Euler equation can be derived minimizing the noninteracting kinetic energy T s [] with the constraints that the density is fixed (the same as the interacting one) and that the density is normalized to the number of electrons N:
T s [ ] +
v KS (r)(r) dr − μ
(r) dr.
(2)
The Lagrange multipliers v KS (r) and μ are the Kohn–Sham potential and the chemical potential, respectively. The variation of this expression with respect to the density leads to the Euler equation of the density functional theory:
δ T s [ ] + v KS (r) = μ. δ (r)
(3)
Though Eq. (1) gives the formal definition of the non-interacting kinetic energy, its exact form as a functional of the density is unknown. There are, however, several approximations for it. The simplest one is the Thomas–Fermi functional [11]:
(r)5/3 dr,
T TF [] = C TF
(4)
with C TF =
3 2 2/3 3π .
(5)
10
The Thomas–Fermi functional was generalized for arbitrary dimension [12–16]. The kinetic energy of a D-dimensional system has the form
(r)1+2/ D dr,
T D [ ] = C D
(6)
with CD =
D 2( D + 2)
D 2K D
2/ D .
(7)
The factor K D can be given by the recurrence relation K D +2 =
KD 2π D
,
(8)
with
*
Corresponding author. E-mail address: [email protected] (E. Romera).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.01.004
K1 =
1
π
(9)
Á. Nagy, E. Romera / Physics Letters A 373 (2009) 844–846
and 1
K2 =
(10)
.
2π
The maximum Rényi entropy principle is a generalization of the maximum entropy principle developed by Jaynes [17]. In Jaynes’ principle the Shannon entropy [18]
Sf =−
f (r) ln f (r) dr
(11)
is maximized under proper conditions including the normalization of the probability density f (r)
f (r) dr = 1.
(12)
The maximum Rényi entropy principle has been used recently for systems with power-law Hamiltonians [19]. The Rényi entropy of order α for a D-dimensional probability density function f (r1 , . . . , r D ) normalized to one is defined as R αf =
1 1−α
f α (r) dr,
ln
(13)
where r stands for r1 , . . . , r D The limit α → 1 gives the Shannon entropy. Now we seek the extremum of the Rényi entropy of order α under the conditions: 1. The density is normalized to 1. A Lagrange multiplicator ν is introduced. 2. The density is kept fixed. This requirement is taken into account by a local potential w (r). This constraint is used to ensure that the density of the non-interacting system be equal to that of the interacting one. Then, the extremum of the functional
R αf − ν
f (r) dr +
w (r) f (r) dr
(14)
is considered. The variation leads to the equation:
δ R αf
− ν + w (r) = 0.
δf
(15)
The functional derivative of the Rényi entropy (13) is
δ R αf
=
δf
α 1−α
f α −1 f α dr
.
(16)
Then Eq. (15) takes the form:
α 1−α
f α −1
+ w (r) = ν
f α dr
(17)
or it can also be written as
˜ (r) = ν˜ c f α −1 + w with
˜ = w
ν˜ =
c (1 − α )
α c (1 − α )
α
(18)
w
f α dr,
(19)
ν
f α dr
(20)
and c is any constant. Comparing Eq. (18) with Eq. (4) we immediately see that the Thomas–Fermi case recovers with the choice α = 5/3 and c = cTF . Eq. (18) is a generalization of the Thomas– Fermi model derived from the maximum Rényi entropy principle. The Thomas–Fermi model was worked out for the free-electron gas and can be applied in systems with slowly changing densities. It is
845
frequently used even today, for example, to study the structure of the new artificial atoms or quantum dots [20], Bose–Einstein condensates in traps [21], systems under pressure [22], simulations in chemical and biological systems [23]. Here we use the Thomas–Fermi kinetic energy functional not the original Thomas–Fermi model itself, that is, the Thomas–Fermi model in a more general sense. Comparing Eq. (18) with Eq. (6) we can recover the D-dimensional Thomas–Fermi model. We are led to the simple relationship between the dimension D and the parameter α
α=1+
2 D
(21)
.
In this case there is a physical meaning of the Rényi parameter α as it is related to the dimension. The relationship between the kinetic energy and Shannon information entropy (that is, α = 1) was studied earlier. Massen and Panos [25] fitted numerical values for several nuclei and atomic clusters and found that S = aT + bT ln (cT ),
(22)
where a, b and c are constants. Now, we extend this kind of investigation to cover several other values of α . The Thomas–Fermi model is a crude approximation for atoms (though it is correct in the limit of large atomic numbers). Applying the generalization presented here, we studied the relation between the kinetic energy functional and the Rényi entropy of order α for neutral atoms with Z = 1–36. The Rényi entropy has been numerically calculated with the density obtained from ground-state Roothaan–Hartree–Fock (RHF) wave functions [24]. The approximate relation between the kinetic energy and the Rényi entropy is written as Rα ρ = a(α ) ln T + b(α ),
(23)
where the parameters are given in Table 1. The fit is not very accurate for values of α about or below 1.5. Let us point out that limα →∞ R α ρ = − ln ||ρ ||∞ = − ln ρ (0), be-
cause ||ρ || p ≡ ( ρ p dr)1/ p and ||ρ ||∞ ≡ sup |ρ | [26]. We make use of the fact that the ground state density of atoms is a nonincreasing function of r. Consequently, one finds that when α goes to infinity expression (23) gives us − ln ρ (0) = a(∞) ln T + b(∞), so we have obtained a relation between the value of the density at the nucleus ρ (0) and the kinetic energy ρ (0) = Z T −a(∞) e −b(∞) . Figs. 1 and 2 present the Rényi entropy (13) and the approximate expression for the Rényi entropy (23) as a function of atomic number for the values of α ∞, 10, 4, 5/3 and 1.2. Approaching to the value of 1, the shell structure is getting more apparent in the Rényi entropy, which is not showed by the approximate expression (23). We would like to emphasize that the true Roothaan– Hartree–Fock densities were applied in these calculation. In summary, the central result of this Letter is the derivation of the generalized Thomas–Fermi model using the maximum Rényi Table 1 Fitted parameters a(α ) and b(α ) for Eq. (23). The fit is very good for 0.999) and no so good for 4 α 1.5 (0.998 c.c. 0.996).
α ∞ 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 5.0 4.0
a(α ) −0.902 −0.873 −0.870 −0.867 −0.863 −0.858 −0.851 −0.843 −0.831 −0.813 −0.784
b(α ) 0.299 0.936 0.972 1.014 1.061 1.116 1.182 1.262 1.362 1.489 1.660
α 3.5 3.0 2.5 2.3 2.1 2.0 1.9 1.8 1.7 1.6 1.5
a(α ) −0.760 −0.726 −0.674 −0.646 −0.612 −0.593 −0.573 −0.550 −0.526 −0.501 −0.475
α 4 (c.c. b(α ) 1.771 1.908 2.089 2.182 2.296 2.364 2.441 2.530 2.633 2.753 2.894
846
Á. Nagy, E. Romera / Physics Letters A 373 (2009) 844–846
tained an approximate expression between the Rényi entropy and the logarithm of the kinetic energy of atoms and as a special case an approximate relation between the density at the nucleus and the kinetic energy. Acknowledgements Á.N. acknowledges grant OTKA No. T67923. E.R. acknowledges the Spanish project FQM-2725 (Junta de Andalucía). References
(α )
α = 4, 10, ∞ (from top
(α )
α = 1.2, 5/3 (from top
Fig. 1. Plot of R ρ (+ points) and a(α ) ln T + b(α ) (line) for to bottom) vs atomic number.
Fig. 2. Plot of R ρ (+ points) and a(α ) ln T + b(α ) (line) for to bottom) vs atomic number.
entropy principle. In this case there is a physical meaning of the Rényi parameter α as it is related to the dimension. Finally, we ob-
[1] A. Rényi, in: Proceedings of Fourth Berkeley Symposium on Mathematics, Statistics and Probability, vol. 1, Univ. California Press, Berkeley, 1961, p. 547. [2] O. Gühne, M. Lewenstein, Phys. Rev. A 70 (2004) 022316; G. Adesso, A. Serafini, F. Illuminati, Phys. Rev. A 70 (2004) 022318; A. Bovino, G. Castagnolli, A. Ekert, P. Horodecki, C.M. Alves, A.V. Serfienko, Phys. Rev. Lett. 95 (2005) 240407. [3] R. Renner, N. Gisin, B. Kraus, Phys. Rev. A 72 (2005) 012332; V. Giovanetti, S. Lloyd, Phys. Rev. A 69 (2004) 062307. [4] P. Lévay, S. Nagy, J. Pipek, Phys. Rev. A 72 (2005) 022302. [5] D.G. Arbo, C.O. Reinhold, J. Burgdörfer, A.K. Pattanayak, C.L. Stokely, W. Zhao, J.C. Lancaster, F.B. Dunning, Phys. Rev. A 67 (2003) 063401. [6] E. Romera, Á. Nagy, Phys. Lett. A 372 (2008) 4918. [7] B.R. Frieden, Am. J. Phys. 57 (1989) 1004. [8] Á. Nagy, J. Chem. Phys. 119 (2003) 9401. [9] R. Nalewajski, Chem. Phys. Lett. 372 (2003) 28. [10] M. Levy, Proc. Natl. Acad. Sci. USA 76 (1979) 6002; M. Lieb, Int. J. Quantum Chem. 24 (1982) 243. [11] L.H. Thomas, Proc. Cambridge Philos. Soc. 23 (1927) 542; E. Fermi, Z. Phys. 48 (1928) 73. [12] G.F. Kventsel, J. Katriel, Phys. Rev. A 24 (1981) 2299. [13] N.H. March, J. Math. Phys. 26 (1985) 554. [14] A. Holas, N.H. March, Philos. Mag. 69 (1994) 787. [15] N.H. March, S. Kais, Int. J. Quantum Chem. 65 (1997) 411. [16] B.K. Shivamoggi, Physica A 248 (1998) 195. [17] E.T. Yanes, Phys. Rev. A 106 (1957) 620; E.T. Yanes, Phys. Rev. A 107 (1957) 171. [18] C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. [19] A.G. Bashrirov, Phys. Rev. Lett. 93 (2004) 1306017. [20] J. Sanudo, A.F. Pacheco, Can. J. Phys. 84 (2006) 833. [21] P. Schuck, X. Vinas, Phys. Rev. A 61 (2000) 043603. [22] E. Cappelluti, L. Delle Site, Physica A 303 (2002) 481. [23] M. Hodak, W. Lu, http://meetings.aps.org/link/BAPS.2006.MAR.V27.8. [24] T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Int. J. Quantum Chem. 71 (1999) 491; T. Koga, K. Kanayama, S. Watanabe, T. Imai, A.J. Thakkar, Theor. Chem. Acc. 104 (2000) 411. [25] S.E. Massen, C.P. Panos, Phys. Lett. A 280 (2001) 65. [26] L. Debnath, P. Mikusinski, Introduction to Hilbert Spaces with Applications, Elsevier, 2005.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Maximum Rényi entropy principle and the generalized Thomas–Fermi model Á. Nagy a , E. Romera b,∗ a b
Department of Theoretical Physics, University of Debrecen, H-4010 Debrecen, Hungary Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Fuentenueva s/n, 18071 Granada, Spain
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 10 December 2008 Received in revised form 27 December 2008 Accepted 2 January 2009 Available online 8 January 2009 Communicated by V.M. Agranovich
The Thomas–Fermi model is generalized using the maximum Rényi entropy principle. A simple relation between the dimension and the Rényi parameter is emphasized. The Rényi entropy of atoms is approximately expressed with the logarithm of the kinetic energy. An approximate relation between the value of the density at the nucleus and the kinetic energy is found. © 2009 Elsevier B.V. All rights reserved.
Recently, there has been a growing interest in using information concepts in several fields of science. Although Rényi entropy has been introduced by Rényi [1] as early as 1961, this kind of entropy has only recently obtained a wide range of application e.g. in the analysis of quantum entanglement [2], quantum communication protocols [3], quantum correlations [4] or localization properties [5]. The present authors have recently studied the Rényi information of atoms both in position and momentum spaces [6]. In this Letter we explore how the maximum Rényi entropy principle can be applied to generalize the Thomas–Fermi model. Earlier the principle of extreme physical information [7] was used to derive the Euler and the Kohn–Sham equations of the density functional theory [8,9]. There are several ways to derive the Euler equation of the density functional theory. Consider now a system of N electrons moving in a local external potential v. Imagine a system of noninteracting electrons having the same density (r) as that of the original interacting electrons. The kinetic energy T s [] of the noninteracting system is defined by the Levy–Lieb [10] constraint search as
T s [] = min Φ| Tˆ |Φ = Φ[] Tˆ Φ[] , Φ→
(1)
where Tˆ is kinetic energy operator and the minimum is searched over the non-interacting wave functions Φ having the density . Then the Euler equation can be derived minimizing the noninteracting kinetic energy T s [] with the constraints that the density is fixed (the same as the interacting one) and that the density is normalized to the number of electrons N:
T s [ ] +
v KS (r)(r) dr − μ
(r) dr.
(2)
The Lagrange multipliers v KS (r) and μ are the Kohn–Sham potential and the chemical potential, respectively. The variation of this expression with respect to the density leads to the Euler equation of the density functional theory:
δ T s [ ] + v KS (r) = μ. δ (r)
(3)
Though Eq. (1) gives the formal definition of the non-interacting kinetic energy, its exact form as a functional of the density is unknown. There are, however, several approximations for it. The simplest one is the Thomas–Fermi functional [11]:
(r)5/3 dr,
T TF [] = C TF
(4)
with C TF =
3 2 2/3 3π .
(5)
10
The Thomas–Fermi functional was generalized for arbitrary dimension [12–16]. The kinetic energy of a D-dimensional system has the form
(r)1+2/ D dr,
T D [ ] = C D
(6)
with CD =
D 2( D + 2)
D 2K D
2/ D .
(7)
The factor K D can be given by the recurrence relation K D +2 =
KD 2π D
,
(8)
with
*
Corresponding author. E-mail address: [email protected] (E. Romera).
0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.01.004
K1 =
1
π
(9)
Á. Nagy, E. Romera / Physics Letters A 373 (2009) 844–846
and 1
K2 =
(10)
.
2π
The maximum Rényi entropy principle is a generalization of the maximum entropy principle developed by Jaynes [17]. In Jaynes’ principle the Shannon entropy [18]
Sf =−
f (r) ln f (r) dr
(11)
is maximized under proper conditions including the normalization of the probability density f (r)
f (r) dr = 1.
(12)
The maximum Rényi entropy principle has been used recently for systems with power-law Hamiltonians [19]. The Rényi entropy of order α for a D-dimensional probability density function f (r1 , . . . , r D ) normalized to one is defined as R αf =
1 1−α
f α (r) dr,
ln
(13)
where r stands for r1 , . . . , r D The limit α → 1 gives the Shannon entropy. Now we seek the extremum of the Rényi entropy of order α under the conditions: 1. The density is normalized to 1. A Lagrange multiplicator ν is introduced. 2. The density is kept fixed. This requirement is taken into account by a local potential w (r). This constraint is used to ensure that the density of the non-interacting system be equal to that of the interacting one. Then, the extremum of the functional
R αf − ν
f (r) dr +
w (r) f (r) dr
(14)
is considered. The variation leads to the equation:
δ R αf
− ν + w (r) = 0.
δf
(15)
The functional derivative of the Rényi entropy (13) is
δ R αf
=
δf
α 1−α
f α −1 f α dr
.
(16)
Then Eq. (15) takes the form:
α 1−α
f α −1
+ w (r) = ν
f α dr
(17)
or it can also be written as
˜ (r) = ν˜ c f α −1 + w with
˜ = w
ν˜ =
c (1 − α )
α c (1 − α )
α
(18)
w
f α dr,
(19)
ν
f α dr
(20)
and c is any constant. Comparing Eq. (18) with Eq. (4) we immediately see that the Thomas–Fermi case recovers with the choice α = 5/3 and c = cTF . Eq. (18) is a generalization of the Thomas– Fermi model derived from the maximum Rényi entropy principle. The Thomas–Fermi model was worked out for the free-electron gas and can be applied in systems with slowly changing densities. It is
845
frequently used even today, for example, to study the structure of the new artificial atoms or quantum dots [20], Bose–Einstein condensates in traps [21], systems under pressure [22], simulations in chemical and biological systems [23]. Here we use the Thomas–Fermi kinetic energy functional not the original Thomas–Fermi model itself, that is, the Thomas–Fermi model in a more general sense. Comparing Eq. (18) with Eq. (6) we can recover the D-dimensional Thomas–Fermi model. We are led to the simple relationship between the dimension D and the parameter α
α=1+
2 D
(21)
.
In this case there is a physical meaning of the Rényi parameter α as it is related to the dimension. The relationship between the kinetic energy and Shannon information entropy (that is, α = 1) was studied earlier. Massen and Panos [25] fitted numerical values for several nuclei and atomic clusters and found that S = aT + bT ln (cT ),
(22)
where a, b and c are constants. Now, we extend this kind of investigation to cover several other values of α . The Thomas–Fermi model is a crude approximation for atoms (though it is correct in the limit of large atomic numbers). Applying the generalization presented here, we studied the relation between the kinetic energy functional and the Rényi entropy of order α for neutral atoms with Z = 1–36. The Rényi entropy has been numerically calculated with the density obtained from ground-state Roothaan–Hartree–Fock (RHF) wave functions [24]. The approximate relation between the kinetic energy and the Rényi entropy is written as Rα ρ = a(α ) ln T + b(α ),
(23)
where the parameters are given in Table 1. The fit is not very accurate for values of α about or below 1.5. Let us point out that limα →∞ R α ρ = − ln ||ρ ||∞ = − ln ρ (0), be-
cause ||ρ || p ≡ ( ρ p dr)1/ p and ||ρ ||∞ ≡ sup |ρ | [26]. We make use of the fact that the ground state density of atoms is a nonincreasing function of r. Consequently, one finds that when α goes to infinity expression (23) gives us − ln ρ (0) = a(∞) ln T + b(∞), so we have obtained a relation between the value of the density at the nucleus ρ (0) and the kinetic energy ρ (0) = Z T −a(∞) e −b(∞) . Figs. 1 and 2 present the Rényi entropy (13) and the approximate expression for the Rényi entropy (23) as a function of atomic number for the values of α ∞, 10, 4, 5/3 and 1.2. Approaching to the value of 1, the shell structure is getting more apparent in the Rényi entropy, which is not showed by the approximate expression (23). We would like to emphasize that the true Roothaan– Hartree–Fock densities were applied in these calculation. In summary, the central result of this Letter is the derivation of the generalized Thomas–Fermi model using the maximum Rényi Table 1 Fitted parameters a(α ) and b(α ) for Eq. (23). The fit is very good for 0.999) and no so good for 4 α 1.5 (0.998 c.c. 0.996).
α ∞ 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 5.0 4.0
a(α ) −0.902 −0.873 −0.870 −0.867 −0.863 −0.858 −0.851 −0.843 −0.831 −0.813 −0.784
b(α ) 0.299 0.936 0.972 1.014 1.061 1.116 1.182 1.262 1.362 1.489 1.660
α 3.5 3.0 2.5 2.3 2.1 2.0 1.9 1.8 1.7 1.6 1.5
a(α ) −0.760 −0.726 −0.674 −0.646 −0.612 −0.593 −0.573 −0.550 −0.526 −0.501 −0.475
α 4 (c.c. b(α ) 1.771 1.908 2.089 2.182 2.296 2.364 2.441 2.530 2.633 2.753 2.894
846
Á. Nagy, E. Romera / Physics Letters A 373 (2009) 844–846
tained an approximate expression between the Rényi entropy and the logarithm of the kinetic energy of atoms and as a special case an approximate relation between the density at the nucleus and the kinetic energy. Acknowledgements Á.N. acknowledges grant OTKA No. T67923. E.R. acknowledges the Spanish project FQM-2725 (Junta de Andalucía). References
(α )
α = 4, 10, ∞ (from top
(α )
α = 1.2, 5/3 (from top
Fig. 1. Plot of R ρ (+ points) and a(α ) ln T + b(α ) (line) for to bottom) vs atomic number.
Fig. 2. Plot of R ρ (+ points) and a(α ) ln T + b(α ) (line) for to bottom) vs atomic number.
entropy principle. In this case there is a physical meaning of the Rényi parameter α as it is related to the dimension. Finally, we ob-
[1] A. Rényi, in: Proceedings of Fourth Berkeley Symposium on Mathematics, Statistics and Probability, vol. 1, Univ. California Press, Berkeley, 1961, p. 547. [2] O. Gühne, M. Lewenstein, Phys. Rev. A 70 (2004) 022316; G. Adesso, A. Serafini, F. Illuminati, Phys. Rev. A 70 (2004) 022318; A. Bovino, G. Castagnolli, A. Ekert, P. Horodecki, C.M. Alves, A.V. Serfienko, Phys. Rev. Lett. 95 (2005) 240407. [3] R. Renner, N. Gisin, B. Kraus, Phys. Rev. A 72 (2005) 012332; V. Giovanetti, S. Lloyd, Phys. Rev. A 69 (2004) 062307. [4] P. Lévay, S. Nagy, J. Pipek, Phys. Rev. A 72 (2005) 022302. [5] D.G. Arbo, C.O. Reinhold, J. Burgdörfer, A.K. Pattanayak, C.L. Stokely, W. Zhao, J.C. Lancaster, F.B. Dunning, Phys. Rev. A 67 (2003) 063401. [6] E. Romera, Á. Nagy, Phys. Lett. A 372 (2008) 4918. [7] B.R. Frieden, Am. J. Phys. 57 (1989) 1004. [8] Á. Nagy, J. Chem. Phys. 119 (2003) 9401. [9] R. Nalewajski, Chem. Phys. Lett. 372 (2003) 28. [10] M. Levy, Proc. Natl. Acad. Sci. USA 76 (1979) 6002; M. Lieb, Int. J. Quantum Chem. 24 (1982) 243. [11] L.H. Thomas, Proc. Cambridge Philos. Soc. 23 (1927) 542; E. Fermi, Z. Phys. 48 (1928) 73. [12] G.F. Kventsel, J. Katriel, Phys. Rev. A 24 (1981) 2299. [13] N.H. March, J. Math. Phys. 26 (1985) 554. [14] A. Holas, N.H. March, Philos. Mag. 69 (1994) 787. [15] N.H. March, S. Kais, Int. J. Quantum Chem. 65 (1997) 411. [16] B.K. Shivamoggi, Physica A 248 (1998) 195. [17] E.T. Yanes, Phys. Rev. A 106 (1957) 620; E.T. Yanes, Phys. Rev. A 107 (1957) 171. [18] C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. [19] A.G. Bashrirov, Phys. Rev. Lett. 93 (2004) 1306017. [20] J. Sanudo, A.F. Pacheco, Can. J. Phys. 84 (2006) 833. [21] P. Schuck, X. Vinas, Phys. Rev. A 61 (2000) 043603. [22] E. Cappelluti, L. Delle Site, Physica A 303 (2002) 481. [23] M. Hodak, W. Lu, http://meetings.aps.org/link/BAPS.2006.MAR.V27.8. [24] T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Int. J. Quantum Chem. 71 (1999) 491; T. Koga, K. Kanayama, S. Watanabe, T. Imai, A.J. Thakkar, Theor. Chem. Acc. 104 (2000) 411. [25] S.E. Massen, C.P. Panos, Phys. Lett. A 280 (2001) 65. [26] L. Debnath, P. Mikusinski, Introduction to Hilbert Spaces with Applications, Elsevier, 2005.