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Our reference: PLA 21677 P-authorquery-v7 AUTHOR QUERY FORM Journal: PLA Please e-mail or fax your responses and any corrections to: E-mail: [email protected] Article Number: 21677 Fax: +1 61 9699 6735 Dear Author, Please check your proof carefully and mark all corrections at the appropriate place in the proof (e.g., by using on-screen annotation in the PDF file) or compile them in a separate list. Note: if you opt to annotate the file with software other than Adobe Reader then please also highlight the appropriate place in the PDF file. To ensure fast publication of your paper please return your corrections within 48 hours. For correction or revision of any artwork, please consult http://www.elsevier.com/artworkinstructions Any queries or remarks that have arisen during the processing of your manuscript are listed below and highlighted by flags in the proof. Click on the ‘Q’ link to go to the location in the proof. Location in article Q1 Query / Remark: click on the Q link to go Please insert your reply or correction at the corresponding line in the proof Please confirm that given names and surnames have been identified correctly and are presented in the desired order. (p. 1/ line 15) Please check this box if you have no corrections to make to the PDF file Thank you for your assistance. Page 1 of 1 JID:PLA AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.1 (1-5) Physics Letters A ••• (••••) •••–••• 1 67 Contents lists available at SciVerse ScienceDirect 2 68 3 69 Physics Letters A 4 70 5 71 6 72 www.elsevier.com/locate/pla 7 73 8 74 9 75 10 76 11 12 13 77 Local coordinate, wave vector, Fisher and Shannon information in momentum representation 78 79 14 15 Q1 16 17 18 19 20 80 Á. Nagy a,∗ b , E. Romera , S.B. Liu c 81 82 a 83 Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary b 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 c Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA 84 85 86 21 22 87 a r t i c l e i n f o 88 a b s t r a c t 23 24 25 26 27 28 89 Article history: Received 9 August 2012 Accepted 5 November 2012 Available online xxxx Communicated by V.M. Agranovich The formalism of Luo [Int. J. Theor. Phys. 41 (2002) 1713] on local values of quantum observables is generalized for N-electron systems both in coordinate and momentum spaces. It is shown that the imaginary part of the total local coordinate (momentum) is the half of the local wave vector (or the half of the gradient of the local momentum(coordinate)-space Shannon information per particle). © 2012 Elsevier B.V. All rights reserved. 90 91 92 93 94 29 95 30 96 31 97 32 33 98 1. Introduction 34 99 2. Local functions of quantum observables in one-electron systems 100 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 101 Nowadays, there is a growing interest in applying information theoretical concepts in several fields of physics and chemistry. Local information quantities attracted less attention. Here we show that local Shannon and Fisher information can be related to local quantum observables. Local values of quantum observables were derived by several authors [1–3]. Here the formalism of Luo [1] is applied and generalized for N-electron systems. Luo showed that the real part of the local value of a quantum observable is the expectation value of the quantum observable, while the imaginary part comprises the fluctuation closely related to the Fisher information [4]. In this Letter a relation to the local Shannon information [5] and the local wave vector is explored. In the following section the formalism of Luo [1] is summarized. The method is generalized for N-electron systems in Section 3. We show that the imaginary part of the total local momentum is the half of the local wave vector or the half of the gradient of the local Shannon information per particle. Local functions of quantum observables in N-electron systems in momentum space are revealed in Section 4. We point out that the imaginary part of the total local coordinate is the half of the local momentum-space wave vector or the half of the gradient of the local momentumspace Shannon information per particle. Local Shannon and Fisher momentum densities in neutral atoms are presented in Section 5. 60 61 62 63 64 65 66 * Corresponding author. Fax: +36 52415102. E-mail addresses: [email protected] (Á. Nagy), [email protected] (E. Romera), [email protected] (S.B. Liu). 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.11.018 Consider a quantum mechanical observable a and a quantum system at a state ψ . The local value of a can be decomposed into two local contributions: 102 103 104 105 a(r) = ā(r) + iã(r). (1) 106 107 The real part ā(r) gives the expectation value of a 108 109 ā(r)|ψ|2 dr. aψ = (2) 110 111 112 The imaginary part ã(r) is related to the variance: V arψ (a − ā) = 2 ã(r) |ψ|2 dr. 113 (3) 114 115 116 Luo showed that ā(r) = Re and aψ ψ (r) 117 118 (4) 119 120 121 ã(r) = Im aψ ψ 122 (r). (5) Bohórquez and Boyd [6] obtained the local momentum in coordinate representation. The wave function can be written in polar form ψ(r) = (r)1/2 e ib(r) , (6) 123 124 125 126 127 128 129 130 131 132 JID:PLA AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics 1 2 3 where (r) = |ψ(r)|2 is the density. The momentum operator in coordinate representation is p = −i ∇ . (Atomic units are used throughout this Letter.) Then the local momentum [6] has the form 4 5 6 7 8 9 10 11 12 [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.2 (1-5) Á. Nagy et al. / Physics Letters A ••• (••••) •••–••• 2 p(r) = ∇r b(r) − i ∇r (r) (r) 2 (7) , that is, the average value is p̄ (r) = ∇r b(r) and the fluctuation part is p̃ (r) = −1/2(∇r (r))/(r). We mention by passing that p̃ is proportional to Nelson’s [7] osmotic velocity −∇r ln (r). Quantum properties of Fisher information were discussed by Hall [8]. Fisher information for central potential was calculated by Romera et al. [9]. 13 14 15 3. Local functions of quantum observables in N -electron systems in coordinate space 16 17 18 19 20 Now we generalize the concept of local quantities to N-electron systems. Consider a quantum mechanical observable A acting on the wave function Ψ (r1 , . . . , r N ). Now we define a local quantity as 21 22 23 24 25 26 A (r) = Ā (r) + i à (r). (8) The real part Ā (r) gives the expectation value of a A Ψ = Ā (r)(r) dr, (9) 27 28 29 30 (r) is defined as 2 (r1 ) = N Ψ (r1 , . . . , rN ) dr2 . . . drN . 34 35 36 37 Ā (r1 ) = (r1 ) à (r1 ) = 40 41 AΨ 46 49 VarΨ ( A − Ā ) = 52 à (r)2 55 56 57 58 59 P= (12) N p j. (14) The wave function can be written in coordinate representation in polar form as Ψ (r1 , . . . , r N ) = K (r1 , . . . , r N )e i F (r1 ,...,rN ) , 64 65 66 (15) where K (r1 , . . . , r N ) and F (r1 , . . . , r N ) are real functions. Then we immediately obtain that Re p jΨ Im = ∇r j F Ψ q (r) = − P̃(r) = 2 p jΨ Ψ =− ∇r j K (16) (20) K 75 76 77 78 79 81 82 q (r). (21) In a recent paper [12] we established relationships between the local wave-number vector, the local Shannon and Fisher information. The Shannon information is defined in coordinate space as s (r) dr (22) with the local Shannon information 83 84 85 86 87 88 89 90 91 92 93 94 96 97 (23) i (r) dr 98 99 100 (24) 101 102 104 105 [∇r (r)]2 i (r) = . (r) (25) q = ∇r s 108 109 (26) (17) The real part of the local quantity P in state Ψ (r1 , . . . , r N ) takes the form 110 111 112 and q2 = 106 107 113 i (27) . P̃(r) = 1 2 114 115 116 That is, the local wave vector q is the gradient of the Shannon information per particle and the square of the local wave vector is the Fisher information per particle. Using these results we obtain that the imaginary part of the total local momentum is the half of the gradient of the local Shannon information per particle: ∇r s . 118 119 120 121 122 (28) 124 125 VarΨ (P − P̄) = = 117 123 . 74 80 ∇r (r) , (r) has the dimension of wave-number. Comparing Eqs. (19) and (20) we obtain that the imaginary part is the half of the local wave vector: 1 72 73 Therefore the variance of the total momentum can be written as and 62 63 71 (19) . Some years ago Nagy and March [10] introduced the ratio of the density gradient to the electron density as a local wavenumber to characterize the ground state of atoms and molecules. Independently, Kohout, Savin and Preuss [11] also investigated the role of the quantity |∇r /| in the shell structure of atoms. The local wave vector (13) j =1 60 61 (r) 2 70 We showed that (r) dr. 68 103 53 54 1 ∇r (r) with the local Fisher information Consider the total momentum operator P as a sum of oneelectron operators p j , 50 51 P̃(r) = − I = 2 67 69 (11) It is related to the variance: 47 48 (18) The Fisher information, on the other hand, has form: . 44 45 , s (r) = −(r) ln (r). Im( Ψ )|Ψ | dr2 . . . dr N . (r1 ) 42 43 (r1 ) The imaginary part à (r) can be written as 38 39 Re( AΨΨ )|Ψ |2 dr2 . . . dr N K 2 ∇r1 F dr2 . . . dr N 95 (10) The real part Ā (r) has the form 33 N while the imaginary part is S = where the density 31 32 P̄(r1 ) = 2 P̃(r) 1 4 127 128 (r) dr 2 s (r) ∇r dr. 126 129 130 (29) 131 132 JID:PLA AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.3 (1-5) Á. Nagy et al. / Physics Letters A ••• (••••) •••–••• 1 2 4. Local functions of quantum observables in N -electron systems in momentum space 3 Consider the total coordinate operator R as a sum of oneelectron operators r j = i ∇ p j , 3 4 5 6 9 10 11 12 13 14 γ (p1 ) = N 17 18 19 20 21 22 23 24 ∇ p γ (p) qγ (p) = − . γ (p) (30) (31) R= as 70 r j. Φ(p1 , . . . , p N ) = L (p1 , . . . , p N )e iG (p1 ,...,pN ) , sγ (p) dp (32) with the local Shannon information r jΦ (45) Im r jΦ = Φ R̄(p1 ) = − ∇p j L L N L 2 ∇ p 1 Gdp2 . . . dp N γ (p1 ) , (34) 39 40 41 42 43 44 45 46 47 48 49 50 (35) 1 ∇ p γ (p) qγ = ∇ p 53 sγ (36) γ iγ (37) . γ That is, the momentum-space local wave vector qγ is the gradient of the momentum-space Shannon information per particle and the square of the momentum-space local wave vector is the momentum-space Fisher information per particle. Consider a quantum mechanical observable B acting on the momentum-space wave function Φ(p1 , . . . , p N ) and define a local quantity as B (p) = B̄ (p) + i B̃ (p). 58 59 60 61 62 (38) B̄ (p1 ) = Re( BΦΦ )|Φ|2 dp2 . . . dp N γ (p1 ) . (39) B̃ (p1 ) = Im( BΦΦ )|Φ|2 dp2 . . . dp N γ (p1 ) 65 . (40) B̄ (p)γ (p) dp, (41) 66 while the imaginary part is related to the variance: V arΦ ( B − B̄ ) = 100 sγ 101 (50) . γ That is, the imaginary part of the total local coordinate is minus the half of the gradient of the local Shannon information per particle. Therefore the variance can be written as VarΨ (R − R̄) = = 1 4 R̃(p)2 γ (p) dp 2 sγ γ (p) ∇ p dp. γ 102 103 104 105 106 107 108 109 110 (51) 111 112 113 5. Local Shannon and Fisher momentum densities in neutral atoms 114 115 Now, the spherically averaged local Shannon and Fisher informations B̃ (p)2 γ (p) dp. 117 118 119 (52) 120 121 63 64 98 99 and The average of B is given by the real part B Φ = (49) sγ ( p ) = −γ ( p ) ln γ ( p ) The imaginary part B̃ (p) can be written as 96 116 The real part B̄ (p) has the form 94 97 R̃(p) = − ∇ p 2 56 57 90 95 or with the local Shannon information 1 54 55 γ (p) 2 R̃(p) = − qγ (p) 2 and q2γ = 89 93 (48) . 1 After elementary calculation we are led to the relationship 51 52 (47) 2 36 38 85 92 The latter can also be expressed with the local wave vector 37 81 87 while the imaginary part is [∇ p γ (p)] i γ (p) = . γ (p) 35 80 91 i γ (r) dr 29 34 78 88 R̃(p) = 33 77 86 with the local Fisher information 32 76 84 (46) . 28 31 75 83 (33) The Fisher information, on the other hand, takes form: 74 82 The real part of the local quantity R in state Φ reads sγ (p) = −γ (p) ln γ (p). 73 79 = −∇ p j G Φ 27 30 (44) where L (p1 , . . . , p N ) and G (p1 , . . . , p N ) are real functions. Then we immediately obtain that 71 72 and Sγ = (43) The wave function can be written in momentum representation as Re Iγ = N j =1 The Shannon information in momentum space can be written 25 26 Φ(p1 , . . . , p N )2 dp2 . . . dp N , where Φ(p1 , . . . , p N ) is the momentum-space wave function. The momentum-space local wave vector is defined as 15 16 68 69 All results obtained in the previous section in coordinate space can be derived in the momentum space. The momentum-space density γ (p) reads as 7 8 67 (42) (γ ( p ))2 iγ ( p) = γ ( p) 122 123 (53) are analyzed. The spherically averaged momentum density γ ( p ) is calculated with the ground state momentum densities for neutral atoms with nuclear charge 1 Z 103 by means of the accurate Roothan–Hartree–Fock wave functions of Koga and co-workers [13]. We have shown that attending to the number of local extrema and the zeros in the open interval (0, ∞), sγ ( p ) can be classified into three types: 124 125 126 127 128 129 130 131 132 JID:PLA 4 AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.4 (1-5) Á. Nagy et al. / Physics Letters A ••• (••••) •••–••• 1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 23 88 Fig. 1. Examples of each type of sγ : Z = 10 (Type I), Z = 14 (Type II), Z = 17 (Type III) and Z = 46 (Type IV). Atomic units are used. 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 48 49 50 51 52 53 54 55 56 57 58 Fig. 2. Examples of each type of the density i γ : Z = 24 (Type I), Z = 30 (Type II) and Z = 36 (Type III). Atomic units are used. • Type I: sγ has a local maximum for Z = 1, 2, 7–10. Fig. 1 shows the local Shannon entropy for one of these atoms, Ne ( Z = 10). • Type II: sγ has a local maximum and a zero for Z = 11–14, 19–32, 37–45, 47–50, 55, 82, 87–103. Fig. 1 shows the local Shannon entropy for Si ( Z = 14). • Type III: sγ has a minimum, a zero and a maximum for Z = 15–18, 33–36, 51–54, 83–86. Fig. 1 shows the local Shannon entropy for Ar ( Z = 17). 59 60 61 62 63 64 65 66 There are two exceptions to the above classification: sγ has two relative minima and a relative maximum for Pd ( Z = 46) (Fig. 1 shows the local Shannon entropy for this element) and sγ has just a minimum and a maximum for Ar Z = 18. We can point out that Type II atoms correspond to groups 1–14 except H and Pd, Type II atoms are the norm for groups 15–18 (except He N, O, F, and Ne), and the rest of atoms cor- 113 114 115 responds to Type I (expect Pd and Ar as we have mentioned above). Analysis of the local fisher density i γ in terms of the extrema and zeros shows a complex schedule (up to ten types of densities). In this case we can do the classification attending to the number of zeros (always in the open interval (0, ∞)), emerging three different types of densities: • Type I: The 65 neutral atoms with Z = 1–7, 11–25, 31, 37–42, 49, 50, 55–74 and 87–103 have an i γ with no zeros. • Type II: The 21 neutral atoms with Z = 8–10, 15–18, 32–36, 46, 51–54, 83–86 have a density with a zero. • Type III: The 16 neutral atoms with Z = 26–30, 43, 45, 47 and 48 have an i γ with two zeros. Fig. 2 presents i γ ( p ) for three atoms, one of each type: Z = 24 (Type I), Z = 30 (Type II) and Z = 36 (Type III). Obviously, the 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 JID:PLA AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.5 (1-5) Á. Nagy et al. / Physics Letters A ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 emergence of zeros in the local fisher information in connected with the existence of a relative maximum or minimum in the momentum density, and we have checked that this classification is in agreement with the classification of the spherically averaged momentum density studied in [14–18]. As in the case of the momentum density, these types of structures are determined by the relative contribution of the outermost s, p, and d subshells (see [14–18] for a detailed analysis of the momentum density structure). 10 11 6. Summary 12 13 14 15 16 17 18 19 20 21 22 23 In summary, we generalized the formalism of Lou for Nelectron systems both in coordinate and momentum spaces. We studied two local quantum observables: the total local momentum and the total local coordinate. It turned out that the imaginary part of the total local momentum is the half of the local wave vector (or the half of the gradient of the local Shannon information per particle) and the imaginary part of the total local coordinate is the half of the local momentum-space wave vector (or the half of the gradient of the local momentum-space Shannon information per particle). Finally, we analyzed the local Shannon and Fisher momentum densities of neutral atoms. 24 25 Acknowledgements 26 27 28 29 This work is supported by the Projects FIS2011-24149 and CEI-BIOTIC-20F12.41. The work is also supported by the TAMOP 4.2.1/B-09/1/KONV-2010-0007 and TAMOP 4.2.2/B-10/1-2010-0024 5 projects. The project is co-financed by the European Union and the European Social Fund. Grant OTKA No. K 100590 is also gratefully acknowledged. 67 68 69 70 References 71 72 [1] S. Luo, Int. J. Theor. Phys. 41 (2002) 1713. [2] R.A. Mosna, I.P. Hamilton, L. Delle Site, J. Phys. A 38 (2005) 3869; I.P. Hamilton, R.A. Mosna, L. Delle Site, Theor. Chim. Acta 118 (2007) 407; I.P. Hamilton, R.A. Mosna, J. Comp. Appl. Math. 233 (2010) 1542. [3] C. Salesi, E. Recami, Found. Phys. 28 (1998) 763. [4] R.A. Fisher, Proc. Cambridge Philos. Soc. 22 (1925) 700. [5] C.E. Shannon, Bell Syst. Tech. J. 27 (1948) 379. [6] H.J. Bohórquez, R.J. Boyd, J. Chem. Phys. 129 (2008) 024110. [7] E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985. [8] M.J.W. Hall, Phys. Rev. A 62 (2000) 012107. [9] E. Romera, P. Sánchez-Moreno, D. Dehesa, Chem. Phys. Lett. 414 (2005) 468. [10] Á. Nagy, N.H. March, Mol. Phys. 90 (1997) 271. [11] M. Kohout, A. Savin, H. Preuss, J. Chem. Phys. 95 (19) (1997) 28. [12] Á. Nagy, S.B. Liu, Phys. Lett. A 372 (2008) 3256. [13] T. Koga, K. Kanayama, S. Watanabe, A.J. Thakkar, Int. J. Quantum Chem. 71 (1999) 491; T. Koga, K. Kanayama, T. Watanabe, T. Imai, A.J. Thakkar, Theor. Chem. Acc. 104 (2000) 411. [14] W.M. Westgate, A.M. Simas, V.H. Smith Jr., J. Chem. Phys. 83 (1985) 4054; W.M. Westgate, R.P. Sagar, A. Farazdel, V.H. Smith Jr., A.M. Simas, A.J. Thakkar, At. Data Nucl. Data Tables 48 (1991) 213. [15] E. Romera, T. Koga, J.S. Dehesa, Z. Phys. D 42 (1997) 251. [16] T. Koga, H. Matsuyama, H. Inomata, E. Romera, J.S. Dehesa, A.J. Thakkar, J. Chem. Phys. 109 (1998) 1601. [17] T. Koga, H. Matsuyama, E. Romera, J.S. Dehesa, A.J. Thakkar, Phys. Rev. A 59 (1999) 4805. [18] T. Koga, E. Romera, J.S. Dehesa, H. Matsuyama, A.J. Thakkar, Theor. Chem. Acc. 103 (1999) 70. 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132 JID:PLA AID:21677 /SCO Doctopic: Atomic, molecular and cluster physics [m5Gv1.5; v 1.86; Prn:14/11/2012; 10:18] P.6 (1-5) Physics Letters A ••• (••••) •••–••• 1 67 Contents lists available at SciVerse ScienceDirect 2 68 3 69 Physics Letters A 4 70 5 71 6 72 www.elsevier.com/locate/pla 7 73 8 74 9 75 10 76 11 77 12 78 13 14 15 16 17 18 19 20 21 79 Local coordinate, wave vector, Fisher and Shannon information in momentum representation Physics Letters A ••••, •••, ••• 80 81 Á. Nagya,∗ , E. Romerab , S.B. Liuc 82 a 84 Department of Theoretical Physics, University of Debrecen, H–4010 Debrecen, Hungary b 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 c Research Computing Center, University of North Carolina, Chapel Hill, NC 27599-3420, USA 83 85 86 87 22 88 23 89 24 Highlights 90 25 We studied total local momentum and coordinate for N-electron systems. The total local momentum’s imaginary part is the half of the local wave vector. The total local coordinate’s imaginary part is half of Shannon entropy’s gradient. 91 26 27 92 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132