Complexity in atoms: an approach with a new analytical density

Complexity in atoms: An approach with a new analytical density Jaime Sañudo* and Ricardo López-Ruiz** * Departamento de Física, Facultad de Ciencias, Universidad de Extremadura, E-06071 Badajoz, Spain ** DIIS and BIFI, Facultad de Ciencias, Universidad de Zaragoza, E-50009 Zaragoza, Spain 2 Abstract In this work, the calculation of complexi ty on atom ic systems is considered. In order to unveil the increasing of this statis tical magnitude with the ato mic number due to the relativistic effects, recently reported in [A. Borgoo, F. De Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett., 444 (2007) 186], a new analytical density to describe neutral atoms is proposed. This density is in spired in the Tietz potential model. The parameters of this density are determined from the normalization condition and from a variational calculation of the energy, which is a functional of the density. The density is non-singular at the origin and its specific form is selected so as to fit the results coming from non-relativistic Hartree-Fock calcula tion s. The main ingredients of the ene rgy functional are the non-relativis tic kinetic en ergy , the nuclear-elect ron attraction energy and the classical term of the electron repulsion. The relativis tic correction to the kinetic energy and the Weizsacker term are also taken into account. The Dirac and the correlation terms are shown to be less important than the other term s and they have been discarded in this study. When the statistical measure of co mplexity is calculated in position space with th e analytical density deri ved from this m odel, the increasing trend of this magnitude as the atomic number increases is also found. PACS numbers: 31.30.Jv; 31.15.Bs [KEY WORDS: Com plexity; Density functional, Relativ istic effects] * Electronic address: jsr@unex.es ** Electronic address: rilopez@unizar.es 3 1. Introduction Despite a universal measure of comple xity is yet unknown, the application of complexity to atomic system s is a topic of great scientific in terest [1]. In a recent work [2], it has been shown the marked influence th at the consid eration of relativistic effects in atoms has on a statistical measure of com plexity, LMC C , defined in Refs. [3, 4]. The main ingredient for the calculation of the complexity is the electron dens ity in the atom, ) ( r n r . Different electron densities have been proposed through the years [5, 6]. Using the non-relativistic Hartree-Fock wave functions, the analysis of the electronic structural complexity for atom s with atomic num ber 54 2 − = Z has shown an slight increase of complexity when the atom ic number incr eases [1, 7, 8]. W hen the Dirac-Fock relativistic wave functions are use d, the in creasing trend of the complexity with the atomic number, 103 1 − = Z , is enhanced [2]. Knowing th ese facts, we can wonder if the inclusion of the relativistic terms in sim p ler atomic mo dels also causes this kind of behaviour. In this work, our aim is to unveil this possibility in a modified Thomas-Ferm i model [6, 9]. In order to reach this goal, this study is twofold: first, to obtain an analytical density that incorpora tes the rela tivistic effects in a perturbative manner and that is well behaved at the origin. This is pr esented in Section 2. And second, to use this density to check the influence of the re lativistic eff ects on the complexity. This calculation is performed in Section 3. The conclusions are included in Section 4. 4 2. The Analytical Density Hohenberg and Kohn [10] show that the ground-state energy of a quantum- mechanical system can be written a s a density functional, [ ] n E . However, it is not established the specific form of that functi onal. Thus, for our pur poses, the ingredients of the energy functional for the atom are pres ented in Section 2.1. In Section 2.2, an analytical expression for the density of the atom is justified. All the parameters of this density are determined by using the minimi zation of the energy and the norm alization condition, and by fitting the density at th e orig in to Hartree-Fock calculations. 2.1 The Functional E[ n ] The total energy, E , for a point-like nucleus of atom ic number Z surrounded by an electron cloud, can be obtained from the energy-density functional [11, 12], [] [ ] [ ] [ ] n E n E n E n E ee eN kin + + = , (1) where [] n E kin is the kinetic energy (whose expression is given below), [] n E eN is the electron-nuclear attraction energy [] ∫ − = r d r r n Ze n E eN r r ) ( 2 , (2) e being the charge of the electron, and [ ] n E ee is the electron-electron repulsion energy. This last term can be divided into two parts, [ ] [ ] [ ] n K n J n E ee + = , (3) where [] n J is the classical Coulomb re pulsion between the electrons 5 [] ∫∫ ′ ′ − ′ = r d r d r r r n r n e n J r r r r r r ) ( ) ( 2 2 , (4) and [] n K is called the exchange-correlation energy, that, in turn, can also be separated into [] [ ] { } [ ] { } L L + + + = 1 1 n K n K n K NR Corr NR D . (5) The first term in (5) is the exchange energy, and here, we adopt for it the non-relativistic ( NR ) homogeneous electron gas approximation of Dirac [11, 12], [] ∫ = r d r n e n K D NR D r r 3 / 4 2 ) ( λ , (6) where 3 / 1 π λ 3 4 3 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − = D . The second term in (5) is the NR correlation energy obtained by Ceperley and Adler [13], and precisely f itted by Barbiellini-Amidi [ 14] by means of the expression [] ∫ = r d r n e n K B a Cep NR Corr r r 6 / 7 2 / 1 2 ) ( λ , (7) with 2 2 me a B h = the Bohr radius, m the electron mass, and 6 / 1 3 4 0635 . 0 π λ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − Cep . It should be noted that in Eq. (5) the ellipsis inside the cu rly brackets is meaning the relativistic correc tions [12] that we are not considering here. For the kinetic energy we adopt the firs t two terms of the gradient expansion [11, 12], [ ] [ ] [ ] { } L L + + + = 1 2 0 n T n T n E NR kin , (8) where the quantity 6 [] ( ) ∫ ∇ = r d n n m n T W NR r r h 2 2 λ 9 1 2 (9) is one-ninth the inhomogen eity correction introduced by W eizsacker, with 8 1 λ = W , and the ellipsis inside the curly brackets in Eq. (8) has the same meaning as said above. On the other hand, the quantity [] n T 0 can be split in [] [] [] n T n T n T R NR 0 0 0 + = , (10) where the term [] n NR T 0 is the Thomas-Fermi kinetic energy [6,9] given by [] () ∫ = r d n m n T NR r h 5/3 2 3 / 2 2 π 3 10 3 0 , (11) and [] n R T 0 is the relativistic co rrections, to the first order, for the homogeneous electron gas [6], [] () ) α ( ο α π 4 7/3 2 2 2 3 / 4 2 3 280 15 0 + − = ∫ r d n a m n T B R r h , (12) c α 2 h e = being the fine structure constant. The density we are looking for is the solution that minimizes the energy [ ] n E , given in Eq. (1), taking into account the cons train fo r the normalization of the density , ∫ = r d r n N r r ) ( , (13) where N denotes the number of electrons. Here, we consider the neutral atom, N = Z . Within the limits of the problem , we pro ceed to do that in the next section. 7 2.2 The Density and the Minimization of the Energy Let us start by recalling th e well kno wn Thomas-Fermi solution for Eq. (1). This is obtained when the energy is minimized by avoiding the rela tivis tic correction () 0 α ≡ , the Weizsacker term ( ) 0 λ ≡ W and the exchange- correlation energy ( ) 0 , 0 λ λ ≡ ≡ Cep D with the constrain given by (13). This solution, namely the Thomas-Fermi density for neutral atom s with a point-like n ucleus at zero temperature [6, 9], is 2 / 3 3 2 3 ) ( χ π ) ( 9 32 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = x x a Z x n B , with B a Z b 3 / 1 3 / 1 2 128 9 π − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = and bx r = . The potential ) ( χ x can not be found analytically. The Thomas-Ferm i model has also been studied in the relativistic case for non point-like nucleus in [ 15] and at finite tem perature in [16], and also in these cases the soluti on can be only obtained numerically. Tietz [17] proposed an analytic expres sion for that potential, () 2 β ) ( χ 1 1 x x + = , that fits very well the exact Thom as-Fermi solution in the range 0 1 0 ≅ ≤ x [18]. The parameter β can be determined by minim izing the energy or by us ing Eq. (13), but, evidently, it is not possible to reach the fulfilment of both conditions with only one parameter. Inspired in the Tietz potential mode l, we propose the following density: () ( ) 2 / 3 3 3 2 3 γ β ε π ) ( 1 9 32 + + = x x a Z x n B , (14) to describe neutral atom s, with three parameters, ε and β γ , to be determined. The parameter γ allows the density to be non singular at the origin. Kato [19] proved the important result that in a quantum system the density should fulfil the condition 8 ) ( - ) ( 0 2 0 = = = r n a Z dr r dn B . With the density given in Eq. (14), the la st condition implies that () 3 / 4 3 / 2 3 / 2 ο π γ 3 4 2 3 − − + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Z Z . Therefore, we choose for γ the form 3 / 2 3 / 2 λ π γ 3 4 2 3 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = Z , (15) where λ is a new parameter not very far from 1 that will be determined latter. The parameter ε in the density (14) is fixe d by means of the n ormalization condition (13). We obtain ) β γ ( β ε 2 / 3 ⋅ = F , (16) where () () () ( ) () () () () ⎪ ⎪ ⎭ ⎪ ⎪ ⎬ ⎫ ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ⋅ ⋅ ⋅ − ⋅ + ⋅ + − + ⋅ + ⋅ ⋅ + + ⋅ − ⋅ ⋅ − = ⋅ β γ - β γ β γ β γ β γ β γ - β γ β γ π β γ β γ - β γ ) β γ ( 1 1 8 8 1 1 1 14 2 1 8 8 1 4 1 2 / 7 2 3 2 / 1 2 / 7 2 arctan F . (17) For β γ ⋅ <<1, the behaviour of ) β γ ( ⋅ F is () ( ) 2 β γ ο β γ π π ) β γ ( 16 9 8 1 ⋅ + ⋅ − = << ⋅ F . (18) Therefore, the only parameter to be determined at th is point is β . This will be found with the minimization of the energy. We substitute Eqs. (16) and (18) in th e density (14), and plug it in Eq. (1), [] [] [] [] [] [] [] [] n NR Corr NR D eN R NR NR K n K n J n E n T n T n T n E + + + + + + = 0 2 0 . (19) For β γ ⋅ << 1, the power expansion of the different term s is: 9 () () () {} β γ ο β γ . β . 2 / 1 2 3 / 7 2 209 6 1 764 2 0 ⋅ + ⋅ − = Z B NR a e T , (20) () () {} 2 / 1 2 / 3 1/2 3 / 5 2 β γ ο β - γ λ 1 8663 . 0 2 ⋅ + = Z W B NR a e T , (21) () ( ) () {} 2 / 1 2 / 3 1/2 3 / 5 2 2 β γ ο β - γ α 1 582 . 2 0 ⋅ + − = Z Z B R a e T , (22) () () () {} β γ ο β γ . β 2 / 1 3 / 7 2 395 3 1 389 . 3 ⋅ + ⋅ − − = Z B eN a e E , (23) () () {} β γ ο β 1 4735 . 0 3 / 7 2 ⋅ = + Z B a e J , (24) () () {} 2 / 1 3 / 5 2 β γ ο β 1 λ 5631 . 0 ⋅ + − = Z D B NR D a e K , (25) () () {} 2 / 1 1/2 3 / 4 2 β γ ο β 1 λ 6410 . 0 ⋅ + = Z Cep B NR Corr a e K . (26) We advance that it will be obtained that β ≈ 0.5, then the former expressions (20- 26) are power expansions in γ . Keeping in mind that γ is proportional to 3 / 2 − Z , see Eq. (15), then Eqs. (20-26) display the ty pical development of the energy terms in decreasing powers of 3 / 1 Z [6, 9]. So, for NR T 0 , eN E and J the dominant power is 3 / 7 Z , for both NR T 2 and R T 0 the dominant power is 2 Z . In this last term, R T 0 , the factor Z α has been taken as a constant of order 1 due to the fact that the rela tivistic eff ects in 10 which we are interested are mainly important when Z>>1. Th e term NR D K goes as 3 / 5 Z , whereas NR Corr K as 3 / 4 Z . Now, the parameter β can be obtained by minim izing the energy, 0 β = d dE . In order to do zero all the coe fficients of the powers of Z in that derivative, we find that β must be expressed in decreasing powers of 3 / 1 Z as ( ) { } γ γ β β 2 / 1 1 0 ο + + = a . (27) Equalling to zero the coefficient of the power 3 / 7 Z , 0 β is obtained 5272 0 0 . β = . (28) In the same manner, f rom equalling to zero the coefficient of the te rm in 2 Z , we find that () { } W a λ . α Z . λ . 3837 0 139 1 1 336 1 2 − + = . (29) The last param eter to be fi xed in our density model is λ . From (14), the behaviour of our density model at origin reads () 3/2 3 3 3 γ ε π ) ( 9 32 0 Z Z a n B = . (30) Substituting ε and γ given by Eqs. (15) and (16) into (30), and taking into account (27)- (29), we find that the right -hand side of (30) is only dependent of the parameter λ . After taking 0 ≡ α , we can fit the right-hand side of Eq. (30) with the non-relativistic Hartree- Fock calculation of 3 3 / ) ( 0 Z a n B due to Fischer [20, 21]. The value obtained for λ is 2/3 3 / 1 714 1 3388 0 3960 0 . . . λ − − + − = Z Z . (31) 11 This fit is shown in Fig.1 where we have represented 3 3 ) ( 0 Z a n B versus Z . The continuous line shows Eq. (30) with λ given by Eq. (31), and the dots represent the non-relativistic Hartree-Fock calcu lation due to Fischer. Let us note that the fit is excellent. We should indicate at this point that the procedure becomes cum bersome if one tries to perform the calculations f or lower orders th an 2 Z . Hence, we have cut the development of β to the zero and first order in powers of γ 2 / 1 in Eq. (27). Also, let us note that the exchange-correlations terms give n in Eqs. (25-26) are of order lower than 2 Z , and therefore, negligible until the orde r of our approximation. A further calculation to obtain more terms could be reported elsewhere. Let us remark the goodness of this dens ity (14) to calculate the energy of a non- relativistic neutral ato m. According to Eq. (19), this yields ( ) 2 3 / 7 ο . 7684 0 Z E Z + = , where it has been taken the order zero in the development of the parameters: ε = 0.9748 , β = 0.5272 and 3 / 2 3355 0 . γ − = Z . Knowing that the exact Thom as-Fermi energy [6, 9] is 3 / 7 7688 . 0 Z TF E = and the energy derived from th e original Tietz density gives 3 / 7 7682 . 0 Z Tietz E = (obtained by doing ε = 1 and 0 γ = , whereas β i s 0.5632 when the normalization condition is used), the precision of our calculation is notab le. Furthermore, the density (14) here proposed allows us to incorporate the relativistic effects with the possibility to see the ir influence in different m agnitudes, such as we will show with the complexity in th e next section. 12 3. Statistical Complexity Now, we calculate w ith the density (14) several m agnitudes related with statistical complexity. A s it has been reported in [2] when Dirac-Fock relativ istic wave functions are used, here we al so show the influence of the relativistic eff ects in those magnitudes, in particular the incr easing tr end of the c omplexity with the a tomic number. The measure of complexity C , the so-called LMC complexity [3,4], is defined as D H C LMC • = , (32) where H represents the information [4, 22] content of the system 3 / 2 2 1 e π r S H e = , (33) r S being the Shannon information entropy [23] in position spa ce, () r d r n a log r n S B r r r r ) ( ˆ ) ( ˆ 3 ∫ − = , (34) and D is calculated as the density expectation value [3, 4] r d r n a D B r r ∫ = ) ( ˆ 2 3 . (35) In Eqs. (34-35) the electron den sity is norm alized to unity, therefore Z n n ≡ ˆ , and n is given in Eq. (14). Let us recall at this point th at C has been quantif ied in different contexts (see Ref. [24] and references ther ein). It has been shown that C is a useful indicator to 13 successfully discern many situations regarded as complex in systems out of equilibrium [24]. Thus, C identifies the entropy or information H stored in a system and its disequilibrium D , that in the discrete case is the di stance from its actual state to another probability distribution of equilibrium, as th e two basic ingredients for calcu lating the complexity of a system. In the case of a continuous support in the distributions, the magnitudes are redefined as indicated in Eqs. (33-35) (see [4] for a detailed discussion). In Fig. 2, D versus Z is plotted. The continuous lin e represents the calculation including the relativistic correct ion to the kinetic energy, it is to say, the influence of the term given in Eq. (12 ). The dotted line is a similar c alculation but om itting the relativistic correc tion, that is, when 0 ≡ α . We can see the importance of taking into account the relativistic co rrection to the kinetic energy. Th is result qualitatively agrees with that of the Ref. [ 2], where Dirac-Fo ck relativistic wave functions are used. In Fig. 3, we plot H versus Z . The continuous line and the dotted line represent, as before, the relativistic and non-relativ istic calculations, respectively. Qualitatively speaking, the relativistic influence here is slightly bigger th an in the results pr esented in Ref. [2]. Finally, in Fig. 4, we show C versus Z . The meaning of co ntinuous line and the dotted line are the same as before. Let us obser ve that the influence of the relativistic effects here are qualitatively less important than those obtained in Ref. [2]. All these results can be also com pared with the non -relativistic calc ulations performed in Ref. [25]. There, several stat istical complexities were obtained by using 14 the universal electron density found by Gáspár [ 26], and, in general, an increasing trend of these magnitudes with increasing atomic number was reported. This tendency is also found in Fig. 4 with the difference that our approach incorporat es the influence of relativistic correc tions to the kinetic energy. 15 4. Conclusions In this work, a new analytical density to describe neut ral atoms has been proposed. The specific form of this density is inspired in the Tietz potential m odel. This density, which is not singular at the origin, has three parameters th at are fixed by means of three constraints: the norm alization cond ition, the minim ization of the energy as a functional of the density and the fit of the density at the origin with non-relativistic Hartree-Fock calculations. We have used as ingredients for the energy functional the non-relativistic kinetic ener gy, the attractive nuclea r-ele ctron energy, the class ical repulsive electron interaction, the rela tivis tic correction to the kinetic energy, the Weizsacker term, and the Dirac and correlation terms. Up to order 2 Z in the energy functional, the last two terms, nam ely Dir ac and correlation ones, are negligible and therefore they have been om itted. After minim izing the energy functional, we have obtained the density. The calculation of the energy with this density yields ( ) 2 3 / 7 ο . 7684 0 Z E Z + = . This result is comparable with the exact Thomas-Ferm i energy, 3 / 7 7688 . 0 Z TF E = , and with the energy derived from the original Tietz den sity, 3 / 7 7682 . 0 Z Tietz E = . Moreover, this density has allowed us to see the qualitative influence of the relativistic ef fects in the statistical complexity. Thus, we have calculated with this new density different statistical magnitudes: H , which denotes the Shannon information, D , which represents the disequilibrium, and LMC C that is a statistical measure of co m plexity. We have made m anifest the 16 qualitative influence of the relativistic co rre ctions in all the se magnitudes in agreem ent to the behaviour found in Ref. [2], where the increasing trend of the complexity with the atomic number was put in evidence. Thus, Borgoo et al . [2] used the Dirac-Fock relativistic wave functions to unmask this behaviour, whereas our result has been obtained in an analytical manner. Let us conclude by saying that some simp le analytical approaches can reproduce the qualitative behaviour of different statisti cal magnitudes in atomic systems, and so, to help our physical intuitio n in trying to discern complexity at a quantum level. 17 Acknowledgements The authors acknowledge some fina ncial support by grant DGICYT-FIS2005- 06237. 18 References [1] See, for example, C.P. Panos, K.Ch. Ch atzisavvas, Ch.C. Moustakidis, and E.G. Kyrkow, Phys. Lett . A 363 (2007) 78; and references therein. [2] A. Borgoo, F. De Proft, P. G eerlings, K.D. Sen, Chem. Phys. Lett. 444 (2007) 186. [3] R. López-Ruiz, H.L. Mancin i, and X. Calbet, Phys. Lett . A 209 (1995) 321. [4] R. Catalán, J. Garay, a nd R. López-Ruiz, Phys. Rev. E 66 (2002) 011102. [5] I.M. Torrens, Interatomic Potentials (Academic Press, New York, 1972). [6] S. Eliezer, A. Ghatak and H. Hora, An Introduction to Equations of State (Cambridge University Press, Camb ridge, 1986), and references therein. [7] K. Ch. Chatzisavvas, Ch. C. Mousta kidis, and C.P. Panos, J. Chem. Phys. 123 (2005) 174111. [8] J.C. Angulo and J. Antolin, J. Chem. Phys. 128 (2008) 164109. [9] N.H. March, Self-Consistent Fields in Atom s (Pergamon Press, Oxford, 1975). [10] P. Hohenberg, and W. Kohn, Phys. Rev. 136 (1964) B864; W. Kohn, and J.S. Sham, Ibid. 140 (1965) A1113; N.D. Lang, and W. Kohn, Phys. Rev . B1 (1970) 4555. [11] R.G. Parr, and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford U.P., Oxford, 1989). [12] R.M. Dreizler, and E.K.U. Gross, Density Functional Theory (Springer-Verlag, Berlin, 1990). [13] D.H. Ceperley and B.J. Adle r, Phys. Rev. Lett. 45 (1980) 566. [14] B. Barbiellini-Am idi, Phys. Lett A134 (1989) 328. [15] L.J. Boya, J. Sañudo, A.F. P acheco, and A. Seguí, Phys. Rev. A 32 (1985) 1299. [16] M. Membrado, A.F. Pacheco, and J. Sañudo, J. Phys. A : Math. Gen. 24 (1991) 3605. [17] T. Tietz, J. Chem. Phys. 22 (154) 2094. 19 [18] S. Flügge, Practical Quantum Mecan i cs (Springer, Berlin, 1974). [19] T. Kato, Comm. Pure Appl. Math. 10 (1957) 151. [20] C.F. Fischer, The Hartree-Fock Method for Atoms: A Numerical Approach (Wiley, New York, 1977). [21] C.F. Fischer, Computer Phys. Comm., 14 (1978) 145. [22] A. Dembo, T.A. Cover, and J.A. Thomas, IEEE Trans. Inf. Theory 37 (1991) 1501. [23] C.E. Shannon, Bell Sys. Tech. J . 27 (1948) 379; ibid . 27 (1948) 623. [24] R. Lopez-Ruiz, Biophys. Chem., 115 (2005) 215. [25] J.B. Szabó, K.D. Sen, and A. Nagy, Phys. Lett. A 372 (2007) 2428. [26] R. Gáspár, Acta Phys. Hung. 3 (1954) 263 (in German), J. Mol. Struct. Theochem. 501-502 (2000) 1 (English translation T. Gál). 20 Figure Captions Fig. 1.- Atomic density at the origin as a function of Z . The continuous line represents our density with λ given by Eq. (31) after taking α = 0. The dots are the non- relativistic Hartree-Fock calculati ons of Fischer (see the text). Fig. 2.- The disequilibrium, D , versus the atomic number, Z , as given in Eq. (35). The dotted line represents the non -relativistic calcula tion ( 0 ≡ α ), whereas the continuous line is the rela tivistic case (see the text). Fig. 3.- The Shannon entropy, H , versus the atomic number, Z , as given in Eq. (33). The comments done in Fig. 2 are also valid here. Fig. 4.- The LMC complexity measure, LMC C , as a function of atomic number, Z, as given in Eq. (32). The comments in Fig. 2 are also valid here. 21 Fig.1 22 Fig.2 23 Fig.3 24 Fig.4