Alternative evaluation of statistical indicators in atoms: the non-relativistic and relativistic cases

Alternativ e ev aluation o f statistic al indicato rs in atoms: the no n-relativis tic and relativis tic case s Jaime Sa ˜ nudo a and Ricardo L´ op ez-Rui z b a Dep artamento de F ´ ısic a, F acultad de Ciencias, Universidad de Extr emadur a, E-06071 Badajoz, Sp ain, and BIFI, Universidad de Zar agoza, E-50009 Zar agoza, Sp ain b DIIS and BIFI, F acultad de Ciencias, Universidad de Zar agoza, E-50009 Zar agoza, Sp ain Abstract In this w ork, the calculation of a statistical measure of complexit y an d th e Fish er- Shannon information is p erf ormed for a ll the atoms in the p erio d ic table. N on- relativistic and relativistic cases are considered. W e follo w the metho d suggested in [C.P . P anos, N.S. Nik olaidis, K . C h . Chatzisa vv as, C.C. Tsouros, arXiv:0812. 3963v1] that uses the fractional o ccupation pr obabilities of electrons in atomic orbitals, in- stead of the con tin uous electronic w av e functions. F or the order of shell filling in the relativistic case, w e tak e in to accoun t the effect due to electronic sp in-orbit interac- tion. Th e increasing of b oth magnitudes, th e statistica l complexit y and the Fisher- Shannon information, with th e atomic num b er Z is ob s erv ed. The shell structur e and the irregular shell filling is well disp la yed by the Fisher-Shannon information in the relativistic case. Key wor ds: Statistical C omplexit y; Fisher-Shann on information; At oms; Sh ell Structure P ACS: 31.15.-p, 89.75.Fb. In the last y ears, the application of differen t informat io n-theoretic magnitudes, suc h as Shannon, Fisher, R´ en yi, and Tsallis en tropies and statistical complex- ities, in quantum systems has taken a g r owing intere st [1,2]. P articularly , the use of these indicators in the study of the electronic structure of atoms has receiv ed sp ecial attention [3,4,5,6,7,8]. Email addr esses: j sr@unex. es (Jaime Sa ˜ nudo), rilo pez@uniz ar.es (Ricardo L´ op ez-Ruiz). Preprint submitted to Elsevier 28 J une 2021 The basic ingredien t to calculate these statistical magnitudes is the electron probabilit y density , ρ ( ~ r ), that can b e obta ined from t he nu merically deriv ed Hartree-F o c k atomic w a v e f unction in the non- relativistic case [2,4], and f rom the Dirac- F o c k atomic w av e function in the relativistic case [6]. The b ehav ior of these statistical quan tifiers with the atomic nu m b er Z has rev ealed a con- nection with phy sical measures, suc h a s t he ionization p o ten tia l and the static dip ole p olarizability [5]. All of t hem, theoretical and phys ical magnitudes, are capable o f unv eiling t he shell structure of atoms, sp ecifically the closure of shells in the noble gases. Also, it has b een observ ed that statistical complexit y fluctuates around an a v erage v alue that is non-decreasing a s the atomic num- b er Z increases in the non- relativistic case [4,6]. This av erage v alue b ecomes increasing in the relativistic case [6]. This trend has also b een confirmed when the a tomic electron density is obta ined with a differen t approach [9]. An a lternativ e metho d to calculate the statistical mag nitudes can b e used when the at o m is seen as a discrete hierar chical organization. The atomic shell structure can also b e captured b y the fractiona l o ccupation probabilities of electrons in the differen t atomic o rbitals. This set of probabilities has b een emplo ye d in [10] to ev aluate all these quan tifiers for the non-relativistic ( N R ) case. On one hand, a non-decreasing trend in complexit y as Z increases is reco ve red. On the o t her hand, the closure of shells for some noble gases is now mask ed. In order to complemen t t he results giv en in [10] for the N R case, here w e un- dertak e the calculation for the relativistic ( R ) case b y a lso using the fractional o ccupation probabilities o f electrons in a t omic orbitals. F or the N R case, eac h electron shell of t he a tom is giv en b y ( nl ) w [11], where n denotes the principal quantum num b er, l the o rbital a ng ular momentum (0 ≤ l ≤ n − 1 ) and w is the num b er of electrons in the shell (0 ≤ w ≤ 2(2 l + 1)). F or the R case, due to the spin-orbit in teraction, each shell is split, in general, in tw o shells: ( nl j − ) w − , ( nl j + ) w + , where j ± = l ± 1 / 2 (fo r l = 0 only one v alue of j is p ossible, j = j + = 1 / 2 ) and 0 ≤ w ± ≤ 2 j ± + 1 [12]. As an example, w e explicitly giv e the electron configuration of Ar ( Z = 18 ) in b oth cases, Ar ( N R ) : (1 s ) 2 (2 s ) 2 (2 p ) 6 (3 s ) 2 (3 p ) 6 , (1) Ar ( R ) : (1 s 1 / 2) 2 (2 s 1 / 2) 2 (2 p 1 / 2) 2 (2 p 3 / 2) 4 (3 s 1 / 2) 2 (3 p 1 / 2) 2 (3 p 3 / 2) 4 . (2) F or eac h ato m, a fractional o ccupation probability distribution of electrons in atomic orbitals { p k } , k = 1 , 2 , . . . , ν , b eing ν the n um b er of shells of the atom, can b e defined. This norma lized probability distribution { p k } ( P p k = 1) is easily calculated b y dividing the sup erscripts w ± (n umber of electrons in eac h shell) b y Z , the total num b er o f electrons in neutral atoms, whic h is the case we are considering here. It should also b e mentioned that the order of shell filling 2 dictated b y nature [11] has b een chos en. Then, from this probability distri- bution, the differen t statistical magnitudes (Shannon entrop y , disequilibrium, statistical complexit y and Fisher-Shannon en trop y) can b e calculated. In this w ork, w e calculate the so-called LMC complexit y [13,1 4], a statistical measure of complexit y , C , that has b een recen tly used to enligh t en differen t questions on the hierarc hical organization of some few-b o dy quan tum systems [15,16,17,18] and also in the case of man y- b o dy quantum systems [2,4,6,7,1 0]. It is defined as C LM C = H · D , (3) where H , that represen t s the info r mation conten t of the system, is in this case the simple exp onen tial Shannon en trop y [14,19], H = e S , (4) S b eing the Shannon information entrop y [20], S = − Z ρ ( x ) log ρ ( x ) dx , (5) where ρ is the electron densit y normalized to unit y . D gives an idea of ho w m uch concen trated is its spatial distribution and it is calculated as the densit y exp ectatio n v alue [13,14] D = Z ρ 2 ( x ) dx . (6) The discrete vers ions of expressions (5) and (6) used in o ur calculations are giv en b y S = − ν X k =1 p k log p k , (7) D = ν X k =1 ( p k − 1 /ν ) 2 . (8) The Fisher-Shannon information, P , has also b een applied in atomic systems [15,16,17,18,21,22]. This quan tit y is giv en by P = J · I , (9) where J is a v ersion of t he p o w er Shannon en tropy [1 9] H = 1 2 π e e 2 S/ 3 , (10) whereas I is the so-called Fisher information measure [23], t ha t quan tifies the 3 narro wness of the probability densit y and it is giv en b y I = Z |∇ ρ ( ~ r ) | 2 ρ ( ~ r ) d ~ r . (11) In order to apply the presen t metho d, the same discrete v ersion of I as in [10] is used I = ν X k =1 ( p k +1 − p k ) 2 p k , (12) where p ν +1 = 0. The statistical complexit y , C , a s a function of the ato mic n um b er, Z , fo r the N R and R cases f or neutral atoms is g iven in Fig s. 1 and 2, resp ectiv ely . W e can observ e in b oth figures that this magnitude fluctuates around a n increasing a v erage v alue with Z . This increasing trend reco ve rs the b eha vior obtained in [4,6] b y using the con tinuous quantum-mec hanical w av e functions, a lt ho ugh it is not the same as found in [1 0]. This different tendency can b e explained b y the f act that we hav e used H = e S , whereas in [10] the authors tak e H = S . A shell-lik e structure is also unv eiled in this approac h b y lo oking at the minim um v alues of C taken on the noble gases p o sitions (the dashed lines in the figures) with the exception of N e ( Z = 10) and Ar ( Z = 18). The Fisher-Shannon en tropy , P , as a function o f Z , for the N R and R cases in neutral atoms is given in Figs. 3 and 4, resp ectiv ely . The shell structure is again displa ye d, esp ecially in the R case (Fig. 4) where P tak es lo cal maxima for all the noble gases (see the dashed lines on Z = 2 , 10 , 18 , 36 , 54 , 82). The irregular filling (i.f.) of s and d shells [11] is also detected b y p eaks in the magnitude P , mainly in the R case. In particular, see the elemen ts C r and C u (i.f. o f 4 s and 3 d shells); N b , M o , R u , Rh , and Ag (i.f. of 5 s and 4 d shells); and finally P t and Au (i.f. of 6 s and 5 d shells). P d also has a n irregular filling, but P do es no t displa y a p eak on it b ecause the shell filling in this case do es not fo llow the same pro cedure as the b efore elemen t s (the 5 s shell is empt y and the 5 d is full). Finally , the increasing trend of P with Z is clearly manifested. 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S tatistica l complexit y , C , vs. Z in the relativistic case ( C R ). Th e commen ts giv en in Fig. 1 are also v alid here. 7 0 10 20 30 40 50 60 70 80 90 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 P NR = (2 S e) -1 * e 2S NR /3 * I NR Z Fig. 3. Fisher-Sh annon entrop y , P , vs. Z , in the non relativistic case ( P N R ). T h e dashed lines in dicate the p osition of noble gases. F or details, see the text. 0 10 20 30 40 50 60 70 80 90 100 0.0 0.1 0.2 0.3 P R = (2 S e) -1 * e 2S R /3 * I R Z Cr Cu Nb Mo Ru Rh Ag Pt Au Fig. 4. Fisher-Shannon ent rop y , P , vs. Z , in the relativistic case ( P R ). Th e commen ts giv en in Fig. 3 are also v alid here. 8