Discontinuity of Shannon information entropy for two-electron atoms

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1 Chemial Physis 39 (25) Disontinuity of Shannon information entropy for two-eletron atoms Qiun Shi, Sabre Kais * Department of Chemistry, Purdue University, 1393 Brown Building, West Lafayette, IN 4797, USA Reeived 12 Deember 23; aepted 22 August 24 Available online 29 September 24 Abstrat Atomi Shannon information entropies are omputed using Hyllerass-type basis funtions for the helium isoeletroni series. As one varies the nulear harge Z for the two-eletron atoms, finite size saling analysis shows that the system exhibits a ritial point of the ground state energy at k =1/Z = ±.5 with a ritial exponent a =1,E(k) (k k ) a. At the ritial point, the ground state energy beomes degenerate with the hydrogeni threshold. The Shannon information entropy develops a step-like disontinuity at k. Further analysis indiates that the entropy as a funtion of k is proportional to the first derivative of the energy with respet to k. The ritial exponent for the entropy a s =,SðkÞ ðk k Þ as. Ó 24 Elsevier B.V. All rights reserved. 1. Introdution Shannon information entropy measures the extent, spread, shape, unertainty, and the information ontent of the underlying distribution from whih it was derived [1]. Shannon proposed that the information entropy for a system with a ontinuous probability distribution P(x) in one dimension ould be haraterized as [1] Z Z S ¼ PðxÞ ln PðxÞdx; PðxÞdx ¼ 1: ð1þ For atomi systems one an define the Shannon entropy in position spae S q, where the probability distribution is the eletroni harge density q(r). The momentum spae Shannon entropy S p of the eletroni momentum density p(p) is defined in a fully analogous way. The importane of the entropy sum S t = S q + S p is firmly established by noting that a stronger version of HeisenbergÕs unertainty priniple may be formulated for any quantum N-eletron system [2,3], * Corresponding author. Tel.: ; fax: address: kais@purdue.edu (S. Kais). S t P 3Nð1 þ ln pþ 2N ln N: ð2þ Reently, numerial alulations demonstrate that the total entropy is invariant to saling and ould be used to measure basis set quality [3,4]. The total entropy also measures orrelations in many-eletron systems [5] and nulei [6]. It has linear dependene on the logarithm of the number of partiles in atoms, nulei, atomi lusters [7,8] and even in orrelated boson systems [6]. Shannon entropy in atomi alulations has further been related to various properties suh as atomi ionization potential [9], moleular geometri parameters [1], hemial similarity of funtional groups [11], harateristis of orrelation methods for global deloalizations [12], moleular reation paths [13], orbital-based kineti theory [14], and highly exited states of single-partile systems [15]. In previous studies [16,17], using finite size saling, we have shown that two-eletron atoms exhibit a ritial point as one varies the nulear harge Z. At the ritial point k =1/Z = ±.4 the ground state energy beomes degenerate with the hydrogeni threshold with a ritial exponent a =1, E(k) (k k ) a. In the present work we alulate the atomi Shannon 31-14/$ - see front matter Ó 24 Elsevier B.V. All rights reserved. doi:1.116/j.hemphys

2 128 Q. Shi, S. Kais / Chemial Physis 39 (25) information entropies using Hyllerase-type basis funtions for the helium isoeletroni series. At the ritial point, the Shannon information entropy develops a step-like disontinuity at k with a ritial exponent a s =, SðkÞ ðk k Þ as. 2. Finite size saling for two-eletron atoms E/Z 2 (a.u.) The Hamiltonian for two-eletron atoms in the saled variable, r! r/z, is given by H ¼ 1 2 r r þ k ; ð3þ r 1 r 1 r 12 where k =1/Z is the inverse of the nulear harge Z and measures the strength of the intereletroni repulsion. By varying the parameter k one an study the behavior of the energy for the helium isoeletroni sequene [16] starting from k = 1, whih orresponds to the stable H anion [18]. For this Hamiltonian, a ritial point means the value of k for whih a bound state energy beomes degenerate with the hydrogeni threshold. To apply the finite size saling one has to hoose a onvenient basis set to obtain the lowest two eigenvalues, E (k) and E 1 (k). Sine our fous on the ground state 1 S, we an hoose the following Hylleraas-type basis sets [19,2]: W ¼ X 1 C i;j;k p ffiffi ðr i 1 i;j;k 2 rj 2 e ar 1 br 2 þ r j 1 ri 2 e br 1 ar 2 Þr k 12 ; ð4þ in whih i,j,k are indies defining the Pekeris shell [21] with 6 i,j,k 6 N and i + j + k 6 N. N is the radius of the shell whih determines the expansion length M(N) of Eq. (4). Our numerial results show that M(N) N Suh a systemati expansion of the wave funtion in the Hylleraas-type basis sets has been used extensively in alulations of the ground and exited states of the two-eletron atoms [22 24,17]. In Eq. (4) a and b are the two hydrogeni exponents. For the ground state of helium in Eq. (3) we hoose a = 2 and b 2 and a =1andb 1 for the H anion at k = 1. As we inrease the parameter k to simulate the eletron ionization we have found that b =.21 is the optimal hoie for the wave funtion. Putting the trial wave funtion into the Shrödinger equation and diagonalizing the energy matrix, we obtain the eigenvalues and the eigenvetors C, the expansion oeffiients in Eq. (4). All the numerial alulations were performed using 64-bit Fortran 95 odes. Fig. 1 shows the behavior of the ground state energy as a funtion of k for different values of N =5,6,7,...,13. The results are in omplete agreement with our previous results [16,25]. In the present work, a = 1 and b =.21 are fixed and the energy eigenvalues are obtained by systematially inreasing the order N for the expansion in Eq. (4). InFig. 1 the energy lines for = 1/Z Fig. 1. The ground state saled energy, E/Z 2, as a funtion of k =1/Z, where Z is the nulear harge, for different basis set orders,6,..., 13. The dashed line orresponds to the threshold energy, E th =.5 a.u. eah order N gradually go up at k < 1.14 and bend over sharply at k = As N!1 the true ground state energy beomes degenerate with the lowest ontinuum E = 1/2 at k [25]. Now, the finite size saling analysis an be used to obtain the ritial point. The phenomenologial renormalization equation for finite systems of sizes N and N +1 is given by [16] N E 1 ðnþ ¼ E Nþ1 1ðN þ 1Þ ð5þ E ðnþ E ðn þ 1Þ and has a fixed point at k ðnþ. It is expeted that the suession of the rossing points k ðnþ in the limit of infinite size to onverge to the true k. Fig. 2 shows the rossing points, whih are the fixed points of Eq. (5), for N =5,6,7,...,13. The values of the fixed points as a funtion of N an be extrapolated to the limit N!1 by using the Bulirsh and Stoer algorithm [17]. We obtain k = as shown in the window of Fig. 2. Next, we examine the onditional probability in order to visualize the attrative and repulsive fores whih influene the system lose to the ritial point. This funtion represents the probability distribution seen by one eletron given that the other eletron is fixed at a ertain distane. We define the onditional probability to find one eletron at distane r 1 when the other is fixed at r 2 =1by Pðr 1 Þ r2 ¼1 ¼ 2p Z r1 þ1 jwj 2 r 12 dr 12 : ð6þ r 1 jr 1 1j Fig. 3 shows the three onditional probability urves orresponding to k < k, k = k, k > k at k = 1.15, 1.155, 1.16, respetively. In order to ompare the three urves, they were normalized suh that the first maximum is one. For k < k the leading maximum is at small distane, r 1 2 a.u., for k! k the two maxima E th

3 Q. Shi, S. Kais / Chemial Physis 39 (25) [ ] N E 1 () E () reahes the same height and for k > k, the seond maxima is getting larger and the nulear harge annot bind two eletrons. Thus one eletron move to infinity and beome a free eletron for k > k. 3. Critiality of Shannon information entropy Information theory is the framework in whih one attempts to measure the amount of information inherent in a system. Shannon information entropy for the twoeletron systems with eletroni harge distribution q(r) is given by S ¼ Z (N) /N qðrþ ln qðrþ4pr 2 dr; = ð7þ where the eletron density q(r 1 ) an be obtained from the wave funtion W(r 1,r 2,r 12 ) by averaging over the other oordinates, Fig. 2. The ratio between the ground state energy, E, and the seond lowest eigenvalue, E 1, raised to a power N as a funtion of k ¼ 1=Z for,6,..., 13. The small window shows the pseudoritial points, k ðnþ as a funtion of 1/N. The value of the extrapolated k = is also shown as a dot. Z 1 Z r1 þr 2 qðr 1 Þ¼ 2p jwj 2 r 2 dr 2 r 12 dr 12 ; r 1 jr 1 r 2 j in whih q(r) is normalized to 1, Z 1 qðrþ4pr 2 dr ¼ 1: ð8þ ð9þ q(r) is ontinuous over 6 r 6 1, representing a radial orrelation of eletron 1 with eletron 2 bounded to the nuleus. Fig. 4 shows the alulated entropy S from Eq. (7) as a funtion of k for different values of N. For k < k, the entropy is about S 6.5 and slowly inreased for larger values of k up to the ritial point. At the ritial point, the entropy develops a step-like disontinuity and the entropy jumps to higher value, S 1.7. This behavior resemble the behavior of the first derivative of the energy with respet to k, whih develops a step-like disontinuity at k. Fig. 5 shows the ratio of the two onseutive entropies raised to power N, (S N +1 /S N ) N, as a funtion of k. Eah of the orresponding urves exhibits a sharp peak. The height of the peak is proportional to the entropy gradient. The peak inreases rapidly as N beomes large while the position of the peak approahes k = 1.9 whih ompares well with the most aurate estimate of k = Now, let us determine the ritial exponent a s for the entropy SðkÞ ðk k Þ as : ð1þ In our previous studies we have shown that the ritial exponent for the energy is 1, EðkÞ ¼Eðk Þþ:235ðk k Þ 1 ; ð11þ where E(k )= 1/2. Our numerial results indiate that the entropy is proportional to the first derivative of the ground state energy with respet to k. Fig. 6 show the first derivative of energy with respet to k (solid lines) and the fitted entropy (dots) assuming the form 3. N = 12 = P(r 1 ) r2 = (12) = S(N) 8. = r 1 (a.u.) Fig. 3. Conditional probability distribution, Pðr 1 Þ r2 ¼1, as a funtion of r 1 for k = 1.15,1.155,1.16 with basis set order N = 12. The three urves are saled to 1 at r 1 = Fig. 4. Shannon information entropy S(N) as a funtion of k for N =5,6,...,12.

4 13 Q. Shi, S. Kais / Chemial Physis 39 (25) [ S(N+1) ] N S(N) N = Fig. 5. The ratio between two onseutive entropies raised to power N, ðfsðn þ 1Þg=SðNÞÞ N as a funtion of k for N =5,6,...,13. de (N)/d.2.1 The Shannon information entropies for the two eletron atoms were alulated using a highly aurate wave funtion based on expansion in Hyllerase-type basis set. At the ritial point k =1/Z = 1.976, the entropy develops a step-like disontinuity whih resemble the behavior of the first derivative of the energy with respet to k. Numerial analysis shows that the entropy is indeed proportional to the first derivative of the ground state energy with respet to k and thus its ritial exponent a s =. This physial piture is onsistent with the definition of Shannon information entropy whih measures the deloalization or the lak of struture in the respetive distribution. Thus S is maximal for uniform distribution that for an unbound system and is minimal when the unertainty about the struture of the distribution is minimal. For two eletron atoms the entropy S is minimal when the two eletrons are bound in the ground state k < k then develops a step-like disontinuity at k and jumps to maximal value S 1.7 for k > k. Reently, we have established the analogy between symmetry breaking of eletroni struture onfigurations and quantum phase transitions [26]. In partiular for two eletron atoms, the mapping between symmetry breaking and mean-field theory of phase transitions was shown by allowing the nulear harge Z, the parameter whih tunes the phase transition, to play a role analogous to temperature in lassial statistial mehanis. In this study we have shown that one an add the entropy to this mapping. Here the Shannon information entropy develops a step-like disontinuity as a funtion of k whih resemble the step-like disontinuity of the thermodynami entropy as a funtion of the temperature in a first order phase transition. SðkÞ ¼a b de ð12þ dk with a = 1.93 and b = For large values of N, the fitted results are good agreement with alulated first derivative of the ground state energy. From this numerial fit, the ritial exponent for the entropy a s = as k! k. 4. Conlusions Fig. 6. The first derivative of energy, de (N)/dk (solid lines) and the fitted derivative from the entropy S(N) = dE (N)/dk (dots) as a funtion of k for N =5,6,...,13. Note that the entropy, S(k), is proportional to the first derivative of the energy with respet to k. Aknowledgments We aknowledge the finanial support of the National Siene Foundation (NSF) and the Amerian Chemial Soiety, Petroleum Researh Fund (PRF). Referenes [1] R.D. Levine, in: R.D. Levine, M. Tribus (Eds.), The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 1979, and referenes therein. [2] I. Bialyniki-Birula, J. Myielski, Commun. Math. Phys. 44 (1984) 129. [3] S.R. Gadre, S.B. Sears, S.J. Chakravorty, R.D. Bendale, Phys. Rev. A 32 (1985) 262. [4] S.R. Gadre, Phys. Rev. A 3 (1984) 62. [5] N.L. Guevara, R.P. Sagar, R.O. Esquivel, Phys. Rev. A 67 (23) [6] S.E. Massen, Phys. Rev. C 67 (23) [7] S.E. Massen, C.P. Panos, Phys. Lett. 246 (1998) 53. [8] C.P. Panos, S.E. Massen, C.G. Koutroulos, Phys. Rev. C 63 (21) [9] M. Ho, R.P. Sagar, H.L. Shmider, D.F. Weaver, V.H. Smith Jr., Int. J. Quantum Chem. 53 (1995) 627. [1] M. Ho, R.P. Sagar, D.F. Weaver, V.H. Smith Jr., Int. J. Quantum Chem. S29 (1995) 19. [11] M. Ho, V.H. Smith Jr., D.F. Weaver, C. Gatti, R.P. Sagar, R.O. Esquivel, J. Chem. Phys. 18 (1998) [12] M. Ho, V.H. Smith Jr., D.F. Weaver, R.P. Sagar, R.O. Esquivel, S. Yamamoto, J. Chem. Phys. 19 (1998) 162.

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