Finite Size Scaling for the Atomic Shannon-Information Entropy

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 12 22 SEPTEMBER 2004

Finite size scaling for the atomic Shannon-information entropy Qicun Shi and Sabre Kais Department of Chemistry, Purdue University, West Lafayette, Indiana 47907 ͑Received 4 May 2004; accepted 1 July 2004͒ We have developed the ﬁnite size scaling method to treat the criticality of Shannon-information entropy for any given quantum Hamiltonian. This approach gives very accurate results for the critical parameters by using a systematic expansion in a ﬁnite basis set. To illustrate this approach we present a study to estimate the critical exponents of the Shannon-information entropy Sϳ(␭ Ϫ Ϫ ␣S Ϫ ␣E Ϫ ␯ ␭c) , the electronic energy Eϳ(␭ ␭c) , and the correlation length ␰ϳ͉␭ ␭c͉ for atoms with the variable ␭ϭ 1/Z, which is the inverse of the nuclear charge Z. This was realized by approximating the multielectron atomic Hamiltonian with a one-electron model Hamiltonian. This model is very accurate for describing the electronic structure of the atoms near their critical points. For several atoms in their ground electronic states, we have found that the critical exponents ϭ ϭ ϭ ϭ ϭ (␣E ,␯,␣S) for He (Z 2), C (Z 6), N (Z 7), F (Z 9), and Ne (Z 10), respectively, are ͑1, ϭ 0, 0͒. At the critical points ␭c 1/Zc , the bound state energies become absorbed or degenerate with continuum states and the entropies reach their maximum values, indicating a maximal delocalization of the electronic wave function. © 2004 American Institute of Physics. ͓DOI: 10.1063/1.1785773͔

I. INTRODUCTION is, for an unbound system, and is minimal when the uncertainty about the structure of the distribution is minimal. In statistical physics, the finite size scaling method pro- Shannon proposed that the information entropy for a system vides a systematical way to extrapolate information available with a probability distribution P(x) in one dimension could from a finite system to the thermodynamic limit.1,2 Finite be characterized by size scaling crossing points, the pseudocritical points, are the information needed in order to extrapolate to the thermody- SϭϪ P͑x͒ln P͑x͒dx; P͑x͒dxϭ1. ͑1͒ namic limit, where singularities or phase transitions occur. ͵ ͵ Recently we have developed the finite size scaling methods For atomic systems one may define the spatial Shannon en- to study phase transitions that are taking place at the absolute tropy S␳ using the electronic density ␳ and its momentum 3 zero of temperature in atoms and molecules. These transi- space analog S␲ . Such definitions result in that the sum St ϭ ϩ tions are not driven by temperature as in classical phase tran- S␳ S␲ provides a stronger version of the Heisenberg’s sitions but rather by quantum fluctuations as a consequence uncertainty principle for any N-electron system,16,17 of the Heisenberg’s uncertainty principle.4,5 In this regard, S у3N 1ϩln ␲͒Ϫ2N ln N. ͑2͒ the finite size corresponds not to the spatial dimension, as in t ͑ statistical physics, but to the number of elements in a com- The total entropy has been used in recent years to mea- plete basis set used to expand the wave functions of a given sure correlations in many-electron systems18 and nuclei.19 Hamiltonian. Here the ‘‘thermodynamic limit’’ is achieved Also, it is invariant to scaling and has been used to measure by extrapolation to the infinite basis set limit. The phase basis set quality.17,20 Furthermore, it has a linear dependence transitions occur by tuning parameters in the Hamiltonian. on the logarithm of the number of particles in atoms, nuclei, Different finite size scaling methods were recently re- atomic clusters,21,22 and in correlated boson systems.19 viewed in quantum mechanics6 and were successfully ap- Shannon entropy has further been related to various proper- plied to atoms,7 molecules,8 three-body systems,9 crossover ties such as molecular geometric parameters,23 chemical phenomena and resonances10 and critical conditions for similarity of different functional groups,24 ionization stable dipole and quadrupole bound anions.11,12 Critical phe- potential,25 global delocalizations,26 molecular reaction nomena described by energy operators are examined through paths,27 orbital-based kinetic theory,28 and highly excited varying the nuclear charges’ or molecular fragments’ separa- states of single-particle systems.29 tion distances. For atoms, the phase transitions at a critical Recently, we studied the criticality of energy operator for point exploit a fundamental process of an electron leaving a given quantum-mechanical system of either known atoms the atom, the ionization process.7,11 For molecules and clus- and molecules or models.7–13 The theoretical calculations ters the phase transitions are closely related to the symmetry provided energetic information and made it possible to ex- breaking in the geometrical configurations of the molecular tract the critical exponent using the scaling function of en- fragments.8,9,13 ergy. Most recently we revisited the simplest two-electron Shannon-information entropy14 measures the delocaliza- Helium atom in the Hylleraas coordinate.30 The Hylleraas tion or the lack of structure in the respective distribution.15 expansion approach to the wave function was even con- Thus the entropy S is maximal for uniform distribution, that firmed as the most accurate method in the Hilbert space

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(N) (ϱ) (ϱ) ϭ Ϫ Ϫ␯ spanned by a single-exponent Slater-type basis set. After cal- ͗O͘ ͑␭͒ϳ͗O͘FO͑N/␰ ͑␭͒͒, ␰ ͑␭͒ ͉␭ ␭c͉ , ͑7͒ culating both energy and electron density ͑and therefore en- with a different scaling function FO for each different opera- tropy͒ and then ®tting those accurate data we found an ap- (ϱ) proximately linear relation between the entropy and the ®rst tor but with a unique scaling exponent ␯. ␰ (␭) is the cor- → derivative of the energy. In particular, we noted that close to relation length at N ϱ. O (N) the critical point such linear relation becomes true. These Since ͗ ͘ (␭) is analytical in ␭, the scaling function at → results show that the exponent of the entropy for helium is ␭ ␭c must behave like Ϫ ␮O /␯ the exponent of the energy minus 1. FO͑x͒ϳx , ͑8͒ In this paper, we develop the ®nite size scaling method where xϭN(␭Ϫ␭ )␯→0. It follows that O (N)(␭) at ␭ to obtain the critical parameters related to the atomic c ͗ ͘ c depends on N as a power law Shannon entropies. In Sec. II we review the scaling functions (N) Ϫ ␮O /␯ and establish relations among the critical exponents for the ͗O͘ ͑␭c͒ϳN ; N→ϱ. ͑9͒ atomic Shannon entropy, the correlation length, and the elec- Using Eq. ͑9͒ one may extrapolate the exponent ␮ and tronic energy. In Sec. III we describe a one-electron model O ␯ ͑Ref. 31͒ from the crossings of the pseudocritical expo- Hamiltonian used to approximate the multielectron Hamil- (N,NЈ) (N,NЈ) tonian. In Sec. IV we present results and also the integrals nents ␮O and ␯ for two consecutive expansions N, Јϭ ϩ ϩ 32 needed for calculating the different energy matrix elements. N N 1 ͑or N 2 if parity effects involve ͒, Ј Discussion and conclusions follow in Sec. V. Atomic units (N,NЈ) ln͑͗O͘(N)/͗O͘(N )͒ ␮O ␭c ␭c are used throughout the paper unless otherwise speci®ed. ϭ . ͑10͒ ␯(N,NЈ) ln͑NЈ/N͒ II. FINITE SIZE SCALING FOR THE CRITICALITY At this point we may de®ne the energy E exponent ␣ as OF SHANNON-INFORMATION ENTROPY E E ϳ ͑␭Ϫ␭ ͒␣E ͑11͒ The Hamiltonian for many interesting problems in quan- ␭ ϩ c ␭→␭c ϳ tum mechanics, in particular for atomic and molecular and obtained from Eq. ͑10͒ by replacing OϵH systems,8 can be transformed to the following general form: (N,NЈ) HϭH ϩV ␣E 0 ␭ , ͑3͒ ϭ ⌬H͑␭c ;N,NЈ͒, ͑12͒ ␯(N,NЈ) where H0 is ␭ independent while V␭ is a ␭ dependent term and ␭ represents a free parameter. For electronic structure of where the notation ⌬H is used for the right side of Eq. ͑10͒. atoms, ␭ is the inverse of the nuclear charge Z. At the critical For unique exponents we start from Eq. ͑11͒ and take the 32 point ␭c , a bound state described by the SchroÈdinger equa- derivative tion H⌿ϭE⌿ becomes absorbed or degenerate with a con- E Ϫץ ␭ Ϫ ␣E 1 tinuum. ϳ ͑␭ ␭c͒ . ͑13͒ ␭ → ϩץ For a ␭-independent complete and orthonormal basis set ␭ ␭c 2 ͕⌽n͖, spanning an in®nite-dimensional Hilbert space L , the Using Hellmann-Feynman theorem bound state can be expanded as V␭ץ Hץ E␭ץ ϱ ϭ ϭ ͑14͒ ʹ ␭ץ ␭ ʹ ͳץ ␭ ͳץ ϭ ␭ ␭ ⌿␭ ͚ Cn͑␭͒⌽n , ͑4͒ n Oϭ V VЈ ␭ ϵ ␭ in Eq. ͑10͒ we haveץ/ ␭ ץ and taking where n is an adequate set of quantum indices and ͕Cn(␭)͖ Ј ␣(N,N )Ϫ1 the expansion coef®cients which are generally ␭ dependent. E ϭ ⌬V Ј͑␭c ;N,NЈ͒. ͑15͒ In calculating the energy matrix, the expansion has to be ␯(N,NЈ) ␭ truncated at order N, i.e., expanding up to a ®nite length, M(N) which equals a ®nite dimension in the Hilbert space. Given the electron probability ␳, atomic Shannon- For a ®nite basis set, the expectation value of any quantum information entropy S may be de®ned as the expectation I operator O is given by value of the operator ␳ in ϭ Ϫ M(N) S␭ ͗ ln ␳͑r͒͘␭ϵ͗I␳͘␭ . ͑16͒ O ͑N͒ ϭ (N) (N) O ͗ ͘ ͑␭͒ ͚ Ci ͑␭͒C j ͑␭͒ i, j , ͑5͒ Its exponent ␣ is then de®ned by i, j S Ϫ ␣S O O S␭ ϳ ϩ͑␭ ␭c͒ . ͑17͒ where i, j are the matrix elements of in the basis set ␭→␭ O c ͕⌽n͖. In general, the mean value of the operator is not OϭI ϭ By replacing ␳ in Eq. ͑10͒, one obtains analytical at ␭ ␭c , and has the form Ј ␮ (N,N ) O ϳ ␭Ϫ␭ O ͑ ͒ ␣S ͗ ͘ ϩ͑ c͒ , 6 ϭ ␭→␭ ⌬I ͑␭c ;N,NЈ͒. ͑18͒ c ␯(N,NЈ) ␳ where ␮O is the critical exponent. The ®nite size scaling hypothesis31 assumes the exis- From Eqs. ͑12͒, ͑15͒, and ͑18͒ we can obtain the pseud- (N,NЈ) (N,NЈ) (N,NЈ) tence of a scaling function for the truncated expansion such ocritical exponents ␣E , ␯ , and ␣S by de®ning that the following functions:

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2 ⌬H p Ј ϭ ϭ N ϩ ⌫␣ ͑␭,N,N ͒ , h V͑r0N ,r01 ,r02 ,...,r0(NϪ1)͒, ͑24͒ E ⌬HϪ⌬V Ј 2m ␭ N 1 and ϭ ⌫␯͑␭,N,NЈ͒ , ͑19͒ ⌬HϪ⌬V Ј V͑r ,r ,r ,...,r Ϫ ͒ ␭ 0N 01 02 0(N 1) NϪ1 ⌬I 1 1 Ј ϭ ␳ ϭϪ ϩ␭ ⌫␣ ͑␭,N,N ͒ . r ͚ϭ r S ⌬HϪ⌬V Ј 0N i 1 iN ␭ Ϫ 1 ␭ N 1 1 These functions are independent of the values of N and ϭϪ ϩ Ј ϭ ͚ Ϫ , ͑25͒ N at the critical point ␭ ␭c and give the critical exponents r0N r0N iϭ1 ͉rÃ0N ͉r0i͉/r0N͉ ␣ ϭ⌫ ͑␭ ,N,NЈ͒; ␯ϭ⌫ ͑␭ ,N,NЈ͒; E ␣E c ␯ c where rÃ0N stands for the unit vector of r0N . ϭ ͑20͒ V(r0N ,r01 ,r02 ,...,r0(NϪ1)) is not a central potential and fur- ␣S ⌫␣ ͑␭c ,N,NЈ͒. S ther approximation is needed. If we examine the limit r0N Ӷ The ®nite size scaling equations are valid as an asymptotic r0i , as the electron N moves closer to the nucleus, expression, N→ϱ. However, with a ®nite basis set, pseud- Ј Ј 1 (N,N ) (N,NЈ) (N,N ) V͑r ,r ,r ,...,r Ϫ ͒→Ϫ . ͑26͒ ocritical exponents ␣E , ␯ , and ␣S can be ob- 0N 01 02 0(N 1) r tained as a succession of values as a function of (N,NЈ). In 0N Ӷ order to obtain the extrapolated values ␣E , ␯, and ␣S at N However, in the limit r0i r0N , the electron N is closer to →ϱ we used the algorithm of Bulirsch and Stoer.33 the ionization region, 1 ␥ Ϫ ϩ V͑r0N ,r01 ,r02 ,...,r0(NϪ1)͒→ , r0N r0N III. ONE-ELECTRON APPROXIMATION FOR ATOMIC HAMILTONIAN ␥ϭ͑NϪ1͒␭. ͑27͒

A direct way to treat multielectron systems is to use ab Generally, r0iрr0N for all i electrons and the potentials initio electronic structure methods with Gaussian basis-set should include a Coulomb repulsive term less than ␥/r0N . functions. This approach is widely applied in modern quan- This may be approximated by a short-range interaction of the tum chemistry computations. We have recently examined Yukawa type potential such Gaussian basis functions using ®nite size scaling V͑r ,r ,r ,...,r Ϫ ͒ method for simple molecular systems.8 The problem of using 0N 01 02 0(N 1) Gaussian functions in ®nite size scaling analysis is that they 1 ␥ Ϫ ϭϪ ϩ Ϫ ␦r0N ϷV͑r0N͒ ͑1 e ͒, ͑28͒ do not form a complete basis set and are more localized than r0N r0N the Slater functions. In the present work, we apply the one-electron approxi- where ␦ expresses the interaction range and may be extrapolated using data available either from experiments or other mation to the Hamiltonian ͓Eq. ͑3͔͒ to avoid the complexity 34 of the multielectron problem while producing the relevant theories according to the following equation: main physics near the critical points. Let us start by consid- Ϫ Ϫ Ϫ ␦0͑␥ ␥1͒ ␦1͑␥ ␥0͒ ering an N-electron Hamiltonian for neutral atoms with in®- ␦ϭ . ͑29͒ ͑␥ Ϫ␥ ͒ nite nuclear mass,8 0 1 N 2 N NϪ1 N Here (␦0 ,␥0) and (␦1 ,␥1) are parameters corresponding to pi 1 1 Hϭ͚ Ϫ͚ ϩ␭ ͚ ͚ , ͑21͒ the neutral atom and its isoelectronic negative ion ͑if the iϭ1 2mi iϭ1 r0i iϭ1 jϭ2,iϽ j ri j negative ion does not exist, the parameters corresponding to the positive ion are used͒, respectively. where p is the momentum of electron i with mass m , r is i i 0i Solving the one-electron model in H⌿ϭE⌿ and utiliz- the distance between electron i and the nucleus ͑index 0͒, ing the solution ⌿ we may construct the electron spatial and r are the interelectron distances. i j density ␳(r )ϭ ⌿(r ) 2. Under spherically averaging ap- We decompose the Hamiltonian H into two parts: one N ͉ N ͉ proximation with electron radial density ␳(r ), the atomic represents the highest occupied orbital for electron N and the N Shannon entropy takes30 the following form: other for the atomic core with the remaining NϪ1 electrons, ϭ ϩ ϱ H Hcore h. ͑22͒ SϭϪ ͵ ␳͑r͒ln ␳͑r͒4␲r2dr ͑30͒ ϭ 0 If we assume that the the core is frozen, ⌿ ⌽core␺, and is ϭ in its ground state, Hcore⌽core Ecore⌽core , with with normalization Ϫ Ϫ Ϫ Ϫ N 1 p2 N 1 1 1 N 2 N 1 1 ϱ ϭ i Ϫ ϩ ␳͑r͒4␲r2drϭ1, ͑31͒ Hcore ͚ ͚ ͚ ͚ , ͑23͒ ͵ iϭ1 2mi iϭ1 r0i Z iϭ1 jϭ2,iϽ j ri j 0 ϭ Ϫ then h must satisfy h␺ (E Ecore)␺ in which where rϵrN and ␳(r) is continuous over 0рrрϱ.

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IV. RESULTS

For the one-electron problem, we expand the wave function in H⌿ϭE⌿,

ϭ ⌿͑r͒ ͚ Rn,l͑r͒Y l,m͑⍀͒ ͑32͒ n,l,m with 1 R ͑r͒ϭ eϪr/2L(2) ͑r͒, ͑33͒ n,l ͓͑nϩ1͒͑nϩ2͔͒1/2 nϪl where n is the principal quantum number, l is the angular ϭϪ Ϫ ϩ Ϫ (2) quantum number, and m l, l 1,...,l 1,l. LnϪl(r) is the generalized Laguerre polynomial of order 2 and degree (nϪl). Here the order 2 is chosen only for convenience in the calculations. ͕Rn,l(r)Y l,m(⍀)͖ constitutes a complete and orthogonal basis set for lрnϪ1. In the spherical aver- age, Eqs. ͑30͒ and ͑31͒, the spherical harmonic functions Y l,m(⍀) are normalized to 4␲. Ј FIG. 1. The crossing points de®ning the pseudocritical exponents ␣(N,N ) , A. Yukawa potential E (N,NЈ) (N,NЈ) ϭ ␯ , and ␣S for the Yukawa potential (s,l 0) as a function of Let us consider a system involving one electron moving parameter ␭ϭ1/␦ for (N,NЈ)ϭ(8,10),(12,14),...,(56,58). in the Yukawa potential V(r)ϭϪeϪ␦r/r, which is obtained from Eq. ͑27͒ by setting ␥ϭ1. The corresponding Hamil- ϭϪ 1 2Ϫ Ϫ␦r 2 tonian is H 2ٌ e /r. By scaling ␦r→r and H/␴ 1 1 ␭ hϭϪ ٌ2Ϫ ϩ ͑1ϪeϪ␦r͒, ͑35͒ →H, the scaled Hamiltonian becomes 2 r r 1 1 ϭ ϭ ϭϪ 2Ϫ Ϫr ϭ where ␦ 0.22 is calculated from Eq. ͑29͒ with ␦0 1.066 H ٌ ␭e /r; ␭ . ͑34͒ ϭ 34 2 ␦ and ␦1 0.881. h may approximate the full Hamiltonian Ј Next, we calculate the pseudocrossing points ␣(N,N ) , 1 1 1 1 ␭ E HϭϪ ٌ2ϩ Ϫ ٌ2Ϫ Ϫ ϩ , ͑36͒ (N,NЈ) (N,NЈ) 2 1 ͫ 2 2 r ͬ r r ␯ , and ␣S using Eq. ͑19͒ between two consecutive 2 1 12 sizes N and NЈϭNϩ2, with basis-set expansion Nϭ2,4,..., if Hcore , the bracketed term in Eq. ͑36͒, is simply replaced 58. By extrapolating N to in®nity,33 we obtain the critical Ϫ by 0.5 a.u, the hydrogen ground-state energy, and r1 by r. exponents for the Shannon entropy ␣S , the energy ␣E , and Figure 3 shows the pseudocritical exponent curves for the correlation length ␯. Nϭ8 ± 58. From the ®gures we obtained the critical point Figure 1 presents the pseudocritical exponent crossings for an s-state electron lϭ0 in the Yukawa potential as a function of ␭ϭ1/␦. The extrapolated values of the three Ϫ critical exponents (␣E ,␯,␣S) are (2,1, 0.23), respectively. The energy and correlation length exponents are in full agreement with previous results.32 The system exhibits a ``continuous phase transition'' as the s-state electron moves to the continuum at a threshold Eϭ0. The entropy is singular ϭ Ϫ at ␭c 0.84 with an exponent 0.23. Figure 2 displays the pseudocritical exponent crossings for a p-state electron lϭ1 with Nϭ26± 68. The three expo- ϭ ϭ 1 ϭ nents are (␣E 1,␯ 2,␣S 0). The energy phase transition ϭ is of ``®rst order'' with ␭c 4.54. All the exponent values are listed in Table I.

B. One-electron atoms For one-electron atoms we expand the wave functions using Eqs. ͑32͒ and ͑33͒. First, we describe the calculations for the heliumlike atoms and show how the one-electron model approximates the two-electron systems. According to (N,NЈ) FIG. 2. The crossing points de®ning the pseudocritical exponents ␣E , Ј Eqs. ͑24͒ and ͑28͒ the approximate Hamiltonian includes an ␯(N,NЈ), and ␣(N,N ) for the Yukawa potential (p,lϭ1) as a function of ϩ S ionic core He and a 1s electron, parameter ␭ϭ1/␦ for (N,NЈ)ϭ(26,28),(28,30),...,(58,60).

TABLE I. Critical charges Zc and the extrapolated critical exponents ␣E , ␯, ϭ ␣S for atomic systems with nuclear charges Z 2,6,7,9,10, and s state and p state for Yukawa potential.

Yukawa potential

N nl ␭c ␣E ␯ ␣S 1 1s 0.84 2 1 Ϫ0.23 1 2p 4.54 1 0.5 0

Atoms ϭ Ϫ Ϫ1 N( Z) nl Zc (Z 1) ␣E ␯ ␣S 2͑He͒ 1s 0.91 HϪ1 1 0 0 6͑C͒ 2p 4.97 BϪ1 1 0 0 7͑N͒ 2p 5.87 CϪ1 1 0 0 9͑F͒ 2p 7.87 OϪ1 1 0 0 10͑Ne͒ 2p 8.74 FϪ1 1 0 0

ϭ ␭c 1.096. This value is in agreement with an exact calcu- ϭ lation ␭c 1.097 using the full Hamiltonian and Hylleraas- 30 (N,NЈ) type functions as basis sets. The corresponding critical FIG. 4. The crossing points de®ning the pseudocritical exponents ␣E , ϭ (N,NЈ) (N,NЈ) ϭ charge Zc 1/␭c is 0.911. With this critical charge in mind ␯ , and ␣S for neonlike atoms as a function of parameter ␭ 1/Z we may draw a picture for the two-electron systems: at the for (N,NЈ)ϭ(8,10),(10,12),...,(58,60). critical point one of the 1s electrons is bounded by the nucleus while the other 1s electron moves away to in®nity. For ZϾZ , the system is stable and the two electrons are For two-electron atoms the entropy S is minimal when the c Ͻ Ͻ two electrons are bound in the ground state ␭ ␭c , and then bounded. However, for Z Zc , the system is unstable. This analysis is useful to describe stability for a given isoelec- develops a steplike discontinuity at ␭c and jumps to maximal Ͼ tronic atomic series.34 value SϷ10.7 for ␭ ␭c . ϭ Next we calculate the critical parameters for C ͑carbon͒, At the critical point ␭c 1.096, the extrapolated values ϭ ϭ ϭ N ͑nitrogen͒, F ͑¯uorine͒, and Ne ͑neon͒. C, N, and F have of the three critical exponents are ␣E 1, ␯ 0, ␣S 0 as ϭ an open 2p orbital while Ne has a closed 2p orbital. Our shown in Fig. 3. The entropy exponent ␣S 0 con®rms our previous analysis.30 This physical picture is consistent with calculations show that all these systems at their ground states the de®nition of Shannon-information entropy which mea- undergo a ®rst-order phase transitions. The critical points ␭c sures the delocalization or the lack of structure in the respec- are 0.2013, 0.1704, 0.1270, and 0.1145, with the correspond- tive distribution. Thus S is maximal for uniform distribution, ing critical charges 4.97, 5.87, 7.87, and 8.73 for C, N, F, and ϭ Ϫ that is, for an unbound system, and is minimal when the Ne respectively. The excess charges ⌬Z Z 1/␭c are 1.03, uncertainty about the structure of the distribution is minimal. 1.13, 1.13, and 1.27 for 6-, 7-, 9-, and 10-electron atoms, respectively. This con®rms the existence of stable anionic species BϪ1, CϪ1, OϪ1, and FϪ1.32 However, there are no stable dianions because it requires an excess charge of 2.32 For atoms with ®lled 2p orbitals, the three critical exponents ϭ ϭ ϭ ϭ are ␣E 1, ␯ 0, ␣S 0 as shown in Fig. 4 with N 8 ± 58. All the critical parameters are summarized in Table I.

V. DISCUSSION AND CONCLUSIONS

We presented a complete set of critical exponents ␣E , ␯, and ␣S using ®nite size scaling analysis for the electronic energy, correlation length, and Shannon entropy,

Ϫ ␣E E ϳ ͑␭ ␭c͒ , ͑37͒ → ϩ ␭ ␭c Ϫ Ϫ␯ ␰ ϳ ͉␭ ␭c͉ , ͑38͒ → ϩ ␭ ␭c

Ϫ ␣S S ϳ ͑␭ ␭c͒ . ͑39͒ → ϩ ␭ ␭c

(N,NЈ) These exponents describe the singularity of the different FIG. 3. The crossing points de®ning the pseudocritical exponents ␣E , (N,NЈ) (N,NЈ) quantities at the phase transition of the electron from a bound ␯ , and ␣S for heliumlike atoms as a function of parameter ␭ ϭ1/Z for (N,NЈ)ϭ(8,10),(10,12),...,(58,60). to a continuum state. These results were obtained by reduc-

Downloaded 21 Sep 2004 to 132.205.24.88. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5616 J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Q. Shi and S. Kais ing a multiparticle Coulomb potential to a one-electron one- center atomic potential.8 We examined the phase transition near the threshold for s-state (lϭ0) and p-state (lϭ1) Yukawa atoms, and atoms He (Zϭ2), C ͑6͒, N ͑7͒, F ͑9͒, and Ne ͑10͒. The method is general and can be used to treat the rest of the elements in the periodic table provided a good basis set and electron density. Our data in Table I and Figs. 1±4 indicate the correspon- dence between the critical exponents and the interaction type. For all atoms with nuclear charge Zϭ2,6,7,9,10 their correlation length exponents are all 0, corresponding to a long-range Coulomb interaction as a dominant force. In con- trast, the atoms with an electron in the Yukawa potential have positive exponents, corresponding to a short-range force which is more localized in the atomic scale than the Cou- lomb forces. The atomic Shannon entropy exponents follow the trend for the correlation length exponents. For Coulomb atoms FIG. 5. Upper panel: Free energy G, entropy S, and speci®c heat C p as a function of temperature for a typical ®rst-order liquid-gas phase transition they have a common characteristic: at the critical point ␭c ͑Ref. 35͒. Lower panel: Ground state energy E, Shannon entropy S, and the entropy exponent is 0; as ␭ deviates from ␭c the entropy ``electronic speci®c heat'' C⑀ as a function of the inverse of the nuclear exponent varies dramatically. This shows a steplike, or dis- charge Z, ␭ϭ 1/Z, for two-electron atoms in zero external ®eld ⑀. continuous, change in the Shannon entropy of the atoms. Therefore, we conclude that the critical point is a point where atomic Shannon entropy reaches a maximum. At this ACKNOWLEDGMENT point we have a delocalization of the electronic wave function. Such a property of the atomic Shannon entropy was We would like to acknowledge the ®nancial support of 30 recently described for the two-electron heliumlike atoms. the National Science Foundation ͑NSF Grant No. 501 1393- Examining the critical exponents for the energy indicates 0650͒. that for the Yukawa atoms with an s-state electron it is of a continuous type while for p-state Yukawa atoms and atomic elements with nuclear charge Zϭ2, 6, 7, 9 and 10 it is of a ®rst-order type. Combining the data of the ground state en- APPENDIX: ENERGY MATRIX ELEMENTS ergy and Shannon entropy for atomic systems we can sys- In order to calculate the energy matrix elements one tematically map the quantum-mechanical quantities to their needs to evaluate an integral as a product of four functions: analogous classical quantities, (␣) (␣) two generalized Laguerre polynomials Ln (x) and LnЈ (x), Ϫx nuclear charge Zϭ1/␭↔temperature T one exponential function e , and one power law function x␣. The integrals with four power indices (␣,␣ϩ1,␣Ϫ1,␣ electronic energy E↔free energy G Ϫ2) were presented in Atomic, Molecular and Optical Handbook.36 For convenience we list them with our correc- ↔ number of basis functions thermodynamic limit tions to the case (␣Ϫ2). All were evaluated with numerical methods. atomic Shannon entropy S↔thermodynamical entropy S. ͑40͒ ϱ ϩ ϩ Ϫ ␣ ⌫͑n ␣ 1͒ Figure 5 presents an analogy between the behavior of the x␣e xL(␣)͑x͒L( )͑x͒dxϭ ␦ , ͵ n nЈ n! n,nЈ different thermodynamic quantities typical of liquid-gas ®rst- 0 order phase transition35 and the critical behavior of the corresponding quantities for two-electron atoms. At ®rst-order ϱ x␣ϩ1eϪxL(␣)͑x͒L(␣)͑x͒dx phase transition, the free energy G bends sharply as a func- ͵ n nЈ 0 tion of the temperature T at the transition temperature Td ⌫͑nϩ␣ϩ1͒ which leads to a discontinuity of the entropy S and in®nite ϭ Ϫ ϩ ϩ ϩ ͓ n␦n,nЈϩ1 ͑2n ␣ 1͒ heat capacity C p at constant pressure P. In analogy, the n! ground state energy for two-electron atoms E bends sharply ϫ Ϫ ϩ ϩ ϭ ϭ ␦n,nЈ ͑n ␣ 1͒␦n,nЈϪ1͔, at the transition point ␭d* ␭c 1/Zc ; this leads to discontinuity of the Shannon information entropy and in®nite ``heat ␭) with a zero external ®eld. This ϱץ/Sץ) capacity,'' C ϭ␭ ⑀ x␣Ϫ1eϪxL(␣)͑x͒L(␣)͑x͒dx analogy is very exciting and might give us a tool of classi- ͵ n nЈ 0 fying the electronic structure of atoms. Research is underway to classify all elements according to their types of ``phase ⌫͑nϽϩ␣ϩ1͒ ϭ , ␣Ͼ0, transition'' and their consequences in chemical reactions. nϽ!␣

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