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 2004! 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 S;(l 2 2 aS 2 aE 2 n lc) , the electronic energy E;(l lc) , and the correlation length j;ul lcu for atoms with the variable l5 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 5 5 5 5 5 (aE ,n,aS) for He (Z 2), C (Z 6), N (Z 7), F (Z 9), and Ne (Z 10), respectively, are ~1, 5 0, 0!. At the critical points lc 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: 10.1063/1.1785773# I. INTRODUCTION is, for an unbound system, and is minimal when the uncer- tainty 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- S52 P~x!ln P~x!dx; P~x!dx51. ~1! namic limit, where singularities or phase transitions occur. E E 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 Sr using the electronic density r and its momentum 3 zero of temperature in atoms and molecules. These transi- space analog Sp . Such definitions result in that the sum St 5 1 tions are not driven by temperature as in classical phase tran- Sr Sp 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 >3N 11ln p!22N ln N. ~2! 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 0021-9606/2004/121(12)/5611/7/$22.00 5611 © 2004 American Institute of Physics Downloaded 21 Sep 2004 to 132.205.24.88. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5612 J. Chem. Phys., Vol. 121, No. 12, 22 September 2004 Q. Shi and S. Kais (N) (`) (`) 5 2 2n spanned by a single-exponent Slater-type basis set. After cal- ^O& ~l!;^O&FO~N/j ~l!!, j ~l! ul lcu , ~7! culating both energy and electron density ~and therefore en- with a different scaling function FO for each different opera- tropy! and then ®tting those accurate data we found an ap- (`) proximately linear relation between the entropy and the ®rst tor but with a unique scaling exponent n. j (l) is the cor- ! derivative of the energy. In particular, we noted that close to relation length at N `. O (N) the critical point such linear relation becomes true. These Since ^ & (l) is analytical in l, the scaling function at ! results show that the exponent of the entropy for helium is l lc must behave like 2 mO /n the exponent of the energy minus 1. FO~x!;x , ~8! In this paper, we develop the ®nite size scaling method where x5N(l2l )n!0. It follows that O (N)(l) at l 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) 2 mO /n and establish relations among the critical exponents for the ^O& ~lc!;N ; N!`. ~9! atomic Shannon entropy, the correlation length, and the elec- Using Eq. ~9! one may extrapolate the exponent m and tronic energy. In Sec. III we describe a one-electron model O n ~Ref. 31! from the crossings of the pseudocritical expo- Hamiltonian used to approximate the multielectron Hamil- (N,N8) (N,N8) tonian. In Sec. IV we present results and also the integrals nents mO and n for two consecutive expansions N, 85 1 1 32 needed for calculating the different energy matrix elements. N N 1 ~or N 2 if parity effects involve !, 8 Discussion and conclusions follow in Sec. V. Atomic units (N,N8) ln~^O&(N)/^O&(N )! mO lc lc are used throughout the paper unless otherwise speci®ed. 5 . ~10! n(N,N8) ln~N8/N! II. FINITE SIZE SCALING FOR THE CRITICALITY At this point we may de®ne the energy E exponent a as OF SHANNON-INFORMATION ENTROPY E E ; ~l2l !aE ~11! The Hamiltonian for many interesting problems in quan- l 1 c l!lc ; tum mechanics, in particular for atomic and molecular and obtained from Eq. ~10! by replacing O[H systems,8 can be transformed to the following general form: (N,N8) H5H 1V aE 0 l , ~3! 5 DH~lc ;N,N8!, ~12! n(N,N8) where H0 is l independent while Vl is a l dependent term and l represents a free parameter. For electronic structure of where the notation DH is used for the right side of Eq. ~10!. atoms, l is the inverse of the nuclear charge Z. At the critical For unique exponents we start from Eq. ~11! and take the 32 point lc , a bound state described by the SchroÈdinger equa- derivative tion HC5EC becomes absorbed or degenerate with a con- ]E 2 l 2 aE 1 tinuum. ; ~l lc! . ~13! ]l ! 1 For a l-independent complete and orthonormal basis set l lc 2 $Fn%, spanning an in®nite-dimensional Hilbert space L , the Using Hellmann-Feynman theorem bound state can be expanded as ]El ]H ]Vl ` 5 5 ~14! ]l K ]l L K ]l L 5 l l Cl ( Cn~l!Fn , ~4! n O5 V V8 and taking ] l /]l [ l in Eq. ~10! we have where n is an adequate set of quantum indices and $Cn(l)% 8 a(N,N )21 the expansion coef®cients which are generally l dependent. E 5 DV 8~lc ;N,N8!. ~15! In calculating the energy matrix, the expansion has to be n(N,N8) l truncated at order N, i.e., expanding up to a ®nite length, M(N) which equals a ®nite dimension in the Hilbert space.