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Finite Jellium Models. I. Restricted Hartree–Fock Calculations

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Finite Jellium Models. I. Restricted Hartree–Fock Calculations THE JOURNAL OF CHEMICAL PHYSICS 122, 154108 ͑2005͒ Finite jellium models. I. Restricted Hartree–Fock calculations Sankha Ghosh School of Chemistry, University of Nottingham, Nottingham NG7 2RD, United Kingdom ͒ Peter M. W. Gilla Research School of Chemistry, Australian National University, Canberra, Australian Capital Territory 0200, Australia ͑Received 26 April 2004; accepted 26 January 2005; published online 20 April 2005͒ Restricted Hartree–Fock calculations have been performed on the Fermi configurations of n electrons confined within a cube. The self-consistent-field orbitals have been expanded in a basis of N particle-in-a-box wave functions. The difficult one- and two-electron integrals have been reduced to a small set of canonical integrals that are calculated accurately using quadrature. The total energy and exchange energy per particle converge smoothly toward their limiting values as n increases; the highest occupied molecular orbital–lowest unoccupied molecular orbital gap and Dirac coefficient converge erratically. However, the convergence in all cases is slow. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1873552͔ I. INTRODUCTION finite space. Ellert et al., for example, have rationalized17 the optical absorption spectra of small, open-shell sodium clus- Jellium is the most popular electronic structure model in ters by considering n electrons in a spheroidal cavity, Koski- solid-state physics and also constitutes the starting point for nen, Lipas, and Manninen have studied n electrons in a re- many of the density functional theories that are now widely laxed background charge density,18 Holas et al. have 19 used in computational chemistry.1 Its success stems largely examined n electrons in a magnetic field, and Alavi’s group from its surpassing simplicity: an infinite uniform gas of has recently published studies of n=2 highly correlated elec- 20 21 electrons characterized by a single number, its electron den- trons in a cube and a ball. sity ␳. The original idea, particulate electrons moving in a Inspired by these studies, we have embarked upon a sys- background of uniform positive charge, was introduced2 by tematic study of the properties of n electrons confined within a cube, with a background of uniform positive charge, a neu- Kelvin soon after the discovery of the electron and was tral system that we will call “finite jellium.” In this paper, we quickly adopted by Thomson in his “plum pudding” model report the results of restricted Hartree–Fock ͑RHF͒ calcula- of atomic structure.3 Although it fell out of favor following tions where the orbitals are expanded in a basis of N particle- Rutherford’s discovery of the nucleus, it was subsequently in-a-box functions. We have examined the behavior of sev- revived in the “statistical” quantum mechanical models of ͓ 4 5 6 7 eral properties the total energy, the Fock exchange energy, Thomas, Fermi, Bloch, and Dirac who determined the the Dirac exchange energy, the density variance, and the exact kinetic energy and exchange energy of jellium, as func- HOMO-LUMO gap ͑HOMO and LUMO represent highest tions of ␳. The ␳ dependence of the correlation energy of occupied and lowest unoccupied molecular orbitals, respec- jellium proved more difficult to determine and produced an tively͔͒ as functions of n and N, and we have examined the impressive body of work by Wigner,8 Gell-Mann and convergence of these properties as n and N increase. Brueckner,9 Carr,10,11 and Nozières and Pines12 before the The following section describes the physical model and definitive quantum Monte Carlo calculations13,14 of Ceperley basis set that we have chosen and outlines the symmetry and Alder and their parametrization by Vosko, Wilk, and aspects of our calculations. After this, we present the RHF Nusair.15 As a result, the properties of jellium are now well theory of our system and propose a numerical approach to established, in the low-density limit,8,10,12 the high-density the difficult Coulomb integrals that arise. We conclude with a limit,9,11 and for intermediate densities in both ferromagnetic discussion of some of the results that we have obtained. and paramagnetic states.13–15 Atomic units are used throughout. Jellium has continued to underpin a wide array of simple models for complex physical and chemical systems and is widely used, for example, in theoretical electrochemistry.16 II. MODEL, BASIS SET, AND SYMMETRY However, there has also been a resurgence of interest in the quantum mechanical treatment of Kelvin–Thomson models Our system consists of n electrons in a cube of length ␲ wherein a finite number n of electrons move freely within a within which there is uniformly distributed positive charge sufficient to make the system neutral. Thus, the Hamiltonian ͒ a Electronic mail: [email protected] is 0021-9606/2005/122͑15͒/154108/12/$22.50122, 154108-1 © 2005 American Institute of Physics Downloaded 20 Apr 2005 to 150.203.2.222. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 154108-2 S. Ghosh and P. M. W. Gill J. Chem. Phys. 122, 154108 ͑2005͒ ͕ ͖ 2 ⌫ TABLE I. Irreducible representations of Oh spanned by a set a,b,c of TABLE II. Energies m and symmetries of the particle-in-a-box wave basis functions. functions. a,b ␾ 2 ⌫ Parities Representations i i m ͕ ͖ EEE A1g 1 1,1,1 3 A1g ͕ ͖ EEEЈ A1g +Eg 2–4 2,1,1 6 T1u ͕ ͖ EEЈ EЉ A1g +A2g +2Eg 5–7 2,2,1 9 T2g ͕ ͖ EOO T2g 8–10 3,1,1 11 A1g +Eg ͕ ͖ EOOЈ T1g +T2g 11 2,2,2 12 A2u ͕ ͖ 12–17 3,2,1 14 T1u +T2u OOO A2u ͕ ͖ 18–20 3,2,2 17 T2g OOOЈ A +E 2u u 21–23 ͕4,1,1͖ 18 T Ј Љ 1u OO O A1u +A2u +2Eu ͕ ͖ 24–26 3,3,1 19 A1g +Eg OEE T1u 27–32 ͕4,2,1͖ 21 T +T Ј 1g 2g OEE T1u +T2u ͕ ͖ 33–35 3,3,2 22 T1u a ͕ ͖ E and O represent even and odd Cartesian parity, respectively. 36–38 4,2,2 24 A2u +Eu b ͕ ͖ Single and double primes imply distinct quantum numbers. 39–44 4,3,1 26 T1u +T2u ͕ ͖ 45 3,3,3 27 A1g 46–48 ͕5,1,1͖ 27 A +E n 1g g 1 1 ͑n/␲3͒2 49–54 ͕4,3,2͖ 29 T +T Ј 1g 2g ͵͵ 2ٌ ˆ H =− ͚ i + dr dr ͕ ͖ 2 2 ͉r − rЈ͉ 55–60 5,2,1 30 T1u +T2u i=1 V ͕ ͖ 61–63 4,4,1 33 T2g n n ͕ ͖ ͑n/␲3͒ 1 64–66 5,2,2 33 T2g ͵ ͑ ͒ ͕ ͖ − ͚ dr + ͚ , 2.1 67–69 4,3,3 34 T1u ͉r − r͉ Ͻ ͉r − r ͉ ͕ ͖ i=1 V i i j i j 70–75 5,3,1 35 A1g +A2g +2Eg ͕ ͖ 76–78 4,4,2 36 A2u +Eu ͕ ͖ where V indicates the region within the cube. The first term 79–84 5,3,2 38 T1u +T2u ͕ ͖ arises from the kinetic energy of the electrons, the second 85–87 6,1,1 38 T1u ͕ ͖ 88–93 6,2,1 41 T1g +T2g from the self-interaction of the positive charge, the third ͕ ͖ 94–96 4,4,3 41 T2g from the attraction between the positive charge and the elec- ͕ ͖ 97–102 5,4,1 42 T1u +T2u trons, and the fourth from interelectronic repulsions. The ͕ ͖ 103–105 5,3,3 43 A1g +Eg self-interaction term does not depend on the electron coordi- ͕ ͖ 106–108 6,2,2 44 A2u +Eu ͕ ͖ nates and, like the nuclear repulsion energy in a molecular 109–114 5,4,2 45 T1g +T2g ͕ ͖ calculation, can be treated as a constant. 115–120 6,3,1 46 T1u +T2u ͕ ͖ The Schrödinger equation associated with Eq. ͑2.1͒ can- 121 4,4,4 48 A2u ͕ ͖ not be solved exactly but it can be treated approximately 122–127 6,3,2 49 T1g +T2g ͕ ͖ using RHF theory within a finite basis set. The natural basis 128–133 5,4,3 50 T1u +T2u ͕ ͖ functions are the normalized particle-in-a-box wave func- 134–136 5,5,1 51 A1g +Eg ͕ ͖ tions 137–139 7,1,1 51 A1g +Eg ͕ ͖ 140–145 6,4,1 53 T1g +T2g ͕ ͖ 146–151 7,2,1 54 T1u +T2u ␾ ͑ ͒ ͑ ␲͒3/2 ͑ ͒ ͑ ͒ ͑ ͒ ͕ ͖ m r = 2/ sin mxx sin myy sin mzz 152–154 5,5,2 54 T1u ͕ ͖ ͑ ഛ ഛ ␲͒͑͒ 155–157 6,3,3 54 T1u 0 x,y,z 2.2 ͕ ͖ 158–163 6,4,2 56 A1u +A2u +2Eu ͕ ͖ 164–166 7,2,2 57 T2g ͑ ͕ ͖ where the quantum numbers mx, my, and mz are positive 167–169 5,4,4 57 T2g ͕ ͖ integers͒ and it is convenient to order these functions accord- 170–172 5,5,3 59 A1g +Eg ͕ ͖ ing to their energies 173–178 7,3,1 59 A1g +A2g +2Eg ͑ 2 2 2͒ 2 ͑ ͒ 2Em = m + m + m = m . 2.3 x y z The theory below is easily extended to cubes of any size.
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