Stability of the Negative Hydrogen Ion: Variational Approach with Electron Correlation Author: Ignacio L´opez de Arbina de Frutos Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Advisor: Mario Centelles Aixal`a Abstract: We present a study on the negative hydrogen ion, particularly following the works of Chandrasekhar [1, 2]. The conditions for the stability of this system are discussed as well as the methods used to achieve it. We also present the full analytical expression of the variational energy for a Chandrasekhar correlated wave function. Further, we stress the special structure concerning this ion in comparison with the other two-electron systems. I. INTRODUCTION gen, it is easier to accelerate to high energies [10]. The present work is organized as follows: in Sec. II The negative hydrogen ion (H−) is an atomic system we present the key concepts regarding stability and the composed of two electrons bound to a proton. It belongs use of the Rayleigh-Ritz method to guarantee it; in Sec. to the so-called helium isoelectronic sequence or two- III different variational approaches are discussed using electron systems, which we may denote as (Z; −1; −1), different trial wave functions; and in Sec. IV the shape where Z is the nuclear charge. At first sight it seems of H− is described. that the negative hydrogen could be described by the two-electron atoms formalism typically studied in intro- ductory courses in atomic physics. However, perturba- tion and mean field approximations, such as the well- known Hartree-Fock method, fail completely when ap- II. STABILITY AND VARIATIONAL METHOD plied to H− because the ground state (g.s.) energies ob- tained with these methods do not predict its stability. To claim stability it is necessary to specify a method As it will be argued along the work this is due to the that ensures that a two-electron system (Z; −1; −1) will relevance of the interelectronic correlation in front of the not dissociate into a one-electron atom or ion (Z; −1). In Coulomb attraction with the nucleus. this case, hydrogen-like systems bring out the threshold − The theoretical importance of the study of H , which of stability with a non-relativistic energy of the ground only has a bound state and no excited states [3], is that 2 state Eth = −Z =2, assuming a nucleus of infinite mass. it shapes the prototype of a three-body atomic system This expression is given in atomic units (a.u.) where ~ = in the same way as the hydrogen atom is the prototype e = me = 1. Hence, the energies are proportional to the for two-body atomic systems. Therefore, one expects to 4 2 Hartree Eh = e me=~ ' 27:2114 eV and the distances benefit from the results in order to describe other non- 2 2 −9 to the Bohr radius a0 = ~ =(e me) ' 5:2918 × 10 cm. trivial three-body systems with strong dependence in the The method used to compute the energies of the g.s. of interelectronic distance [4]. Historically, Bethe [5] and these two-electron systems is described straightaway. Hylleraas [6] were the first authors to discover the sta- bility of H− using the frame of the variational method If H is the Hamiltonian describing the system and with electron correlation against perturbation and mean Ψn the set of orthonormal eigenfunctions, then the ex- field approximations. Since them, an extended study of act energies are the eigenvalues of the Hamiltonian, i.e. this system has been done, like in the seminal works of En = hΨnjHjΨni. Usually, the set of Ψn are unknown Chandrasekhar [1, 2] and the accurate calculations of and the problem remains to seek an appropriate func- Frankowski and Pekeris [7], or Frolov [8] , just to mention tion which differs from Ψn by an infinitesimal quantity some. δ , such that the energy E[ ] gives a value as close as possible to E . This condition can be obtained by im- The H− ion was a necessity in astrophysics in order to n posing E[ ] to be stationary to first-order variations, i.e. explain the continuous absorption spectra of the Sun and δE[ ] = 0. It can be shown [11] that if = Ψ + δ the other stellar atmospheres [1, 2, 9]. The concept of a hy- n difference between E[ ], under the stationary condition, drogen atom with a weakly bound second electron helped and the exact energy E is second order in δ . More- to describe the opacity of some stellar atmospheres with n over, the functional E[ ] gives an upper bound to the photon-absorption between 0:75 − 4 eV [9]. Besides, it is exact ground state energy E , i.e. E[ ] ≥ E [11]. remarkable that the H− ion plays a central role in linear 0 0 accelerators and in ion beams for nuclear fusion. Because A typical procedure to find out the best E[ ] is to as- of its extra electron, in comparison with neutral hydro- sume a function depending on some variational parame- ters, i.e. [α1; : : : ; αN ] usually called trial wave function, and to minimize E[ ] with respect to these parameters in order to get the best approximation to E0. This ap- ∗Electronic address: [email protected] proach is known as Rayleigh-Ritz variational method and STABILITY OF THE NEGATIVE HYDROGEN Ignacio L´opez de Arbina 0 Variational Energies Ψ[Zeff ] [a; b] [a; b; c] Z −E[Zeff ] −E[a; b] −E[a; b; c] −Enr ZZeff a b a b c H− 1 0:472 0:51330 0:52592 0:52775 1 0.68 1.03922 0.28322 1.07487 0.47745 0.31255 He 2 2:848 2:87566 2:90142 2:90372 2 1.68 2.18317 1.18853 2.20842 1.43624 0.29271 Li+ 3 7:223 7:24875 7:27717 7:27991 3 2.68 3.29498 2.07091 3.29943 2.36182 0.27697 Be2+ 4 13:598 13:62396 13:65255 13:65557 4 3.68 4.38973 2.98472 4.37442 3.29336 0.26880 B3+ 5 21:973 21:99754 22:02776 22:03097 5 4.68 5.47341 3.90126 5.43992 4.23120 0.26385 C4+ 6 32:348 32:37227 32:40290 32:40625 6 5.68 6.54910 4.82568 6.49888 5.17423 0.26054 N5+ 7 44:723 44:74709 44:77799 44:78145 7 6.68 7.61870 5.75614 7.55296 6.12144 0.25817 O6+ 8 59:098 59:12195 59:15306 59:15660 8 7.68 8.68348 6.69140 8.60320 7.07210 0.25639 TABLE I: Ground state variational energies (in a.u.) of the TABLE II: Variational parameters that minimize the energies first eight ions of the helium isoelectronic sequence. The en- for the simple variational method and the two- and three- ergies correspond to the simple variational method (5), the parameters of Chandrasekhar wave functions and the case two- and three-parameters Chandrasekhar wave functions us- with a frozen electron to a = 1. ing (8) and the best non-relativistic energy [7]. is spin independent, the total wave function factorizes as can be summarized as h jHj i Ψ(r1; r2; σ1; σ2) = Ψ(r1; r2)χ00(σ1; σ2): (3) E[αk] = min ≥ E0; (1) αk h j i where E[αk] is the energy obtained with the trial wave A. Simple variational method function [αk] after minimizing it with respect the set of chosen variational parameters fαkg. Henceforth, the −1 Rayleigh-Ritz variational method can be used to study The simplest attempt is to consider r12 as a per- stability. If one is able to propose a trial wave function turbation and solve (2) for two independent electrons. depending on some variational parameters that describes Thus, one can use 1s hydrogenic orbitals to describe the system and fulfills E[α ] < E = −Z2=2, then sta- each electron, i.e. '(r) = u(r)Y00(θ;p φ), where u(r) = k th 3=2 bility is ensured. In the problem of the stability of the 2Z exp(−Zr) and Y00(θ; φ) = 1= 4π. Additionally, negative hydrogen it is enough to find out a certain if one assumes that both electrons feel the same effec- tive nuclear charge, Z , the resulting spatial normalized such that E[αk] is lower than the energy of the neutral eff hydrogen atom −0:5 a:u: wave function reads 3 Ψ(r1; r2) = (Zeff /π) exp(−Zeff (r1 + r2)): (4) III. GROUND STATE OF HELIUM ISOELECTRONIC SEQUENCE 2 Then, one finds that hΨjHjΨi = Zeff − 2ZZeff + 5Zeff =8 [11], and the Rayleigh-Ritz variational method lead to a The Hamiltonian of two-electron systems with a point- g.s. energy like, infinite mass nucleus in the electrostatic approxima- 2 tion can be written as E[Zeff ] = minhΨjHjΨi = −Zeff ; (5) Zeff r2 r2 r1 r2 Z Z 1 H = − − − − + ; (2) where Z = Z − 5=16. The energies concerning this 2 2 r1 r2 r12 eff model and the effective nuclear charges are given in where r1 and r2 are the distances of each electron to the columns E[Zeff ] of Table I and [Zeff ] of Table II. For nucleus and r12 = jr1 −r2j is the separation between elec- Z ≥ 2 the energies are rather good in comparison with trons. The first two terms in eq. (2) represent the kinetic the best non-relativistic calculations [7], column Enr of energy of each electron, the following terms correspond Table I, taking into account the simplicity of the method. to the Coulomb attraction between the electrons and the This is because for increasing values of Z Coulomb at- nucleus of charge Z, and to the Coulomb repulsion be- traction with the nucleus becomes more important and −1 tween the electrons.