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Heteroisotopic Molecular Behavior. the Valence-Bond Theory of the Positronium Hydride

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Heteroisotopic Molecular Behavior. the Valence-Bond Theory of the Positronium Hydride Brazilian Journal of Physics, vol. 34, no. 3B, September, 2004 1197 Heteroisotopic molecular behavior. The Valence-Bond Theory of the Positronium Hydride Fl´avia Rolim, Tathiana Moreira, and Jos´e R. Mohallem Laborat´orio de Atomos´ e Mol´eculas Especiais, Departamento F´ısica, ICEx, Universidade Federal de Minas Gerais, PO Box 702, 30123-970, Belo Horizonte, MG, Brazil Received on 10 December, 2003 We develop an adiabatic valence-bond theory of the positronium hydride, HPs, as a heteroisotopic diatomic molecule. Typical heteronuclear ionic behaviour comes out at bonding distances, yielded just by finite nuclear mass effects, but some interesting new features appears for short distances as well. 1 Introduction hydride, HPs (H=H+e− + Ps=e+e−), which can be seen as an extreme homonuclear but heteroisotopic isotopomer of In theory of diatomic molecules, the concepts of homo and H2. The large difference between the proton and positron hetero nuclearity have been developed in a natural way, to masses yields a considerable asymmetry, that stimulates us recognize whether a molecule is made up of equal or dif- to investigate how it affects the electronic distribution, in ferent atoms, respectively. Since atoms differ from each comparison with the common heteronuclear case (which other by their atomic charge Z, which is contained in the we call here the Z − M analogy). As a matter of fact, potential energy part of the Hamiltonian operator, the Born- Saito [8] has already pointed out that the electronic den- Oppenheimer (BO) theory of molecules [1], based on a sity of HPs, obtained from four-body correlated calcula- clamped-nuclei electronic Hamiltonian, is sufficient to ac- tions, shows a visual approximate molecular behavior, but count for these features and to explore all the consequences this kind of approach can give no further structural details, of point group symmetries displayed by the electronic wave- however. We are, on the other hand, interested in explor- functions. Any further features yielded by coupling of ing possible Z − M analogy effects with a theory in which electronic and nuclear motion are treated as non-adiabatic the ”nuclear ” (proton and positron) and electronic motions effects, whose calculations involve many BO states [1]. are adiabaticaly separated, that is, with the same theoretical For polyatomic molecules, the concept and terminology of treatment used with standard isotopomers, in order to better ”equivalent atoms ” are introduced, meaning units of the understand their isotopic properties. same chemical element that transform among each other by It is well known that the valence-bond (VB) theory is symmetry operations (for example, the two hydrogen atoms appropriate to analyse in detail the bonding of a typical di- in the water molecule). atomic molecule. Particularly for a heteronuclear one, a On the other hand, recent developments have shown that mixed covalent-ionic behavior of the wavefunction points to a variational adiabatic (non-BO) approach for the electronic a polarization of the electronic distribution toward the more problem is able to account for another property of atoms, electronegative atom, measured by the relative value of the their mass M (the isotopic effect), in a somewhat analogous linear coefficient of the ionic structure. Furthermore, the ex- way of Z [2]-[6]. This introduces the idea of homo and ponents of the atomic orbitals measure the effective charge heteroisotopic molecular behavior. For example, being D of each nucleus felt by the electrons and become indicative the symbol for deuterium, H2O and D2O are homoisotopic of the same above effect, as well. Classical chemical con- but HDO is heteroisotopic, the last one having its symmetry cepts as, for example, electronegativity, have immediate in- terpretation in terms of the VB output. broken from C2v to Cs [6]. The consequent isotope shifts, dipole moments and new spectroscopical transitions, are real Here, we check whether a VB calculation of HPs is able and measurable [7]. to study the consequences of its mass asymmetry on the electron distribution and deeply explore the Z − M anal- This new way of symmetry classification of molecules ogy. This is done in the following section. may look strange, however, as it is generated by kineti- cal terms of the total Hamiltonian, instead of by a rigid framework of nuclear charges. Furthermore, for typical 2 The valence-bond theory of HPs molecules, the effects are too small to be visualized in elec- tronic density maps, as usually done in the common homo and heteronuclear cases. The theoretical signature of sym- 2.1 Methodology metry breaking is just the non-commutability of some sym- Atomic units (au) and conventional notation are used metry operators with the adiabatic Fock operator [6]. throughout. Our approach to include the finite nuclear mass Fortunately, there is the singular case of the positronium effects in molecular electronic structure calculations [2]-[4] 1198 Fl´avia Rolim, Tathiana Moreira, and Jos´e R. Mohallem has been to look for electronic wavefunctions Φ k that are assures that the corrections will be properly placed in the eigenfunctions of the total Hamiltonian, instead of the BO hamiltonian matrix. In the present case, the ground state one, that is, (gs) correction with wavefunction (2), becomes simply HΦk = kΦk. (1) Q = c2(Q + Q )+c2(2Q )+c2(2Q ) Different from the common adiabatic approximation [9], 1 A B 2 A 3 B that corrects just the electronic energy (or the potential en- +c1c2Q a|b + c1c3Q b|a . (4) ergy curves, PEC), this approach allows the electronic wave- A B functions to account for the nuclear motion as well. A Note that if the one-electron atoms are set apart, only the product wavefunction is assumed for any state k, that is, first term survives and the correction becomes exact. For fi- Ψ =Φχ χ k k k, where is a nuclear wavefunction. The so- nite R, the correction performs quite well, as shown in refs. lution of equation (1) is done in a variational sense, which [3],[4]. The same result of equation (4) could be obtained explains the terminology ”variational adiabatic ”. Further with the model Hamiltonian for the FNMC [10] as well. approximations are explained bellow. − − The atomic orbitals and the linear coefficients are opti- The system, ABe e , is treated as being heteroiso- mized in order to minimize the adiabatic electronic energy, topic, which means here that the positive nuclei A (H +) and + B (e ), separated by the distance R, have different masses, (R)= (R)+Q (R), k BOk k (5) MA and MB (MB =1ua), but the same charge, Z =+1 au. which yields the PECs for each state k. The results of equa- For the definition of our electronic basis, we consider tions (4) and (5) express our approximate solution of equa- a covalent (Heitler-London, HL), and two ionic VB struc- tion (1). tures, so that a spatial symmetric state can be written as the superposition 2.2 The VB basis Φ = c [a(1)b(2) + a(2)b(1)] k k1 Before concerning the complete expansion of the wavefunc- +ck2a(1)a(2) + ck3b(2)b(2). (2) tion in the VB basis, it is interesting to verify how this non- a b 1s orthogonal basis can be devised. Our treatment resembles Here, and are normalized orbitals centered on nu- an old one by Zener [11] for ionic molecules, but with some clei A and B, with exponents ζA and ζB, respectively, being R additional complications due to the need of variational de- all the linear and non-linear parameters dependent on .To termination of each state (see bellow). We ignore for while save notation, we refer, in what follows, to the HL, A-ionic B B the -ionic structure, since, in view of the smaller energy of and -ionic structures, with obvious meaning, and call this the corresponding dissociation products, H ++Ps−, it must set the VB basis. not correspond to a low-lying state of the system. On the We first attempt to solve equation (1) to obtain the PECs contrary, the HL structure and the A-ionic structure are ex- 1 pected to correlate to lower states of HPs. U (R)= (R)+ . k k R With just the HL structure as the electronic wavefunc- tion, the FNMC (4) becomes Q = Q + Q , and a vari- On this level, the problem has been treated, in our lab- A B ational calculation of the PEC yields curve 1 of Fig. 1A. It oratory, within a modified-electron-mass approach [2],[3], advances the dissociation of the system in H + Ps, giving which works well for one and two-electron homonuclear di- the exact separated-atom (SA) energy, E(∞)=−0.74973 atomic molecules. For more complicated systems we re- au. With the A-ionic structure, the correction is Q =2Q sorted to an empirical correction [4] for which a model A and curve 2 is obtained, which advances the dissociation in Hamiltonian has been developed later [10]. Summarizing − + H +Ps , with the independent particle approximation en- the prescription of ref. [4], first assume that for an electron ergy of E(∞)=−0.53858 au. Curve 1 must thus cor- occupying the atomic orbital a centered on a nucleus A with relate better to the electronic ground state. As a matter of mass M , the finite nuclear mass correction (FNMC) to its A fact, both curves, corresponding to non-orthogonal states, energy is (unormalized), mimic the ground state, for intermediate values of R. In the a(1) −∇2 a(1) united-atom (UA) limit, the HL structure tend to imitate the Q = 1 . A 2M (3) ionic one, which is far more appropriate to this limit, so that A the curves close approach and cross (not an actual crossing, This correction is exact for an isolated non-relativistic however).
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