PHYSICAL REVIEW A VOLUME 57, NUMBER 3 MARCH 1998

:states 0,1,2؍Stability and annihilation of L A quantum Monte Carlo study

Dario Bressanini* Istituto di Scienze Matematiche, Fisiche e Chimiche, Universita` degli Studi di Milano, sede di Como, via Lucini 3, 22100 Como, Italy

Massimo Mella† and Gabriele Morosi‡ Dipartimento di Chimica Fisica ed Elettrochimica, Universita` degli Studi di Milano, via Golgi 19, 20133 Milano, Italy ͑Received 9 September 1997͒ States of positronium hydride having different angular momenta have been studied by means of quantum Monte Carlo techniques. Explicitly correlated wave functions for different states have been obtained using the variational Monte Carlo optimization method. These wave functions have been used in variational Monte Carlo and diffusion Monte Carlo ͑DMC͒ simulations to compute energies, annihilation rates, and other observables. Our DMC results compare well with the best published variational ground-state binding energy, and show that positronium hydride has metastable states with angular momentum Lϭ1 and 2 above the ground-state disso- ciation threshold. The values of the other observables for the ground state are comparable with the best variational calculations. The results for the Lϭ1 and 2 states are used to discuss a proposed model for the annihilation of in alkali crystals. ͓S1050-2947͑98͒09502-X͔

PACS number͑s͒: 36.10.Ϫk, 02.70.Lq

I. INTRODUCTION not include electron-electron and electron-positron correla- tion. It relies on a ‘‘mean-field’’ description, missing the During the last few years, attention has been paid to the local behavior of the exact wave function near particle coa- properties of systems containing one or more positrons with lescence points; as a consequence it underestimates both pos- the aim to elucidate the problem of their stability and the itron affinity and positronium binding energy. To correct the annihilation behavior of positrons in ordinary matter. Among SCF deficiencies, correlation has been introduced by means the systems studied, there are the positronium anion PsϪ of standard molecular-orbital methods, like the Moller- Plesset perturbation theory ͓26͔ and the many-body- ͓1,2͔, the dipositronium Ps2 ͓3–5͔, the positronium hydride PsH ͓6–18͔ and other small atomic and model sys- perturbation theory ͓16,27͔. Recently, some attempts have been made to explore the possibility of using configuration- tems ͓19͔. These issues are still wide open, and there is a interaction techniques ͓28,29͔, but convergence of energy growing mass of experiments and experimental techniques and other mean values has proved to be painfully slow. On that can accurately probe the interaction between matter and the other hand, to correct for the lack of a description of the antimatter, and which need theoretical support to be inter- positron-electron cusp in DFT, a pair correlation correction preted. So far, few attempts have been devoted to solving the has been developed ͓24͔: by means of an enhancement factor problems of stability and annihilation in a systematic way: one is able to estimate the annihilation properties simply Cade and Farazdel ͓20͔, Patrick and Cade ͓21͔, Kurtz and using the positron and electron densities, but this is rather a Jordan ͓22͔, and Gol’danskii, Ivanova, and Prokop’ev ͓8͔ heuristic approach to the problem since it has been devel- proposed applying the Hartree-Fock method to compute the oped starting from model systems and extrapolating their ϩ Ϫ positron affinity and positronium (e ,e ) binding energy of behavior to larger ones. atoms, ions, and ; the authors of Refs. ͓10,12͔, The best theoretical approach used so far to describe this ͓14,15͔, and ͓13͔ used explicitly correlated trial wave func- class of systems exploits explicitly correlated trial wave tions to compute energies for small atomic systems contain- functions to compute accurate values of the observables. Ma- ing positrons; attempts have been made to apply density- trix elements between explicitly correlated wave functions functional theory ͑DFT͒ to the calculation of average values are not easy to compute for systems containing more than for first-row atoms, ions, and solids ͓17,23–25͔. Although two particles, unless one uses correlated Gaussians ͓30,4͔. the cited methods can help to rationalize the increasing Since these functions do not reproduce the cusp conditions, amount of experimental data, each one has his own draw- i.e., the behavior of the exact wave function at small inter- backs: DFT theory lacks knowledge of the exact energy particle distances, very large basis sets must be employed, functional, and its results are dependent on the particular and also a careful and computationally expensive optimiza- choice of the exchange and correlation potential used in the tion of the nonlinear parameters of the trial wave function is computation, while self-consistent-field ͑SCF͒ theory does required in order to obtain an accurate description. Monte Carlo techniques are flexible and powerful meth- ods to solve the Schro¨dinger equation for small atoms and *Electronic address: [email protected] molecules, even if they contain exotic particles like positrons †Electronic address: [email protected] or muons ͓31–38͔, or if one is interested in observables dif- ‡ Electronic address: [email protected] ferent from the energy ͓39͔. In this work we present a nu-

1050-2947/98/57͑3͒/1678͑8͒/$15.0057 1678 © 1998 The American Physical Society 57 STABILITY AND POSITRON ANNIHILATION OF . . . 1679 merical study of various states of PsH having different total The breakup of this system in a three-particle cluster plus angular momentum by means of variational and diffusion a free particle is not possible because the Coulomb attraction Monte Carlo ͑VMC and DMC͒ methods. The ground state of between the two fragments binds them together: so the low- this system has been studied in the past using various stan- est energy dissociation threshold for this system is given by dard methods, namely, SCF and Hartree-Fock methods ͓41͔ ͓8,20,22,23͔, DFT ͓17͔, and DMC ͓33,36͔. Also, explicitly correlated trial functions have been used ͓6,7,10–15͔, the E ϭE ϩE ϭϪ 1 Ϫ 1 ϭϪ 3 , ͑3͒ thr Hgs Psgs 2 4 4 most recent and accurate calculation being the one performed by Frolov and Smith ͓18͔ using a linear combination of ex- where PsH dissociates in two neutral fragments, both in their plicitly correlated Gaussians. A compilation of the main re- ground state. The stability of higher angular momentum sults on the PsH ground state was published by Yoshida and states is relative to their own dissociation threshold, i.e., Miyado ͓36͔ in their paper on the stability of PsH and where the two fragments are resting at infinity and the posi- ͓Li,eϩ͔. Although a lot of computational effort has been tronium carries the total angular momentum L of the system: spent on the ground state of this system, and accurate infor- mation is now available, the results obtained so far need to L 1 1 be checked by independent calculations exploiting different E ϭEH ϩEPs ϭϪ Ϫ . ͑4͒ thr gs ͑L͒ 2 2 methods, especially regarding the annihilation behavior of 4͑Lϩ1͒ PsH. This position can be justified noticing that even the 396 The other possibilities, in which the atom carries a term wave function by Ho ͓15͔, a reference calculation in nonzero angular momentum, or when for L 2 the angular this field, does not seem to be well converged to definitive у momentum is shared between the two fragments, always results, if compared with the other values of the shorter lin- have a higher energy. To approximate the wave functions of ear combinations published in the same paper and with the states having different total angular momenta, we propose to results published in Ref. ͓18͔. The ground state of PsH is a use a linear combination of explicitly correlated functions good candidate for the application of our methods since it 38,42,43 contains all the relevant physics of this class of systems: ͓ ͔ without a complete description of the correlation between the Nterms particles, no meaningful information could be obtained, es- ⌿Tϭ ͚ ci⌽i , ͑5͒ pecially for properties like the annihilation rates that are iϭ1 strongly dependent on the quality of the description of the correlated motion. Our work is therefore motivated by the where need to compute accurate values for these observables. In ⌽ ͑R,k ͒ϭ ͕f ͑R͒ addition, we applied Monte Carlo methods to higher angular i i A i momentum states of this system, since these have been much ϫeϪki,1r1Ϫki,2r2Ϫki,3raϩki,4r12ϩki,5r1aϩki,6r2a less studied than the ground state and could be of relevant 1,2 a interest in scattering experiments with positron beams ͓40͔. ϫ⌰0,0⌰1/2,1/2͖. ͑6͒ Furthermore these results will allow to test the model pro- posed in Ref. ͓8͔ to explain the annihilation behavior of slow In this equation is the antisymmetrization operator; f i(R) positrons in alkali hydrides. It is also worth noting that dur- is a function thatA explicitly contains the dependence on the ing the last few years, experiments ͓40͔ with positron beams spatial coordinates of the particles ͑we refer to this as the opened the possibility to detect positron systems in the gas preexponential part of the trial wave function͒, but does not 1,2 phase, giving rise to a new amount of experimental informa- contain any variational parameter; ⌰0,0 is the spin eigenfunc- tion needing theoretical counterparts to be interpreted mean- a tion for the two electrons, while ⌰1/2,1/2 is the spin function ingfully. for the positron; and ki is a vector of parameters for the ith term of the linear expansion. The first three components of II. TRIAL WAVE-FUNCTION FORM this vector were forced to have only positive values for all AND MONTE CARLO SIMULATION the terms of the linear combination. As already stated in our previous work ͓43͔, it is possible to write this trial wave In the following an alphabetic subscript denotes a posi- function in a spin-free formalism by means of a linear com- tron, while a numerical subscript denotes an electron. In bination of spatial terms with exchanged particle indices, i.e., atomic units, the Hamiltonian operator for PsH system in the

Born-Oppenheimer approximation has the form Np R,k ϭ sym f R 1 2 2 2 ⌽i͑ i͒ ͚ j ͓ i͑ ͒ ϭϪ ٌ͑ ϩٌ ϩٌ ͒ϩV͑R͒, ͑1͒ jϭ1 P H 2 1 2 a Ϫk r Ϫk r Ϫk r ϩk r ϩk r ϩk r where V(R) is the Coulomb interaction potential, ϫe i,1 1 i,2 2 i,3 a i,4 1a i,5 2a i,6 12͔, ͑7͒

sym 1 1 1 1 1 1 where j are the exchange operators generated by acting V͑R͒ϭϪ Ϫ ϩ ϩ Ϫ Ϫ , ͑2͒ with Pon Eq. ͑6͒ and collecting all the terms with the same r1 r2 ra r12 r1a r2a spin productA ͓e.g., ␣(1)␤(2)␣(a)͔. This trial wave function, between the four-unit-charge particles ͑i.e., two electrons, which has the correct spin and space symmetry, describes the the positron, and the nucleus͒, and R is a point in configu- correlation between the electrons and the positron by means ration space. of the exponential part depending on the explicit electron- 1680 DARIO BRESSANINI, MASSIMO MELLA, AND GABRIELE MOROSI 57 positron and electron-electron distances. This analytical form 1 2 2 allows an accurate description of the correct behavior of the ͗␦͑rij͒͘ϭϪ ⌿ ͑Rٌ͕͒ ͓ln⌿T͑R͔͒ 2␲͵ T ri exact wave function at the coalescence point for equal and opposite sign charges. Satisfying the cusp condition usually 1 ϩ2ٌ͓ ln⌿ ͑R͔͒2͖ dR, ͑12͒ accelerates the convergence ͓44͔ of the linear expansion and ri T r reduces its length for a chosen accuracy. This is useful to ij reduce the computational cost of the optimization of the pa- where ⌿T is assumed to be normalized. Using the differen- rameters in the trial wave function, usually a quite heavy tial identity task. Moreover, for the specific system we are dealing with, the trial wave function must explicitly contain the positron- ٌ2 f ͑R͒ ri electron distance, otherwise the annihilation behavior and the ϭٌ2 lnf ͑R͒ϩٌ͓ lnf͑R͔͒2, ͑13͒ r ri ‘‘pile-up’’ effect of the electron density due to the positron f ͑R͒ i are not correctly described. Eq. ͑12͒ can be written as The chosen form for the trial wave function makes it very difficult, if not impossible, to compute analytically the ma- 2 r ⌿T͑Rٌ͒ 1 trix elements of the Hamiltonian operator of the system. A 2 i ͗␦͑rij͒͘ϭϪ ⌿T͑R͒ numerical method must be used to obtain the energy mean 2␲͵ ͭ ⌿T͑R͒ value and other observables for a given trial wave function. The variational Monte Carlo method ͓45͔ is well suited for 1 ϩٌ͓ ln⌿ ͑R͔͒2 dR, ͑14͒ ri T this goal since it requires only the evaluation of the wave ͮrij function, its gradient, and its Laplacian. Since the VMC and other Monte Carlo methods are well described in the litera- which is easy to implement in a VMC and DMC code, and ture ͓45͔, we only summarize the main points relevant to this has been used in this work to estimate the collision probabil- work. ity and the annihilation time of the PsH system. The mean value of a local operator over a given trial wave To optimize the linear and nonlinear parameters in the function is computed using trial wave function, we minimized the function

Nconf 2 2 2 ⌿T͑R͒ loc͑R͒dR ␮ ͑Er͒ϭ ͚ ͓Eloc͑Rj͒ϪEr͔ , ͑15͒ ͵ O jϭ1 ͗ ͘ ϭ , ͑8͒ O VMC ⌿2͑R͒dR ͵ T where ͕Rj , jϭ1,Nconf͖ is a set of Nconf fixed configurations 2 sampled from ⌿T , and Er is an approximation to the true where value of the energy for the system. This method, proposed by Frost ͓46͔ and Conroy ͓47͔, has been described in detail by Umrigar, Wilson, and Wilkins ͓48͔ and by Mushinski and ⌿T͑R͒ loc͑R͒ϭ O . ͑9͒ Nightingale ͓49͔ and has been proved to be much more O ⌿T͑R͒ stable than the optimization of the energy itself.

2 The optimized trial wave functions can be used to com- Here ⌿T(R) is interpreted as a probability distribution and it pute approximated mean values of the observables of the is sampled using Metropolis or Langevin algorithms ͓45͔.As studied systems and to give upper and lower bounds to their an explicit example, the expectation value of the Hamil- energy. Unfortunately, the mean values strongly depend not tonian is written as only on the analytical form of the function employed, but also on the optimization method used ͓50͔ to define the trial ⌿2͑R͒ ͑R͒dR wave function. In order to obtain the exact ground-state en- ͵ T Hloc ergy and improve the accuracy of the other mean values, the ϭ , ͗ ͘VMC DMC method ͓51,45͔ is employed to simulate the time- H 2 ͵ ⌿T͑R͒dR dependent Schro¨dinger equation in imaginary time as a dif- fusion equation having source and sink terms. This method samples the distribution f (R)ϭ⌿T(R)⌿0(R), where ⌿0(R) ⌿T͑R͒ loc͑R͒ϭ H . ͑10͒ is the ground-state wave function of the system with the H ⌿T͑R͒ same nodal surface of the trial wave function if we use the fixed node approximation to sample an antisymmetrized Other operators can be converted to form ͑9͒; for example, wave function. The value of the energy can be computed the mean value for the Dirac’s delta operator ␦(rij) can be using the ‘‘mixed estimator’’ computed using the identity

͐⌿T͑R͒⌿0͑R͒ loc͑R͒dR 1 1 ͗ ͘DMCϭ H . ͑16͒ 2 H ͐⌿T͑R͒⌿0͑R͒dR rij͒ϭϪ ٌr . ͑11͒͑␦ 4␲ irij If the ground state has no nodes, or ⌿T(R) has the correct Introducing Eq. ͑11͒ into Eq. ͑8͒, and integrating by parts nodal structure, this equation gives the correct ground-state one obtains energy; otherwise one obtains an upper bound to it. To com- 57 STABILITY AND POSITRON ANNIHILATION OF . . . 1681

TABLE I. Mean energy and various values and observables for the ground state 2,1S of the positronium hydride.

N ͗E͗͘V͗͘rϩϪ͗͘rϪϪ͗͘rϪ͗͘rϩ͗͘␦ϩϪ͘ ␹ 3 Ϫ0.778 001(26) Ϫ1.5332(2) 3.708 64͑61͒ 4.3337͑11͒ 2.7497͑12͒ 4.3209͑11͒ 0.022 00͑10͒ 0.0115͑3͒ 4 Ϫ0.782 189(24) Ϫ1.5466(3) 3.789 33͑57͒ 4.2071͑10͒ 2.6647͑11͒ 4.2071͑10͒ 0.021 64͑21͒ 0.0117͑3͒ 5 Ϫ0.784 848(11) Ϫ1.5587(2) 3.586 62͑34͒ 3.972 88͑55͒ 2.545 40͑65͒ 3.972 88͑55͒ 0.024 70͑9͒ 0.0071͑3͒ 6 Ϫ0.786 310(11) Ϫ1.5562(2) 3.554 87͑30͒ 3.757 13͑47͒ 2.427 20͑50͒ 3.862 14͑49͒ 0.023 74͑17͒ 0.0107͑2͒ 7 Ϫ0.786 788(12) Ϫ1.5514(3) 3.576 36͑45͒ 3.739 90͑59͒ 2.412 73͑66͒ 3.834 84͑66͒ 0.023 19͑23͒ 0.0145͑3͒ DMC Ϫ0.789 175(10) Ϫ1.5652(3) 3.5236͑8͒ 3.6550͑9͒ 2.361͑10͒ 3.743͑1͒ 0.02438͑20͒ SOE Ϫ1.5895(3) 3.4708͑8͒ 3.5701͑9͒ 2.308͑1͒ 3.651͑1͒ 0.02556͑30͒

Ho a Ϫ0.778 45 0.0243 Ho b Ϫ0.788 945 3.849 3.556 3.644 0.0244 FS c Ϫ0.789 1794 Ϫ1.578 38 3.479 3.573 2.311 3.661 0.024 134 0.000 0131 aHo’s 126-term wave function ͓15͔. bHo’s 396-term wave function ͓15͔. cFrolov and Smith’s wave function ͓18͔. pute the mean values of dynamical observables whose opera- ried out using a stochastic selection procedure ͓42,43,52,53͔ tors do not commute with the Hamiltonian, i.e., the general for the nonlinear parameters of the new term, and choosing case, one can still exploit the ‘‘mixed estimator’’ substituting the linear coefficient cnew that minimizes the value of 2 the new operator to the Hamiltonian, but the value so com- ␮ (Er). Usually one or two thousand nonlinear parameter puted is biased by the errors of the trial wave function. A sets were randomly chosen, and the one that gave the lowest simple strategy to partially correct for this bias is to use the 2 ␮ (Er), once the linear parameter had been optimized, was quantity ͓45͔ kept as the new trial term. Usually, this procedure gives a small reduction of the variance of the local energy and a ͗ ͘ ϭ2͗ ͘ Ϫ͗ ͘ , ͑17͒ O SOE O DMC O VMC fairly good starting point for the following optimization. where During all the optimization procedures, a set of 8000 con- figurations in the nine-dimensional configuration space was ͐⌿T͑R͒⌿0͑R͒ loc͑R͒dR used. This set was chosen after sampling the best wave func- ͗ ͘DMCϭ O . ͑18͒ tion previously optimized ͑e.g., the one-term function given O ͐⌿T͑R͒⌿0͑R͒dR by Ref. ͓7͔ for the first step of the optimization͒. After three Equation ͑17͒ gives an estimate of ͗ ͘ that is second order or four optimization steps this ensemble was updated by ͑SOE͒ on the error of the trial waveO function. means of a VMC run, usually 5ϫ104 steps long, sampling the square of the new wave function. These VMC simula- III. VARIATIONAL AND DIFFUSION MONTE CARLO tions are long enough to give reliable average values for all RESULTS the observables. The variational results of the optimized trial 2,1 2,1 wave functions for the S ground state of PsH are reported A. S ground state of PsH in Table I for a number of terms Nϭ3Ϫ7, together with the The 2,1S ground state of PsH was chosen as starting point results obtained by Ho ͓15͔ using Hylleraas-type wave func- of our investigation on systems containing positron. For this tions, and the results by Frolov and Smith ͓18͔. In this table, state the trial wave function must be written using only pre- ͗V͘ is the expectation value for the potential energy, while exponential terms having spherical symmetry or containing ␹ϭ͉1ϩ(͗V͘/2͗E͘Ϫ2͗V͘)͉ is the so-called virial parameter. explicitly the interparticle distances. Since the exponential This parameter, since we did not optimize the expectation part of the trial wave function already possesses an analytical value of the energy, does not have to be equal to zero for a form that should be able to describe the interparticle corre- full optimized wave function written as a linear combination lation correctly, the preexponential part of the analytical an- of a small number of terms. These results show that our satz has been used only to modify the shape of the positronic expansion is quickly convergent toward an accurate value of density, expressing it as function of the positron-nucleus dis- the total energy, the three-term wave function already recov- tance only. This choice makes it possible for the trial wave ering more than 70%, and the seven-term one more than 93% function to describe the positron-nucleus cusp condition cor- of the binding energy computed by Frolov and Smith. Com- rectly, reducing the probability to find the positron in the paring the other mean values for our seven term, and Ho’s nuclear region. 396-term trial wave function, it is worth noting that the mean To optimize the wave function, the simple one-term func- distance between an electron and a positron is shorter for our tion given in Ref. ͓7͔ was used as starting point: its param- trial wave functions than for that of Ho, while the mean 2 eters were fully optimized minimizing the quantity ␮ (Er) distance between the two electrons is larger. These differ- given by Eq. ͑15͒. Starting from this new wave function, ences might be due to incomplete convergence of the two more terms were added to reduce the variance of the local wave functions of different analytical forms, or to the opti- energy and to lower the mean energy. These steps were car- mization method used. Our mean distance between an elec- 1682 DARIO BRESSANINI, MASSIMO MELLA, AND GABRIELE MOROSI 57

TABLE II. Parameters of the optimized seven-term trial wave function.

N ci ki,1 ki,2 ki,3 ki,4 ki,5 ki,6 f i

1 1.00000 Ϫ0.36659 Ϫ1.00580 Ϫ0.34328 Ϫ0.37344 Ϫ0.04846 0.03263 ra 2 0.35442 Ϫ0.12301 Ϫ0.75828 Ϫ0.14497 Ϫ0.30421 Ϫ0.81432 Ϫ0.05546 1 3 0.45760 Ϫ1.58378 Ϫ0.95978 Ϫ0.39332 Ϫ0.18145 0.11324 0.18718 1

4 Ϫ0.65857 Ϫ0.77919 Ϫ0.46739 Ϫ0.32126 Ϫ0.03521 Ϫ0.45572 Ϫ0.39182 ra 5 0.72787 Ϫ0.46927 Ϫ1.26244 Ϫ0.28601 Ϫ0.89774 Ϫ0.10067 0.15699 1

6 0.23700 Ϫ0.86864 Ϫ0.22488 Ϫ0.38221 Ϫ0.11085 Ϫ0.79638 0.03429 ra 7 Ϫ0.12385 Ϫ0.53677 Ϫ0.73638 Ϫ0.60591 Ϫ0.10452 0.00031 Ϫ0.03292 1

tron and the positron is in better agreement with the one one that leads to the fragmentation of PsH in H(Lϭ0) and computed by Frolov and Smith ͓18͔, being slightly larger Ps(Lϭ1) both at rest, and the nodal surfaces due to a total like our mean electron-nucleus and positron-nucleus dis- angular momentum equal to 1. Using this analytical form for tances; that is, our VMC wave function describes a system ⌽i(R,ki) in the linear expansion, a two-term wave function less compact than Frolov and Smith ͓18͔ do. On the whole, was optimized, as already explained for the ground-state cal- our seven-term expansion, whose optimized parameters are culations. As an initial guess for the one-term wave function, shown in Table II, is a very compact and good quality wave we wrote the total wave function as the product between the function, well suited as guide function in a DMC calculation. exact functions of the two fragments, modified introducing DMC simulations were performed to project out the re- some correlation between the particles belonging to the two maining components of the excited states of the same sym- different fragments, in a way to produce a trial wave function metry of the ground state. Five simulations using different that is square integrable: its parameters quickly rearranged Ϫ time steps spanning the range 0.001–0.012 hartree 1 have into a pair of optimization steps to give a much better trial been carried out to obtain extrapolated zero time step values wave function, whose energy was below the dissociation for the energy and other observables. These results ͑DMC͒ threshold for this state. The second term was added using the and their second-order estimates are shown in Table I. It is previously described stochastic selection procedure, fol- worth noting that our value for the ground-state energy of lowed by full reoptimization of the wave function. PsH, Ϫ0.78917(1) hartree, is 0.23͑1͒ m hartree lower than The VMC, DMC, and SOE results for the 2,1P state, ob- the value computed by Ho: this is not a surprising result tained using this function, are shown in Table III. DMC since the mean energy values published by Ho did not appear simulations were carried out using three different time steps, as conclusive, due to their slow convergence. Our energy is spanning the range 0.005–0.015 hartreeϪ1, to check for the in good agreement with the two recently published DMC time step bias: this appears smaller than the statistical error calculations of Ϫ0.7885(5) ͓33͔ and Ϫ0.789(1) hartree of the ͗E͘ values, as shown in Fig. 1. ͓36͔, but it is more accurate, due to the smaller standard For the 2,1D state the preexponential part has the form deviation obtained in our simulation. Finally, we found com- plete agreement with the latest result Ϫ0.789179 hartree by f i͑R͒ϭ͑x1Ϫxa͒͑y 1Ϫy a͒, ͑20͒ Frolov and Smith ͓18͔. Both DMC and SOE values show a decrease of their ab- and a one-term trial wave function has been optimized using solute values when compared with the VMC results: this the same procedure as for 2,1P state. The VMC, DMC, and effect can be attributed to a better description of the correla- SOE results for this state are shown in Table IV. tion between the various particles in the system. The SOE correction improves the agreement with the results of Ref. IV. DISCUSSION ͓18͔, but it appears that higher-order corrections are needed to obtain accurate predictions of these observables. In this work we studied the PsH system, computing the energy and some mean values for the states carrying a total B. Lower 2,1P and 2,1D states of PsH angular momentum Lϭ0, 1, and 2. This has been done by 2,1 optimizing explicitly correlated trial wave functions, written As to the P state, we adopted the preexponential term as linear combinations of different numbers of correlated f ͑R͒ϭz Ϫz , ͑19͒ terms, by means of the variational Monte Carlo technique, i 1 a followed by diffusion Monte Carlo simulations. to describe both the relevant dissociative channel, i.e., the Since a trial wave function always contains some errors TABLE III. Mean energy and other observables for the 2,1P state of positronium hydride: two-term trial wave function.

͗E͗͘rϩϪ͗͘rϪϪ͗͘rϪ͗͘rϩ͗͘␦ϩϪ͘ VMC Ϫ0.592 11(5) 9.472͑4͒ 4.202͑1͒ 2.574͑1͒ 9.212͑3͒ 0.001 25͑5͒ DMC Ϫ0.615 28(5) 8.377͑3͒ 3.953͑2͒ 2.448͑2͒ 8.172͑5͒ 0.000 90͑3͒ SOE 7.822͑4͒ 3.704͑2͒ 2.332͑2͒ 7.132͑5͒ 0.000 55͑5͒ 57 STABILITY AND POSITRON ANNIHILATION OF . . . 1683

TABLE IV. Mean energy and other observables for the 2,1D state of positronium hydride: one-term trial wave function.

͗E͗͘rϩϪ͗͘rϪϪ͗͘rϪ͗͘rϩ͗͘␦ϩϪ͘ VMC Ϫ0.543 40(2) 12.937͑2͒ 4.5517͑9͒ 2.788͑1͒ 12.380͑3͒ 0.000 066͑13͒ DMC Ϫ0.566 72(3) 14.615͑5͒ 4.147͑1͒ 2.569͑1͒ 14.345͑5͒ 0.000 017͑2͒ SOE 16.293͑5͒ 3.742͑1͒ 2.350͑1͒ 16.310͑5͒ due to the incompleteness of the basis set used, the results ences between VMC and DMC values, usually of the order will be incorrect by some amount. The DMC technique ap- of 10% of the DMC values. This means that the observables pears as a very useful tool to correct partially for this bias, computed for these states can be used only as indicators of since it is able to sample the correct wave function if the the order of magnitude of the exact ones. state has no nodal surfaces or if the trial function used to Although the mean values of the Dirac’s delta operator guide the simulation has the correct nodal structure. For the ␦ϩϪ for the two excited states are not as accurate as for the 2,1S state of PsH, our optimized seven term trial wave func- ground state, they can be usefully employed to discuss anni- tion allows one to compute accurate values for both the en- hilation of positrons in alkali hydrides ͓54͔. To interpret the ergy and the other observables shown in Table I: for this experimental results, Ref. ͓8͔ proposed a model where 2,1S state the relative error (͗ ͘Ϫ͗ ͘)/͗ ͘ is al- and 2,1P states of PsH are assumed as responsible of the two ODMC OVMC ODMC ways less than 0.04. Although it is not generally possible to different extrapolated annihilation rates ⌫1ϭ2.347(110) and Ϫ1 estimate the quality of the DMC and SOE corrections to the ⌫2ϭ1.149(66) ns that are seen experimentally. While VMC mean values, some relevant hints could be obtained their results based on Hartree-Fock calculations were not using some model systems as a benchmark. In this regard, conclusive due to some computational error ͑see Ref. ͓10͔͒, for the two systems Ps and XZ Ps ͓19,33͔, i.e., the system their model could not be definitely ruled out, and the prob- composed by a fixed fractional charge 0рZр1andaPs lem is still waiting for a definitive answer. As already stated, Ϫ1 atom, we found that for an order of magnitude of the relative accurate 2.463 and 2.4361-ns values for the ⌫2␥ annihila- error similar to the one we obtain for PsH, SOE values give tion rate of the PsH ground state have been obtained respec- an accurate approximation to the correct mean values, being tively by Ho and Frolov and Smith, but for the 2,1P state no in error by less than 1%. Therefore we estimate the values explicitly correlated results are available to compare with. 2,1 reported in Table I to have a similar accuracy. Our ͗␦ϩϪ͘ value for P appears to be at least an order of Using Eq. ͑3͒ of Ref. ͓15͔ and our mean values for the magnitude less than the one for the ground state: assigning 2,1 ␦ϩϪ operator, we obtain 2.341͑17͒, 2.461͑20͒, and 2.580͑30͒ the ⌫1 annihilation rate to S, it appears obvious that the Ϫ1 2,1 ns , respectively, for the VMC, DMC, and SOE two- computed VMC ⌫2␥ rate for the P state is too small by a photon annihilation rate ⌫2␥ . These results are in good factor of 10 at least. We strongly believe that this discrep- agreement with the value of 2.463 nsϪ1 published by Ho ancy cannot be resolved by means of a more accurate trial ͓15͔, and with the one of 2.4361 nsϪ1 computed by Frolov wave function: the DMC method is supposed to improve all and Smith, but the SOE value seems to suggest a faster an- the mean values computed, reducing the error in the sampled nihilation in PsH than the previously published results. walker distribution, but it further decreases the ͗␦ϩϪ͘ value, Due to the shorter length of the linear combinations used indicating the need to further reduce the overlap between the to approximate the states having higher angular momentum, positronic and electronic densities. the VMC results for these states are less accurate than the The VMC and DMC binding energies for the different L ones for the ground state. This fact is stressed by the differ- states,

L ϭEL ϪEL ͑21͒ EVMC,DMC VMC,DMC thr are shown in Table V. 2,1P and 2,1D states are bound if compared with the relevant threshold. The only previous re- sult for 2,1P excited state of PsH was published by Kurtz and Jordan ͓22͔: their SCF total energy was Ϫ0.5873 hartree, giving a binding energy ͓Eq. ͑21͔͒ of 0.0248 hartree, i.e.,

TABLE V. Binding energy ͑hartree͒ of PsH. VMC energies are computed using the best available wave function.

2,1S 2,1P 2,1D

0.036 79͑1͒ 0.029 61͑5͒ 0.015 62͑2͒ EVMC 0.039 17͑1͒ 0.052 78͑5͒ 0.039 97͑3͒ EDMC 0.039 179 4 a 0.0248 b Eanalytic FIG. 1. DMC energy vs time step in the simulation of the 2,1P aFrolov and Smith’s result ͓18͔. state of PsH. bKurtz and Jordan’s result ͓22͔. 1684 DARIO BRESSANINI, MASSIMO MELLA, AND GABRIELE MOROSI 57

FIG. 3. Positronic distribution for the 2,1S, 2,1P, and 2,1D states.

being polarized outward by the positron-electron interaction, both the SOE value of ͗rϩ͘ and the SOE positron distribu- tions shown in Fig. 3 for different L values show that posi- tron carries almost all the total angular momentum of the system. This physical picture was already pointed out by FIG. 2. Difference of the electronic distribution of the 2,1P and Kurtz and Jordan ͓22͔ using SCF calculations, even for the 2,1D states with the electronic distribution of the 2,1S ground state. highly polarizable hydride anion. Using the SOE positron-electron distribution d(rϩϪ), one 47% of our DMC result. Our two-term trial wave function is could obtain useful information about the analytical form already better than the SCF description, recovering 56% of that should be used to describe the correlation between these the DMC binding energy. two particles: for all three states these distributions can be As far as it is concerned with the experiment carried out accurately fitted by means of the function in Ref. ͓40͔ by means of the reaction Nd ϩ 2 Ϫ␣irϩϪ e ϩCH4ϭCH3ϩPsH͑L͒ ͑22͒ d͑rϩϪ͒ϭrϩϪ cie , ͑24͒ i͚ϭ1 whose energetic balance is given by

using Ndϭ2, showing that a good way to describe the cor- 6.8025 ͑L͒ relation between light particles having opposite charges is 14.35 eVϪ Ϫ PsH ͑23͒ ͑Lϩ1͒2 E given by an exponential form, such as the one used in this work. our energy values do not change the interpretation of these results, but suggest the onset of the production of PsH in ACKNOWLEDGMENTS higher angular momentum states at the threshold energy of 11.22 eV for the P state, and of 12.51 eV for the D state. Financial support by MURST is gratefully acknowledged. Unfortunately these regions of kinetic energy of the positron The authors are indebted to the Centro CNR per lo Studio beam were not explored in the experiment ͓40͔. delle Relazioni tra Struttura e Reattivita’ Chimica for grants While the electronic distribution changes only slightly, of computer time. Also, M.M. would like to thank CNR for increasing the angular momentum, as is shown in Fig. 2, financial support.

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