32nd ICPIG, July 26-31, 2015, Iai, Romania DIFFUSION MONTE CARLO MODELS OF CONFINED HYDROGEN SPECIES 1 1,2,3 4 G. Micca LongoP P, S. Longo P, D. GiordanoP P 1CNR-IMIP, via Amendola 122/D, 70126 Bari, Italy 2Department of Chemistry, University of Bari, via Orabona 4, 70126 Bari, Italy 3INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy 4ESA-ESTEC, Aerothermodynamics Section, Kepleerlaan 1, 2201 AZ, Noordwijk, The Netherlands As already known from the past, the concept of confined atom and molecule is a useful analogy to understand the behaviour of species under strong pressure or dense plasma conditions. We report on recent results obtained by the authors to develop and use a versatile code, based on diffusion Monte Carlo (DMC) method, for the determination of low lying eigenvalues of confined hydrogen + species. The method is applied to hydrogen atom and to molecular ion H2 under different geometries. Excited states are addressed by specifying appropriate nodal surfaces. Results for + eigenvalues of the H atom and H2 molecule confined inside barriers of different shape as a function of barrier size are obtained. In the case of the molecular system, a new result is found: the 2 + lowest excited state ( Σu ), which is antibonding in the vacuum, presents a minimum due to barrier. 1. Introduction The problem of confined physical systems has 2. Method and results been the subject of many studies in the past [1-7] in The method used for calculation is a fixed node view of its importance as a basic theoretical problem DMC calculation of the state of lowest energy with and as a model for systems in astrophysics (interior given symmetry. Details of the numerical method of stars), geophysics (giant planets), nuclear science are reported in [17-19]. (compression techniques for nuclear fusion). In our simulations, the hydrogen atom is placed in + Also confined H2 , the simplest molecular an impenetrable spherical box and the proton is system, has been the subject of many investigations assumed fixed in the center of the box, while the two as a model for the analysis of molecular systems protons of the dihydrogen cation are placed along under pressure [8-15]. Most of these previous the x axis at ! = ! 2 and ! = − ! 2 respectively (d studies assume prolate ellipsoidal confinement: this being the intenuclear distance). The single electron, condition is suggested by the fact that, as known, the which is present in both systems, is replaced by an + Schrödinger equation for H2 factorizes using prolate ensemble of fictitious particles called walkers [19]; ellipsoidal coordinates. the number N of walkers becomes a parameter of the In view of the perspectives opened by such a simulation. The Coulomb potential acts on every wide range of studies and applications, it is clear that walker. The confining barrier is a sphere of radius r0 a more straightforward approach is desirable. A with centre at the origin of the axis. methodology is required to produce results for the The propagator that acts on every single walker is ground state and the lowest excited states of the given by the approximate Green's function: hydrogen atom and the hydrogen molecular ion within cavities of different shape and with the !! ! ! ! − ! − ! ! !, !!, ! = 2!" ! !"# possibility to displace the nucleus/i easily. Such a 2! procedure can be formulated using the diffusion ! ! − ! !! − ! Monte Carlo (DMC) method, which allows to !"# −! ! (1) consider confining walls of any geometry, 2 preserving the use of Cartesian coordinates, and to include any distribution of positive charge within the which governs the diffusion and the branching cavity. Low lying excited states can be considered process of each walker. τ is the time step, V is the by including appropriate symmetry-based nodal potential energy and ET is the so called energy offset, + surfaces, an approach demonstrated for H2 states for that controls the total population of the walkers and confined systems in a previous paper by the present is adjusted to keep the average walker population authors [16]. roughly constant [20]: 32nd ICPIG, July 26-31, 2015, Iai, Romania !!!! The DMC method is able to reproduce the best !!! = !!!!! + ! ln (2) !! data with good accuracy, in spite of its simple implementation in cartesian coordinates. where ETi-1 is the energy value at time step i-1, α The effects of a cubical confinement on the is a small positive parameter and Ni-1 and Ni are hydrogen atom are studied here in order to respectively the number of walkers at time step i-1 demonstrate the DMC code efficiency with respect and the actual one. to a change of geometry. Keeping the same code Although DMC is principally a ground state structure and cartesian coordinates, only details of method, it can still be used for low excited states the subroutine for the calculation of the potential, as with distinct symmetry. The starting point is the a function of the walker position, are updated. fixed-node DMC method [21-24]: a trial wave Figure 2 shows the ground state and the 2p state function is chosen and used to define a trial node energies of the hydrogen atom confined in a cubical surface that reproduces the symmetries of the box, as a function of box side dimension l, as excited state. Here the hydrogen atom 2p state and obtained by DMC code. As in the previous cases, 2 + 3 the dihydrogen cation Σu state eigenvalues are about 10 walkers have been used, and τ values -2 reproduced. An extra absorbing boundary is ranged between 10 for free cases and weak -4 introduced at the centre of the box, along the x confinement and 10 for strong confinements and direction, in order to represent the symmetry of these excited state. particular states. Thus, a DMC code has been written consistently with the preceding description. 2.1. Confined hydrogen atom Figure 1 shows the ground state and the 2p state energies of the three-dimensional confined hydrogen atom as a function of box radius r0, as obtained by DMC code and those published by Aquino et al. [7]. Here results are presented in atomic units. About 103 walkers have been used in the simulations, while τ values ranged between 10-2 for free cases and weak confinement and 10-4 for strong confinements and excited state. Figure 2. Cubical confined hydrogen atom These results for the cubical confinement are reported for the first time, and, besides providing an illustration of the proposed procedure, may be a useful reference for studies in the field of atomic and semiconductor physics. 2.2. Confined hydrogen molecular ion + Here results for the H2 electronic ground state and the first excited state are reported. About 103 walkers have been used, while values ranged between 10-3 for strong confinement and the excited -2 Figure 1. Spherical confined hydrogen atom state and 10 for free case. The total energy (presented in eV) as a function of the internuclear separation d (presented in atomic units) is determined. 32nd ICPIG, July 26-31, 2015, Iai, Romania Preliminary (see figure 3), the calculations are validated by comparing the results for the two states with results from the literature, for the case of an unconfined ion, since no result is available for the confined system. + Figure 5. Confined H2 : Σu A very interesting feature emerges from figure 5: 2 + the Σu state, which is antibonding in the case of the + free H2 ion, has pronounced minimum in the case of the compressed ion. These minima are mostly due to + Figure 3. Free H2 a partial release of the compression energy when the nuclei are displaced from the barrier surface towards In figures 4 and 5 we can see the effect of the center, leaving more effective space to the different confinement dimensions. electron cloud in the Coulomb potential well of the nuclei. 3. Conclusions In this paper we have summarized the work in progress by our group to develop a practical and versatile technique to calculate eigenvalues of hydrogen species confined by potential wells of different geometry. A numerical code has been developed based on Diffusion Monte Carlo method with symmetry-based state selection: basically, the eigenvalue extracted is the lowest one between those sharing a selected eigenfunction symmetry. The method allows using cartesian coordinates, and moving nuclei in any position (thereby allowing to determine potential energy surfaces). This study in progress is bound to produce in the long term a new view of the quantum behaviour of plasma species under extreme conditions which are also of interest of the community. + Figure 4. Confined H2 : Σg 4. Acknowledgements This research activity has been supported by the General Studies Programme of the European Space Agency under grant 4200021790CCN2. 32nd ICPIG, July 26-31, 2015, Iai, Romania 5. References [1] A. Michels, J. D. Boer and A. Bijl, Physica 4 (1937) 981. [2] S. A. De Groot and C. D. Ten Seldam, Physica Amstersdam 12 (1946) 669. [3] D. Suryanarayana and J. A. Weil, J. Chem. Phys 64 (1976) 510. [4] S. J. Yngve, Am. J. Phys. 54 (1986) 1103. [5] S. J. Yngve, J. Math. Phys. 29 (1988) 931. [6] B. L. Burrows and M. Cohen, Int. J. Quantum Chem. 106 (2006) 478. [7] N. Aquino, G. Campoy and H. E. Montgomery Jr., Int. J. Quantum Chem. 107 (2007) 1548. [8] T. L. Cottrell, Trans. Faraday Soc. 47 (1951) 337. [9] K. K. Singh, Physica 30 (1964) 211. [10] E. Ley-Koo and S. A. Cruz, J. Chem. Phys. 74 (1981) 4603. [11] J. Goreki and W. Byers Brown, J. Chem. Phys. 89 (1988) 2138. [12] S. Mateos-Cortés, E.