The unbiased diffusion Monte Carlo: a versatile tool for two-electron systems confined in different geometries

THE EUROPEAN PHYSICAL JOURNAL D Eur. Phys. J. D (2021)75:83 https://doi.org/10.1140/epjd/s10053-021-00095-7 Regular Article - Molecular Physics and Chemical Physics The unbiased diffusion Monte Carlo: a versatile tool for two-electron systems confined in different geometries Gaia Micca Longo1,2,a , Carla Maria Coppola1,3,b , Domenico Giordano4,c , and Savino Longo1,2,d 1 Department of Chemistry, Università degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy Istituto per la Scienza e Tecnologia dei Plasmi – Consiglio Nazionale delle Ricerche, Bari Section, Via Amendola 122/D, 70125 Bari, Italy 3 Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, Florence, Italy 4 European Space Agency – ESTEC (Retired), Noordwijk, The Netherlands 2 Received 23 November 2020 / Accepted 17 February 2021 © The Author(s) 2021 Abstract. Computational codes based on the diffusion Monte Carlo method can be used to determine the quantum state of two-electron systems confined by external potentials of various natures and geometries. In this work, we show how the application of this technique in its simplest form, that does not employ complex analytic guess functions, allows to obtain satisfactory results and, at the same time, to write programs that are readily adaptable from one type of confinement to another. This adaptability allows an easy exploration of the many possibilities in terms of both geometry and structure of the system. To illustrate these results, we present calculations in the case of two-electron hydrogen-based species (H2 and H+ 3 ) and two different types of confinement, nanotube-like and octahedral crystal field. 1 Introduction The fundamental relevance and concrete applications of the confined quantum systems have been established since the early days of quantum physics [13,28,41], up to the present day [23,35,39,40]. Simple confined physical systems have a very long tradition in physics because of their importance as basic theoretical problems [4,7,24–27], and as prototypes to gain insights in semiconductor physics (quantum dots [8,33], quantum wires and quantum wells [17,30]). Atoms imprisoned in zeolite traps, clusters and fullerene cages provide good examples of confined systems in atomic physics and inorganic chemistry [6,10,18,42]. Many of these studies on the physical–chemical properties of confined systems focus on systems with a small number of electrons: for example, species with small molecules containing hydrogen, helium and lithium. Furthermore, confinement is produced by means of an infinite potential well with different geometries, in order to produce analogies with different real systems without considering their details. In recent years, our group has developed a very simple and versatile methodology to study this kind of systems [24–27]. It is a modification of the diffusion Monte Carlo (DMC) method: since the number of electrons is low, no variance reduction technique requesting a e-mail: e-mail: c e-mail: d e-mail: (corresponding author) b 0123456789().: V,-vol a trial wavefunction is employed; instead, some efficient solutions to accelerate calculations and reduce noise are employed. Thanks to this method, it is possible to study different systems, with different orientations, and even quickly change the geometry of the potential well, from spherical to elliptical, from cylindrical to cubical, using Cartesian coordinates and implementing few modifications to the starting code. Our previous works [24–27] focused on the effects of various confinement geometries on the excited states of monoelectronic systems, such as H and H+ 2 . In this paper, we show the application of the method to the ground state of two-electron systems (H2 and H+ 3 ) with non-trivial confinement geometries. An exhaustive report on the many-electron confined system can be found in the work by Jaskólski [19]: Different methods of analysis and description of spatial confinement effects are reviewed, with a particular attention to their importance in semiconductor structures. Free H2 is one of the simplest molecular systems, and its electronic properties represent a keypoint in understanding larger molecular systems; therefore, its confinement has received much attention during the last decades. The molecular hydrogen confined in a rigid spheroidal box, with fixed and not fixed nuclear positions, was studied with variational calculations and quantum Monte Carlo techniques [9,14,15,21,22,31], also with Monte Carlo methods beyond the Born– Oppenheimer approximation [37]. 123 83 Page 2 of 7 Eur. Phys. J. D (2021)75:83 Many studies concerning the ground state and the potential energy surfaces of the H+ 3 can be found in the literature ([1,11,20,29,32,34]; to cite a few), but almost no information is available about the quantum confinement of this interesting molecular system. Quantum calculations based on diffusion method for the study of the free H+ 3 in equilateral triangle configuration trace back to the works by Anderson [2,3]. In this paper, we present the first results concerning the confined H2 inside a six-negative-charge cage and the collinear H+ 3 enclosed in a nanotube. All results are obtained by diffusion Monte Carlo method. 2 Method Quantum Monte Carlo (QMC) is a class of computer algorithm that is able to simulate quantum systems and to compute the electronic ground state of atoms, molecules and solids. Among several quantum Monte Carlo methods available [16], the diffusion Monte Carlo (DMC) method offers the possibility to consider confining walls of any geometry while retaining the use of Cartesian coordinates and to include any distribution of positive charge within the cavity. Basically, DMC is a stochastic projector method that makes use of the similarity between the imaginary time Schrödinger equation and a generalized diffusion equation, which can be solved using stochastic calculus and simulating a random walk. The diffusion process of numerous “walkers” in the phase space is simulated. A walker is a mobile mathematical point in a set {Ri } of phase coordinates R in a 3N -dimensional space, where N is the number of physical electrons in the system, different from the number of walkers M , which is an integer numerical variable, M ∼ 103 . An evolution is performed by repeated application of the short-time diffusion–reaction propagator ⎡     −3N  ⎢− R − R G R ← R , τ ≈ (2πτ ) 2 exp ⎣ 2τ 2 ⎤ ⎥ ⎦ ⎤    −τ V (R)−V R − 2ET ⎦ exp ⎣ 2 ⎡ (1) where τ is a numerical step, and ET is the so-called energy offset that controls the walker total population. The kinetic energy term of the wave equation is connected with the diffusion process (the first exponential term of Eq. (1), including the normalizing constant), while the potential energy term (the second exponential term) controls the so-called birth–death algorithm, that is the destruction or the multiplication of walkers [16]. According to this second factor, a walker may randomly disappear, or duplicate itself, in a nu (...truncated)