Helium shows fascinating quantum phenomena unseen in any other element. In its liquid phase, it is the only known superfluid. The smallest aggregates of helium, the dimer (He 2 ) and the trimer (He 3 ) are, in their predicted structure, unique natural quantum objects. While one might intuitively expect the structure of 4 He 3 to be an equilateral triangle, a manifold of predictions on its shape have yielded an ongoing dispute for more than 20 years. These predictions range from 4 He 3 being mainly linear to being mainly an equilateral triangle. Here we show experimental images of the wave functions of 4 He 3 and 3 He 4 He 2 obtained by Coulomb explosion imaging of mass-selected clusters. We propose that 4 He 3 is a structureless random cloud and that 3 He 4 He 2 exists as a quantum halo state.
The stability of the ground and excited states of Positronium-atom complexes ͓A, Ps͔, Ps ϭ͓e ϩ ,e Ϫ ͔, has been explored for AϭLi, B, C, O, F using variational and diffusion Monte Carlo techniques. From the numerical results of our simulations it turns out that the ground state of the complexes ͓Li, Ps͔ 2 S, ͓C, Ps͔ 3 S, ͓O, Ps͔ 1 P, and ͓F, Ps͔ 2 S is stable against the break up in the two neutral fragments A and Ps, while the ground state of ͓B, Ps͔ 2 P has an energy above the same dissociation threshold. As to the excited states, the only possible candidate, ͓F, Ps͔ 2 P, has a total energy statistically equal to the lower dissociation threshold, i.e. it does not seem to be stable against the dissociation.
The stability of the ground-state of positron-polar molecule complexes ͓M,e ϩ ͔ has been explored for MϭLiH,HF,H 2 O,BeO,LiF using variational and diffusion Monte Carlo techniques. Our simulations show that the ground-state of the complexes ͓LiH,e ϩ ͔ 2,1 ⌺ ϩ , ͓BeO,e ϩ ͔ 2,1 ⌺ ϩ , and ͓LiF,e ϩ ͔ 2,1 ⌺ ϩ is stable against the dissociation either in the two fragments M and e ϩ or in the other two fragments M ϩ and Psϭ͓e ϩ ,e Ϫ ͔, while the ground-state of ͓H 2 O,e ϩ ͔ 2,1 A 1 , and of ͓HF,e ϩ ͔ 2,1 ⌺ ϩ has an energy equal to the dissociation threshold, M and e ϩ. We also compare the predicted vertical positron affinity ͑PA͒ with high quality vertical electron affinity ͑EA͒ and discuss the relevant difference between the two values.
The energy variance optimization algorithm over a fixed ensemble of configurations in variational Monte Carlo often encounters problems of convergence. Being formally identical to a problem of fitting data, we re-examine it from a statistical maximum-likelihood point of view. We show that the assumption of an underlying Gaussian distribution of the local energy, implicit in the standard variance minimization scheme, is not theoretically nor practically justified, and frequently generates convergence problems. We propose alternative procedures for optimization of trial wave functions in quantum Monte Carlo and successfully test them by optimizing a trial wave function for the helium trimer
Structure and energetics of small helium clusters: Quantum simulations using a recent perturbational pair potential Small helium ( 4 He) clusters containing the lighter isotope 3 He are studied by means of quantum Monte Carlo methods. Accurate ground state energies and structural properties are obtained using accurate trial wave functions and the Tang-Tonnies-Yiu ͑TTY͒ helium-helium pair potential. The dimer 4 He-3 He is not bound; as well as the trimer 4 He 3 He 2 . The smallest cluster containing 3 He is 4 He 2 3 He with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the 4 He 3 trimer, is able to bind another 3 He atom, forming the tetramer 4 He 2 3 He 2 , which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single 4 He atom in a pure cluster with a 3 He atom leads to an energetic destabilization, as the pair 4 He-3 He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining 4 He atoms, which retain the high floppiness already found in the pure clusters. As the number of atoms increases the isotopic impurity has the marked tendency to stay on the surface of the cluster. This behavior is consistent with the formation of the so-called ''Andreev states'' of a single 3 He in liquid 4 He helium and droplets, where the impurity tends to form single-particle states on the surface of the pure 4 He.
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