Session B23: Focus Session: Explicitly Correlated Methods and Quantum Few-Body Systems

Applications of the explicitly correlated Gaussian approach to cold few-atom systems

The explicitly correlated Gaussian basis set expansion approach has been applied to a wide range of few-body systems relevant to particle, nuclear, atomic and molecular physics. This talk summarizes our theoretical investigations of universal aspects of ultracold gaseous and liquid few-atom systems using explicitly correlated Gaussians. In the ultracold regime, where the de Broglie wave length is large compared to the range of the interaction potentials, the collisions between particles become so slow that the details of the interactions are, to leading order, neglegible. In this regime, the dynamics of few-atom systems is governed by just a few ``effective parameters,'' such as the s-wave scattering length, and largely independent of the details of the underlying two-body potentials. As a first example, we present results for harmonically trapped few-fermion systems with infinitely large interspecies s-wave scattering length consisting of up to ten particles. As a second example, we investigate the behavior of weakly-bound Bose droplets with a light impurity and elucidate how the properties of these droplets are related to the three-body Efimov effect in heavy-heavy-light trimers.   

Very accurate variational non-relativistic non-Born-Oppenheimer atomic \& molecular spectra predictions employing explicitly correlated Gaussian basis functions

Due to the fast increasing capabilities of modern computers it now becomes feasible to calculate spectra of small atom and molecules with accuracy which matches the accuracy of high-resolution measurements. The algorithms for the calculations are directly derived from the first principles of quantum mechanics. The Hamiltonian operator used in the approach is called the internal Hamiltonian and is obtained by rigorously separating out the center-of-mass motion from the laboratory-frame Hamiltonian. Algorithms for determining the isotopic energy shifts of L=0 and M=0 states of atoms were implemented and tested in the calculations of the ground $^4$S state of the nitrogen atom. Bound states of diatomic molecules corresponding to the total angular momentum quantum number equal to one (N=1) was derived and implemented and was tested in the calculations of the N=1, v=0, . . . , 22 states of the HD$^+$ ion and in the calculations of the ortho-para spin isomerization of the hydrogen molecule in its all bound vibrational states. This has lead to the development of a new studying of muonic molecules (dp$\mu$, tp$\mu$ and td$\mu$). The algorithms for calculating rovibrational states of small molecules is currently being extended to H$_3^+$ using $sin$ and $cos$ ECGs.   

Challenges and advances in calculations of highly non-adiabatic systems employing the explicitly correlated Gaussian functions

Accurate calculations of the highly non-adiabatic systems have been drawing attention for several decades. While most accurate modern methods allow outstanding accuracy, they are limited to only three or four particles. In the current work we have developed a highly accurate method which is not limited by the number of particles in the system but is only limited by the computer resource available. Examples of calculations of exotic systems with\footnote{N. Kirnosov, K. L. Sharkey and L. Adamowicz, submitted to J.~Chem.~Phys.} and without\footnote{D. B. Kinghorn and L. Adamowicz, J.~Chem.~Phys. {\bf 110}, 7166 (1999).} rotational excitation and of conventional electronic molecules\footnote{K. L. Sharkey, N. Kirnosov and L. Adamowicz, J.~Chem.~Phys. {\bf 139}, 164119 (2013).}$^,$\footnote{K. A. Jones, N. Kirnosov, K. L. Sharkey and L. Adamowicz, submitted to J.~Chem.~Phys.} are presented and the future development of the method is discussed.   

High precision variational calculations of five-electron systems: S-states of boron

We have performed benchmark variational calculations of the lowest two S-states of the boron atom. The spatial part of the wave function has been expanded in terms of all-particle explicitly correlated Gaussians, whose nonlinear variational parameters were extensively optimized. We have also computed leading relativistic corrections and various expectation values for both states. This work demonstrates that the level of accuracy achievable in calculations of five-electron atoms is now approaching the one previously seen only in three- or four-electron systems.   

Calculation of energies of three-electron systems in a strong magnetic field using Explicitly Correlated Gaussian Basis

Strong magnetic fields can significantly alter the properties of atoms and allow the formation of stable negative ions such as He$^-$. We have calculated the energies of systems comprised of three electrons in the presence of strong magnetic fields by using the Stochastic Variational Method with deformed Explicitly Correlated Gaussian basis.\footnote{J. A. Salas and K. Varga, \textbf{Phys. Rev. A} 89, 052501 (2014)} This approach yields accurate values for three-electron systems and predicts that the He$^-$ ion in a strong magnetic field has stable states, within the non-relativistic framework, in the infinite nuclear mass approximation. The energy spectrum and the properties of three-electron systems as a function of the strength of the magnetic field show the effect of the rivalry between the Coulomb interaction and the magnetic confinement.   

High-order Path Integral Monte Carlo methods for solving strongly correlated fermion problems

In solving for the ground state of a strongly correlated many-fermion system, the conventional second-order Path Integral Monte Carlo method is plagued with the sign problem. This is due to the large number of anti-symmetric free fermion propagators that are needed to extract the square of the ground state wave function at large imaginary time. In this work, I show that optimized fourth-order Path Integral Monte Carlo methods, which uses no more than 5 free-fermion propagators, in conjunction with the use of the Hamiltonian energy estimator, can yield accurate ground state energies for quantum dots with up to 20 polarized electrons. The correlations are directly built-in and no explicit wave functions are needed.   

Beyond the Born-Oppenheimer approximation with quantum Monte Carlo

We develop tools that enable the study of non-adiabatic effects with variational and diffusion Monte Carlo methods. We introduce a highly accurate wave function ansatz for electron-ion systems that can involve a combination of both clamped ions and quantum nuclei. We explicitly calculate the ground state energies of H$_{2}$, LiH, H$_{2}$O and FHF$^{-}$ using fixed-node quantum Monte Carlo with wave function nodes that explicitly depend on the ion positions. The obtained energies implicitly include the effects arising from quantum nuclei and electron-nucleus coupling. We compare our results to the best theoretical and experimental results available and find excellent agreement.   

Physical Sputtering vs. Gas Assisted Etching of Silicon Dioxide with a Gallium Focused Ion Beam: Elucidating Experiments via Monte Carlo Simulations

In order to increase ion beam nanomachining precision and improve imaging resolution, fine tuning of the ion beam profile is absolutely necessary. To understand the effects of ion beam tails, experiments and Monte Carlo simulations were conducted with a 40 keV gallium beam with and without gas assisted chemical etching. A gallium ion beam was scanned in an area of 25x25 nm$^{2}$ on a silicon dioxide film with and without a localized XeF$_{2}$ gas at 1pA current. Four different ion doses (0.23, 0.9, 1.8 and 3.6 nC/$\mu$m$^{2}$) were experimentally considered to study the sputtered and etched via profiles. Monte Carlo simulations using EnvizION program was performed to elucidate the sputtered and gas-assisted etch process. New features including gas-assisted etching by secondary electrons and a binary collision model to dissociate the precursor molecules were introduced. Sputtered via and gas assisted etching (XeF$_{2}$ precursor gas) via profiles with various gas-assist pressures were studied to understand the experimental temporal behavior. Various contributions including sputtering from primary, forward scattered, backscattered ions as well as etching by recoiled atoms and secondary electrons will be discussed.   

Fully Exponentially Correlated Wave Functions for Few-Body Systems

Analytical methods now make practical the study of three- and four-body problems using wave functions in which all the interparticle distances (and not just their squares) occur as exponentials. This type of basis yields wave functions that exhibit superior initial convergence toward exact results and that facilitate the accurate treatment of systems in which no one of the particles is far more massive than the others. Progress in the practical use of this formulation is reviewed.   

Calculating Properties of Finite Mass Atoms

Most atomic calculations assume that the mass of the nucleus is finite. If one is interested in evaluating atomic properties to high precision then this approximation cannot be made. We have developed a simple method that includes the kinetic energy of the nucleus into atomic calculations and does not increase the time or the complexity of these calculations. Our results for a variety of properties for several different atoms will illustrate some of the advantages of this method.   

Electron shake-off and recoil following 6He beta decay

When the helium isotope $^6$He undergoes beta decay in the process $^6$He $\rightarrow$ $^6$Li $+ e^- + \bar\nu$, the atomic electrons suddenly find themselves in a $^6$Li$^+$ environment. The electrons subsequently redistribute themselves over all possible states of the $^6$Li$^+$ ion, including the continuum leading to $^6$Li$^{++}$ and $^6$Li$^{3+}$. There is currently considerable interest in studying the recoil ions in connection with experiments to look for evidence of new physics as revealed by angular correlations between the electron and the antineutrino [1]. We will present calculations employing Stieltjes imaging techniques in Hylleraas coordinates to study the probabilities for the shake-up and shake-off mechanisms, and especially the additional recoil accompanying the emission of the shake-off electrons.\\[4pt] [1] C. Couratin et al., Phys. rev. Lett. {\bf 108} 243201 (2012).   

Coulombic few-body systems in the adiabatic hyperspherical representation

We study few-body systems consisting of charged particles in free space using the adiabatic hyperspherical representation. We use a correlated Gaussian basis at a fixed hyperradius with efficiently calculated matrix elements [1] to generate the adiabatic potentials and non-adiabatic couplings as a function of the hyperradius. [1] K. M. Daily and Chris H. Greene, Phys. Rev. A 89, 012503 (2014).   

Low-energy effective interactions beyond cRPA by the functional renormalization group

In the derivation of low-energy effective models for solids targeting the bands near the Fermi level, the constrained random phase approximation (cRPA) has become an appreciated tool to compute the effective interactions. Here we present applications of a constrained functional renormalization group (cfRG) scheme to two simple multi-band systems and compare the resulting effective interactions to the cRPA. The employed wick-ordered fRG scheme generalizes the cRPA approach by including all interaction channels in an unbiased way. First we consider a multiband model for graphene, where we integrate out the $\sigma$-bands to get an effective theory for $\pi$-bands. It turns out that terms beyond cRPA are strongly suppressed by $xy$-plane reflection-symmetry of the bands and that in our model, the cRPA stays qualitatively correct even if one breaks this symmetry slightly. The second example is a model for a Cu-O-chain, where we consider an effective theory for the Cu 3d-orbital. Here the fRG data points to relevant corrections compared to the cRPA results.