Gaussian Wave Functions Research Articles

Neural network variational Monte Carlo for positronic chemistry

Quantum chemical calculations of the ground-state properties of positron-molecule complexes are challenging. The main difficulty lies in employing an appropriate basis set for representing the coalescence between electrons and a positron. Here, we tackle this problem with the recently developed Fermionic neural network (FermiNet) wavefunction, which does not depend on a basis set. We find that FermiNet produces highly accurate, in some cases state-of-the-art, ground-state energies across a range of atoms and small molecules with a wide variety of qualitatively distinct positron binding characteristics. We calculate the binding energy of the challenging non-polar benzene molecule, finding good agreement with the experimental value, and obtain annihilation rates which compare favourably with those obtained with explicitly correlated Gaussian wavefunctions. Our results demonstrate a generic advantage of neural network wavefunction-based methods and broaden their applicability to systems beyond the standard molecular Hamiltonian.

Decoherence in neutrino oscillation between 3D Gaussian wave packets

There is renewed attention to whether we can observe the decoherence effect in neutrino oscillation due to the separation of wave packets with different masses in near-future experiments. As a contribution to this endeavor, we extend the existing formulation based on a single 1D Gaussian wave function to an amplitude between two distinct 3D Gaussian wave packets, corresponding to the neutrinos being produced and detected, with different central momenta and spacetime positions and with different widths. We find that the spatial widths-squared for the production and detection appear additively in the (de)coherence length and in the localization factor for governing the propagation of the wave packet, whereas they appear as the reduced one (inverse of the sum of inverse) in the momentum conservation factor. The overall probability is governed by the ratio of the reduced to the sum.

Time evolution of a decaying quantum state: Evaluation and onset of nonexponential decay

The method developed by van Dijk, Nogami, and Toyama for obtaining the time-evolved wave function of a decaying quantum system is generalized to potentials and initial wave functions of noncompact support. The long time asymptotic behavior is extracted and employed to predict the timescale for the onset of nonexponential decay. The method is illustrated with a Gaussian initial wave function leaking through Eckart's potential barrier on the half-line.

Anharmonicity-induced phase transition of spin–orbit coupled Bose–Einstein condensates

In the mean-field framework, using variational analysis and numerical simulation, we investigate the effect of anharmonic trap and atomic interaction on the ground-state phases of spin-orbit (SO) coupled Bose–Einstein condensates (BECs) in the harmonic plus quartic potential. Then, the Gaussian wave function is selected to predict the analytical conditions of the phase transition boundary of the SO coupled BECs by using the variational method. We found that the anharmonicity of the external potential induces the SO coupled BECs to undergo a phase transition between the zero-momentum phase and plane-wave phase, which is more pronounced in the cases of weak harmonic potential or strong interspecies interaction. Since the potential energy of the system modified by anharmonicity competes with other energies of the system, the anharmonicity changes the critical SO coupling strength and Raman coupling strength when the phase transition occurs. At the same time, the critical anharmonic coefficients are also affected by interspecies interaction and harmonic potential. Finally, the correctness of the theoretical results is verified by numerical simulation of the Gross-Pitaevskii equation.

OpenMP Fortran programs for solving the time-dependent dipolar Gross-Pitaevskii equation

OpenMP Fortran programs for solving the time-dependent dipolar Gross-Pitaevskii equation

Effect of Wigner rotation on estimating the unitary-shift parameter of a relativistic spin-1/2 particle

We obtain the accuracy limit for estimating the expectation value of the position of a relativistic particle for an observer moving in one direction at a constant velocity. We use a specific model of a relativistic spin-1/2 particle described by a Gaussian wave function with a spin down in the rest frame. To derive the state vector of the particle for the moving observer, we use the Wigner rotation that entangles the spin and the momentum of the particle. Based on this wave function for the moving frame, we obtain the symmetric logarithmic derivative (SLD) Cram\'er-Rao bound that sets the estimation accuracy limit for an arbitrary moving observer. It is shown that estimation accuracy decreases monotonically in the velocity of the observer when the moving observer does not measure the spin degree of freedom. This implies that the estimation accuracy limit worsens with the observer's increasing velocity, but it is finite even in the relativistic limit. We derive the amount of this information loss by the exact calculation of the SLD Fisher information matrix in an arbitrary moving frame.

Elastic Form Factors and Matter Density Distributions of Some Neutron-Rich Nuclei

The ground-state properties of exotic 18N and 20F nuclei, including the neutron, proton and matter densities and related radii are investigated using the two-body model of within Gaussian (GS) and Woods Saxon (WS) wave functions. The long tail is evident in the computed neutron and matter densities of these nuclei. The plane wave Born approximation (PWBA) is calculate the elastic form factors of these exotic nuclei. The variation in the proton density distributions due to the presence of the extra neutrons in 18N and 20F leads to a major difference between the elastic form factors of these exotic nuclei and their stable isotopes 14N and 19F. The reaction cross sections for these nuclei are investigated using the Kox and Glauber models. Furthermore, the Glauber model is employed to calculate the matter radii of these exotic nuclei. The calculated results for the selected exotic nuclei are in a good agreement with the experimental data.

Quantum assimilation-based data augmentation for state of health prediction of lithium-ion batteries with peculiar degradation paths

Lithium-ion batteries with more rapid capacity loss or “peculiar degradation paths” are usually hard to completely avoid in production given complex electrochemical systems and diverse failure mechanisms. These “abnormal” batteries typically account for only a small proportion in production, and their degradation paths may be very different from most “normal” batteries in the existing dataset. Therefore, it is difficult to accurately predict their subsequent capacity degradation trends using models trained by the existing dataset. To solve this problem, this paper proposes a novel data augmentation technique based on quantum assimilation to accurately predict the state of health of lithium-ion batteries with peculiar degradation paths. The quantum assimilation algorithm has the outstanding advantage of revealing the distribution of the samples by constructing the potential energy surface of the existing data with a nonlinear Gaussian wave function, which provides a novel feature space for data augmentation. The early-cycle data of newly emerged abnormal samples can also be located on this potential energy surface, and the possible degradation increment of each subsequent cycle can be inferred based on the principle that particles tend to gather at the position with the lowest potential energy. Based on this, possible degradation paths of abnormal samples can be generated to enhance the training dataset and the machine learning model trained using it can better predict the abnormal lithium-ion batteries. The proposed method is verified on an actual lithium-ion battery dataset with mean percentage error of the prediction results less than 0.5%, which is at least 44% lower than that of the other four conventional methods.

Variational polaron equations applied to the anisotropic Fröhlich model

Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic non-degenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\Gamma$-point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.

Finite-volume pionless effective field theory for few-nucleon systems with differentiable programming

Finite-volume pionless effective field theory provides an efficient framework for the extrapolation of nuclear spectra and matrix elements calculated at finite volume in lattice QCD to infinite volume, and to nuclei with larger atomic number. In this work, it is demonstrated how this framework may be implemented via a set of correlated Gaussian wave functions optimized using differentiable programming and via solution of a generalized eigenvalue problem. This approach is shown to be significantly more efficient than a stochastic implementation of the variational method based on the same form of correlated Gaussian wave functions, yielding comparably accurate representations of the ground-state wave functions with an order of magnitude fewer terms. The efficiency of representation allows such calculations to be extended to larger systems than in previous work. The method is demonstrated through calculations of the binding energies of nuclei with atomic number $A\ensuremath{\in}{2,3,4}$ in finite volume, matched to lattice QCD calculations at quark masses corresponding to ${m}_{\ensuremath{\pi}}=806\text{ }\text{ }\mathrm{MeV}$, and infinite-volume effective field theory calculations of $A\ensuremath{\in}{2,3,4,5,6}$ systems based on this matching.

Matter Density Distributions and Reaction Cross Sections for 8Li and 22N Exotic Nuclei

The Harmonic Oscillator (HO) and Gaussian (GS) wave functions within the Binary Cluster Model (BCM) were employed to investigate neutron, proton and matter densities of the ground state as well as the elastic proton form factors of one neutron 8Li and 22N halo nuclei. The long tail is a property that is clearly shown in the neutron density. The existence of a long tail in the neutron densities of 8Li and 22N indicates that these nuclei have a neutron halo structure. Moreover, the matter rms radii and the reaction cross section of these nuclei were calculated using the Glauber model.

Two-body problems in the orthogonal condition model

In this work we investigate the 0+, 2+, 4+, 6+, 8+ and 10+resonant states of the + system and each advantage of Gaussian and Harmonic Oscillator basis wave functions in the complex scaled orthogonal condition model (CSOCM).

General memory kernels and further corrections to the variational path integral approach for the Bogoliubov-Fröhlich Hamiltonian

The celebrated variational path integral approach to the polaron problem shows remarkable discrepancies with diagrammatic Monte Carlo for the Bogoliubov-Fr\"{o}hlich Hamiltonian which describes an impurity weakly coupled to a Bose condensed atomic gas. It has been shown both via a renormalization group approach and by the method of correlated Gaussian wavefunctions that the model has a subtle UV divergence caused by quantum fluctuations, which are not captured within Feynman's approach. In this work we address the issues with Feynman's approach and show that by extending the model action to a more general form, and by considering higher order corrections beyond the Jensen-Feynman inequality, a good agreement with diagrammatic Monte Carlo can be obtained.

Bottomonium spectroscopy using Coulomb plus linear (Cornell) potential

The Coulomb plus linear (Cornell) potential is used to investigate the mass spectrum of bottomonium. Gaussian wave function is used in position and momentum space to estimate values of potential and kinetic energies, respectively. Based on our calculations, we study newly observed $${\varUpsilon {(10860)}}$$ as an admixture of $$4^{3}S_{1}$$ with $$4^{3}D_{1}$$ and $${\varUpsilon {(10753)}}$$ as an admixture of $$6^{3}S_{1}$$ with $$4^{3}D_{1}$$ ; also, we try to assign $${{\varvec{\Upsilon }}{(11020)}}$$ as a pure $$4^{3}D_{1}$$ bottomonium state. We also study the Regge trajectories in the $$(J,M^{2})$$ and $$(n_r,M^{2})$$ planes to help prove our association. We estimate the pseudoscalar and vector decay constants, the radiative (electric and magnetic dipole) transition rates, and the annihilation decay width for bottomonium states.

Production of light (anti)nuclei in pp collisions at sqrt{s} = 13 TeV

Understanding the production mechanism of light (anti)nuclei is one of the key challenges of nuclear physics and has important consequences for astrophysics, since it provides an input for indirect dark-matter searches in space. In this paper, the latest results about the production of light (anti)nuclei in pp collisions at sqrt{s} = 13 TeV are presented, focusing on the comparison with the predictions of coalescence and thermal models. For the first time, the coalescence parameters B2 for deuterons and B3 for helions are compared with parameter-free theoretical predictions that are directly constrained by the femtoscopic measurement of the source radius in the same event class. A fair description of the data with a Gaussian wave function is observed for both deuteron and helion, supporting the coalescence mechanism for the production of light (anti)nuclei in pp collisions. This method paves the way for future investigations of the internal structure of more complex nuclear clusters, including the hypertriton.

Study of Positron Impact Scattering from Methane and Silane Using an Analytically Obtained Static Potential with Correlation Polarization

A detailed study of positron impact elastic scattering from methane and silane is carried out using a model potential consisting of static and polarization potentials. The static potential for the molecular target is obtained analytically by using accurate Gaussian molecular wavefunctions. The molecular orbitals are expressed as a linear combination of Gaussian atomic orbitals. Along with the analytically obtained static potential, a correlation polarization potential is also added to construct the model potential. Utilizing the model potential, the Schrödinger equation is solved using the partial wave phase shift analysis method, and the scattering amplitude is obtained in terms of the phase shifts. Thereafter, the differential, integrated and total cross sections are calculated. These cross-section results are compared with the previously reported measurements and theoretical calculations.

The Influence of the Symmetry of Identical Particles on Flight Times.

In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains as well as in the scattering by a potential -barrier. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two sets of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a ‘screen’ is calculated in each case from the density and flux. Fermions display broader distributions as compared with either distinguishable particles or bosons, leading to earlier and later arrivals for all the cases analyzed here. The symmetry of the wave function seems to speed up or slow down the propagation of particles. Due to the cross terms, certain initial conditions lead to bimodality in the fermionic case. Within the nonrelativistic domain, and when the short-time survival probability is analyzed, if the cross term becomes important, one finds that the decay of the overlap of fermions is faster than for distinguishable particles which in turn is faster than for bosons. These results are of interest in the short time limit since they imply that the well-known quantum Zeno effect would be stronger for bosons than for fermions. Fermions also arrive earlier and later than bosons when they are scattered by a -barrier. Although the particle symmetry does affect the mean tunneling flight time, in the limit of narrow in momentum initial Gaussian wave functions, the mean times are not affected by symmetry but tend to the phase time for distinguishable particles.

Study of the matter density distributions of halo nuclei 6He and 16C using the binary cluster model

The harmonic oscillator (HO) and Gaussian (GS) wave functions within the binary cluster model (BCM) have been employ to investigate the ground state neutron, proton and matter densities as well as the elastic form factors of two-neutron 6He and 16C halo nuclei. The long tail is a property that is clearly revealed in the density of the neutrons since it is found in halo orbits. The existence of a long tail in the neutron density distributions of 6He and 16C indicating that these nuclei have a neutron halo structure. Moreover, the matter rms radii and the reaction cross section (σr) of these nuclei have been calculated using the Glauber model.

Relativistic and QED corrections for the ground state lithiumlike ionization energies

The ground state ionization energies of Z ⩽ 10 lithiumlike ions are calculated using fully correlated Gaussian wavefunctions. Leading-order relativistic corrections are evaluated, while QED corrections are established with small uncertainties by directly calculating the Araki–Sucher energy and expanding the three-electron Bethe logarithm in 1/Z. The non-relativistic α6 level shifts have also been calculated, and we have used these energies to recommend ionization energies, which include estimates of the influence of the relativistic portion of the α6 energy. The results emphasize the importance of the direct computation of the complete α6 correction, but also the need for new, higher accuracy experimental ionization limits.

A new approach to study electron and positron scattering from acetylene

Studies on electron and positron scattering from acetylene are very important for their applications in the plasma and astronomical environments. In view of such importance, we have performed the cross section calculations for electron and positron scattering from acetylene molecule using the optical model potential method with an analytically obtained static potential. In the present study, the charge cloud of acetylene is represented through multi-center Gaussian wavefunctions and using the same, static potential seen by the projectile electron/positron is evaluated analytically with a new approach. A complex optical potential consisting of our analytical static potential with the appropriate exchange, polarization and absorption potentials has been obtained. Using this complex potential in the Schrodinger equation and utilizing partial wave analysis technique, the scattering phase shifts are evaluated for the electron and positron collisions with the acetylene molecule. Differential, integral, momentum transfer, absorption and total cross sections have been calculated for the incident projectile energies in the range 10−500 eV. The obtained results are compared with the previous available theoretical and experimental results, which show good agreement and thus demonstrate the applicability of our method.