Accurate Trial Wave Functions Research Articles

Benchmark Phaseless Auxiliary-Field Quantum Monte Carlo Method for Small Molecules.

We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO2, and O2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

Optimal series expansion method for bound states of quantum systems

An optimal series expansion method is developed to improve variational wave functions for the bound state calculation of quantum systems. With one variational parameter residing in the exponential decay factor, the finite power series expansion of the remaining factor in the trial wave function is determined with the criterion that the Schrödinger equation be satisfied as well as possible. In this way, more and more accurate trial wave functions can be obtained without undesirably increasing the number of variational parameters. Thus, instead of the standard variational principle of minimizing the energy expectation value, we utilize an alternative variational principle with which one minimizes the deviation of the trial wave function from being a true eigenstate of the Hamiltonian. The variational principle applies to both the ground state and the excited states. We then employ the optimal series expansion method to calculate the bound states of several quantum systems, including the one-dimensional double well potential, the Morse oscillator, the two-dimensional double well potential, and the vibrational states in the lowest electronic surface of methyl iodide. Computational results indicate that this method can yield excellent results for both energy eigenvalues and eigenfunctions. Therefore, the optimal series expansion method provides an alternative approach to the bound state calculation of quantum systems.

Taming the Sign Problem in Auxiliary-Field Quantum Monte Carlo Using Accurate Wave Functions.

We explore different ways of incorporating accurate trial wave functions into free projection auxiliary-field quantum Monte Carlo (fp-AFQMC). States employed include coupled-cluster singles and doubles, multi-Slater, and symmetry-projected mean-field wave functions. We adapt a recently proposed fast multi-Slater local energy evaluation algorithm for fp-AFQMC, making the use of long expansions from selected configuration interaction methods feasible. We demonstrate how these wave functions serve to mitigate the sign problem and accelerate convergence in quantum chemical problems, allowing the application of fp-AFQMC to systems of substantial sizes. Our calculations on the widely studied model Cu2O22+ system show that many previously reported isomerization energies differ substantially from the near-exact fp-AFQMC value.

Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines.

The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground-state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground-state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.

Ground-state calculations of confined hydrogen molecule H2 using variational Monte Carlo method

ABSTRACTThe variational Monte Carlo method is used to evaluate the ground-state energy of a confined hydrogen molecule H2. Accordingly, we considered the.me case of hydrogen molecule confined by a hard prolate spheroidal cavity when the nuclear positions are clamped at the foci (on-focus case). Also, the case of off-focus nuclei in which the two nuclei are not clamped to the foci is studied. This case provides flexibility for the treatment of the molecular properties by selecting an arbitrary size and shape for the confining spheroidal box. A simple chemical analysis concerning the catalytic role of enzyme is investigated. An accurate trial wave function depending on many variational parameters is used for this purpose. The obtained results for the case of clamped foci exhibit good accuracy compared with the high precision variational data presented previously. In the case of off-focus nuclei, an improvement is obtained with respect to the most recent uncorrelated results existing in the literature.

Ground-state properties of 4He and 16O extrapolated from lattice QCD with pionless EFT

We extend the prediction range of Pionless Effective Field Theory with an analysis of the ground state of 16O in leading order. To renormalize the theory, we use as input both experimental data and lattice QCD predictions of nuclear observables, which probe the sensitivity of nuclei to increased quark masses. The nuclear many-body Schrödinger equation is solved with the Auxiliary Field Diffusion Monte Carlo method. For the first time in a nuclear quantum Monte Carlo calculation, a linear optimization procedure, which allows us to devise an accurate trial wave function with a large number of variational parameters, is adopted. The method yields a binding energy of 4He which is in good agreement with experiment at physical pion mass and with lattice calculations at larger pion masses. At leading order we do not find any evidence of a 16O state which is stable against breakup into four 4He, although higher-order terms could bind 16O.

Ground state calculations of the confined molecular ions H2+ and HeH++ using variational Monte Carlo method

The ground state energy of hydrogen molecular ion [Formula: see text] confined by a hard prolate spheroidal cavity is calculated. The case in which the nuclear positions are clamped at the foci is considered. Our calculations are based on using the variational Monte Carlo method with an accurate trial wave function depending on many variational parameters. The results were extended to also include the HeH++ molecular ion. The obtained results are in good agreement with the most recent results.

Ground states of the hydrogen molecule and its molecular ion in the presence of a magnetic field using the variational Monte Carlo method

ABSTRACTBy using the variational Monte Carlo (VMC) method, we calculated the 1sσg-state energies, the dissociation energies, and the binding energies of the hydrogen molecule and its molecular ion in the presence of an aligned magnetic field regime between 0 and 10 a.u. The present calculations are based on using two types of compact and accurate trial wave functions, which are put forward for consideration in calculating energies in the absence of a magnetic field. The obtained results are compared with the most recent accurate values. We conclude that the applications of the VMC method can be successfully extended to cover the case of molecules under the effect of a magnetic field.

Ground States of Helium Atom and Hydrogen Negative Ion in the Presence of Magnetic Field Using Variational Monte Carlo Technique

Ground States of Helium Atom and Hydrogen Negative Ion in the Presence of Magnetic Field Using Variational Monte Carlo Technique S.B. Doma, M O. Shaker, a M. Farag and F.N. El-Gammal Mathematics Department, Faculty of Science, Alexandria University, Alexandria, Egypt Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt Mathematics Department, Faculty of Science, Meno a University, Shebin El-Kom, Egypt (Received January 29, 2014; in nal form June 15, 2014) Using variational Monte Carlo method, we calculated the 10 state energies, the derivatives of the total energy and the ionisation energies of the helium atom, and hydrogen negative ion in the presence of magnetic eld regime between 0 a.u. and 10 a.u. Our calculations are based on using two types of compact and accurate trial wave functions used before to calculate energies in the absence of magnetic eld. Our results are in good agreement with the most recent previous accurate values and also with the exact values. DOI: 10.12693/APhysPolA.126.700

Delocalized Davydov D1 Ansatz for the Holstein polaron

An efficient, yet very accurate trial wave function, constructed from projecting the well-known Davydov D1 Ansatz onto momentum eigenstates, is employed to study the ground state properties of the generalized Holstein Hamiltonian with simultaneous diagonal and off-diagonal coupling. Ground-state energies have been obtained with a precision matching that of the computationally much more demanding density-matrix renormalization group method. The delocalized D1 Ansatz lowers the ground-state energies at the Brillouin zone boundary significantly compared with the Toyozawa and Global-Local Ansätze in the weak coupling regime, while considerable improvement is demonstrated to have been achieved over the entire Brillouin zone in the strong coupling regime. Unique solutions are obtained with the delocalized D1 for different initial conditions when the transfer integral is 20 times the phonon frequency at the zone center, implying the absence of any self-trapping discontinuity. The scaled correlation variance is found to fit satisfactorily well with the predictions of the perturbation theories.

Atomic Properties of the Two-Electron System using Variational Monte Carlo Technique

Using variational Monte Carlo method we calculated the lowest order relativistic corrections for the ground state energies of the helium-like atoms, up to Z = 10, and also for some excited state energies of the helium atom. These relativistic corrections include: mass velocity e ect, orbit orbit interaction, spin magnetic and dipole moments of the two electrons and the Darwin e ect. Moreover, correction due to the nucleus motion has been also calculated. Our results were obtained by using two new types of compact and accurate trial wave functions for the helium ions. For excited states we used simple trial wave functions of good quality and accurate results. The obtained results are in good agreement with the most recent previous accurate values and also with the exact values.

Application of variational Monte Carlo method to the confined helium atom

A new application of variational Monte Carlo method is presented to study the helium atom under the compression effect of a spherical box with radius (rc). The ground-state energies of the helium atom were calculated for different values of rc. Our calculations were extended to include Li + and Be 2+ ions. The calculations were based on the use of a compact accurate trial wave function with five variational parameters. To optimize variational parameters, we used the steepest descent method. The obtained results are in good agreement with previous results.

Anomaly-free singularities in the generalized Kohn variational method

We have carried out an analysis of singularities in Kohn variational calculations for low-energy (e+–H2) elastic scattering. Provided that a sufficiently accurate trial wavefunction is used, we argue that our implementation of the Kohn variational principle necessarily gives rise to singularities which are not spurious. We propose two approaches for optimizing a free parameter of the trial wavefunction in order to avoid anomalous behavior in scattering phase shift calculations, the first of which is based on the existence of such singularities. The second approach is a more conventional optimization of the generalized Kohn method. Close agreement is observed between the results of the two optimization schemes; further, they give results which are seen to be effectively equivalent to those obtained with the complex Kohn method. The advantage of the first optimization scheme is that it does not require an explicit solution of the Kohn equations to be found. We give examples of anomalies which cannot be avoided using either optimization scheme but show that it is possible to avoid these anomalies by considering variations in the nonlinear parameters of the trial function.

Chemical Solutions in a Quantum Solvent: Anionic Electrolytes in 4He Nanodroplets

Variational and diffusion Monte Carlo (VMC and DMC) calculations are presented for anionic electrolytes solvated in (4)He. The electrolytes have the general structure X(-)(He)(N), with X=F, Cl, Br and I, and N varying up to 40 (41 for I(-)). The overall interaction potential is obtained from accurate ab initio data for the two-body components and then using the sum-of-potentials approximation. Our computational scheme is a robust procedure, giving us accurate trial wavefunctions that can be used to perform high-quality DMC calculations. The results indicate very marked delocalization and permanence of the liquid-like quantum features of the solvent adatoms surrounding the anionic impurities. This finding stands in contrast to the more structured, solid-like behavior of the quantum solutions with alkali metal cations embedded in He nanodroplets. While other negatively charged species such as H(-) have shown an overall repulsive interaction with He, the present calculations clearly indicate that the halogen anions remain solvated within liquid-like solvent "bubbles" of species-dependent size.

A self-consistent method for the generation of configuration interaction coefficients using variational Monte Carlo

We have developed a new method for calculating configuration interaction coefficients for trial wavefunctions used in quantum Monte Carlo calculations of molecular structure. These numerical calculations can be carried out with optimized Jastrow functions included in the wavefunction. These calculations produce coefficients different from those obtained through methods using analytical integration without the Jastrow functions and lead to more accurate trial wavefunctions. We tested the method on the beryllium atom and found that the VMC energy obtained with improved coefficients (—14.6615 hartrees) was 0.9 millihartrees lower than the energy obtained using coefficients from analytical calculations (—14.6606 hartrees). This energy difference corresponds to about 1% of the correlation energy.

Quantum Monte Carlo investigation of small He4 clusters with a He3 impurity

Small helium (4He) clusters containing the lighter isotope He3 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 (TTY) helium–helium pair potential. The dimer He4–3He is not bound; as well as the trimer He4He23. The smallest cluster containing He3 is He24He3 with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the He34 trimer, is able to bind another He3 atom, forming the tetramer He24He23, which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single He4 atom in a pure cluster with a He3 atom leads to an energetic destabilization, as the pair He4–3He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining He4 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 He3 in liquid He4 helium and droplets, where the impurity tends to form single-particle states on the surface of the pure He4.