Variational Monte Carlo Techniques Research Articles

Distortion effects and clustering of isotopic impurities in solid molecularpara-hydrogen from variational Monte Carlo simulations with shadow wave functions

We employed a fully optimized Shadow Wave Function (SWF) in combination with Variational Monte Carlo techniques to investigate the properties of HD molecules and molecular ortho-deuterium (o-D_2) in bulk solid para-hydrogen (p-H_2). Calculations were performed for different concentrations of impurities ranging from about 1% to 25% at the equilibrium density for the para-hydrogen crystal. By computing the excess energy both for clustered and isolated impurities we tried to determine a limit for the solubility of HD and o-D_2 in p-H_2.

Gutzwiller–RVB theory of high-temperature superconductivity: Results from renormalized mean-field theory and variational Monte Carlo calculations

We review the resonating valence bond (RVB) theory of high-temperature superconductivity using Gutzwiller projected wave functions that incorporate strong correlations. After a general overview of the phenomenon of high-temperature superconductivity, we discuss Anderson's RVB picture and its implementation by renormalized mean-field theory (RMFT) and variational Monte Carlo (VMC) techniques. We review RMFT and VMC results with an emphasis on recent developments in extending VMC and RMFT techniques to excited states. We compare results obtained from these methods with angle-resolved photoemission spectroscopy (ARPES) and scanning tunnelling microscopy (STM). We conclude by summarizing recent successes of this approach and discuss open problems that need to be solved for a consistent and complete description of high-temperature superconductivity using Gutzwiller projected wave functions.

Improved wave functions for Hubbard model: Superconductivity and Mott transition

We extend the doublon–holon binding wave function, which was recently often used to consider Mott transitions and dx2-y2-wave superconductivity, for the half-filled-band Hubbard model on square lattices with the diagonal hopping t′. We introduce two new features: (1) In the doublon–holon binding factor, more accurate configuration correlations are included. (2) The band renormalization effect owing to the electron correlation is introduced within the third-neighbor hopping, including the anisotropy between x- and y-directions in the nearest-neighbor hopping. Using an optimization variational Monte Carlo technique, we draw some definite answers to the topics concerning the above features.

Quantum Monte Carlo method using a stochastic Poisson solver

The quantum Monte Carlo (QMC) technique is an extremely powerful method to treat many-body systems. Usually the quantum Monte Carlo method has been applied in cases where the interaction potential has a simple analytic form, like the 1/r Coulomb potential. However, in a complicated environment as in a semiconductor heterostructure, the evaluation of the interaction itself becomes a nontrivial problem. Obtaining the potential from any grid-based finite-difference method for every walker and every step is infeasible. We demonstrate an alternative approach of solving the Poisson equation by a classical Monte Carlo calculation within the overall quantum Monte Carlo scheme. We have developed a modified "walk on spheres" algorithm using Green's function techniques, which can efficiently account for the interaction energy of walker configurations, typical of quantum Monte Carlo algorithms. This stochastically obtained potential can be easily incorporated with variational, diffusion, and other Monte Carlo techniques. We demonstrate the validity of this method by studying a simple problem, the polarization of a helium atom in the electric field of an infinite capacitor.

Variational Monte Carlo study of the current carried by a quasiparticle

With the use of Gutzwiller-projected variational states, we study the renormalization of the current carried by the quasiparticles in high-temperature superconductors and of the quasiparticle spectral weight. The renormalization coefficients are computed by the variational Monte Carlo technique, under the assumption that quasiparticle excitations may be described by Gutzwiller-projected BCS quasiparticles. We find that the current renormalization coefficient decreases with decreasing doping and tends to zero at zero doping. The quasiparticle spectral weight Z_+ for adding an electron shows an interesting structure in k space, which corresponds to a depression of the occupation number k just outside the Fermi surface. The perturbative corrections to those quantities in the Hubbard model are also discussed.

Vibrational–rotational energies of all H2 isotopomers using Monte Carlo methods

AbstractUsing variational Monte Carlo techniques, we have computed several of the lowest rotational–vibrational energies of all the hydrogen molecule isotopomers (H2, HD, HT, D2, DT, and T2). These calculations do not require the excited states to be explicitly orthogonalized. We have examined both the usual Gaussian wave function form as well as a rapidly convergent Padé form. The high‐quality potential energy surfaces used in these calculations are taken from our earlier work and include the Born–Oppenheimer energy, the diagonal correction to the Born–Oppenheimer approximation, and the lowest‐order relativistic corrections at 24 internuclear points. Our energies are in good agreement with those determined by other methods. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Spectroscopic constants of [formula omitted] using Monte Carlo methods

We use variational Monte Carlo techniques to calculate molecular energies at 26 internuclear distances as well as several energy derivatives at the equilibrium of the hydrogen molecular ion. With these values we compare three methods of computing four spectroscopic constants of this system and discuss their applicability to larger systems. Our best results for this molecule are obtained from a fit to the potential energy surface: ω e = 2323.98(8) cm −1, ω e x e = 67.3(2) cm −1, α e = 1.597(2) cm −1 and B e = 29.9626(2) cm −1.

Wave function optimization in the variational Monte Carlo method

An appropriate iterative scheme for the minimization of the energy, based on the variational Monte Carlo (VMC) technique, is introduced and compared with existing stochastic schemes. We test the various methods for the one-dimensional Heisenberg ring and the two-dimensional $t\text{\ensuremath{-}}J$ model and show that, with the present scheme, very accurate and efficient calculations are possible, even for several variational parameters. Indeed, by using a very efficient statistical evaluation of the first and the second energy derivatives, it is possible to define a very rapidly converging iterative scheme that, within VMC, is much more convenient than the standard Newton method. It is also shown how to optimize simultaneously both the Jastrow and the determinantal part of the wave function.

A ground state potential energy surface for H2 using Monte Carlo methods

Using variational Monte Carlo and a simple explicitly correlated wave function we have computed the Born-Oppenheimer energy of the H2 ground state (X 1Sigmag+) at 24 internuclear distances. We have also calculated the diagonal correction to the Born-Oppenheimer approximation and the lowest-order relativistic corrections at each distance using variational Monte Carlo techniques. The nonadiabatic values are evaluated from numerical derivatives of the wave function with respect to the nuclear coordinates. With this potential energy surface we have computed several of the lowest vibrational-rotational energies for this system. Our results are in good agreement with the best values found in the literature.

Spectroscopic constants of H2 using Monte Carlo methods

AbstractWe have computed the spectroscopic constants we, wexe, αe, and Be of the ground state of H2 using two methods. In the first method, a highly accurate potential energy surface was generated using explicitly correlated wave functions and variational Monte Carlo techniques. We calculated the vibrational‐rotational energies from this surface and then obtained the spectroscopic constants by fitting these energies to a simple equation. In the second method, the spectroscopic constants were determined from local derivatives of the energy with respect to the internuclear distance. These derivatives have also been calculated using variational Monte Carlo techniques. For this system we find that the first method yields spectroscopic constants that are in excellent agreement with experiment. We also find that the second method, although slightly less accurate, is extremely well suited to Monte Carlo calculations. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Variational Monte Carlo study of the ground-state properties and vacancy formation energy of solid para-H2using a shadow wave function

A Shadow Wave Function (SWF) is employed along with Variational Monte Carlo techniques to describe the ground state properties of solid molecular para-hydrogen. The study has been extended to densities below the equilibrium value, to obtain a parameterization of the SWF useful for the description of inhomogeneous phases. We also present an estimate of the vacancy formation energy as a function of the density, and discuss the importance of relaxation effects near the vacant site.

Transcorrelated method for electronic systems coupled with variational Monte Carlo calculation

A Jastrow–Slater-type wave function is often used as a trial function for precise calculations of the total energy of electronic systems, where the correlation effect is taken into account by the Jastrow factor that directly depends on the distance between electrons. Since many-body integrals are inevitable there, the calculation totally depends on Monte Carlo sampling, and so, except for very simple cases, it is very difficult to optimize one-body wave functions in the Slater determinant which determine the nodal surfaces of the total wave function. Here we propose and demonstrate that the total wave function is efficiently optimized by coupling an ordinary variational Monte Carlo (VMC) technique with the transcorrelated method, in which the one-body wave functions are definitely obtained by solving Hartree–Fock-type self-consistent-field (SCF) equations derived from the similarity-transformed Hamiltonian. It is shown that the present method reproduces about 90% of the correlation energy for helium-like two-electron systems (H−, He, Li+, and Be2+) and gives much better results than the conventional VMC method using the Hartree–Fock orbitals for a Li atom, a Be atom, and a H2 molecule. It is also shown that the orbital energy appearing in the SCF equations gives a good approximation to the ionization potential.

Variational Monte Carlo calculations ofΛ5Hehypernucleus

We perform a realistic study of ${}_{\ensuremath{\Lambda}}^{5}\mathrm{He}$ hypernucleus using variational Monte Carlo technique. The Hamiltonian for ${}^{4}\mathrm{He}$ nuclear core of the hypernucleus is written using Argonne ${v}_{18}$ $\mathrm{NN}$ potential and Urbana model-IX $\mathrm{NNN}$ potential, where N stands for nucleon. For the strange sector, we use phenomenological $\ensuremath{\Lambda}N$ potential having central, spin, and exchange components and $\ensuremath{\Lambda}\mathrm{NN}$ potential which includes spin-dependent dispersive force and two-pion exchange force. Using this Hamiltonian and a fully correlated variational wave function, we reproduce the experimental $\ensuremath{\Lambda}$ binding energy. Without three-body $\ensuremath{\Lambda}\mathrm{NN}$ potential in the Hamiltonian and its corresponding correlation in the wave function, the hypernucleus is overbound by 0.56(4) MeV, which is about a quarter of the previous reported values of 2--3 MeV due to the use of central forces. We present the detailed energy breakdown of the hypernucleus and also show the effect of $\ensuremath{\Lambda}\mathrm{NN}$ correlation on it. The one-body density profiles for nucleon and $\ensuremath{\Lambda}$ in the hypernucleus and in its nuclear core have been critically examined. The nuclear core polarization due to presence of $\ensuremath{\Lambda}$ is precisely determined.

Increasing d-wave superconductivity by on-site repulsion.

We study, through the variational Monte Carlo technique, an extended Hubbard model away from half filled band density which contains two competing nearest-neighbor interactions: a superexchange J favoring d-wave superconductivity and a repulsion V opposing it. We find that the on-site repulsion U effectively enhances the strength of J while suppressing that of V, thus favoring superconductivity. This result shows that attractions which do not involve charge fluctuations are very well equipped against strong electron-electron repulsion so much to get advantage from it.

Wigner molecules in quantum dots: A quantum Monte Carlo study

We study two-dimensional quantum dots using the variational quantum Monte Carlo technique in the weak-confinement limit where the system approaches the Wigner molecule, i.e., the classical solution of point charges in an external potential. We observe the spin-polarization of electrons followed by a smooth transition to a Wigner-molecule-like state as the confining potential is made weaker.

Radiativeα-capture cross sections from realistic nucleon-nucleon interactions and variational Monte Carlo wave functions

We report the first calculations of cross sections for the radiative capture reactions ${}^{3}\mathrm{H}(\ensuremath{\alpha},\ensuremath{\gamma}{)}^{7}\mathrm{Li}$ and ${}^{3}\mathrm{He}(\ensuremath{\alpha},\ensuremath{\gamma}{)}^{7}\mathrm{Be}$ below 2 MeV that use wave functions derived from realistic nucleon-nucleon interactions by the variational Monte Carlo technique. After examining several small corrections to the dominant $E1$ operator, we find energy dependences for the low-energy S factors that agree reasonably with experimental measurements. There is no contradiction with the previous theoretical understanding of these processes, but the zero-energy derivative of the ${}^{3}\mathrm{H}(\ensuremath{\alpha},\ensuremath{\gamma}{)}^{7}\mathrm{Li}$ S factor is smaller than that in most models. While this method can, in principle, predict cross section normalizations, the normalizations of our results are mostly too low.

Bose-Einstein condensation in trapped bosons: A variational Monte Carlo analysis

Several properties of trapped hard-sphere bosons are evaluated using variational Monte Carlo techniques. A trial wave function composed of a renormalized single-particle Gaussian and a hard-sphere Jastrow function for pair correlations is used to study the sensitivity of condensate and noncondensate properties to the hard-sphere radius and the number of particles. Special attention is given to diagonalizing the one-body density matrix and obtaining the corresponding single-particle natural orbitals and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single-particle orbital with the highest occupation. The effect of interaction on other quantities such as the ground-state energy, the mean radial displacement, and the momentum distribution is calculated as well. Results are compared with mean-field theory in the dilute limit.

Stability of the striped phase in the two-dimensional t– J model: A variational Monte Carlo study

By using a variational Monte Carlo technique, stability of the striped phases is studied for the two-dimensional t– J model with the three-site term near 1/8 hole doping. It is found that both diagonally and vertically striped phases are comparable in energy with the d x 2− y 2 -wave superconducting state in the small- J/ t region. Hole-`filled’ domain walls prefer diagonal stripes, whereas `half-filled’ domain walls vertical ones, in accordance with the situation observed in nickelates and cuprates.

Variational wave function for a two-electron quantum dot

We have applied the variational quantum Monte Carlo technique to a two-electron quantum dot. A simple variational wave function is presented and we compare the results obtained using it with exact numerical diagonalizations. The comparison shows that almost exact results are obtained for lowest energy states of different relative angular momenta.

Positron chemistry by quantum Monte Carlo. II. Ground-state of positron-polar molecule complexes

The stability of the ground-state of positron-polar molecule complexes [M,e+] has been explored for M=LiH,HF,H2O,BeO,LiF using variational and diffusion Monte Carlo techniques. Our simulations show that the ground-state of the complexes [LiH,e+]2,1Σ+, [BeO,e+]2,1Σ+, and [LiF,e+]2,1Σ+ is stable against the dissociation either in the two fragments M and e+ or in the other two fragments M+ and Ps=[e+,e−], while the ground-state of [H2O,e+]2,1A1, and of [HF,e+]2,1Σ+ has an energy equal to the dissociation threshold, M and e+. We also compare the predicted vertical positron affinity (PA) with high quality vertical electron affinity (EA) and discuss the relevant difference between the two values.