We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps $τ\sim O(1)$ and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.

arXiv:1810.00864Oct 1, 20182 citations

Variational Monte Carlo

James E. Gubernatis, Naoki Kawashima, Philipp Werner

Cambridge University Press eBooksJan 31, 20161 citation

Speeding Up and Quantifying Approximation Error in Continuum Quantum Monte Carlo Solid-State Calculations

William D. Parker

Quantum theory has successfully explained the mechanics of much of the microscopic world.However, Schrödinger's equations are difficult to solve for many-particle systems.Meanfield theories such as Hartree-Fock and density functional theory account for much of the total energy of electronic systems but fail on the crucial correlation energy that predicts solid cohesion and material properties.Monte Carlo methods solve differential and integral equations with error independent of the number of dimensions in the problem.Variational Monte Carlo (VMC) applies the variational principle to optimize the wave function used in the Monte Carlo integration of Schrödinger's time-independent equation.Diffusion Monte Carlo (DMC) represents the wave function by electron configurations diffusing stochastically in imaginary time to the ground state.Approximations in VMC and DMC make the problem tractable but introduce error in parameter-controlled and uncontrolled ways.The many-electron wave function consists of single-particle orbitals.The orbitals are combined in a functional form to account for electron exchange and correlation.Plane waves are a convenient basis for the orbitals.However, plane-wave orbitals grow in evaluation cost with basis-set completeness and system size.To speed up the calculation, polynomials approximate the plane-wave sum.Four polynomial methods tested are: Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline approximation.The polynomials all increase speed by an order of the number of particles.B-spline approximation most consistently maintains accuracy in the seven systems tested.However, polynomials increase the memory needed by a factor of two to eight.B-spline approximation with a separate approximation for the Laplacian of List of TablesTable Page 5.1 DMC number of steps vs. statistical uncertainty . . . . . . . . . . . . . . .5.2 Percent VMC time evaluating plane waves in Si . . . . . . . . . . . . . . . .5.3 Orbital, energy and fluctuation error in three metals . . . . . . . . . . . . .5.4 Orbital, energy and fluctuation error in two semiconductors . . . . . . . . .5.5 Orbital, energy and fluctuation error in two insulators . . . . . . . . . . . .6.1 DFT, VMC and DMC total & formation energies vs. plane-wave cutoff . .6.2 DMC energy vs. configuration population . . . . . . . . . . . . . . . . . . .6.3 DMC energy vs. time step . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4 DMC total and formation energies at L-and Baldereschi points . . . . . . .6.5 DMC total and formation energies with finite-size corrections . . . . . . . .6.6 DMC total and formation energies vs. number of atoms . . . . . . . . . . .6.7 DMC total and formation energies with different pseudopotential methods .6.8 DFT & DMC total energies with different exchange-correlation potentials .6.9 DMC formation energies for varying Jastrow factors and pseudopotentials .6.10 DMC error summary . . . . . . . . . . . . . . .Quantum theory has successfully explained the mechanics of much of the microscopic world.However, Schrödinger's equations are difficult to solve for many-particle systems.Meanfield theories such as Hartree-Fock and density functional theory account for much of the total energy of electronic systems but fail on the crucial correlation energy that predicts solid cohesion and material properties.Monte Carlo methods solve differential and integral equations with error independent of the number of dimensions in the problem.Variational Monte Carlo (VMC) applies the variational principle to optimize the wave function used in the Monte Carlo integration of Schrödinger's time-independent equation.Diffusion Monte Carlo (DMC) represents the wave function by electron configurations diffusing stochastically in imaginary time to the ground state.Approximations in VMC and DMC make the problem tractable but introduce error in parameter-controlled and uncontrolled ways.The many-electron wave function consists of single-particle orbitals.The orbitals are combined in a functional form to account for electron exchange and correlation.Plane waves are a convenient basis for the orbitals.However, plane-wave orbitals grow in evaluation cost with basis-set completeness and system size.To speed up the calculation, polynomials approximate the plane-wave sum.Four polynomial methods tested are: Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline approximation.The polynomials all increase speed by an order of the number of particles.B-spline approximation most consistently maintains accuracy in the seven systems tested.However, polynomials increase the memory needed by a factor of two to eight.B-spline approximation with a separate approximation for the Laplacian of List of TablesTable Page 5.1 DMC number of steps vs. statistical uncertainty . . . . . . . . . . . . . . .5.2 Percent VMC time evaluating plane waves in Si . . . . . . . . . . . . . . . .5.3 Orbital, energy and fluctuation error in three metals . . . . . . . . . . . . .5.4 Orbital, energy and fluctuation error in two semiconductors . . . . . . . . .5.5 Orbital, energy and fluctuation error in two insulators . . . . . . . . . . . .6.1 DFT, VMC and DMC total & formation energies vs. plane-wave cutoff . .6.2 DMC energy vs. configuration population . . . . . . . . . . . . . . . . . . .6.3 DMC energy vs. time step . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.4 DMC total and formation energies at L-and Baldereschi points . . . . . . .6.5 DMC total and formation energies with finite-size corrections . . . . . . . .6.6 DMC total and formation energies vs. number of atoms . . . . . . . . . . .6.7 DMC total and formation energies with different pseudopotential methods .6.8 DFT & DMC total energies with different exchange-correlation potentials .6.9 DMC formation energies for varying Jastrow factors and pseudopotentials .6.10 DMC error summary . . . . . . . . . . . . . . .

Jan 1, 20100 citations

Time-Dependent Variational Monte Carlo

Federico Becca, Sandro Sorella

Cambridge University Press eBooksNov 29, 20170 citations

Variational Monte Carlo studies of atoms

Håvard Sandsdalen

Jan 1, 20102 citations

τ\sim O(1)

and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.