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arXiv:cond-mat/0407037 · Physical Review B

Efficient localized basis set for quantum Monte Carlo calculations on condensed matter

Dario Alfè, Michael J. Gillan

Published Oct 11, 2004• 155 citations
arXivHTMLPDF

Abstract

We present an efficient scheme for representing many-body wave functions in quantum Monte Carlo (QMC) calculations. The scheme is based on $B$ splines (blip functions), which consist of localized cubic splines centered on the points of a regular grid. We show that blip functions are unbiased, systematically improvable, and conveniently obtained from any standard plane-wave density functional theory (PW-DFT) code, and therefore provide a convenient and natural interface between PW-DFT and QMC calculations. We present tests on a 16-atom system of $\mathrm{Si}$ in the $\ensuremath{\beta}$-tin structure, and on 2- and 8-atom systems of $\mathrm{MgO}$ in the $\mathrm{NaCl}$ structure. We show that already with such small systems the speed-up of blip functions with respect to plane waves is between one and two order of magnitudes, without compromising the accuracy.

DOI
10.1103/physrevb.70.161101

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We present an efficient scheme for representing many-body wave functions in quantum Monte Carlo (QMC) calculations. The scheme is based on BBB splines (blip functions), which consist of localized cubic splines centered on the points of a regular grid. We show that blip functions are unbiased, systematically improvable, and conveniently obtained from any standard plane-wave density functional theory (PW-DFT) code, and therefore provide a convenient and natural interface between PW-DFT and QMC calculations. We present tests on a 16-atom system of Si\mathrm{Si}Si in the \ensuremathβ\ensuremath{\beta}\ensuremathβ-tin structure, and on 2- and 8-atom systems of MgO\mathrm{MgO}MgO in the NaCl\mathrm{NaCl}NaCl structure. We show that already with such small systems the speed-up of blip functions with respect to plane waves is between one and two order of magnitudes, without compromising the accuracy.
The scale and complexity of the quantum system to which real-space quantum Monte Carlo (QMC) can be applied in part depends on the representation and memory usage of the trial wavefunction. B-splines, the computationally most efficient basis set, can have memory requirements exceeding the capacity of a single computational node. This situation has traditionally forced a difficult choice of either using slow internode communication or a potentially less accurate but smaller basis set such as Gaussians. Here, we introduce a hybrid representation of the single particle orbitals that combine a localized atomic basis set around atomic cores and B-splines in the interstitial regions to reduce the memory usage while retaining the high speed of evaluation and either retaining or increasing overall accuracy. We present a benchmark calculation for NiO demonstrating a superior accuracy while using only one eighth of the memory required for conventional B-splines. The hybrid orbital representation therefore expands the overall range of systems that can be practically studied with QMC.
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Quantum Monte Carlo (QMC) methods have received considerable attention over the last decades due to their great promise for providing a direct solution to the many-body Schrodinger equation in electronic systems.Thanks to their low scaling with number of particles, QMC methods present a compelling competitive alternative for the accurate study of large molecular systems and solid state calculations.In spite of such promise, the method has not permeated the quantum chemistry community broadly, mainly because of the fixed-node error, which can be large and whose control is difficult.In this Perspective, we present a systematic application of large scale multi-determinant expansions in QMC, and report on its impressive performance with first row dimers and the 55 molecules of the G1 test set.We demonstrate the potential of this strategy for systematically reducing the fixed-node error in the wave function and for achieving chemical accuracy in energy predictions.When compared to traditional quantum chemistry methods like MP2, CCSD(T), and various DFT approximations, the QMC results show a marked improvement over all of them.In fact, only the explicitly-correlated CCSD(T) method with a large basis set produces more accurate results.Further developments in trial wave functions and algorithmic improvements appear promising for rendering QMC as the benchmark standard in large electronic systems.
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 . . . . . . . . . . . . . . .
to develop quantum Monte Carlo (QMC) approaches for excited states and for more systematic and accurate treatment of ground states of strongly correlated sytems and heavier elements. We use the auxiliary-field quantum Monte Carlo (AFQMC) framework, which is a many-body total energy method that looks like many coupled mean-field calculations. I will discuss this method’s relations and dierences with traditional density-functional and quantum chemistry methods, and with other forms of QMC. The AFQMC method uses importance-sampled random walks in the space of nonorthogonal Slater determinants to project out the many-body ground state, or excited state. The random walks are constrained by a trial wave function (TWF) to remove the so-called sign problem, giving approximate (but non-perturbative) results. With singledeterminant TWFs from Hartree-Fock or density-functional theory (DFT), this approach can be thought of as a post-DFT method with no parameter. In this form, the accuracy is typically comparable to the quantum chemistry coupled-cluster CCSD(T) method near equilibrium geometries, and better than CCSD(T) when bonds are stretched. The computational cost of AFQMC scales as N 3 with characteristic size of the system. Its accuracy can be further improved at the cost of increased computing (releasing the constraint) or better TWF. With AFQMC, we can systematically target selected orbitals to correlate. This allows many-body calculations in solids with a frozen core, with down-folded Hamiltonians, or possibly embeded in a DFT calculations to extend scales. Results from a few on-going applications will be presented, including band structures in simple solids, magnetic phases in MnO, and Cobalt adsoption on graphene.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.