Communication: Fixed-node errors in quantum Monte Carlo: Interplay of electron density and node nonlinearities

Rasch, Kevin; Hu, Shuming; Mitáš, Luboš

doi:10.1063/1.4862496

Cited by 36 publications

(52 citation statements)

References 20 publications

(31 reference statements)

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“…However, unlike the present case, the magnitude of the fixed-node errors has to be addressed and it has been shown that large differences in the fixed-node errors can appear depending on the degree of node nonlinearity [91]. In the case of large fixed-node errors, the performance of the FCI-QMC method developed by Alavi and coworkers [92][93][94][95] will be investigated.…”

Section: Discussionmentioning

confidence: 99%

Generalized local-density approximation and one-dimensional finite uniform electron gases

Loos

2014

Phys. Rev. A

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We explicitly build a generalized local-density approximation (GLDA) correlation functional based on one-dimensional (1D) uniform electron gases (UEGs). The fundamental parameters of the GLDA -a generalization of the widely-known local-density approximation (LDA) used in density-functional theory (DFT) -are the electronic density ρ and a newly-defined two-electron local parameter called the hole curvature η. The UEGs considered in this study are finite versions of the conventional infinite homogeneous electron gas and consist of n electrons on an infinitely thin wire with periodic boundary conditions. We perform a comprehensive study of these finite UEGs at high, intermediate and low densities using perturbation theory and quantum Monte Carlo calculations. We show that the present GLDA functional yields accurate estimates of the correlation energy for both weakly and strongly correlated one-dimensional systems and can be easily generalized to higher-dimensional systems.

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Section: Discussionmentioning

confidence: 99%

Generalized local-density approximation and one-dimensional finite uniform electron gases

Loos

2014

Phys. Rev. A

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“…The purpose of B i s is to qualitatively reflect the expected rise of FN bias with higher bond multiplicity 47 . In linear HCN dimer, for instance, we set B 1 = 3 and B 2 = 1 since the hydrogen bond (contact) consists of triply-bound nitrogen and hydrogen.…”

Section: Definitionsmentioning

confidence: 99%

“…We thus propose to use another measure, based on δ, reflecting also the fact that, in general, FN errors depend on electron density 47 . Although this requires a deeper study, as an example, we define here an empirical measure κ that progressively depends on a maximum bond multiplicity of terminal atoms constituing noncovalent contact/s (Eq.…”

Section: Small Noncovalent Complexesmentioning

confidence: 99%

“…Such a high level of accuracy obtained with far-from-accurate SJ trial wave functions, was attributed to a high degree of FN bias cancellation operative in systems dominated by "sufficiently" weak interactions 40,43,44 , taking place in low-density [45][46][47] intermolecular regions with a predominant s character where the exchange contributions are negligible 48 . Bias cancellation limits of such an approach are however unknown, that somewhat hampers widespread usage of FN-DMC as a benchmark method.…”

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confidence: 99%

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Bias cancellation in one-determinant fixed-node diffusion Monte Carlo: Insights from fermionic occupation numbers

Dubecký

2017

Phys. Rev. E

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Accuracy of the fixed-node diffusion Monte Carlo (FN-DMC) depends on the node location of the best available trial state ΨT . The practical FN-DMC approaches available for large systems rely on compact yet effective ΨT s containing explicitly correlated single Slater determinant (SD). However, SD nodes may be better suited to one system than to another, which may possibly lead to inaccurate FN-DMC energy differences. It remains a challenge, how to estimate inequivalency or appropriateness of SDs. Here we use the differences of a measure based on Euclidean distance between the natural orbital occupation number (NOON) vector of the Slater determinant (SD) from the exact solution in the NOON vector space, that can be viewed as a measure of SD inequivalency and a measure of the expected degree of nondynamic-correlation-related bias in FN-DMC energy differences. This is explored on a set of small noncovalent complexes and covalent bond breaking of Si2 vs. N2. It turns out that NOON-based measures well reflect the magnitude and sign of the bias present in the data available, thus providing new insights to the nature of bias cancellation in SD FN-DMC energy differences. I. MOTIVATIONFixed-node diffusion Monte Carlo (FN-DMC) is a projector many-body electronic structure method 1-3 , promising for its accuracy, massive parallelism, loworder CPU cost scaling and direct treatment of extended models 4,5 . For a given Hamiltonian H, FN-DMC projects out the ground-state Ψ that has non-zero overlap with the antisymmetric trial state Ψ T , in imaginary time τ :In real space, FN-DMC captures symmetric correlations exactly, and the accuracy of Fermi states is limited by the location of the supplied approximate node, i.e., a subset of electron positions R where Ψ T (R) = 0. The total FN-DMC energy is an upper bound to the exact energy 6 (which would result if the node would be exact) and the related bias, FN bias, scales quadratically with the nodal displacement error 7 . In principle, Ψ T can be systematically improved so that the related FN bias becomes negligible 8,9 , however, with increasing system size, such an approach eventually becomes unreliable. The practical FN-DMC approaches available for large systems 10,11 therefore rely on compact yet effective Ψ T s like Slater-Jastrow (SJ) ansätze 12 , containing explicitly correlated single Slater determinant (SD) 13 (or a small number of them). Such an approach makes FN-DMC somewhat empirical, because the nodes of SJ Ψ T s are not converged and related FN bias is hard to control. Thus, the method performance must be mapped a priori for a representative class of systems considered 14,15 . In general, nevertheless, FN-DMC using SJ wave functions is predictive and reaches acceptable accuracy in a number of important systems where it has no competitors, e.g., transition metal oxides at high pressure 16 , magnetic states in solids 17-21 or large noncovalent systems 10,11,22-24 (for more examples, see reviews 4,5,25-34).In small noncovalent systems, surprisingly accurate single-point...

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“…[13][14][15] Recently, Mitas and coworkers have discovered a interesting relationship between electronic density, degree of node nonlinearity and fixed-node error. [16][17][18] Here, we study the topology and properties of the nodes in a class of systems composed of electrons located on the surface of a sphere. Due to their high symmetry and their mathematical simplicity, these systems are the ideal "laboratory" to study nodal hypersurface topologies in electronic states.…”

Section: Introductionmentioning

confidence: 99%

Nodal surfaces and interdimensional degeneracies

Loos

Bressanini

2015

The Journal of Chemical Physics

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The aim of this paper is to shed light on the topology and properties of the nodes (i.e. the zeros of the wave function) in electronic systems. Using the "electrons on a sphere" model, we study the nodes of two-, threeand four-electron systems in various ferromagnetic configurations (sp, p 2 , sd, pd, p 3 , sp 2 and sp 3 ). In some particular cases (sp, p 2 , sd, pd and p 3 ), we rigorously prove that the non-interacting wave function has the same nodes as the exact (yet unknown) wave function. The number of atomic and molecular systems for which the exact nodes are known analytically is very limited and we show here that this peculiar feature can be attributed to interdimensional degeneracies. Although we have not been able to prove it rigorously, we conjecture that the nodes of the non-interacting wave function for the sp 3 configuration are exact.

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Communication: Fixed-node errors in quantum Monte Carlo: Interplay of electron density and node nonlinearities

Abstract: Perspective: Explicitly correlated electronic structure theory for complex systems

Cited by 36 publications

References 20 publications

Generalized local-density approximation and one-dimensional finite uniform electron gases

Generalized local-density approximation and one-dimensional finite uniform electron gases

Bias cancellation in one-determinant fixed-node diffusion Monte Carlo: Insights from fermionic occupation numbers

Nodal surfaces and interdimensional degeneracies

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