Density dependence of fixed-node errors in diffusion quantum Monte Carlo: Triplet pair correlations

Kulahlioglu, Adem Halil; Rasch, Kevin; Hu, Shuming; Mitáš, Luboš

doi:10.1016/j.cplett.2013.11.033

Cited by 14 publications

(37 citation statements)

References 20 publications

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“…The determinantal part of Ψ T fixes the node locus, and, convincing evidence shows that FN-DMC recovers dynamic correlations exactly within the node constraint and the nodes are directly responsible for the remaining missing nondynamic correlations (see, e.g., refs. 45,46,51). We therefore expect that systems where SD Ψ T causes large FN bias, d will be also large in magnitude, and vice versa.…”

Section: Definitionsmentioning

confidence: 99%

“…In multireference systems, it is known that SDs do not suffice for 46 , and, SJ Ψ T s should contain additional dominant configurations so that the FN bias well cancels out [49][50][51][52][53] . On the other hand, in closedshell systems, the effect of additional determinants is expected to be negligible.…”

Section: Motivationmentioning

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.…”

mentioning

confidence: 99%

“…Bias cancellation limits of such an approach are however unknown, that somewhat hampers widespread usage of FN-DMC as a benchmark method. Deviations of the order of 0.3 kcal/mol (relative errors exceeding 7%), unacceptable at the benchmark level (relative errors should not exceed say 2-3%), of unknown nature found in complexes with multiply bound hydrogen bond acceptors like HCN dimer or formaldehyde (FD) dimer 41 start to delineate its limits and motivate our further elaborations in this field.In multireference systems, it is known that SDs do not suffice for 46 , and, SJ Ψ T s should contain additional dominant configurations so that the FN bias well cancels out [49][50][51][52][53] . On the other hand, in closedshell systems, the effect of additional determinants is expected to be negligible.…”

mentioning

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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Section: Definitionsmentioning

confidence: 99%

Section: Motivationmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

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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“…[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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Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus: III. Spin‐quartet state (1s2s3s) of the lithium sequence

Nader

Valle

Vieyra

et al. 2022

Int J of Quantum Chemistry

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As a continuation of Part I, dedicated to the ground state of He‐like and Li‐like isoelectronic sequences for nuclear charges Z≤20, and Part II, dedicated to two excited states of He‐like sequence, two ultra‐compact wave functions in the form of generalized Guevara–Harris–Turbiner functions are constructed for Li‐like sequence. They describe accurately the domain of applicability of the non‐relativistic Quantum Mechanics of Coulomb Charges (QMCC) for energies (2–3 significant digits [s.d.]) of the spin‐quartet state 140+ of Li‐like ions (in static approximation with point‐like, infinitely heavy nuclei). Variational parameters are fitted in Z by 2nd degree polynomials. The most accurate ultra‐compact function leads to the absolute accuracy ∼10−3 a.u. for energy, and ∼10−4 for the normalized electron‐nuclear cusp parameter for Z≤20. Critical charge Z=ZB, where the ultra‐compact trial function for the 140+ state loses its square‐integrability, is estimated, ZB()140.1em0+∼1.26−1.30. As a complement to Part I, square integrability for the compact functions constructed for the ground, spin‐doublet state 120.1em0+ of the Li‐like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as ZB()120.1em0+=1.62−1.65. It implies that at Z=2—the negative helium ion He−—both states, 120.1em0+ and 140.1em0+, exist as states embedded in the continuum.

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Density dependence of fixed-node errors in diffusion quantum Monte Carlo: Triplet pair correlations

Cited by 14 publications

References 20 publications

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

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

Nodal surfaces and interdimensional degeneracies

Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus: III. Spin‐quartet state (1s2s3s) of the lithium sequence

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