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# Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation

## Abstract

A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. Here, the method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015 *E _{h}* of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10

^{–5}

*E*without an importance sampling or Jastrow factor.

_{h}- Authors:

- Univ. of Illinois at Urbana-Champaign, IL (United States)

- Publication Date:

- Research Org.:
- Univ. of Illinois at Urbana-Champaign, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1594980

- Grant/Contract Number:
- SC0006028

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physical Review E

- Additional Journal Information:
- Journal Volume: 101; Journal Issue: 1; Journal ID: ISSN 2470-0045

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS

### Citation Formats

```
Kunitsa, Alexander A., and Hirata, So. Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation. United States: N. p., 2020.
Web. doi:10.1103/PhysRevE.101.013311.
```

```
Kunitsa, Alexander A., & Hirata, So. Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation. United States. doi:10.1103/PhysRevE.101.013311.
```

```
Kunitsa, Alexander A., and Hirata, So. Mon .
"Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation". United States. doi:10.1103/PhysRevE.101.013311.
```

```
@article{osti_1594980,
```

title = {Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation},

author = {Kunitsa, Alexander A. and Hirata, So},

abstractNote = {A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. Here, the method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015 Eh of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10–5 Eh without an importance sampling or Jastrow factor.},

doi = {10.1103/PhysRevE.101.013311},

journal = {Physical Review E},

number = 1,

volume = 101,

place = {United States},

year = {2020},

month = {1}

}

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