# Self-healing diffusion quantum Monte Carlo algorithms: methods for direct reduction of the fermion sign error in electronic structure calculations

## Abstract

We develop a formalism and present an algorithm for optimization of the trial wave-function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project-out a multi-determinant expansion of the fixed node ground state wave function and (ii) to define a cost function that relates the interacting-ground-state-fixed-node and the non-interacting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust towards the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multi-determinant expansions of the trial wave function. The method can be generalized to other wave function forms such as pfaffians. We test the method in a model system where benchmark configuration interaction calculationsmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 956841

- Report Number(s):
- LLNL-JRNL-409707

Journal ID: ISSN 1098-0121; TRN: US1002135

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. B, Condensed Matter and Materials Physics; Journal Volume: 79; Journal Issue: 19

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ALGORITHMS; BENCHMARKS; CONFIGURATION; CONFIGURATION INTERACTION; DIFFUSION; DISTRIBUTION; ELECTRONIC STRUCTURE; EXACT SOLUTIONS; FERMIONS; GROUND STATES; HAMILTONIANS; OPTIMIZATION; PROBABILITY; SAMPLING; WAVE FUNCTIONS

### Citation Formats

```
Reboredo, F A, Hood, R Q, and Kent, P C.
```*Self-healing diffusion quantum Monte Carlo algorithms: methods for direct reduction of the fermion sign error in electronic structure calculations*. United States: N. p., 2009.
Web. doi:10.1103/PhysRevB.79.195117.

```
Reboredo, F A, Hood, R Q, & Kent, P C.
```*Self-healing diffusion quantum Monte Carlo algorithms: methods for direct reduction of the fermion sign error in electronic structure calculations*. United States. doi:10.1103/PhysRevB.79.195117.

```
Reboredo, F A, Hood, R Q, and Kent, P C. Tue .
"Self-healing diffusion quantum Monte Carlo algorithms: methods for direct reduction of the fermion sign error in electronic structure calculations". United States. doi:10.1103/PhysRevB.79.195117. https://www.osti.gov/servlets/purl/956841.
```

```
@article{osti_956841,
```

title = {Self-healing diffusion quantum Monte Carlo algorithms: methods for direct reduction of the fermion sign error in electronic structure calculations},

author = {Reboredo, F A and Hood, R Q and Kent, P C},

abstractNote = {We develop a formalism and present an algorithm for optimization of the trial wave-function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project-out a multi-determinant expansion of the fixed node ground state wave function and (ii) to define a cost function that relates the interacting-ground-state-fixed-node and the non-interacting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust towards the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multi-determinant expansions of the trial wave function. The method can be generalized to other wave function forms such as pfaffians. We test the method in a model system where benchmark configuration interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal non-interacting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.},

doi = {10.1103/PhysRevB.79.195117},

journal = {Physical Review. B, Condensed Matter and Materials Physics},

number = 19,

volume = 79,

place = {United States},

year = {Tue Jan 06 00:00:00 EST 2009},

month = {Tue Jan 06 00:00:00 EST 2009}

}