# The fixed hypernode method for the solution of the many body Schroedinger equation

## Abstract

We propose a new scheme for an approximate solution of the Schroedinger equation for a many-body interacting system, based on the use of pairs of walkers. Trial wavefunctions for these pairs are combinations of standard symmetric and antisymmetric wavefunctions. The method consists in applying a fixed-node restriction in the enlarged space, and computing the energy of the antisymmetric state from the knowledge of the exact ground state energy for the symmetric state. We made two conjectures: first, that this fixed-hypernode energy is an upper bound to the true fermion energy; second that this bound would necessarily be lower than the usual fixed-node energy using the same antisymmetric trial function. The first conjecture is true, and is proved in this paper. The second is not, and numerical and analytical counterexamples are given. The question of whether the fixed-hypernode energy can be better than the usual bound remains open.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 877847

- Report Number(s):
- UCRL-CONF-218419

Journal ID: ISSN 1947--5918; TRN: US0601715

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

- Resource Type:
- Conference

- Resource Relation:
- Journal Volume: 953; Conference: Presented at: Pacifichem 2005, Honolulu, HI, United States, Dec 15 - Dec 20, 2005

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; FERMIONS; GROUND STATES; SCHROEDINGER EQUATION

### Citation Formats

```
Pederiva, F, Kalos, M H, Reboredo, F, Bressanini, D, Guclu, D, Colletti, L, and Umrigar, C J.
```*The fixed hypernode method for the solution of the many body Schroedinger equation*. United States: N. p., 2006.
Web. doi:10.1021/bk-2007-0953.ch007.

```
Pederiva, F, Kalos, M H, Reboredo, F, Bressanini, D, Guclu, D, Colletti, L, & Umrigar, C J.
```*The fixed hypernode method for the solution of the many body Schroedinger equation*. United States. doi:10.1021/bk-2007-0953.ch007.

```
Pederiva, F, Kalos, M H, Reboredo, F, Bressanini, D, Guclu, D, Colletti, L, and Umrigar, C J. Tue .
"The fixed hypernode method for the solution of the many body Schroedinger equation". United States.
doi:10.1021/bk-2007-0953.ch007. https://www.osti.gov/servlets/purl/877847.
```

```
@article{osti_877847,
```

title = {The fixed hypernode method for the solution of the many body Schroedinger equation},

author = {Pederiva, F and Kalos, M H and Reboredo, F and Bressanini, D and Guclu, D and Colletti, L and Umrigar, C J},

abstractNote = {We propose a new scheme for an approximate solution of the Schroedinger equation for a many-body interacting system, based on the use of pairs of walkers. Trial wavefunctions for these pairs are combinations of standard symmetric and antisymmetric wavefunctions. The method consists in applying a fixed-node restriction in the enlarged space, and computing the energy of the antisymmetric state from the knowledge of the exact ground state energy for the symmetric state. We made two conjectures: first, that this fixed-hypernode energy is an upper bound to the true fermion energy; second that this bound would necessarily be lower than the usual fixed-node energy using the same antisymmetric trial function. The first conjecture is true, and is proved in this paper. The second is not, and numerical and analytical counterexamples are given. The question of whether the fixed-hypernode energy can be better than the usual bound remains open.},

doi = {10.1021/bk-2007-0953.ch007},

journal = {},

number = ,

volume = 953,

place = {United States},

year = {Tue Jan 24 00:00:00 EST 2006},

month = {Tue Jan 24 00:00:00 EST 2006}

}