# The Exact Boundary Condition to Solve the Schrodinger Equation of Many Electron System

## TL;DR

This paper introduces an exact boundary condition based on the exclusion principle for solving the Schrödinger equation in many-electron systems, improving the accuracy of quantum Monte Carlo simulations.

## Contribution

It derives the Coulomb-Exchange nodal surface as an exact boundary condition, enabling more precise solutions for atomic and molecular ground states.

## Key findings

- Ground state energies agree well with exact estimates.
- The boundary condition effectively bypasses the sign problem.
- Applicable to various atoms and molecules.

## Abstract

In an attempt to bypass the sign problem in quantum Monte Carlo simulation of electronic systems within the framework of fixed node approach, we derive the exclusion principle "Two electrons can't be at the same external isopotential surface simultaneously" using the first postulate of quantum mechanics. We propose the exact Coulomb-Exchange nodal surface i.e. the exact boundary condition to solve the non-relativistic Schrodinger equation for the non-degenerate ground state of atoms and molecules. This boundary condition was applied to compute the ground state energies of N, Ne, Li2, Be2, B2, C2, N2, O2, F2, and H2O systems using diffusion Monte Carlo method. The ground state energies thus obtained agree well with the exact estimate of non-relativistic energies.

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Source: https://tomesphere.com/paper/0704.0383