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Updated: Jun 26, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
How accurately does the free complement wave function of a helium atom satisfy the Schrödinger equation?
Hiroyuki Nakashima1, Hiroshi Nakatsuji
1Quantum Chemistry Research Institute, JST, CREST, Kyodai Katsura Venture Plaza 106, Goryo Oohara 1-36, Kyoto 615-8245, Japan.
This study rigorously tested the accuracy of calculated wave functions for helium atoms using local energy and H-square error methods. The free complement method accurately determined the helium ground state energy to 32 digits.
Area of Science:
- Quantum Chemistry
- Atomic Physics
Background:
- The accuracy of calculated wave functions is crucial in quantum mechanics.
- Local energy and H-square error serve as stringent tests for wave function accuracy.
Purpose of the Study:
- To examine the accuracy of a helium atom's wave function calculated via the free complement method.
- To validate the reliability of local energy and H-square error as accuracy metrics.
Main Methods:
- Utilized the free complement method to solve the Schrödinger equation for a helium atom.
- Employed local energy (Hpsi/psi) and H-square error (sigma2) as measures of wave function accuracy.
- Applied a modified Temple's formula to establish a lower bound for the exact energy.
Main Results:
- The calculated wave function for the helium atom demonstrated high accuracy.
- The free complement method, combined with energy bounds, yielded a precise value for the helium ground state energy.
- The helium fixed-nucleus ground state energy was determined to be -2.903,724,377,034,119,598,311,159,245, 194,4 atomic units (a.u.), accurate to 32 digits.
Conclusions:
- The free complement method is a reliable approach for accurate quantum mechanical calculations.
- Local energy and H-square error are effective diagnostics for evaluating wave function quality.
- The precise determination of the helium ground state energy validates the employed computational methodology.
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