Approximate Methods for Calculating Ground-State Energy in Helium Atom

In this paper, several main approximate methods are discussed, including variational principle, Hartree-Fock method, configuration interaction method, quantum Monte Carlo method and parametric perturbation method. These methods solve the complexity of electron-electron correlation and interaction from different angles, and each has its advantages and disadvantages. The variational principle estimates the upper limit of ground state energy by testing the wave function, which effectively captures the electron correlation. Hartree-Fock method uses the mean field approximation to simplify the calculation but ignores the dynamic correlation. The quantum Monte Carlo method provides high-precision results through random sampling where other methods do not perform well, while the configuration interaction method further improves the Hartree-Fock results by considering multiple electron configurations. The parametric perturbation method performs well in iterative approximation, and its calculated results are close to the experimental values. At the end of this paper, the advantages of variational method are analyzed, and how to solve the ground state of helium atom by optimizing the wavefunction and minimizing the energy is shown, which provides theoretical and practical support for the development of materials science and quantum mechanics.