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In this report, a numerical method for finding the electron correlation energy for the ground state of helium. Specifically, we develop a wave function for the electrons that is no longer spherical. We do this by altering the ground state energy wave function for hydrogenic like atoms. This original hydrogenic wave function is spherical, that is the equipotential surfaces are balls around the nucleus. When a second electron is introduced these equipotential surfaces are shifted and distorted. We modify the wave functions for these electrons by considering this distortion. We can do this by altering the Bohr radius that appears in the hydrogenic wave function and making it a function of the placement of both electrons. Doing this allows us to make sure that the electrons have equal probability to appear anywhere on a given equipotential surface. Once we find and normalize the electron correlated wave functions, we can then find the electron correlated energy. All of the calculations are done numerically on Jupyter Notebook. The usefulness of this research is that we may find a theoretical value for the energy of the ground state helium atom that closely resembles experimental values. The experimental value for the energy of the ground state helium atom is -78.8eV = -5.8Ry . We theoretically find the energy of the atom by a perturbation on the electron correlation energy. That is, we solve for the energy of the atom not including the electron correlation energy. This allows us to use the hydrogenic like wavefunctions. This value is commonly known to be -8Ry . Therefore, we hope to get an electron repulsion energy term of roughly 2.2Ry. The method described gave me an electron correlation energy of about 3.2Ry which is quite off the expected value. I believe that improvements on this method and using a more powerful programming language like Fortran could help improve the value for the electron correlation energy.
 J.J. Sakurai, ‘Modern Quantum Mechanics’, 1994, Chapter 6
Doig, Conor, "Electron Repulsion Corrections to the Energy for the Ground State of Helium." (2020). Honors Theses. 3245.
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