Hylleraas wavefunction for He

 The reason why the Hartree method cannot reproduce the exact solution is due to the inability of the Hartree wave-function to account for electron correlation. We know that the two electrons in He repel one another and absent any other factor they would stay as far apart as possible so that the probability of finding them close to one another would be small. 

 

https://www2.chemistry.msu.edu/faculty/harrison/web_docs_He/Hylleraas_wavefunction.pdf