Electron repulsion orbitals, an approximation

Besides being antisymmetric, the wavefimction for a polyelectronic atom must also reflect the ever present electron-electron repulsion. In Section 5.1, the electron repulsion was ignored in order to fadhtate the analysis of the Pauli principle. However, huge errors can be produced when electron repulsion is neglected. For example, the ionization potential so calculated for He would be 5250 kJ mol (4 hartree), which is the value for He , but the experimental value is 2372 kJ moP  

The interelectronic repulsions are represented in the wave equation by the corresponding potential energy term 

The way out lies in the following approximation each electron, as far as the calculation is concerned, is considered as interacting with a smoothed-out averaged density of the other electrons. For example, the description of the 11th electron of Na - electron 3s in Eq. (5.2) - is based on an average field due to the remaining 10 electrons (besides the nucleus attraction). In this way, a central field is simulated and one-electron functions, i.e. orbitals, reappear, this time as an approximation. 

There is an obvious vicious circle in this approach if the spatial distribution of each electron is one of the unknowns, how can we speak of averaged distributions The answer is an iterative numerical calculation as demonstrated originally by the British physicist Douglas Hartree in 1928. In 1930, the method was improved by the Russian physicist Vladimir Fock who adapted the method to antisymmetric wavefunctions as required by the Pauli principle. The Hartree-Fock method is a numerical calculation that can be summarized in the following steps  

These terms should also be calculated and included in the determination of the new set of orbitals and the new value for the energy. They can be seen as a new potential (although not corresponding to any new force or to a multiplicative operator like Vj) called an exchange potential. 

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