Many-electron wave functions the Hartree-Fock equation

Many-electron wave functions the Hartree—Fock equation 

In section 6.2.2 we introduced the Pauli principle and showed how, in the case of a two-electron system like the helium atom, the wave function could be written in the form of a determinant, called a Slater determinant (6.19). In a two-electron system the spin and spatial parts of the wave function can be separated, but for more than two electrons the general form of the wave function, written as a Slater determinant, is 

The complete Hamiltonian for the many-electron system is written 

Since the determinantal wave function (6.36) is normalised to unity and the spin functions are orthonormal, the variation function (6.26) takes a simple form, given by, 

Note that i and j run over the orbitals in this closed shell case. There are three different types of integral in equation (6.39), and they are defined in the manner shown below. 

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