Highly-accurate solutions of the Schrodinger equation

For systems of two or more electrons, we are not able to obtain analytical solutions in closed form, as in the case of hydrogen. Fortunately, a variational principle exists for the Schrodinger equation. According to this principle, any trial for the ground state always yields an upper bound to the exact energy E  

The equality holds only when the exact wave function is used. Therefore the variational principle allows us to choose a basis set of functions gj such that a linear combination of them will tend to the exact solution. Thus the problem is reduced to finding the best set of coefficients that will minimize the right-hand side of eq. (9). This implies the calculation of a large number of integrals of the form, 

Given the large number of integrals that are required in a calculation, it is 

The trial T has to fulfill certain constraints two of the trivial ones are that it should be square integrable, and it should be normalized to unity. A more important and difficult constraint for T is that it must be antisymmetric with respect to the exchange of any two electrons in order to satisfy the Pauli exclusion principle. 

The antisymmetry feature of the trial wavefunction is required because the Schrodinger equation does not exclude those trial functions that do not fulfill the antisymmetry requirement. Therefore, not all solutions to eq. (1) are acceptable the search for a solution must be restricted to those trial wavefunctions that are antisymmetric with respect to the exchange of any two electrons. This implies that the basis set should have a form that allows the wavefunction to be antisymmetric to the interchange of any two electrons. As a result, the Vee energy has two parts a direct term, 

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