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The independent-particle approximation.

When the Hamiltonian has terms that depend on two or electrons (e.g., the Coulomb interaction between electrons) then

Y(1,2,...,N) = y₁(1)y₂(2)...y_N(N).

will not be a solution of the Schrödinger equation. However, we might try and see if we can get a good approximation of the Schrödinger equation that we can write in that form. For that we use the variation principle.

To use the variation principle we first have to determine the expectation value of the Hamiltonian, or, as is often simpler stated, we have to determine the (electronic) energy of the molecule when the electrons are described by the wave function Y above. Then we have to minimize this energy by varying the orbitals y_n. This leads to a lengthy procedures that's not really easy and only interesting for the mathematical gourmet. The result is important, however. We get an equation for the orbitals of the same form as without interaction between electrons;

h_ny_n = e_ny_n.

The operator h_n consists of the three operators; the kinetic energy of an electron, the interaction of an electron with the nuclei, and the interaction of an electron with all other electrons. This latter term is quite complicated, because it depends on all orbitals y_i with i different from n. In fact, it should be even more complicated, because the form of the all-electron wave function Y that we have used is fundamentally flawed.

The independent-particle model in quantum chemistry.

Electrons that don't interact.
The variation principle.
The independent-particle approximation (this page).
The Fock equation without exchange.
The Pauli principle and Slater determinants.
The Fock equation with exchange.
Spin orbitals, spatial orbitals, and the Pauli exclusion principle.