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Approximation methods

is the electron-electron interaction. H is just the sum of the two hydrogenic Hamiltonians: H 0 = − 1 2 m p 1 2 − 1 2 m p 2 2 − k Z e 2 r 1 − k Z e 2 r 2 = H 1 + H 2 {\displaystyle H_{0}=-{\frac {1}{2m}}\mathbf {p} _{1}^{2}-{\frac {1}{2m}}\mathbf {p} _{2}^{2}-{\frac {kZe^{2}}{r_{1}}}-{\frac {kZe^{2}}{r_{2}}}=H_{1}+H_{2}} where H i = − 1 2 m p i 2 − k Z e 2 r i {\displaystyle H_{i}=-{\frac {1}{2m}}\mathbf {p} _{i}^{2}-{\frac {kZe^{2}}{r_{i}}}} are independent Coulomb field Hamiltonian of each electron. Since the unperturbed Hamiltonian is a sum of two independent Hamiltonians (ie. are separable), the wavefunction must be of form | ψ ⟩ = | ψ 1 ⟩ | ψ 2 ⟩ {\textstyle |\psi \rangle =|\psi _{1}\rangle |\psi _{2}\rangle } where | ψ 1 ⟩ {\textstyle |\psi _{1}\rangle } and | ψ 2 ⟩ {\textstyle |\psi _{2}\rangle } are eigenkets of H 1 {\textstyle H_{1}} and H 2 {\textstyle H_{2}} respectively. However, the spatial wavefunction of the form ψ (r 1, r 2) = ψ a (r 1) ψ b (r 2) {\textstyle \psi (\mathbf {r} _{1},\mathbf {r} _{2})=\psi _{a}(\mathbf {r} _{1})\psi _{b}(\mathbf {r} _{2})} need not correspond to physical states of identical electrons as per the symmetrization postulate. Thus, to obtain physical solutions symmetrization of the wavefunctions | ψ 1 ⟩ {\textstyle |\psi _{1}\rangle } and | ψ 2 ⟩ {\textstyle |\psi _{2}\rangle } is carried out.