 Solution

ID: past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2014/paper-1/2/b/solution

The central field approximation replaces the anisotropic many-electron environment by an average spherical potential, giving one-electron orbitals labelled by

n, l, m

. Screening changes the effective radial attraction and removes the pure Coulomb degeneracy between different

l

The Hartree method improves the potential self-consistently. For normalized occupied orbitals,

[- \frac{ℏ ^{2} \nabla ^{2}}{2 m _{e}} - \frac{Z e ^{2}}{4 π ε _{0} r} + \sum_{j \neq = i} \frac{e ^{2}}{4 π ε _{0}} \int \frac{∣ ϕ _{j} ( r ^{'} ) ∣ ^{2}}{∣ r - r ^{'} ∣} d^{3} r^{'}] ϕ_{i} = ϵ_{i} ϕ_{i} .

Guess orbitals, construct their mean fields, solve the Hartree equations, and iterate to convergence. The

j \neq = i

sum excludes self-interaction in this product-state model. A spherical average can enforce the central-field specialization. A Hartree product state omits fermionic exchange and dynamical correlation; Hartree-Fock method adds antisymmetry and exchange through a Slater determinant. Converged orbital eigenvalues must not simply be added as the total energy, because that double counts pair interactions.^[ref]

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