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By March, Norman Henry

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Additional resources for Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms

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In the Thomas-Fermi theory, we have to evaluate the kinetic energy T via eqn. 6) from a knowledge of g(r), the number of electrons per unit volume, related to the total potential energy K(r) by eqn. 4). To evaluate eqn. 6) for the self-consistent Thomas-Fermi theory by direct manipulation is possible, the rather lengthy derivation being due, essentially, to Milne (1927). We show below that ---^(è)>+K)'^ Γ / N\2 1 1 1 e* = 0-4841^(0)+(l--)-]z3-. 19) and we obtain the Thomas-Fermi result for atomic binding energies E(Z, Z) = -0-7687Z* —.

This property can be seen from Fig. 1 for s andp states. In multi-electron atoms, we need to decide how the field in which a given electron moves is to be set up before we can tackle the problem of the spatial extent of the electron cloud. It is natural enough to try to answer this question by looking at the next simplest atom, helium, with two electrons. As we remarked above, this problem has not yet been solved exactly. 2. Wave equation for helium atom The Schrödinger equation for this case must evidently involve kinet­ ic energy terms for two electrons, with positional coordinates ri and r 2 (again we assume the nucleus to be fixed).

12) which satisfy the condition φ(0) = 1, show that there is an expansion for small x of the form φ(χ) = 1 + a2x+a3x2 4-a±x2+ ... Determine a3 and Û4. (This expansion is due to Baker (1930). Note that it will generate, for small x, the various solutions shown schematically in Fig. ) 2. Express the average value of r2 for an electron in the Thomas-Fermi atom in terms of the dimensionless solution φ(χ) satisfying the conditions φ(0) = 1 and φ -* 0 at infinity. 3. 3, find the energy below which just two electrons are to be found.

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