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qmmanyelec_v1_04212011 - Helium Atom Many-Electron Atoms...

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Helium Atom, Many-Electron Atoms, Variational Principle, Approximate Methods, Spin 21st April 2011 I. The Helium Atom and Variational Principle: Approximation Methods for Complex Atomic Systems The hydrogen atom wavefunctions and energies, we have seen, are deter- mined as a combination of the various quantum ”dynamical” analogues of classical motions (translation, vibration, rotation) and a central-force inter- action (i.e, the Coulomb interaction between an electron and a nucleus). Now, we consider the Helium atom and will see that due to the attendant 3-body problem for which we cannot determine a closed-form, first-principles analytic solution, we will have to find recourse in approximate methods. The Helium atom has 2 electrons with coordinates r 1 and r 2 as well as a single nucleus with coordinate R . The nucleus carries a Z = +2 e charge. The Schrodinger equation is: - ¯ h 2 2 M 2 - ¯ h 2 2 m e 2 1 - ¯ h 2 2 m e 2 2 ! ψ ( R , r 1 , r 2 ) + - 2 e 2 4 π o | R - r 1 | - 2 e 2 4 π o | R - r 2 | + e 2 4 π o | r 1 - r 2 | ! ψ ( R , r 1 , r 2 ) = ( R , r 1 , r 2 ) where the symbol ”nabla”, when squared, is given by: 2 = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 Keep in mind that the R , r 1 , and r 2 represent the Cartesian coordinates of each paticle. This is a 3-body problem and such problems are not solved exactly. Thus, the problem will be reformulated in terms of coordinates of two particles, the electrons. The first approximation: M >> m e , fix the nu- cleus at the origin ( R ) = 0 . This is more rigorously shown by transforming 1
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the origin to the center of mass of the system. For the two electron-nucleus coordinates, this is much like what we have seen for the hydrogen atom electron-nucleus formulation from earlier discussion. Thus, the Schrodinger equation in relative variables is: ¯ h 2 2 m e -∇ 2 1 - ∇ 2 2 ψ ( r 1 , r 2 ) - 2 e 2 4 π o 1 r 1 + 1 r 2 + e 2 4 π o | r 2 - r 1 | ψ ( r 1 , r 2 ) = ( r 1 , r 2 ) The 2 terms represent the kinetic energy of the two electrons. The 1 r 1 and 1 r 2 terms represent the nucleus-electron Coulomb interaction. The last term on the left hand side of the equation represents the electron-electron repulsion taken as a Coulomb interaction based on the absolute value of the electron-electron separation. NOTE: The electron-nucleus Coulomb interaction is a radially symmetric potential. It depends on the radial positions of the electrons from the nu- cleus taken as the origin. The electron-electron repulsion does not possess inherent symmetry (radial or otherwise). It depends on the absolute value of the separation between electrons. Recall that the 2 , representing the kinetic energy operator, in spherical polar coordinates is: 1 r 2 1 ∂r 1 r 2 1 ∂r 1 + 1 r 2 1 sinθ 1 ∂θ 1 sinθ 1 ∂θ 1 + 1 r 2 1 sin 2 θ 1 2 ∂φ 2 1 The Independent Electron Approximation to Solving the Helium Atom Schrodinger Equation If we neglect electron-electron repulsion in the Helium atom problem, we can simplify and solve the effective 2-body problem.
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