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Unformatted text preview: Helium Atom, ManyElectron 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 centralforce 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 3body problem for which we cannot determine a closedform, firstprinciples 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 ! ( R , r 1 , r 2 ) + 2 e 2 4 o  R r 1  2 e 2 4 o  R r 2  + e 2 4 o  r 1 r 2  ! ( R , r 1 , r 2 ) = E ( 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 3body 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 ) = . This is more rigorously shown by transforming 1 the origin to the center of mass of the system. For the two electronnucleus coordinates, this is much like what we have seen for the hydrogen atom electronnucleus formulation from earlier discussion. Thus, the Schrodinger equation in relative variables is: h 2 2 m e 2 1 2 2 ( r 1 , r 2 ) 2 e 2 4 o 1 r 1 + 1 r 2 + e 2 4 o  r 2 r 1  ( r 1 , r 2 ) = E ( 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 nucleuselectron Coulomb interaction. The last term on the left hand side of the equation represents the electronelectron repulsion taken as a Coulomb interaction based on the absolute value of the electronelectron separation. NOTE: The electronnucleus 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 electronelectron 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 sin 1 1 sin 1 1 + 1 r 2 1 sin 2 1 2 2 1 The Independent Electron Approximation to Solving the Helium Atom Schrodinger Equation If we neglect electronelectron repulsion...
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.
 Fall '08
 Dybowski,C
 Physical chemistry, Electron, pH

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