Lecture18 - The helium atom Consists of a nucleus and two...

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1 Physical Chemistry Lecture 18 The Helium Atom’s Spatial States The helium atom Consists of a nucleus and two electrons Hamiltonian has four kinds of terms Nuclear kinetic energy Electronic kinetic energy Nuclear-electron Coulombic potential energy Electron-electron Coulombic potential energy | | 1 4 | | 1 | | 1 4 2 2 2 2 2 1 0 2 2 1 0 2 2 2 2 1 2 2 1 2 1 r r r r  e e m p m p m p V V V T T T H e e e e n n ee ne ne e e n Center of mass separation Focus on the relative problem Center of mass is assumed to be at the nucleus Good approximation as nuclei get heavier Determine only relative energies of a single atom Center-of-mass problem again gives a particle-in- a-box solution; translational energies | | 1 4 | | 1 | | 1 4 2 2 2 2 1 0 2 2 1 0 2 2 2 2 2 1 2 1 r r r r e e m p m p V V V T T H e e e e ee ne ne e e Simplification of the helium- atom problem Consider the problem with neglect of the electron-electron repulsion Decompose into two hydrogen-like problems 2 1 2 2 1 1 2 1 2 2 2 2 2 2 1 1 1 1 1 1 ) ( ) ( ) , ( ) ( ) ( ) ( ) ( E E E E H E H r r r r r r r r | | 1 4 2 2 | | 1 4 2 2 2 0 2 2 2 1 0 2 2 1 2 1 2 1 r r e m p e m p V T V T H e e e e ne e ne e Hydrogen-like energies and wave functions Difference between the approximate helium atom and the hydrogen atom Z = 1 for hydrogen Z = 2 for helium Nuclear-charge difference
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Lecture18 - The helium atom Consists of a nucleus and two...

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