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Unformatted text preview: Physics 4605, Wave mechanics II Project No. 3 Due April 23, 2001 The He atom: Self-consistent field and multi-electron orbitals References See Liboff Section 13.10. You should plan on reading about the Hartree method or Hartree-Fock method in at least one other textbook from the library. Goals To calculate the wavefunctions and ground-state energies of 2 (and possibly 3) electron atoms. Theory For simplicity, we are going to assume that the nucleus has infinite mass and is therefore immobile. The charge of the nucleus is given to be +Ze; our results are valid for any nuclear charge orbitted by 2 electrons (to the level of approximation that we use). We know that the electronic configuration of the ground-state for He is 1s2, and for Be is 1s22s1. We are going to assume that the multi-electron wavefunction is separable, and that the final ground-state charge distribution of the exact wavefunctions is spherically symmetric (it is). The only thing we are going to neglect is a part of the electron-electron repulsion that is not spherically symmetric, called the correlation energy, which is beyond our capabilities in this course. The wavefunction is assumed to have the product wavefunction form ( 29 ( 29 ( 29 = z S S r r r r , | , 2 1 2 1 & where the spin part is assumed to be the case of paired spins, S=0 (it is). The two electrons are identical, and each interacts with the nucleus through an attractive electrostatic potential. Electron number 1 also interacts with electron number 2 through coulomb repulsion, and vice- versa. This is a new term in the Hamiltonian. Each one electron wavefunction satisfies a TISE that is written ( 29 ( 29 j o j j j j j j V H H r E r H + = = where H is the hydrogenic potential of the electron in the field of the nucleus, and V j is the electron-electron repulsion term. Explicitly,electron-electron repulsion term....
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