Instead, the single electron wavefunction that best represents the interactions is analysed, specifically, a potential that includes the potential of the ions U-ion and the electric fields from the other electrons in the system U-el (the approximation is made that this is given by a smooth distribution of negative charge). The contributions from all the electrons to the potential energy of the electron considered can then be worked out (A&M pg. 330-331) and plugged into the Schrodinger equation for that particular electron. If such an equation is written for each electron, the set of equations is known as the Hartree Equations. The Hartree equations are solved computationally by proposing a form of the potential U-el, solving the equation to obtain the wavefunction, from which the next U-el is found, and so forth until the potential reaches a certain degree of accuracy (does not change very much between successive iterations).
5/3/11
Hartree Approximation
The Hartree approximation arises from a first attempt to account for the effects of electron-electron interactions, which are neglected when using the independent electron approximation (as in the bloch model etc). The single-electron wavefunction seems insufficient to properly describe how the electrons in an N-particle system interact. So the Schrodinger equation for an N-particle system is considered, i.e. we want a wavefunction for the N electron system. However, attempts to solve this equation prove to be futile.
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actually it's gitting more sofesticsted than before due to the intraction between elctron-electron
ReplyDeleteThe major failing of the Hartree approximation is that it does not take the Pauli exclusion principle into account. When the N-electron wavefunction is constructed from the N single-electron wavefunctions, the sign does not change when two electron wavefunctions are switched in the product.
ReplyDeleteThe way to achieve this sign change, and therefore obey the Pauli exclusion principle, is to represent the N-electron wavefunction as a Slater Determinant, which is the determinant of the NxN matrix consisting of the single electron wavefunctions. Since the determinant of a matrix changes sign under a switch of 2 columns or rows, it follows that interchanging two electrons will change the sign of the wavefunction, thus obeying the Pauli Principle.
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