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Replaced asymmetric with antisymmetric (doh!)
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Max Lein
Max Lein
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The Hartree-Fock approximation makes an assumption on the state, namely that your $N$-electron wave function is the asymmetricantisymmetric product of single-electron states; if you drop the asymmetryantisymmetry (i. e. you forget that electrons are fermions), you get the Hartree approximation. You can implement the asymmetrizationantisymmetrization procedure by plugging the single-particle wave functions into a Slater determinant.

To find an approximate ground state of the system, you can use a variational principle under the constraint that all the single-particle wave functions are orthogonal (that's the Pauli principle again). The Hartree-Fock equations now come out as the Euler-Lagrange equations to that variational principle. Note that the ground state is approximate, because instead of minimizing over all asymmetricantisymmetric wave functions, you are only admitting asymmetrizedantisymmetrized product states.

In summary, you approximate the true ground state of the system by an asymmetrizedantisymmetrized product state of single-particle wave that solve a non-linear equation. The true ground state will not be a product state, but in many situations it is energetically close to a product state.

The Hartree-Fock approximation makes an assumption on the state, namely that your $N$-electron wave function is the asymmetric product of single-electron states; if you drop the asymmetry (i. e. you forget that electrons are fermions), you get the Hartree approximation. You can implement the asymmetrization procedure by plugging the single-particle wave functions into a Slater determinant.

To find an approximate ground state of the system, you can use a variational principle under the constraint that all the single-particle wave functions are orthogonal (that's the Pauli principle again). The Hartree-Fock equations now come out as the Euler-Lagrange equations to that variational principle. Note that the ground state is approximate, because instead of minimizing over all asymmetric wave functions, you are only admitting asymmetrized product states.

In summary, you approximate the true ground state of the system by an asymmetrized product state of single-particle wave that solve a non-linear equation. The true ground state will not be a product state, but in many situations it is energetically close to a product state.

The Hartree-Fock approximation makes an assumption on the state, namely that your $N$-electron wave function is the antisymmetric product of single-electron states; if you drop the antisymmetry (i. e. you forget that electrons are fermions), you get the Hartree approximation. You can implement the antisymmetrization procedure by plugging the single-particle wave functions into a Slater determinant.

To find an approximate ground state of the system, you can use a variational principle under the constraint that all the single-particle wave functions are orthogonal (that's the Pauli principle again). The Hartree-Fock equations now come out as the Euler-Lagrange equations to that variational principle. Note that the ground state is approximate, because instead of minimizing over all antisymmetric wave functions, you are only admitting antisymmetrized product states.

In summary, you approximate the true ground state of the system by an antisymmetrized product state of single-particle wave that solve a non-linear equation. The true ground state will not be a product state, but in many situations it is energetically close to a product state.

Source Link
Max Lein
Max Lein
  • 955
  • 4
  • 9

The Hartree-Fock approximation makes an assumption on the state, namely that your $N$-electron wave function is the asymmetric product of single-electron states; if you drop the asymmetry (i. e. you forget that electrons are fermions), you get the Hartree approximation. You can implement the asymmetrization procedure by plugging the single-particle wave functions into a Slater determinant.

To find an approximate ground state of the system, you can use a variational principle under the constraint that all the single-particle wave functions are orthogonal (that's the Pauli principle again). The Hartree-Fock equations now come out as the Euler-Lagrange equations to that variational principle. Note that the ground state is approximate, because instead of minimizing over all asymmetric wave functions, you are only admitting asymmetrized product states.

In summary, you approximate the true ground state of the system by an asymmetrized product state of single-particle wave that solve a non-linear equation. The true ground state will not be a product state, but in many situations it is energetically close to a product state.