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hf - 1 Hartree-Fock Approximation N N We have a system of N...

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1 Hartree-Fock Approximation We have a system of N interacting fermions with a Hamiltonian H = N i =1 h ( i ) + N i<j V int ( i, j ) . (1) Here, h ( i ) is the “single-particle Hamiltonian” for the particle- i and V int ( i, j ) is the part of the Hamiltonian describing the interaction between particles i and j . By the “single-particle Hamiltonian” we mean the terms that would be present if there were only one fermion in the system. For the case of N electrons in an atom with proton number Z at the nucleus, these terms would be h ( i ) = p 2 i 2 m - Ze 2 r i , V int ( i, j ) = e 2 | r i - r j | . Again we ignored the rest of the terms like spin-orbit interaction which would introduce spin-dependent terms into the Hamiltonian. Note that in Eq. (1), the interaction part is summed over each particle pair only once. As a result, there are N 2 = N ( N - 1) 2 terms in the last sum. Such problems cannot be solved exactly in general (there are a few unin- teresting exceptions). As a result, we need a general approximation procedure for solving such many-body equations. Obviously, what we want to calcu- late is important in choosing the approximation procedure. Hence, there are various methods developed to cope with different aspects of such kind of problems. The very first step should be obtaining a “feel” for the energy levels for those problems. That certainly means that we don’t need accurate numerical values. Independent particle picture is this first step in visualizing the energy levels of most of the many-body problems. This is the language used in describing the electronic states of atoms in Atomic Physics courses. When we are saying that in Lithium, there are two electrons in the 1s orbital and 1
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one electron in the 2s orbital, we are using that language. However, we say, in those atoms, the orbital wavefunctions and their energies are different than those of the hydrogenic atoms. Our purpose in here is to formalize those kind of statements. Therefore, the question we would like to answer is “What is the best independent-particle approximation for an interacting system of fermions?” And the answer can be provided by the variational theory. We should form a many-body wavefunction of the atom as a Slater determinant using some one-particle states ϕ 1 , . . . , ϕ N . And, we should choose these one-particle states as the ones that minimize the expectation value of the Hamiltonian in Eq. (1). This is the Hartree-Fock approximation. 2 Expectation Values Let us suppose that we have N arbitrary one-particle states ϕ 1 , . . . , ϕ N . We should choose them as an orthonormal set, since this will make the calculation of expectation values easier. So, we have ϕ i | ϕ j = δ i,j i, j = 1 , . . . , N. (2) As a result, we have N 2 equations giving us a restriction on the one-particle states. (Some may notice that, there are only N ( N + 1) / 2 equations in Eq. (2), but the number of real quantities fixed to a certain value is N 2 . So, there are N 2 real equations.) We form the Slater determinant from these N states as Ψ(1 , 2 , . . . , N ) = 1 N !
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