By Professor Dr. Franz Schwabl (auth.)

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4b) i=l i,j L L N (il U Ij) (1j;1 alaj 11j;) = i,j L i=l whilst the two-particle contributions are found as (1j;1 ara}amak 11j;) = (1j;1 (JimJjkatnal + JikJjmalatn)amak 11j;) = (JikJjm - JimJjk) 8(m, k E 1, ... ,N) . 4c) The first factor implies that the expectation value vanishes whenever the creation and annihilation operators fail to compensate one another. The second 11 QM I ap. eit. 3 Hartree-Fock Equations for Atoms 51 implies that both the operators a m and ak must be present in the set a1 ...

We thus define the creation operators by a! 5a) 18 1. 5a) and the commutator of two operators A and Bare defined by {A,B} == [A,B]+ == AB+BA [A,B] == [A,B]_ == AB - BA. 6) Given these preliminaries, we can now address the precise formulation. If one wants to characterize the states by means of occupation numbers, one has to choose a particular ordering of the states. This is arbitrary but, once chosen, must be adhered to. 7) The effect of the operator a! must be a! I... , ni,"') = (1 - ni)( _l)L:;<;n; I··.

These must be defined such that the result of applying them twice is zero. Furthermore, the order in which they are applied must play a role. We thus define the creation operators by a! 5a) 18 1. 5a) and the commutator of two operators A and Bare defined by {A,B} == [A,B]+ == AB+BA [A,B] == [A,B]_ == AB - BA. 6) Given these preliminaries, we can now address the precise formulation. If one wants to characterize the states by means of occupation numbers, one has to choose a particular ordering of the states.