For a long time I did not understand the connection between density matrices as shown in my other post and density matrices as they are used in quantum chemical programs. On first sight it is not quite apparent that you can get the matrix representation of the 1-particle density matrix in an orbital basis in the following way:
The operators in the second expression are the creation and annihilation operators from second quantization. The expression means: you take an electron out of orbital j, put it in orbital i, and overlap this wave function with the original wave function. Note that the first scalar product is in 1-particle space, the second one in n-particle space.
To reach this conclusion you just have to write it out explicitely and rotate the integrals a little bit. With the definition from the other post, it looks like this.
As a next step you also write out the n-1 integrations that are involved in the density matrix definition.
You rearrange it a little bit:
The crucial point is the interpretation of the terms in the parentheses. To do this, we expand our wave function in Slater determinants constructed from an orthonormal orbital set including φi and φj. Formally we can construct any wave function in this way in a complete CI expansion.
The question reduces to: How does multiplying with an orbital and integration change one such n-particle determinant? - If the determinant contains the orbital, one obtains the (n-1)-particle determinant where this orbital is taken out. If it does not contain the orbital (and therefore only orthonormal orbitals) one obtains 0. This operator is usually called annihilation operator and written as ai.
We may rewrite this equation with the adjoint operator.
This offers an expression that can be efficiently evaluated. In a determinant basis all you would have to do is check whether the orbitals are present for every pair of determinants - no explicit integrations or things of that sort are needed.
 The adjoint operator is usually called creation operator. It adds an orbital to a determinant or makes it 0 if the orbital is already there. The explicit definition would be something like multiplying with the orbital and antisymmetrizing the expression.
Another interesting point here is that we have scalar products (= matrix elements) with different numbers of particles. For every k, the k-particle space is constructed as a Hilbert space, i.e. you can form scalar products. It is not possible to have a scalar product between wavefunctions of different particle numbers. Therefore the direct sum of all k-particle spaces (called Fock space) is indeed a vector space but not a Hilbert space. edit: you can construct the Fock space as a hilbert space if scalar products with different numbers of particles are just zero.
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