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I was reading this paper by Per-Olov Löwdin and it discusses how density matrices can be used to represent/interpret the wavefunction. And, I had a question regarding how the eigenvalues and eigenvectors of these matrices can be used to interpret any physical meaning.

Within first quantization, the first-order case is the one-body density matrix is defined as, $$ \rho(x,x') = \int_{-\infty}^{\infty} \delta(x-x_1) \delta(x' - x_2)\lvert \Psi\left(x_1, \ldots x_A \right) \rvert^2 dx_1 \ldots dx_A $$ whose eigenvalues are the occupation numbers and eigenvectors are the natural orbitals. The eigenvalues lie in range $0 \leq n_i \leq 1$ and sum to the number of particles in the system. They represent how 'occupied' a given orbital is which in the non-interacting case is 1 or 0, and in the interacting case can be any value in between 0 and 1.

The natural orbitals I'm not 100% sure of, I believe they're the optimal orbital when constructing a wavefunction in accordance to Full Configuration Interaction? Or are they more similar to the orbitals of a Slater determinant in an optimal basis?

The next ordered density matrix is the two-body density matrix which is defined as, $$ \Gamma\left(x'_1, x'_2, x_1, x_2 \right) = \int \Psi^*\left(x'_1, x'_2, \ldots, x_A \right) \Psi\left( x_1, x_2, \ldots, x_A \right) dx_3 \ldots dx_A $$

which results in a function of 4 co-ordinates. Löwdin then states that the diagonal of this function holds special importance, $$ \Gamma\left( x'_1 = x_1, x'_2 = x_2 \right)$$

This is in a similar way to how the diagonal of the one-body density matrix is the classical probability distribution of finding a particle within a finite volume. However, for this function, it corresponds to finding the number of pairs within a finite volume instead.

Now, here's my question, what is the interpretation of the eigenvalues and eigenvectors for this diagonal of the two-body density matrix? Could the eigenvalues be like an occupation of the number of pairs within 2 given states? The eigenvectors I have no idea about but perhaps they are related to Geminals in some capacity? Although, as you can tell I'm not 100% sure about this!

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