# What is a relaxed reduced density matrix?

I am trying to understand how to calculate electronic properties using reduced density matrices from electronic structure calculation. So far, I understand (I think) that for the expectation value of an n-electron operator, you would need the n-electron reduced density with $$\langle \hat{O}\rangle = \mathrm{Tr}(\Gamma^{(n)} O ).\tag{1}$$ I have also found that these n-electron RDMs can be obtained, for example, from PySCF, as described here.

When trying to understand the practicality on how to use these quantities, however, I've encountered the distinction between relaxed and unrelaxed RDMs, for example, in this question, where it is stated that the trace of the 2RDM is incorrect in MP2, not giving $$N(N-1)$$ as it should, but it should be correct in FCI, and the reason for this is that the matrix is unrelaxed. I tried to find an explanation to what exactly is the difference between relaxed and unrelaxed RDMs - apart from the intuition that some terms are missing -, but I have found no success. This concept seems to be beyond normal textbook knowledge, while maybe also being obvious for researchers in the field so that it is not really explained in papers/dissertations.

What is exactly the difference between a relaxed and unrelaxed nRDMs? I assume I can not just relax an unrelaxed quantity as post-processing, but do I need to do this at all? Or are quantities obtained from unrelaxed RMDs accurate enough?

• +1 but you wrote: "I've encountered the distinction between relaxed and unrelaxed RDMs, for example, in this question, where it is stated that the trace of the 2RDM is incorrect in MP2, not giving N(N−1) as it should, but it should be correct in FCI, and the reason for this is that the matrix is unrelaxed," but the word "relax" does not appear in that question's thread at all apart from "This is from the definition of the MP2 density matrix (it's not the relaxed one)" and "The code for MP2 1- and 2-RDMs comes from the MP2 Hylleraas functional without orbital relaxation." Commented Jan 3 at 18:17