# Is there a connection between the definitions of one- and two-particle reduced density matrices?

In quantum chemistry, there are concepts about one-particle reduced density matrix (1-RDM) and similarly, the two-particle reduced density matrix (2-RDM). Generally, for an $$n$$ particle wavefunction $$|\Psi\rangle$$ and the corresponding density matrix is $${}^nD=|\Psi\rangle\langle\Psi|$$. Then there are two definitions of the 1-RDM and 2-RDM. The first is: $$$$\tag{1} {}^1D= \begin{pmatrix} 1 \\ n \end{pmatrix} tr_{2,3,\cdots,n} {}^nD,$$$$ in which will resulting $$C_n^1=n$$ density matrix. Another method is to define the element of 1-RDM as: $$$$\tag{2} {}^1D_{ij} = \langle \Psi|a_i^{\dagger} a_j|\Psi\rangle,$$$$ in which $$i$$ and $$j$$ run over $$1$$ to $$n$$ and the matrix will be a $$n\times n$$ one. Is there any connection or difference between the two representations?

• Can you specify what you mean by $a^{\dagger}_{i}$ and $a_{j}$? Are they ladder operators?
– JSdJ
Dec 14, 2020 at 9:59
• There are creation and annihilation operators.
– 刘环宇
Dec 14, 2020 at 10:47
• You'll find more people that know what you're talking about here: mattermodeling.stackexchange.com Dec 14, 2020 at 19:53
• If you are interested in moving the question to Matter Modeling, I'm a mod there and we would be open to having it migrated. I can contact the mods here to have it moved. Dec 15, 2020 at 23:12
• OK. Thanks a lot.
– 刘环宇
Dec 16, 2020 at 8:12