I am reading the book on MONTE CARLO METHODS IN AB INITIO QUANTUM CHEMISTRY by Hammond and Lester. (Chapter 5-Variational Trial Functions, Section 5.3-Hartree Fock and Beyond, Sub-Section 5.3.3 Hartree Fock and Correlation Energies)In section 5.3.3, the book says the following,

"In Monte Carlo, we may factor the determinant into electron spin components, i.e., $\psi_{D}=\psi_{\alpha}\psi_{\beta}$. This factorization is justified because all possible permutations of this term that are present in the full antisymmetric $\psi$ are operationally equivalent: each term simply corresponds to a relabeling of the electrons. Expectation values, and even local values of any observable, are unchanged."

I couldn't understand how these two are equal. Is it really possible to factorize this into two slater determinants? How does one justify that this is true?


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