  We propose a new scheme to implement the self-consistent optimization of the
trial wave-function in constrained path auxiliary field Quantum Monte Carlo
(CP-AFQMC) in the framewok of natural orbitals. In this scheme, a new trial
wave-function in the form of Slater determinant is constructed from the
CP-AFQMC results by diagonalizing the mixed estimator of the one-body reduced
density matrix. We compare two ways (from real and mixed estimators in
CP-AFQMC) to calculate the one-body reduced density matrix in the
self-consistent process and study the ground state of doped two dimensional
Hubbard model to test the accuracy of the two schemes. By comparing the local
density, occupancy, and ground state energy we find the scheme in which
one-body reduced density matrix is calculated from mixed estimator is
computational more efficient and provides more accurate result with less
fluctuation. The local densities from mixed estimator scheme agree well with
the numerically exact values. This scheme provides a useful tool for the study
of strongly correlated electron systems.
