## Definitions and notations

The expectation value of $$\hat{H}_0$$ $$\int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = A! \int \Phi_H^*\hat{A}\hat{H}_0\hat{A}\Phi_H d\mathbf{\tau}$$ is readily reduced to $$\int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = A! \int \Phi_H^*\hat{H}_0\hat{A}\Phi_H d\mathbf{\tau},$$ where we have used Eqs. (5) and (6). The next step is to replace the antisymmetrization operator by its definition and to replace $$\hat{H}_0$$ with the sum of one-body operators $$\int \Phi^*\hat{H}_0\Phi d\mathbf{\tau} = \sum_{i=1}^A \sum_{p} (-)^p\int \Phi_H^*\hat{h}_0\hat{P}\Phi_H d\mathbf{\tau}.$$