Definitions and notations

It is defined as $$$$\hat{A} = \frac{1}{A!}\sum_{p} (-)^p\hat{P}, \tag{4}$$$$ with $$p$$ standing for the number of permutations. We have introduced for later use the so-called Hartree-function, defined by the simple product of all possible single-particle functions $$\Phi_H(x_1,x_2,\dots,x_A,\alpha,\beta,\dots,\nu) = \psi_{\alpha}(x_1) \psi_{\beta}(x_2)\dots\psi_{\nu}(x_A).$$