Three-body

In the same spirit of the pairing determinant we built a three-body factor as:

$$U = \exp \left ( \sum^{N_{elec}}_{i,j} u(\vec{r}_i,\vec{r}_j) \right )$$ (B.24)

$$u(\vec{r}_i,\vec{r}_j) = \sum^{N_{orb}}_{m,n}\lambda_{mn}\phi_m(r_i)\phi_n(r_j)$$ (B.25)

When you move an electron $r_k$ the ratio between the two three-body factor is given by:

$$\frac{U(r_k')}{U(r_k)} = \exp \left( \sum_{m,n} { \left [ \sum_{(... ... \right ) + \phi_m(r_k') \phi_n(r_k')-\phi_m(r_k) \phi_n(r_k) \right] } \right)$$ (B.26)

and if you accept the move to update the value of three-body you have only to update $N_{orb}$ orbitals.