Pairing determinant

Let us define the matrix $A_{ij}$ as:

$$A_{ij} = \Phi(r_i,r_j)=\sum_{l,m}{\lambda_{l,m}\phi_{l}(r_i)\phi_{m}(r_j)}$$ (B.8)

where $i$ are coordinates of spin up electrons and $j$ of spin down electrons. For polarized system is possible extend the definition of the geminal wave-function. This generalization was first proposed by Coleman (110). In practise if $N^\uparrow > N^\downarrow$ we can define a $N^\uparrow x N^\uparrow$ matrix $A_{ij}$ in the following way:

$$A_{ij} = \Phi(r^\uparrow_i,r^\downarrow_j) j = 1,N^\downarrow$$ (B.9)

$$= \bar \phi_j(r_i^\uparrow) j=N^\downarrow+1,N^\uparrow$$ (B.10)

When we move an electron to ratio between the old and the new determinant will be given for a spin down electron:

$$\left\vert A(r_i,r'_k) \right\vert = \sum_i \Phi(r_i,r'_k) \left\vert A(r_i,r_k) \right\vert A_{ik}^{-1}$$

$$\frac{\left\vert A(r_i,r'_k) \right\vert }{ \left\vert A(r_i,r_k) \right\vert } = \sum_i \Phi(r_i,r'_k) A_{ik}^{-1}$$

$$\frac{\left\vert A(r_i,r'_k) \right\vert }{ \left\vert A(r_i,r_k) \right\vert } = \sum_i \sum_{l,m}{\lambda_{l,m} \phi_{l}(r_i)\phi_{m}(r_k')} A_{ik}^{-1}$$ (B.11)

and for spin up:

$$\left\vert A(r'_i,r_k) \right\vert = \sum_k \Phi(r'_i,r_k) \left\vert A(r'_i,r_k) \right\vert A_{ik}^{-1}$$

$$\frac{\left\vert A(r'_i,r_k) \right\vert }{ \left\vert A(r'_i,r_k) \right\vert } = \sum_k \Phi(r'_i,r_k) A_{ik}^{-1}$$

For updating the inverse matrix $A^{-1}$ , used here, after a move we follow the simple formula used for the Slater determinant(38) with indices that depend from the spin of the electron.