Majorana, Bartlett, Heisenberg Exchanges & Spin Interactions

[Petar Mali]

$$\hat{\mathbb{P}}_M\psi(\vec{r}_1,\xi_1;\vec{r}_2,\xi_2)=\psi(\vec{r}_2,\xi_1;\vec{r}_1,\xi_2)$$

Majorana exchange
$$\hat{\mathbb{P}}_B\psi(\vec{r}_1,\xi_1;\vec{r}_2,\xi_2)=\psi(\vec{r}_1,\xi_2;\vec{r}_2,\xi_1)$$

Bartlett exchange
$$\hat{\mathbb{P}}_H\psi(\vec{r}_1,\xi_1;\vec{r}_2,\xi_2)=\psi(\vec{r}_2,\xi_2;\vec{r}_1,\xi_1)$$

Heisenberg exchange

Where $$\vec{r}_1,\vec{r}_2$$ are spatial and $$\xi_1, \xi_2$$ are the spin coordinates of two electrons.

How to show that

$$\hat{\mathbb{P}}_B=\frac{1}{2}(1+4\hat{\vec{S}}_1 \cdot \hat{\vec{S}}_2)$$ and how to show

$$\hat{H}=-\sum_{i,j}I_{i,j}\hat{\vec{S}}_i \cdot \hat{\vec{S}}_j$$

 

[BigJDubs]

Where \hat{\vec{S}}_i is the Pauli spin operator of the i-th electron and I_{i,j} is the exchange integral between the i-th and j-th electrons.