3.3.1 Exchange Energy

The exchange energy for one Cartesian component is given by

$$E_{\mathrm{exch}}= \int_\Omega \sum_j A (\nabla u_j \eta_j)^2 d{v} \quad.$$ (3.15)

For the gradient we obtain

$$\frac{\partial E_\mathrm{exch}}{\partial u_i}= \int_\Omega... ...{\partial u_i} \left( \nabla u_j \eta_j \right)^2 d{v}$$ (3.16)

$$\frac{\partial}{\partial u_i} \left( \nabla u_j \eta_j \right)^2 = 2 u_j \nabla \eta_j \cdot \frac{\partial u_j \nabla \eta_j}{\partial u_i} =$$ (3.17)

$$= 2 u_j \nabla \eta_j \cdot \nabla \eta_j \delta_{ij}=$$ (3.18)

$$= 2 u_j \nabla \eta_j \cdot \nabla \eta_i \quad.$$ (3.19)

Finally, the gradient of the exchange energy is given by

$$\frac{\partial E_\mathrm{exch}}{\partial u_i }= 2 A \int_\... ...a \sum_j u_j \nabla \eta_j \cdot \nabla \eta_i d{v} \quad,$$ (3.20)

which can be written as a linear system of equations with the coefficient matrix

$$G_{\mathrm{exch},ij}=2 A \int_\Omega \nabla \eta_j \cdot \nabla \eta_i d{v} \quad.$$ (3.21)

The gradient can then be simply calculated as

$$\boldsymbol{g}_\mathrm{exch} = G_\mathrm{exch} \cdot \boldsymbol{u} \quad.$$ (3.22)

The expressions for the $x$, $y$, and $z$-component are identical and there are no mixed terms.

This exchange energy matrix is proportional to the stiffness matrix of the Laplacian operator Eq. (2.15), which is also obvious from Brown's equation Eq. (3.4) and the effective field Eq. (3.5).