Three-body Derivatives

The derivative respect to a parameter $\alpha_m$ of an orbital $m$ is given by:

$$\frac{\partial \ln U}{\partial \alpha_m}= \sum_{i,j} \sum_{l} \le... ...\lambda_{ml} \frac{\partial \phi_m(r_i)}{\partial \alpha_m} \phi_l(r_j) \right)$$ (B.29)

and the derivative respect to $\lambda_{ab}$ is

$$\frac{\partial \ln U}{\partial \lambda_{ab}} = \sum_{i,j} \phi_a(r_i)\phi_b(r_j)+ \phi_b(r_i)\phi_a(r_j)$$ (B.30)

because $\lambda$ matrix is symmetric.