Gradients and Laplacian

The gradients and laplacian of the logarithm of three-body term are given by:

$$\vec{\nabla}_k \ln U = \sum_{j}\sum_{m , n} \lambda_{mn} \vec{\nabla}_k \phi_m(r_k) \phi_n(r_j)$$ (B.27)

$$\nabla^2_k \ln U = \sum_{j}\sum_{m , n} \lambda_{mn} \nabla^2_k \phi_m(r_k) \phi_n(r_j)$$ (B.28)