In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as Gauss-Legendre quadrature will not be adequate for say the computation of the energy of a many-body system.
Here we have defined the vector \( \boldsymbol{R} = [\boldsymbol{r}_1,\boldsymbol{r}_2,\dots,\boldsymbol{r}_n] \) as an array that contains the positions of all particles \( n \) while the vector \( \boldsymbol{\alpha} = [\alpha_1,\alpha_2,\dots,\alpha_m] \) contains the variational parameters of the model, \( m \) in total.