We write the Hamilton operator, or Hamiltonian, in a generic way $$ \hat{H} = \hat{T} + \hat{V} $$ where \( \hat{T} \) represents the kinetic energy of the system $$ \hat{T} = \sum_{i=1}^A \frac{\mathbf{p}_i^2}{2m_i} = \sum_{i=1}^A \left( -\frac{\hbar^2}{2m_i} \mathbf{\nabla_i}^2 \right) = \sum_{i=1}^A t(x_i) $$ while the operator \( \hat{V} \) for the potential energy is given by $$ \begin{equation} \hat{V} = \sum_{i=1}^A \hat{u}_{\mathrm{ext}}(x_i) + \sum_{ji=1}^A v(x_i,x_j)+\sum_{ijk=1}^Av(x_i,x_j,x_k)+\dots \tag{2} \end{equation} $$ Hereafter we use natural units, viz. \( \hbar=c=e=1 \), with \( e \) the elementary charge and \( c \) the speed of light. This means that momenta and masses have dimension energy.