  Hamiltonian Truncation Methods are a useful numerical tool to study strongly
coupled QFTs. In this work we present a new method to compute the exact
corrections, at any order, in the Hamiltonian Truncation approach presented by
Rychkov et al. in Refs. [1-3]. The method is general but as an example we
calculate the exact $g^2$ and some of the $g^3$ contributions for the $\phi^4$
theory in two dimensions. The coefficients of the local expansion calculated in
Ref. [1] are shown to be given by phase space integrals. In addition we find
new approximations to speed up the numerical calculations and implement them to
compute the lowest energy levels at strong coupling. A simple diagrammatic
representation of the corrections and various tests are also introduced.
