With the realization that the intruder state problem manifests itself when the energy denominator in Equation ( ) vanishes, a possible solution would be to add a small value ε > 0 to ∆ i , such that when ∆ i → 0, T i → -V i /ε. This is the simple idea behind level shift techniques. It can be formally implemented in any perturbation theory approach by modifying the partitioning of the Hamiltonian as follows