When Kleinman-Bylander pseudopotentials are used the Hamiltonian operator is given by $$\hat{H} = -\frac{1}{2}\nabla^2+V_{\textrm{local}}+\delta \hat{V}_{\textrm{NL}}$$ where $$\hat{V}_{\textrm{NL}} = \sum_{lm}\frac{\vert\chi^{\textrm{PS}}_{lm}\rangle\langle{}\chi^{\textrm{PS}}_{lm}\vert} {\langle{}\chi^{\textrm{PS}}_{lm}\vert \psi^{\textrm{PS}}_{lm}\rangle} $$ See [1]. What happens if we have two orbitals with same $l$ and $m$ but different principal quantum number?

[1] Richard M. Martin. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press (2004).


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