Simplified diagrammatic expansion for effective operator
  For a quantum many-body problem, effective Hamiltonians that give exact eigenvalues in reduced model space usually have different expressions, diagrams and evaluation rules from effective transition operators that give exact transition matrix elements between effective eigenvectors in reduced model space. By modifying these diagrams slightly and considering the linked diagrams for all the terms of the same order, we find that the evaluation rules can be made the same for both effective Hamiltonian and effective transition operator diagrams, and in many cases it is possible to combine many diagrams into one modified diagram. We give the rules to evaluate these modified diagrams and show their validity. 
Chang-Kui Duan, Yungui Gong, Hui-Ning Dong and Michael F. Reid