and are equal, as are the diagrams and ; they differ only in that the integration variables and are interchanged, whereas the COULOMB potential is symmetric under this substitution. It is therefore sufficient to retain just one diagram of each type, simultaneously omitting the factor in front of (3.38). For the th-order perturbation there are ! possible interchanges of integration variables. Therefore, the repetition of the same diagrams cancels the factor in (3.31).
Diagram and contain sub-units that are not connected by any lines to the rest of the diagram. Feynman diagrams in which all parts are not connected are called disconnected diagrams. Equation (3.38) shows that such diagrams are typically have GREEN's function and interactions whose arguments close on themselves. As a result the contribution of this sub-unit can be factored out of the expression for . The same procedure can be applied for the denominator. In this cases, the second term of the expansion includes only two non-vanishing terms which are only disconnected diagrams of (3.38), namely (a) and (d).
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors