General solution to nonlinear optical quantum graphs using Dalgarno–Lewis summation techniques

We develop an algorithm to apply the Dalgarno-Lewis (DL) perturbation theory to quantum graphs with multiple connected edges. We use it to calculate the nonlinear optical hyperpolarizability tensors for graphs and show that it replicates the sum over states computations but executes 10 to 50 times faster. DL requires only knowledge of the ground state of the graph, eliminating the requirement to determine all possible degeneracies of a complex network. The algorithm is general and may be applied to any quantum graph.