Theory of double quantum two-dimensional electron spin resonance with application to distance measurements

A formulation is presented for calculating double quantum two dimensional electron spin resonance (DQ-2D ESR) spectra in the rigid limit that correspond to recent experimental DQ-2D ESR spectra obtained from a nitroxide biradical. The theory includes the dipolar interaction between the nitroxide moieties as well as the fully asymmetric $g$ and hyperfine tensors and the angular geometry of the biradical. The effects of arbitrary pulses (strong but not truly nonselective pulses) are included by adapting the recently introduced split Hamiltonian theory for numerical simulations. It is shown how arbitrary pulses in magnetic resonance create “forbidden” coherence pathways, and their role in DQ-2D ESR is delineated. The high sensitivity of these DQ-2D ESR signals to the strength of the dipolar interaction is demonstrated and rationalized in terms of the orientational selectivity of the “forbidden” pathways. It is further shown that this selectivity also provides constraints on the structural geometry (i.e., the orientations of the nitroxide moieties) of the biradicals. The theory is applied to the recent double quantum modulation (DQM) experiment on an end-labeled poly-proline peptide biradical. A distance of 18.5 Å between the ends is found for this biradical. A new two pulse double quantum experiment is proposed (by analogy to recent NMR experiments), and its feasibility for the ESR case is theoretically explored.