Coherent two-dimensional electronic and infrared crystallography

Abstract

The two-dimensional electronic and infrared spectroscopy of oriented single crystals is sensitive to structure and point group symmetry. The third order response of crystals is generally different from measurements of isotropic solutions because each coherence path that contributes to the measured field scales to the ensemble average of the four-point correlation functions of the four field-dipole interactions involved in the respective Feynman paths. An analytical evaluation of 2D optical crystallography which depends on the crystal symmetry, laboratory orientation as well as the orientation in the crystallographic frame is presented. Applying a symmetry operator in the basis of the allowed polarised radiation modes provides a method for evaluation of non-zero fourth rank tensor elements alternative to direct inspection methods. Uniaxial and biaxial systems are distinguished and the contributions to the rephasing and non-rephasing directions are evaluated for isolated and coupled oscillators. By exploiting coordinate analysis the extension of non-linear electronic and infrared crystallography for coupled oscillators demonstrates the structural, directional and symmetry dependent selection of coherences to the four-wave mixing signal.

Advances and opportunities in ultrafast X-ray crystallography and ultrafast structural optical crystallography of nuclear and electronic protein dynamics

“…A more complicated case would present itself if more than one dipole was present in the asymmetric unit, and coupling exists between them. In that case, the theoretical calculation has shown that depending on the crystal orientation and the polarization that is prepared, particular coherences and Liouville pathways become symmetry disallowed 40 . This has been calculated and shown for four-wave mixing conditions but is also true for the interaction of rank of two that is sufficient for the generation of interstate coherence.…”

“…In all other cases and assuming transparency, birefringence will produce two orthogonal polarizations which prepare a sum of two field-dipole interactions. The directions of these fields are given by the solutions to Maxwell's polarized wave equations and are in practice found either by evaluating the indicatrix or quadratic representation, method or the wave-surface method which are covered elsewhere 40,41 . In the presence of birefringence, the field decomposes as E)(|z=truebold-italice^1E1e−ik1z+truebold-italice^2E2e−ik2z, k1=ωμoϵ1=k0ne1, k2=ωμoϵ2=k0ne2,where E1 and E2 are the amplitudes of the fields of in truebold-italice^1& truebold-italice…”

“…The propagation of light in crystals and anisotropic media has been established starting with the Fresnel equations two centuries ago 104–109 . By combining knowledge from X-ray crystallographic coordinates and associated transition dipole gradients with the ensemble averaged field-dipole correlation of a given rank which becomes the sum of crystallographic symmetry progressions, a structural sensitivity results that can be directly visualized in the molecular basis 40 …”

“…As with the isotropic case, each contributing response function is weighted by the ensemble average of the field dipole interaction to the fourth rank 110,112 . The ensemble average of the dipole term is found for crystals according to the following equation: 40 ⟨true(μ^⋅e^true)h⟩=1n∑k=1ntrue(Ok~⋅μ^⋅e^true)h,which applies the symmetry operator O k with n-fold symmetry and a rank of h for the number of interactions.…”

“…In order to add a structural sensitivity in the molecular frame, ultrafast spectroscopy measurements of oriented single crystals have recently been successfully demonstrated 39 . An exciting prospect is that methods developed by the nonlinear spectroscopy community, when applied to oriented single crystals, should likewise yield a real-space analysis of coherences 40 . The two different techniques of ultrafast X-ray crystallography and ultrafast optical crystallography measure different aspects of structural dynamics but are married through the necessary application of crystal spectroscopy and crystal optics analysis in both cases 40,41 …”