A Unified Treatment of Selection Rules and Symmetry Relations for Sum-Frequency and Second Harmonic Spectroscopies

Abstract

A simplifying treatment is developed for describing the molecular origins of electric dipole allowed sumfrequency generation (SFG) and second harmonic generation (SHG). The full sum-over-states expressions for the nonlinear polarizability simplify tremendously at or near resonance to straightforward formulas easily connected to intuitive molecular properties. For resonance enhancement at the sum or second harmonic frequency, the molecular nonlinear polarizability tensor is shown to be the direct product of the transition moment and the two-photon absorption (TPA) polarizability tensor. To our knowledge, this is the first rigorous mathematical demonstration indicating such a simple relationship directly connecting second harmonic generation with TPA, providing a link between the two fields of inquiry. Under resonance enhancement with one of the incident frequencies, the SFG and SHG nonlinear polarizability tensors similarly are given by the products of the transition moments and the anti-Stokes Raman polarizability tensors (a reasonably wellknown result for SFG). Under double-resonance conditions (i.e., resonant with one of the incident frequencies and the sum frequency), the two descriptions for the nonlinear polarizability become mathematically equivalent. Nonlinear optical character tables for both SHG and SFG under all resonance conditions have been compiled for chromophores of C s , C 2 , C 2V , and C 3V symmetries. Explicit evaluation of the corresponding orientational averages for each allowed transition in each character table assuming a uniaxial macroscopic orientation distribution reveals numerous relationships connecting the microscopic symmetry with the macroscopic nonlinear response. The approaches developed in this work are sufficiently general to allow their use in interpreting electronic, vibrational, and vibronic spectroscopic measurements by SHG and SFG.

IM30 triggers membrane fusion in cyanobacteria and chloroplasts

“…To account for homogeneous broadening, we convolute the result with a Lorentzian line shape. The response is transformed from the molecular frame to the lab frame by averaging the Euler transformation between the two frames over the orientational distribution of the molecules 64,65 , which gives the nonlinear susceptibility w with PLR being the protein oscillator-to-lipid oscillator ratio near the interface. Thus, the calculated VSFG response is dependent on the orientation and magnitude of the transition dipole moment l and Raman polarizability tensor a of the local modes of the system, the protein and lipid orientation distribution, the line widths of the Lorentzians of the protein and lipid normal modes, G protein amide-I and G lipid ester-C ¼ O , the overall frequency offsets for the amide-I and ester C ¼ O stretch modes, n 0, protein amide-I and n 0, lipid ester-C ¼ O , and on the PLR.…”

Contrasting organic aerosol particles from boreal and tropical forests during HUMPPA-COPEC-2010 and AMAZE-08 using coherent vibrational spectroscopy

“…Conversely, methylene CH stretching modes can appear in vibrational SFG spectra at the following frequencies: asymmetric CH 2 stretching modes appear in the range 2920-2940 cm −1 and symmetric CH 2 stretching modes appear between 2840 and 2870 cm −1 . In this present work, we present the first vibrational SFG spectra of atmospherically relevant terpenes and secondary organic aerosol material formed from them, which we assign by combining the existing literature data with the general practice that symmetric stretches and Fermi resonances are 180 • out of phase with respect to their corresponding asymmetric stretches when they are probed using the SSP polarization (Lu et al, 2004;Moad et al, 2004). The following section provides a brief overview of the theoretical background of our spectroscopic measurement technique.…”

Detection of chiral sum frequency generation vibrational spectra of proteins and peptides at interfaces in situ

“…By assuming a molecular coordinate system for an antiparallel ␤-sheet as shown in Fig. 4, the SFG hyperpolarizability tensor elements (40,46,74,75) can be written as…”