Consequences of Kramers–Kronig relations on Kleinman antisymmetric spectral observables: Chiral-specific sum frequency generation spectroscopy

Investigating water structures at aqueous interfaces has been a central focus in the field of nonlinear surface spectroscopy over decades. This large body of work leads to a conclusion that asymmetric OH stretches should be largely silent in sum frequency generation (SFG) spectra. Nonetheless, our recent studies show chiral-specific SFG response of water originating from the first hydration shell of proteins and DNA arising from a sum of equal-magnitude and opposite-phase symmetric and asymmetric OH stretches, resulting in an oppositely signed couplet. Motivated initially by this apparently inconsistent behavior for chiral-specific vs conventional (achiral) SFG, we demonstrate herein a fundamental argument for the appearance of the oppositely signed couplets in chiral-specific vibrational SFG. The interplay between two foundational optical relations, Kramers-Kronig relations and index-interchange symmetry arising in the adiabatic zero-frequency limit (often referred to as Kleinman symmetry), is shown to require the imaginary resonant contributions to the tensors describing chiral-specific SFG responses to collectively sum to zero. In brief, all indices within the surface susceptibility must become interchangeable in the degenerate, zero-frequency adiabatic limit. From Kramers-Kronig relations connecting the real and imaginary susceptibility, this asymptotic limit in the real susceptibility can only be met if the imaginary-valued resonant contributions over the relevant spectral range sum to zero. These symmetry constraints were found to agree with density functional theory calculations on small water clusters and with experimental and computational phase-resolved chiral-specific SFG spectra. These symmetry requirements provide constraints for analyzing phase-resolved chiral-specific SFG spectra for extracting structural information about chiral molecules at interfaces.