The Dipole-Quadrupole theory of Surface Enhanced Hyper Raman Scattering (SEHRS), created by the authors is expounded in details. Peculiarities of the behavior of electromagnetic field on rough metal surfaces are considered. It is demonstrated that there is an enhancement of the dipole and quadrupole light-molecule interaction near the places with a large curvature. The expression for the SEHRS cross-section of symmetrical molecules, which consists of several contributions is obtained. Selection rules for the scattering contributions are obtained and a qualitative classification of the contributions after an enhancement degree is performed. Analysis of experimental spectra of pyrazine and phenazine, and also some another molecules is performed too. It is demonstrated a full coincidence of experimental regularities in these spectra with the theory suggested.
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