A unified approach to surface-enhanced Raman spectroscopy

We present a unified expression for surface-enhanced Raman spectroscopy (SERS). The expression contains a product of three resonance denominators, representing the surface plasmon resonance, the metal-molecule charge-transfer resonance at the Fermi energy, and an allowed molecular resonance. This latter resonance is that from which intensity is borrowed for charge transfer, and when the molecular resonance is active it is responsible for surface-enhanced resonance Raman spectroscopy. We examine this expression in various limits, to explore the relative contribution or each resonance. First, we look at the situation in which only the surface plasmon resonance is active and examine the various contributions to the Raman signal, including the surface selection rules. Then we examine additional contributions from charge-transfer or molecular resonances. We show that the three resonances are not totally independent, since they are linked by a product of four matrix elements in the numerator. These linked matrix elements provide comprehensive selection rules for SERS. One involves a harmonic oscillator in the observed normal mode. This is the same mode which appears in the. vibronic coupling operator linking one of the states of the allowed molecular resonance to the charge-transfer state. The charge-transfer transition moment is linked to the surface plasmon resonance by the requirement that the transition dipole moment be polarized along the direction of maximum amplitude of the field produced by the plasmon (i.e., perpendicular to the metal surface). We show that these selection rules govern the observed SERS spectral intensities and apply these to the observed spectra of several molecules. We also suggest a quantitative measure of the degree to which charge transfer contributes to the overall SERS enhancement.