10.8.5 Resonance-Raman intensities

The theory of resonance-Raman spectroscopy is fully described by the Kramers-Heisenberg-Dirac dispersion formalism based on the Raman polarizability tensor⁵⁵⁸

between initial state $|i⟩$ and final state $|f⟩$. Here, ${\omega }_{\text{L}}$ and ${\omega }_{\text{S}}$ are the frequencies of the laser (incident photon) and of the scattered photon, respectively. Eq. (10.23) is inconvenient due to the sum over intermediate states $\upsilon$ (vibrational levels on all accessible electronic states), and the usual procedure is to expand the static molecular polarizability as a Taylor series in the normal coordinates,^{34, 1109} which allows the Raman intensity to be decomposed into Franck–Condon (or “$A$-term”) contributions and coordinate-dependent Herzberg–Teller (“$B$”- and “$C$”-term) contributions.^{34, 1109, 558} Nevertheless, each term contains sums over intermediate vibrational states and becomes difficult to evaluate for large molecules with numerous vibrational modes.

As such, in most cases only the lowest-lying Raman-active excited electronic state is considered in computing the RR spectrum. In principle one should consider the effects of Duschinsky rotation,¹⁰²⁹ i.e., the fact that the normal modes are different in each electronic state. Neglecting this effect for simplicity and thus using ground-state normal modes only, one arrives at the “independent-mode, displaced harmonic oscillator” (IMDHO) model,⁸⁸⁶ in which resonant enhancements to the vibrational intensities (for modes 1 and 2, say) are expressed as ratios^{448, 800, 250}

In this equation, ${\omega }_1^{\text{g}}$ and ${\omega }_2^{\text{g}}$ represent the ground-state vibrational frequencies for normal modes $Q_1$ and $Q_2$ and $\omega$ is the electronic excitation energy. The first equality in Eq. (10.24), written as an approximation here, is exact within the IMDHO model. The quantity

evaluated at the ground-state geometry ($𝐐=\mathrm{𝟎}$), is the slope of the excited-state potential energy surface along mode $k$. This leads to the second equality in Eq. (10.24).

The time-dependent picture provides means to derive this expression.⁴⁴⁸ In this approach, the requisite polarizability tensor elements involving different electronic states are expressed as the Fourier transformation of the time-evolving overlap between initial- and final-state electronic wave functions:

Here, “NRT” indicates the non-resonant terms that are neglected in RR spectroscopy. Large molecules likely spend no more than 10–20 fs in the Franck-Condon region and the overlap integral is likely only significant on that timescale.^{448, 800, 250} Within a model that considers only two electronic states, the RR intensity that one obtains is

where ${\mu }_k$ is the reduced mass of the $k$th normal mode.

Assuming identical force constants for $Q_k$ in both the ground and excited electronic state, one obtains a linear transformation between the displacement ${\mathrm{\Delta }}_k$ of the equilibrium position of this mode, expressed in normal coordinates, and the displacements ${\tilde{\mathrm{\Delta }}}_i$ expressed in Cartesian coordinates:

In matrix form this is

where $\lambda$ is the eigenvalues of mass-weighted Hessian matrix, M defines the matrix of atomic masses and $V_X$ is the energy gradient in Cartesian coordinate Raman intensities are related to the dimensionless displacements

JOBTYPE set to RRAMAN invokes the calculation of resonance-Raman intensities.