10.7.6 Resonance Raman intensities

The theory of resonance Raman spectroscopy is fully described by the Kramers-Heisenberg-Dirac dispersion formalism based on the Raman polarizability tensor ⁶⁴⁰ Kelley A. M.
J. Phys. Chem. A
(2008), 112, pp. 11975. Link

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. (10.45) 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, ³⁵ Albrecht A. C.
J. Chem. Phys.
(1961), 34, pp. 1476. Link 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. ³⁵ Albrecht A. C.
J. Chem. Phys.
(1961), 34, pp. 1476. Link ^{, 640} Kelley A. M.
J. Phys. Chem. A
(2008), 112, pp. 11975. Link 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, ¹¹⁶³ Sharp T. E., Rosenstock H. M.
J. Chem. Phys.
(1964), 41, pp. 3453. Link 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, ¹⁰⁰⁸ Petrenko T., Neese F.
J. Chem. Phys.
(2007), 127, pp. 164319. Link in which resonant enhancements to the vibrational intensities (for modes 1 and 2, say) are expressed as ratios ⁵¹³ Heller E. J., Sundberg R. L., Tannor D.
J. Phys. Chem.
(1982), 86, pp. 1822. Link ^{, 908} Myers A. B.
Chem. Rev.
(1996), 96, pp. 911. Link ^{, 285} Dasgupta S., Rana B., Herbert J. M.
J. Phys. Chem. B
(2019), 123, pp. 8074. Link

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. (10.46), 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. (10.46).

The time-dependent picture provides means to derive this expression. ⁵¹³ Heller E. J., Sundberg R. L., Tannor D.
J. Phys. Chem.
(1982), 86, pp. 1822. Link 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. ⁵¹³ Heller E. J., Sundberg R. L., Tannor D.
J. Phys. Chem.
(1982), 86, pp. 1822. Link ^{, 908} Myers A. B.
Chem. Rev.
(1996), 96, pp. 911. Link ^{, 285} Dasgupta S., Rana B., Herbert J. M.
J. Phys. Chem. B
(2019), 123, pp. 8074. Link 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

Setting JOBTYPE = RRAMAN invokes the calculation of resonance Raman intensities.