10.9.3 Resonance-Raman intensities

The theory of resonance-Raman spectroscopy is fully described by the Kramers-Heisenberg-Dirac dispersion formalism based on the Raman polarizability tensor⁴⁶⁷

$${\alpha }_{\sigma \tau }({\omega }_{\text{L}},{\omega }_{\text{S}})=\sum _{\upsilon }\left[\frac{⟨f|{\widehat{r}}_{\sigma }|\upsilon ⟩⟨\upsilon |{\widehat{r}}_{\tau }|i⟩}{\mathrm{\hslash }{\omega }_{\text{L}}-\mathrm{\hslash }{\omega }_{\upsilon i}+i{\mathrm{\Gamma }}_{i\upsilon }}+\frac{⟨f|{\widehat{r}}_{\tau }|\upsilon ⟩⟨\upsilon |{\widehat{r}}_{\sigma }|i⟩}{\mathrm{\hslash }{\omega }_{\upsilon i}+\mathrm{\hslash }{\omega }_{\text{S}}+i{\mathrm{\Gamma }}_{i\upsilon }}\right]$$

(10.19)

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. (10.19) 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,^{15, 947} 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.^{15, 947, 467} 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,⁸⁷⁶ 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,⁷⁴⁷ in which resonant enhancements to the vibrational intensities (for modes 1 and 2, say) are expressed as ratios^{368, 675, 204}

$$\frac{I_1}{I_2}\approx {\left(\frac{{\omega }_1^{\mathrm{g}}{\mathrm{\Delta }}_1}{{\omega }_2^{\mathrm{g}}{\mathrm{\Delta }}_2}\right)}^2.$$

(10.20)

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. (10.20), written as an approximation here, is exact within the IMDHO model. The quantity

$${\mathrm{\Delta }}_k={\left(\frac{{\omega }_k}{\mathrm{\hslash }}\right)}^{1/2}\mathrm{\Delta }{Q}_k$$

(10.21)

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. (10.20).

The time-dependent picture provides means to derive this expression.³⁶⁸ 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:

$$\alpha ({\omega }_{\text{L}})\propto {\int }_0^{\mathrm{\infty }}{e}^{i{\omega }_{\text{L}}t-\mathrm{\Gamma }t}⟨{\psi }_f|{\psi }_i(t)⟩𝑑t+\text{NRT}.$$

(10.22)

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.^{368, 675, 204} Within a model that considers only two electronic states, the RR intensity that one obtains is

$$I_k\propto {\omega }_{\text{L}}{({\omega }_{\text{L}}-{\omega }_k)}^3{({\omega }_k{\mathrm{\Delta }}_k)}^2$$

(10.23)

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:

$${\tilde{\mathrm{\Delta }}}_i=\sum _{k=1}^{3N-6}\left(\frac{L_{ik}}{m_i^{1/2}}\right){\mathrm{\Delta }}_k.$$

(10.24)

In matrix form this is

$${𝚫}_Q={\lambda }^{-1}{𝐋}^{†}{𝐌}^{1/2}{𝐕}_X$$

(10.25)

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

$${\mathrm{\Delta }}_k={\left(\frac{{\lambda }_k}{m_e}\right)}^{1/4}{\mathrm{\Delta }}_{Qk}.$$

(10.26)

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

$molecule
0 1
  C         1.8288506578   -0.1219336002    0.0000000000
  C         0.6155951063    0.3987918905    0.0000000000
  C        -0.6155955606   -0.3987931260    0.0000000000
  C        -1.8288502653    0.1219348794    0.0000000000
  H         2.7085214046    0.4909328271    0.0000000000
  H         1.9881851899   -1.1843222290    0.0000000000
  H         0.4885913610    1.4671254626    0.0000000000
  H        -0.4885933454   -1.4671268234    0.0000000000
  H        -1.9881816088    1.1843239478    0.0000000000
  H        -2.7085226822   -0.4909289672    0.0000000000
$end

$rem
 method          hf
 jobtype         RRAMAN !Jobtype resonance Raman
 basis           3-21G
 cis_n_roots 1
 cis_state_derivative 1 ! Excited state
$end