10.9 Harmonic Vibrational Analysis 10.9.2 Isotopic Substitutions 10.9.4 Partial Hessian Vibrational Analysis

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

$\alpha_{\sigma\tau}(\omega_{\text{L}},\omega_{\text{S}})=\sum_{\upsilon}\left[% \frac{\langle f|\hat{r}_{\sigma}|\upsilon\rangle\langle\upsilon|\hat{r}_{\tau}% |i\rangle}{\hbar\omega_{\text{L}}-\hbar\omega_{\upsilon i}+i\Gamma_{i\upsilon}% }+\frac{\langle f|\hat{r}_{\tau}|\upsilon\rangle\langle\upsilon|\hat{r}_{% \sigma}|i\rangle}{\hbar\omega_{\upsilon i}+\hbar\omega_{\text{S}}+i\Gamma_{i% \upsilon}}\right]$

(10.19)

between initial state $|i\rangle$ and final state $|f\rangle$ . 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}

$\frac{I_{1}}{I_{2}}\approx\left(\frac{\omega^{\rm g}_{1}\Delta_{1}}{\omega^{% \rm g}_{2}\Delta_{2}}\right)^{\!2}\;.$

(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

$\Delta_{k}=\left(\frac{\omega_{k}}{\hbar}\right)^{1/2}\Delta Q_{k}$

(10.21)

evaluated at the ground-state geometry ( $\mathbf{Q}=\bm{0}$ ), 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:

$\alpha(\omega_{\text{L}})\propto\int_{0}^{\infty}e^{i\omega_{\text{L}}t-\Gamma t% }\langle\psi_{f}|\psi_{i}(t)\rangle dt+\text{NRT}\;.$

(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

$I_{k}\propto\omega_{\text{L}}(\omega_{\text{L}}-\omega_{k})^{3}(\omega_{k}% \Delta_{k})^{2}\;$

(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 $\Delta_{k}$ of the equilibrium position of this mode, expressed in normal coordinates, and the displacements $\widetilde{\Delta}_{i}$ expressed in Cartesian coordinates:

$\widetilde{\Delta}_{i}=\sum_{k=1}^{3N-6}\left(\frac{L_{ik}}{m_{i}^{1/2}}\right% )\Delta_{k}\;.$

(10.24)

In matrix form this is

$\mathbf{\Delta}_{Q}=\lambda^{-1}\mathbf{L}^{\dagger}\mathbf{M}^{1/2}\mathbf{V}% _{X}$

(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

$\Delta_{k}=\left(\frac{\lambda_{k}}{m_{e}}\right)^{\!1/4}\Delta_{Qk}\;.$

(10.26)

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

RR_NO_NORMALISE
       Controls whether frequency job calculates resonance-Raman intensities
TYPE:
       LOGICAL
DEFAULT:
       False
OPTIONS:
       False Normalise RR intensities True Doesn’t normalise RR intensities
RECOMMENDATION:
       False

Example 10.19 Calculating 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

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10.9.3 Resonance-Raman intensities