10.8.6 Vibrationally-Resolved Electronic Spectra and Resonance Raman Simulations

On the basis of perturbation theory, the transition rate of one- or two-photon transition processes from the initial state $|I⟩$ to the final state $|F⟩$ is proportional to $k_{IF}={|⟨F|\widehat{M}|I⟩|}^2\delta (\mathrm{\Delta }\omega )$, where $\delta (\mathrm{\Delta }\omega )$ is the line shape function with $\mathrm{\Delta }\omega ={\omega }_i-{\omega }_{FI}$ for a one-photon transition and $\mathrm{\Delta }\omega ={\omega }_1+{\omega }_2-{\omega }_{FI}$ for a two-photon process. Here ${\omega }_i$, ${\omega }_1$, and ${\omega }_2$ denote the incident photon frequencies and the operator $\widehat{M}$ is given by

In the two-photon case there are intermediate or “virtual” states $|L⟩$.

Raman scattering is a two-photon process. In this process, one photon with the frequency ${\omega }_i$ is absorbed, another photon with the frequency ${\omega }_S$ is emitted, and the transition from the initial to the final vibrational states takes place. Based on perturbation theory, the transition rate of the Raman process is proportional to $S({\omega }_i,{\omega }_S)={|⟨F|\widehat{M}|I⟩|}^2\delta (\mathrm{\Delta }\omega )$, where

and $\mathrm{\Delta }\omega ={\omega }_S-{\omega }_i+{\omega }_{FI}$. The differential photon scattering cross section is given by^{34, 717}

RRS spectroscopy is a type of vibrational Raman spectroscopy in which the incident laser frequency is close to an electronic transition of the molecule or crystal studied. As the adiabatic energy gap ${\omega }_{LI}$ between the $L$ state and the initial $I$ state is close to the laser frequency ${\omega }_i$, the intermediate $L$ state will make the dominant contribution to RRS. Under the “resonant” condition, the contributions from the non-resonant electronic states can be neglected.

One may evaluate $M_{IF}=⟨{\mathrm{\Phi }}_F|\widehat{M}|{\mathrm{\Phi }}_I⟩$ by making use of the Herzberg-Teller (HT) expansion, i.e., one expands the integrals about the nuclear equilibrium configuration $Q=0$. Writing the pure-spin Born–Oppenheimer (psBO) functions as products of an electronic wavefunction $\mathrm{\Psi }$ and a vibrational wavefunction $\mathrm{\Lambda }$

we have

The second term in Eq. (10.35) origins from the HT expansion. If we truncate the expansions after the lowest-order non-vanishing term, $M_{IF}$ can be written as

If the first term, the direct transition, vanishes, this process is orbitally forbidden.

To evaluate the vibrational terms in the remaining part of the vibronic matrix elements, we can use the harmonic oscillator approximation. Then the vibrational part of the wave function is written as ${\mathrm{\Lambda }}_n={\prod }_{k=1}^N{\chi }_k^{(n)}({\nu }_n)$, where $N$ is the total number of normal modes ${\chi }_k$ and ${\nu }_n$ the vibrational quantum number associated with mode $k$ in state $|n⟩$.

The delta function $\delta (\mathrm{\Delta }\omega )$ can be expressed as the Fourier integral

and then the transition rate from the initial state to the final state becomes

where $\gamma$ is a damping factor and

Here $\beta =1/k_{B}T$, ${\omega }_0={\omega }_i$ in one-photon absorption and emission processes, ${\omega }_0={\omega }_i-{\omega }_S$ in Raman scattering process. The notation $\text{tr}(\mathrm{\cdots })$ represents a trace over nuclear and electronic degrees of freedom, and $\widehat{M}=|{\mathrm{\Lambda }}_i⟩M_{IF}⟨{\mathrm{\Lambda }}_f|+|{\mathrm{\Lambda }}_f⟩M_{FI}⟨{\mathrm{\Lambda }}_i|$. The quantities ${\widehat{H}}_i$ and ${\widehat{H}}_f$ denote the nuclear Hamiltonians of electronic ground and excited states, respectively.

The Hamiltonian of vibrational motions on the ground and excited states can be written as

where $P$ and $Q$ are the momenta and coordinates of vibrational normal modes, respectively. The normal mode coordinates of ground and excited states are correlated by the Duschinsky rotation matrix $\overline{D}$,²⁹⁸ with $Q_e=\overline{D}{Q}_g+\overline{\mathrm{\Delta }}$. The quantity $\overline{\mathrm{\Delta }}$ is the displacement of normal mode coordinates between ground and excited states, i.e., the same quantity that appears in the IMDHO theory of Section 10.8.5, Eq. (10.25). The dimensionless forms are correspondingly ${\mathrm{\Delta }}_j={({\omega }_j^e)}^{1/2}{\overline{\mathrm{\Delta }}}_j$ and $D_{ij}={({\omega }_i^e/{\omega }_j^g)}^{1/2}{\overline{D}}_{ij}$. From the above, the transition rate can be calculated directly in the time domain using the correlation function approach. This time-dependent approach has been implemented to calculate vibronic spectra.^{724, 689, 725, 686}

It is obvious that ground and excited electronic states have different potential energy surfaces (PES) which lead to different vibrational frequencies and normal modes. The relation between mass-weighted Cartesian displacement coordinates $x$ and normal mode coordinates $Q$ is given by

where $x_0^g$ and $x_0^e$ are the equilibrium structures of ground and excited states. For an ideal $N$-dimensional harmonic oscillator, the normal mode coordinates of ground and excited states are related by

The minimum points at the PES and the Hessian matrix are required to calculate the Duschinsky rotation matrix and displacement vector. It can be time-consuming to calculate the excited state PES, especially for large molecules. The linear coupling model (LCM), which is also known as the vertical gradient (VG) approximation, has been proposed to avoid this issue.¹⁷⁸ Assuming that the excited state PES is approximated by a shift in the ground state PES, namely ${\omega }_j^e={\omega }_j^g$ and $L_e=L_g$, the displacement of $Q_j$ can be calculated by the excited state energy gradient ${\left(\frac{\partial E}{\partial Q}\right)}_j$, and ${\mathrm{\Delta }}^{\text{VG}}$ can be written as

The VG approximation is equivalent to the IMDHO approximation that is discussed in Section 10.8.5.^{547, 405, 406}

Generally, the Franck-Condon (FC) approximation is accurate enough for strongly one- or two-photon allowed transitions, while it breaks down for forbidden or weakly allowed transitions, and the FC term becomes nearly zero. In this situation, a correction to this deviation should be introduced by including the Herzberg-Teller (HT) or non-Condon effect.^{692, 845, 34, 33, 1050} HT-type vibronic coupling comes from the normal mode-coordinate dependence of the transition moments. When these quantities are expanded in terms of the normal mode coordinates, the contribution of the linear-coordinate-dependent terms is commonly called HT effect.^{34, 33} Many works, whether or not they account for the mode-mixing or Duschinsky rotation (DR) effect or not, have shown the importance of the HT effect in OPA, OPE, and RRS spectra.⁶⁸⁶

To predict OPA, OPE and RRS spectra, electronic structure calculations on ground and excited states should be performed. The necessary jobs at different level of approximation are summarized in the following:

• •

  FC. This is available for OPA, OPE, and RRS spectra.

  1. 1.

     $x_0^g\ne x_0^e$, $L_g=L_e$, and ${\omega }^g={\omega }^e$. Geometry optimization on excited state PES is performed, followed by ground state optimization and frequency analysis.

  2. 2.

     $x_0^g\ne x_0^e$, $L_g\ne L_e$, and ${\omega }^g\ne {\omega }^e$. Geometry optimization and frequency calculation are needed on both ground and excited states.

• •

  FCHT. This is available for OPA, OPE, and RRS spectra.

  • •

    $x_0^g\ne x_0^e$, $L_g\ne L_e$, and ${\omega }^g\ne {\omega }^e$. It is similar to the second kind of FC calculation, in which transition dipole derivative is obtained via frequency calculation on excited state. Geometry optimization and frequency calculation are needed on both ground and excited states.

• •

  VG. This is available for OPA and RRS spectra.

  • •

    $x_0^g=x_0^e$, $L_g=L_e$, and ${\omega }^g={\omega }^e$. Only the geometry optimization and frequency calculation of the ground state is involved. Frequencies and normal modes of excited state are assumed to be the same as ground state. The displacement vector is approximated by Eq. (10.43), in which the gradient of excited state PES is produced by excited state force job. Of course VG model has only contribution from FC term.

Since both ground state and excited state parameters are required, the routines to predict vibronic spectra are designed to have two steps. Firstly excited state calculation is performed and information about excited state will be saved in $QCSCRATCH/savename. Then the vibronic spectra utility is called to simulate the requested spectra after frequency analysis on ground state. SYM_IGNORE should be set to TRUE in order to prevent the molecular geometry being transformed to the standard orientation. Therefore the vibronic spectra job input can be set up in two ways. First, the multiple jobs can be separated by the string @@@ as described in Section 3.5. Or, jobs can be separated into individual inputs using $QCSCRATCH/savename as described below and given as examples 10.8.6.3 and 10.8.6.3.

qchem infile_excited_state outfile_excited_state savename
qchem infile_ground_state outfile_ground_state savename

There are two $rem variables and one section $vibronic involved in vibronic spectra calculations.

Variables in the $vibronic section: