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# 10.8.6 Vibrationally-Resolved Electronic Spectra and Resonance Raman Simulations

(February 4, 2022)

Optical spectroscopy is the study of the interaction between the light and matter, and the study that encompasses a wide range of physical and chemical behavior, which can be directly recorded by the spectrometers. Contemporary spectroscopic techniques have been applied to widespread research fields and have served as a popular tool to obtain the information of structural and dynamical features of the matter. However, the experimentally-measured spectra can’t straightforwardly give the microscopic information of the matter. The theoretical calculations of the spectra can serve as a supplementary tool to the experimental measurements and provide a deeper understanding on the underlying physical and chemical phenomena.866, 62, 61 One can easily determine geometrical, electronic and dynamical features of matters through a comparison between the experimental results and the calculated values. Moreover, the role of different effects in spectroscopic properties can also be quantified by the calculations.

Vibrationally-resolved one-photon absorption (OPA) and emission (OPE) spectra and resonance Raman scattering (RRS) spectra, each of which involves simultaneous changes in the vibrational and electronic states of a molecule, can reveal a reliable molecular structure–property relationship. Theoretical prediction of these spectra needs to combine both the electronic structure theories and quantum dynamics methods to obtain the structure parameters and describe quantum dynamics, respectively.1060, 1185, 194, 800, 801, 1174 For RRS simulation using the IMDHO model (which neglects Duschinsky rotation), see Section 10.8.5.

## 10.8.6.1 Time-dependent approach to simulating spectra

On the basis of perturbation theory, the transition rate of one- or two-photon transition processes from the initial state $|I\rangle$ to the final state $|F\rangle$ is proportional to $k_{IF}=|\langle F|\hat{M}|I\rangle|^{2}\delta(\Delta\omega)$, where $\delta(\Delta\omega)$ is the line shape function with $\Delta\omega=\omega_{i}-\omega_{FI}$ for a one-photon transition and $\Delta\omega=\omega_{1}+\omega_{2}-\omega_{FI}$ for a two-photon process. Here $\omega_{i}$, $\omega_{1}$, and $\omega_{2}$ denote the incident photon frequencies and the operator $\hat{M}$ is given by

 $\hat{M}=\begin{cases}\hat{\mu}&\text{(one-photon transition)}\\ \sum_{L}\left[\frac{{\bf\mu}\cdot\hat{e}_{2}|L\rangle\langle L|{\bf\mu}\cdot% \hat{e}_{1}}{(\omega_{1}-\omega_{LI})}+\frac{{\bf\mu}\cdot\hat{e}_{1}|L\rangle% \langle L|{\bf\mu}\cdot\hat{e}_{2}}{(\omega_{2}-\omega_{LI})}\right]&\text{(% two-photon transition)}\end{cases}\;.$ (10.31)

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

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 $S(\omega_{i},\omega_{S})=|\langle F|\hat{M}|I\rangle|^{2}\delta(\Delta\omega)$, where

 $\displaystyle\hat{M}=\sum_{L}\left[\dfrac{\mu\cdot\hat{e}_{2}|L\rangle\langle L% |\mu\cdot\hat{e}_{1}}{(\omega_{i}-\omega_{LI})}-\dfrac{\mu\cdot\hat{e}_{1}|L% \rangle\langle L|\mu\cdot\hat{e}_{2}}{(\omega_{S}+\omega_{LI})}\right]$ (10.32)

and $\Delta\omega=\omega_{S}-\omega_{i}+\omega_{FI}$. The differential photon scattering cross section is given by34, 717

 $\sigma(\omega_{i},\omega_{S})\propto\frac{4\omega_{i}\omega_{S}^{3}}{9c^{4}}S(% \omega_{i},\omega_{S}).$ (10.33)

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 $M_{IF}=\langle\Phi_{F}|\hat{M}|\Phi_{I}\rangle$ 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 $\Psi$ and a vibrational wavefunction $\Lambda$

 $\Phi_{n}(q,Q)=\Psi_{n}(q,Q)\Lambda_{n}(Q)\;,$ (10.34)

we have

 \displaystyle\begin{aligned} \displaystyle\Psi_{n}(q,Q)&\displaystyle=\Psi_{n}% (q,0)+[\partial{\Psi_{n}(q,0)}/{\partial Q}]_{Q=0}Q+\cdots\;,\\ \displaystyle M_{IF}&\displaystyle=M_{IF}(Q=0)+[\partial{M_{IF}}/{\partial Q}]% _{Q=0}Q+\cdots\;.\end{aligned} (10.35)

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

 $\displaystyle M_{IF}$ $\displaystyle=\langle\Psi_{F}(q,0)|\hat{M}|\Psi_{I}(q,0)\rangle\langle\Lambda_% {F}(Q^{\prime})|\Lambda_{I}(Q)\rangle$ $\displaystyle\qquad+[(\partial/\partial Q)\langle\Psi_{F}(q,0)|\hat{M}|\Psi_{I% }(q,0)\rangle]_{Q=0}\langle\Lambda_{F}(Q^{\prime})|Q|\Lambda_{I}(Q)\rangle\;.$ (10.36)

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 $\Lambda_{n}=\prod^{N}_{k=1}\chi_{k}^{(n)}(\nu_{n})$, where $N$ is the total number of normal modes $\chi_{k}$ and $\nu_{n}$ the vibrational quantum number associated with mode $k$ in state $|n\rangle$.

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

 $\delta(\Delta\omega)=\frac{1}{2\pi\hbar}\int_{-\infty}^{+\infty}\mathrm{e}^{(i% \Delta\omega)t/\hbar}\,dt\;,$ (10.37)

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

 $k_{IF}=\int_{-\infty}^{+\infty}\exp[i(\omega_{0}+E_{i}-E_{f})t/\hbar-\gamma t]% C_{i}(t)dt$ (10.38)

where $\gamma$ is a damping factor and

 $C_{i}(t)=\frac{\mbox{tr}[e^{-\beta\hat{H}_{i}}\mathrm{e}^{i\hat{H}_{i}t/\hbar}% \hat{M}e^{-iH_{f}t/\hbar}\hat{M})}{\mbox{tr}[\mathrm{e}^{-\beta\hat{H}_{i}}]}\;.$ (10.39)

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 $\mbox{tr}(\cdots)$ represents a trace over nuclear and electronic degrees of freedom, and $\hat{M}=|\Lambda_{i}\rangle M_{IF}\langle\Lambda_{f}|+|\Lambda_{f}\rangle M_{% FI}\langle\Lambda_{i}|$. The quantities $\hat{H}_{i}$ and $\hat{H}_{f}$ 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

 $\displaystyle H_{g}$ $\displaystyle=\frac{1}{2}\sum\limits_{j}^{N}\big{[}(P_{g,i})^{2}+(\omega_{j}^{% g}Q_{g,j})^{2}\big{]}\;,$ (10.40a) $\displaystyle H_{e}$ $\displaystyle=\frac{1}{2}\sum\limits_{j}^{N}\big{[}(P_{e,i})^{2}+(\omega_{j}^{% e}Q_{e,j})^{2}\big{]}\;,$ (10.40b)

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 $\bar{D}$,298 with $Q_{e}=\bar{D}Q_{g}+\bar{\Delta}$. The quantity $\bar{\Delta}$ 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 $\Delta_{j}=(\omega_{j}^{e})^{1/2}\bar{\Delta}_{j}$ and $D_{ij}=(\omega_{i}^{e}/\omega_{j}^{g})^{1/2}\bar{D}_{ij}$. 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

 \displaystyle\begin{aligned} \displaystyle x^{g}-x_{0}^{g}&\displaystyle=L_{g}% Q_{g}\;,\\ \displaystyle x^{e}-x_{0}^{e}&\displaystyle=L_{e}Q_{e}\;,\end{aligned} (10.41)

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

 \displaystyle\begin{aligned} \displaystyle Q_{e}&\displaystyle=(L_{e})^{T}L_{g% }Q_{g}+(L_{e})^{T}(x_{0}^{g}-x_{0}^{e})\\ &\displaystyle=\bar{D}Q_{g}+\bar{\Delta}\;.\end{aligned} (10.42)

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 $\Delta^{\text{VG}}$ can be written as

 $\Delta_{j}^{\text{VG}}=(\omega_{j}^{g})^{-3/2}\left(\frac{\partial E}{\partial Q% }\right)_{j}=\sum\limits_{i}(\omega_{j}^{g})^{-3/2}\left(\frac{\partial E}{% \partial x_{i}}\right)L^{ij}_{g}.$ (10.43)

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.

$x_{0}^{g}\neq x_{0}^{e}$, $L_{g}=L_{e}$, and $\omega^{g}=\omega^{e}$. Geometry optimization on excited state PES is performed, followed by ground state optimization and frequency analysis.

2. 2.

$x_{0}^{g}\neq x_{0}^{e}$, $L_{g}\neq L_{e}$, and $\omega^{g}\neq\omega^{e}$. Geometry optimization and frequency calculation are needed on both ground and excited states.

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

• $x_{0}^{g}\neq x_{0}^{e}$, $L_{g}\neq L_{e}$, and $\omega^{g}\neq\omega^{e}$. 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.

## 10.8.6.2 Job Control

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.

SAVE_VIBRONIC_PARAMS

SAVE_VIBRONIC_PARAMS
Save information about excited state which is requested in vibronic spectra simulation.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE
RECOMMENDATION:
TRUE

VIBRONIC_SPECTRA

VIBRONIC_SPECTRA
Specifies which type of vibronic spectra will be predicted. Should be used in a frequency job (jobtype = Freq).
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0 No vibronic spectra is predicted. 1 OPA spectra is calculated. 2 OPE spectra is calculated. 3 RRS spectra is calculated.
RECOMMENDATION:
Use the default.

Variables in the $vibronic section: MODEL Specifies which kind of model will be used to simulate the vibronic spectra. INPUT SECTION:$vibronic
TYPE:
INTEGER
DEFAULT:
-1
OPTIONS:
1 FC. 2 FCHT. 3 VG.
RECOMMENDATION:
User defined

TEMPERATURE
Specifies the temperature in the vibronic spectra simulation.
INPUT SECTION: $vibronic TYPE: FLOAT DEFAULT: $298.15$ OPTIONS: $t$ temperature, in K. RECOMMENDATION: User defined FREQ_RANGE Specifies the frequency range of vibronic spectra. INPUT SECTION:$vibronic
TYPE:
FLOAT
DEFAULT:
$1.0\ 40000.0\ 10.0$
OPTIONS:
$\nu_{\text{min}}\ \nu_{\text{max}}\ \delta\nu$ minimum, maximum and step size, in $\text{cm}^{-1}$.
RECOMMENDATION:
User defined

TIME_RANGE
Specifies the step size and the number of steps in time domain propagation.
INPUT SECTION: $vibronic TYPE: FLOAT and INTEGER DEFAULT: $1.0\ 40000$ OPTIONS: $\delta t\ n_{\text{step}}$ time step size in a.u., and the number of steps. RECOMMENDATION: User defined DAMPING Specifies the damping factor. INPUT SECTION:$vibronic
TYPE:
FLOAT
DEFAULT:
$300.0$
OPTIONS:
$\gamma$ damping factor, in $\text{cm}^{-1}$.
RECOMMENDATION:
User defined

FREQ_SCALE_FACTOR
Specifies the frequency scale factors.
INPUT SECTION: $vibronic TYPE: FLOAT DEFAULT: $1.0\ 1.0\ 1.0\ 1.0$ OPTIONS: $\lambda_{\text{H}}^{g}\ \lambda_{\text{H}}^{e}\ \lambda_{\text{ZPE}}^{g}\ % \lambda_{\text{ZPE}}^{e}$ scale factor for ground state harmonic frequency, for excited state harmonic frequency, for ground state zero-point energy, and for excited state zero-point energy RECOMMENDATION: User defined EPSILON Specifies the spectral broadening factor. It is available only for RRS spectra simulation. INPUT SECTION:$vibronic
TYPE:
FLOAT
DEFAULT:
$25.0$
OPTIONS:
$\varepsilon$ broadening factor, in $\text{cm}^{-1}$.
RECOMMENDATION:
User defined

## 10.8.6.3 Vibronic Job Examples

Example 10.24  Input files for OPA spectra in the FCHT approximation of formaldehyde corresponding to the $S_{0}\rightarrow S_{1}$ transition. In the first step, frequency analysis at the $S_{1}$ equilibrium geometry. Then run a ground state frequency analysis on the $S_{0}$ ground state optimized structure.

$molecule 0 1 O -0.0367447359 -0.0007590817 0.6963163574 C 0.1461299638 0.0026846285 -0.5839700302 H -0.0732270514 0.9340547891 -1.1138640182 H -0.0391581765 -0.9359803358 -1.1140167891$end

$rem JOBTYPE freq METHOD b3lyp BASIS def2-TZVP CIS_STATE_DERIV 1 CIS_SINGLETS true CIS_TRIPLETS false CIS_N_ROOTS 10 SYM_IGNORE true SAVE_VIBRONIC_PARAMS true ! enables saving information of S1 state$end

@@@

$molecule 0 1 O 0.0000000000 0.0000000000 0.6637077571 C 0.0000000000 0.0000000000 -0.5351027012 H 0.0000000000 0.9394749352 -1.1220697679 H 0.0000000000 -0.9394749352 -1.1220697679$end

$rem JOBTYPE freq METHOD b3lyp BASIS def2-TZVP SYM_IGNORE true VIBRONIC_SPECTRA 1 !enables vibronic_spectra and reads saved information$end

$vibronic model 2 freq_range 20000. 60000. 10. time_range 1. 40000 damping 40.$end


View output

Example 10.25  Vibrationally resolved fluorescence, i.e., OPE with the first kind of FC model, is calculated as following. The emission from $D_{1}$ to $D_{0}$ of p-fluorobenzyl radical is used as an example. This is the first job of the total vibronic spectra simulation, by running the excited state geometry optimization retaining information in $QCSCRATCH/savename. Information from this job will be needed to complete the simulation in Example 10.8.6.3. $molecule
0 2
C         1.4840482200    0.0000338155    0.0000000000
C         0.7160497031    0.0000524901   -1.2119311870
C        -0.7159596058    0.0000542629   -1.2126930961
C        -1.4043236629    0.0000543088    0.0000000000
C        -0.7159596058    0.0000542629    1.2126930961
C         0.7160497031    0.0000524901    1.2119311870
C         2.8748450131    0.0000120580    0.0000000000
H         1.2370230923    0.0000896598   -2.1693859212
H        -1.2717579173    0.0000412967   -2.1497435961
H        -1.2717579173    0.0000412967    2.1497435961
H         1.2370230923    0.0000896598    2.1693859212
H         3.4346492051    0.0000003003   -0.9330758768
H         3.4346492051    0.0000003003    0.9330758768
F        -2.7508602624    0.0000394216    0.0000000000
$end$rem
JOBTYPE                opt
METHOD                 b3lyp
BASIS                  def2-SVP
CIS_STATE_DERIV        1
CIS_N_ROOTS            10
SYM_IGNORE             true
SAVE_VIBRONIC_PARAMS   true !saved into $QCSCRATCH/savename$end


View output

Example 10.26  The final job for obtaining vibrationally resolved fluorescence of the $D_{1}$ to $D_{0}$ transition of p-fluorobenzyl radical using OPE with the first kind of FC model from Example 10.8.6.3.

$molecule 0 2 C 1.4578807306 0.0130092784 0.0000000000 C 0.7102753558 0.0082793447 -1.2194714816 C -0.6772053823 -0.0007923729 -1.2210832164 C -1.3603507249 -0.0052928605 0.0000000000 C -0.6772053834 -0.0007923730 1.2210832170 C 0.7102753546 0.0082793447 1.2194714789 C 2.8669152234 0.0219746232 0.0000000000 H 1.2502372081 0.0119050313 -2.1697312391 H -1.2498963812 -0.0045285858 -2.1510495277 H -1.2498963822 -0.0045285858 2.1510495285 H 1.2502372136 0.0119050313 2.1697312323 H 3.4299577819 0.0255536038 -0.9358170827 H 3.4299577814 0.0255536038 0.9358170860 F -2.7010161586 -0.0142661666 0.0000000000$end

$rem JOBTYPE freq METHOD b3lyp BASIS def2-SVP SYM_IGNORE true VIBRONIC_SPECTRA 2$end

$vibronic model 1 temperature 0. freq_range 1. 40000. 10. time_range 1. 40000 damping 20.$end


View output

Example 10.27  RRS spectra of phenoxyl radical ($D_{0}\rightarrow D_{3}$ transition) with the VG approximation. Therefore the first job calculates the $D_{3}$ state force at the ground state optimized geometry, followed by the ground state frequency analysis. The excited state forces and ground state frequencies are calculated in the ground state equilibrium geometry.

$molecule 0 2 C 0.0000000000 1.2271514002 -1.0879472096 C 0.0000000000 0.0000408897 -1.7873074655 C 0.0000000000 -1.2270324440 -1.0880160727 C 0.0000000000 -1.2409681161 0.2924435676 C 0.0000000000 -0.0000313560 1.0551142042 C 0.0000000000 1.2409428316 0.2924458686 H 0.0000000000 2.1656442172 -1.6487551860 H 0.0000000000 -0.0001767539 -2.8803293768 H 0.0000000000 -2.1655968771 -1.6487220344 H 0.0000000000 -2.1715667156 0.8648121894 H 0.0000000000 2.1714692701 0.8649813475 O 0.0000000000 0.0001236541 2.3063351676$end

$rem JOBTYPE force METHOD b3lyp BASIS def2-SVP CIS_STATE_DERIV 3 CIS_N_ROOTS 10 SYM_IGNORE true SAVE_VIBRONIC_PARAMS true$end

@@@

$molecule read !VG approximation uses the same geometry for ground and excited state$end

$rem JOBTYPE freq METHOD b3lyp BASIS def2-SVP SYM_IGNORE true VIBRONIC_SPECTRA 3$end

$vibronic MODEL 3 TEMPERATURE 0. FREQ_RANGE 1. 4000. 1. TIME_RANGE 1. 40000 DAMPING 100. EPSILON 25.$end


View output