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.
On the basis of perturbation theory, the transition rate of one- or two-photon transition processes from the initial state to the final state is proportional to , where is the line shape function with for a one-photon transition and for a two-photon process. Here , , and denote the incident photon frequencies and the operator is given by
In the two-photon case there are intermediate or “virtual” states .
Raman scattering is a two-photon process. In this process, one photon with the frequency is absorbed, another photon with the frequency 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 , where
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 between the state and the initial state is close to the laser frequency , the intermediate 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 by making use of the Herzberg-Teller (HT) expansion, i.e., one expands the integrals about the nuclear equilibrium configuration . Writing the pure-spin Born–Oppenheimer (psBO) functions as products of an electronic wavefunction and a vibrational wavefunction
The second term in Eq. (10.35) origins from the HT expansion. If we truncate the expansions after the lowest-order non-vanishing term, 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 , where is the total number of normal modes and the vibrational quantum number associated with mode in state .
The delta function can be expressed as the Fourier integral
and then the transition rate from the initial state to the final state becomes
where is a damping factor and
Here , in one-photon absorption and emission processes, in Raman scattering process. The notation represents a trace over nuclear and electronic degrees of freedom, and . The quantities and 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 and 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 ,298 with . The quantity 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 and . 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 and normal mode coordinates is given by
where and are the equilibrium structures of ground and excited states. For an ideal -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 and , the displacement of can be calculated by the excited state energy gradient , and can be written as
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.
, , and . Geometry optimization on excited state PES is performed, followed by ground state optimization and frequency analysis.
, , and . Geometry optimization and frequency calculation are needed on both ground and excited states.
FCHT. This is available for OPA, OPE, and RRS spectra.
, , and . 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.
, , and . 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:
$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
$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
$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
$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