VG-FC Resonance Raman

The vertical gradient Franck-Condon (VG-FC) method, also called the Independent Mode Displaced Harmonic Oscillator (IMDHO) model, we use to calculate vibrationally resolved absorption spectra can also be applied to the calculation of resonance Raman spectra. In resonance Raman spectroscopy a molecule is excited from its ground state to some electronically excited state. After a short period of time, the molecule then relaxes back to its electronic ground state. However, when doing so, it might end up in a different vibrational state than it started off in. The result is an energy difference between the incident and emmitted photon. One can then plot the intensity for different energy differences to produce what is known as a Raman spectra. Resonance Raman spectroscopy uses incident light with a wavelength close to that of an electronic transition.

AMS supports the calculation of such spectra by modeling the vibronic coupling of electronic transitions using the VG-FC model. This model is discussed also on the Vibronic-Structure documentation page. Here we will discuss the modifications necessary to use the VG-FC model for resonance Raman spectroscopy. It is worth noting that the resonance Raman application currently does not support the mode selective analysis options that are available for absorption spectra (see Vibronic-Structure Tracking and Vibronic-Structure Refinement). As a result the VG-FC Resonance Raman application will always first perform a full frequency analysis to obtain its normal modes.

Theory

While the basic theory behind the VG-FC model is explained in detail on the Vibronic-Structure documentation page, we will briefly summarize the most imporant points here, as well as the modifications necessary for its application to resonance Raman spectroscopy. It applies the harmonic approximation to both the ground state and excited state PES and then goes on to assume that neither frequency changes nor normal mode rotations occur. Thus the excited state PES is a shifted version of the ground state PES. We do not include temperature effects (so all initial states will be ground states) and work at the Franck-Condon point. Under these assumptions, the Raman polarizability of a particular excited state n, for a transition between initial and final vibrational states I and F can be written as:

\[(\alpha_{n,ij})_{F\leftarrow I} = \mu_{n,i}\mu_{n,j} \int_{0}^{\infty} \langle F|I_n(t)\rangle e^{i[\omega -(E_{n,0}-E_{m,0})]t}\cdot e^{-\Gamma t}dt\]

Here, \(i,j\) label the components of the polarizability tensor and \(\langle F|I_n(t)\rangle\) denotes the overlap of the initial state I, propagated along the excited state PES with the final state F. Under the assumptions of the VG-FC model, this overlap is equal to:

\[\langle F|I_n(t)\rangle = \prod_{j=1}^{N_{modes}}\left\{\frac{(-1)^{m_j}\Delta^{m_j}}{2^{m_j/2}m_j!}(1 - e^{-i\omega_jt})^{m_j}\right\}\exp\bigg[-\frac{\Delta_{n,j}^2}{2}(1-e^{-i\omega_jt})\bigg]\]

Where the \(m_j\) denote the excitation number of mode j in final state F. For a more detailed discussion, we refer to [1]. The only parameters that appear in our expression are the dimensionless oscillator displacements \(\Delta_{n,j}\) that represent displacement of the excited state PES along normal mode j. Under the simplifying assumptions of the VG-FC, these can be obtained from the ground state normal modes and a single excited state gradient. The Raman intensity is then proportional to the square of the polarizabilities:

\[\sigma(\omega)_{F\leftarrow I} \propto \sum_{i,j}|\sum_n (\alpha_{n,ij})_{F\leftarrow I}|^2\]

A spectrum is then generated by including various different final states F, which are defined by different combinations of normal mode excitation numbers, and assigning a relative intensity to each transition equal to the above expression. AMS only supports spectra which display relative intensities so the results are plotted in arbitrary units and are normalized such that the largest peak reaches an intensity of 1.

Input

The calculation setup for resonance Raman spectra largely proceeds as it does for Absorption Spectra. We need a set of ground state normal modes as well as an excited state gradient. The former are calculated at the start using the selected AMS engine, or, in case the user has a pre-calculated set of normal modes, these can be read from a .rkf file using the ModeFile key in the NormalModes sub-block. In this latter case, the engine is not used. The ModeSelect block can be used to select specific modes from the full set of normal modes for which the spectrum should be calculated. For details see the Mode Select documentation on the main page. If one simply wants the spectrum for the full set of normal modes, the Full key in the ModeSelect block can be set to True. The excited state information is passed to the application via the ExcitationSettings block.

Another point to note is that since our states are labeled by discrete indices we will be calculating stick spectra (which can be homogeneously broadened in adfspectra). By contrast, the absorption spectra produced by VibronicStructure are raw x,y data. Due to this difference in nature of the Raman spectrum compared to the absorption spectrum, this method uses the ResonanceRaman block for input options related to its spectrum (as opposed to the AbsorptionSpectrum block).

The ExcitationSettings block is discussed on the Vibronic-Structure page. One important difference with the latter is that Resonance Raman calculations are supported for more than one excitation at once. This is more important for the case of Raman spectra as the intensity associated with a set of transitions is not equal to the sum of their individual intensities (we sum over electronic states n before we square the polarizabilities). Here we will address settings specific to the Raman spectrum, all of which can be found in the ResonanceRaman block. A short example of how a typical input file might look is included at the end of this section.

Task VibrationalAnalysis
VibrationalAnalysis
   Task ResonanceRaman
   ResonanceRaman
      IncidentFrequency float
      LifeTime float
      RamanOrder integer
      RamanRange float_list
      MaximumStates integer
      FrequencyBins integer
   End
   ...
End
IncidentFrequency float
Frequency of incident light.
LifeTime float
sets the value of \(\Gamma\) (in Hartree) that controls the exponential damping in our integral. This phenomenological parameter can be interpreted as the (inverse) life time of the Raman excited state and can be used to help the results agree with experiment. The default value of 4.5e-4 is on the low end of reasonable values but should provide a good starting point for most cases.
RamanOrder integer
determines the set of final states and overtones to be included in the spectrum. It is an integer and the application considers only final states such that the sum of excitation numbers of all normal modes is less than or equal to this number. Setting this to 1 means we only include the fundamental band.
RamanRange float_list
this keyword specifies the frequency range (in \(cm^{-1}\)) the Raman shift is restricted to lie in. This prevents us from including excessively many states and overtones for high frequency modes. The default is [0, 2000] \(cm^{-1}\) but this can be changed to whatever is desired.
MaximumStates integer
Expert key. Due to the combinatorial explosion of included final states that occurs for combinations of large values of the raman order, large molecules and wide spectrum ranges, there is a maximum number of final states that can be included in the spectrum. This is to prevent the program from using excessive amounts of memory/computation times. The user can set this number using the MaximumStates key but this should be done with caution.

Finally we give an example of a typical VibrationalAnalysis block for a resonance Raman calculation. This also gives an idea of how the settings that were not explicitly mentioned above work:

VibrationalAnalysis
   Type ResonanceRaman
   NormalModes
     ModeSelect
       Full True
     End
   End
   ExcitationSettings
     ExcitationInfo File
     ExcitationFile ./your_excitation/dftb.rkf
     Singlet
       A 1 2 4
     End
   End
   ResonanceRaman
     RamanOrder 3
     RamanRange 0.0 3000.0
   End
End

References

[1]
  1. Petrenko and F. Neese, Analysis and prediction of absorption band shapes, fluorescence band shapes, resonance Raman intensities, and excitation profiles using the time-dependent theory of electronic spectroscopy The Journal of Chemical Physics 127, 164319 (2007)