Theoretical backgound of COSMOSS

COSMOSS currently can be used to simulate:

• Fourier Transform Infrared spectrum (FTIR)
• Sum-Frequency Generation spectrum (SFG)
• Two-dimensional Infrared spectrum (2D IR)
• Two-dimensional Sum-Frequency Generation spectrum (2D SFG)

No matter which type of the spectral simulation mentioned above, their signal can all be calculated in the following framework:

$$Signal=E\otimes J\otimes L\otimes ⟨R⟩\otimes \mathrm{\Gamma }$$

• $\mathrm{\Gamma }$ : Molecule response in microscopic molecular frame
• $⟨R⟩$ : Ensemble average of the molecule distribution in the macroscopic lab frame
• L : Fresnel coefficient of the interface
• J : Incident angles of laser beams
• E : Polarizations of laser beams
• $\otimes$: tensor product

Molecular response

The most complicated part is calculating the molecular response (

$\mathrm{\Gamma }$). On a programmatic level, COSMOSS determines this through the following steps:

1. Generation of a model structure
   COSMOSS contains several structure generation options for different molecules/systems which generate the atomic identities and positions of a particular system. For the purposes of spectral simulation, the generation of a model structure also creates the following relevant information for all transition moments (dipoles or quadropoles, aka Raman tensors) within the system.

   • (X,Y,Z) coordinates and angular orientation.
   • Transition Dipole Moment Frequency, Anharmonicity, and Strength

2. Selection of a Coupling Model
   The off-diagonal matrix elements of the one-exciton system Hamiltonian (

   $\hat{H}$) is determined by the selected coupling model.

   Current implemented models are:

   • TDC: Transition Dipole Coupling model
   • NN_Mix_TDC: Nearest-Neighbor modified TDC
   • Jansen_TDC: Jansen
     $\varphi /\psi$ angle map modified TDC
   • Zero: Zero coupling for testing purpose

   Details of each model can be found in the note of Coupling models

3. Generation of two-exciton Hamiltonian
   We now have information on the geometric arrangement of the transition dipole moments and a model for how they couple to one another. The next step is to create a Hamiltonian from this information with the associated eigenvalues and eigenvectors describing the energies and transition dipole moments of the system.

   The molecule response (

   $\mathrm{\Gamma }$) is simulated with the Frankle Exiton Model, where each of the locally excited mode is treated as a harmonic oscillator (
   ${\omega }_i$) that coupled (
   ${\beta }_{ij}$) to each other. To simulate 2D spectrum, we add the anharmonic shift (
   $\mathrm{\Delta }$) in the Hamiltonian (
   $\hat{H}$).

$$\hat{H}=\sum _i\hslash {\omega }_i{b}_i^{†}{b}_i+\sum _{i,j}{\beta }_{ij}(b_i^{†}{b}_j+b_i{b}_j^{†})-\frac{\mathrm{\Delta }}{2}\sum _i{b}_i^{†}{b}_i^{†}{b}_i{b}_i$$

4. Generation of the Feynman pathways
   To be written..

   I found a note from Thomas la Cour Jansen is quite useful for further introductory reading.

Ensemble average

To be written..

Fresnel coefficient

To be written..

Incident angles

To be written..

Polarizations

To be written..

More about SFG and 2D SFG

The most relevant paper for the 2D SFG simulation is the JPCA paper published by Laaser and Zanni 2013. Although this paper covers both SFG and 2D SFG, the computation details of the COSMOSS was not mentioned (since the COSMOSS was not yet even being written!). For better understanding of the COSMOSS code, please read the attached slides from Garth Simpson. This slides summarized the coordinate conventions for all the parts (S = EJLRβ) of simulated signal. I would also recommend materials on the annual Chataque conference hosted by Garth. I personally benefit a lot from the conference!

tags: COSMOSS Theory