Can HEOM solver Provided by QuTiP be used in simulating absorption spectrum of molecule?

[Jason Eu]

Dear all,

I am going to simulate the absorption spectrum of molecule beyond the Born-Oppenheimer approximation. The system is modeled by a two-level system interacting with some vibrational modes. 

Can I simulate this by using QuTiP? Is QuTiP suitable for this case?

I think I can treat the vibration as environment and do the system dynamics using HEOM solver. Am I right? If HEOM is what I need, how to set the coupling of vibrational modes? 

In fact my hamiltonian is almost the same as the system-bath coupling Hamiltonian given in the here[1] and I have the value of all bosonic modes of frequency \omega and coupling strength g, 

Best,

Jason

[1] https://github.com/qutip/qutip-notebooks/blob/master/examples/HEOM-TLS.ipynb

[Neill Lambert]

Hi Jason,

The parameters in the HEOM come from the choice of spectral density, which then sets the bath (vibrational modes) correlation functions.  So in the example note book the spectral density is set by a coupling parameter (lambda) and a width (gamma).  In addition you have to put in the bath temperature, and some convergence parameters, as shown in the example notebook.

The code currently available in qutip only works for the Drude-Lorentz spectral density, and only returns back the system dynamics.  We are currently working on a big extension/update which overcomes these limitations.

In particular, for your needs, because the current implementation only solves for system dynamics, if you want to get the absorption spectrum it requires a bit of work. You can either modify the solver to calculate two-time correlation functions of the system, and then do the Fourier transform, or you can find it from the pseudo-inverse and other approaches (basically as is done with the normal master equation solver in the functions in https://github.com/qutip/qutip/blob/master/qutip/countstat.py and https://github.com/qutip/qutip/blob/master/qutip/correlation.py )

We just did this a few weeks ago in our personal HEOM code, so if this is something you want to try I can probably give some tips/help (our new code is not quite ready to be released yet, soon I hope).

Thanks

Neill

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[Jason Eu]

Dear Neill,

It will be great if I can try the unreleased HEOM code. 

The purpose of my simulation with HEOM is to compare the results given by methods which treat coupling as a closed finite-dimentional system, with discrete vibrational degrees of freedom (e.g. MCTDH method and variational Davydov ansatz I used in my work). I am not familiar with the theory of HEOM, so it is hard to implement this by myself. Will be good if I can simply use the ready-made code as some sort of black box. 

Besides, the spectrum part seems not to be a problem from my side, if I understand it correctly, after I get the system dynamics, I can calculate the two-time correlation functions and then do the Fourier transform using the wavefunctions of each timesteps.

Therefore, only the implementation of HEOM that work with discrete vibrational modes is what I need.    

Thanks,

Jason

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[Brian Rolczynski]

Hello,

I had a similar question, which was whether it was possible to specify our own spectral density function instead of using Drude-Lorentz. 

For instance, suppose I estimate a spectral density based on an QM/MM simulation, like in the publication:  Rosnik, A. M.; Curutchet, C. J. Chem. Theory Comput. 2015, 11, 5826.

I have read in the above comments that it was not possible as of Apr. 2020, but that a big update was hopefully coming soon that would make it possible. Has that update been released? If so, is it now possible? Is there an example posted somewhere that I could follow? I appreciate it.

Thank you,

Brian

[Neill Lambert]

hi brian, 

The latest version of the heom solver in qutip also lets you specify the bath correlation functions as the pre-factors and exponents of a sum of exponentials. This lets you do fitting of arbitrary bath correlation functions with a sum of exponentials. Alternatively, one can directly fit arbitrary spectral densities with underdamped brownian motion spectral densities (in the literature the latter is sometimes called the Tannor-Meier decomposition), and do standard matsubara or pade decompositions of each term in the fit.

We put a short example here of both approaches, showing how to do some fitting of Ohmic spectral densities with exponential cut-offs

https://github.com/qutip/qutip-notebooks/blob/master/examples/heom/heom-1d-spin-bath-model-ohmic-fitting.ipynb

This is also explained a little in this preprint https://arxiv.org/abs/2010.10806  but it's a bit out of data, and should be updated soon.

How the fitting is done is left up to the user, as demonstrated in the notebook above. In practice each case is different, and whether you can capture the general spectral density well with this kind of fitting really varies.

all the best

neill

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[ahmed ghareeb]

Dear all 

Bofin so valuable improvement for qutip community

It allow you to run heom under many spectral densities

 

Advantage of qutip implementation

If you have analytical expression of correlation function ,you can fit it directly to exp form also this done for every change of temperature

But this has nice convergence performance over analytical(complex integration) decomposition from spectral density directly ,

See

https://github.com/berkelbach-group/pyrho/blob/master/heom/heom.py

See also ref paper

https://aip.scitation.org/doi/abs/10.1063/1.4870035

In summary

1_ easy matsubara like drude Lorentz

2_fitting known form of correlation function but for  every change in temperature 

3_fitting spectral density itself  ,

Fitting done once and temperature just a parameter

(Miere tannor method)

May be optain same expential power for different prefactor 

Qutip has option to deal with it.

To lower computational cost

If you want know about new computational spectral decomposition

See

https://github.com/tatsushi-ikeda/pyheom

With best regards

Ahmed 

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[Brian Rolczynski]

Hello Neill and Ahmed,

Thank you very much. That was extremely helpful!

Brian