QFiND

This repository contains Python codes for constructing an effective discrete representation of a system-bath model. In other words, this codes provides an approximation of the bath correlation function $C(t)$ for a given spectral density $J(\omega)$

$$ \begin{aligned} C(t)&=\frac{1}{2\pi} \int_{-\infty}^{\infty} \mathrm{d}\omega J(\omega)\left[\mathrm{coth}\left(\frac{\beta \omega}{2}\right)+1\right] \mathrm{e}^{-i \omega t}\\ &\approx \sum_{k=1}^M g_k^2 \mathrm{e}^{-i\omega_k t} \end{aligned} $$

where $\omega_k,g_k \in ℝ \backslash \{0\}$. The code allows for the estimation of frequencies and coefficients in the system plus bosonic bath model using Interpolative Decomposition (ID) and Non-negative Least Squares (NNLS).

Usage

1. Set the required parameters in an input file input.txt (see below).
2. Run the main script:

   python ./src/qfind.py input.txt
   

3. The output will include the estimated frequencies and coefficients (saved as omega_g.txt), along with a plot of the resultant BCF (saved as bcf.png).

Parameter Configuration

Setting Parameters

To customize the simulation, you need to adjust certain parameters in the following files:

• input.txt: This file contains important global parameters such as:
  • method: Discretization method, such as:
    • BSDO: BSDO method
    • ID: ID approach
    • LOG: Logarithmic discretization
    • MDM: Mode density method
  • temperature: Specifies the temperature of the system in $[\mathrm{K}]$.
  • Tc (double): Cutoff time in $[\mathrm{fs}]$.
  • Omega_min (double): Minimum cutoff frequency in $[\mathrm{cm}^{-1}]$.
  • Omega_max (double): Maximum cutoff frequency in $[\mathrm{cm}^{-1}]$.
  • N_t (integer): Number of sample points in the time domain.
  • N_w (integer): Number of sample points in the frequency domain.
  • wmax_quad: The maximum frequency cutoff used in the numerical integration.
  • eps (double): Threshold for the ID.
  • frank (integer): Rank for the ID. When frank is set to a value larger than 0 (frank>0), ID is performed based on the rank.
  • stype: The type of spectral density $J(\omega)$ (PWR, TM, BO). The program supports several types of spectral density, such as:
    • Power-law with exponential cutoff (PWR) $$J(\omega)=\pi\alpha\omega_c^{1-s}\omega^s\mathrm{e}^{-\omega/\omega_c}$$
    • Sum of Tannor-Meyer type spectral densities (TM) $$J(\omega)=\sum_{j=1}^n \frac{4\Gamma_j\lambda_j(\Omega_j^2+\Gamma_j^2)\omega}{\left[(\omega+\Omega_j)^2+\Gamma_j^2\right]\left[(\omega-\Omega_j)^2+\Gamma_j^2\right]}$$
    • Sum of Brownian spectral densities (BO) $$J(\omega)=\sum_{j=1}^n 2\lambda_j\frac{\Gamma_j \Omega_j^2\omega}{(\omega^2-\Omega_j^2)^2+\Gamma_j^2\Omega_j^2}$$
  • Parameters for specific spectral density types, such as:
    • s, alpha, gamc for Power-law Exponential (PWR).
    • Omg, Gam, Lam for Tannor-Meyer type (TM) and Brownian Oscillator (BO).

Example Parameter Settings

See the directory ./examples.

Cite QFiND

If you find the framework useful in your research, we would be grateful if you could cite our publications:

• H. Takahashi and R. Borrelli, J. Chem. Phys. 161, 151101 (2024). (https://doi.org/10.1063/5.0232232)
• H. Takahashi and R. Borrelli, J. Chem. Theory Comput. (2025). (https://doi.org/10.1021/acs.jctc.4c01728)

Here are the bibtex entries:

@article{TakahashiBorrelli2024JCP,
  title = {Effective modeling of open quantum systems by low-rank discretization of structured environments},
  author = {Takahashi, Hideaki and Borrelli, Raffaele},
  year = {2024},
  month = oct,
  journal = {The Journal of Chemical Physics},
  volume = {161},
  number = {15},
  pages = {151101},
  issn = {0021-9606},
  doi = {10.1063/5.0232232}
}

@article{TakahashiBorrelli2025JCTC,
  title = {Discretization of {{Structured Bosonic Environments}} at {{Finite Temperature}} by {{Interpolative Decomposition}}: {{Theory}} and {{Application}}},
  shorttitle = {Discretization of {{Structured Bosonic Environments}} at {{Finite Temperature}} by {{Interpolative Decomposition}}},
  author = {Takahashi, Hideaki and Borrelli, Raffaele},
  year = {2025},
  month = feb,
  journal = {Journal of Chemical Theory and Computation},
  publisher = {American Chemical Society},
  issn = {1549-9618},
  doi = {10.1021/acs.jctc.4c01728}
}

Authors

Hideaki Takahashi ([email protected])

License

This project is distributed under the BSD 3-clause License.