🔬 materials science

QR $^2$ -code: An open-source program for double resonance Raman spectra

The paper introduces QR $^2$ -code, an open-source program that utilizes first-principles calculations from Quantum ESPRESSO and EPW to compute single-resonance, double-resonance, and defect-induced Raman spectra, providing tutorials and validation through graphene examples.

Original authors: Jianqi Huang, Renhui Liu, Ye Zhang, Nguyen Tuan Hung, Huaihong Guo, Riichiro Saito, Teng Yang

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jianqi Huang, Renhui Liu, Ye Zhang, Nguyen Tuan Hung, Huaihong Guo, Riichiro Saito, Teng Yang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a tiny, magical drum made of a single layer of atoms (like graphene). When you hit this drum with a specific color of light (a laser), it doesn't just vibrate; it sings a very specific song called a Raman spectrum. Scientists use these songs to figure out what the drum is made of and how its atoms are arranged.

However, some of these songs are incredibly complex. They involve the drum vibrating in two different ways at the same time, or the light bouncing off a tiny crack in the drum. Calculating these complex songs on a computer is like trying to predict the weather for every single square inch of the Earth simultaneously—it takes a massive amount of time and power.

This paper introduces a new, free computer program called QR2-code that acts like a super-efficient "songwriter" for these atomic drums. Here is a breakdown of what it does, using simple analogies:

1. The Problem: The "Double-Resonance" Puzzle

Previously, scientists had a tool (called QERaman) that could predict the simple songs (Single-Resonance) where the light hits the drum and it vibrates once. But many materials, especially flat ones like graphene, produce "Double-Resonance" songs.

The Analogy: Imagine a simple song is a drummer hitting the drum once. A "Double-Resonance" song is like the drummer hitting the drum, the sound bouncing off a wall, hitting the drum again, and then making a sound. To predict this, you have to track the sound bouncing all over the entire drum surface.
The Issue: Doing this calculation for every possible spot on the drum was too slow for old computers.

2. The Solution: QR2-code

The authors built QR2-code, a new open-source program that speeds this up significantly. It works by taking a "shortcut" map (called Wannier functions) to predict how the electrons and vibrations interact, rather than calculating every single step from scratch.

Think of it like this:

Old Method: Trying to walk every single path in a giant forest to find the exit.
QR2-code: Using a GPS that knows the shortcuts and the terrain, getting you to the answer much faster.

3. What Can QR2-code Do?

The paper shows that this program can handle three types of "songs":

The Simple Song (Single-Resonance): It can still calculate the basic vibrations, just like the old program. The authors tested this on graphene and found the results matched the old program closely, proving it works correctly.
The Complex Echo (Double-Resonance): This is the main feature. It can calculate the "two-phonon" songs where the light interacts with the material twice.
- Real-world example: In graphene, this program successfully predicted the famous "2D" and "G'" peaks that scientists see in experiments. It figured out that these peaks are caused by specific pairs of vibrations (like a "TO" vibration teaming up with an "LA" vibration).
The Broken Drum (Defect-Induced): Real materials aren't perfect; they have tiny cracks or missing atoms (defects).
- The Analogy: If you have a crack in your drum, it makes a weird, extra noise (the "D" and "D'" bands).
- The Program's Trick: The program assumes that the interaction between the electron and the defect is a simple, constant value (like assuming the crack is a fixed size). Even with this simplification, it successfully predicted the "D" and "D'" peaks in graphene, which are crucial for identifying defects in materials.

4. How It Works (The Recipe)

The paper provides a "cookbook" (tutorials) for how to use this program. It doesn't work alone; it's an add-on for two other famous, free scientific tools:

Quantum ESPRESSO (QE): The main chef that prepares the basic ingredients (the electron and atom positions).
EPW: The sous-chef that prepares the specific "flavor" (how electrons and vibrations talk to each other).
QR2-code: The final assembler that takes those ingredients and cooks the specific "Raman song."

5. The Results

The authors tested this on graphene (a single layer of carbon atoms).

They showed that the program can predict the exact pitch (frequency) of the peaks seen in real experiments.
They showed that the program can tell you which vibrations are making the sound (the "assignment" of the peaks).
They proved that by using a denser "grid" (checking more points on the drum), the predictions become more accurate, eventually matching the real world perfectly.

Summary

In short, QR2-code is a free, open-source tool that helps scientists predict the complex "music" of tiny materials. It makes it possible to calculate difficult "double-resonance" and "defect" spectra quickly, allowing researchers to better understand the structure of materials like graphene, MoS2, and boron nitride without needing a supercomputer for years. The authors have shared the code, the examples, and the instructions on GitHub for anyone to use.

Technical Summary of QR2-code: An Open-Source Program for Double Resonance Raman Spectra

Problem Statement
While low-dimensional materials frequently exhibit resonance Raman effects essential for assigning phonon modes and characterizing electronic structures, existing open-source tools have limitations. The authors' previous program, QERaman, successfully calculated single-resonance Raman (SRR) spectra using the Quantum ESPRESSO (QE) package but lacked support for double-resonance Raman (DRR) spectra. DRR spectra, which involve two phonons with opposite momenta ( $q$ and $-q$ ) and are critical for analyzing materials like graphene, MoS $_2$ , and carbon nanotubes, require electron-phonon coupling matrix elements across the entire Brillouin zone (BZ). Calculating these elements via standard Density Functional Theory (DFT) is computationally prohibitive compared to SRR calculations, which only require the $\Gamma$ point.

Methodology
The paper introduces QR2-code, an open-source Fortran program designed to compute SRR, two-phonon DRR, and defect-induced DRR spectra using first-principles calculations. The code integrates with the Quantum ESPRESSO (QE) suite (v7.3.1) and the Electron-Phonon Wannier (EPW) code (v5.8.1).

The core methodology relies on time-dependent perturbation theory to calculate Raman scattering amplitudes:

Data Source: Electron and phonon energy dispersions are derived from QE.
Matrix Elements: The electron-phonon matrix elements, required as a function of the phonon wavevector $q$ over the entire BZ, are obtained via the EPW code using maximally localized Wannier functions. This approach significantly reduces computational time compared to direct DFT methods.
Scattering Processes:
- SRR: Involves a single phonon scattering event near the $\Gamma$ point ( $q=0$ ).
- DRR: Involves two phonons with wavevectors $q$ and $-q$ ( $q \neq 0$ ). The code calculates the fourth-order perturbation form of the Raman tensor.
- Defect-Induced DRR: Modeled as a special case of DRR where one electron-phonon scattering event is replaced by elastic scattering from a lattice defect. The current version assumes the electron-defect matrix element for elastic scattering is a constant, allowing the calculation of both D and D' bands in graphene, though it does not quantify the relative intensity between defect-induced and two-phonon DRR spectra.
Workflow: The process involves four main steps: (1) generating electron and phonon databases using QE (pw.x, ph.x); (2) obtaining phonon dispersions and ordering modes (q2r.x, phonon_sort.x); (3) calculating electron-phonon matrix elements via Wannierization (epw.x); and (4) performing the resonance Raman calculation and post-processing (epw.x with Raman flags, raman_pp.x).

Key Contributions

Open-Source Implementation: QR2-code is released under the GNU General Public License 3.0, with source code, examples, and scripts available on GitHub.
Expanded Capabilities: Unlike QERaman, QR2-code supports DRR and defect-induced Raman spectra, enabling the analysis of complex scattering pathways (e.g., $ee$, $hh$, $eh$, $he$ processes).
Efficiency: By utilizing the Wannier interpolation method via EPW, the code enables DRR calculations within a reasonable computational timeframe, overcoming the bottleneck of full BZ sampling in standard DFT.
Physical Features: The code accounts for Fröhlich electron-phonon coupling (interaction with long-wavelength longitudinal optical phonons) and allows for the assignment of specific two-phonon modes to observed Raman peaks.

Results and Validation
The authors validated QR2-code using graphene as a primary example, with additional examples for MoS $_2$ monolayers and bulk h-BN available in the repository.

SRR Comparison: For single-resonance spectra at laser energies of 1.96 eV and 2.33 eV, QR2-code produces results similar to QERaman. At higher energies (2.54 eV), QR2-code shows lower intensity than QERaman, which the authors attribute to QERaman's inclusion of more bands in the resonance condition via direct DFT, whereas QR2-code relies on Wannier interpolation with fewer bands.
DRR Analysis: The code successfully reproduces experimental DRR peaks in graphene at 1.96 eV, identifying the G* (2456.1 cm $^{-1}$ ), G' (2707.8 cm $^{-1}$ ), and 2D' (3199.7 cm $^{-1}$ ) modes. The analysis reveals that the $eh$ and $he$ scattering pathways (involving both electrons and holes) dominate the DRR intensity at this energy, while $ee$ and $hh$ pathways are suppressed by destructive quantum interference.
Defect-Induced Spectra: The code successfully calculates the D (1355.7 cm $^{-1}$ ) and D' (1600.0 cm $^{-1}$ ) bands in graphene, consistent with experimental observations of defect-activated peaks.
Convergence: Convergence tests indicate that a dense interpolated grid of $150 \times 150 \times 1$ for both $k$ and $q$ points is sufficient for accurate results.

Significance
The paper positions QR2-code as a practical tool for the scientific community to analyze resonance Raman spectra in low-dimensional materials. By providing an open-source solution that combines the accuracy of first-principles calculations with the efficiency of Wannier interpolation, the authors aim to facilitate the assignment of phonon modes and the understanding of electronic structures in materials where double-resonance effects are prominent. The authors explicitly invite user feedback to further improve the program, acknowledging that while the current defect model is an approximation (constant matrix element), it provides a functional starting point for studying defect-induced spectra.