Extending numerical simulations in SIMPSON: Electron paramagnetic resonance, dynamic nuclear polarisation, propagator splitting, pulse transients, and quadrupolar cross terms

This paper introduces a C++-based next generation of the SIMPSON software package featuring novel capabilities for simulating advanced NMR, EPR, and pulsed DNP experiments, including propagator splitting, pulse transients, and quadrupolar cross terms, while improving computational speed and facilitating community contributions.

The Gist

Imagine you are trying to understand the secret language of atoms. Scientists use powerful tools called NMR (Nuclear Magnetic Resonance) and EPR (Electron Paramagnetic Resonance) to "listen" to how tiny particles like electrons and atomic nuclei spin and interact. It's like trying to hear a whisper in a hurricane; the signals are incredibly faint and complex.

To make sense of these whispers, scientists use a digital "simulator" called SIMPSON. Think of SIMPSON as a flight simulator for atoms. Before they build a real experiment, they fly a virtual plane in the computer to see if the controls work, if the engine holds up, and what the view will look like.

This paper announces a massive upgrade to SIMPSON (version 6.0). Here is what's new, explained with everyday analogies:

1. The Engine Upgrade: From C to C++

The Old Way: The previous version of SIMPSON was built with an older programming language (C). It was like a reliable, sturdy truck that got the job done but was getting a bit rusty and hard to modify.
The New Way: The new version is built with C++, a modern, object-oriented language.

• The Analogy: Imagine swapping that old truck for a modular, high-tech electric vehicle. The engine is faster, the parts are easier to swap out, and most importantly, it's built so that other mechanics (the scientific community) can easily pop the hood, add new features, and fix things without needing a PhD in engineering. It's now easier for anyone to contribute to the software.

2. The New Passengers: Electrons and Polarization

The Old Way: SIMPSON was great at simulating "nuclei" (the center of atoms) but struggled with "electrons" (the tiny particles orbiting them).
The New Way: This update allows SIMPSON to simulate electrons and a special technique called Dynamic Nuclear Polarization (DNP).

• The Analogy: Imagine DNP as a megaphone. Nuclei are usually very quiet (low sensitivity). Electrons are loud and energetic. DNP is a way to take the loud energy from the electrons and "borrow" it to amplify the quiet nuclei, making the signal 100x or 1000x stronger.
• The Upgrade: Now, SIMPSON can simulate this "megaphone" effect. It can model how to transfer that energy efficiently, helping scientists design experiments that see things they couldn't see before.

3. The "Split-Second" Trick: Propagator Splitting

The Problem: Simulating how atoms move over time is like trying to calculate the path of a billion billiard balls bouncing off each other. Doing this step-by-step is incredibly slow and computationally expensive.
The Solution: The new version uses a technique called Propagator Splitting.

• The Analogy: Imagine you need to walk across a huge, bumpy field.
  • The Old Way: You try to calculate the exact path of every single step, every pebble, and every gust of wind at once. It takes forever.
  • The New Way (Splitting): You break the journey into two parts. First, you calculate the easy, straight-line walking (the "control" part). Then, you calculate the bumps and wind (the "interaction" part). You do these separately and stitch them together.
  • The Result: It's like using a high-speed train for the straight parts and a hiking boot for the rough parts. You get to the destination much faster without losing accuracy.

4. Fixing the "Static": Pulse Transients

The Problem: When scientists send a radio signal to the atoms, the equipment doesn't always respond perfectly instantly. It's like turning on a faucet; the water doesn't hit full pressure immediately, and there might be a splash or a delay. In the lab, these "splashes" (called transients) distort the signal.
The Solution: The new software can now model these imperfections.

• The Analogy: Imagine you are trying to paint a perfect circle with a spray can. But your spray can is slightly clogged, so the paint comes out in a jagged line.
  • The Old Way: You tried to draw the perfect circle, ignoring the clogged can. The result was messy.
  • The New Way: SIMPSON now knows the can is clogged. It calculates exactly how the paint will splatter and then pre-distorts your drawing in the opposite direction. When you spray it, the splatter cancels out the distortion, and you get a perfect circle. This allows scientists to design pulses that work perfectly even with imperfect hardware.

5. The "Heavy Lifter": Quadrupolar Nuclei

The Problem: Some atoms (like Nitrogen-14) are "heavy" and spin in a wobbly, chaotic way. They are like a spinning top that is slightly unbalanced. Simulating them is a nightmare because their math is incredibly complex.
The Solution: The update includes better tools to handle these "wobbly" atoms, specifically looking at how they interact with their neighbors.

• The Analogy: If regular atoms are like smooth marbles, these quadrupolar nuclei are like spiky sea urchins. The new software has a special "glove" that can handle the spikes, allowing scientists to simulate how these difficult atoms interact with others, which is crucial for studying things like battery materials and proteins.

Summary

This paper is about giving scientists a super-powered, user-friendly, and versatile flight simulator for the atomic world.

• It's faster (thanks to the splitting tricks).
• It's smarter (it can handle electrons and "megaphone" effects).
• It's more realistic (it accounts for equipment glitches).
• And it's open for collaboration (built in a way that lets the whole scientific community help build the next version).

With these tools, scientists can design better experiments to understand everything from new medicines to the materials that will power our future batteries.

Technical Summary

1. Problem Statement

The field of magnetic resonance (NMR, EPR, and the hybrid discipline of Dynamic Nuclear Polarization, DNP) is rapidly evolving toward more sophisticated pulse sequences, larger spin systems, and higher precision requirements. While the SIMPSON software package has been a standard for solid-state NMR simulation since 2000, it faced several limitations in the context of modern experimental demands:

• Scope Limitations: It lacked native support for electron spins, making it difficult to simulate pulsed EPR and DNP experiments which require coupling electron and nuclear spin dynamics.
• Computational Bottlenecks: As spin system dimensions increase (e.g., multi-spin ensembles in DNP or large quadrupolar systems), calculating time-propagators via matrix exponentials becomes computationally prohibitive, especially for iterative optimal control design.
• Physical Realism: Standard simulations often assume ideal pulses. They frequently neglect experimental imperfections such as RF/microwave field inhomogeneities, pulse transients (ring-down effects), and high-order cross-terms in quadrupolar systems.
• Software Architecture: The previous versions were written in C, which, while fast, made community contributions and object-oriented extensions difficult compared to modern C++ standards.

2. Methodology and Technical Innovations

The authors present SIMPSON version 6.0, a major overhaul that transitions the core engine from C to C++ while retaining TCL as the user scripting interface. The key methodological contributions include:

A. Electron Spin Dynamics and DNP Integration

• Formalism: The software now treats electron spins ($S$) alongside nuclear spins ($I$) using irreducible tensor operators and Wigner transformations.
• Hamiltonians: New keywords were introduced to define:
  • Zeeman interaction: $g$-tensors (isotropic, anisotropic, asymmetry).
  • Hyperfine coupling: Fermi-contact and dipolar interactions between electrons and nuclei.
  • Electron-Electron interactions: Dipolar coupling and Heisenberg exchange.
• Reference Frames: A new DNPframe method was implemented, allowing nuclear spins to be treated in the laboratory frame while electron spins are in their rotating frame, essential for accurate DNP and EPR simulations.

B. Propagator Splitting (Time-Propagation)

To address the computational cost of matrix exponentials ($e^{-iHt}$), the authors implemented propagator splitting methods:

• Concept: The total Hamiltonian $H$ is split into a control part $A$ (single-spin terms, analytically solvable via Euler-Rodrigues formulas) and an interaction part $B$ (multi-spin terms).
• Algorithms:
  • Lie-Trotter Splitting: First-order approximation.
  • Strang Splitting: Second-order symmetric splitting ($e^{Bt/2}e^{At}e^{Bt/2}$).
  • Higher-Order Splittings: Up to 6th-order palindromic chains to achieve high accuracy without full matrix diagonalization.
• Benefit: This allows for significantly faster simulations of complex multi-spin systems and optimal control problems by avoiding expensive matrix exponentials for the full Hamiltonian at every time step.

C. Pulse Transients and Distortion Operators

• Modeling: The software models the physical pulse shape $\upsilon(t)$ as a convolution of the ideal input shape $\Upsilon(t)$ with an impulse response function $h(t)$ (representing hardware distortions like ring-down).
• Optimal Control with Transients: A distortion operator ($\phi_{nm}$) is defined to map the coarse input pulse parameters to the fine-grained transient-compensated pulse.
• Gradient Reconstruction: For gradient-based optimal control (GRAPE), the authors implemented a mathematical framework to transform the gradient calculated on the distorted (fine) grid back to the input (coarse) grid using the transpose of the distortion operator. This allows for the design of pulses that are robust against specific hardware imperfections.

D. Quadrupolar Cross-Terms

• New capabilities were added to simulate second-order cross-terms between quadrupolar interactions and other interactions (Chemical Shielding Anisotropy - CSA, or Dipolar couplings).
• This enables accurate simulation of residual dipolar splittings in $^1$H spectra coupled to quadrupolar nuclei (e.g., $^{14}$N) and complex 2D STMAS (Satellite Transition Magic Angle Spinning) spectra.

E. Software Architecture and Visualization

• C++ Core: The shift to C++ utilizes object-oriented principles to improve code modularity, speed, and community extensibility.
• Visualization: Integration with EasyNMR (web-based workflow) and SimView (Python-based GUI) for easier data visualization and fitting, moving away from the legacy SIMPLOT binary limitations.
• Distribution: Available via GitLab (source), Docker, and NMRbox to facilitate accessibility.

3. Key Results and Performance

The paper validates these new features through several benchmark simulations:

• EPR and DNP Simulations: Successfully simulated 2-pulse and 3-pulse ESEEM (Electron Spin Echo Envelope Modulation) and various pulsed DNP sequences (NOVEL, BEAM, PLATO, cRW-OPT1). The software accurately reproduced offset dependencies and mixing time profiles, demonstrating the ability to handle electron-nuclear spin systems.
• Optimal Control Efficiency:
  • SORDOR Pulses: The authors designed SORDOR (Second Order Phase Dispersion by Optimised Rotation) pulses using the new splitting methods.
  • Speedup: Propagator splitting (specifically split_order 2 and 3) reduced wall-clock times significantly compared to exact diagonalization (diag). For example, a DNP contact-time simulation took ~19s with split_order 2 vs ~67s with the exact method, with negligible loss in accuracy for ensemble averaging.
• Pulse Transient Compensation:
  • Simulations showed that optimizing pulses while accounting for transient distortions (using the distortion operator) yields pulses with superior bandwidth and fidelity compared to those optimized assuming ideal rectangular pulses.
  • The gradient reconstruction method successfully converged to optimal solutions despite the non-linear distortion landscape.
• Quadrupolar Systems:
  • Simulations of 2D STMAS spectra with quadrupolar-CSA and quadrupolar-dipolar cross-terms were performed.
  • Performance: For a $16 \times 16$ Hamiltonian (quadrupolar spin-3/2 coupled to another spin-3/2), the split_order 2 method completed in 287 seconds, whereas full diagonalization took 3556 seconds (a >10x speedup).

4. Significance and Impact

• Bridging Disciplines: SIMPSON v6.0 effectively bridges the gap between NMR, EPR, and DNP, providing a unified platform for simulating hybrid quantum systems. This is crucial for the emerging field of quantum sensing and advanced materials characterization.
• Computational Efficiency: The introduction of propagator splitting makes the simulation of large spin ensembles and high-dimensional optimal control problems feasible on standard workstations, removing a major bottleneck in pulse sequence design.
• Experimental Realism: By incorporating pulse transients and hardware-specific response functions, the software allows researchers to design pulses that are robust against real-world experimental imperfections, leading to better reproducibility and performance in actual experiments.
• Community Development: The transition to C++ and open-source distribution via GitLab and Docker lowers the barrier for community contributions, ensuring the software remains adaptable to future theoretical and experimental advancements in magnetic resonance.

In conclusion, this paper represents a substantial evolution of the SIMPSON package, transforming it from a specialized solid-state NMR tool into a versatile, high-performance engine capable of handling the complex, multi-spin, and hardware-constrained simulations required by modern magnetic resonance research.