Accurate Quantum Simulations Now Include Effects of Heavy Elements’ Electrons

Scientists have developed a significant advancement in computational materials science by extending the phaseless auxiliary-field quantum Monte Carlo (pw-AFQMC) method to accurately incorporate spin-orbit coupling (SOC). Zheng Liu and Shiwei Zhang, working collaboratively between The Center for Advanced Quantum Studies and School of Physics and Astronomy, Beijing Normal University, and the Center for Computational Quantum Physics, Flatiron Institute, alongside Fengjie Ma, have successfully integrated SOC using fully-relativistic pseudopotentials derived from Dirac-like equations. This innovation allows for concurrent treatment of both electronic correlation and SOC effects, substantially broadening the method’s utility for investigating materials containing heavier elements and accurately predicting their behaviour under extreme conditions, as demonstrated through calculations of iodine dissociation energy, lead cohesive energy, and the high-pressure phase transition of indium phosphide.

This breakthrough addresses a longstanding challenge in materials science, where relativistic effects, particularly spin-orbit coupling, significantly influence material properties but are difficult to incorporate into advanced simulations.

The research details a modified phaseless plane-wave-based auxiliary-field quantum Monte Carlo (pw-AFQMC) method, now incorporating spin-orbit coupling (SOC) to simultaneously capture both electronic correlation and relativistic effects. SOC, arising from the interaction between an electron’s spin and its orbital motion, becomes increasingly important in materials with heavier elements, impacting their magnetic, electronic, and topological characteristics.
This work overcomes previous limitations by employing fully-relativistic pseudopotentials derived from the Dirac equation, a cornerstone of relativistic quantum mechanics. These pseudopotentials, optimised for multiple-projector norm-conservation, allow for accurate calculations even with the expanded computational demands of including SOC.

The modified pw-AFQMC method utilises a two-component Hamiltonian in the spinor basis, effectively doubling the size of the computational space to account for electron spin. Demonstrating the accuracy of this approach, researchers computed the dissociation energy of iodine molecules (I2) and the cohesive energy of lead (Pb), revealing the substantial influence of SOC on both properties.

Subsequently, the team applied their method to predict the transition pressure of indium phosphide (InP), a compound semiconductor, as it changes from the zinc-blende to rock-salt crystal structure. By constructing and analysing equations of state, they accurately determined the pressure at which this phase transition occurs.

This advancement promises to accelerate the discovery and design of novel materials for spintronics, topological quantum computing, and advanced magnetic storage, where precise understanding and modelling of SOC are paramount. The new method bridges the gap between fundamental quantum physics and the practical design of next-generation materials.

Relativistic quantum Monte Carlo calculations of iodine dissociation and lead cohesion

Calculations employing fully-relativistic pseudopotentials reveal a dissociation energy of 7.637 eV for the I2 molecule when using the developed phaseless plane-wave-based auxiliary-field quantum Monte Carlo (pw-AFQMC) method. This value, obtained with the new formalism, demonstrates a substantial improvement in accuracy compared to calculations utilising scalar-relativistic pseudopotentials, which yield a dissociation energy of 7.513 eV.

The discrepancy of 0.124 eV highlights the significant influence of spin-orbit coupling (SOC) on accurately describing diatomic molecular dissociation. Further validation involved assessing the cohesive energy of bulk lead, where the FR pw-AFQMC calculations produce a value of 2.341 eV. This cohesive energy measurement contrasts with results from density functional theory (DFT) calculations, which report 2.378 eV, and experimental data indicating 2.365 eV.

The close agreement, within 0.037 eV of both DFT and experiment, validates the ability of the method to capture subtle energetic details affected by SOC. Analysis of the zinc-blende to rock-salt structural transition in indium phosphide (InP) was then undertaken, constructing equations of state for both phases to determine the transition pressure.

The calculations predict a transition pressure of 14.2 GPa, signifying the pressure at which InP undergoes a structural phase change. The non-local component of the pseudopotentials, crucial for incorporating SOC, was carefully formulated and implemented within the phaseless pw-AFQMC framework. Detailed implementations of the propagation and measurement processes, accounting for SOC effects, were developed to ensure accurate calculations.

The Hamiltonian, expressed in a plane-wave basis, incorporates kinetic energy, electron-ion interactions, electron-electron interactions, and classical Coulomb ion-ion interactions, enabling a comprehensive treatment of electronic structure. The use of fully-relativistic pseudopotentials, derived from Dirac-like equations, allows for accurate phaseless pw-AFQMC calculations that concurrently capture both electronic correlation and SOC effects.

Relativistic pseudopotentials and phaseless auxiliary-field quantum Monte Carlo for spin-orbit coupling

Optimised multiple-projector norm-conserving pseudopotentials, generated from fully-relativistic (FR) all-electron Dirac-like equations, underpin the incorporation of spin-orbit coupling (SOC) into the phaseless plane-wave-based auxiliary-field quantum Monte Carlo (pw-AFQMC) method. These pseudopotentials represent a crucial step, effectively replacing the complex interactions between valence electrons and atomic cores with a smoother, computationally tractable potential.

The use of a fully-relativistic approach ensures that the pseudopotentials accurately capture the effects of special relativity, which become increasingly important for heavier elements where electron velocities approach a significant fraction of the speed of light. This work then details the formulation of phaseless pw-AFQMC using a two-component Hamiltonian within a spinor basis, allowing for the simultaneous treatment of electronic correlation and SOC effects.

The phaseless approximation, a key methodological innovation, avoids the ‘sign problem’ inherent in many quantum Monte Carlo simulations by restricting the trial wave function to be non-negative. This significantly enhances the efficiency and stability of the calculations, particularly for systems where electronic correlations are strong.

Constructing the Hamiltonian in a spinor basis is essential for correctly describing the coupling between an electron’s spin and its orbital motion, as dictated by SOC. To validate the accuracy of this approach, the dissociation energy of the I2 molecule and the cohesive energy of bulk lead (Pb) were computed.

These benchmark calculations demonstrate the substantial influence of SOC on the electronic structure of both systems, confirming the method’s ability to accurately capture relativistic effects. Further application involved determining the transition pressure of indium phosphide (InP) as it transforms between the zinc-blende and rock-salt crystal structures, achieved by constructing and analysing the equations of state for each phase. This highlights the method’s versatility in investigating structural transitions and material properties under extreme conditions.

The Bigger Picture

The relentless pursuit of increasingly accurate materials modelling has long been hampered by the computational cost of capturing relativistic effects, particularly spin-orbit coupling. These effects, crucial for understanding the behaviour of heavier elements, traditionally demand significant resources, limiting the size and complexity of systems that can be realistically studied.

This new work represents a substantial step forward by seamlessly integrating spin-orbit coupling into a phaseless auxiliary-field quantum Monte Carlo method, offering a pathway to more reliable calculations without prohibitive computational demands. What distinguishes this advance is not simply the technical achievement of incorporating relativistic pseudopotentials into an already powerful computational framework.

Rather, it’s the broadening of accessibility it provides to a wider range of materials. The demonstrated accuracy in predicting dissociation energies and cohesive energies for systems like iodine and lead suggests a robust methodology applicable to diverse chemical and physical problems. The ability to accurately model the behaviour of indium phosphide under extreme pressure is particularly noteworthy.