Solids Simulated with Greater Accuracy and Efficiency Now

Researchers have long sought accurate and scalable methods for simulating the quantum mechanical behaviour of materials. Jinghong Zhang, Meng-Fu Chen, and Adam Rettig, working with colleagues at the Department of Chemistry and Chemical Biology, Harvard University, present a significant advance in this field with their development of an improved ab initio auxiliary-field quantum Monte Carlo (AFQMC) method. Their work overcomes previous computational limitations by combining tensor hypercontraction and point-group symmetry to achieve reduced scaling in both computational cost and memory requirements. This breakthrough enables direct and simultaneous calculations in both the thermodynamic limit and complete basis set, applicable to a wide range of solids without relying on approximations or composite schemes, establishing AFQMC as a viable and systematically improvable alternative to established techniques like diffusion Monte Carlo and coupled-cluster methods for predictive materials simulations.

Calculating the properties of materials has long been hampered by the limitations of computer power and mathematical approximations. Now, a new computational technique overcomes these hurdles, allowing for precise modelling of complex solids without relying on workarounds or estimations. This advance promises more accurate predictions of material behaviour and accelerates the discovery of new substances.

Scientists are continually refining methods for accurately simulating the behaviour of materials, a necessity driving advances across diverse fields including condensed matter physics, materials science, and chemistry. Accurate modelling of solids presents a considerable challenge, with current techniques often balancing computational cost against the need to capture complex quantum mechanical effects.

Kohn-Sham density functional theory (DFT) remains a widely used approach, offering a reasonable compromise between accuracy and expense, typically scaling as O(N 3) with system size (N), yet it struggles with strongly correlated systems and suffers from self-interaction errors. However, DMC relies on approximations like pseudopotentials and fixed-node constraints, introducing potential biases that are difficult to fully quantify. Similarly, coupled-cluster (CC) methods, while highly accurate for certain systems, scale as O(N 7) and require extrapolations to the CBS and TDL limits, often necessitating local approximations.

These local approximations, while reducing computational demands, can introduce further inaccuracies. By combining tensor hypercontraction with k-point symmetry, they have achieved a reduction in computational scaling to O(N 3) and memory requirements to O(N 2), matching the efficiency of DMC.

This advancement allows for direct and simultaneous calculations in both the TDL and CBS limits for a wide range of materials, including insulators, metals, and strongly correlated solids, without the need for approximations or composite schemes. Now, AFQMC emerges as a viable alternative to both DMC and CC methods, offering a pathway to more accurate and predictive simulations of solid-state systems.

Unlike many existing techniques, this approach enables the calculation of energies and magnetic properties within a single, unified framework. This progress stems from a novel application of tensor factorization, specifically interpolative separable density fitting, coupled with efficient k-point sampling. As a result, direct and simultaneous thermodynamic-limit and complete-basis-set AFQMC calculations are now feasible for insulating, metallic, and strongly correlated solids, eliminating the need for embedding, local approximations, or empirical corrections.

Previous AFQMC applications to solids were hampered by computational demands, often restricting calculations to coarse k-point sampling. The work demonstrates a substantial reduction in computational cost, moving from an initial O(N4) cost to O(N3), alongside a decrease in memory requirements from O(N3) to O(N2). By comparison, established methods like coupled-cluster with singles, doubles, and perturbative triples (CCSD(T)) exhibit an O(N7) scaling with O(N4) storage, highlighting the improved efficiency of this new AFQMC approach.

Achieving accurate simulations necessitates addressing challenges like strong correlation and self-interaction error, problems traditionally tackled by density functional theory. Unlike diagrammatic methods which require extrapolation to the complete basis set and thermodynamic limits, this AFQMC implementation directly accesses these limits. Specifically, researchers combined tensor hypercontraction, a technique for efficiently calculating many-body integrals, with interpolative separable density fitting to represent electron densities. This pairing diminished the computational scaling of AFQMC for solids to O(N3), where N represents system size, and reduced memory scaling to O(N2).

As a result, direct calculations approaching both the thermodynamic limit, where finite-size effects become negligible, and the complete-basis-set limit, where the basis set used to describe atomic orbitals is exhaustive, became feasible. Traditional methods often rely on embedding techniques, local approximations, or empirical corrections to manage system size and basis set truncation.

Instead, this work enabled simultaneous access to both the thermodynamic and complete-basis-set limits without these compromises, allowing for more accurate and unbiased simulations. K-points represent points in reciprocal space used to sample the Brillouin zone, and leveraging symmetry reduces the number of independent calculations needed. Calculations were performed on systems exhibiting insulating, metallic, and strongly correlated behaviours to demonstrate the method’s broad applicability.

The team carefully considered the limitations of existing methods like DMC, which can suffer from pseudopotential and fixed-node errors, and coupled-cluster methods, which often require extrapolations to achieve the complete basis set limit. The researchers sought to address the inherent difficulties in accurately simulating strongly correlated materials.

Unlike some approaches that focus on localized impurity problems, this work aimed for a general-purpose method applicable across diverse solid-state systems. Since the method avoids locality errors, increasing the impurity size is not necessary, and the computational scaling remains manageable. Beyond simply achieving a lower computational cost, the work prioritised a systematic approach to improving accuracy, allowing for reliable predictions of energies and magnetic properties.

Tensor hypercontraction unlocks accurate solid-state material simulations

For decades, calculating the properties of materials with absolute precision has remained a major challenge in computational physics and chemistry. Existing methods often rely on approximations that, while practical, introduce uncertainties and limit predictive power. Accurate modelling of solids is becoming increasingly feasible. This advance stems from a combination of tensor hypercontraction and symmetry exploitation, reducing the scaling of calculations with system size.

As a result, simulations can now approach the theoretical limits of both basis set completeness and infinite system size, previously unattainable without resorting to workarounds. The elimination of these approximations is not merely a technical achievement; it opens doors to predicting material behaviour with a level of confidence previously reserved for experiment.

Extending this approach to more complex systems, those with defects, surfaces, or disordered structures, will demand further algorithmic refinement. Assessing the impact of the remaining, albeit smaller, sources of error requires careful consideration. The method’s true potential will be revealed by its application to materials where experimental data is scarce or difficult to obtain, such as those under extreme conditions.

This work signals a shift in how materials are modelled. By providing a systematically improvable framework, it allows researchers to focus on refining the underlying physics rather than compensating for computational limitations. Unlike many current methods, this approach promises a clear route toward reducing uncertainty and building truly predictive models. For the field of materials science, this represents a substantial step toward designing new materials with tailored properties, potentially accelerating discoveries in energy storage, catalysis, and beyond.