Our main goal was to develop a new capability for the accurate and efficient study from first principles of strongly correlated elemental materials with high atomic numbers. We developed a novel implementation of the auxiliary-field quantum Monte Carlo (AFQMC) method that is accurate and efficient, with the ability to treat complicated materials and take advantage of modern large scale computational resources. As a result of this work, new efforts have been funded within the Department of Energy’s Basic Energy Science and Advanced Scientific Computing to further develop the AFQMC method. The capabilities developed in this project are being used within Lawrence Livermore National Laboratory to study the properties of materials at extreme conditions, including high pressure equations of state and phase diagrams of multiple materials.

The main goal of the project was the development of a first principles simulation method capable of producing results with high predictive capability on materials that contain strong electron correlation and high atomic numbers. These materials are central to many of the critical missions of Lawrence Livermore and are at the core of the theoretical high-energy-density science and stockpile stewardship science done at the Laboratory. As such, accurate theoretical methods are critical to the success of these Laboratory missions. Many applications require equations of state that must span up to nine orders of magnitude in density, well beyond the reach of available experimental capabilities. Accurate simulation methods fill the knowledge gap in those regimes inaccessible to experiments.

First-principles simulation methods are those that attempt to obtain a complete description of a material without any input from experiments and with little to no empiricism. This is achieved by attempting to solve directly the fundamental equations that describe matter at the atomic scale, the Schrödinger equation, using powerful numerical techniques. While the field of first-principles simulations has been quite successful at obtaining reasonable predictive capabilities for many materials at modest computational costs, certain materials have posed a significant challenge to traditional methods. These materials—typically described as “strongly correlated materials” because of the large influence that strong electronic correlation has in their electronic structure—are typically quantitatively and qualitatively incorrectly described by traditional first-principles approaches. To obtain an adequate description of these materials, novel methods must be developed that can successfully solve the many-body quantum mechanical system at the heart of the problem.

In this project we developed a new implementation of the AFQMC method, a quantum many-body approach capable of describing strong electron correlation and able to handle the complexity presented by real materials. The main objectives of the project involved: (1) a high-performance and scalable implementation of the AFQMC method, (2) development of additional features within the AFQMC framework to enable the study of real materials with strong electron correlation and high atomic numbers, (3) development of interfaces to standard electronic structure codes, and (4) the study of several materials across the periodic table. Most of the objectives were successfully accomplished. A new implementation of AFQMC was developed and particularly careful attention was placed on the parallelization and efficiency of the code, which was necessary to mitigate the high computational cost of the approach. We implemented new approaches to improve the efficiency and accuracy of the method in the study of highly correlated matter through the development of (1) novel wave-functions (Chang 2016), (2) efficient optimization techniques, (3) new understanding of the role of symmetry in strong electron correlation (Degroote 2016), and (4) new connections between strong electron correlation and traditional wave-function methods typically used in quantum many-body approaches (Wahlen-Strothman 2017), among others. We developed several interfaces to standard electronic structure codes and investigated a selection of materials.

Auxiliary-field quantum Monte Carlo is an orbital-based quantum many-body method. Similar to most projector Monte Carlo methods, it is based on the stochastic projection of a trial wave-function in imaginary-time. It is easy to show that such an imaginary-time projection will lead to one of the eigenstates of the system, typically the ground state. The wave function is represented as an ensemble of “walkers," each representing a Slater determinant. The method is implemented in orbital space and the only uncontrolled approximation in the method is encapsulated in the phaseless, or constrained path, approximation (Zhang 2003), depending only on the trial wave function. The approximation eliminates the sign or phase instability associated with all fermion Monte Carlo methods and restores low-power (typically to the third power of system size) computational scaling. Applications to a variety of systems have shown that the method is very accurate, even with simple trial wave functions taken directly from mean-field calculations (Purwanto 2009, Chang 2010).

Four properties of the AFQMC method make it particularly attractive as a tool for the study of strongly correlated materials with high atomic numbers. First, the problem can be formulated in any single-particle basis set with a very flexible treatment of core electrons (all-electron, frozen-core, pseudopotential, etc.). Second, AFQMC enables easy incorporation of non-collinear magnetism and relativistic effects, which is essential for the study of materials containing heavy elements. Third, AFQMC offers efficient and scalable implementation, enabling the use of modern large scale computational resources. Finally, AFQMC leads to systematically improvable results when accurate trial wave-functions are employed. The implementation of the method developed in this project thoughtfully exploits all of these features to produce a new, accurate capability for the study of correlated materials with high atomic numbers.

Our project supports the Laboratory's high-energy-density science core competency and the NNSA's goal to strengthen its science, technology, and engineering base. The capability developed offers a significant new component to the study of materials at high pressures and to the development of equations of state. It not only serves as a primary tool for studying materials that pose great difficulty for traditional approaches, but it also helps in the validation of results produced by simpler approaches. This approach will enable new and exciting research in areas such as strongly correlated materials, material-by-design, and critical materials.

The project was successful in creating a new implementation of the AFQMC method, with appropriate extensions for the study of strongly correlated materials with high atomic numbers. In addition, the research performed within the project successfully led to two externally funded efforts within the Laboratory, which will support the further development of this capability. These efforts are the “Center for Predictive Simulation of Functional Materials” project funded by the Department of Energy's Basic Energy Sciences and the “QMCPACK: A Framework for Predictive and Systematically Improvable Quantum-Mechanics Based Simulations of Materials” project funded by the DOE's Advanced Scientific Computing Research (ASCR) program. The latter is part of ASCR’s Exascale Computing Project initiative. The developed tool is already being employed within the Laboratory's equation of state effort as a new addition to their theoretical and computational capabilities.

Chang, C. C., and S. Zhang. 2010. “Spin and Charge Order in the Doped Hubbard Model: Long-Wavelength Collective Modes." *Phys. Rev. Lett.* 104(116402). doi:10.1103/PhysRevLett.104.116402.

Chang, C.C., et al. 2016. “Auxiliary-Field Based Trial Wave Functions in Quantum Monte Carlo Calculations." *Phys. Rev. B* 94(235144). doi:10.1103/PhysRevB.94.235144. LLNL-JRNL-687063.

Degroote, M., et al. 2016. “Polynomial Similarity Transformation Theory: A Smooth Interpolation Between Coupled Cluster Doubles and Projected BCS Applied to the Reduced BCS Hamiltonian.” *Phys. Rev. B* 93(125124). doi:10.1103/PhysRevB.93.125124. LLNL-SR-744061.

Purwanto, W., et al. 2009. “Pressure-Induced Diamond to Beta-Tin Transition in Bulk Silicon: A Quantum Monte Carlo Study." *Phys. Rev. B* 80(214116). doi:>10.1103/PhysRevB.80.214116.

Wahlen-Strothman, J. M., et. al. 2017. "Merging Symmetry Projection Methods with Coupled Cluster Theory: Lessons from the Lipkin Model Hamiltonian." *J. Chem. Phys*. 146(054110). doi:10.1063/1.4974989. LLNL-SR-744060.

Zhang, S., and H. Krakauer. 2003. “Quantum Monte Carlo Method Using Phase-Free Random Walks with Slater Determinants." *Phys. Rev. Lett*. 90(136401).

Chang, C.C., et al. 2016. “Auxiliary-Field Based Trial Wave Functions in Quantum Monte Carlo Calculations." *Phys. Rev. B* 94(235144). doi:10.1103/PhysRevB.94.235144. LLNL-JRNL-687063.

Degroote, M., et al. 2016. “Polynomial Similarity Transformation Theory: A Smooth Interpolation Between Coupled Cluster Doubles and Projected BCS Applied to the Reduced BCS Hamiltonian.” *Phys. Rev. B* 93(125124). doi:10.1103/PhysRevB.93.125124. LLNL-SR-744061.

Morales, M. A. 2016. “Non-Orthogonal Slater Determinant Expansions in Quantum Monte Carlo." MESBA 2016: Molecular Electronic Structure Conference, Buenos Aires, Argentina, 19–23 September 2016. LLNL-PRES-703332.

Wahlen-Strothman, J. M., et. al. 2017. "Merging Symmetry Projection Methods with Coupled Cluster Theory: Lessons from the Lipkin Model Hamiltonian." *J. Chem. Phys*. 146(054110). doi:10.1063/1.4974989. LLNL-SR-744060.