Rubenstein Group Research:
Stochastic Electronic Structure Theory, Alternative Computing, and Biomolecular Simulation

Stochastic electronic Structure

For decades, quantum chemists have been forced to make an oftentimes humbling choice in their day-to-day work: to use highly accurate, many-body methods that are too slow to apply to realistic quantum systems, or, to use faster one-body methods that are significantly less accurate (see Figure 1). This fundamental compromise has glaringly limited the impact of quantum chemistry. Indeed, while most of modern experimental chemistry is focused upon synthesizing complex molecules and designing novel nano- and bulk materials, most modern quantum chemistry techniques are hard-pressed to even approach the scales necessary to answer many of the most pivotal experimental questions about these systems. Our group is focused on developing electronic structure methods that are at once highly accurate and scale well with system size to help bridge this divide and enable theory-driven materials design.




Figure 1: Traditional deterministic quantum chemistry techniques such as coupled cluster theory and configuration interaction are highly-accurate, yet computationally expensive. Density Functional Theory is significantly cheaper, but can yield inaccurate results for strongly-correlated systems. Quantum Monte Carlo methods lie in the methodological sweet spot because they at once scale gracefully with system size and are highly reliable for a variety of systems. 

The Specifics

Of key interest to the group is the development and application of stochastic methods to meet the long-standing need for rigorous techniques capable of accessing experimentally relevant scales. Because they make careful use of random numbers, quantum Monte Carlo methods are not only highly accurate (they are the gold standard in much of condensed matter physics), but scale gracefully with system size (see Figure 1).1,2,3,4 They can therefore address a wide variety of experimentally motivated questions inaccessible to equally accurate deterministic methods and be used to improve existing electronic structure methods including Density Functional Theory. Problems of primary interest to the group fall into three main categories:

 

Modeling of bulk transition metal oxides, transition metal-based nanoclusters, and lanthanide/actinide compounds

Electrons in transition metals occupy highly degenerate, strongly-correlated d- and f-orbitals that are particularly difficult to model using one-body theories.The group’s recent advances developing stochastic techniques capable of modeling hundreds of electrons in hundreds of orbitals opens the door to simulating a variety of transition metal-based compounds and materials (see Figure 2).6




Figure 2: A comparison of the equation of states produced using Coupled Cluster Theory Singles and Doubles (CCSD) vs. Auxiliary-Field Quantum Monte Carlo (AFQMC) for diamond. Although substantially cheaper than CCSD calculations, AFQMC estimates of the energy per atom are in strong agreement with CCSD results.6 

Quantum Monte Carlo simulations of transition metal oxides will enable the high accuracy prediction of band gaps and high temperature, high pressure phase transitions, which will pave the way toward an improved understanding of the phase diagrams of complex solids. In concert with these modeling efforts, the group has been and will continue to be engaged in a substantial theoretical effort aimed at developing the highly-accurate trial wave functions that underpin these simulations (see Figure 3).7

 




Figure 3: The Group is engaged in an extensive effort to develop new ways of generating trial wave functions that can then be used as a starting point for more accurate projector quantum Monte Carlo techniques, such as Auxiliary-Field Quantum Monte Carlo. Left: The error of the energy predicted using Constrained Path Quantum Monte Carlo (CPMC) for a 1D Hubbard Model is significantly reduced using a random sampling of determinants from a full multideterminant Gutzwiller wave function (green) compared with using just a single free electron wave function (red). Right: The reduction in the CPMC error on the energy as more determinants from the full multideterminant Gutzwiller expansion are used.

Development of new methods capable of accurately capturing relativistic effects, including spin-orbit coupling

Being able to properly model relativistic effects is critical to predicting the behavior of heavy elements including Se and Te, which comprise topological insulators and important semiconductors, and the lanthanides and actinides, which assume a pivotal role in the nuclear fuel cycle and exhibit intriguing volume collapse transitions in the bulk.8,9,10 Our group is currently exploiting quantum Monte Carlo’s unique ability to obtain the ground state wave functions of multi-component Hamiltonians to explore the full hierarchy of relativistic Hamiltonians and their accuracy in quantum chemical calculations.


Design and testing of finite temperature electronic approaches to model hot electrons 

Few electronic structure approaches are capable of modeling materials under conditions in which electrons are excited well above the ground state. This currently limits our understanding of photochemistry, plasma physics, and astrophysical processes such as star formation. Our group is developing density matrix-based methods (see Figure 4) that will enable the direct prediction of molecular behavior at high temperatures using both stochastic methods and more conventional finite temperature density functional theories.11




Figure 4: Path integral methods provide a means for describing a variety of "warm" systems. Left: A path integral simulation of a superglassy mixture of Kob-Andersen Lennard-Jones particles (red and yellow).20 Right: The potential energy and condensate fraction with decrease temperature (inverse beta) for a Bose-Fermi mixture.11

Advances made in these directions will establish the foundations necessary for rigorous simulations of a variety of materials currently outside the scope of modern-day exploration, dramatically expanding the predictive power of quantum chemical techniques. Students engaged in this work will develop a deep understanding of both quantum and statistical mechanics and will become familiar with modern high performance computing paradigms.

 

Other Research Interests and Projects

 

Pinpointing Sources of Anomalous Noise in Quantum Computers

Because of their prospects for miniaturization and scalability, trapped ions represent a promising implementation of qubits for quantum computers. However, their widespread adoption into quantum computing architectures has been severely hindered by the effect of anomalous heating, an observed phenomenon of ion heating rates orders of magnitude higher than the Johnson noise limit.12 Despite a number of recent experiments aimed at better characterizing this phenomenon, the microscopic sources of anomalous heating are still not well understood. Over the past few years, the group has been involved in multi-institution effort to better understand these noise sources through a combination of experiment and high-accuracy ab initio modeling. Our work has shown that hydrocarbon adsorbates are some of the dominant contributors to anomalous heating (see Figure 5).13 Future work will revolve around the development of even more predictive theories and simulations of ion traps to address this important technological problem. 
 




Figure 5: Previous research performed by the group has demonstrated that hydrocarbons adsorbed onto electrode surfaces may substantially contribute to heating within ion traps. Left: The adsorbate-electrode binding potentials as a function of the adsorbate distance above the electrode. Right: Benzene adsorbed onto a model gold surface.13

Biomolecular Simulation

Biology is filled with problems, ranging from protein and RNA folding14 to flocking patterns,15 that can now be answered using a combination of statistical and quantum mechanics. The group has a longstanding interest in developing and applying methods that enable the exploration of such biological questions. Past work has focused on creating course-grained models that can be solved using conventional statistical mechanics to better understand such phenomena as tumor growth and protein binding (see Figure 6).16,17 Future work will delve into improved modeling of multi-metallic protein active sites.18,19 

 




Figure 6: The Rubenstein Group has a long history of developing course-grained models of biological systems simple enough to study, yet complex enough to be informative about real-world problems. Left: A simulation of glioblastoma multiforme tumor (green and purple) growth atop a collagen (yellow) substrate.16 Right: Three model proteins (red) designed to fold and bind in different ways to substrates (black) amidst crowding polymers (blue).17 

References
1W.M.C. Foulkes, L. Mitas, R.J. Needs, and G. Rajagopal. Quantum Monte Carlo simulations of solids. Reviews of Modern Physics. 73, 33 (2001). 
2D.M. Ceperley. Path integrals in the theory of condensed helium. Reviews of Modern Physics. 67, 279 (1995).
3S.R. White, D. J. Scalapino, R.L. Sugar, E.Y. Loh, J.E. Gubernatis, and R.T. Scalettar. Numerical study of the two-dimensional Hubbard model. Physical Review B. 40, 506 (1989). 
4G.H. Booth, A.J.W. Thom, and A. Alavi. Fermion Monte Carlo without fixed nodes: A game of life, death, and annihilation in Slater determinant space. The Journal of Chemical Physics. 131, 054106 (2009).
5G. Kotliar and D. Vollhardt. Strongly correlated materials: Insights from dynamical mean-field theory. Physics Today. 53, March 2004; M. Imada, A. Fujimori, and Y. Tokura. Metal-insulator transitions. Reviews of Modern Physics. 70(4), 1039 (1998).
6E. Landinez-Borda, D. Matthews, B.M. Rubenstein, and M.A. Morales-Silva. Auxiliary-field quantum Monte Carlo for Strongly-Correlated Solids. In preparation for JCP (2016). 
7C.C.-Chang, B.M. Ruenstein, and M.A. Morales-Silva. Auxiliary-field based trial wave functions in quantum Monte Carlo calculations. arXiv:1604.00345.
8M.J. Lipp, D. Jackson, H. Cynn, C. Aracne, W.J. Evans, and A.K. McMahan. Thermal Signatures of the Kondo Volume Collapse in Cerium. Physical Review Letters. 101, 165703 (2008).
9X. Cao and M. Dolg. Relativistic energy-consistent ab initio pseudopotentials as tools for quantum chemical investigations of actinide systems. Coordination Chemistry Reviews. 250, 900 (2006).
10P.J. Hay and R.L. Martin. Computational studies of actinide chemistry. Los Alamos Science. 26, 391 (2000). 
11B.M. Rubenstein, S. Zhang, and D.R. Reichman. Finite-temperature auxiliary-field quantum Monte Carlo technique for Bose-Fermi mixtures. Physical Review A. 86, 053606 (2012). 
12M. Brownnutt, M. Kumph, P. Rabl, and R. Blatt. Ion-trap measurements of electric-field noise near surfaces. Reviews of Modern Physics. 87, 1419 (2015). 
13K.G. Ray,* B.M. Rubenstein,* and V. Lordi. An ab initio study of ion-trap electric field noise caused by electrode surface adsorbates. Submitted to Physical Review Letters (2016). *Authors contributed equally to this work. 
14C.M. Dobson, A. Sali, and M. Karplus. Protein folding: A perspective from theory and experiment. Angewandte Chemie. 37, 868 (1998).
15W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, and A.M. Walczak. Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences of the United States of America. 109, 4786 (2012). 
16B.M. Rubenstein and L.J. Kaufman. The role of extracellular matrix in glioma invasion: A cellular Potts model approach. Biophysical Journal. 95, 5661 (2008). 
17B.M. Rubenstein, I. Coluzza, and M.A. Miller. Controlling the folding and substrate-binding of proteins using polymer brushes. Physical Review Letters. 108, 208104 (2012). 
18D.E. Wilcox. Binuclear metallohydrolases. Chemical Reviews. 96, 2435 (1996). 
19J.A. Sigman, H.K. Kim, X. Zhao, J.R. Carey, and Y. Lu. The role of copper and protons in heme-copper oxidases: Kinetic study of an engineered heme-copper center in myoglobin. Proceedings of the National Academy of Sciences. 100, 3629 (2003).
20B.M. Rubenstein. Novel Quantum Approaches for Quantum Liquids. Ph.D. Thesis, Columbia University (2013).