Quantum Monte Carlo for Ab Initio calculations of energy-relevant materials

Authors


Abstract

Humanity faces one of its greatest challenges in the move from fossil-fuel based energy sources to alternative sources that do not produce greenhouse gases. New materials design is an important facet of the overall solution, since designed materials have the potential to increase efficiency in areas ranging from solar electricity generation to energy storage and distribution technologies. In that context, it is vital to be able to predict the properties of materials from basic physical principles. While traditional electronic structure techniques such as the ubiquitous density functional theory (DFT) are very important in this goal, there are many cases where current implementations of DFT fail in a design-important way. Among other solutions, quantum Monte Carlo techniques have emerged as a practical way to obtain predictive power for challenging materials. This perspective highlights some recent advances in this field, concentrating in particular on the effect that quantum Monte Carlo methods have and will have on our energy challenge. © 2013 Wiley Periodicals, Inc.

Introduction

In our lifetimes, clean and efficient energy resource management is one of the largest challenges facing humanity. Plentiful energy has led to the quality of life that those of us in developed countries enjoy and offers a path for the developing world to increase their quality of life. It is our responsibility to make sure that this process can occur without severely affecting the capability of the earth to support us. As it now seems clear that CO2 emissions in particular are affecting our climate, new solutions for energy are imperative. Many of the pieces of a clean energy package involve the manipulation of electrons in matter, from photovoltaic electricity generation to hydrogen storage to water splitting to carbon capture, among others. Some of these pieces would benefit from materials with very unique properties.

While experimental work has been critical in the exploration of new materials, computational design has been building momentum over the past few decades. This has been due to a number of factors. First, computer power in terms of floating point operations per second or per watt have been increasing exponentially, far beyond the span of years predicted by Moore.[1] Second, the theory of electrons in matter has been developed dramatically, with an amazing assortment of approximations to the solution of the many-body Schrödinger equation. Finally, implementation of these theories as computer program packages is now fairly mature, and many of them can be used in a way similar to any other reasonably user friendly scientific software. The fact that these programs and theories are now well mapped out means that they can be used in bulk to perform materials design, as evidenced in the rise of high-throughput materials design and the materials genome initiative[2] in the United States.

Most of the development in predictive electronic structure calculations has been within the context of density functional theory (DFT). While modern approximations to the unknown density functional are often quite accurate, they encounter problems in a number of instances. These errors can be large enough to affect the design of new materials. Because of these defects in DFT, a number of methods (often termed beyond DFT methods) have surged in popularity. In this perspective, we will highlight the quantum Monte Carlo class of methods, which offer explicit many-body treatment of the electronic problem, and are able to obtain extremely high accuracy results with good scaling with system size.

In the coming years, as the speed of sequential execution on modern processors levels out and computational resources become more parallel in nature, state of the art calculations will need to take advantage of this parallelism to continue addressing relevant problems. Quantum Monte Carlo techniques are particularly relevant in this environment because they are able to take advantage of many forms of parallelism, from graphics processing unit (GPU) devices[3] to multicore systems.[4]

A Brief Summary of Electronic Structure Methods

Supposing that we wish to solve the first-principles electronic structure problem, the problem we face is that the many-electron wave function is a tremendously complicated object. We can divide the approaches into a few broad categories. The first category consists of reduction based methods, such as DFT and density matrix theory, while the second consists of methods that directly deal with the many-electron wave function. A third category consists of methods based on perturbation theory, such as the GW approximation, which we will not discuss here.

Reduction based methods avoid dealing with the many-electron wave function and instead focus on calculating only some reduced quantity, such as the density. The tradeoff for this simplification is that the Hamiltonian becomes a functional in the case of DFT, and density matrices must be representable by a many-electron wave function. This unknown functional and/or representability problem are quite challenging problems to solve.

Conversely, wave function methods deal directly with the many-body Schrödinger equation. The question in this case then becomes how to represent this many-dimensional object and how to evaluate properties. Quantum chemical methods make the choice of relatively easy integral evaluation but slow convergence to the exact solution by expanding the many-electron wave function in a basis of Slater determinants. Quantum Monte Carlo approaches instead allow the wave function to contain explicit electron–electron dependence by using Monte Carlo techniques to evaluate the integrals. These broad classes are not completely orthogonal: particularly in recent years the r12 method[5] in quantum chemistry and the FCI-QMC/AFQMC[6, 7] methods from the Monte Carlo side have blurred the lines between these two approaches.

It is important to keep in mind that methods based on reduction and methods based on the many-electron wave function each have areas of utility. As in reduction-based methods, there is typically a constraint that is not systematically convergable, their applicability is stronger when they can be checked against either experiment or converged wave function-based calculations. In contrast, wave function methods have a clear path toward exact solutions, when the eigenvalues and eigenvectors of inline image have been obtained; the nonrelativistic quantum mechanical problem has been solved. The advantage of using reduction based methods is that they are typically much faster than wave function based methods, and so can access length and time scales otherwise unattainable. In many important problems, an approximate answer at relevant length and time scales may be better than a more accurate one that does not describe the relevant physics.

Quantum Monte Carlo Methods

There are many resources that summarize quantum Monte Carlo methods very well. The most commonly used method is a combination of Variational Monte Carlo (VMC) and fixed node diffusion Monte Carlo (FN-DMC), so we will summarize those first and some other flavors briefly later.

Variational Monte Carlo

VMC is a direct implementation of the variational method using Monte Carlo techniques to evaluate the integrals. VMC was used very early in the history of computational physics to study He4.[8] We first parametrize the many-electron trial wave function inline image, where R is the set of all electronic coordinates and {p} is a set of variational parameters. For any given {p}, the energy expectation value is

display math(1)

If E0 is the (unknown) ground state energy, then it is straightforward to show by spectral expansion that inline image for any value of {p}. We thus can obtain an upper bound for the ground state energy by minimizing inline image with respect to variations in {p}.

Most studies have been performed using the Slater-Jastrow wave function:

display math(2)

where the one-particle orbitals inline image are taken from some mean-field calculation, such as Hartree-Fock or DFT. In the most common version of the Slater-Jastrow wave function, these orbitals are kept fixed and not optimized. The u term, called the Jastrow factor, allows for explicit dependence on the distances between electrons. This term can be parameterized in many different ways; one common and effective parameterization is as follows:

display math(3)

where α is the index of a nucleus, i and j are electron indices, inline image is the electron–nucleus distance, rij is the electron–electron distance, and a and b are basis functions in one spacial dimension.

Different variational methods differ by how the wave function is parameterized and how the parameters are optimized. In the past decade, a flurry of activity in this subject has resulted in a very robust optimization method[9] that can minimize many parameters efficiently and precisely. Having a robust optimization method also allows one to go beyond the Slater-Jastrow parameterization, which we will discuss later in this article.

Diffusion Monte Carlo

The general method of DMC, of which FN-DMC is an approximation, is based on particle-based simulation of the imaginary time Schrödinger equation. It is straightforward to show that in the long time limit this equation,

display math(4)

converges to the ground state of the Hamiltonian inline image, under fairly general conditions. This method is essentially the matrix power method for finding extremal eigenvalues, where the matrix is the operator inline image. If τ is small enough, the largest eigenfunction of this operator is the eigenfunction of inline image with the lowest eigenvalue. For small numbers of particles, Eq. ((4)) can be solved using standard differential equation solvers; however, as the number of particles increases, the dimensionality of inline image also increases, and deterministic solutions become exponentially inefficient. DMC is based on a Feynman-Kac isomorphism between the partial differential equation in Eq. ((4)) and a stochastic process. For common Hamiltonians that include a kinetic energy and a local potential, the corresponding stochastic process is a combination of diffusion of particles combined with a birth/death process. We can thus use an ensemble of particles to integrate Eq. ((4)), with the density of particles representing the magnitude of inline image as a function of time. This approach does not suffer from the curse of dimensionality, making the method applicable to systems with large number of particles.

The DMC approach described in the previous paragraph suffers from two main deficiencies. First, the lowest energy eigenfunction of most Hamiltonians is bosonic in nature, while we are typically interested in calculating properties of fermions for electronic structure. This can be solved by defining a reference nodal structure which is not allowed to change, called FN-DMC. If this nodal structure enforces fermionic symmetry, this restriction limits the dynamics of Eq. ((4)) to a subspace of the potential wave functions, all of which have fermionic symmetry. The resulting solution is thus an upper bound to the exact lowest energy eigenfunction with fermionic symmetry. The fixed node restriction can also be used to study excited states by excluding states with the symmetry of the ground state. The second deficiency of Eq. ((4)) is that in practice one finds a simulation on realistic systems to be highly inefficient. This can be partially resolved by performing a similarity transformation using a trial wave function, also called importance sampling. This transformation is described in detail elsewhere.[10, 11] The importance transformation improves the efficiency dramatically and automatically gives a set of nodes with which to enforce the fixed node condition. The final FN-DMC algorithm is surprisingly elegant and simple, reducing to straightforward rules of moving walker positions.

The only major uncontrolled approximations in FN-DMC are the nodes from the trial wave function and the pseudopotential approximation. The nodal error can be optimized variationally, and the pseudopotentials must be tested very carefully to ensure accurate results. It is also worth noting that the closely related reptation Monte Carlo[12] method allows more accurate evaluation of properties other than energy.

Fixed node error: The importance of the one-particle orbitals

For systems that are not strongly correlated, experience has shown that the quantum Monte Carlo results are not very sensitive to the one-particle orbitals inline image in the single Slater determinant. However, this situation changes dramatically for transition metal oxide systems[14] and potentially other transition metal-containing systems. In these systems, the amount of d-p hybridization varies from very little in the case of orbitals from Hartree-Fock to a quite hybridized state in the case of DFT in the local density approximation (Fig. 1).

Figure 1.

A case in which the one particle orbitals are very important for accuracy of the FN-DMC calculation. This is an isosurface of the d-p hybridization orbital for the TiO molecule, calculated in Hartree-Fock (above) and DFT in the B3LYP approximation (below). The oxygen atom is in red on the right and the titanium in green on the left. The uncorrelated Hartree-Fock solution overionizes the p orbital and thus causes large fixed node errors. Figure reproduced with permission from Ref. [13]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 2.

Predicted ranges at which metal hydride nanoparticles consisting of aluminum and a 50/50 alloy of magnesium and aluminum have the desired desorption energy for hydrogen storage purposes, with FN-DMC results compared to a number of DFT methods. Note the logarithmic scale on the y axis. The DFT methods vary wildly in their size prediction. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 3.

(Figure reproduced with permission from Ref. [67]). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Because the orbitals determine the nodes of the trial wave function, the energy resulting from the FN-DMC varies with the d-p hybridization. One of the fundamental advantages of the FN-DMC method is that it is variational and so the amount of d-p hybridization can be varied to find the lowest energy FN-DMC result. This can easily be done by varying the amount of exchange in a hybrid calculation. Recent advances in wave function optimization should also allow the orbitals to be optimized directly, which have been done for TiO.[15] The FN-DMC method does partially correct the hybridization toward the correct value, but it is not able to fully correct the density given a poor trial wave function. This effect has been demonstrated both in molecules and solids.[16] It is also possible and fruitful to use multireference trial wave functions, which we will discuss below.

Pseudopotentials

There are now several sets of pseudopotentials[15, 17, 18] specifically designed for highest accuracy with quantum Monte Carlo methods. For the first row transition metal series, it is tempting to reduce the expense of the calculation by including only the 4s and 3d electrons in the valence; however, this approximation reduces the accuracy of the calculation significantly. This can be seen clearly in the ionization energy in Mn and Fe, which has an error of ∼0.5 eV when the semicore states 3s and 3p are ignored.[19] When using standard DFT codes that use the Kleinman-Bylander approximation, it is thus important to take great care in their treatment of the pseudopotential. Codes that use the semilocal representation directly do not suffer from this problem.

In the projection based methods such as FN-DMC and FN-RMC, there are several ways of treating the nonlocal term in the Hamiltonian. Many older calculations use the locality approximation,[20] which makes a generally small approximation to simplify the projection. The locality approximation is usually accurate if the VMC wave function is accurate. The variational approximation can be restored by allowing for nonlocal moves,[21] as well as a slight stabilization of the algorithm.

Implementations of quantum Monte Carlo techniques

There are a number of packages that at a minimum implement variational and DMC techniques for solids and molecules. These include CASINO,[22] QWalk,[4] QMCPACK,[23] and CHAMP.[24] These packages vary in their feature list and what DFT/quantum chemistry programs with which they interface. However, all of them are capable of performing the calculations that are presented here.

The Current State of the Art

To summarize, there is a commonly used, roughly standard, recipe as follows:

  1. Choose an appropriate set of pseudopotentials (likely smooth Dirac-Fock potentials).
  2. Obtain a DFT solution of the system.
  3. Optimize a Jastrow factor using VMC.
  4. Perform DMC using the optimized Slater-Jastrow wave function.
  5. If the system contains transition metals, repeat steps 2–4 with different levels of hybridization between DFT and exact exchange to variationally optimize the FN-DMC energy.

At the level of a single Slater determinant describing the wave function nodes, the energies calculated by FN-DMC are a large improvement over that of DFT. This version of QMC costs around 100 times as much as DFT, but scales the same (∼ inline image, where Ne is the number of electrons) and can efficiently use up to hundreds of thousands of cores if those cores are available. Thus with a massively parallel machine, this version of QMC becomes straightforward for systems with less than approximately 1000 electrons.

There are three main defects with the “standard recipe” presented earlier. These three areas are for the most part under active investigation in the QMC community. They are in the areas of excited states, forces, and systematic convergence to the exact wave function.

Excited state spectra

While FN-DMC is often thought of as a ground state method, it has been applied in a number of cases[25-30] to excited states. A rigorous upper bound to some excited states[31] can be obtained, but in practice even when an upper bound is not assured, the estimations of the excited state energies are accurate. FN-DMC calculates properties only about a single quantum state, which is prohibitively expensive for an entire excitation spectrum. There has been a substantial amount of work in improving this state of the art, particularly by Filippi and coworkers on organic molecules.[28-30]

Forces

While calculating the derivative of the energy with respect to nuclear coordinates seems like a simple request, it is not trivial to do this in FN-DMC. There have been a large number of advances in this subject[32-39] over the past 10 years, yet a generally applicable method to calculate forces is still not broadly available. For light atoms, the situation is a bit better; for example, molecular dynamics using forces from VMC has been performed for hydrogen systems,[40] vibrational spectra have been computed using VMC forces,[41] and minimum energy pathways have been determined from using correlated sampling approaches.[42] However, this is still a far cry from a general method.

Because of the challenging nature of forces in FN-DMC, several researchers have investigated other methods to obtain information that would normally be obtained using forces. Of particular note are a geometry optimization method that can build on DFT forces to obtain FN-DMC geometries,[43] and a method that allows for classical and quantum Monte Carlo simulations of the nuclear degrees of freedom using quantum Monte Carlo energies.[44] Also of note is the continuous DMC method,[45] which allows a FN-DMC simulation to efficiently follow a predefined molecular dynamics trajectory. With these methods, many problems can be solved without needing forces. However, in cases where dynamics are needed, FN-DMC methods still need development.

Beyond Slater-Jastrow

In principle, for a sufficiently flexible parameterization of the wave function, the nodes can be optimized to obtain accurate results for the ground state and selected excited states.[30] This has been demonstrated for small molecules, but not for bulk systems. In VMC, the choice is made to evaluate the many-body integral in Eq. ((1)) using Monte Carlo techniques. This choice is convenient because only the value and derivatives of inline image need to be evaluated given some point in R. There is thus a significant amount of flexibility in how the wave function is parameterized. Some of the most common forms are summarized here.

In a backflow wave function,[46] the Slater determinant is written in terms of dressed particle coordinates inline image, where inline image and inline image is the coordinate of the ith particle. In homogeneous systems, this is the next order of correction after the Jastrow factor.[47] While this functional form can change the nodes, if it is used in a local form, it is difficult to describe nondynamic correlations that can be important in transition metal systems. The backflow wave function is also quite expensive.

Another route beyond the Slater-Jastrow wave function is the pairing wave function. This type of wave function is similar to the BCS pairing wave function and is written as a determinant or a Pfaffian of two-electron pairing functions inline image. In general systems, the pairing function by itself suffers from severe size consistency errors, but when paired with a Jastrow factor, these errors can in principle be removed.[48] However, so far when tested approaching the thermodynamic limit for graphene, the pairing wave function has not had significant contribution to the total energy.[49]

Finally, the multi-Slater-Jastrow is of particular interest recently, since Nukala and Kent[50] and later Clark et al.[51] have formulated efficient ways of evaluating many Slater determinants with minimal additional effort over a single Slater determinant. This form is also promising because it converges systematically in the number of determinants and the multiple Slater determinants are complementary to the correlation described by the Jastrow factor. The Jastrow factor is very good at describing the electron cusp and short-range correlations, while the Slater determinant basis is better suited for long-range correlations.

The main issue with the multi-Slater-Jastrow wave function is size-consistency. It is well known that truncated configuration interaction wave functions are not size consistent and in principle are not applicable to the thermodynamic limit. Coupled cluster wave functions, while size consistent, are not straightforward to evaluate in quantum Monte Carlo. However, it is still an open area of study how the truncated wave functions interact with the Jastrow factor and whether the Jastrow factor can ameliorate the size consistency error. More information can be found in, for example, Refs. [9, 48, 50-53].

Applications to Energy Research

One of the most appealing and straightforward uses of quantum Monte Carlo methods is the accurate calculation of the ground state electronic energy and selected excited states. This quantity is fundamental in a number of quantities as we shall see. What follows is a set of applications that leverage these accurate energies to provide insight into energy-related problems.

Hydrogen storage

Hydrogen has a very high energy/mass ratio when combusted or used in a hydrogen fuel cell, but the energy/volume ratio is very low. Materials-based hydrogen storage offers the possibility to trade off some of the energy/mass performance to significantly improve the energy/volume performance. This requires a very particular material, however, since the adsorption energy must fall in a precise range, about 20 kJ/mol, to allow recharging under moderate pressures, and to allow desorption under moderate temperatures. The accuracy of traditional DFT methods is not sufficient to pin down energetics to this range. Quantum Monte Carlo methods have been applied to metal hydrides,[54, 55] calcium ion-based materials,[56, 57] and metal organics.[58] In several of these studies, the QMC calculations were compared either to experiment or very high level quantum chemistry techniques, which reinforced the accuracy of the energetics. In particular, in Ref. [55], we were able to provide scaling curves for the desorption energy of several different intermetallic alloys as a function of nanoparticle sizes. These curves were shown not to be accurately calculated (Fig. 2) using DFT based methods. This study used several hundred high precision FN-DMC calculations of energies of molecules and solids, so it is an emphatic demonstration of the potential of using FN-DMC in high-throughput design of energy relevant materials.

Photovoltaics

Recently, FN-DMC has been applied to the problem of band alignment,[59, 60] which is critical for designing new solar cell materials and for surface physics. Typically used density functionals have particular difficulty when comparing two systems with very different electronic character; for example, as considered in Wu et al.,[59] the case of a molecule interacting with a surface. Because FN-DMC obtains a solution much closer to the exact solution of the Schrödinger equation, it is able to more accurately estimate the band alignment between the different materials. In the case considered by Wu et al, the predicted type of junction was corrected by FN-DMC, with DFT methods predicting a Type II junction suitable for heterojunctions, and FN-DMC predicting a Type I junction, which would eliminate that combination from consideration for solar materials. While in the above case, FN-DMC obtained the “boring” result, this is very important for computational materials design; much effort could have been wasted on attempting to make the predicted device experimentally.

For the closely related problem of doping a semiconductor, the accuracy of FN-DMC has been checked carefully[61-65] with quite encouraging results. In particular, in Ref. [66], the FN-DMC calculations aided significantly in the interpretation of an experimental attempt to produce intermediate band defect states. The experiment detected a large increase in light absorption in Se hyperdoped silicon, which could be attributed to either a metal-insulator transition or to intermediate bands. Using just DFT, it is difficult to distinguish between these two scenarios because of band gap underestimation. However, the metal-insulator transition was seen in FN-DMC, which provided confidence that the light absorption was not due to an intermediate band.

Having simultaneous accurate barrier heights, excited states, and defect energy estimation was used by Wagner and Grossman[67] to study models of amorphous silicon. Hydrogenated amorphous silicon (a-Si:H) is a well-established thin-film technology that is used in applications from flat panel displays to solar cells. A-Si:H has a much higher absorption coefficient than crystalline silicon, which allows for the production of very thin and even flexible devices, with lower processing costs because of the lack of need for energetically costly crystallization. However, one of the major disadvantages of a-Si:H is that the hole transport is dispersive in nature, and thus very slow. This is further exacerbated by the Staebler-Wronski effect,[68] in which the hole transport further degrades upon exposure to light. The nature of this degradation has been a puzzle for many years. Wagner and Grossman performed large-scale DFT and FN-DMC calculations on computer models of amorphous silicon to help unravel this mystery. They found a new degradation channel that involved bond rotations (Fig. 3) to create stressed regions of silicon bonds that then trapped holes. Having quantitatively accurate calculations of the various energies involved in this process was critical to providing this insight into a 30-year old problem.

Wet chemistry

Weak interactions such as van dar Waal's forces are fundamentally a long-range electron correlation effect that is captured in standard FN-DMC methods.[69, 70] This ability, coupled with the applicability of QMC methods to larger systems, has the potential to make a significant contribution in the development of effective classical potentials (e.g., see Ref. [71]) for nanofluidic applications. These potentials in turn can be used to design desalination and water purification devices[72] at the nanoscale.

For applications such as water splitting and other catalysis reactions, the calculations of barriers are a large challenge. Here again the accurate energetics[42, 73] obtained by QMC methods could likely be very important, although to my knowledge there have not been significant advances in this area. One of the barriers to entry is that many reactions of this type occur on surfaces, which require simulation of many atoms with small error bars.

Finally, becuase quantum Monte Carlo approaches are quite expensive, incorporating the effects of solvents are rather important. For the time being, explicit solvation is too expensive. However, recently, there have been significant efforts[74, 75] in incorporating solvent models into quantum Monte Carlo simulations of molecules.

Strongly correlated systems

On a more forward looking note, quantum Monte Carlo calculations have promise in the field of strongly correlated electrons. The strong correlation in these systems allows for electronic states very different from the traditional dichotomy of metal or insulator seen in one-particle band structure. These states such as giant magnetoresistance and superconductivity are potentially game-changing for energy applications. Similarly, there are many energy dense materials for batteries that are strongly correlated systems. However, as understanding of these states is quite poor, designing these materials to be practical for large-scale energy applications is very challenging.

In this field, first-principles quantum Monte Carlo calculations are invaluable, since they are able to obtain accurate simulations of the electronic structure without any fitting parameters, as shown by a number of pilot studies.[13, 14, 16, 76, 77] Similarly, strongly correlated metal oxide systems can be used for catalysis.[78] This field represents an electronic structure theorist's nightmare of the interaction of excited states, transition barriers, and strongly correlated electrons. Any one of these subjects poses a challenge, and together, powerful methods such as quantum Monte Carlo will be needed to provide atomic-level insight.

Summary

From a pragmatic perspective, in the standard recipe of FN-DMC with a Slater-Jastrow factor, energies can be obtained that are much more accurate than standard DFT calculations. The extra computational cost over that of DFT strongly depends on the quantity to be calculated and the desired stochastic uncertainties, but can be anywhere between a factor of 100 up to a factor of 10,000. However, if a highly parallel resource is available, then more cores can be brought to bear on the QMC solution to mitigate the extra cost. For some applications in energy, this extra accuracy is well worth the additional cost; however, the method must be used judiciously so as not to waste computer resources.

The standard recipe also lacks a few features that one would ideally wish to have in an electronic structure solution tool. Substantial progress is being made in these directions, and it is quite possible many of these issues will be resolved in the coming years. There has been an impressive output from a relatively small community.

Looking forward, a major direction in materials research is the nanoscale design of heterogeneous materials, for example, superlattices,[79] often consisting of several materials where the electron interactions are very important. These materials offer unprecedented flexibility in their electronic-level function and thus tremendous opportunity for new energy-related devices. This large flexibility also emphasizes the need for predictive calculations to avoid the necessity of laboriously creating trial materials in the lab. However, many standard electronic structure methods rely on a strong cancellation of errors, as pointed out in Ref. [59]. That is, one expects the accuracy of standard DFT calculations to decrease as the complexity of the material increases. For this new frontier, methods such as quantum Monte Carlo that come closer to the exact quantum mechanical solution are crucial.

Biography

  • Image of creator

    Lucas K. Wagner is a research scientist at the University of Illinois at Urbana-Champaign. He obtained his B.S. in physics and applied mathematics at N.C. State University in 2002, followed by a PhD at the same institution in 2006. During this time, he developed the open-source quantum Monte Carlo code QWalk (http://qwalk.org). Dr. Wagner then spent a postdoc each at University of California at Berkeley and the Massachusetts Institute of Technology on applying electronic structure methods to energy-relevant problems, before moving to Illinois. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]