Jason D. Goodpaster

A Simple, Exact Density-Functional-Theory Embedding Scheme

Density functional theory (DFT) provides a formally exact framework for quantum embedding. The appearance of nonadditive kinetic energy contributions in this context poses significant challenges, but using optimized effective potential (OEP) methods, various groups have devised DFT-in-DFT methods that are equivalent to Kohn–Sham (KS) theory on the whole system. This being the case, we note that a very considerable simplification arises from doing KS theory instead. We then describe embedding schemes that enforce Pauli exclusion via a projection technique, completely avoiding numerically demanding OEP calculations. Illustrative applications are presented using DFT-in-DFT, wave-function-in-DFT, and wave-function-in-Hartree–Fock embedding, and using an embedded many-body expansion.

Exact nonadditive kinetic potentials for embedded density functional theory

We describe an embedded density functional theory (DFT) protocol in which the nonadditive kinetic energy component of the embedding potential is treated exactly. At each iteration of the Kohn-Sham equations for constrained electron density, the Zhao-Morrison-Parr constrained search method for constructing Kohn-Sham orbitals is combined with the King-Handy expression for the exact kinetic potential. We use this formally exact embedding protocol to calculate ionization energies for a series of three- and four-electron atomic systems, and the results are compared to embedded DFT calculations that utilize the Thomas-Fermi (TF) and the Thomas-Fermi-von Weisacker approximations to the kinetic energy functional. These calculations illustrate the expected breakdown due to the TF approximation for the nonadditive kinetic potential, with errors of 30%-80% in the calculated ionization energies; by contrast, the exact protocol is found to be accurate and stable. To significantly improve the convergence of the new protocol, we introduce a density-based switching function to map between the exact nonadditive kinetic potential and the TF approximation in the region of the nuclear cusp, and we demonstrate that this approximation has little effect on the accuracy of the calculated ionization energies. Finally, we describe possible extensions of the exact protocol to perform accurate embedded DFT calculations in large systems with strongly overlapping subsystem densities.

Bimetallic Pd-Cu oxygen reduction electrocatalysts

A series of Vulcan carbon-supported Pd-Cu catalysts with various molar ratios of Pd to Cu was prepared by co-impregnation followed by a reduction in a hydrogen atmosphere at three different temperatures. The degree of alloying between the two metals, alloy composition, and particle size and size distribution were characterized by X-ray diffraction, transmission electron microscopy, and energy-dispersive X-ray spectroscopy. The electrocatalytic activity for the oxygen reduction reaction (ORR) for these various compositions was determined using the thin-film rotating disk electrode technique. Our study reveals that the Pd-Cu bimetallic electrocatalysts, with a suitable degree of alloying, offer a greatly enhanced ORR activity compared to the Pd monometallic electrocatalyst. The best electrocatalytic activities were observed for the bimetallic catalysts that showed alloy nanoparticles with a Pd-Cu molar ratio of approximately 1:1.

Embedded density functional theory for covalently bonded and strongly interacting subsystems.

Embedded density functional theory (e-DFT) is used to describe the electronic structure of strongly interacting molecular subsystems. We present a general implementation of the Exact Embedding (EE) method [J. Chem. Phys. 133, 084103 (2010)] to calculate the large contributions of the nonadditive kinetic potential (NAKP) in such applications. Potential energy curves are computed for the dissociation of Li(+)-Be, CH(3)-CF(3), and hydrogen-bonded water clusters, and e-DFT results obtained using the EE method are compared with those obtained using approximate kinetic energy functionals. In all cases, the EE method preserves excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures. We also demonstrate an accurate pairwise approximation to the NAKP that allows for efficient parallelization of the EE method in large systems; benchmark calculations on molecular crystals reveal ideal, size-independent scaling of wall-clock time with increasing system size.

Density functional theory embedding for correlated wavefunctions: Improved methods for open-shell systems and transition metal complexes

Density functional theory (DFT) embedding provides a formally exact framework for interfacing correlated wave-function theory (WFT) methods with lower-level descriptions of electronic structure. Here, we report techniques to improve the accuracy and stability of WFT-in-DFT embedding calculations. In particular, we develop spin-dependent embedding potentials in both restricted and unrestricted orbital formulations to enable WFT-in-DFT embedding for open-shell systems, and develop an orbital-occupation-freezing technique to improve the convergence of optimized effective potential calculations that arise in the evaluation of the embedding potential. The new techniques are demonstrated in applications to the van-der-Waals-bound ethylene-propylene dimer and to the hexa-aquairon(II) transition-metal cation. Calculation of the dissociation curve for the ethylene-propylene dimer reveals that WFT-in-DFT embedding reproduces full CCSD(T) energies to within 0.1 kcal/mol at all distances, eliminating errors in the dispersion interactions due to conventional exchange-correlation (XC) functionals while simultaneously avoiding errors due to subsystem partitioning across covalent bonds. Application of WFT-in-DFT embedding to the calculation of the low-spin/high-spin splitting energy in the hexaaquairon(II) cation reveals that the majority of the dependence on the DFT XC functional can be eliminated by treating only the single transition-metal atom at the WFT level; furthermore, these calculations demonstrate the substantial effects of open-shell contributions to the embedding potential, and they suggest that restricted open-shell WFT-in-DFT embedding provides better accuracy than unrestricted open-shell WFT-in-DFT embedding due to the removal of spin contamination.

Accurate and systematically improvable density functional theory embedding for correlated wavefunctions

We analyze the sources of error in quantum embedding calculations in which an active subsystem is treated using wavefunction methods, and the remainder using density functional theory. We show that the embedding potential felt by the electrons in the active subsystem makes only a small contribution to the error of the method, whereas the error in the nonadditive exchange-correlation energy dominates. We test an MP2 correction for this term and demonstrate that the corrected embedding scheme accurately reproduces wavefunction calculations for a series of chemical reactions. Our projector-based embedding method uses localized occupied orbitals to partition the system; as with other local correlation methods, abrupt changes in the character of the localized orbitals along a reaction coordinate can lead to discontinuities in the embedded energy, but we show that these discontinuities are small and can be systematically reduced by increasing the size of the active region. Convergence of reaction energies with respect to the size of the active subsystem is shown to be rapid for all cases where the density functional treatment is able to capture the polarization of the environment, even in conjugated systems, and even when the partition cuts across a double bond.

Identification of Possible Pathways for C–C Bond Formation during Electrochemical Reduction of CO2: New Theoretical Insights from an Improved Electrochemical Model

We have carried out a periodic Kohn-Sham density functional theory investigation of the pathways by which carbon-carbon bonds could be formed during the electrochemical reduction of CO2 on Cu(100) using a model that includes the effects of the electrochemical potential, solvent, and electrolyte. The electrochemical potential was set by relating the applied potential to the Fermi energy and then calculating the number of electrons required by the simulation cell for that specific Fermi energy. The solvent was included as a continuum dielectric, and the electrolyte was described using a linearized Poisson-Boltzmann model. The calculated potential of zero charge for a variety of surfaces agrees with experiment to within a mean average error of 0.09 V, thereby validating the assumptions of the model. Analysis of the mechanism for C-C bond formation revealed that at low-applied potential, C-C bond formation occurs through a CO dimer. However, at high applied potentials, a large activation barrier blocks this pathway; therefore, C-C bond formation occurs through reaction of adsorbed CHO and CO. Rate parameters determined from our calculations were used to simulate the kinetics of ethene formation during the electrochemical reduction of CO over a Cu(100) surface. An excellent match was observed between previously reported measurements of the partial current for ethene formation as a function of applied voltage and the variation in the partial current for C-C bond formation predicted by our microkinetic model. The electrochemical model reported here is simple, fairly easy to implement, and involves only a small increase in computational cost over calculations neglecting the effects of the electrolyte and the applied field. Therefore, it can be used to study the effects of applied potential and electrolyte composition on the energetics of surface reactions for a wide variety of electrochemical reactions.

Accurate basis set truncation for wavefunction embedding

Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size of the MO virtual space. In this study, we extend a recently introduced projection-based embedding method [F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput. 8, 2564 (2012)] to allow for the systematic and accurate truncation of the embedded subsystem basis set. The approach is applied to both covalently and non-covalently bound test cases, including water clusters and polypeptide chains, and it is demonstrated that errors associated with basis set truncation are controllable to well within chemical accuracy. Furthermore, we show that this approach allows for switching between accurate projection-based embedding and DFT embedding with approximate kinetic energy (KE) functionals; in this sense, the approach provides a means of systematically improving upon the use of approximate KE functionals in DFT embedding.

The Role of Surface Defects in CO Oxidation, Methanol Oxidation, and Oxygen Reduction on Pt ( 111 )

Some surface reactions of interest to electrocatalysis in alkaline media are promoted by crystalline defects, while others occur preferentially on defect-free terraces. Different forms of structure sensitivity, and the underlying causes of this structure sensitivity, have been examined using several fuel-cell-relevant surface reactions in alkaline media as model reactions. Oxidation of CO serves as a model for defect favored reactions, while reduction of oxygen serves as a model for terrace favored reactions. More complicated reactions, such as methanol oxidation, can be interpreted as containing multiple steps that are either defect favored or terrace favored. The role of defects in each of these reactions was interpreted in terms of geometric and electronic effects, with different types of defects kink type and step type showing different effects for the different electrocatalytic processes. CO oxidation is promoted by both step-type and kink-type defects, as a result of electronic structure, but methanol dehydrogenation is promoted only by step-type defects, as a result of geometric structure.

Mechanistic insights into electrochemical reduction of CO2 over Ag using density functional theory and transport models

Significance Chemical storage of solar energy can be achieved by electrochemical reduction of CO2 to CO and H2, and subsequent conversion of this mixture to fuels. Identifying optimal conditions for electrochemical cell operation requires knowledge of the CO2 reduction mechanism and the influence of all factors controlling cell performance. We report a multiscale model for predicting the current densities for H2 and CO formation from first principles. Our approach brings together a quantum-chemical analysis of the reaction pathway, a microkinetic model of the reaction dynamics, and a continuum model for mass transport of all species through the electrolyte. This model is essential for identifying a physically correct representation of product current densities dependence on the cell voltage and CO2 partial pressure. Electrochemical reduction of CO2 using renewable sources of electrical energy holds promise for converting CO2 to fuels and chemicals. Since this process is complex and involves a large number of species and physical phenomena, a comprehensive understanding of the factors controlling product distribution is required. While the most plausible reaction pathway is usually identified from quantum-chemical calculation of the lowest free-energy pathway, this approach can be misleading when coverages of adsorbed species determined for alternative mechanism differ significantly, since elementary reaction rates depend on the product of the rate coefficient and the coverage of species involved in the reaction. Moreover, cathode polarization can influence the kinetics of CO2 reduction. Here, we present a multiscale framework for ab initio simulation of the electrochemical reduction of CO2 over an Ag(110) surface. A continuum model for species transport is combined with a microkinetic model for the cathode reaction dynamics. Free energies of activation for all elementary reactions are determined from density functional theory calculations. Using this approach, three alternative mechanisms for CO2 reduction were examined. The rate-limiting step in each mechanism is **COOH formation at higher negative potentials. However, only via the multiscale simulation was it possible to identify the mechanism that leads to a dependence of the rate of CO formation on the partial pressure of CO2 that is consistent with experiments. Simulations based on this mechanism also describe the dependence of the H2 and CO current densities on cathode voltage that are in strikingly good agreement with experimental observation.