Near-Quantitative Agreement of Model-Free DFT-MD Predictions with

Aug 29, 2012 - Charged Transition-Metal Ions. John L. Fulton,. †. Eric J. Bylaska,. † ... and John H. Weare*. ,§. †. Chemical and Materials Sci...
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Near-Quantitative Agreement of Model-Free DFT-MD Predictions with XAFS Observations of the Hydration Structure of Highly Charged Transition-Metal Ions John L. Fulton,† Eric J. Bylaska,† Stuart Bogatko,§ Mahalingam Balasubramanian,‡ Emilie Caueẗ ,§ Gregory K. Schenter,† and John H. Weare*,§ †

Chemical and Materials Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, California 92093, United States ‡ Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, United States §

S Supporting Information *

ABSTRACT: First-principles dynamics simulations (DFT, PBE96, and PBE0) and electron scattering calculations (FEFF9) provide near-quantitative agreement with new and existing XAFS measurements for a series of transition-metal ions interacting with their hydration shells via complex mechanisms (high spin, covalency, charge transfer, etc.). This analysis does not require either the development of empirical interparticle interaction potentials or structural models of hydration. However, it provides consistent parameter-free analysis and improved agreement with the higher-R scattering region (first- and second-shell structure, symmetry, dynamic disorder, and multiple scattering) for this comprehensive series of ions. DFT+GGA MD methods provide a high level of agreement. However, improvements are observed when exact exchange is included. Higher accuracy in the pseudopotential description of the atomic potential, including core polarization and reducing core radii, was necessary for very detailed agreement. The first-principles nature of this approach supports its application to more complex systems. SECTION: Molecular Structure, Quantum Chemistry, and General Theory

T

scattering theory and, therefore, provides a parameter-free method to interpret XAFS data. With this consistent method of analysis, these data for this diverse series of ions provide a comprehensive interpretation of the structure of the hydration shell as a function of features of ion interactions. In addition, the quantitative and comprehensive nature of the data provides a unique benchmark for testing first-principles dynamics simulations and the accuracy of the theoretical representation of the electronic structure of the hydration region. The results presented here illustrate that the first-principles MD-XAFS method provides a highly accurate view of the hydration structure of these ions, including (i) the bond distance, (ii) the symmetry and (iii) disorder of the first hydration shell, and (iv) the structure of the second hydration shell. In addition, they demonstrate that the use of the efficient but approximate density-functional-based first-principles method of simulation (forces based on DFT+GGA and exact exchange)3 leads to near-quantitative agreement with the

he hydration structure surrounding ions greatly affects their chemical properties such as reaction rates, electron transport, activity of ion cofactors in enzyme catalysis (proton donation), and reductive respiration. In this Letter, we report results from the synergistic use of high-resolution XAFS (X-ray absorption fine structure) observations (some data reported for the first time) and first-principles-based simulations to analyze the hydration shell properties of a number of strongly interacting aqueous divalent and trivalent metal ions (Ca2+, Cr3+, Mn2+, Fe3+, Co2+, Ni2+, and Zn2+). This series includes ions (important to many of these applications) having d-orbital occupancies from d0 to d10, zero-ligand field stabilization energies (low- and high-spin occupations) such as d0 (Ca2+), d3 (Cr3+), d5 (Fe3+, Mn2+), and d10(Zn2+) species, as well as several in which a significant amount of charge transfer is expected (e.g., Fe3+, Cr3+, and Ni2+). The interaction of this series of ions with their hydration shells is expected to extend from intermediate (e.g., Ca2+ intermediate charge and large ion size) to very strong (Cr3+ high charge and small ion size). The first-principles MD-XAFS (molecular dynamics-XAFS) method that we have implemented uses on-the-fly first-principles DFTMD simulations as input to the direct calculation of the XAFS scattering function using the well-validated FEFF91,2 ab initio © 2012 American Chemical Society

Received: June 29, 2012 Accepted: August 28, 2012 Published: August 29, 2012 2588

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on fitting of eq 2 to the XAFS data and an assumed average atomic structure.

observed XAFS scattering intensity [herein, the DFT+GGA calculations are performed using ab initio molecular dynamics (AIMD), and calculations using Hybrid exchange are performed using quantum mechanics/molecular mechanics (QM/MM) calculations (see below and in the Supporting Information (SI))]. Increasing the accuracy of the pseudopotential representation of the self-consistent atomic potential used in the DFT ab initio MD simulations and inclusion of exact exchange in the DFT simulation increased the quantitative agreement with data. This level of agreement has not been obtained by other approaches using empirical interaction potentials, which for each of these species would require a major effort to develop. In addition, because this method is based on first-principles calculations of the interactions in the system, they are straightforward to apply to arbitrary temperature, pressure, and composition conditions. They therefore support the detailed interpretation of experimental studies in more complex systems and conditions. In this work, the XAFS scattering data are analyzed in two ways. In the complete first-principles MD-XAFS method, DFTbased (+GGA and Hybrid methods)3 MD simulations are used to generate a large ensemble of structural snapshots or frames. These snapshots provide the atom coordinates that are used by the ab initio scattering code (FEFF9)1,2 to generate the theoretical scattering spectra from eq 1 that are then compared directly to the experimental XAFS spectra. χ (k ) =

1 i

frames paths

∑ ∑ S02 i

j

Fij(k) kR ij2

paths

χ ̅ (k ) =

∑ S02Nn n

Fn̅ (k) kR̅ n2

2 2

e−2R̅ n/ λne−2k σn

⎛ ⎞ 4 sin⎜2kR̅ n + φn̅ (k) − C3, nk3⎟ ⎝ ⎠ 3

(2)

where Nn, σ2n, and C3,n are the coordination number, the Debye−Waller factor (DWF), and the third cumulant, respectively. The third cumulant is used to account for the anharmonicity in the pair potential. The remaining terms are as defined in eq 1. As described below, typically 6−8 characteristic scattering paths are used to describe the average structures and disorder parameters. The resultant structural parameters can then be used to conduct a simple comparison between the averaged DFT-MD and experimental structures. In order to obtain the detailed agreement reported here for some of these systems, it was necessary to improve the pseudopotentials commonly used in the DFT ab initio MD simulations (see the SI). In developing pseudopotentials for an element, only atomic calculated data were used. However, there were decisions as to what level of detail of the pseudopotential representation of the self-consistent single (isolated) atom potential was required to compute the electronic structure of the interacting system to the level of accuracy observed in the data. (These issues are discussed in detail in the SI.) The level of agreement is also a function of the DFT formalism used (e.g., the inclusion of “exact exchange”). In our application of the DFT method, this was not considered to be a parameter because only well-established exchange functionals were used (PBE9611 GGA local exchange functional and the PBE012 hybrid functional; see the SI). The hybrid DFT (PBE0) calculations are computationally expensive; therefore, a QM/ MM method that included only a QM first hydration shell was implemented (identified as PBE0 below; see the SI). For Ca2+, rapid exchange between the first and second hydration shell was observed, and QM/MM PBE0 calculations were not attempted. The previously unpublished experimental Fe3+, Cr3+, and Co2+ K-edge XAFS spectra used in these studies were collected in transmission mode on the bending magnet beamline (PNC/ XSD, Sector 20) at the Advanced Photon Source, Argonne National Laboratory. The spectra for Mn2+, Ni2+, Zn2+, and Ca2+ were reported previously. (see the SI) As described in much more detail in the SI, the samples were prepared using metal salts of weakly complexing counterions such as ClO4−, with additional checks to be sure that complexed or hydrolysis species were not present. The background removal and fitting of the XAFS data was accomplished using the Athena and Artemis analysis programs.13,14 There is only one adjustable parameter, E0 (the absorption edge energy), that is used to compare the MD-XAFS to the experimental spectra (see the SI). In Figure 1, we compare the magnitude and imaginary parts of the real space Fourier transform (FT) of χ(k) calculated from eq 1 using the full first-principles MD-XAFS to that of the XAFS data for the strongly interacting ion Cr3+(aq). In an effort to illustrate the structural resolution of the XAFS method, five of the most significant scattering processes are schematically represented with their regions of influence in the lower part of the Figure 1. As a +3 ion, with a relatively small ionic radius,

e−2R ij / λij sin(2kR ij + φij(k)) (1)

In this equation, Fij(k), φij(k) and λij are the scattering amplitudes, the photoelectron phase shifts, and the photoelectron mean-free paths, respectively, as calculated by FEFF9. S20 is the core−hole lifetime amplitude factor, and Rij are the atom−atom distances for a particular path. The momentum space coordinate is given by k = (2me(E − E0)/ℏ2)1/2, where E and E0 are the X-ray energy and absorption edge energy, respectively. From each simulation frame or snapshot, we retain approximately 103 relevant scattering paths or events. Subsequently, several thousand frames are processed to produce the ensemble average of approximately 106 scattering events. The essential feature of this method4,5 is its ability to represent all of the relevant photoelectron scattering events (∼106) including single and multiple scattering processes that uniquely define certain hydration symmetries.6,7 Through the inclusion of the multiple scattering signals, we are able to go beyond the traditional route of comparing individual g(r)’s from MD trajectories to the single scattering contributions of the XAFS spectra. This MD-XAFS approach has been previously implemented using empirical interaction potentials5,8 and more recently for DFT-based simulations for +1 and +2 charged ions.6,9,10 The first-principles MD-XAFS analysis implements a parameter-free estimation of the interactions. Remarkably, as we will illustrate below and discuss in more detail in the SI, the agreement between the calculated XAFS spectra and the data in this approach is almost quantitative. This agreement extends well beyond that of the usual XAFS analysis (see the SI). A second method of XAFS analysis is based on a more traditional approach to obtain average values of the first and second hydration shell structural and disorder parameters based 2589

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Figure 1. k2-weighted |χ̃(R)| (solid lines) and Im[χ̃(R)] (dashed line) for hydrated Cr3+. The highlighting of different spectral regions shows the assignments to different types of single and multiple scattering events, as illustrated. The inset expands the view for the longer multiple scattering region.

Figure 2. The XAFS k2-weighted Im[χ̃(R)] plots for hydrated Cr3+. The experimental spectrum (points with line) is compared to a leastsquares refinement (green line) of a structural model, eq 2. The individual scattering paths (blue lines) that are used in the structural refinement are shown.

Cr3+ is expected to have a well-structured first and weaker second hydration shell (e.g., strong interactions). This is consistent with the analysis illustrated in Figure 1 and reveals the influence of both multiple scattering and second-shell structure. We would like to emphasize three important points illustrated by this figure. (1) The level of agreement between the calculated and observed spectra is considerably higher than has been obtained in prior XAFS analysis. (2) This was obtained from a parameter-free simulation with no fitting of the interaction potentials to any data (developing force fields for ions is a very labor intensive problem). (3) Additional information about hydration shell structure is now available because of this more detailed agreement. Figure 2 shows an approximate breakdown of the individual FT scattering channels to Im[χ̃(R)] as predicted from the average structure model, eq 2. We note from Figure 2 that the strong single scattering from the first shell dominates the scattering in the region from 1 to 2 Å. The multiple scattering events from the Cr−O−O, Cr−O−Cr−O and Cr−O−O−Cr−O geometries (forth through seventh spectra in Figure 2) in the region from 2 to 4 Å are sensitive to first-shell symmetry and disorder. Finally, the second-shell contributions have identifiable signatures that also accumulate in the 3−4 Å region. This overlap reduces the ability to separately resolve second-shell contributions, but the contribution to the scattering for strong interaction systems (e.g., Cr3+) can be demonstrated by the more complete analysis given in the SI (see Figure S6). For Fe3+ and Cr3+, approximately 30−40% of the scattering intensity in the 3−4 Å region comes from second-shell scattering. The remarkable agreement of the full first-principles MD-XAFS calculation of both the magnitude and imaginary part of Im[χ̃(R)] for R > 2 Å is illustrated in the expanded view in Figure 1. The agreement of the full ab initio model (no adjusted parameters) with the

data is better than that calculated from the structural model (using eq 2), which is fit to the experimental data. This strongly supports the future application of this general first-principles DFT-MD method to more complex systems with poorly understood structures. Agreement between the observed and calculated |χ̃(R)| similar to that of Figure 1 is illustrated for the entire series of ions in Figure 3. We note that the presence of second-shell scattering and evidence for identifiable multiple scattering contributions roughly correlates with calculations and other measures of the stiffness of the hydration shell across this series of ions.15 The high agreement of the full first-principles XAFS analysis of multiple scattering of Zn2+ and Ca2+ is discussed further in Figures S7 and S8 of the SI. We note that consistent parameter-free interpretation of the data for this series of very different ions using this method greatly improves the reliability of the interpretation over more traditional approaches that require the development of empirical potentials (often of differing quality) for each species. The symmetry and disorder of the hydration region are more difficult to quantify because of the many scattering paths that contribute to this region (R > 2 Å). In contrast to the strongly interacting +3 ions Fe3+ and Cr3+, the Ca2+ ion has a large ion size and is believed to have a relatively weak interaction with its hydration shell with a coordination number of around 6.8.16,17 For such a system, the collinear multiple and second-shell scattering will be suppressed. This is consistent with the lack of features in scattering intensity for Ca2+ beyond 2.5 Å, as illustrated in Figure 3 (Also see Figure S7, SI). We note that the agreement in the multiple scattering region for the ions Co2+ and Ni2+ is not nearly as quantitative as that for Cr3+, Fe3+, and Mn2+ (QM/MM PBE0). For these systems, default pseudopotentials were used in the calculations. We believe that 2590

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Figure 4. A series of XAFS k2-weighted χ(k) plots generated from AIMD and QM/MM MD for the series of ions. The MD-XAFS result for a classical potential (dashed purple line) is included in the plot for Ca2+.

Figure 3. k2-weighted MD-XAFS-generated |χ̃(R)| plots for various metal ions are compared to experimental data. The sets of spectra have been offset for clarity. The MD-XAFS result for a well-parametrized classical potential8 (dashed purple line) is included in the plot for Ca2+.

in this approach, all ions are treated at the same level by a parameter-free theory. To more succinctly represent our results (including the differences in the DFT level of calculation; see below), we also analyzed the data using the more standard approach given by eq 2. These calculations yielded the structural characterizations of the first- and second-shell structures that are summarized in Table 1S (SI). Selected results for the first hydration shell are given in Figure 5. As discussed in the following, significant improvements between the structures calculated from MD and those from experiments were realized through improvements in the atomic pseudopotentials for Cr3+, Fe3+, and Mn2+. For these three ions, the average difference between the simulated and measured first-shell bond distance is an unprecedented low value of 0.006 Å (from a parameter-free analysis). For the other ions, in which the default pseudopotentials were used, an average difference of 0.031 Å was obtained. In addition, the σ2 values (DWFs) for Cr3+, Fe3+, and Mn2+ (QM/MM PBE0) are remarkably close to the experimental value, although slightly smaller. We note that the DWF predictions of the PBE0 calculation (hybrid DFT with exact exchange included) are significantly better than the DFT+GGA results. The magnitude and direction of the slight remaining deviations may be due to quantum zero-point-energy effects. At this high level of agreement for a comprehensive series of ions with differing interaction strengths, it is possible to explore the systematic errors related to the level of theory that has been used in the calculations. The efficient implementation of DFT leads to approaches that are both computationally tractable and provide a reliable and flexible description of the interactions in

improvements in the representation of the pseudopotentials at the atomic level (eq S1, SI) would improve the scattering in this region as it does for Fe3+, Cr3+, and Mn2+. In Figure 4, plots of k2χ(k) versus k are illustrated for the series of ions. Only the k region above approximately 2 Å−1 is treated in the simulated XAFS because a different type of scattering theory (X-ray absorption near-edge, XANES) would be required for the region below 2 Å−1. As in the prior figures, the calculated spectra are in nearly quantitative agreement with observations. This representation emphasizes the decay of the primary oscillations as k becomes large (see eq 2), which is due to disorder in the first solvent shell, as is commonly represented by the DWF. The first-principles MD-XAFS method (eq 1) explicitly includes broadening due to dynamic disorder. We note that efforts have been made to obtain the proper decay function using empirical potentials in MD without a great deal of success (an exception is the highly parametrized hydrated ion potential of Merkling et al.18,19). We provide an example of this problem using a classical potential8 for Ca2+ as the dashed line in Figures 3 and 4. Although the classical potential adequately describes the average ion−water distance (the frequency in Figure 4), it predicts a first hydration shell whose disorder is too low (σ2 (DWF) is too small, causing the diverging amplitude at high k in Figure 4). We emphasize that it is dramatically simpler to use existing first-principles software to generate structures that replicate the XAFS spectra than it is to generate the highly accurate empirical potentials that would be required for the empirical-model-based approach. In addition, 2591

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long first-shell residence times (see the SI). As shown in Figure 5 (and Figures S13−S15, SI), there is a general improvement of between 0.02 and 0.05 Å in the M−O first-shell bond length upon the inclusion of exact exchange. Remarkably, a considerably larger improvement in the DWFs as illustrated in Figure 4 is found from the inclusion of exchange. We would like to emphasize, however, that the use of the much more efficient DFT+GGA still provides remarkably good results. For systems in which there is considerable reorganization in the solvation shells or for which the structures are poorly known, QM/MM with PBE0 methods are difficult to implement. An example is the Ca2+ ion, shown in Figures 3 and 4. (3) Treatment of Second-Shell Water−Water Interactions: The QM/MM PBE0 results use a phenomenological potential to describe the water−water interaction (see the SI) in the second shell. Direct comparisons between QM/MM using either PBE96 or PBE0 are given in Figures S13−S15 (SI). As expected, there are some differences particularly in the description of the second-shell structure. We note, however, that these differences are not large. It might be expected that the QM/MM would be better because the second-shell MM potential probably provides a better water−water description than DFT+GGA. Even within this limitation, we note that for the Ca2+ ion, in which there are far too many intershell transfers to allow a QM/MM calculation, the AIMD simulation is very good (see Figures 3 and 4), emphasizing that methods based on DFT+GGA simulations can be used to analyze data in more complex problems.

Figure 5. Plots of bond distances and the DWFs for the first-shell waters about various ions obtained from eq 2 through model fitting to experimental data and from simulated MD-XAFS using AIMD (DFT +GGA) and QM/MM (PBE0, Hybrid DFT) (see Table 1S, SI). Both default and improved pseudopotentials were used as indicated.

solutions. Nevertheless, there are well-documented problems with the accuracy of DFT approximations for the study of water interactions.20 In this study, the areas of concern are (1) the accuracy of the pseudopotential, (2) the accuracy of the level of exchange in the theory, and (3) the sensitivity of the calculation to the quality of the description of the water−water interaction in the second shell. (1) Evaluation of the Pseudopotential: As more fully described in the SI, the objective of a pseudopotential is to replace the full atomic self-consistent potential used in the condensed phase by a much weaker smoothly varying potential having essentially the same atomic wave function (scattering properties) outside of some core region. In forming the pseudopotential, it is necessary to decide which orbitals need to be included as the valence orbitals in the atom and condensed phase (see the SI). If important orbitals are left out of the calculation, the pseudopotential will not reproduce the behavior of the wave functions in the condensed phase. Examples of the sensitivity of the full first-principles MD-XAFS calculations to this decision are given in Figures S11 and S12 of the SI. In the case of Fe3+ and Cr3+, our default pseudopotentials originally included only the 3d orbitals. By also including the 3s and 3p functions as part of the active space in the pseudopotential, thereby better describing polarization, there was a large, 0.1 Å, improvement in the predicted first-shell bond lengths (a reduction of the core radius in the pseudopotential description was also necessary). We emphasize that the pseudopotential development is all based on atomic-level calculations (see the SI). (2) Inclusion of Exact Exchange in the DFT Calculation: The deficiencies of DFT+GGA are well-known.20 However, because this is a relatively efficient calculation, this is the level of calculation used in essentially every ab initio molecular dynamics simulation. Results suggest that inclusion of exact exchange improves the general accuracy of DFT+GGA (e.g., PBE0). Unfortunately, the cost of the exact exchange calculation is high. In order to add this contribution, we developed a QM/MM-type calculation for ions having very

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ASSOCIATED CONTENT

S Supporting Information *

A comprehensive description of (i) the DFT-MD methods, including pseudopotential development, (ii) the XAFS theory, spectra analysis, and fitting methods, and (iii) comparison between different levels of DFT MD-XAFS and experiment. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences. Pacific Northwest National Laboratory (PNNL) is operated for the U.S. DOE by Battelle. E.J.B. and J.H.W. acknowledge U.S. DOE ASCR Petascale tools program for algorithm development. A portion of the research was performed using EMSL, a national scientific user facility sponsored by the U.S. DOE’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory. Research at PNC/XSD at the Advanced Photon Source is supported by the U.S. DOE (Contract No. DE-AC02-06CH11357), NSERC (Canada), and its founding institutions.

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