Comparison of proton conduction in KTaO3 and SrZrO3Maria A. Gomez, Saryu Jindal, Katharyn M. Fletcher, Leigh S. Foster, Nanna Dufie Addo et al. Citation: J. Chem. Phys. 126, 194701 (2007); doi: 10.1063/1.2735592 View online: http://dx.doi.org/10.1063/1.2735592 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v126/i19 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Comparison of proton conduction in KTaO3 and SrZrO3

Maria A. Gomez,a� Saryu Jindal, Katharyn M. Fletcher, Leigh S. Foster,Nanna Dufie A. Addo, Debbie Valentin, Cristina Ghenoiu, and Abigail HamiltonDepartment of Chemistry, Mount Holyoke College, South Hadley, Massachusetts 01075

�Received 13 February 2007; accepted 5 April 2007; published online 15 May 2007�

In the present paper, the authors focus on proton conduction pathways in a cubic perovskite KTaO3

and an orthorhombic perovskite SrZrO3. Density functional theory with a generalized gradientapproximation is used to find proton binding sites. The nudged elastic band method is used to findtransition states between minima. With this potential energy map of binding and transition states,adjacency matrices and their analogs identify four types of conduction paths in KTaO3. Distortionsfrom these paths are seen in SrZrO3. In both cases, the lowest energy path has an intraoctahedraltransfer rate-limiting barrier. A Fourier analysis of the OH stretch in ab initio molecular dynamicssimulations revealed a strongly redshifted OH stretch in SrZrO3 relative to KTaO3. Hence, anorthorhombic system with a lowest energy conduction path limited by an intraoctahedral barrier canexhibit a redshifted OH stretch. © 2007 American Institute of Physics. �DOI: 10.1063/1.2735592�

I. INTRODUCTION

Proton conducting perovskite oxide ceramics haveshown potential for use in fuel cell technologies, humiditysensors, and batteries.1 Recent work2,3 has shown that smalldistortions from cubic structure �shown in Fig. 1 of Ref. 3�can affect proton trajectories significantly by creating newpathways for proton transfer. Münch et al.2 also noted thatthe greater the deviation from cubic structure, the more com-petitive transfer and rotational barriers become. By findingbinding sites and transition states directly, Gomez et al.3 areable to compare specific paths for proton conduction. Octa-hedral tilting in perovskites facilitates interoctahedral trans-fer by bringing oxygens on different octahedra closer to-gether. This was suggested as an explanation for the redshiftin the OH stretching frequency in spectra with small toler-ance parameters, i.e., those with high potential for octahedraltilting. In the present paper, we focus on two perovskites,KTaO3 and SrZrO3, to show that the perovskite with greatertilting does indeed exhibit greater redshifting in the OHstretching frequency while maintaining an intraoctahedralproton transfer rate-limiting step.

KTaO3 has a stable cubic structure at room temperatureand remains cubic at even lower temperatures.4,5 SrZrO3, onthe other hand, exhibits significant octahedral tilting and isorthorhombic at room temperature. It undergoes a transitionto a tetragonal phase at 750 °C and becomes cubic at1070 °C.6,7 The contrasting structures provided by KTaO3

and SrZrO3 permit a study of how proton conduction path-ways change with octahedral tilting. Tilting creates a greatervariety of proton binding sites. The variety of proton bindingsites results in different conduction pathways and OH stretchspectra.

The basic structures of KTaO3 and SrZrO3 are presentedin Sec. II. Sec. III shows the proton binding sites and transi-tion states in each perovskite. A map or graph of these sites

can be used to find conduction paths. Section IV shows howadjacency matrices and their analogs can be used to findconduction paths in these perovskites. Molecular dynamicstrajectories for OH vibrations in each perovskite are dis-cussed in Sec. V. The results are summarized in a final dis-cussion in Sec. VI.

All the simulations in this paper treat the proton classi-cally. Perovskites are high temperature proton conductors. Atthese high temperatures, the delocalized nature of the protonis likely to be a small effect in the proton conduction path-ways. However, the IR spectra probing the effect of structureon OH stretching frequencies are generally taken at roomtemperature and below. The classical IR spectra discussed inSec. V are only an approximation to the real spectra, whichcould show some quantum effects.

II. LATTICE SIZE AND DISTORTIONS

Since previous studies4–7 characterize KTaO3 as cubic4,5

and SrZrO3 as orthorhombic6,7 at room temperature, oursimulations started with these structures. For the cubicKTaO3, the lattice parameter of a single cubic unit cell was

a�Electronic mail: magomez@mtholyoke.edu

FIG. 1. The dominant binding site in cubic KTaO3 is shallow with respectto rotations in the face of the cube. Arrows are used to indicate transfers androtations to equivalent binding sites. The large dark ions are oxygen ions.The smaller corner ions are the B-type ions. The lone ion in the center is anA-type ion. Hydrogen is shown in white.

THE JOURNAL OF CHEMICAL PHYSICS 126, 194701 �2007�

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varied to find the optimum lattice size. To find the optimizedstructure of SrZrO3, an orthorhombic unit cell was allowedto change shape and size.

More specifically, the Vienna ab initio simulationpackage8 �VASP� implementation of density functionaltheory was used with PBE, PW91, and LDA functionalswithin generalized gradient approximations �GGAs�. Theprojector augment wave �PAW� method9,10 was used with thevalence states of K �3s, 3p, 4s�, Ta �5p, 6s, 5d�, Sr �4s, 4p,5s, 4d�, Zr �4s, 4p, 5s, 4d�, and O �2s, 2p�.11 The actualoccupancies are adjusted during the self-consistent field elec-tronic cycles while maintaining a fixed number of electrons.

A plane wave basis set was generated on an 8�8�8and 4�4�2 Monkhorst-Pack k-point mesh with zero shift12

for cubic KTaO3 and orthorhombic SrZrO3, respectively.The partial wave occupancies are calculated with Gaussiansmearing. Each band was optimized iteratively using a pre-conditioned residual minimization method with direct inver-sion in the subspace. Ten non-self-consistent steps that keepthe initial Hamiltonian are done at the start of each run sincethe initial wave function is random. For all the lattice calcu-lations, the cutoff energy for the plane wave basis set was600 eV; the fast fourier transform meshes include all G vec-tors up to those twice as large as the longest wave in thebasis set; and the support grids representing the augmenta-tion charges, charge densities, and potentials have twice thenumber of points as the wave function grids. The latter twocriteria are obtained by setting VASP �Ref. 8� keywordPREC as “accurate.”

All electronic optimization stops when the total systemenergy and the band structure energy between successivesteps both differ by less than 10−4 eV. Projection operatorswere evaluated in reciprocal space. Geometry optimizationson shape and size used a conjugate gradient method. Thegeometry optimization stops when successive energy differ-ences are less than 10−3 eV.

The optimum lattice parameters for KTaO3 and SrZrO3

using LDA, PW91, and PBE functionals are reported inTable I. Increasing the energy cutoff and k-point mesh didnot change the parameters. The optimized lattice parametersfor SrZrO3 with the lower energy cutoff of 400 eV and usingthe PW91 functional were 5.80, 5.86, and 8.22 Å. These re-sults are consistent with Shi et al.13 However, the more con-

verged higher cutoff lattice parameters have increased errorwhen compared to experiment.5,6 All the functionals and cut-offs considered gave reasonable agreement with experiment.The slightly different lattice sizes are unlikely to change con-duction path trends.

III. BINDING SITES AND TRANSITION STATES

The conjugate gradient method was used to find the low-est energy proton binding sites starting from configurationsfound in other perovskites.3 The PBE functional was used forall binding site calculations. However, some binding siteswere also found using the PW91 and LDA functionals. Rela-tive binding site energies show similar trends for these func-tionals. This is consistent with our observations in other per-ovskite calculations.3

Calculations involving a proton used the same methodsdescribed in Sec. II with some additions and changes. H�1s�was treated as a valence state. However, the total number ofelectrons was kept as the sum of the valence electrons of thenonhydrogen species. A homogeneous background electrongas was used to maintain charge neutrality. Simulation boxescontaining a 2�2�2 cubic and a 2�2�1 orthorhombicunit cells were used for KTaO3 and SrZrO3, respectively.This represents 40 and 80 atoms, respectively. A plane wavebasis set was generated on a 2�2�2 Monkhorst-Packk-point mesh with zero shift12 for both KTaO3 and SrZrO3.

Medium precision and a 400 eV cutoff energy were usedin optimization calculations involving the proton. ForKTaO3/H calculations, 64�64�64 grid points were used inthe fast Fourier transform mesh. This ensured that wrap-around errors were avoided. For SrZrO3/H calculations thedefault medium precision fast Fourier transform mesh of39�39�27 points was used. In this system a 96�96�70mesh would have avoided wrap-around error but the runswould take significantly longer. Proton binding site calcula-tions at accurate level �i.e., avoiding wrap-around error� andwith a 600 eV plane wave cutoff showed that the relativeenergies of proton binding sites did not change by more than0.01 eV compared to calculations where the cutoff and levelof precisions were reduced to 400 eV and medium precision.This lower precision also allows us to perform nudged elasticband calculations.

Minimum energy paths between neighboring protonbinding sites were found using the nudged elastic band14

�NEB� method with six images. Initial images for transferpaths were simple linear interpolations of the initial and finalbinding sites. In rotational paths, the OH distance was keptconstant in the initial images while the bond was rotated. Theimages are connected with harmonic springs and their posi-tions are optimized. This optimization allows motion of theimages in all degrees of freedom. In fact, final paths areusually curved. The force used in the optimization is thespring force along the path direction plus the true force alongthe direction perpendicular to the path. The former preventsthe images from sliding down to the minima, while the latterensures that the lowest energy path is found. The resultingpath was reoptimized with the climbing NEB method. In thiscase, the force on the highest energy image is the true force

TABLE I. The lattice constants for KTaO3 and SrZrO3 found with DFT/PW91, PBE, and LDA compare well to those found in experiment �Refs. 5and 6�. For SrZrO3, the optimized lattice sizes using PW91 with a cutoff of400 eV are 5.80, 5.86, and 8.22 Å, in agreement with Shi et al. �Ref. 13�.Increasing the cutoff to 600 eV yields the values in the table. Not muchchange is seen for cutoffs in excess of 600 eV.

Perovskite

Latticeconstants

�DFT/PW91���

Latticeconstants

�DFT/PBE���

Latticeconstants

�DFT/LDA���

Latticeconstants

�Experimenta���

KTaO3 �cubic� 4.03 4.03 3.96 4.00SrZrO3 5.84 5.84 5.73 5.79�orthorhombic� 5.91 5.91 5.80 5.82

8.28 8.29 8.13 8.20

aReferences 5 and 6.

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with the component along the path inverted. This force di-rects the highest energy image to climb to the saddle.14 Thetransition state reported is this climbing image. The transi-tion state calculations use the PBE functional only, as non-local effects are important for getting accurate transitionstates.15 The pattern in proton binding and transition stateenergies in this study is very similar to that found for otherperovskites in our earlier work3 using the PW91 functional.This suggests that the patterns are independent of functionalused.

The only binding site type in cubic KTaO3 found withboth GGA/PBE and LDA is shown in Fig. 1. Arrows areused to indicate possible transfers and rotations to equivalentbinding sites. The barrier for transfer is 0.37 eV with GGA/PBE. The rotational barrier is 0.21 eV. Experimental activa-tion energies vary from 0.73 to 1 eV depending on dopant.16

The GGA/PBE transfer barrier is significantly smaller thanthis range. However, including a dopant is expected to sig-nificantly increase activation barriers.17

Figure 2 shows the binding sites and transition state en-ergies for SrZrO3 found using GGA/PBE. The figure is aflattened cube with Sr ions left out for clarity. On each oxy-gen, the energy for a proton binding at that oxygen is shownin eV. The actual proton binding site is like that shown inFig. 1, i.e., the OH bond is perpendicular to the edge of thecube. All energies in the figure are relative to the lowestenergy binding site in the SrZrO3 perovskite. Transfer tran-sition state energies are written between the oxygens in-volved in the transfer. Rotational transition state energies arewritten between the faces involved in the rotation and havean R in front of them for clarity. The transition states in thecenter of the cube are transition states for interoctahedralproton transfer between the closest oxygens. To obtain bar-rier energies, the energy of the binding site must be sub-tracted from the energy of the transition state. Since SrZrO3

contains such a wide diversity in binding sites and transitionstates, conduction paths need to be considered to find theconduction rate-limiting barrier.

IV. PROTON CONDUCTION PATHS

The binding site and transition state energies found inSec. III provide a map of possible proton trajectories withinthe perovskite. This map is a graph that allows cycle and treesearches in traditional graph theory approaches.18 Consider-ing each binding site as a vertex in the graph and each tran-sition state as a connection or edge in the graph, an adja-cency matrix A can be constructed. Aij =1 when binding sitesi and j �or its periodic image� are connected by a transitionstate, and Aij =0 when the binding sites i and j are too far tobe connected by a single transition state. WheneverAik1

,Ak1k2, . . . ,Aknj is nonzero, the path i ,k1 ,k2 , . . . ,kn , j is

possible. The problem of finding conduction paths throughthe periodic system is the challenge of finding paths thattraverse the length of the simulation box and end on chemi-cally equivalent sites. For systems such as KTaO3 whereevery site is chemically equivalent, it is convenient to definetraversing the simulation box as ending in a periodic imageof the starting site. This restriction limits the paths searchedto paths with the periodicity of the simulation box. Thesepaths can be extended by periodically replicating the simu-lation box. We hope to relax this restriction in future work.

The simplest simulation boxes are cubic. For simplicityour first application of the adjacency matrix idea will con-sider cubic simulation boxes with binding sites on box edges.Since our adjacency matrix takes periodicity into account,conduction paths traversing the simulation box as definedabove are simple cycles that start and end at the same borderbinding site. These can be found by scanning the cycle pos-sibilities i ,k1 ,k2 , . . . ,kn , i that lead to nonzero Aik1

Ak1k2. . .Akni

and contain no repeating binding sites except for the bordersite i. The highest energy barriers of this cycle in clockwiseand counterclockwise directions can be found with a map ofbinding site and transition state energies. This provides a setof conduction paths and their most energetically difficultsteps. Sorting the paths by limiting barrier energy reveals thelowest energy paths that allow conduction through the simu-

FIG. 2. Proton binding sites and tran-sitions states in SrZrO3 are shown onan unfolded perovskite cubic unit. Thesimulation box is comprised of four ofthese units placed together as de-scribed in Ref. 3. The proton bindingsite energy is shown on the boundoxygen. The actual geometry of theOH bond in the proton binding site issimilar to that shown in Fig. 1 for thecubic case. Transition state energiesare shown between oxygens. The tran-sition state energies in the center of thecube are for interoctahedral protontransfer. Rotational transition state en-ergies from one face to an adjacentface are written on the cube edges. Allenergies are in eV. To aid comparisonwith other papers, an arrow highlightsthe row of O1 sites.

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lation box. Paths whose periodicity is not contained in thesimulation box will not be found using this scheme.

Figure 3 shows the cubic conduction paths found usingthe adjacency matrix formalism. The shortest path that trav-els through a cubic unit cell consists of two rotations fol-lowed by two transfers, notated as RRTT �Fig. 3�a��. Thenext shortest periodic paths within the unit cell are RRTRRT�Fig. 3�b��, RTTRTT �Fig. 3�c��, and RTRTRT �Fig. 3�d��.There were no five-step paths and no searches beyond six-step paths were considered. Paths in non-cubic systems areexpected to be distortions of these paths.

It is possible to simulate an orthorhombic system with a2�2�2 cubic unit cell system which is just large enough toallow the octahedral tilting characteristic of the orthorhombicface.19 This simplifies our first application of adjacency ma-trices to find proton conduction pathways. As in the cubicperovskite �see Fig. 1�, the OH bonds point towards the cen-ter of each cube face. Since each edge is shared by fourcubes, each OH bond has four orientations, namely, towardsthe center in each of four cubes. Hence, each oxygen hasfour proton binding sites on the faces of the four cubes shar-ing the oxygen containing edge. Figure 2 shows binding siteenergies on each face’s oxygen projections. To make the 2�2�2 cubic box system represent the orthorhombic box,eight of the cubes shown in Fig. 2 are attached such that faceB overlaps a 180° rotation of face B� on the adjacent cube.The same is true for faces C and C�. For correct A/A� faceoverlap, rotational barriers out of the plane should also beinverted. The shortest and lowest energy path found usingadjacency matrix path searches takes place in the plane com-prised of B and C� faces as in Fig. 4. It contained seven stepsand six intermediates. The limiting barrier was 0.32 eV inthe forward and backward directions. An equivalent path isseen in the B� /C plane. The equivalent path in the A/A�plane has a limiting barrier of 0.34 eV. The backward barrieris 0.01 eV higher. However, a simple switch in intraoctahe-dral transfers yields a backward path with a limiting barrierof 0.34 eV. All of these paths are a limiting case of the cubic

Fig. 3�a� path. To traverse two cubes with the Fig. 3�a� path,an RRTTRRTT trajectory is needed. The orthorhombic tra-jectory replaces TT with the new type of proton transfer,namely, one interoctahedral transfer Tn. This yields theseven-step RRTnRRTT. The barrier to intraoctahedral trans-fer remains the limiting barrier for this shortest path travers-ing the entire system. However, the lowest energy B/C� pathcontains a rotational barrier within 0.02 eV of the largestintraoctahedral transfer barrier. Inspection of the eight-cubesystem shows that paths avoiding intraoctahedral transfer arenot possible. Other seven-step paths were of higher energy.

The path shown in Fig. 5 was found in an earlier paper3

FIG. 3. The adjacency matrix formalism identifies one four-step path and three six-step paths that lead to conduction through the whole cubic perovskite. Thefour-step path, RRTT, is shown in �a�. The six-step paths are �b� RRTRRT, �c� RTTRTT, and �d� RTRTRT.

FIG. 4. The lowest energy and shortest conduction path for SrZrO3 isshown. The seven-step path begins with a rotation �R� to a binding site on aperpendicular plane followed by a rotation back to the main plane in the Bface �R�. Interoctahedral transfer �Tn� in the B face is followed by tworotations �RR� moving the proton to a perpendicular face and back to theplane in the C’ face. Finally two intraoctahedral transfers �TT� complete thepath. This path is denoted by TnRRTTRR.

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by visual inspection of the cubes. It has the periodicity of thelarger orthorhombic simulation box and hence cannot befound with the current adjacency matrix search scheme. Thisten-step path has a limiting intraoctahedral transfer barrier of0.34 eV in the forward direction and 0.35 eV in the back-ward direction. It is both longer and higher in energy than thelowest energy path found using the adjacency matrix searchscheme. However, it still has a limiting intraoctahedral bar-rier.

V. OH STRETCH

To capture the dynamics at the most probable protonbinding sites, molecular dynamics trajectories of a proton inKTaO3 and SrZrO3 were started in VASP �Ref. 8� from thelowest energy minima in each perovskite. The single protonbinding site type in KTaO3 is shown in Fig. 1. The lowestenergy proton binding site in SrZrO3 is the B 0.00 site in Fig.2. This site is contained in the lowest energy conduction pathshown in Fig. 4. Other low energy conduction paths containsimilar binding sites.

For the molecular dynamics run, the VASP �Ref. 8� pre-cision was set to low. The time step was 0.5 fs. Equilibra-tions consisted of 1000 steps. During the equilibration, ve-locities were rescaled every ten steps to maintain atemperature of 300 K. Note that VASP �Ref. 8� defines itstemperature as an average of the kinetic energy over all theions while maintaining a constant center of mass position.Hence, to obtain the actual simulation temperature, the ki-netic energy needs to be rescaled by Nions / �Nions−1�, whereNions is the total number of ions in the system. The constantenergy data collection trajectory was 1000 and 2000 steps forKTaO3 and SrZrO3, respectively. The average temperaturesduring the PBE/molecular dynamics run were 246 and 309 Kfor KTaO3 and SrZrO3, respectively. The absolute value ofthe ratio of the standard deviation to the average energy,

���E2� − �E�2

�E�� ,

was 5�10−5 for both KTaO3 and SrZrO3.The proton does not change the binding sites during the

molecular dynamics trajectory. Rather, a vibration coupledwith a rotation within the binding site is seen. From therelative OH positions found in the molecular dynamics tra-jectory, a position autocorrelation was calculated. TheWiener-Khintchine theorem20 connects the position autocor-relation function with the vibrational spectral density via aFourier transform. The OH displacement autocorrelationfunction shows both a fast oscillation �OH stretch� and aslower to decay motion, likely a vibrational-rotational cou-pling. Computational resources prevented running moleculardynamics trajectories long enough to result in a fully de-cayed correlation function, minimizing the ringing in thespectrum. However, the major spectral peak is evident. Fig-ure 6 shows that the H/SrZrO3 spectra �dashed� are red-shifted relative to the H/KTaO3 spectra �solid�. The OHstretch is at about 3530 and 3110 cm−1 in the H/KTaO3 andH/SrZrO3, respectively. Interestingly, a short molecular dy-namics run starting at the B 0.06 SrZrO3 binding site shownin Fig. 3 also yields a OH stretch peak near 3100 cm−1.

Experimental spectra for KTaO3 show a OH stretch peakrange of 3472–3500 cm−1 depending on preparationconditions.21 The corresponding OH vibration in experimen-tal H/SrZrO3 spectra is at about 3200 cm−1 and has a weakerabsorption band at about 2500 cm−1. However, the weakband decreases as the size of the dopant ion gets closer to thezirconium ion size.22 The OH stretches in the calculatedspectrum are in excellent agreement with the main experi-mental peaks. However, no weaker feature is seen in theH/SrZrO3 spectra since there is no dopant in our simula-tions.

VI. CONCLUDING DISCUSSION

Proton conduction paths in cubic KTaO3 and orthorhom-bic SrZrO3 have been searched through the use of an adja-cency matrix formalism. Conduction paths in the cubic per-ovskite were comprised of rotations �R� and intraoctahedraltransfers �T� in the combinations RRTT, RRTRRT, RTTRTT,

FIG. 5. The ten-step path shown below was found in a previous paper �Ref.3�. As in Fig. 4, all rotation arrows involve a rotation to a binding site on aperpendicular plane followed by a rotation back to the main plane. Rotationsover and under the plane are dotted and dashed, respectively. This path issummarized by RRTTnRRTTnRRTTnRRTTn. The thick square indicates thearea included in our adjacency matrix. This path is periodic in the largerbox. It is also higher in energy and longer than the path shown in Fig. 4.

FIG. 6. The spectra of SrZrO3 �dashed� is redshifted relative to the KTaO3

spectra �solid�. The spectra were normalized by area for plotting in the samescale.

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and RTRTRT. The orthorhombic proton conduction paths aresimple combinations of the cubic paths with addition of anew type of transfer. The new interoctahedral transfer �Tn�allows cutting across a face by replacing two intraoctahedraltransfers �TT�. The lowest energy path found has an intraoc-tahedral transfer rate-limiting barrier of 0.32 eV. The rota-tional barriers on this path range from 0.05 to 0.30 eV withthe latter being very competitive with the intraoctahedraltransfer barrier. Based on binding site energies in the lowestenergy paths, the proton spends a significant amount of timebetween oxygens of different octahedra. The lowest of thesebinding sites was the starting point for a molecular dynamicstrajectory. During the 1000 fs trajectory, the OH bond vi-brated and rotated within the binding site. The vibrationalspectra calculated from this trajectory found significant red-shifting of the OH stretch in SrZrO3 relative to KTaO3. Thisstudy shows that a strong redshifted component in the spec-trum is possible even with an intraoctahedral rate-limitingstep. While doping is a critical component of proton conduct-ing perovskites, for simplicity, it has been left out from thisfirst application of the adjacency matrix formalism. Studiesincluding dopants are underway.

ACKNOWLEDGMENTS

The authors would like to thank Lynn Fletcher for herextensive editing advice. This research was supported by anACS PRF Type G Starter Grant and NSF RUI �Grant No.CHE-0608813�. Computational resources were provided inpart by the MERCURY supercomputer consortium �http://mars.hamilton.edu� under NSF MRI �No. CHE-0116435�.

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