FULL TEXT - Alfredo Di Nola

12 Pages·1996·0.23 MB·Italian

An extended x-ray absorption fine structure study by employing molecular
dynamics simulations: Bromide ion in methanolic solution
P. D’Angelo, A. Di Nola, M. Mangoni, and N. V. Pavela)
DipartimentodiChimica,Universita` degliStudidiRoma‘‘LaSapienza,’’P.leAldoMoro5,
00185Roma,Italy
(cid:126)Received 20 May 1995; accepted 17 October 1995(cid:33)
X-rayabsorptionspectroscopyiswidelyemployedinthestructuralanalysisofdisorderedsystems.
In the standard extended x-ray absorption fine structure (cid:126)EXAFS(cid:33) analysis the coordination of the
photoabsorber is usually defined by means of Gaussian shells. It is known that this procedure can
lead to significant errors in the determination of the coordination parameters for systems which
present anharmonic thermal vibrations or interatomic asymmetric pair distribution functions. An
efficient method has been recently employed in the study of the hydration shells of bromide and
rubidium ions and brominated hydrocarbon molecules in diluted aqueous solutions. According to
this method, pair distribution functions [g(r)] obtained from molecular dynamics simulations can
be used as relevant models in the calculation of the EXAFS signals. Moreover, asymmetric shells
modeled on the g(r) first peaks, have been employed in the EXAFS analysis and the parameters
defining the asymmetric peaks have been optimized during the minimization procedure. In the
(cid:50)
presentpaperthisnewprocedurehasbeenusedtoinvestigatethecoordinationofBr inmethanol.
The analysis of this system is particularly interesting due to the presence of three well separated
coordination shells. We show that the inclusion of the hydrogen signal is essential to perform a
reliableanalysis.AcomparisonoftheanalysiswithasymmetricandGaussianshellsshowshowthe
accuracy of the EXAFS data analysis is improved by using asymmetric shells. © 1996 American
Institute of Physics. (cid:64)S0021-9606(cid:126)96(cid:33)50404-4(cid:35)
I. INTRODUCTION sorber atom as compared to x-ray or neutron diffraction. A
limitation of the XAS technique which has to be mentioned
X-raydiffraction(cid:126)XRD(cid:33)andneutrondiffractionarevery
isitslowsensitivitytothelargedistances.Infact,duetothe
powerful techniques in the structural investigation of solid
broadcorrelationfunctiontowardsthelargedistancesandto
and liquid disordered systems and they are employed to de-
the finite mean free path of the photoelectron, the sensitivity
termine the radial distribution functions g(r) of the system
oftheXAStechniqueislimitedtotheneighbourhood(cid:126)about
under study. The structural information contained in the
5–7 Å(cid:33) of the photoabsorber atom. The experimental deter-
structure factor for a single component system is complete
minationoftheg(r) overthefullrangeofdistancesisham-
and includes short-range as well as medium and long-range
pered by this short range sensitivity, providing no informa-
information. In the case of multicomponent systems the in-
tion on the g(r) long distance tail. However the XAS signal
determination increases and the diffraction techniques, fre-
quently, do not allow the assignment of specific pair distri- isverysensitivetotheshort-distancefeaturesoftheg(r) and
bution functions which contribute to the diffraction signal. inparticulartotheshapeofitsfirstriseallowinginformation
The increasing interest in characterizing disordered sys- on the short-range order. In the standard extended x-ray ab-
tems has stimulated the development of complementary sorption fine structure (cid:126)EXAFS(cid:33) analysis the coordination of
structural techniques. Among these, x-ray absorption spec- the photoabsorber is usually defined by means of Gaussian
troscopy (cid:126)XAS(cid:33) has acquired a central role due to its atomic shells. Earlier studies have shown limitations in this
selectivity. By analyzing the oscillating behavior of the ab- procedure5,6whichcanleadtosignificanterrorsinthedeter-
sorption cross section above the excitation threshold, the lo- mination of the coordination parameters for systems which
cal structure around the photoabsorber atom can be deter- present anharmonic thermal vibrations or interatomic asym-
mined both for crystalline1 and disordered systems.2,3,4 This metric pair distribution functions. The cumulant method has
technique is more selective than x-ray and neutron diffrac- been employed to detect and measure the effects of anhar-
tion since the partial distribution functions around a single monicityandofasymmetricdistancedistributions.4,7–12Sev-
atomic type can be determined. For an N-component system eral EXAFS studies have been devoted to the determination
the g(r) extracted from the XAS signal contains the overlap oftheasymmetricdistributionfunctionsinliquidmetalsand
ofN independentpairdistributionfunctions,againsttheN(N amorphous solids.3,4,6,8,13–16Acomparison of radial distribu-
(cid:49)1)/2 pair distribution functions overlapped in the XRD tionfunctionsobtainedfromMonteCarlo(cid:126)MC(cid:33)andmolecu-
data.Therefore,itiseasytoextractthestructuralinformation lardynamics (cid:126)MD(cid:33)simulationsorXRDandneutrondiffrac-
related to different coordination shells around the photoab-
tion with the asymmetric peaks used in the EXAFS analysis
allows one to verify the reliability of the refined parameters.
a(cid:33)Authortowhomcorrespondenceshouldbeaddressed. In recent years MC and MD simulations have been widely
J.Chem.Phys.104(5),1February1996 0021-9606/96/104(5)/1779/12/$10.00 ©1996AmericanInstituteofPhysics 1779
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1780 D’Angeloetal.:EXAFSofBr(cid:50)inmethanol
developed. Structural aspects of solids, liquids, and TABLEI. ChargesandLennard-Jonesparameters.
solutions17,18 can be predicted and the results can be com-
q (cid:115)(cid:126)Å(cid:33) (cid:101)(cid:126)kJmol(cid:50)1(cid:33)
paredwithNMR,XRD,XASandneutrondiffractionexperi-
mental data. MC and MD simulations have been used to Br(cid:50) (cid:50)1.000 4.634 0.452
determine radial distribution functions of aqueous salt H 0.398 0.000 0.000
solutions19–23 which have been compared with XRD (cid:126)Ref. O (cid:50)0.548 2.955 0.849
CH 0.150 3.786 0.754
24(cid:33) and neutron diffraction experiments.25 3
Computer simulations are less frequently employed in
the EXAFS data analysis. A ‘‘free style EXAFS fit algo-
The paper is organized as follows: in Sec. II the results
rithm’’ has been used to search for the best agreement be-
of the MD simulations are presented, Sec. III is devoted to
tweentheexperimentalandcalculatedsignal,startingfroma
thedescriptionoftheexperimentalconditions,inSec.IVthe
generic radial distribution function with free shape and vari-
able coordination number.26 A reverse Monte Carlo EXAFSdataanalysisispresentedandtheresultsaregivenin
procedure27 which allows the fit of g(r) functions by gener- Sec. V.
ating a three-dimensional structural model, has been applied
to the EXAFS analysis.28 Comparison between MD simula- II. MOLECULAR DYNAMICS COMPUTATIONAL
PROCEDURE
tions and EXAFS experiments have been performed in the
study of high-temperature and high-pressure solids9,12,14,16 The MD simulations were performed using a rigid three
and metal clusters.29 sitemethanolmodel.TheequilibriumdistancesforC–Oand
Recently,ithasbeenshownthatadeeperinsightintothe O–H were 1.43 Å and 0.95 Å, respectively, with a bond
structural organization of disordered systems can be pro- angleof108.53°.AnNPTensemblewasconsideredusingan
vided by employing, in the EXAFS data analysis, some ex- isothermal–isobaric algorithm.37 Weak coupling to an exter-
ternal input on the distribution functions derived from dif- nal temperature bath of 300 K with a coupling constant of
fraction data or theoretical simulations. Radial distribution 0.1ps,andtoanexternalpressurebathof1.0(cid:51)105Pawitha
functions extracted from XRD and neutron diffraction mea- coupling time constant of 0.5 ps was used to maintain con-
surements have been employed to fit realistic g(r) functions stant temperature and pressure. Model potential and geom-
to the EXAFS spectra of monoatomic systems such as Hg,30 etry for methanol was taken from the GROMOS software
Ga,31 Cu, and Pd (cid:126)Ref. 32(cid:33) liquid metals. Similarities and package library.38 The applied empirical potential energy
differencesbetweenthestructuralinformationobtainedfrom function contains terms representing bond angle bending,
EXAFS and diffraction signals are discussed in Ref. 32.An van der Waals, and electrostatic interactions.39 The function
(cid:50)
efficient method which has been employed in the study of used in the present study for Br was reported by Straatsma
the hydration shells of Br(cid:50) and brominated hydrocarbon and Berendsen.21 The SHAKE algorithm40 was used to con-
moleculesindilutedaqueoussolutions,usespairdistribution strainbondlengths.Adielectricpermittivity,(cid:101)(cid:53)1,andatime
functions obtained from MD simulations as relevant models step of 2.0 fs were used. The cutoff distance for the non-
in the calculation of the EXAFS signals.33 This procedure bonded interactions was 10 Å. Charges and Lennard-Jones
hasbeensuccessfullyappliedtothestudyofthestructureof parameters are reported in Table I. Simulations were carried
micellar aqueous solutions of rubidium salts of bile acids.34 out using a cubic box consisting of one Br(cid:50) and 124 metha-
Inthepresentpaperwehaveusedthisnewprocedureto nol molecules subjected to periodic boundary conditions.
investigate the coordination of Br(cid:50) in methanol (cid:126)BMOH(cid:33). It Theinitiallengthoftheboxedgewassetsoastoagreewith
is important to stress that this approach becomes extremely the experimental density. Configurations were saved every
useful when complex systems containing several pair distri- 25 steps and after equilibration, 2000 configurations were
bution functions are considered. In the presence of many used to calculate ion–solvent and solvent–solvent pair dis-
coordination shells around the photoabsorber atom, g(r) tribution functions. The g (r), g (r), and g (r) pair
BrH BrO BrMe
models obtained from MD simulations can represent an es- distributionfunctionsareshowninFig.1,wheretherunning
sential starting point for the EXAFS analysis. By combining integrationnumbersarealsoreported.Theseare6.2,6.3,and
EXAFS and MD information it is possible to perform an 7.5 up to the corresponding first minima (cid:126)3.2 Å, 4.1 Å, and
accurate analysis of complex systems. Such direct informa- 4.8 Å(cid:33) of g (r), g (r), and g (r), respectively. The
BrH BrO BrMe
tion on the structural organization of disordered systems Me solvation number is larger than those of H and O, indi-
would be difficult to achieve by means of diffraction tech- cating an interpenetration of outer molecules into the first
niques. The spatial separation among the pair distribution solvation shell, as found in the case of chloride ion.35,36 Re-
functionfirstpeaksisafavourableconditionforasuccessful cently, an EXAFS study of bromide ion in various solvents
analysisofapolyatomicsystem.PreviousMDsimulationsof provided a low value (cid:126)3.7(cid:33) for the Br–O coordination num-
MgCl and NaCl in methanol35,36 have shown that the first ber in methanol, on the basis of a single Gaussian shell
2
peaksofthepairdistributionfunctionsrelatedtothehalogen (cid:126)Br–O(cid:33) analysis.41
anion are well separated. Since the high absorption coeffi- The orientation of the methanol molecules in the first
cient of the solvent hampers the acquisition of reliable data solvationshellcanbededucedfromthedistributionofcos(cid:99),
at low concentration at the chlorine K-edge, we decided to showninFig.2(cid:126)the(cid:99)angleisdefinedintheinsertion(cid:33).Most
(cid:50)
focus our attention on the bromine K-edge. of the methanol molecules lie in an O–H–Br linear con-
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D’Angeloetal.:EXAFSofBr(cid:50)inmethanol 1781
FIG.1.Br–H,Br–O,andBr–Mepairdistributionfunctionsasderivedfrom
MDsimulationsforBr(cid:50)inmethanol(cid:126)leftscale(cid:33)andcorrespondingrunning FIG.3. O–H,H–H,andO–O(cid:126)upperpanel(cid:33),Me–O,Me–H,andMe–Me
integrationnumber(cid:126)rightscale(cid:33). (cid:126)lowerpanel(cid:33)pairdistributionfunctionsformethanol–methanolinteractions
asderivedfromMDsimulationsforBr(cid:50)inmethanol.
(cid:50)
figurationdemonstrating,asinthecaseofCl inmethanol,a
2.75, 3.48, and 4.1 Å, respectively. The Me–O first peak
strong preference for a linear hydrogen bond formation of
(cid:50) yields four oxygens within a distance of 4.2 Å. Intermolecu-
the first solvation shell molecules with Br . On the other
hand, the cos(cid:99)distribution explains the sharpness of the lar bonding energy distribution of methanol monomers,
shown in Fig. 4, is in agreement with the MC data,42 where
g (r) and g (r) first peaks. These are indicative of a
BrH BrO
a bimodal profile was found.
tightly structured first solvation shell.
The solvent–solvent pair distribution functions are re-
ported in Fig. 3. The first peaks of the g (r), g (r), and III. EXPERIMENT
OO OH
gHH(r) reflect the hydrogen bonding. The locations of the Tetramethylammoniumbromide(cid:126)Flukapurum(cid:33)wasused
firstmaximaare2.73,1.82,and2.48Å,respectively,andthe
to prepare a 0.15 M methanolic solution.
corresponding integrals are 2.2, 1.0, and 2.3, respectively.
The XAS spectrum above the bromine K-edge has been
These values are in good agreement with earlier studies on
recorded at room temperature, in the transmission mode at
rigidliquidmethanolobtainedbyMCsimulations.42Thefirst
the ROMO II station of the Hamburger Synchrotronstrahl-
peaks of the gMeH(r), gMeO(r), and gMeMe(r) are located at ungslabor HASYLAB. The ROMO II beamline (cid:126)bending
magnet source, Doris storage ring(cid:33) was equipped with a Si
FIG.2. Distributionofcos(cid:99)forthemethanolmoleculesinthefirstsolva- FIG.4. Distributionofthebindingenergyofmethanolmonomersasderived
tionshellofBr(cid:50). fromMDsimulationsforBr(cid:50)inmethanol.
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1782 D’Angeloetal.:EXAFSofBr(cid:50)inmethanol
(cid:126)311(cid:33) double crystal monochromator controlled by a special been used to calculate the phase shifts. The values of the
feedback system.43The ionization chambers were filled with muffin-tin radius are 1.58 Å, 0.40 Å, 0.90 Å, and 0.70 Å for
argon at atmospheric pressure.The storage ring was running the Br(cid:50), hydrogen, oxygen, and carbon of the methyl group,
at an energy of 4.45 GeV with an electron current between respectively.Ithasbeenverifiedthatdifferentconfigurations
80 and 20 mA.Athird ionization chamber allowed possible give the same result. A Hedin–Lundqvist plasmon-pole ap-
shifts in energy to be checked by recording a reference sub- proximation is used for the self-energy part of the optical
stancetogetherwiththeinvestigatedsample.Theanalysisof potential.44Inelasticlossesofphotoelectronsinthefinalstate
the edge region, performed as described in a recent paper,33 are accounted for intrinsically by complex potentials.45 The
allowed the determination of the monochromator resolution imaginarypartalsoincludesaconstantfactoraccountingfor
(cid:126)1.1 eV(cid:33). the known core–hole width.
The total (cid:120)(k) theoretical curve is given by the sum of
the (cid:120)(k) signals associated with different atoms of the sol-
IV. XAS CALCULATIONS
vent. These have been calculated by means of Eq. (cid:126)1(cid:33), start-
As shown in Sec. II, the solvent molecule g(r) distribu- ing from the pair distribution functions obtained from the
tion function is made up of three pair distribution functions, MD simulations. Fits of the XAS spectra have been per-
ascalculatedbymeansofMDsimulations.Foragiveng(r), formeddirectlyontherawdatausingtheFITHEOprogram.46
the (cid:120)(k) signal can be calculated by means of a well-
For an accurate XAS analysis it is necessary to take into
established equation in the EXAFS field,4 which is often ap-
account the presence of discrete resonances and slope
plied to the study of disordered systems,
changes in the atomic background associated with the onset
(cid:69)
(cid:96) of multielectron excitation channels. The KN , KN ,
(cid:120) (cid:126)k(cid:33)(cid:53) dr4(cid:112)(cid:114)r2g(cid:126)r(cid:33)A(cid:126)k,r(cid:33)sin(cid:64)2kr(cid:49)(cid:102)(cid:126)k,r(cid:33)(cid:35), 2,3 1
pair KM , and KM edges, corresponding to the simultaneous
0 4,5 2,3
(cid:126)1(cid:33) excitation of the 1s4p, 1s4s, 1s3d, and 1s3p electrons,
respectively, have been detected in the absorption spectra of
where A(k,r) and (cid:102)(k,r) are the amplitude and phase
several brominated compounds. The background functions
functions,1 respectively. In spite of the apparent upper inte-
have been modeled accounting for the double-excitation
gration limit of infinity in Eq.(cid:126)1(cid:33), the sensitivity of the(cid:120)(k)
edges by means of step shaped functions as previously
signal is limited to the neighborhood of the photoabsorber.
described,47 with energy onset values equal to those found
This is due to the finite mean-free path (cid:108)(k) of the photo-
for gaseous HBr within the reported errors.48
electron which generates an exponential decay of the type
As shown previously,47 the background parameters are
exp[(cid:50)r/(cid:108)(k)]. This effect, as well as the additional damp-
practically uncorrelated with the others. Three important
ing due to the monochromator resolution which mainly af-
nonstructuralparametersrelatedtothe(cid:120)(k) signalareE ,S2
fects the low-k region of the spectrum, is included in the 0 0
and the resolution of the monochromator. E is the photon
A(k,r) function. Several EXAFS investigations on disor- 0
dered systems assume that the (cid:120)(k) signal is sensitive to the energyrequiredforthetransitiontothecontinuumthreshold
and allows the theoretical and experimental energy scales to
first coordination shell around the photoabsorber, only. The
be compared. S2 accounts for a uniform reduction of the
first-neighbor peak is often modeled with a Gaussian distri- 0
signal associated with many-body effects. As previously
bution although, by considering the low-distance shape of a
shown47thisparameterisequalto1,whendouble-excitation
pairdistributionfunction,itisevidentthatthefirstpeakcan-
effects are properly included in the atomic background. The
not be well described by one or more Gaussian functions.A
value of the monochromator resolution has been determined
method which employs Gamma like distribution curves to
from the experimental spectrum as explained in Sec. III.
describetheshapeoftheg(r) firstpeakhasbeenpreviously
described.33 According to this method each model peak is During the fitting procedure these three parameters were
definedbythemeandistanceR,standarddeviation(cid:115),asym- minimized. The monochromator resolution showed varia-
metryindex(cid:98),andcoordinationnumberNc.Thisfunctionis tionsoflessthan0.1eVandS02waspracticallyequaltoone.
usedtocalculatetheEXAFSsignalassociatedwiththeg(r) All the minimizations have been performed in the same
first peak. The asymmetric peak is subtracted from the MD k range (cid:126)3(cid:60)k(cid:60)16 Å(cid:50)1(cid:33), including 418 experimental points.
g(r) obtaining a long-distance tail whose EXAFS signal is The fit index Ri is defined by Eq. (cid:126)5(cid:33) of Ref. 47 and a k
calculated by means of Eq. (cid:126)1(cid:33). This contribution is kept weightingvalueof2.5wasapplied.Thenumberofthestruc-
fixed during the minimization, while the four parameters de- tural parameters used in the fitting procedure was 3 for the
scribing the asymmetric peak are refined in order to achieve analysis performed using the MD g(r) functions, 9 for the
the best fit to the experimental spectrum. In this work three analysisintheGaussianapproximation,and12fortheanaly-
asymmetricpeaks,representingtheBr(cid:50)methanolpairdistri- sis with three asymmetric shells.
bution functions, have been employed in the EXAFS analy- Aroughestimateofthefreeparametersthatcanbefitted
sis. in the EXAFS data analysis49 is given by the 2(cid:68)k(cid:68)r/(cid:112)(cid:49)2
Phase shifts have been calculated in the muffin-tin ap- relation,50whichsupportsthepresentleast-squarefittingpro-
proximation starting from overlapped spherically averaged cedure.(cid:68)k isthek-spacerangeoverwhichthe(cid:120)(k) signalis
relativisticatomicchargedensities.Oneofthemoleculardis- fitted and (cid:68)r is the width of the r space Fourier filter win-
tributions obtained from the MD simulations of BMOH has dow. In our case (cid:68)k is approximately 13 Å(cid:50)1 and (cid:68)r is
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D’Angeloetal.:EXAFSofBr(cid:50)inmethanol 1783
limited by the mean-free path only, since we are not Fourier
filtering the data.
Parameter correlations and standard deviations can be
determined, for the p free parameters used in the minimiza-
tion, by tracing constant chi-square boundaries, also known
as correlation maps.51 R is calculated around the minimum
i
as a function of two parameters, keeping fixed the others
(cid:110)(cid:53)p(cid:50)2 parameters. Three contours are drawn for (cid:68)R con-
stant shifts from the minimum R (cid:51)(N(cid:50)p)/N and corre-
i
spondtothe68.3%(cid:126)(cid:54)(cid:115)(cid:33),95.4%(cid:126)(cid:54)2(cid:115)(cid:33),and99.73%(cid:126)(cid:54)3(cid:115)(cid:33)
confidence regions. (cid:68)R is defined by the equation
(cid:68)R(cid:53)(cid:120)2(cid:51)R /(N(cid:50)p). (cid:120)2 represents the value of the re-
(cid:110) N (cid:110)
duced chi-square corresponding to a defined confidence
level,andR isanestimateoftheexperimentalnoisecalcu-
N
lated in the FITHEO program, as described elsewhere.46
The confidence region ellipses that contain 68.3% of
normally distributed data were used to determine the stan-
dard deviations, which were obtained by projecting the
higherdimensionalregionontothelower-dimensionspace.51
From the correlation maps, drawn for different couples of
parameters, standard deviation values can be obtained. The
FIG.5. FitoftheBMOHexperimentalspectrum.Upperpanel:fromtopto
largest ones have been chosen to define the standard devia- bottomthe(cid:120) (k),(cid:120) (k),and(cid:120) (k)theoreticalsignalscorresponding
BrH BrO BrMe
tion magnitudes associated with the parameters obtained totheMDpairdistributionfunctions,theirsumcomparedwiththeexperi-
mental spectrum, and the residual are shown. Bottom panel: fit performed
from the XAS analysis.
withoutincludingthe(cid:120) (k) signal.Fromtoptobottomthetotaltheoreti-
BrH
cal signal compared with the experimental spectrum, and the residual are
shown.
V. XASANALYSIS OF Br(cid:50) IN METHANOL
(cid:50)
The XAS analysis of the Br methanolic solution has
beencarriedoutstartingfromtheresultsdescribedinSec.II. less, the inclusion of this low-frequency contribution has
The outstanding observation of this analysis concerns the been found to be essential to properly reproduce the experi-
(cid:50)
strong amplitude of the Br –H signal. Due to the well or- mental spectrum.
dered structure of the solvent molecules, it was possible to Proof of the importance of the Br–H signal was also
identify, for the first time, the contribution of the hydrogen obtained by excluding this contribution from the fitting pro-
atoms of the solvent to the x-ray absorption cross section. cedure. The lower panel of Fig. 5 shows the comparison
Generally, the low scattering amplitude prevents the hydro- between the total theoretical signal, including the Br–O and
gen signal to be identified, especially in the case of disor- Br–Me contributions only, and the experimental spectrum,
deredsystems.ItisclearfromtheMDpairdistributionfunc- together with the residuals. In this case, the agreement be-
(cid:50)
tions that the presence of the negative charge of the Br tween experiment and theory is unsatisfactory
enforces the methanol molecules to take up a well ordered (cid:126)R (cid:53)1.030(cid:51)10(cid:50)6(cid:33) and the residual curve contains a fre-
i
structure with the hydrogen atoms directed towards the ion. quency and amplitude component which is similar to the
This occurrence explains the unusually strong amplitude of (cid:120) (k) signal shown in the upper panel of Fig. 5.
BrH
the hydroxyl hydrogen signal. The second step of the analysis was the substitution of
The first step of the analysis involved the calculation of the pair distribution functions with three asymmetric shells
the theoretical signal by means of Eq. (cid:126)1(cid:33), starting from the representing the g(r) first peaks. Each peak is narrow
g(r) pair distribution function obtained from the MD simu- enough to be described by one asymmetric peak, only. The
lations.Afitting procedure was applied in order to improve, difference between the MD g(r) and the asymmetric peak
asfaraspossible,theagreementwiththeexperimentalspec- definesalong-distancetail33whosecontributioniscalculated
trum.TheupperpanelofFig.5showstheBr–H,Br–O,and by means of Eq. (cid:126)1(cid:33). The three tail signals have been found
Br–Me theoretical signals, the comparison between the sum tobenegligibleinthewholek-rangeandthereforetheyhave
of these contributions and the experimental spectrum, and not been included in the calculated curve. This result con-
the residuals (cid:126)from top to bottom, respectively(cid:33). The agree- firms the low sensitivity of the XAS technique to the large
ment between the experimental and the theoretical signal is distances. The (cid:120)(k) signals associated with the asymmetric
satisfactory and a R (cid:53)0.495(cid:51)10(cid:50)6 has been obtained. From peaks were calculated by using the previously described
i
the minimization no significant shifts of the g(r) functions procedure.33Duringtheminimization,theshapeoftheasym-
have been observed. From the upper panel of Fig. 5 it is metricpeakshasbeenoptimizedbyvaryingthepeakparam-
evident that the total signal is dominated by the Br–O and eters.TheupperpanelofFig.6showstheBr–H,Br–O,and
Br–Me contributions, while the Br–H signal is weaker and Br–Me theoretical signals, the comparison between the total
mainly affects the low-k region of the spectrum. Neverthe- (cid:120)(k) signalandtheexperimentalspectrum,andtheresiduals.
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1784 D’Angeloetal.:EXAFSofBr(cid:50)inmethanol
TABLE II. First solvation shell parameters of Br(cid:50) in methanol, for the
minimizations with asymmetric (cid:126)AS(cid:33) and Gaussian (cid:126)GS(cid:33) shells. The stan-
darddeviationsaregiveninparenthesesfortheminimizationsincludingthe
hydrogenshell.R representstheaveragedistanceinÅ,(cid:115)2isthevibrational
variation in Å2, (cid:98)is the asymmetry parameter, and N is the coordination
number.
AS AS(cid:126)withoutH(cid:33) GS GS(cid:126)withoutH(cid:33)
R 2.46 (cid:126)0.03(cid:33) ••• 2.34 (cid:126)0.04(cid:33) •••
H
(cid:115)2 0.072 (cid:126)0.013(cid:33) ••• 0.031 (cid:126)0.009(cid:33) •••
H
(cid:98) 1.58 (cid:126)0.15(cid:33) ••• ••• •••
H
N 6.8 (cid:126)0.9(cid:33) ••• 5.9 (cid:126)1.7(cid:33) •••
H
R 3.40 (cid:126)0.01(cid:33) 3.41 3.28 (cid:126)0.01(cid:33) 3.30
O
(cid:115)2 0.058 (cid:126)0.005(cid:33) 0.059 0.023 (cid:126)0.002(cid:33) 0.027
O
(cid:98) 1.18 (cid:126)0.04(cid:33) 1.11 ••• •••
O
N 6.1 (cid:126)0.2(cid:33) 7.2 4.8 (cid:126)0.3(cid:33) 6.3
O
R 4.23 (cid:126)0.03(cid:33) 4.21 3.92 (cid:126)0.02(cid:33) 3.93
Me
(cid:115)2 0.124 (cid:126)0.016(cid:33) 0.125 0.067 (cid:126)0.013(cid:33) 0.047
Me
(cid:98) 1.32 (cid:126)0.09(cid:33) 1.15 ••• •••
Me
N 7.0 (cid:126)0.7(cid:33) 7.4 9.0 (cid:126)2.8(cid:33) 7.6
Me
FIG.6. FitoftheBMOHexperimentalspectrumperformedwithasymmet- and 6% can be observed for the oxygen and methyl coordi-
ric shells. Upper panel: from top to bottom the (cid:120) (k), (cid:120) (k), and nationnumbers,respectively.Thestandarddeviationsarenot
BrH BrO
(cid:120)BrMe(k) asymmetric peak contributions, their sum compared with the ex- reported, as the low agreement with the experimental spec-
perimental spectrum, and the residual are shown. Bottom panel: fit per-
trum does not allow a measure of reliable errors from the
formedwithoutincludingthe(cid:120) (k) signal.Fromtoptobottomthetotal
BrH
theoretical signal compared with the experimental spectrum, and the re- correlation maps.
sidualareshown. The EXAFS analysis of disordered systems is often car-
riedoutbyusingGaussianpeakstomodelthephotoabsorber
coordination shells. The reliability of this approach can be
The agreement between the calculated and the experimental checked by comparing the results of this method with those
signal is satisfactory and this result is confirmed by the fit obtained from the asymmetric peak analysis. The minimiza-
index value (cid:126)R (cid:53)0.490(cid:51)10(cid:50)6(cid:33), 1.6 times greater than R . tion performed in the Gaussian approximation gives rise to
i N
Notice that the residual curve shows a high frequency oscil- more dumped calculated signals and to a poorer agreement
lation which is above the noise of the spectrum. The high between the total calculated and the experimental signal
frequencyofthissignalcouldbeassociatedwithalong-path (cid:126)R (cid:53)0.608(cid:51)10(cid:50)6(cid:33), compared to the previous analysis. Sig-
i
contribution and in particular with multiple scattering (cid:126)MS(cid:33) nificant differences have been observed in the structural pa-
effects due to well-defined three body arrangements. This rameters obtained from the two methods. The parameters
oscillating behavior is less evident, but still detectable in the describing the Gaussian shells and standard deviations are
residual curve shown in the upper panel of Fig. 5.The pres- listed in Table II. The iteration procedure has been carried
ence of triplet correlations in the hydration shell of aquaions out starting from fitting parameters similar to those obtained
has been already evidenced in other systems.52 This finding from the asymmetric shell analysis.
suggeststhenecessityofadeeperinvestigationonthetriplet As in the above-mentioned cases the fit index of the
correlation contributions to the XAS spectra of liquid sys- analysis omitting the hydrogen contribution was
tems. R (cid:53)0.998(cid:51)10(cid:50)6 and the structural parameters are given in
i
The parameters describing the three asymmetric shells Table II.Acomparison of the structural parameters obtained
together with the standard deviations are shown in Table II. fromtheanalysiswiththeinclusionandtheexclusionofthe
The standard deviations of the refined parameters are ob- hydrogenshell,showssignificantvariationoftheoxygenand
tained from the correlation maps, as explained in Sec. IV. methyl coordination numbers. These values are similar to
The importance of the hydrogen contribution has been those obtained from the asymmetric shell analysis. This co-
pointed out also by performing a minimization without the incidence is due, merely, to the low sensitivity of the mini-
Br–H signal. The results of this analysis are shown in the mizationproceduretothevariationofthecoordinationnum-
lower panel of Fig. 6. The agreement between theory and bers. Minimizations carried out using different starting
experiment is not satisfactory (cid:126)R (cid:53)1.093(cid:51)10(cid:50)6(cid:33) and the re- coordination numbers failed to show any significant shift
i
sidual curve contains a low-frequency oscillation which is from the initial values.
due to neither the oxygen nor the methyl coordination shell. Amore direct description of the differences between the
The Br–O and Br–Me asymmetric shell parameters, shown coordinationshellparametersobtainedfromthetwoEXAFS
in Table II, are similar to those obtained from the previous analysiscanbeobtainedbycomparingtherefinedasymmet-
analysis with the exception of the coordination numbers. In ric and Gaussian peaks. In Fig. 7 these peaks are shown
the absence of the hydrogen shell, an increase of about 18% togetherwiththeMDg(r)’s.Strongdifferencesbetweenthe
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D’Angeloetal.:EXAFSofBr(cid:50)inmethanol 1785
carbon atoms, such shells give rise to non distinguishable
EXAFS signals.
Thestatisticalsignificanceoftheinclusionofthehydro-
gen signal can be checked by performing the F-test53 for the
three cases under study.54 In all these cases, the F-test ap-
pliedforthe95%confidencelevel,assessesthatthedecrease
ofthefittingindexaftertheadditionofthehydrogenshell,is
statistically significant.
At this point, the problem of the accuracy of the struc-
turalparametersshowninTableII,needstobeaddressed.By
neglecting systematic errors in the experimental data and in
the theory, the errors affecting the fitted values can be esti-
matedonthebasisofstandardstatisticalconcepts.Neverthe-
less,correlationeffectscanincreasethestandarddeviationof
the parameters and a deeper insight into this problem can be
provided by using contour maps for each couple of
parameters.51 In Figs. 8–11 some of the most meaningful
correlationmapscalculatedfortheasymmetricandGaussian
shells can be compared.
The correlation between E and the shell distances is
0
showninFig.8.E appearstobestronglycorrelatedwiththe
0
shell distances. The errors on E and on the hydrogen and
0
oxygen distances, shown in Table II, have been obtained
fromtheRE correlationmaps.Thisresultisconsistentwith
0
thewell-knownindeterminationondistancesproducedbyE
0
(cid:126)see for example, Refs. 1–3(cid:33).
The R(cid:115)2 correlation maps are shown in Fig. 9. The dis-
tance and the Debye–Waller factors are strongly correlated
FIG.7. Fromtoptobottom:asymmetric(cid:126)dashed(cid:33)andGaussian(cid:126)full(cid:33)peaks in the case of the asymmetric shells. However, it has to be
obtainedfromtheEXAFSanalysiscomparedwiththeg (r),g (r),and
BrH BrO stressedthattheFWHMofaGaussianshellisdeterminedby
g (r),respectively,obtainedfromtheMDsimulations.
BrMe the (cid:115)value, only, while the width of an asymmetric shell is
determined by two parameters (cid:126)(cid:115)and (cid:98)(cid:33). The narrow confi-
dence regions of the oxygen shells point out that the accu-
Gaussian and the asymmetric shells are evident from this
racyoftheminimizationdependsmainlyontheoxygencon-
figure.Inparticular,theGaussianshellsdonotreproducethe
tribution.
shape and the integration numbers of the MD g(r) first
The correlations between the coordination numbers, as
peaks. The Gaussian methyl peak is shifted towards lower
shown in Fig. 10, are more pronounced in the case of the
distances, giving rise to a partial overlap with the oxygen
hydrogen and methyl shells, both in the Gaussian and in the
shell,whiletheintegrationnumberandthefull-widthathalf-
maximum(cid:126)FWHM(cid:33)valuesarebothincreasedforthemethyl asymmetric shell analysis. The narrow confidence region of
the correlation maps associated with the oxygen shell is due
shell, as opposed to the decreased values observed for the
toboththelargedifferenceofthescatteringamplitudeofthe
oxygen one. From these results it is evident that the analysis
hydrogen and oxygen atoms, and the long Br–Me distance.
performed with Gaussian shells does not allow a good de-
scription of the distribution of methanol molecules around The existence of a strong correlation between the coor-
the bromine anion. dination numbers and the Debye–Waller factors is a well
Noticethattheconstraintrule(cid:68)N(cid:53)0(cid:126)(cid:68)N representsthe established phenomenon in the EXAFS data analysis. This
variation of the sum of the coordination numbers during the effect is responsible for the large uncertainty on the coordi-
minimization(cid:33) suggested for the refinement of monatomic nation numbers which is a characteristic of EXAFS per-
systems with more coordination shells,32 cannot be applied formed in the Gaussian approximation, and in general of
in the case of a polyatomic system, in particular in the case diffraction techniques. A strong correlation between Nc and
of the Gaussian approximation. In the present case, the total (cid:115)2 is evident from the maps associated with the Gaussian
coordinationnumbervariesof0.2only(cid:126)goingfrom19.9,for shells shown in Fig. 11. On the other hand from the correla-
the asymmetric shell analysis, to 19.7 for the Gaussian shell tion maps associated with the asymmetric shells a smaller
analysis(cid:33) in spite of the difference between the agreement correlation between these parameters can be observed. In
indexes. It is evident that for systems with coordination addition,theconfidenceregionsarenarrower,showingabet-
shells constituted by atoms with similar phases and scatter- ter reliability of the refined parameters.The most interesting
ing amplitudes, the constraint rule (cid:68)N(cid:53)0 cannot be applied conclusion which can be drawn from a comparison of the
in the minimization procedure.As in the case of oxygen and correlation maps is that the correlations between the Gauss-
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1786 D’Angeloetal.:EXAFSofBr(cid:50)inmethanol
FIG.8. CorrelationmapsoftheRE couplesforasymmetricandGaussianshellsminimizations(cid:126)leftandrightcolumn,respectively(cid:33).Thecontourscorrespond
0
tothe68.3%(cid:126)(cid:54)(cid:115)(cid:33),95.4%(cid:126)(cid:54)2(cid:115)(cid:33),and99.73%(cid:126)(cid:54)3(cid:115)(cid:33)confidenceregions.Fortheasymmetricshells,Rrepresentstheaveragevalueshelldistance.
ian parameters are not equivalent to those of the asymmetric importanceofrefiningEXAFSdatausingrealisticg(r) mod-
shells. Moreover, these maps allow the identification of the els. In the case of polyatomic disordered systems, a proper
most correlated parameters and of the couples which give description of the photoabsorber coordination shells is diffi-
rise to the maximum standard deviation for each parameter. culttoachievebymeansofthestandardEXAFSanalysis.In
These results provide new insight into the limits of the EX- particular, the strong correlation between the parameters
AFS data analysis of polyatomic systems. hampers an unerring set of coordination shells to be identi-
The last remark we would like to make concerns the fied. Minimizations carried out using peak parameters far
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D’Angeloetal.:EXAFSofBr(cid:50)inmethanol 1787
FIG.9. CorrelationmapsoftheR(cid:115)2couplesforasymmetricandGaussianshellsminimizations(cid:126)leftandrightcolumn,respectively(cid:33).Thecontourscorrespond
tothe68.3%(cid:126)(cid:54)(cid:115)(cid:33),95.4%(cid:126)(cid:54)2(cid:115)(cid:33),and99.73%(cid:126)(cid:54)3(cid:115)(cid:33)confidenceregions.Fortheasymmetricshells,Rrepresentstheaveragevalueshelldistance.
from the MD ones failed to provide a definite description of (cid:126)1(cid:33) Theimportanceofusingrealisticg(r) modelsasastart-
the studied system. ingpointfortheEXAFSanalysisofdisorderedsystems,
has been shown. This approach becomes essential when
VI. CONCLUSIONS complex disordered systems are considered.
The results presented in this paper represent a step for- (cid:126)2(cid:33) It has been shown that the inclusion of the hydrogen
ward in the EXAFS analysis of disordered systems. Some signal is essential to perform a reliable analysis of the
outstanding results can be deduced. studied system.
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1788 D’Angeloetal.:EXAFSofBr(cid:50)inmethanol
FIG. 10. Correlation maps of the N N couples for asymmetric and Gaussian shells minimizations (cid:126)left and right column, respectively(cid:33). The contours
c c
correspondtothe68.3%(cid:126)(cid:54)(cid:115)(cid:33),95.4%(cid:126)(cid:54)2(cid:115)(cid:33),and99.73%(cid:126)(cid:54)3(cid:115)(cid:33)confidenceregions.
(cid:126)3(cid:33) A proper method, based on well-established statistical andasymmetricpeakanalysishasbeenaccomplished.In
concepts,51hasbeenusedtocalculatestatisticalerrorsof particular, for the first time the most significant correla-
the fitting parameters. Due to the strong correlations be- tion maps obtained from the asymmetric peak and the
tween some parameters the necessity to use correlation Gaussiananalysishavebeencompared.Thisanalysisal-
maps in the EXAFS error evaluation has been pointed lowed the limit of the Gaussian approximation to be as-
out. sessed; in the case of disordered multicomponent sys-
(cid:126)4(cid:33) Acomparison of the results obtained from the Gaussian tems, the accuracy of the EXAFS data analysis is
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