Interplay of computer simulations and x-ray absorption spectra in the study of the bromide hydration structure

Merkling, Patrick J.; Ayala Espinar, Regla; Martínez Fernández, José Manuel; Rodríguez Pappalardo, Rafael; Sánchez Marcos, Enrique

doi:10.1063/1.1603719

Interplay of computer simulations and x-ray absorption spectra in the study of the bromide hydration structure

Merkling, Patrick J.; Ayala Espinar, Regla; Martínez Fernández, José Manuel; Rodríguez Pappalardo, Rafael; Sánchez Marcos, Enrique

Published: 2003
Publisher: American Institute of Physics Publising LLC
DOI: 10.1063/1.1603719
Repository: idus.us.es
Language: en

Abstract

X-ray absorption spectra (EXAFS and XANES) were generated from snapshots of a Monte Carlo (MC) simulation of a bromide ion aqueous solution and from model structures. The MC simulation relies on a recently developed and tested polarizable potential based on ab initio potential energy surfaces. A comparison with the experimental K-edge Br spectrum of a 0.3 M YBr3 aqueous solution was performed. XANES spectra are reproduced acceptably only if statistical fluctuations are included, which is performed in this work by using snapshots from computer simulation. As expected, single scattering BrO contributions are dominant in the case of the EXAFS region. Due to this fact, Br- in water is a good model system for studying the influence of the distribution of distances on the determination of structural parameters. Then, a parallel study of the data analysis procedure of the experimental EXAFS spectrum and those theoretically computed from the structures supplied by the MC simulation, was carried out. The shape of the distribution function and its asymmetry must be taken into account in a practical way to obtain a more accurate determination of the BrO first-shell distance. A further refinement consists in using the computer simulation to extrapolate the BrO distance from the experimental EXAFS spectrum. In this way, a BrO distance of 3.44±0.07 Å and a coordination number of 6±0.5 were determined

Full text

I nt e r p lay of c o m put e r si m u la t i on s a nd x - r ay a b s o r pt i on s p ec t r a i n th e s tud y of th e b r o m i d e h y d r a t i on s t r u c tu r e P at rick J . M e rkli ng, R eg l a A y a l a, J o s e ´M . M a r tı ´ ne z , R afae l R . P appa l a r do, and E n ri que S a ´ n c he z M a rc o s a ) Depa r t amen t o de Qu ı ´m i ca F ı ´ s i ca, Un i ve rs i dad de Sev ill a, 41012-Sev ill a, Spa i n R ece i ved 4 Apr il 2003 ; accep t ed 1 J u l y 2003 X-ray ab s orp ti on s pec t ra EXA FS and XANE S were genera t ed fro m s nap s ho t s of a M on t e C ar l o M C s i m u l a ti on of a bro m i de i on aqueou s s o l u ti on and fro m m ode l s t ruc t ure s . The M C s i m u l a ti on re li e s on a recen tl y deve l oped and t e s t ed po l ar i zab l e po t en ti a l ba s ed on ab i n iti opo t en ti a l energy s urface s . A co m par i s on w it h t he exper i m en t a l K -edge B r s pec t ru m of a 0.3 M Y B r3aqueou s s o l u ti on wa s perfor m ed. XANE S s pec t ra are reproduced accep t ab l y on l y i f s t a ti s ti ca l ﬂ uc t ua ti on s are i nc l uded, wh i ch i s perfor m ed i n t h i s work by u s i ng s nap s ho t s fro m co m pu t er s i m u l a ti on. A s expec t ed, s i ng l e s ca tt er i ng B rO con t r i bu ti on s are do m i nan t i n t he ca s e of t he EXA FS reg i on. Due t o t h i s fac t , B r i n wa t er i s a good m ode l s y s t e m for s t udy i ng t he i n ﬂ uence of t he d i s t r i bu ti on of d i s t ance s on t he de t er m i na ti on of s t ruc t ura l para m e t er s . Then, a para ll e l s t udy of t he da t a ana l y s i s procedure of t he exper i m en t a l EXA FS s pec t ru m and t ho s e t heore ti ca ll y co m pu t ed fro m t he s t ruc t ure s s upp li ed by t he M C s i m u l a ti on, wa s carr i ed ou t . The s hape of t he d i s t r i bu ti on func ti on and it s a s y mm e t ry m u s t be t aken i n t o accoun t i n a prac ti ca l way t o ob t a i n a m ore accura t e de t er m i na ti on of t he B rO ﬁ r s t - s he ll d i s t ance. A fur t her re ﬁ ne m en t con s i s t s i n u s i ng t he co m pu t er s i m u l a ti on t o ex t rapo l a t e t he B rO d i s t ance fro m t he exper i m en t a l EXA FS s pec t ru m . In t h i s way, a B rO d i s t ance of 3.44 0.07 Å and a coord i na ti on nu m ber of 6 0.5 were de t er m i ned. © 2003 Ame r i can In s tit u t e o f Phy s i c s .DOI : 10.1063 / 1.1603719 I. I NTR O DUCT I O N The s t ruc t ure of i on i c s o l u ti on s ha s been a m a tt er of i ncrea s i ng i n t ere s t fro m exper i m en t a l and t heore ti ca l po i n t s of v i ew.1–3 The m a i n exper i m en t a l t echn i que s for s t ruc t ura l s t ud i e s of s uch s y s t e ms are x-ray d i ffrac ti on X R D , neu t ron d i ffrac ti on ND , and x-ray ab s orp ti on s pec t ro s copy XA S . A lt hough X R D and ND can y i e l d s t ruc t ura l i nfor m a ti on abou t d i s ordered s y s t e ms li ke m o s t li qu i d s , t he i r accuracy decrea s e s s t rong l y i n t he ca s e of m u lti co m ponen t s y s t e ms X R D or requ i re s con s i derab l e s yn t he ti c effor t s t o perfor m t he i s o t op i c s ub s tit u ti on s ND .4,5 Th i s fac t re s t r i c t s t he i r u s e, a s doe s t he need for re l a ti ve l y h i gh concen t ra ti on s . Si nce t he 1980 s , one of t he XA S t echn i que s , ex t ended x-ray ab s orpti on ﬁ ne s t ruc t ure EXA FS , ha s been app li ed t o de t er m i ne t he p s eudopar ti a l pa i r corre l a ti on func ti on s of a t o ms i n bo t h cry s t a lli ne and a m orphou s s y s t e ms .6 B ecau s e XA S i s e l e m en t - s pec iﬁ c, t he l oca l s t ruc t ure around one t ype of a t o m can be ob t a i ned by EXA FS . Add iti ona ll y, it can be app li ed t o a l arge range of concen t ra ti on s , rang i ng t yp i ca ll y fro m a few m illi m o l ar t o s evera l m o l ar. X-ray ab s orp ti on s pec t ro s copy ha s t rad iti ona ll y been s ubd i v i ded, accord i ng t o t wo energy reg i on s of a s pec t ru m : x-ray ab s orp ti on near-edge s t ruc t ure XANE S , wh i ch corres pond s t o t he near-edge reg i on t ha t ex t end s fro m t he t hre s h- o l d t o abou t 50 eV above t he edge, and EXA FS , wh i ch ext end s fro m energ i e s of abou t 50 t o m ore t han 1000 eV above t he edge. Si nce t he k i ne ti c energy of t he pho t oe l ec t ron i s l ow i n t he XANE S reg i on, m u lti p l e s ca tt er i ng M S effec t s are do m i nan t , t ha t i n t he par ti cu l ar ca s e of i on i c s o l u ti on s are m o s tl y due t o s ca tt er i ng pa t hway s be t ween a t o ms of t he ﬁ r s t and s econd s o l va ti on s he ll s around t he ab s orber a t o m . In t he XANE S reg i on, bo t h geo m e t r i ca l and e l ec t ron i c de t a il s affec t t he s hape of a s pec t ru m , and t herefore t here i s no genera l , d i rec t , re l a ti on s h i p be t ween s pec t ru m and geo m e t ry a s i n t he ca s e of EXA FS .7Th i s i s why t he t echn i que i s of t en u s ed a s a ﬁ ngerpr i n t for charac t er i z i ng t he ne i ghborhood of t he ab s orber a t o m .8 In t he EXA FS reg i on, t he s ca tt ered pho t oe l ec t ron ha s a h i gh k i ne ti c energy, s o t ha t s i ng l e back s ca tt er i ng proce ss e s are do m i nan t . However, m u lti p l e s ca tt er i ng pheno m ena m ay con t r i bu t e no ti ceab l y t o t he s pec t ru m , m ak i ng it s i n t erpre t ati on m ore d i f ﬁ cu lt . In order t o ex t rac t i nfor m a ti on con t a i ned i n an EXA FS s pec t ru m , a nu m ber of s i m p li fy i ng a ss u m pti on s have t o be m ade. B ear i ng i n m i nd t ha t t he nu m ber and corre l a ti on of var i ab l e s needed i n t he EXA FS ana l y s i s i s of t en s uper i or t o t he i nfor m a ti on con t a i ned i n t he EXA FS s pec t ru m ,7,9–11 t he u s e of i nfor m a ti on der i ved fro m o t her s ource s , li ke co m pu t a ti ona l m e t hod s , i s of s pec i a l i n t ere s t . Th i s procedure ha s proven t o be s ucce ss fu l i n m any s t ud i e s of s t ruc t ura l e l uc i da ti on of s o li d and li qu i d s y s t e ms .12–17 The a i m of t h i s work i s t o ob t a i n t he bro m i de hydra ti on s t ruc t ure by m ean s of t he co m b i na ti on of exper i m en t a l i nform a ti on der i ved fro m XA S t echn i que s and t heore ti ca l da t a fro m co m pu t er s i m u l a ti on s . When dea li ng w it h t he s t ruc t ura l de s cr i p ti on of i on i c s o l u ti on s , t he pa tt ern s for t he ca ti on and an i on hydra ti on are far fro m s i m il ar, m o s tl y due t o t wo reas on s . Fi r s t of a ll , an i on env i ron m en t i s m a i n l y de t er m i ned by t he re s u lt an t force of t he co m pe titi on be t ween i on–d i po l e i naE l ec t ron i c m a il: s anchez@ s i m u l ux.u s .e s J OU R NAL O F C HE M I C AL P HY S I C S VOLU M E 119, NU M B E R 13 1 O C TO B E R 2003 66470021 - 9606/2003/119 ( 13 ) /6647/8/$20.00 © 2003 A m e ric an In s t i tute of P h ysics R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 t erac ti on s , wh i ch t end t o favor s y mm e t r i c hydrogen a t o ms arrange m en t s around t he an i on-oxygen ax i s , and hydrogen bond i ng t o t he an i on, wh i ch favor s an a s y mm e t r i c arrangem en t of it s s urround i ng wa t er m o l ecu l e s t o m ax i m i ze t he hydrogen bond i n t erac ti on. The s econd rea s on i s t ha t t he e l ec t ron den s it y d i s t r i bu ti on i s m ore expanded around t he nuc l eu s i n t he an i on ca s e wha t re s u lt s i n m ore i m por t an t ﬁ r s t - and s econd-order per t urba ti on s w it h re s pec t t o t he l oca l e l ec t r i c ﬁ e l d s . The s e t wo rea s on s an ti c i pa t e no ti ceab l e d i fference s i n t he s o l ven t organ i za ti on around ca ti on s and an i on s , i n par ti cu l ar of m onoa t o m i c or s i m p l e i on s . In add iti on, t he weak bro m i de–wa t er i n t erac ti on s i m p l y ex t re m e l y fa s t s o l ven t -exchange ra t e s of t he ﬁ r s t s o l va ti on s he ll . A ll t he s e rea s on s m ake t he XA S s t udy of t he bro m i de hydra ti on qu it e d i fferen t fro m t he s t udy of h i gh l y charged ca ti on s prev i ou s l y carr i ed ou t by our group.18–25 Obv i ou s l y, t he s en s iti v it y of t he EXA FS t echn i que t o t he d i s order m ake s t he s t ruc t ura l ana l y s i s of t h i s s y s t e m m ore d i f ﬁ cu lt . Thu s , t he u s e of t heore ti ca l i nfor m a ti on i s expec t ed t o s hed li gh t on it . S evera l au t hor s have prev i ou s l y exa m i ned t he bro m i de hydra ti on w it h t he he l p of t heore ti ca l t oo l s . Wa ll en e t a l .26 u s ed m o l ecu l ar dyna m i c s s i m u l a ti on s t o genera t e t heore ti ca l EXA FS s pec t ra w it h t he progra m F E FF and t he evo l u ti on a s a func ti on of pre ss ure and t e m pera t ure of EXA FS and XANE S s pec t ra wa s ana l yzed. D’Ange l oe t a l .27 s t ud i ed a l s o EXA FS s pec t ra w it h t he progra m GNXA S , and co m b i ned M D s i m u l ati on re s u lt s w it h a s hor t range m ode l d i s t r i bu ti on ba s ed on Ga mm a func ti on s t o ﬁt t he EXA FS s pec t ru m . However, none of t he m ca l cu l a t ed t he bro m i de XANE S s pec t ru m . One of t he m o s t co mm on a ss u m p ti on s m ade t o ca l cu l a t e XA S s pec t ra i s t o reduce t he s y s t e m t o a s i ng l e s t ruc t ure. Never t he l e ss , t he con t r i bu ti on fro m m o l ecu l e s and arrangem en t s i n s t an t aneou s l y d i s t or t ed ha s an i m por t an t effec t on t he average s pec t ru m ,22,25 e s pec i a ll y i n l ab il e s t ruc t ure s li ke t he bro m i de i on hydra ti on. In t h i s s en s e, t he po ss i b ilit y of con s i der i ng s t a ti s ti ca l i nfor m a ti on m ake s t he ana l y s i s procedure ea s i er. A recen tl y deve l oped bro m i de–wa t er i n t erac ti on po t en ti a l , wh i ch i nc l uded t he po l ar i zab ilit y of par ti c l e s ba s ed on ab i n iti oda t a and who s e good behav i or ha s recen tl y been checked by co m pu t er s i m u l a ti on s ,28 w ill be e m p l oyed i n t h i s co m b i ned exper i m en t a l – t heore ti ca l s t udy. I t i s wor t h po i n t - i ng ou t t ha t t he t heore ti ca l s pec t ra are co m pu t ed fro m ﬁ r s t pr i nc i p l e s , s i nce no e m p i r i ca l para m e t er s were i n t roduced ne it her i n t he i n t er m o l ecu l ar po t en ti a l s , nor i n t he co m pu t ati on of t he XA S s pec t ra. The XANE S s pec t ru m w ill be co m - pu t ed fro m a repre s en t a ti ve s e t of geo m e t r i e s ex t rac t ed fro m t he s t a ti s ti ca l t ra j ec t ory. W it h re s pec t t o t he EXA FS ana l y s i s , on one s i de t he M D po t en ti a l s ha ll be probed, on t he o t her s i de, de t a il s of t he s t ruc t ura l de t er m i na ti on w ill be i nve s ti - ga t ed. II. M E TH O D O L OG Y Theore ti ca l XANE S and EXA FS s pec t ra of t he B r K -edge 13 474 eV are ca l cu l a t ed by a co m b i na ti on of s t ati s ti ca l and quan t u m - m echan i ca l ca l cu l a ti on s . The s t udy of t he bro m i de hydra ti on wa s perfor m ed u s i ng t hree approache s : i co m pu t a ti on of XANE S s t ruc t ure s fro m quan t u m - m echan i ca ll y m i n i m i zed c l u s t er s w it hnwa t er m o l - ecu l e s (4 n7) ; ii co m pu t a ti on of XANE S s pec t ra t ak- i ng i n t o accoun t t he s t a ti s ti c s t hrough M on t e C ar l o s i m u l ati on s nap s ho t s ; iii ana l y s i s of EXA FS s pec t ra i n t wo way s : on t he one hand, t he exper i m en t a l s pec t ru m wa s co m pared t o a t heore ti ca ll y co m pu t ed one fro m M on t e C ar l o s i m u l a ti on s nap s ho t s ; on t he o t her hand, t he ﬁtti ng procedure of t he EXA FS s pec t ru m wa s carr i ed ou t on bo t h s pec t ra, t he exper i m en t a l one and t he one co m pu t ed fro m M on t e C ar l o s i m u l a ti on s . The co m par i s on of t he ﬁt s wa s carr i ed ou t w it h t he a i m of t e s ti ng t he va li d it y of t he m ode l s t ruc t ure s u s ed t o de s cr i be t he bro m i de hydra ti on. The exper i m en t a l B r K -edge XA S s pec t ra u s ed i n t h i s work for 0.3 M aqueou s s o l u ti on s of Y B r3were m ea s ured a t bea m li ne B L12 C of t he P ho t on F ac t ory i n T s ukuba, J apan. The r i ng curren t wa s 300 m A, and t he r i ng energy 2.5 GeV. A h i gh re s o l u ti on Si 311 channe l -cu t m onochro m a t or wa s u s ed. C a li bra ti on wa s carr i ed ou t w it h t he L 2-edge of a Au fo il 13734 eV . Da t a were co ll ec t ed i n t ran sm i ss i on m ode u s i ng i on i za ti on cha m ber s a s de t ec t or s . The quan t u m - m echan i ca ll y op ti m i zed s t ruc t ure s u s ed for ob t a i n i ng XANE S s pec t ra were t aken fro m a prev i ou s s t udy on t he m i cro s o l va ti on of t he bro m i de i on i n wa t er, m e t hano l , and ace t on it r il e.29 In t h i s s t udy, B r(H2O)nc l u s t er s (1 n8) were op ti m i zed a t t he B 3LY P l eve l w it h doub l eba s i s s e t s aug m en t ed w it h po l ar i za ti on and d i ffu s e func ti on s . St a ti s ti ca l i nfor m a ti on wa s ob t a i ned fro m M on t e C ar l o M C s i m u l a ti on s u s i ng ﬁ r s t -pr i nc i p l e s bro m i de–wa t er i n t erac ti on po t en ti a l s recen tl y deve l oped by our group.28 F or t h i s po t en ti a l , t he M ob il e C harge Den s it y i n Har m on i c O s c ill at or s M ode l wa s e m p l oyed t o de s cr i be bro m i de an i on and wa t er s pec i e s , bo t h of t he m be i ng po l ar i zab l e, and t he wa t er ﬂ ex i b l e. The wa t er–wa t er i n t erac ti on s were de s cr i bed by a M C DHO po t en ti a l prev i ou s l y de s cr i bed by t he C uernavaca group, M ex i co.30 The re li ab ilit y of t he re s u lt s der i ved fro m nu m er i ca l s i m u l a ti on s depend s cr iti ca ll y on t he po t en ti a l m ode l de s cr i b i ng t he i n t erac ti on s a m ong t he s it e s de ﬁ n i ng t he s y s t e m . The i n t erac ti on po t en ti a l u s ed ha s a l ready been t e s t ed i n prev i ou s s t ud i e s ,28 where t he s i m u l a t ed s y s t e ms s how a good agree m en t w it h exper i m en t a l e s ti m a ti on s . The M C s i m u l a ti on s were carr i ed ou t w it h t he M C HANG progra m deve l oped by t he group of C uernavaca.31 The s y s t e m s i m ul a t ed con s i s t s of 1 B r 211 wa t er m o l ecu l e s i n t he NVT en s e m b l e a t 300 K u s i ng t he P OT–2 i n t erac ti on po t en ti a l fro m R ef. 28. P er i od i c boundary cond iti on s were app li ed, and l ong-range i n t erac ti on s were co m pu t ed by t he Ewa l d s u m m e t hod, i nc l ud i ng a t er m t o accoun t for t he ne t charge of t he s y s t e m .32,33 F ur t her de t a il s abou t t he M C s i m u l a ti on s are g i ven i n R ef. 28. K -Edge XANE S s pec t ra of B r i n wa t er were ca l cu l a t ed fro m 40 s nap s ho t s t aken every 120 K (K 103) con ﬁ gurati on s of a M C s i m u l a ti on. The s i m u l a t ed XANE S s pec t ru m i s an average over t he s pec t ra of t he s nap s ho t s . XA S s pec t ra have been co m pu t ed w it h t he F E FF 8.10 progra m code34 deve l - oped by R ehr’ s group. The po t en ti a l ca l cu l a ti on s u s e t he Hed i n–Lundqv i s t s e l f-energy approx i m a ti on.35 A s e l fcon s i s t en t ﬁ e l d procedure S C F wa s e m p l oyed. A 6.0 Å cu t - off rad i u s wa s cho s en for t he S C F t rea t m en t and for t he fu ll - m u lti p l e s ca tt er i ng pa t h s , en s ur i ng t he i nc l u s i on of t he s econd s o l va ti on s he ll i n t he co m pu t a ti on of t he s pec t ra. A m uf ﬁ nti n po t en ti a l wa s app li ed w it h over l ap cr it er i a of 1.0 6648 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 M e rkli ng eta l . R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 t ouch i ng s phere s , 1.15 and 1.15 for B r, O, and H a t o ms , re s pec ti ve l y. The s e cr it er i a were de t er m i ned ad hoc bu t cou l d be j u s tiﬁ ed w it h t he i on i c na t ure of t he ab s orb i ng a t o m . I t wa s found t ha t t he S C F rad i u s cou l d be l owered t o 5.0 Å w it hou t affec ti ng t he re s u lt s . A XANE S s pec t ru m ca l cu l a t ed fro m 200 s nap s ho t s t aken every 250 K con ﬁ gura ti on s and S C F cu t off of 5.0 Å y i e l d s very s i m il ar re s u lt s . In prev i ou s s t ud i e s on ca ti on hydra ti on s t ruc t ure,12,22,24,25 t he i nc l u s i on of hydrogen a t o ms proved nece ss ary for ob t a i n i ng a rea li s ti c back s ca tt er i ng po t en ti a l bu t l ed t o a s en s i b l e degrada ti on of t he s pec t ru m when t hey were i nc l uded a s back s ca tt erer s . D i f ﬁ cu lti e s are m a i n l y a ss oc i a t ed w it h an overe s ti m a ti on of t he back s ca tt er i ng con t r i buti on, bu t t h i s doe s no t affec t t he i r ro l e i n t he s e l f-con s i s t en t ﬁ e l d ca l cu l a ti on of t he po t en ti a l .16 In t he ca s e of bro m i de hydra ti on s t ruc t ure, our re s u lt s u s i ng t he F E FF progra m i nd i - ca t ed t ha t t he s i m u l a t ed s pec t ra are m ore s i m il ar t o t he exper i m en t a l one s bo t h EXA FS and XANE S when hydrogen a t o ms are i nc l uded a s back s ca tt erer s . O t her au t hor s , who do no t u s e t he s a m e m e t hodo l og i ca l approach t o co m pu t e t he EXA FS s pec t ra, work w it h t he a ss u m p ti on of u s i ng hydrogen a t o ms a s back s ca tt erer s i n bo t h an i on and ca ti on s t ud i e s .17 The d i fferen t s t ra t egy hand li ng ca ti on and an i on hydra ti on, apar t fro m it s heur i s ti ca l j u s tiﬁ ca ti on a s s een i n Fi g s . 3 and 4, cou l d be under s t ood by t he pre s ence of a c l o s e s t hydrogen s he ll around t he an i on. The bro m i de i on i nduce s t he pre s ence of a s e t of hydrogen a t o ms of ﬁ r s t - s he ll wa t er m o l ecu l e s i n it s c l o s e s t v i c i n it y, wh i ch are found t o p l ay a s i gn iﬁ can t ro l e i n bo t h t he pho t oe l ec t ron s ca tt er i ng pheno m enon and t he de t er m i na ti on of e l ec t ron i c wave functi on s of t he bro m i de c l u s t er s . A s i m p liﬁ ed expre ss i on of t he EXA FS func ti on i s g i ven by6 k j N j kR j 2S0 2k F j k e 2 j 2k2e2R j / (k) s i n 2kR j j k, 1 where (k) i s t he t o t a l a m p lit ude a s a func ti on of t he vec t or kand t he s u mm a ti on goe s over a ll pa t h s j . N j i s t he coord i - na ti on nu m ber, R j i s ha l f of t he pa t h l eng t h, S0 2(k) i s t he a m p lit ude reduc ti on fac t or, F j (k) i s t he back s ca tt er i ng a m - p lit ude, j 2 i s t he s econd-order cu m u l an t known a s t he Debye–Wa ll er DW fac t or, (k) t he m ean free pa t h of t he pho t oe l ec t ron, and j (k) t he pha s es h i f t . In Eq. 1 , t he d i s order i s de s cr i bed by t he DW fac t or. None t he l e ss , when t heore ti ca l i nfor m a ti on der i ved fro m num er i ca l s i m u l a ti on s i s con s i dered, s t ruc t ura l ﬂ uc t ua ti on s around t he an i on are i m p li c it w it h i n t he s a m p li ng of t he conﬁ gura ti ona l s pace. Then, t he EXA FS func ti on can be a lt erna ti ve l y expre ss ed a s 22 k1 N s i s t ruc j pa t h s N j kR ij 2S0 2k F j k e 2R i j / (k) s i n 2kR i j j k, 2 where N s i s t he nu m ber of s t ruc t ure s con s i dered i n t h i s ca s e 10 000 s t ruc t ure s even l y s paced over 50 M ( M 106) M C con ﬁ gura ti on s , i goe s over t he s t ruc t ure s der i ved fro m t he s t a ti s ti ca l s a m p li ng, and j over t he pa t h s . The ro l e p l ayed by t he exponen ti a l con t a i n i ng t he DW fac t or of Eq. 1 i s rep l aced by an average over s t ruc t ure s i n Eq. 2 . In order t o check t he convergence of t he re s u lt s , a co m pu t a ti on w it h 40000 s t ruc t ure s even l y s paced over 200 M con ﬁ gura ti on s wa s a l s o perfor m ed. No d i fference s were found be t ween bo t h averaged s pec t ra. F ur t her exa m p l e s of t h i s averag i ng over s i m u l a t ed s pec t ra m ay be found i n R ef s . 12, 15, 26, 36, and 37. The cu t -off rad ii s e l ec t ed were t he s a m e a s t ho s e cho s en i n t he XANE S s pec t ra. The con s i dered pa t h s were re s t r i c t ed t o a l eng t h of up t o 7.0 Å and an a m p lit ude of a t l ea s t 9% of t he i n t en s it y of t he s hor t e s t B rH pa t h. St ruc t ura l para m e t er s were ob t a i ned fro m bo t h exper i - m en t a l and t heore ti ca l s pec t ra by m ean s of ﬁtti ng a m ode l s t ruc t ure t o Eq. 1 . C on t ac t i on pa i r s cou l d be exc l uded, a s appeared a s we ll fro m t he ana l y s i s of t he Y K -edge.38 Thu s , t he m ode l s t ruc t ure cho s en i s t he hexahydra t ed c l u s t er cdep i c t ed i n Fi g. 1. EXA FS da t a were ana l yzed w it h t he F E FF IT 2.55 s of t ware package.39 The ﬁtti ng of t he exper i m en t a l EXA FS s i gna l t o Eq. 1 i s ba s ed on a s t andard non li near l ea s t s quare s t echn i que we ll -e s t ab li s hed i n t he lit era t ure.40 III. R ES ULT S AND D I S CU SS I O N A . X AN ES s p ec t r a C a l cu l a t ed XANE S s pec t ra fro m a s er i e s of quan t u m - m echan i ca ll y m i n i m i zed s t ruc t ure s con t a i n i ng be t ween 4 and 7 wa t er m o l ecu l e s are d i s p l ayed i n Fi g. 2 a and co m pared w it h t he exper i m en t a l s pec t ru m . The geo m e t r i e s of t he s t ruct ure s s e l ec t ed are s hown i n Fi g. 1. The ca l cu l a t ed XANE S s pec t ra s how no ti ceab l e d i fference s , e s pec i a ll y w it h re s pec t t o t he re l a ti ve i n t en s it y of t he re s onance s . To t he con t rary, t he energy va l ue s of t he ﬁ r s t t wo re s onance s wh i ch appear a t 13 472 and 13 489 eV, re s pec ti ve l y, are reproduced rea s onab l y we ll by a ll of t he m and are i n agree m en t w it h t he exper i m en t a l s pec t ru m . The co m par i s on w it h t he exper i m en t a l s pec t ru m s how s t ha t t he ra ti o of t he he i gh t s of t he re s onance s i s no t reproduced. Wha t i s m ore, t he m a i n re s onance of t he exper i m en t a l s pec t ru m i s w i der and h i gher t han t he one s of t he co m pu t ed s pec t ra. XANE S s pec t ra are genera ll y con s i dered t o be i n s en s i - ti ve t o d i s order, bu t charac t er i s ti c of t he coord i na ti on geo m - e t ry. In t he ca s e of bro m i de hydra ti on, where t he hydra ti on s t ruc t ure i s no t we ll de ﬁ ned, t he s en s iti vene ss of t he XANE S s pec t ru m t o t he i n s t an t aneou s geo m e t r i e s can be ill u s t ra t ed on t hree s t ruc t ure s p i cked ou t of t he 40 con s i dered t o ca l cu- F IG. 1. Geo m e t r i e s of t he quan t u m - m echan i ca ll y op ti m i zed s t ruc t ure s u s ed t o ca l cu l a t e XANE S s pec t ra of Fi g. 2 a . 6649 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 B r o mi de h y d r at i on s t r u c tu r e R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 l a t e t he average s t a ti s ti ca l s pec t ru m Fi g. 2 b . On t he one hand, and un li ke t he ca s e of t he quan t u m - m echan i ca ll y op ti - m i zed c l u s t er s , s o m e of t he i nd i v i dua l s pec t ra exh i b it m ore i n t en s e re s onance s t han t he exper i m en t a l one. On t he o t her hand, t he i nd i v i dua l XANE S s pec t ra d i ffer m ore be t ween each o t her, t han t ho s e co m pu t ed fro m quan t u m - m echan i ca l s t ruc t ure s . Th i s s en s iti v it y of t he XANE S s pec t ra t o geom e t r i ca l change s fro m t he ﬂ uc t ua ti on s of t he s y s t e m s uppor t s s t rong l y t he need t o i nc l ude s t a ti s ti ca l i nfor m a ti on i n t he s i m u l a ti on of XANE S s pec t ra of bro m i de s o l u ti on s . If a un i que m ode l s t ruc t ure had been t aken i n t o con s i dera ti on, t he d i s crepanc i e s be t ween t heore ti ca l and exper i m en t a l s pect ra wou l d have been a s cr i bed t o t he approx i m a ti on s con s i dered i n t he quan t u m co m pu t a ti on s needed t o genera t e t he s pec t ru m , and even t ua ll y, t he s e m i e m p i r i ca l approx i m a ti on s i n t roduced i n ob t a i n i ng s o m e of t he para m e t er s . In t he s a m e ﬁ gure, t he co m pu t ed s pec t ra u s i ng t he s t ruc t ure s t runca t ed t o t he ﬁ r s t s o l va ti on s he ll are s hown. The i nd i v i dua l s pec t ra are s t rong l y affec t ed by t he t runca ti on, wha t ill u s t ra t e s how XANE S i s s en s iti ve t o t he s o l va ti on beyond t he ﬁ r s t s he ll , a s a l ready ob s erved i n t he ca s e of ca ti on hydra ti on.24,25 In Fi g. 3, t he co m par i s on of t he averaged ca l cu l a t ed s pec t ru m fro m s t a ti s ti ca l i nfor m a ti on and t he exper i m en t a l one i s s hown. B o t h t he po s iti on of t he t wo m a i n re s onance s and t he ra ti o of t he i r he i gh t s agree. A s expec t ed fro m t he qua lit a ti ve ana l y s i s of t he m e t hodo l ogy u s ed, t he w i d t h of t he m a i n re s onance of t he averaged s t a ti s ti ca l s pec t ru m i s m ore s i m il ar t o t he exper i m en t a l one t han t ha t ob t a i ned fro m a un i que s t ruc t ure. Never t he l e ss , t he fac t t ha t t he i n t en s it y of t he m a i n re s onance i ncrea s e s when a grea t nu m ber of s pec t ra der i ved fro m d i fferen t s t ruc t ure s i s i nc l uded, i s s t r i k i ng. In order t o s t udy t he s en s iti v it y of t he XANE S reg i on t o t he ab s orber-back s ca tt erer d i s t ance, t he XANE S s pec t ru m ha s been reca l cu l a t ed con s i der i ng t he s a m e po i n t s of t he conﬁ gura ti ona l s pace, bu t i nc l ud i ng on l y s o l ven t m o l ecu l e s of t he ﬁ r s t s o l va ti on s he ll . Accord i ng t o t he rad i a l d i s t r i bu ti on func ti on of t he M C s i m u l a ti on s ,28 a m o l ecu l e be l ong s t o t he ﬁ r s t s o l va ti on s he ll when t he B rO d i s t ance i s l ower t han 4.1 Å. The averaged s pec t ru m ob t a i ned i s i nc l uded i n Fi g. 3 s hor t da s hed li ne and d i ffer s s li gh tl y fro m t he one con t a i n- i ng m ore t han one hydra ti on s he ll da s hed li ne . Th i s behav- i or i s d i fferen t fro m t he one found by our group i n t he s t udy of ca ti on hydra ti on.22,24,25 In t he l a tt er ca s e, s pec t ra ca l cul a t ed fro m s t ruc t ure s t ha t con t a i ned on l y one s o l va ti on s he ll pre s en t ed a m ore i n t en s e m ax i m u m , and a s li gh t d i s p l acem en t and narrow i ng of t he m a i n re s onance wa s ob s erved. The effec t of t he wa t er m o l ecu l e s beyond t he ﬁ r s t hydra ti on s he ll on t he averaged XANE S s pec t ru m i s sm a ll i n t he ca s e of t he bro m i de hydra ti on, un li ke wha t i s ob s erved when co m pu ti ng t he effec t of h i gher s he ll s for i nd i v i dua l s naps ho t s . Thu s , it doe s no t s ee m po ss i b l e t o re t r i eve non t r i v i a l s t ruc t ura l i nfor m a ti on beyond t he ﬁ r s t hydra ti on s he ll for t h i s an i on i n s t ud i e s of s o l u ti on che m i s t ry. The ca l cu l a t ed XANE S s pec t ru m u s i ng a cu t off rad i u s of 6.0 Å, where hydrogen a t o ms were no t i nc l uded a s back s ca tt erer s , i s a l s o p l o tt ed i n Fi g. 3 do tt ed li ne s . Th i s co m pu t ed s pec t ru m d i ffer s s i gn iﬁ can tl y fro m t he o t her s pec t ra i nc l uded i n t he ﬁ gure. B . EX AF S s p ec t r a The co m par i s on of t he exper i m en t a l and t heore ti ca l EXA FS s pec t ra i n t er ms of (k)k2 i s d i s p l ayed i n Fi g. 4. The agree m en t be t ween bo t h s pec t ra i s s a ti s fac t ory, e s pec i a ll y con s i der i ng t ha t no e m p i r i ca l para m e t er s were i nc l uded i n t he co m pu t a ti on of t he t heore ti ca l s pec t ru m . When H a t o ms are exc l uded a s back s ca tt erer s , t he agree m en t i s found t o be far fro m s a ti s fac t ory, s uppor ti ng t he i m por t ance of hydrogen a t o ms i n t he XA S s pec t ra of t he ha li de aqueou s s o l u ti on s .41–44 A s h i gh li gh t ed i n t he s e prev i ou s work s , t h i s i s F IG. 2. Exper i m en t a l B r K -edge XANE S s o li d li ne and t he one s co m pu t ed fro m quan t u m - m echan i ca ll y op ti m i zed s t ruc t ure s s hown i n Fi g. 1. b Ind i - v i dua l XANE S s pec t ra co m pu t ed for d i fferen t s nap s ho t s , e it her re s t r i c t ed t o ﬁ r s t hydra ti on s he ll da s hed li ne s or no t s o li d li ne s . F IG. 3. Exper i m en t a l B r K -Edge XANE S s o li d li ne and co m pu t ed fro m M C s i m u l a ti on s , e it her u s i ng ﬁ r s t hydra ti on s he ll s hor t da s hed li ne , or up t o 6 Å l ong da s hed li ne . The hypo t he ti ca l ca s e where H a t o ms are no t i nc l uded a s back s ca tt erer s i s s hown i n do tt ed li ne s . 6650 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 M e rkli ng eta l . R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 due t o t he near l y li near a li gn m en t of a t o ms B rHO and a s a con s equence, t he hydrogen a t o m ha s a focu s i ng effec t for t he pho t oe l ec t ron wave. Fi gure 5 a d i s p l ay s t he exper i m en t a l s pec t ru m and it s ﬁt u s i ng a s a m ode l s t ruc t ure c l u s t er ‘‘c’’ of Fi g. 1. In Fi g. 5 b , t he pha s e-correc t ed F our i er t ran s for m F T a l ong w it h it s ﬁt are p l o tt ed. A s u mm ary of t he s t ruc t ura l para m e t er s ob t a i ned i n t he ﬁt procedure i s s hown i n t he ‘‘exper i m en t a l ’’ fra m e of Tab l e I. A s s een i n t he t ab l e, t he va l ue s co m pare we ll w it h exper i m en t a l re s u lt s ob t a i ned by o t her au t hor s . M u lti e l ec t ron exc it a ti on s are expec t ed around 5, 7.3, and 8.22 Å 1.38 They can be de t ec t ed exper i m en t a ll y i f t he d i fference s pec t ru m be t ween exper i m en t a l s pec t ru m and ﬁt exh i b it s peak s approx i m a t e l y a t t he i nd i ca t ed k-va l ue s . Th i s i s no t t he ca s e, and t herefore m u lti e l ec t ron exc it a ti on s were no t i nc l uded i n t he ana l y s i s . S o m e au t hor s s t udy i ng bro m i de conc l uded t ha t m u lti e l ec t ron exc it a ti on s were no t i m por t an t ,38,41 wh il e o t her s reached a h i gher e s ti m a t e of m u lti e l ec t ron exc it a ti on s and reco mm ended t he i r i nc l u s i on i n t he da t a ana l y s i s .45,46 The fac ilit y t o hand l e i ndependen t s t ruc t ura l i nfor m a ti on of s t a ti s ti ca l na t ure on t he s y s t e m under s t udy enab l e s u s t o under t ake a s econd EXA FS da t a ana l y s i s wh i ch corre s pond s t o t he co m pu t ed EXA FS s pec t ru m fro m t he M C s nap s ho t s da s hed li ne i n Fi g. 4 . Th i s t ype of s t ra t egy ha s a l ready been perfor m ed i n prev i ou s work s dea li ng w it h ca ti on hydra ti on.22,25 A t ﬁ r s t s i gh t , t h i s cou l d be env i s aged a s an unna t ura l procedure t o ob t a i n t he s t ruc t ura l i nfor m a ti on, a s t he d i rec t exa m i na ti on of d i s t r i bu ti on func ti on s der i ved fro m M C s ee ms t o offer a m ore d i rec t way t o do it . However, t he a p r i o r i know l edge of t he s t ruc t ura l re s u lt s g i ve s u s an i dea l fra m ework t o check t he u s ua l da t a ana l y s i s procedure t o ext rac t t h i s i nfor m a ti on fro m an EXA FS s pec t ru m . Fi gure s 5 a and 5 b s how t he co m par i s on be t ween t he t heore ti ca ll y co m pu t ed s pec t ru m and it s ﬁt . The ﬁtt ed s pec t ru m and it s F T are i n qu it e good agree m en t w it h t he i r corre s pond i ng t heore ti ca l s pec t ra. F or t he purpo s e of t he ﬁt , s i ng l e s ca tt er i ng pa t h s B rO and B rH and t he m u lti p l e s ca tt er i ng pa t h s B rHO and B rHOH were con s i dered. To reduce t he nu m ber of ﬁt para m e t er s , a ll DW fac t or s for t he B rO and B rH pa t h s were a ss u m ed t o be i den ti ca l , a s s ugge s t ed by t he s i m u l a ti on va l - ue s ( B rO 20.055 i n Tab l e I and B rH 20.059). In t he ca s e of t he t heore ti ca l s pec t ru m , t he coord i na ti on nu m ber wa s ﬁ xed t o t he m ean va l ue of t he s i m u l a ti on. S0 2, t he a m p lit ude reducti on fac t or, wa s s e t t o 1. The ‘‘ t heore ti ca l ’’ va l ue s g i ven i n Tab l e I are a ll ob t a i ned fro m t he M on t e C ar l o s i m u l a ti on re s u lt s . The ‘‘ M C s i m u l ati on’’ li ne g i ve s d i rec t re s u lt s co m pu t ed fro m t he s t a ti s ti ca l s i m u l a ti on. ‘‘ Fit t o t heore ti ca l s pec t ru m ’’ genera t e s an EXA FS s pec t ru m fro m t he s i m u l a ti on s nap s ho t s and ﬁt s t h i s s pec t ru m , u s i ng or no t t he t h i rd order cu m u l an t . The s e t wo ﬁt s run para ll e l t o t he ‘‘ Fit t o exper i m en t a l s pec t ru m ,’’ t he m a i n d i fference be i ng t ha t t he EXA FS func ti on ana l yzed i s t ha t ob t a i ned fro m t he s nap s ho t s of t he M C s i m u l a ti on. In t he t ab l e, t he re s u lt s of t wo ﬁt s of t he t heore ti ca l s pec t ru m can be co m pared : t he ﬁ r s t con s i der s on l y cu m ul an t s up t o s econd order, t he s econd i nc l ude s t he t h i rd order cu m u l an t , wh i ch i s con s t ra i ned w it h t he s econd order cu m ul an t va l ue by ﬁ x i ng t he s kewne ss ra ti o3 / (2)3 / 2] t o a va l ue of 0.37. Th i s nu m ber ar i s e s fro m t he s i m u l a ti on a s exp l a i ned be l ow. The d i fference be t ween t he ﬁtt ed B rO d i s - t ance s are w it h i n 0.1 Å due t o t he i n t roduc ti on of t he t h i rd order cu m u l an t . The s a m e procedure perfor m ed on t he exper i m en t a l s pec t ru m y i e l d s s i m il ar conc l u s i on s . C o m parab l e ﬁ nd i ng s were ob t a i ned by D’Ange l oe t a l . i n a s t udy of brom i de i n m e t hano l .42 The d i s t ance i n t he ﬁt t o t he co m pu t ed s pec t ru m i s 0.1 Å F IG. 4. C o m par i s on of t he exper i m en t a l s o li d li ne and co m pu t ed l ong da s hed li ne k2-we i gh t ed EXA FS func ti on. When H a t o ms are no t i nc l uded a s back s ca tt erer s , t he EXA FS func ti on wou l d be repre s en t ed by t he s hor t da s hed li ne. F IG. 5. a Exper i m en t a l EXA FS s pec t ru m and it s be s t ﬁt w it h t he para m - e t er s i nc l uded i n Tab l e I. b P ha s e-correc t ed F our i er t ran s for m of t he s pect ru m and it s ﬁt . 6651 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 B r o mi de h y d r at i on s t r u c tu r e R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 l arger t han t he corre s pond i ng exper i m en t a l one 3.44 ver s u s 3.34 . On one hand, t h i s d i s crepancy cou l d be a s cr i bed t o t he m o l ecu l ar po t en ti a l m ode l s . On t he o t her hand, t he d i s crepancy cou l d ar i s e fro m t he ﬁtti ng procedure it s e l f a s d i s cu ss ed be l ow. C . An alysis of th e p r o ce du r e to obt ai n th e g e o m e t r ical p a r a m e t e r s f r o m a non r i g i d d is t r i but i on a r ound th e r e f e r e n ce a to m The EXA FS s pec t ru m of t he bro m i de an i on i n wa t er i s a concep t ua ll y i n t ere s ti ng ca s e for m e t hodo l og i ca l s t ud i e s becau s e t he s pec t ru m i s do m i na t ed by t he B rO back s ca tt er i ng con t r i bu ti on. The pre s en t s t udy corrobora t e s t ha t con t r i buti on s fro m m u lti p l e s ca tt er i ng pa t h s excep t ed t he ‘‘focu s - i ng’’ pa t h B rHO and s i ng l e s ca tt er i ng con t r i bu ti on s fro m a t o ms a t d i s t ance s l arger t han 5.0 Å p l ay a m i nor ro l e.26,27 The B rO d i s t r i bu ti on of d i s t ance s i s , however, qu it e broad and a s y mm e t r i c. Therefore, a m ode l s t ruc t ure s hou l d be va li d, i f it wa s ab l e t o reproduce e ss en ti a ll y t he probab ilit y d i s t r i bu ti on func ti on of t he B rO d i s t ance s . In t h i s s ec ti on, t he ana l y s i s w ill be perfor m ed exc l u s i ve l y on t he t heore ti ca ll y co m pu t ed s pec t ru m , s o t ha t exper i m en t a l con s i dera ti on s do no t need t o be t aken i n t o accoun t and t he d i s t r i bu ti on i s known. The a i m i s t o m on it or t horough l y how t he EXA FS de t er m i na ti on can reproduce t he geo m e t r i ca l d i s t r i bu ti on, wha t wa s u s ed rever s e l y t o genera t e t he EXA FS s pec t ru m . A s t r i k i ng fea t ure of t he s t ruc t ura l de t er m i na ti on by EXA FS s pec t ro s copy i s t ha t d i s t ance s are u s ua ll y s li gh tl y s hor t er t han expec t ed, and / or ob t a i ned by o t her t echn i que s .1 S uch a d i fference i s a l s o ob s erved be t ween t he d i s t ance obt a i ned by averag i ng t he M C s i m u l a ti on s nap s ho t s and t he ﬁtt ed d i s t ance t o t he t heore ti ca l s pec t ru m for exa m p l e, R B rO 3.54 ver s u s 3.44 Å . In fac t , t he coord i na ti on d i s t ance ob t a i ned fro m t he EXA FS ﬁt i s nu m er i ca ll y very c l o s e t o t he m ax i m u m of t he g( r ) func ti on s hown for i n s t ance i n R ef. 28. Th i s ob s erva ti on app li e s a l s o t o t he ca s e of t he s econd s he ll of R h3and C r3.24,25 In t he bro m i de s t udy, t he ﬁtt ed DW fac t or s are on l y rough l y ha l f of t he va l ue found i n t he s i m ul a ti on 0.037 ver s u s 0.055 Å2), wh il e t he coord i na ti on nu m - ber i s m a i n t a i ned ﬁ xed. The s y s t e m a ti c undere s ti m a ti on of d i s t ance s even when t he ‘‘correc t ’’ t h i rd order cu m u l an t i s u s ed enab l e s a s li gh t re ﬁ ne m en t of t he e s ti m a t ed exper i m ent a l d i s t ance ba s ed on a t ran s ferab ilit y of t he d i fference obt a i ned fro m t he t heore ti ca l co m pu t a ti on. Th i s s hou l d be warran t ed i f t he s hape of t he d i s t r i bu ti on of t he s i m u l a ti on corre s pond s t o t he rea l one, R t heor RM C Si m u l a ti on R Fit t o t heor w it h3 3.54 3.44 Å 0.1 Å. Then, Rexp– t heor R Fit t o exp w it h3R t heor 3.34 0.1 Å. Therefore, our e s ti m a t e of t he average ﬁ r s t s he ll B rO d i s - t ance fro m t he exper i m en t a l s pec t ru m i s 3.44 0.07 Å. In order t o ﬁ nd ou t m ore abou t t he rea s on s for t h i s d i fference i n t he ﬁ r s t hydra ti on s he ll , s evera l a tt e m p t s were m ade u s i ng on l y t he i nfor m a ti on abou t t he d i s t r i bu ti on i n t h i s s he ll . The s t ruc t ura l i nfor m a ti on u s ed i n a ll t he s e ca l cu l ati on s i s t ha t ob t a i ned fro m t he M C re s u lt s . Tab l e I g i ve s t he i n t egra ti on nu m ber, average B rO d i s t ance and it s DW fac t or (2). F or t he purpo s e of t he ana l y s i s , t he B rO ﬁ r s t peak had t o be i s o l a t ed s o m ewha t arb it rar il y, s i nce t he d i s t r i bu ti on func ti on doe s no t drop t o zero. The cho s en cr it er i u m wa s t o u s e a s w it ch i ng func ti on, cen t ered around t he m i n i m u m of t he d i s t r i bu ti on func ti on, t ha t br i ng s t he d i s t r i bu ti on t o zero arc t angen tl y be t ween 3.8 and 4.2 Å. If a s t r i c t cu t off cr it er i u m i s cho s en i n s t ead, t he qua lit a ti ve conc l u s i on s drawn here are no t affec t ed. The reference d i s t r i bu ti on i n a ll t he s e ca l cu l a ti on s i s g i ven by t he probab ilit y d i s t r i bu ti on func ti on P D F , p l o tt ed i n Fi g. 6 s o li d li ne . Th i s func ti on i s d i rec tl y re l a t ed t o t he rad i a l d i s t r i bu ti on func ti on (g(R)), P D F R B rO N g s wR B rO R B rO 2, 3 where t he s uper s cr i p t ‘‘ s w’’ refer s t o t he s w it ched g(R) and N i s a nor m a li za ti on con s t an t . TA B LE I. B e s t ﬁt d i s t ance s , Debye–Wa ll er fac t or s , and coord i na ti on nu m ber s around t he bro m i de an i on ob t a i ned fro m t he ana l y s i s of t he EXA FS s pec t ra and of t he M C s i m u l a ti on u s i ng 10 000 s nap s ho t s . N OR B rO / Å B rO 2 / Å23 / Å3 Exper i m en t Fit t o exper i m en t a l s pec t ru m 5.84a3.27 0.04 0.035 0.002 Fit t o exper i m en t a l s pec t ru m w it h30.37( B rO 2)3 / 2 5.84a3.34 0.03 0.034 0.002 0.0023 0.0002 R ef. 26 7.2 0.4 3.36 0.01 0.027 0.002 0.0014 0.0006 R ef. 27 6.9 3.34 0.04 0.0065 R ef. 50 6a3.19 0.016 R ef. 48 6a3.20 0.022 0.0034 R ef. 41 5.1 0.5 3.29 0.04 0.026 0.03 0.0020 0.0020 Theory M C s i m u l a ti on 5.84 0.3 3.54 0.01 0.055 0.005 0.0036 0.0002 Fit t o t heore ti ca l s pec t ru m 5.84a3.37 0.03 0.037 0.002 Fit t o t heore ti ca l s pec t ru m w it h30.37( B rO 2)3 / 2 5.84a3.44 0.03 0.036 0.002 0.0025 0.0004 Fit t o 1 s t - s he ll P D F 7.0 0.1 3.50 0.01 0.065 0.001 Fit t o 1 s t - s he ll P D F w it h36.9 0.1 3.54 0.01 0.058 0.001 0.0052 0.0005 a Fi xed va l ue. 6652 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 M e rkli ng eta l . R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 To fo ll ow t he s t ep s of an EXA FS da t a ana l y s i s , t he F our i er t ran s for m of t he P D F wa s u s ed a s i npu t for t he ﬁt of t he fo ll ow i ng expre ss i on : 47 P D F k A exp i R B rOk2k2 / 2 i 3k3 / 6 4 wh i ch y i e l d s ‘‘effec ti ve cu m u l an t s .’’ Tab l e I g i ve s t wo s e t s of re s u lt s , con s i der i ng or no t t he t h i rd order cu m u l an t . ‘‘ Fit t o 1 s t s he ll P D F ’’ u s e s t he B rO d i s t r i bu ti on func ti on and ﬁt s coord i na ti on nu m ber, d i s t ance, and DW fac t or i n a m anner de t a il ed i n t he nex t s ec ti on. ‘‘ Fit t o 1 s t s he ll P D F w it h3’’ add iti ona ll y ﬁt s t he t h i rd order cu m u l an t . The t h i rd order cu m u l an t i s found t o be 0.0052 Å3 s ee Tab l e I , t ha t i s , 0.37( 2)3 / 2. C on s t ra i n i ng t he s kewne ss t o a va l ue of 0.37 i n a ll t he ﬁt s en s ure s t ha t t he d i s t r i bu ti on s found are co m parab l y a s y mm e t r i ca l . Th i rd order cu m u l an t s ﬁ na ll y d i ffer grea tl y be t ween au t hor s and m e t hod s u s ed. A d i rec t ﬁt t o t he EXA FS s pec t ru m i s s ub j ec t t o h i gh uncer t a i n ti e s : Wa ll en e t a l .26 de t er m i ne a nega ti ve va l ue, wh il e Si m one t e t a l .41 ob t a i n a va l ue i n agree m en t w it h t ho s e ob t a i ned by o t her au t hor s u s i ng a lt erna t e approache s .48 D’Ange l oe t a l .27 obt a i n a s kewne ss of 0.81. W it h i n t he i r approach, however, h i gher order cu m u l an t s a l s o p l ay a s i gn iﬁ can t ro l e. In t h i s work, a d i rec t ca l cu l a ti on of 3fro m t he M C s i m u l a ti on y i e l d s 0.0036, t ha t i s a s kewne ss of 0.28, bu t i f we wan t t o neg l ec t h i gher order cu m u l an t s , it i s m ore i n t ere s ti ng t o m ake u s e of an effec ti ve va l ue. When s econd and t h i rd order cu m u l an t s are i nc l uded, t he average d i s t ance de t er m i ned i n t he ﬁt of t he P D F i s equa l t o t he reference one 3.54 Å . The corre s pond i ng d i s t r i bu ti on i s p l o tt ed i n Fi g. 6 do tt ed li ne . When t he ﬁt i nc l ude s on l y s econd order cu m u l an t s , t he d i s t ance de t er m i ned i s 3.50 Å l ong da s hed li ne . Tha t i s , a s hor t en i ng of t he d i s t ance i s ob s erved. Th i s va l ue i s qu it e d i fferen t fro m t he one ob t a i ned by ﬁtti ng t o t he co m pu t ed EXA FS func ti on 3.44 or 3.37 Å, re s pec ti ve l y . Ano t her t e s t of t he m e t hod for t he de t er m i na ti on of t he s t ruc t ure li e s i n t he i nve s ti ga ti on of t he po ss i b l e ar ti fac t a s - s oc i a t ed t o t he genera ti on of t he EXA FS func ti on v i a t he F E FF/F E FF IT m e t hod fro m t he s nap s ho t s of t he M C s i m u l ati on. Two nu m er i ca l exa m i na ti on s have been de s i gned. The ﬁ r s t con s i s t ed i n genera ti ng a t heore ti ca l EXA FS s pec t ru m ar i s i ng s o l e l y fro m s i ng l e s ca tt er i ng con t r i bu ti on s of t he ﬁ r s t s he ll of oxygen a t o ms . The ﬁt i nc l uded on l y B rO s i ng l e s ca t - t er i ng pa t h s and s econd order cu m u l an t s , y i e l d i ng a d i s t ance of 3.36 Å. The s econd exa m i na ti on wa s t he genera ti on of an EXA FS s pec t ru m ba s ed on m ode l s t ruc t ure s of t he geo m e t ry ‘‘c’’ of Fi g. 1, where t he po s iti on of t he wa t er m o l ecu l e s wa s s y mm e t r i ca ll y m od iﬁ ed be t ween 2.8 and 4.2 Å for t he B rO d i s t ance. The s e s pec t ra were we i gh t ed by t he P D F of t he ﬁ r s t peak. Th i s t heore ti ca l s pec t ru m bear s a c l o s e re s e m b l ance t o t he one ob t a i ned by t he averag i ng of s pec t ra fro m M C s naps ho t s , and t he ﬁt para m e t er s are very s i m il ar. Th i s i s a s t rong i nd i ca ti on t ha t t he rea s on for t he d i s crepancy i s no t due t o m u lti p l e s ca tt er i ng effec t s or t he pre s ence of hydrogen a t o ms c l o s e t o t he bro m i de, bu t i s ra t her t o be found i n t he s hape of t he d i s t r i bu ti on and t he i nab ilit y t o ad j u s t it w it h DW fac t or s . I V . C O NCLUD I N G R E MARK S XANE S and EXA FS s pec t ra of B r K -edge of bro m i de i n wa t er were ca l cu l a t ed fro m s i m u l a ti on s nap s ho t s and, i n t he ca s e of XANE S , quan t u m - m echan i ca ll y m i n i m i zed c l u s t er s were a l s o probed. Wherea s t he quan t u m - m echan i ca l c l u s t er s do no t co m pare we ll w it h exper i m en t a l da t a i n s o l u ti on, t he s e t of s i m u l a ti on da t a l ead s t o a good agree m en t . The d i s - crepanc i e s be t ween XANE S s pec t ra ca l cu l a t ed fro m quan t u m - m echan i ca l s t ruc t ure s and M C s i m u l a ti on s s ugge s t t he need for i nc l ud i ng s t a ti s ti ca l i nfor m a ti on i n t he co m put a ti on. In t h i s s en s e, it i s e ss en ti a l t o u s e a s t a ti s ti ca l d i s t r i - bu ti on of wa t er m o l ecu l e s around t he bro m i de an i on der i ved fro m a conden s ed m a tt er s y s t e m where i on–wa t er and wa t er–wa t er i n t erac ti on s p l ay a s i gn iﬁ can t ro l e. B ear i ng i n m i nd t ha t i n t er m o l ecu l ar po t en ti a l s were ﬁtt ed t oab i n iti o ca l cu l a ti on s of s i m il ar l eve l t han t ho s e u s ed for t he s t ruc t ure op ti m i zed quan t u m - m echan i ca ll y, t he m a i n rea s on t o j u s ti fy t he d i fference m u s t be t he s t a ti s ti ca l fac t or s . In t he ca s e of t he EXA FS co m pu t a ti on s of bro m i de hydra ti on, t he m a i n con t r i bu ti on s are due t o s i ng l e s ca tt er i ng fro m t he ﬁ r s t hydra ti on s he ll and, due t o t h i s rea s on, can be hand l ed rea li s ti ca ll y by a one-d i m en s i ona l d i s t r i bu ti on of m ode l s t ruc t ure s . Th i s i s a s i m p liﬁ ed p i c t ure of t he m ore rea li s ti c s ca tt er i ng pheno m enon where B rHO pa t h s and B rH pa t h s are be i ng con s i dered. The broad and a s y mm e t r i c d i s t r i bu ti on of B rO d i s t ance s i n t he s i m u l a ti on wa s de s cr i bed i n t er ms of an a s y mm e t ry para m e t er of 0.37 s kewne ss i n order t o be ab l e t o ob t a i n s econd and t h i rd order cu m u l an t s w it h l ow i nde t er m i na ti on s for t he ﬁt t o t he EXA FS s pec t ru m . Th i s a s y mm e t ry para m - e t er wa s ob t a i ned by ﬁtti ng t he B rO d i s t r i bu ti on of t he s i m ul a ti on i nks pace. The d i s crepanc i e s t ha t appear be t ween t he t heore ti ca l EXA FS s pec t ru m and it s ﬁt u s i ng cu m u l an t s i nd i ca t e t ha t s y s t e ms w it h s i gn iﬁ can t geo m e t r i ca l d i s order i n t he c l o s er env i ron m en t of t he ab s orber a t o m , li ke t he ﬁ r s t hydra ti on s he ll of t he bro m i de an i on, can hard l y be de s cr i bed by cum u l an t s li m it ed t o t h i rd or four t h order.49 Due t o t he charac t er i s ti c s of t he probab ilit y d i s t r i bu ti on func ti on, d i s t ance s ob t a i ned by a conven ti ona l ﬁt procedure t o t he EXA FS s pect ru m are ob s erved t o be c l o s e t o t he m ax i m u m of t he rad i a l d i s t r i bu ti on func ti on when s o l e l y DW fac t or s are con s i dered, F IG. 6. B rO probab ilit y d i s t r i bu ti on func ti on s P D F fro m M C s i m u l a ti on s o li d li ne , fro m ﬁt t o t heore ti ca l EXA FS s pec t ru m w it h3da s hed , and fro m t wo ﬁt s t o t he M C P D F: w it h cu m u l an t s up t o 2 l ong da s hed and up t o 3 do tt ed . 6653 J . C he m . P h ys ., V o l . 119, N o. 13, 1 O c tobe r 2003 B r o mi de h y d r at i on s t r u c tu r e R eu s e of A I P P ub lis h i ng c ontent is s ub j e c t to the te rms : http s ://pub lis h i ng.a i p.o r g/autho rs / ri ght s- and - pe rmissi on s . D o w n l oaded to I P : 150.214.182.116 O n: W ed, 19 O c t 2016 14:53:39 and a l s o fa ll s hor t when t h i rd order cu m u l an t s are i n t roduced. The s y s t e m a ti c undere s ti m a ti on of d i s t ance s even when t he ‘‘correc t ’’ t h i rd order cu m u l an t i s u s ed enab l e s a s li gh t re ﬁ ne m en t of t he e s ti m a t ed exper i m en t a l d i s t ance. If we a ss u m e t ha t t he d i s t r i bu ti on of d i s t ance s fro m t he s i m ul a ti on i s correc t , we ob t a i n t ha t t he average B rO d i s t ance de t er m i ned i n t he exper i m en t a l ﬁt 3.34 Å i s undere s ti m a t ed by 0.1 Å. Therefore, our be s t gue ss of t he exper i m en t a l d i s - t ance i s 3.44 0.07 Å. A co m b i na ti on of exper i m en t a l and t heore ti ca l da t a ana l y s i s g i ve s w i der po ss i b iliti e s t o i nve s ti - ga t e t he geo m e t r i ca l proper ti e s of s y s t e ms pre s en ti ng ill - de ﬁ ned s t ruc t ura l pa tt ern s . ACKN O WL E D G M E NT S The au t hor s t hank Dr. A. M un ˜ oz- P a ´ez for her exper i - m en t a l co ll abora ti on and her he l pfu l d i s cu ss i on s . S ugge s - ti on s fro m t he referee have i m proved t h i s paper. P a t r i ck M erk li ng w i s he s t o t hank t he J un t a de Anda l uc ı ´a and t he Deu t s che F or s chung s ge m e i n s chaf t for ﬁ nanc i a l s uppor t . 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