THE GEOMETRY AND ORIENTATION OF THE WATER MOLECULE IN ICE Ih

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THE GEOMETRY AND ORIENTATION OF THE WATER MOLECULE IN ICE Ih

W. Kuhs, M. Lehmann

To cite this version:

W. Kuhs, M. Lehmann. THE GEOMETRY AND ORIENTATION OF THE WATER MOLECULE IN ICE Ih. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-3-C1-8. �10.1051/jphyscol:1987101�.

�jpa-00226231�

THE GEOMETRY AND ORIENTATION OF THE WATER MOLECULE IN ICE Ih

W.F. KUHS and M.S. LEHMANN

Institut Laue-Langevin, B.P. 156 X , F-38042 Grenoble Cedex, France

BBsum6 - Les resultats dlune 6tude par diffraction des neutrons & basse temperature (15 K) sur des monocristaux de glace d'eau normale et dreau lourde sont pr6sent8s.

Les mouvements thermiques reduits permettent l'analyse detaillee de la nature et de l'amplitude du desordre mol6culaire. Les d6formations des courbes donnant la densit6 des atomes d10xyg8ne montrent des differences significatives entre les glaces Ih Hz0 et D20 indiquant que la glace d'eau lourde est legerement moins d6sordonn8e (plus structuree) que la glace d'eau normale. L'amplitude absolue du desordre peut stre d6duite & partir des donnges spectroscopiques et cristallographiques et de manisre similaire il apparait que la glace dleau lourde presente des plus faibles deplacements atomiques. L'introduction d'un desordre moleculaire conduit .?i des valeurs des distances OH (D) en bon accord avec celles obtenues par des calculs recents de chimie quantique, tandis que lrestimation thgorique des longueurs des liaisons hydrogbne est nettement plus grande que celles observhes indiquant ainsi que l'aspect cooperatif des liaisons hydrogene n l a pas 6t6 entihrement pris en compte.

The r e s u l t s of a law-temperature (15K) single c r y s t a l neutron d i f f r a c t i o n study on l i g h t and heavy i c e Ih a r e presented. The reduced thermal motions allow t h e analysis of t h e nature and magnitude of t h e molecular disorder i n soma d e t a i l . The disorder deformation d e n s i t i e s show s i g n i f i c a n t differences i n H 0 and D 0 i c e Ih indicating t h a t heavy i c e Ih is s l i g h t l y l e s s disordered (more s&ucturd) than l i g h t i c e Ih. The absolute magnitude of t h e disorder can be deduced frciu t h e combination of spectroscopic and crystallographic data, and consistently D20 i c e Ih is found t o show t h e smaller disorder displacements. The introduction of molecular disorder leads t o OH(D) distances i n good agreement with recent quantum chemical calculations, while t h e t h e o r e t i c a l estimate f o r t h e intermolecular hydrogen-bonded distances is c l e a r l y higher than t h e observed values indicating t h a t t h e

cooperativity of t h e hydrogen bonding has not been f u l l y accounted f o r . Introduction

Ice Ih is - despite its high t r a n s l a t i o n a l syaa~etry - a highly disordered material.

The sylnnetry elements of t h e crystallographic space group act only on t h e time- and space-averaged atomic probability density distributions, while t h e atomic

configurations i n one s p e c i f i c u n i t c e l l do not obey t h e synmretry governing t h e t i m e - and space-average. S t a t i c and/or dynamic disorder w i l l produce l o c a l configurations with much lower symmetry. It is d i f f i c u l t t o deduce from the time- and space-averaged picture obtained by d i f f r a c t i o n methods t h e l o c a l atomic arrangements. Independent information from other experimental techniques probing t h e l o c a l atomic arrangements is thus required. By cambining e.g. spectroscopic and nuclear magnetic resonance with high resolution neutron d i f f r a c t i o n data, a three-dimensional p i c t u r e of t h e l o c a l atomic arrangements can be d r ? ~ . Following t h e same procedure8 as i n our e a r l i e r wor& on l i g h t and heavy i c e Ih ' " (done at higher temperatures and t o somewhat lower resolution) we have now performed a W r y

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987101

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high r e s o l u t i o n d i f f r a c t i o n study of H20 and D20 ice lh _at 15K i n o r d e r t o shed some f u r t h e r l i g h t on t h e local atomic arrangements as w e l l as t o d e t e c t the e f f e c t s o f i s o t o p i c exchange on the l o c a l ordering. The thermal Bmearing a t ^15K is considerably reduced, which together with the better r e s o l u t i o n allows t o see f i n e r details o f the atamic p r o m i l i t y d e n s i t y d i s t r i b u t i o n s . This paper focusses on t h e most r e l e v a n t r e s u l t s obtained with the improved r e s o l u t i o n and especially30n t h e d i f f e r e n c e s i n l i g h t and heavy ice Ih ; a f u l l account w i l l be given l a t e r

.

NeuEkon d i f f r a c t i o n data o f H 0 and D 0 ice Ih were c o l l e c t e d on the four-circle diffractometer D9B o f tge ~ n e t i g u t Laue-Langev&n, Grenoble at a temperature o f ^{15K. A} s h o r t neutron wavelength of 0.4797 A was used, which all- a l a r g e number o f independent Bragg i n t e n s i t i e s t o be measured. The i n t e n s i t y d a t a were corrected f o r absorption and secondary e x t i n c t i o n .

The intensities_pleastr$i i n a d i f f r a c t i o n experiment y e proportional t o the norm square F(Q ) .F(Q ) of the s t r u c t u r e amplitude ^{F ( Q ),} which is given a s :

where 6 is the s c a t t e r i n g vector and $ is the vector describing the p o s i t i o n o f atam n. ~ ( 6 ) is the at-c temperature f a c t o r , which describes the weakening of the i n t e n s i t i e s due t o atamic thermal motions atomic disorder. I f e i t h e r anhamonicity of the thermal motions o r unresolved a t e c disorder occur, a flexible formalism o f the atomic temperature f a c t o r is necessary i n o r d e r t o f i t t h e observed i n t e n s i t y d a t a . A very powerful generalised expression is baaed on t h e quasi-momentum (Gram-charlier) expansion of the harmonic temperature f a c t o r

where W(Q) is the harmonic temperature f a c t o r and y and ⁶ are the quasi-moment t e n s o r s oFYank 3 and 4 respectively. The atomic positions, the harmonic and anharmonic thermal parameters are refined i n a least-squares procedure. The r e s u l t i n g atomic p r o b a b i l i t y d e n s i t y function ( p d f ) is calculated from the derived pareimeters as

where pdf(;) is the harmonic (Gaussian) d e n s i t y d i s t r i b u t i o n and ll3(Z) and Ha(;) a r e the H e r m i t e polynomials o f order 3 and 4, respectively.

The least-squares refinements and the c a l c u l a t i o n o f t h e atof4lic p r o b a b i l i t y d e n s i t y functions were done using the PROMETHEUS s u i t e o f program

.

anharmonic as well as models including s p l i t atom p o s i t i o n s were s u c c e s s f u l l y re- fined ; d e t a i l s are given i n Tab. 1. The t o t a l atomic mean-square displacements obtained i n the harmonic refinements are given i n Tab. 2 and t h e refined anharmonic parametera o f oxygen are given i n Tab. 3. Pig. 1 and 2 show the d i s o r d e r deforma- t i o n d e n s i t i e s on the oxygen atoms i n H 0 and D20 calculated from these parameters.

2

Table 1: Data and Refinement Results"

N u m b e r o f observations weighted R-factors and Goodness-of-f i$, t o t a l unique obsened harmonic anharmonic s p l i t

v i t h , 2 a GOF % Gop %r ^G~~

where the sum i s over all s t r u c t u r e amplitudes IF I and 2 W i s t h e weight based e s s e n t i a l l y on counting statistics. The goodness-of-fit, OOP, is

2 2

( m( I Fobs I - FCa, I )/n* ) 'I2 ^{where n and m are} t h e number of observations and parameters, respectively.

** nodel 11 of Ref. 1 (oxygen atoms displaced i n t o t h e molecular b i s e c t r i x ) .

Table 2. ~ ~ t a l * atomic mean Square displacements i n i2 ^{i n i c e} Ih at 15K

* The mean-square displacements quoted include possible disorder displacements and a r e obtained from a refinement including only harmonic terms.

Table ^3. Anharmonic parameters of the oxygen i n i c e * Ih at 15K

* ₃

Third40rder tenus y a r e multiplied by 10 and fourth order terms 6 a r e multiplied by 10

.

As pointed out above it is not possible i n a disordered system t o dsduc%

t h e l o c a l geometry unambiguously f r o m crystallographic d a t a alone. Bowever, d i f f r a c t i o n data, measured t o a resolution comnensurate with t h e disorder

displacement, can give valuable information about the s p a t i a l atomic arrangements.

The man-square disorder displacements cause deviations from the harmonic description of thermal motions and may be visualised i n a dssorder deformation density map as shown i n Fig. 1 and 2. The most prominent feature is the positive deformation ( i.e. a higher atomic probability density as compared t o t h e harmonic case ) a t t h e ( t i m e - and space-averaged) mean-position of oxygen. Such deformations a r e expected f o r s p a t i a l l y disordered oxygen atom w i t h mean-square displacements smaller t h y t h e thermal displacements. Similar features have been found

previously at t h e hydrogen (deuteron) positions indicating an additional d i s o q e r perpendicular t o t h e bond direction. The very high resolution d a t a (& 20 A - ~ ) (which were not available previously) allow t o e s t a b l i s h t h e deformations at t h e

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oxygen and hydrogen (deuteron) p o s i t i o n with*useful precision. An i n e r p r e t a t i o n 1

of t h e oxygen disorder deformation d e n s i t i e s has been given r e c e n t l y . ^{The three}

models proposed have been tested using the 15K d a t a set. Model I1 of ref. 1 (oxygen atoms displaced i n t o the bisectrix of the water molecule with allowance made f o r a n i s o t r o p i c thermalmotions) shows the best agreement, although o t h e r models (oxygen displaced i n d i r e c t i o n s opposite t o the bond d i r e c t i o n s ) cannot be

led o u t d e f i n i t e l y . I n any case, it

Fia. 1. Disorder deformation d e n s i t i e s of the oxygen atom i n H 0 ice Ih at 15K:

( a ) i n the El-0-H2 plane ( b ) i n the 324-HZ plane. The 2 deformations are due t o antisymmetric ^(third o r d e r ) and symmetric ( f o u r t h o r d e r ) terms and give the deviation from the harmonic p r o b a b i l i t y d e n s i t i e s . Note t h a t t h e r e are two d i f f e r e n t sets of e q u i d i s t a n t contour i n t e r v a l s .

A8 discussed i n Ref. 1, these deformations cannot be due t o i n t r i n s i c anharmonic thermal motions o f t h e oxygen atoms.

Fig. 2. Disorder deformation d e n s i t i e s o f the oxygen atom i n D 0 ice Ih at 1 5 K : ( a ) i n the D 1 4 D 2 plane ( b ) i n the D z 4 D 2 plane. Contour i n t e r v a l s 2

i d e n t i c a l t o Fig. 1.

cannot be expected that simple split-modele as discussed i n R e f . 1 describe t r u l y t h e atomic d e n s i t y d i s t r i b u t i o n s , s i n c e many d i f f e r e n t l o c a l arrangements can be derived £ram the l a r g e number o f d i f f e r e n t configurations o f the i c e - l a t t i c e . Split-models are however q u i t e h e l p f u l t o e s t a b l i s h approximate interatomic distances and angles. The seemingly l a r g e covalent OB(D) bondlength i n i c e Ih can be ascribed e n t i r e l y t o the unresolved molecular disorder o f earlier

* Note t h a t due t o the lower r e s o l u t i o n only third order modifications have been included i n t h e previous analysis.

and D20 i c e Ih respectively. After corgection f o r t h e anharmonic s h i f t ( s e e Rez.

2 ) an equilibrium distance of 0.974(4) A f o r H 0 and D20 i c e Ih i s obtained. This value is i d e n t i c a l t o t h e averaged covalentom2ilibrium bondlength i n t h e ordered i c e s of 0.974 with a mean s c a t t e r of 0.007 A . Rlrtgppore, it is i n excellent agreemegt w i t h recent quantum chemical calculations , which predict a s h i f t of +0.015 A f o r t h e equilibrium OB(D) distance when going from t h e isolated water molecule ( r e

-

OH0

( r& = 0.972 A )

.

It should be mentioned t h a t t h e t r a n s l a t i o n a l and rotational molecular disorder i n i c e Ih inevitably r e s u l t s i n a distribution of hydrogen bond dissances, while the mean value quoted above is only marginally increased (by < 0.01 A) ; thg mean s c a t t e r - a s established from the s p l i t models - is approximately 0.02 A . This ⁱ⁸ of the same magnitude as obtained spectroscopically from t h e width of t h e isolated OH ( i n D 0 ) and OD ( i n H20) s t r e t c h frequency ( e .g. 8 ), whxch givgs a spread of the hydrogen2bonded intermolecular distances between 0.019 and 0.041 A f o r H 0 and 0.017 and 0.039 A f o r D 0 i c e Ih a t 15K depending on t h e chosen empiricag f requency-distance relaZion (see Ref. 2 f o r further d e t a i l s ) .

A c l a s s i c a l l y behaving system would not show any s t r u c t u r a l difference between a HZO- and t h e corresponding D 0-phase. Differences may however o c p r due t o quantum e f f e c t s and have been calcugated f o r liquid water i n some d e t a i l

.

Heavy water is predicted t o be t h e more "structured" liquid and correspondingly one could expect a smaller distribution of hydrogen bond angles and intermolecular diatances i n heavy i c e Ih. There was already an indication f o r such a difference from the distribution of hydrogen bonded intermolecular distances discussed above.

Note t h a t although there is considerable overlap i n the quoted range f o r H20 and D20 i c e Ih, t h e l a t t e r has - for a given choice of t h e frequency-distance relationship - always t h e smaller spread. The crystallographic r e s u l t is not accuratg enough t o show t h i s difference i n t h e spread which is i n t h e order of -0.002 A . The analysis of our high-resolution d i f f r a c t i o n data however shows significant differences i n t h e disorder displacements of t h e oxygen atom ( s . Tab.

3 ) ; E 0 i c e Ih has a greater displacement a s indicated by t h e higher disorder d e f o d t i o n denslty a t the mean position shown i n Fig. 1 ampared t o t h e density i n D 0 i c e Ih shown i n Pig. 2. A numerical estimate may be obtained by comparing t h e crystallographically established t o t a l mean-square displacement of t h e oxygen atom 2 ( a . Tab. 2 ) with the t r a n s l a t i o n a l ( p l u s the very small rotationai0and vibrational) mean square displacement obtained from s p e c i f i c heat measurements ., The

differencg i a due t o t h e molecular disorder and amounts t o 0.036(1) ^A f o r H20 and O.032(1) ^A f o r D 0 i c e Ih. Similar r e s u l t s f o r t h e disorder displacement a r e obtained by c d i n g th e crystallographically calculated 0-H(D) s t r e t c h

frequencies with the spectroscopic measurements as outlined in ref. ¹ and 2 . Note t h a t t h e obtained disorder displacements cannot be catpared straightforwardly t o t h e apread i n intramolecular distance. Hawever, from simple geometrical conside- rations it i s c l e a r t h a t t h e greater t h e disorder displacements a r e t h e wider w i l l be t h e distribution of intermolecular distances.

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The g e n e r a l p i c t u r e o f t h e molecular geometry and t h e o r i e n t a t i o n a l d i s o r d e r as e s t a b l i s h e d i n r e f . 2 h a s been confirmed by t h e p r e s e n t work r t h e oxygen and hydrogen at- are disordered and i n g e n e r a l not located at t h e (time- and space-

averaged) mean p o s i t i o n . The r e s u l t i n g r o t a t i o n a l and t r a n e l a t i o n a l molecular d i s o r d e r l e a d s t o a spread o f intermolecular d i s t a n c e s , which w e r e p r e v i o u s l y thought t o be i d e n t i c a l . The mean-value o f t h e hydrogen bond d i s t a n c e between oxygen and 9 hydrogen atom o f t h e neighbouring molecule is s i g n i f i c a n t l y increased t o 1.80( 1 ) A as a comequence of t h e molecular d i s o r d e r . The intramolecular bond d i s t a n c e is i n agreement with r e c e n t quantum chemical c a l c u l a t i o n s and t h e bond angle wks found t o be 107(1)O samewhere between t h e t e t r a h e d r a l angle and t h e bond-angle of t h e i s o l a t e d water molecule. S i g n i f i c a n t d i f f e r e n c e s between l i g h t and heavy ice Ih have been found what concerns t h e magnitude of t h e molecular d i s o r d e r ; D20 ice Ih is more s t r u c t u r e d t h a n R 0 ice Ih. However, some o f t h e f i n e r d e t a i l s of t h e molecular d i s o r d e r are stih not known and conventional experimental work is n o t l i k e l y t o h e l p much. A deeper i n s i g h t may be expected from molecular dynamics o r Monte-Carlo simulation8. A s t r i n g e n t test on t h e consistency o f such a simulation is - amongst o t h e r s - t h e c o r r e c t reproduction of t h e a c c u r a t e l y e s t a b l i s h e d deformation d e n s i t i e s o r t h e corresponding S t r u c t u r e f a c t o r s .

W e thank W . A. Chaillou from Laboratoire de Glaciologic, Grenoble, f o r growing t h e ice crystals.

W.F. Kuhs 6 M.S. mhmann i n Colston Symposium on W a t e r and Aqueous S o l u t i o n s ( e d s . G.W. Neilson 6 J.E. Enderby), B r i s t o l , R i l g e r , 1986, p. 75.

W.F. ICuhs & M.S. Lehmann i n W a t e r Science Reviews Vol. 2 (ed. P. Franks), Cambridge, Cambridge University Press, 1986.

W.F. Kuhs 6 M.S. Lehmann, Acta Cryst. A, i n preparation.

U.R. Zucker, E. Perenthaler, W.P. Kuhs, R. Bachmann 6 8 . Schulz, J. Appl.

Cryst. 16 (1983), 358.

E. Whalley, Mol. phys. 28 (1974), 1105.

B. J. Yoon, K. Morokuma & E.R. Davidson, J. Chem. Phys. 83 ( 1985), 1223.

M.D. Newton, Trans. Am. Cryst. Ass. (1986) i n p r e s s .

A.V. logansen 6 M.Sh. Rozenberg, Opt. Spektroek. 44 (1978). 87.

R.A. Kuharski 6 P.J. Roesky, J. Chem. Phys. 82 (1985). 5164.

A.J. Leadbetter, Proc. R. Soc. Ser. A-02 (1965), 403.