D]

Chemical Physics 306 (2004) 241–251 www.elsevier.com/locate/chemphys

Structural investigation of liquid formic acid by neutron diffraction. II: Isotopic substitution for DCOO[H/D] Imre Bako´ a

a,*

, Ga´bor Schubert a, Tu¨nde Megyes a, Ga´bor Pa´linka´s a, Geoffrey I. Swan b, John Dore b, Marie-Claire Bellisent-Funel c

Theoretical Chemistry, Central Research Center of the Hungarian Academy of Sciences, Institute of Chemistry, Pusztaszeri ut 59-67, P.O. Box 17, Budapest H-1525, Hungary b Physics Laboratory, University of Kent at Canterbury, Canterbury, Kent CT2 7NR, UK c Laboratoire Leon Brillouin (CEA-CNRS), CEN-Saclay, 91191 Gif-sur-Yvette Cedex, France Received 17 May 2004; accepted 28 July 2004 Available online 24 August 2004

Abstract New measurements of neutron diffraction data for four samples involving H/D isotopic substitution on the hydroxyl hydrogen of liquid formic acid at 20
C are reported. The results are combined with earlier measurements on [H/D]COOD to provide a full range of data. The determination of molecular conformation and bond-lengths has been made with a partial form-factor formalism and also using the ‘‘Monte Carlo determination of g(r)’’ technique. The partial real-space correlation functions, RR, RH and HH are evaluated in each case and compared with existing computer simulations. The results confirm the strongly hydrogen-bonded nature of the liquid, but show that current molecular dynamics predictions based on transferable potentials do not give a very good representation of the structure. The observations provide a basis for a more detailed investigation and work is currently in progress. Ab initio quantum chemical calculations showed that the non-planar configuration suggested by Bertagnolli et al. [Ber. Bunsen. Phys. Chem. 88 (1984) 977; Ber. Bunsen. Phys. Chem. 89 (1985) 500], is very unlikely both for formic acid dimers and monomers.  2004 Elsevier B.V. All rights reserved. Keywords: Neutron diffraction; Liquid structure; Formic acid

1. Introduction Neutron diffraction has proved to be a valuable technique for the structural study of molecular liquids and is now used routinely for measurements on a wide range of materials. However, the study of organic liquids has been only a relatively recent development due to the increased complexity of the molecular size and shape coupled to the fact that there are sometimes strongly angle-dependent forces arising from specific interactions such as hydrogen-bonding. One of the simplest mole*

Corresponding author. Tel.: +3614384141; fax: +3613257554. E-mail address: [email protected] (I. Bako´).

0301-0104/$ - see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.07.036

cules exhibiting H-bonding effects is methanol, which was extensively studied using H/D isotopic substitution methods by Montague, Dore and collaborators [1–3] in the early 1980Õs. Other H-bonded systems such as glycerol and tertiary butanol have also been investigated [4] but the increased molecular size adds considerable complications to the interpretation of the results. Another area of interest is the carboxylic acids of which formic acid is the first member in the series. Kratochwill et al. [5] have studied the liquid formic acid using nuclear magnetic resonance technique. They concluded that in liquid state the formic acid molecule exists in trans, non-planar configuration. Continuing this work by Bertagnolli and colleagues [6,7], using neutron diffraction

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technique stated that the formic acid molecule in the liquid state is a molecule with an OH group rotated by about 50
out of the molecular plane. In their work they remarked that the evidences for the validity of their proposed molecular configuration in the liquid state are fairly weak. A subsequent investigation by Swan and Dore [8] using the same technique led to different conclusions, they observed a planar structure of the formic acid molecule. Recent proton NMR study also shows that the molecule in the liquid state exists in planar form [9]. Molecular dynamics and Monte Carlo simulation studies [10–13] showed that the liquid phase of formic acid consists of small, often branching chain formed via OH


O bonds and these chains are connected to each other by CH


O type hydrogen bonds. An X-ray and neutron diffraction study of liquid deuterated formic acid at various temperatures and pressures showed that the strong hydrogen bonded network is weakly affected by temperature and pressure [14]. A partial report of this work has been published separately [8], however, the earlier experiments did not involve a full set of measurements with H/D substitution on the hydroxyl hydrogen so the datasets were, in some sense, incomplete. It was therefore decided to extend the investigation by performing these additional measurements so that more specific information could be obtained on the hydrogen-bonding. The present paper reports these additional measurements and includes a detailed report of the special techniques used in the earlier measurements, which have not previously been reported in a journal presentation. The results of some quantum chemical calculations are also reported.

2. Formic acid: structure and interactions Formic acid is the simplest organic acid and it represents an ideal model compound for understanding more complicated molecules. This molecule exhibits rotational isomerism representing two structures defined by the rotation of the OHa group around CO bond. In the gas phase, the trans conformer is predominant and the energy difference between the two conformers is about 3.8–3.9 kcal/mol [15]. The rotation barrier is 13.8 kcal/mol [16]. The cis conformer of formic acid in the gas phase was also identified by IR method [17]. The basic structure of the formic acid molecule is planar

in gaseous phase and in the condensed phase it is known to be strongly hydrogen-bonded. It readily forms a dimer in the gas phase by the two symmetric hydrogen bonds shown in Fig. 1(a). By contrast, the crystal structure consists of a planar arrangement of hydrogenbonded chains as shown in Fig. 1(b). In the disordered liquid phase there is the additional possibility of forming branched chains [10–14] so the structure is dependent on the inter-play of the molecular shape with the local hydrogen-bonding interactions.

3. Theoretical formalism and molecular representation of neutron diffraction The structural characteristic of a molecular liquid can be divided into correlation functions involving the geometrical configuration of the molecule (intramolecular) and the relationship between molecules (intermolecular). The diffraction measurements give the liquid structure factor, SM(Q), which can be formally written as S M ðQÞ ¼ f1 ðQÞ þ DM ðQÞ;

ð1Þ

where f1(Q) is the molecular form factor and DM(Q) represents the liquid structure function [1]. The total intermolecular correlation function, g(r), in real space is obtained from the transform relation Z 2 1 d L ðrÞ ¼ 4pqM ½
gðrÞ  1 ¼ QDM ðQÞ sin Qr dQ; p 0 ð2Þ where qM is the molecular number density and g(r) is the weighted sum of individual pair correlation functions gab(r) according to P ca cb ba bb gab ðrÞ ab ð3Þ gðrÞ ¼
2 : P c a ba a

For a simple molecule, it is convenient to determine the molecular parameters by a parameter fit to the high-Q diffraction pattern where the DM(Q) function makes a negligible contribution to the diffraction pattern but this procedure is difficult for large molecules because there is a substantial overlap of intra- and intermolecu˚ . Forlar correlations in the intermediate region of 2–5 A mic acid is an interesting intermediate case where the

Fig. 1. Structural arrangements of formic acid molecule: (a) gas phase dimer; (b) chain structure of crystalline phase.

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overall size of molecule is small but local correlations are strong due to hydrogen-bonding. Furthermore, there are two independent sites for H/D isotopic substitution so that systematic effects and self-consistency in the data analysis can be fully investigated. The conventional form-factor approach is described in Section 3.1, with an additional development of internal consistency checks through the introduction of a partial form factor procedure. The more recent method using the Monte Carlo determination of g(r) (MCGR) routine that incorporates certain features of the intramolecular correlations without defining the full configuration is given in Section 3.2. Both procedures are used in the treatment of the present data. 3.1. Partial form factor analysis A description of the partial form factor method has been given elsewhere [18] but is not available in a journal publication. In essence, the H/D substitution on a single site of a molecule, HR where R is a radical or a group of atoms, means that measurements can be made on a range of isotopic mixtures [H/D]R. It is assumed that this substitution does not affect the structural characteristics of R and that the H/D replacement is completely isomorphic. The separation of the diffraction pattern into components R and H already applied for several cases [19–21]. There are therefore three partial correlation functions, RR, RHf and HfHf, which can be separated by making measurements on at least three isotopic mixtures. Normally it is convenient to use the isotopically pure systems HfR and DfR with one (or two) different H/D mixtures. The extraction of the normalized SM(Q) functions from the experimental measurements requires the application of the usual experimental and analytic (Placzek) correction procedures. Due to the large incoherent scattering from hydrogen, the possibility of small errors in the relative intensities of the interference functions for each isotopic mixture cannot be ruled out. If present, these small discrepancies can lead to systematic errors in the final data. The form factor, f1(Q), consists of various terms, which correspond to the internal structure of the group R, and the cross-correlation of the atoms in R with the substituted H/D atoms; as there is only one substituted atom, there are no intramolecular terms corresponding to the HfHf intermolecular function. The contribution to the cross-section for the group, R, remains constant throughout the experiment but the cross-section for the RHf interference terms is proportional to the b value of the H/D isotopic mixture. It therefore follows that the three measurements can be combined in pairs to give an equivalent cross-section corresponding to a fixed H/D composition. At high Q-values where the intramolecular terms dominate, the different combinations should give

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exactly the same pattern and this provides a direct check for the internal consistency of the various datasets and the procedures used to extract the interference function from the observations. This technique was used first by Swan [18] in the treatment of the earlier formic acid measurements. The use of the f1(Q) form factor guarantees that the intramolecular contributions are all correctly weighted in the fit to the high Q-value but some of the parameters are not well determined particularly the Debye– Waller terms defining the vibrational amplitudes. If the molecule is large, the terms involving long intramolecular distances have only a small effect on the pattern at high Q-values. Including constraints on the geometry from the known configuration of the molecule usually accommodates these effects. If there are strong local interactions, as in hydrogen-bonding, there can be well defined correlations arising from short intermolecular distances. Under these circumstances the use of the f1(Q) formalism may lead an over fitting of the observations and give a false impression of the height of intermolecular correlation in the overlap region. The Reverse Monte Carlo [22] technique for the direct analysis of the measurements has received much attention in recent years. This development has shed new light on the information content in the observed data but its use in the analysis of molecular liquids has proved difficult. The fundamental approach makes no assumptions about the presence of molecules and relies on the fact that the diffraction pattern contains this information and should therefore generate the correct three-dimensional geometry in the optimization of the atomic configuration. In practice, this rarely occurs (except for very simple molecules) and some constraints need to be imposed. However, if information about the molecular geometry is to be included as Ôa prioriÕ knowledge there is a problem about the effects of the constraints in the fitting procedure. Furthermore, even the use of a rigid molecule increases the complexity of parameter space by adding rotational features to the molecular ensemble and making the convergence to an optimal three-dimensional configuration difficult and time consuming. In this context an intermediate approach is desirable, which relates more directly to the one-dimensional nature of the diffraction measurements. The recent development of the MCGR routine meets these objectives and is summarized in the following section. Another branch of the inverse method is the potential refinement. In this method each iterative step consists of two parts; a Monte Carlo [23] or molecular dynamics [24] simulation with an assuming pair potential and the modification of this potential according to quantities calculated in the simulation and to the experimental data. These methods have already been applied

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successfully for some simple [25] and hydrogen bonded liquids [26,27] too.

too. In our case, the MCGR method solves an over determined linear equation (4 equations 3 dependent).

3.2. The Monte Carlo determination of g(r) analysis procedure

4. Interpretation of the results

The conversion of the observed interference function, I(Q) or DM(Q), to an r-space representation is normally conducted by numerical Fourier transformation as given in Eq. (2). This process is subject to various inaccuracies arising from statistical fluctuations, limited Q-range and possible systematic errors in the original datasets. The uncertainties in the r-space functions are not simply related to the measured diffraction patterns and are particularly prone to error if isotopic substitution is involved. There have been several attempts to solve this problem, such as the ‘‘maximum entropy method’’ introduced by Soper [28], and the Reverse Monte Carlo method introduced by McGreevy and Pusztai [22], where the primary object is to set up a distribution of atoms or molecules, which is consistent with the measured diffraction pattern. A relatively new approach to solve this problem has now been developed by Pusztai and McGreevy [29] and is called MCGR, which is an acronym for the Monte Carlo treatment of the g(r) function. This program uses an inverse method in which the pair correlation function (total or partial) is generated numerically and modified by a random process until its inverse Fourier transform I(Q) agrees with the experimentally measured Iobs(Q) within its error limits. This algorithm can be described by the following equation for the interference part of the differential scattering cross-section measured as the intensity distribution, Iobs(Q) such that I obs ðQÞ ¼ I M ðQÞ þ a0 þ a1 Q2 þ a2 Q4 þ ða3 Q6 Þ;

ð4Þ

where a0, a1, a2, and (a3) describe the Placzek corrections for the self scattering terms. IM(Q) is the total interference function is generally written as X I M ðQÞ ¼ wab I ab ðQÞ; ð5Þ where wab are the weighting functions defined by the relative concentrations and b-values and Iab(Q) is the partial structure factor, Z 1 QI ab ðQÞ ¼ qM 4pr½gab ðrÞ  1Þ sinðQrÞ dr: ð6Þ 0

Various types of constraint can be applied to gab(r) or g(r) and some of them are built in the MCGR program. These are the following: zero constraints, positivity constraints, coordination constraints and smoothing constraints. A more detailed description of these constraints is found in [32]. In the present work, a zero constraint is used such that the g(r) functions are required to be zero for r values less ˚ . We used the Gaussian smoothing option than 0.75 A

The diffraction results involve two separate experiments with H/D substitution on either the formic hydrogen or the hydroxyl hydrogen. These neutron diffraction experiments were carried out on the hot-source diffractometer 7C2 of the Laboratoire Leo´n Brillouin CEASACLAY. The liquids were placed in a vanadium container of 5 mm diameter in the first case (HfR) and of 6 mm diameter in the second case (HaR) and 0.1 mm wall thickness. The incident neutron wavelength ˚ . The results from the earlier experiment have was 0.70 A already been reported in separate publications [8] and will be summarized in Section 4.1. The more recent experiments have not been previously reported and are discussed in Section 4.2. The interpretation of the complete dataset is considered in a separate section using different methods to extract the individual pair correlation functions for this particular combination of measurements. As we have already remarked, there are some experimental evidences that in liquid phase the trans formic acid exists in its non-planar form [6,7]. In order to estimate the energy difference between the possible non-planar and planar configuration of trans formic acid and its dimer we have performed additional quantum chemical calculations. We have investigated the change of hydrogen bond strength due to the molecule deformation. In order to estimate the effect of further formic acid molecules on the results of the computation, we performed Car–Parrinello Molecular Dynamics simulation on liquid formic acid [30], and it showed that planar molecular geometry is not altered reasonably. 4.1. Ab initio study All calculations were performed within the framework of the ab initio closed shell approximation using the Gaussian 98 [31] package of computer codes. The geometries of cis and trans formic acid molecules and the acyclic dimer structures, which is the second most stable formic acid dimer [32], were optimized at the B3LYP level of theory using the 6-311 + G** basis set. The potential energy curve for the OH rotation around the CAO bond was obtained and the barrier for the cis– trans transition was determined. It was already shown that for hydrogen-bonded systems the B3LYP method gives very similar results than the more sophisticated MP2 results [33]. The optimized structures with some selected geometrical parameters are shown in Fig. 2. According to the calculations, the trans conformer is more stable than the cis conformer with 4.6 kcal/mol.

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strengths of the hydrogen bonds in the non-planar case are nearly the same as in the case of the planar structure. The 7 kcal/mol destabilization caused by the deformation of the molecule is large compared to the hydrogen bond strength. According to this, it is unlikely that the formic acid molecule exists in this non-planar form in liquid state. 4.2. H/D substitution for [H/D]FCOOD

Fig. 2. The optimized structures for formic acid dimers with some selected geometrical parameters.

The strength of the hydrogen bond in the acyclic dimer is 8.8 kcal/mol. The energy difference between the cis and trans conformer of formic acid, and the interaction energy in the planar acyclic dimer is in good agreement with the earlier MP2 results [33]. We carried out constrained optimizations at different O–C–O–H dihedral angles on the formic acid monomer. The calculated potential energy curve for the OH rotation is shown in Fig. 3. It can be seen that the destabilization of the monomer at 50
, which was the suggested configuration in liquid phase by Bertagnolli and colleagues [6,7], is about 7 kcal/mol. We also optimized the geometry of the acyclic dimer with one of the O–C–O–H dihedral angle frozen at 50
. The total energy of the planar dimer is 7 kcal/mol (same as in the case of the formic acid monomer) lower than the non-planar form. It can be concluded that the

The three measurements on the [H/D]FCOOD system correspond to the separation of partial correlation functions corresponding to the formic hydrogen [HF, DF] where the radical or group, R corresponds to the acidic group [COOD]. The three samples were the followings: HCOOD, DCOOD and their mixture (H0.36 + D0.64)COOD. The observed intensities are shown in Fig. 4 after correction for self-attenuation and multiple-scattering. A form-factor fit has been used to define the shape of the curve in the high Q-value region and the dotted line represents the self-scattering contribution with the Placzek fall-off arising from inelasticity effects. In this particular case, it was necessary to include a term to represent the intermolecular O


D hydrogen bond, in addition to the normal intra-molecular terms in the form-factor; details of this procedure are given elsewhere [18]. For the combination of these datasets, it is important that systematic errors do not arise from the corrections applied to the data and in the case of H/D isotopic substitution there is a particular concern because the total scattering from the different samples varies by factor of almost four. For measurements with a cylindrical

Fig. 3. O–C–O–H dihedral angles as a function of the potential energy for the OH rotation.

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C

A B

B C A

Fig. 4. The diffraction pattern for liquid formic acid showing the three samples with different H/D isotopic composition (A, HCOOD; B, (H0.36 + D0.64)COOD; C, DCOOD).

container of fixed diameter, the multiple-scattering corrections are significantly different in the two cases and consequently there is a possibility that the interference functions shown in Fig. 4 are not correctly normalized relative to each other. Fortunately there is a consistency check, which can be applied to the data to ensure that the separate datasets are internally consistent. This procedure, first introduced by Swan [8], is based on the partial form-factor approach described in Section 3.1. The datasets are combined according to the relations given in the following equations: X ½0R ¼ ½R ¼ b2R f1R ðQÞ: ð7Þ This contribution is constant throughout the H/D substitution and can therefore be subtracted from the other measurements, so that: X ½Df R  ½‘0’R ¼ 2bD bk F k ðQÞ; ð8Þ and ½Hf R  ½‘0’R ¼ 2bH

X

bk F k ðQÞ;

Fig. 5. The combined datasets for the partial form factor analysis of (H/DCOOD) showing the agreement in the high Q-value region: (A) [DR]  [Ô0ÕR], (B) [HR]  [Ô0ÕR], (C) [DR]  [HR]. Dashed line represents the fitted partial form factor.

pattern, which can be used for the fitting of the partial form-factor. In the earlier analysis, it was possible to obtain a good fit to a combined set of results and the resulting form-factor is shown as a dashed line. The parameters extracted from this approach will be discussed in Section 4.4. A full treatment of the individual datasets can be conducted by the conventional route, namely by extracting the intermolecular DM(Q) functions and these can be transformed to give the intermolecular pair-correlation functions dl(r), which are shown in Fig. 6. The contribution from the intermolecular O


H hydrogen bond term, which is fitted in the analysis but is not a part of the usual molecular form-factor, f1(Q), has been in˚. cluded in this representation and occurs at 1.8 A

ð9Þ

where the summation over k involves correlations from the substituted atom (labeled) to all other atoms in the molecule and. The high Q-value region will be dominated by the interference function arising from the partial form-factor for the [R] molecule and will therefore be identical for all combinations. The difference in the curves arising from a different relative contribution from the intermolecular HFHF terms will be present at lower Q-values and provides the required information on the intermolecular arrangement. The results of this treatment are shown in Fig. 5 for the three possible combinations of datasets. Only two independent difference curves can be extracted from the data but the three possibilities are shown to illustrate the systematic changes in the low-Q region. It is clear that the three datasets are in good agreement above 5 ˚ 1 and this region provides a consistent interference A

A

B C

Fig. 6. The real-space distribution functions dL(r) corresponding measurements for (H/D)COOD (A, HCOOD; B, (H0.36 + D0.64) COOD; C, DCOOD).

I. Bako´ et al. / Chemical Physics 306 (2004) 241–251

4.3. H/D substitution, DCOO[H/D]A The H/D substitution experiments for the hydroxyl or acidic hydrogen [HA, DA] have been made for four independent compositions and this permits a different approach to the question of systematic errors as the datasets effectively over-determine the three partial functions, RFRF, RFH and HH, where RF is now the formic group [DCOO]. The four samples were the following: DCOOD, DCOOH and their mixtures (H0.36 + D0.64) COOD and (H0.28 + D0.72)COOD. In the last composition the average scattering length of the hydroxyl hydrogen was +3.74 fm, i.e., the negative of that for pure light hydrogen. The MCGR routine has been used to determine these functions and the fits to the QI(Q) datasets are shown in Fig. 7. The agreement is excellent over the whole Q-range except the main diffraction peak where there is a small underestimation of height for all the four datasets. The equivalence of H and D in isotopic substitution experiments has remained a controversial area for over a decade and is still unresolved. There are indications from thermophysics, diffraction and spectroscopic studies, which indicate small but possibly systematic discrepancies between hydrogenated and deuterated systems but there are no definitive results on possible structural variations. It is therefore convenient to make the assumption of equivalence until clearer evidence produced. The consensus is that substitution in a C–H bond is truly isomorphic (except for the vibrational amplitude) but substitution in an OAH bond engaged in Hbonding may not be. A simple substitution experiment

(a)

(b)

(c)

(d)

Fig. 7. The QI(Q) functions for each of the four samples: pluses, extracted experimental function; solid line the generated MCGR function. (a) DCOOD, (b) DCOO(H0.28 + D0.72), (c) DCOO(H0.36 + D0.64), (d) DCOOH.

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involving three compositions is bound to yield a set of partial functions from the solution of the three simultaneous equations provided if they are not ill-conditioned. However, this direct treatment will be incapable of revealing certain types of systematic error arising from scale factors in the different datasets. For the system of four samples with varied isotopic composition studied in the present experiment, there are several consistency checks that can be made. The simplest is the comparison of the data for the special null mixture with ÆbH/Dæ = 0 [(H0.36 + D0.64)COOD] with the average of the two mixtures with ÆbH/Dæ = bH[(H)COOD] and ÆbH/Dæ = jbHj[(H0.28 + D0.72)COOD]. The combination is given as Eq. (5) and indicates that the RfRf and RfHa components should cancel out leaving only the intermolecular HH contributions. As the main HaHa component arises primarily from just one term due to the nearest hydrogen-bonded ˚ which will not be as sharp distance at approximately 4 A as that of the intermolecular terms, it follows that the shape and intensity of the diffraction pattern in the high Q-value region should be identical. Since these samples contain the largest proportion of hydrogen where the Placzek and multiple scattering corrections are significant, the comparison ensures that the interference functions have the correct relative weighting factors. The results are shown in Fig. 8, where the agreement can be seen to be very good. The residual term, which is shown in the inset is proportional to the IHaHa(Q) function that defines correlations between hydrogen atoms in the hydroxyl groups; the information given by this function is discussed in the Section 4.5. Another consistency check across the two different runs is possible by comparing the I(Q) results for the DCOOD sample which have been measured in both experiments. This comparison is also found to be well satisfied even though a different diameter sample cell was used. The combined datasets therefore represent a convenient set of information for a detailed analysis of the structure of liquid formic acid at ambient temperatures. The four I(Q) curves for the DCOO[H/D] case represent an over-determined set which may be studied detail but it is convenient for general purposes to use a different approach to extract the information. This technique presented here is based on the use of the MCGR method, which is briefly reviewed in Section 4.5. 4.4. Molecular structure The diffraction data give information on the conformation of the molecule in the liquid state. A comparison of the bond-length values obtained from the earlier neutron measurements with the results from other techniques has already been given [8]. In this case, the parameters were obtained from the fitting of the formfactor to the high Q-value region of the diffraction

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0.050

2

HaHa

0.025 0.000 -0.025 -0.050

1

-0.075 -0.100

I(Q)

2

0

4

6

8

10

12

14

A

-1

B -2 0

2

4

6

8

10

12

14

-1

Q(Å ) Fig. 8. Consistency check for DCOO(H/D) measurements: (a) solid line, I(Q)DCOOD; (b) pluses, {I(Q)[DCOOH] + I(Q)[DCOO(H0.28 + D0.72)]}/2. The difference function is shown in the inset and relates to the HaHa function.

pattern but it was necessary to include an intermolecular hydrogen-bond term in order to give good agreement with the observed data. The use of the MCGR method on the new data gives a different emphasis to the information content of the diffraction pattern since there is no explicit use of the molecular geometry in the fitting procedure. The only constraint introduced in the fitting process is the relative weighting of the different partial terms in the expression for the scattering pattern. The procedure generates a set of partial g(r) functions, which are optimized to give a good representation of all available data in Q space. This approach therefore avoids the difficulty of assigning and fitting the individual Debye–Waller factors, which affect the oscillation amplitude of the independent terms in the f1(Q) form factor. The long intra-molecular distances are not well defined by the experimental results because the contribution above 6 ˚ 1 is very small. In the MCGR procedures these terms A do not feature in the analysis. Another difference is apparent when there are two distances that have a similar magnitude, such as the CAO single and C@O double bonds in formic acid. For the form factor treatment, specific characteristics must be assigned to each of these separate contributions but in practice they cannot be easily distinguished and remain constant throughout all the H/D substituted samples. The Fourier transform ˚ 1] shows that the over the current Q-range [Qmax = 16 A individual contributions cannot be resolved but allowing a free parameter search across the two peaks allows the individual parameters, particularly the Æu2æ values in the Debye–Waller functions, to adopt unphysical values in order to compensate for minor variations in the experimental results. In effect, there is too much freedom for

the information that is contained in the data. The use of the MCGR technique overcomes both of these problems and ensures that the optimal representation of the observations is obtained. Since the analysis is conducted in a one-dimensional r-space rather than the three-dimensional space of the RMC procedure, there is a much closer match to the information contained within the measurements. The precise shape of the curve in r-space can be used to determine the molecular distances by assuming that each contribution has a Gaussian form. For example, there are clearly defined peaks in the ˚ region for both RR and RH correlations. 0.75–2.5 A ˚ is the intramolecular OH The shortest peak at 0.94 A distance for the hydroxyl group and is slightly shorter ˚ ) due to the molecular rethan the normal value (0.98 A coil effects on the interference term as previously discussed for water [34]. The first peak in the RR ˚ is composed of three overlapdistribution at 1.0–1.5 A ping peaks for the formic C–H and the two CAO bonds. If the composite peak is fitted by three Gaussian functions of defined weighting factors it is possible to determine the separate bond lengths and attribute the appropriate labels. A similar procedure can be carried out for the second peak although it then becomes neces˚ sary to include the intermolecular H-bond peak at 1.8 A in the RH distribution. As long as the fitting is conducted in a self-consistent way, the fundamental parameters defining the molecular conformation can be evaluated. In our work we assumed that, the formic acid molecule exists in trans conformation according to the NMR results [5,9]. It was already discussed, that the main difference between the cis and trans conformer is in the Hf–Ha and Ha–O distances as shown in Fig. 2.

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It can be concluded from these distances in the two conformations, that these give nearly the same scattering pattern so our isotopic substitution method can not give difference between them, due to the almost same coherent scattering length of D and O. The planar configuration of formic acid is more stable than the non-planar one. The energy difference between the two configurations is about 7 kcal/mol as we show in Section 4.1 and the hydrogen bond strength the planar–planar and non-planar–planar configuration is mainly the same. According to this, in our fitting procedure we apply the planar trans formic acid configuration as an initial configuration. The parameters determined from the analysis are given and compared with the results of the earlier evaluation in Table 1. It is clear that there is reasonable agreement across the different experimental and computational techniques. 4.5. Partial pair correlation factors The separation of the RfRf and RfHa partial pair correlation functions is shown in Figs. 9 and 10 The RR correlations include all the intra- and intermolecular correlations for the RF group For substitution on the formic hydrogen the longest intramolecular distance is ˚ for the oxygen–oxygen correlation so there only 2.2 A is a fairly clear separation of the various contributions with only a small overlap in the intra- and intermolecular parts. It is notable that there are long-range correla˚ , which results from the tions extending up to 10 A longer scale correlations between the R units. By contrast, the RHa function has different characteristics. ˚ arises from the intraThe relatively sharp peak at 1 A ˚ commolecular OH bond and the broad peak at 1.8 A prises both the intra-molecular CH distance and the intermolecular O


H hydrogen-bond as discussed in ˚ conthe previous section. The next peak at about 2.7 A tains information about the Hf–Ha and Hf–O1 intramolecular and C–Ha intermolecular distance. There is then a more complex structure arising from local correlations ˚ but beyond that due to the H-bond, extending up to 6 A

249

Fig. 9. The separated partial g(r) functions obtained from the MCGR treatment of the DCOO(H/D) data given in Fig. 5.

value there is less structure. The full interpretation of these results requires more detailed modeling and will be covered in later publications. The HaHa partial pair correlation function has a low weight-factor but it is particular interest as it represents information on the chain or dimer characteristics of the configuration. The results given by the MCGR treatment are shown as a g(r) plot in Fig. 11(a). The oscillatory nature of this curve is unphysical and a smoothed curve has been drawn through the function to give a more realistic representation of the distribution. A closer examination of the residuals arising from the small discrepancies present in the MCGR fit shown in Fig. 7 seem to be connected with these features. These fluctuations are probably present in all the individual partial functions but are only seen clearly in this case. A more direct way to extract this function is from the difference function shown in the inset to Fig. 8, and the corresponding partial is shown in Fig. 11(b). This curve is much smoother and the absence of any intramolecular ˚ ) provides confirmation of peaks at low r-values (<2.5 A the self-consistence in the normalization of the three datasets used for this analysis. The general features

Table 1 Geometrical parameters for the formic acid molecule compared with previous measurements and ab initio quantum mechanical calculation (distances ˚) in A Technique

Ab initio

Electron

Neutron

X-ray

Neutron

Neutron

References C@O C–O O–H C–H OCO\ COH\ O–H

[15] 1.208 1.33 0.978 1.078 125.9 106.9 1.702

[16] 1.22 1.32 1.03 1.1 126.2 107.8 1.72

[5] 1.23 1.38 1.03 1.1 116.0

[8] 1.18 1.36 1.0 1.1 125.2

[7] 1.19 1.31 0.94 1.08 118 115 1.82

New 1.19 1.32 0.94 1.07 127 108 1.78

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I. Bako´ et al. / Chemical Physics 306 (2004) 241–251

HaHa

(a)

(b)

a

Fig. 11. The pair correlation function, gHH(r) for DCOO(H/D): (a) from MCGR analysis, (b) by difference function analysis.

Fig. 10. A dl(r) plot of the partial functions showing the division into intra- intermolecular contributions (solid line, total; dashed line, intramolecular; dotted line, intermolecular terms).

given by the two methods are in satisfactory agreement but further work is needed to study the possible nonequivalence of the H and D atoms in the hydrogenbonding chain. The MD simulations have suggested that there are negligible numbers of cyclic dimers in the liquid. If present, a sharp peak would be presented in the ˚ and there is no evidence of HaHa correlations at 2.8 A this feature in the results. The simulation studies have also indicated the presence of some branched chains where the hydroxyl oxygen acts as an acceptor site for two hydrogen bonds. If it is assumed that a tetrahedral configuration is formed as in water there would be a ˚ . The experimental results broader peak at about 2.4 A show a much broader peak extending to larger r-values in which the initial rise in gHaHa(r) is at this value so it is possible that there are some branched chains but these cannot be large in number. The main characteristic of ˚ is consistent with the a broad peak at centered on 4 A concept of a chain structure as given by the simulation predictions (see Fig. 11).

mined from using OPLS potential model in this work is in poor agreement with the previous results of both X-ray [10] and neutron diffraction measurements. As a result, Turi and Jedlovszky [11,12] have introduced a new potential model for formic acid; the predictions from this model are in better agreement with the earlier neutron results from the [H/D]COOD study and now compared with the new data presented in this paper in Fig. 12. The agreement now is satisfactory. By incorpo-

(a)

(b)

(c)

4.6. Comparison with other studies A molecular dynamics simulation has been reported for liquid formic acid [10–13] using a set of different potential model. The pair correlation function deter-

Fig. 12. The intermolecular radial distribution functions for liquid formic acid. (a) DCOOD, (b) DCOO(H0.36 + D0.64), (c) DCOOH (solid line, calculated from MC simulation [11,12]; pluses, new experiment).

I. Bako´ et al. / Chemical Physics 306 (2004) 241–251

rating data from the X-ray measurement, it should be possible to refine the parameters of the interaction potential.

5. Conclusion The new set of neutron diffraction measurements on DCOO[H/D] completes a detailed experimental investigation of the structure of formic acid at 293 K using Xray and neutron techniques. The use of H/D isotopic substitution on the two sites of the molecule provides the opportunity for an in-depth study of the atomic correlations using new techniques, such as the partial form factor analysis and the MCGR routine. Several consistency checks are incorporated into the data treatment to ensure that systematic errors are reduced to a minimum. The results confirm that formic acid is strongly hydrogen-bonded liquid with well-defined local orientational correlations. The molecular conformation is planar and the bond lengths are similar to those of the gas and solid phases. Different studies [35] have shown that these features are retained as the temperature is changed and there is no correspondence to the structural rearrangement that occurs in water. The general picture to emerge from the measurements is of hydrogenbonded structure with local intermolecular O–H–O configurations arising predominantly from the hydrogen-bonded chains. Ab initio quantum chemical calculations in present work showed that the non-planar configuration suggested by Bertagnolli et al. [6,7], is very unlikely both for formic acid dimers and monomers. Another possibility for further investigation concerns a second stage analysis of the over-determined datasets to check for possible isotopic variation that would indicate a difference between hydrogen and deuterium atoms in the hydrogen bond. Previous investigations, have revealed small differences but it has been impossible to extract any quantitative information due to the inherent difficulties arising from the analytic corrections for the inelasticity effects, The over-determined datasets for the DCOO[H/D] samples provide a suitable combination for a closer investigation of the small residual discrepancies resulting from the evaluation of several functions based on H/D equivalence. However, it is possible that these effects will still be within the uncertainties resulting from statistical accuracy and systematic errors and consequently a further publication on this effect will depend on the outcome of the analysis.

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