The Conformational Distribution in Diphenylmethane Determined By

JOURNAL OF CHEMICAL PHYSICS VOLUME 118, NUMBER 14 8 APRIL 2003

The conformational distribution in diphenylmethane determined by nuclear magnetic resonance spectroscopy of a sample dissolved in a nematic liquid crystalline solvent G. Celebre and G. De Luca Dipartimento di Chimica, Universita della Calabria, 87030 Arcavacata di Rende (Cs), Italy J. W. Emsley and E. K. Foord Chemistry Department, University of Southampton, Southampton SO17 1BJ, United Kingdom M. Longeri, F. Lucchesini,a) and G. Pileio Dipartimento di Chimica, Universita della Calabria, 87030 Arcavacata di Rende (Cs), Italy ͑Received 10 October 2002; accepted 3 January 2003͒ The deuterium decoupled, proton nuclear magnetic resonance spectrum of a sample of diphenylmethane-d3 dissolved in a nematic liquid crystalline solvent has been analyzed to yield a set of dipolar couplings, Dij . These have been used to test models for the conformational ␶ ␶ distribution generated by rotation about the two ring-CH2 bonds through angles 1 and 2 . Conformational distributions, particularly when obtained from a quantum chemistry calculation, are ␶ ␶ usually described in terms of the potential energy surface, V( 1 , 2), which is then used to deﬁne ␶ ␶ a probability density distribution, P( 1 , 2). It is shown here that when attempting to obtain ␶ ␶ P( 1 , 2) from experimental data it can be an advantage to do this directly without going through ␶ ␶ the intermediate step of trying to characterize V( 1, 2). When applied to diphenylmethane this method shows that the dipolar couplings are consistent with a conformational distribution centered ␶ ϭ␶ ϭ Ϯ on 1 2 56.5 0.5°, which is close to the values calculated for an isolated molecule of 57.0°, and signiﬁcantly different from the asymmetric structure found in the crystalline state. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1555631͔

I. INTRODUCTION methods are either of limited applicability, or are at an early stage of development. Here we present a study of diphenyl- Molecules with internal bond rotational possibilities usu- methane by the method which relies on being able to obtain ally adopt a single conformation in the solid state, but have a partially-averaged dipolar couplings, Dij , between pairs of conformational distribution in liquids and gases. The possi- nuclei by analysing the nuclear magnetic resonance ͑NMR͒ bility exists that the global energy minimum changes in spectrum of a sample dissolved in a liquid crystalline phase. structure between the three phases because of the influence This so-called LXNMR method has a wide range of applica- of intermolecular forces. To investigate this possibility it is bility, and has been used for small, rigid molecules such as necessary to determine the structure and conformation in benzene1 in order to make a comparison at high precision each phase. The structure in the solid state can be obtained between structures obtained in solid, liquid, and gaseous with high precision, and relatively easily by x-ray or neutron phases, and for molecules as large as ubiquitin to obtain diffraction, if the compound forms good crystals at an acces- coarse grained structures of this and similar proteins in their sible temperature. These methods may also be applied to folded states.2,3 molecules of large or small molar mass. For the gaseous state A study of the conformation of diphenylmethane is im- the experimental methods are more limited in their range of portant because it is the simplest case of linkage of two aro- application, and there may also be severe experimental diffi- matic rings through a methylene group. Such a linkage ex- culties. An alternative is to calculate an energy surface by ists, for example, in trimethoprim, an important therapeutic solving Schro¨dinger’s equation for the electronic wave func- agent, and in some recently synthesized liquid crystals.4 tions of an isolated molecule. This approach always involves Diphenylmethane also presents several interesting chal- some degree of approximation, which varies with the number lenges to the LXNMR method, which we will address here. of atoms and electrons in the molecule, and at the present First, the proton spectrum of a sample dissolved in a liquid time there is considerable interest in investigating the valid- crystalline solvent is that of 12 interacting spins, and it is too ity of the commonly available computational methods, to- complex to be analyzed without following some spectral gether with their associated computer programs. analysis simplification strategy. Also, the question has to be For the liquid state, which chemically and biologically is asked of whether it is necessary to analyze the spectrum of generally the most interesting, the available experimental the fully protonated molecule or would a smaller data set, obtained from the analysis of selectively deuteriated isoto- a͒Present address: Dipartimento di Chimica e Tecnologie Farmaceutiche e pomers, be sufficiently large to test models for the confor- Alimentari, Universita` di Genova, 16147 Genova, Italy. mational distribution? The approach adopted here is to pre-

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and the deuterated benzene removed by rotary evaporation. The crude product, which was about 80% deuterated, was used without further puriﬁcation.

2. Synthesis of compound 2 ͑ ͒ ͑ Benzyl chloride 3.80 g, 30.0 mmol and benzene-d6 15 ml͒ in a three-necked flask equipped with a calcium chloride guard tube, a magnetic stirrer and a water-cooled condenser were cooled with an ice bath under stirring and tin ͑IV͒ chloride ͑0.40 g, 1.53 mmol͒ was added slowly in five equal amounts. After stirring for 15 min D2O was added (5 ϫ10 ml) to decompose the catalyst. The mixture was washed with water (2ϫ10 ml) and 1 M HCl (2ϫ10 ml), the organic layer was separated and dried over magnesium sul- FIG. 1. Structure of isotopomers and atomic labeling of diphenylmethane. phate. Volatile impurities were removed at 100 °C/3 Torr, leaving behind the desired diphenylmethane-d5 2. pare suitably deuteriated samples and to analyze the simpler, 3. Synthesis of compound 3 1H–͕2H͖ spectra. Second, in order to derive a conformational distribution a. Preparation of bromobenzene-2,4,6-d3. Aniline hy- ͑ ͒ ͑ ͒͑ from a set of Dij it is necessary to have a theoretical model drochloride 6.00 g, 47.4 mmol and D2O 12.00 g Sigma- to treat the coupling between internal and reorientational mo- Aldrich Dϭ99.9%) were introduced into a flask equipped tions, and a parameterized functional form for the conforma- with a stirrer and a water-cooled condenser. The solution was tional surface which is then brought into agreement with the heated at reflux under nitrogen for 24 h and the water was experiment data by varying the parameters in a system- removed under reduced pressure. The procedure was re- atic way. peated four times, then the aniline hydrochloride-2,4,6-d3 It is usual to describe the conformational surface by a was dissolved in 10 ml 10% NaOH solution, the base ex- ϫ potential energy function, most often a Fourier series, plus a tracted into Et2O(3 50 ml) and dried over sodium sul- ͑ term to take into account steric, repulsive interactions. Such phate. Removal of the solvent left the aniline-2,4,6-d3 3.23 an approach is possible for diphenylmethane, but we will g, 52% yield, 2Hϭ98%). ͑ ͒ show that it is simpler to use a direct probability description The aniline-2,4,6-d3 3.0 g, 31.2 mmol was dissolved in of the conformational distribution. This new probability ap- aqueous 8.83 M HBr ͑22 ml͒ and diazotized with 12.2 ml of proach is of a general nature, and has particular advantages aqueous 2.5 M sodium nitrite at TϽ5 °C. This diazonium for cases where rotations about two bonds ͑such as diphenyl- salt solution was poured into a three-necked flask equipped methane͒ or more are of a cooperative nature. with a stirrer and a water-cooled condenser containing cop- Third, diphenylmethane has been studied by x-ray per ͑I͒ bromide ͑6.34 g, 44.2 mmol͒ and 18 ml conc. HBr. diffraction,5 and by several quantum chemistry methods,6,7 The mixture was heated at reflux for 2 h, steam distilled, and and these results act as a useful set of criteria to judge the the product extracted into Et2O. The organic layer was success of the LXNMR method. washed with 10% sodium hydroxide solution, then with water up to neutrality, dried over sodium sulphate, and the sol- II. EXPERIMENT vent removed by rotary evaporation. The crude product was distilled (53 °C/23 Torr) to yield bromobenzene-2,4,6-d3 A general way of extracting dipolar couplings between ͑2.10 g, 50% yield͒. protons is to simplify the spectrum by replacement of 1Hby 2 b. Preparation of diphenylmethanol-2,4,6-d3. Bromo- H, followed by deuterium decoupling. To apply this method ͑ ͒ ͑ benzene-2,4,6-d3 2.00 g, 12.6 mmol in anhydrous Et2O 35 to diphenylmethane required the synthesis of compounds ml͒ was cooled to 0 °C under nitrogen. n-Butyl lithium in ͑ ͒ 1–3 Fig. 1 . hexanes ͑1.68 M, 8.3 ml͒ was added slowly, dropwise, keep- A. Synthesis of isotopically labeled samples ing the temperature below 2 °C. The mixture was stirred at room temperature for 1 h after which was cooled again at 1. Synthesis of compound 1 0 °C and benzaldehyde ͑1.48 g, 14.0 mmol͒ added. The cool- ͑ ͒ Diphenylmethane 0.50 g, 2.97 mmol and benzene-d6 ing bath was removed and the mixture left at room tempera- ͑ ͒ ͑ ͒ 10 ml were introduced into a three-necked flask fitted with ture for 2.5 h. A solution of NH4Cl 10 ml was added, the a calcium chloride guard tube and a water-cooled condenser. organic layer separated, washed with water and dried over The flask was flushed with nitrogen;a1Msolution of ethy- sodium sulphate. Removal of the solvent left the crude prod- laluminum dichloride in hexanes ͑2ml͒͑Aldrich͒ was added uct ͑2.3 g, 98% yield͒, which was used in the next stage and the solution refluxed for 1 h under stirring. The solution without purification. was allowed to cool, opened to the air, and D2O added until c. Preparation of benzophenone-2,4,6-d3. Diphenyl- ͑ ͒ the yellow color in the organic layer disappeared. The or- methanol-2,4,6-d3 2.35 g, 12.5 mmol in dry dichlo- ganic layer was separated, dried over magnesium sulphate, romethane ͑17 ml͒ was added to a stirred mixture of pyri-

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FIG. 3. 200 MHz 1H–͕2H͖ NMR spectrum of a sample of diphenyl- ͑ ͒ methane-d3 compound 3 dissolved in the nematic solvent ZLI 1132 at 300 K. FIG. 2. 200 MHz 1H–͕2H͖ NMR spectrum of a sample of dipheny- ͑ ͒ lmethane-d5 compound 2 dissolved in the nematic solvent ZLI 1132 at 300 K. observed. The first step in obtaining the ‘‘good’’ set of Dij was to record a 1H–͕2H͖ spectrum of 1, which is a doublet with a separation of 3͉D ͉. To proceed further, the 2H dinium chlorochromate ͑4.13 g, 19.1 mmol͒ and sodium ac- 24,25 spectrum of 2 was recorded. This consists of just four lines: etate ͑0.31 g, 3.8 mmol͒ in dry dichloromethane ͑17 ml͒. The an outer pair from the deuterium at position 21, with com- mixture was stirred for2hatroom temperature and then dry bined intensity of unity, and a separation ⌬␯ , and an inner Et O ͑35 ml͒ was added. The organic phase was separated p 2 pair, intensity 4, separation ⌬␯ from the deuteriums at and the solvent was removed using a rotary evaporator to o,m positions 19, 20, 22, and 23. The deuteriums at positions 19 leave an oily product, which was distilled under reduced and 23 are expected to be equivalent and to give one qua- pressure (105 °C/0.9 Torr) to yield the product ͑1.35 g, 58% drupolar doublet, and the equivalent pair of deuteriums at yield͒. positions 20 and 22 should in principle give another, separate d. Preparation of diphenylmethane-2,4,6-d . A solution 3 doublet. The observation that only one doublet is observed of benzophenone-2,4,6-d ͑1.34 g, 7.23 mmol͒, hydrazine 3 shows that the chemical shift difference between these four hydrate ͑1.24 ml, 0.0247 mol͒, and KOH ͑1.85 g, 0.0330 deuteriums is negligible, and that the vectors r and r mol͒ in triethylene glycol ͑4ml͒ was refluxed for 2 h with 9,19 12,22 are collinear, and so too are r and r . vigorous stirring. Pentane ͑20 ml͒ was added at room tem- 10,20 13,23 The quadrupolar splittings are related to local order perature and the organic layer separated, washed with water parameters,8,9 SR ,by and dried with sodium sulphate. The solvent was removed ␣␤ ⌬␯ ϭ͑ ͒ R ͑ ͒ and the residue distilled under reduced pressure p 3/2 qCDScc , 1 (69 °C/2 Torr) to give the desired product 3 ͑0.586 g, 47% ⌬␯ ϭ͑ ͒ ͓ R ͑ 2 ␪ Ϫ ͒ yield͒. o,m 3/4 qCD Scc 3 cos o,m 1 All the compounds have been characterized by NMR ϩ͑SR ϪSR ͒sin2 ␪ ͔, ͑2͒ and the isotopic purity of compounds 2 and 3 was deter- aa bb o,m mined to be Ͼ95%. where it has been assumed that the quadrupolar tensors for the deuteriums at the three ring positions are equal and are B. NMR experiments each axially-symmetric about the bond directions. The axes Samples of compounds 1–3 were prepared by dissolving a and c are fixed in the ring plane, as shown in Fig. 1, and b approximately 10% by weight in the nematic mixture ZLI is the normal to the plane. The angle between the C–D di- 1 2 2 ␪ 1132, obtained from Merck ͑UK͒ Ltd. The H–͕ H͖, and H rections and the c axis, o,m , is taken to be 60°. The qua- spectra were obtained with a Bruker MSL 200 spectrometer drupolar coupling constant, qCD , is assumed to have the at 300 K on samples contained in 5 mm o.d. sample tubes value of 185 kHz. The signs of the quadrupolar splittings are using a probe witha7mmhorizontal solenoid coil. Deute- not determined by the experiments, and so there are four rium decoupling was achieved using a single pulse at the possible solutions for the two order parameters. These are center of the deuterium spectrum and applied only during the used to predict the dipolar couplings between protons in the acquisition period. A delay of 5 s between pulses was al- ring of 2 from lowed in order to minimize the heating effects of the deute- D ϭϪ͑␮ ␥2 h/32␲3r3 ͓͒SR ͑3 cos2 ␪ Ϫ1͒ rium irradiation. ij 0 H ij cc ijc ϩ͑ R Ϫ R ͒͑ 2 ␪ Ϫ 2 ␪ ͔͒ ͑ ͒ Saa Sbb cos ija cos ijb , 3 III. ANALYSIS OF SPECTRA ␮ ␥ where 0 is the permeability of free space and H is the The 1H–͕2H͖ spectra of 2 and 3 are complex, as shown gyromagnetic ratio for the proton. in Figs. 2 and 3, and to analyze them by the conventional The 2H spectrum given by 3 is shown in Fig. 4, and has approach requires that a trial set of Dij are obtained which fine structure on the peaks from dipolar coupling to the pro- can be used to simulate spectra which are close to those tons as shown in the expanded section. The triplet structure

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⌬␯ TABLE I. Chemical shifts ij , and dipolar couplings, Dij , obtained from the analysis of the 200 MHz 1H–͕2H͖ spectrum of a sample of ͑ ͒ diphenylmethane-d5 compound 2 dissolved in ZLI 1132.

a b ij Jij /Hz Dij /Hz 19,20 8.00 Ϫ994 19,21 2.00 Ϫ146 Ϫ 19,22 ¯ 38 19,23 2.00 Ϫ14 Ϫ 19,24 ¯ 429 20,21 8.00 Ϫ311 20,22 2.00 Ϫ15 Ϫ 20,24 ¯ 83 Ϫ 21,24 ¯ 54 24,25 ¯ 2068 ⌬␯ ij /Hz 19,21 Ϫ27 20,21 Ϫ44 24,21 Ϫ614

aW. Brugel, NMR Spectra and Chemical Structure ͑Academic, New York, 1967͒. bThe spectrum was simulated with a total rms error of 7 Hz. The error on each NMR parameter is estimated to be Ϯ0.5 Hz.

ϵ ˜ ϭ ͵ ͑␶ ␷ ͒ ͑␶ ␷ ͒ ␶ ␷ ͑ ͒ Dij DijZZ PLC k , m Dij k , m d kd m , 4 ␶ ␷ where PLC( k , m) is the probability in the liquid crystal phase that the molecule is in a conformation defined by the ␶ bond rotation angles k , and a vibrational state defined by ␷ m . A vibrational state is defined here as involving only FIG. 4. 30.7 MHz spectrum from the deuteriums in a sample of small amplitude displacements of nuclei, and does not there- ͑ ͒ diphenylmethane-d3 compound 3 dissolved in the nematic solvent ZLI fore include bond rotational motion. The definition of 1132 at 300 K: ͑top͒ normal spectrum and, ͑bottom͒ expanded sections ␶ ␷ where the fine structure of multiplets is shown. Dij( k , m)is

⌬␯ of the peak from deuterium at position 16 is from coupling to TABLE II. Chemical shifts, ij , and dipolar couplings, Dij , obtained 1 2 the protons at positions 15 and 17, and this allows a value to from the analysis of the 200 MHz H–͕ H͖ spectrum of a sample of ͑ ͒ ͉ ͉ diphenylmethane-d3 compound 3 dissolved in ZLI 1132; the calculated be obtained for D15,16 . The peaks from deuterium at posi- values were obtained by the probabilistic approach without ͑column A͒ and tion 14 ͑and 18͒ show a doublet from coupling to proton at with ͑column B͒ vibrational corrections to the dipolar couplings. position 15, and a triplet from coupling to protons at positions 24 and 25, and these allow values to be obtained of i, j Dij /Hz ͉D ͉ and ͉D ͉. Calculated 14,15 14,24 Experiment AB These experiments enabled good estimates to be made of all the dipolar couplings within the C H CH -group, with 15,17 Ϫ15.0Ϯ0.1 Ϫ15.1 Ϫ14.1 6 5 2 Ϫ Ϯ Ϫ Ϫ the exception of D , and this was estimated by compari- 15,19 70.0 0.1 70.0 70.2 15,24 15,20 Ϫ40.7Ϯ0.1 Ϫ40.6 Ϫ41.1 son with the set of data obtained for benzyl chloride dis- 15,21 Ϫ35.7Ϯ0.1 Ϫ36.4 Ϫ36.3 10 solved in the same liquid crystalline solvent. With this set 15,24 Ϫ73.0Ϯ0.1 Ϫ73.0 Ϫ73.6 of starting values it was possible to analyze the 1H–͕2H͖ 19,20 Ϫ897.1Ϯ0.1 Ϫ897.0 Ϫ896.5 spectrum of 2 shown in Fig. 2 by the method of Celebre 19,21 Ϫ133.2Ϯ0.1 Ϫ133.2 Ϫ135.0 11 19,22 Ϫ35.4Ϯ0.1 Ϫ35.2 Ϫ35.8 et al. to give the data in Table I. These Dij were then used Ϫ Ϯ Ϫ Ϫ 1 ͕2 ͖ 19,23 15.0 0.1 15.1 14.1 as starting parameters in the analysis of the H– H spec- 19,24 Ϫ391.9Ϯ0.1 Ϫ391.9 Ϫ391.9 trum of compound 3, which is shown in Fig. 3, to give the 20,21 Ϫ281.1Ϯ0.1 Ϫ281.1 Ϫ281.2 data in Table II. 21,24 Ϫ48.6Ϯ0.1 Ϫ50.1 Ϫ51.1 24,25 1847.5Ϯ0.1 1847.5 1847.5 IV. CONFORMATIONAL ANALYSIS: THEORY R/Hz 0.4 1.0 ⌬␯ A. General ij /Hz Ϫ Ϯ The D are averages over all the normal modes of the 15,21 18.0 0.1 ij 19,21 Ϫ50.0Ϯ0.1 molecule relative to the applied magnetic ﬁeld, which in the 24,21 Ϫ628.8Ϯ0.1 present case is parallel to the mesophase director, n. Thus,12

␮ ␥ ␥ h U ͑␤,␥,␶ ͒ϭU ͑␤,␥,␶ ͒ϩU ͑␶ ͒. ͑10͒ ͑␶ ␷ ͒ϭͩ 0 ͪ i j ͓ ͑␶ ␷ ͒ LC k ext k int k D , ␶ ␷ S , ij k m 4␲ ␲2͑ k , m͒3 zz k m ␶ ␤ ␥ ␶ 8 rij The function PLC( k) is related to PLC( , , k), ϫ͑3 cos2 ␪k,mϪ1͒ϩ͑S ͑␶ ,␷ ͒ ijz xx k m P ͑␶ ͒ϭ ͵ P ͑␤,␥,␶ ͒sin ␤d␤d␥ ͑11͒ LC k LC k Ϫ ͑␶ ␷ ͒͒͑ 2 ␪k,mϪ 2 ␪k,m͔͒ Syy k , m cos ijx cos ijy , ϭZϪ1 exp͓ϪU ͑␶ ͒/k T͔ ͑5͒ int k B ␶ ␷ where Szz( k , m), etc. vary with the conformation and vi- ϫ ͵ ͓Ϫ ͑␤ ␥ ␶ ͒ ͔ ␤ ␤ ␥ ͑ ͒ exp Uext , , k /kBT sin d d . 12 brational state. These order parameters are for principal axes, and these axes will also change their orientation in different ␶ The order parameters S␣␣( k) are obtained as conformational and vibrational states. It is usual to assume that the dependence of the order ͑␶ ͒ϭ ͑␶ ͒Ϫ1 ͵ 1͑ 2 ␪k Ϫ ͒ S␣␣ k W k 2 3 cos ␣ 1 parameters on the vibrational state13 is much smaller than n their dependence on the bond rotational angles, and can be ϫexp͓ϪU ͑␤,␥,␶ ͒/k T͔sin ␤d␤d␥ ͑13͒ neglected to a good approximation. Similarly it is argued that ext k B ␶ ␷ ␷ the dependence of Dij( k , m)on m can also be neglected with ␶ when the main interest is in trying to characterize PLC( k), ͑␶ ͒ϭ ͵ ͓Ϫ ͑␤ ␥ ␶ ͒ ͔ ␤ ␤ ␥ ͑ ͒ the conformational distribution, from a set of experimental W k exp Uext , , k /kBT sin d d , 14 dipolar couplings. To support this view it can be noted that ␪k vibrational corrections to Dij in rigid molecules are usually where ␣ n is the angle between the director and principal Ͻ ␶ ␷ ␶ ␣ 5%, whereas the dependence of Dij( k , m)on k is often axis when the molecule is in the conformation specified by ␶ Ͼ100%. Of course, the main reason for neglecting the ef- the values of the set k . fects of vibrations when large amplitude motion is also Making the division shown in Eq. ͑10͒ allows a confor- ␶ present is the difficulty of their calculation. Here the effects mational distribution, Piso( k), to be defined as of vibrations will be neglected at first, but later an approxi- P ͑␶ ͒ϭQϪ1 exp͓ϪU ͑␶ ͒/k T͔ ͑15͒ mate method for their inclusion will be explored. iso k int k B Neglecting vibrational motion reduces Eqs. ͑4͒ and ͑5͒ with to ϭ ͵ ͓Ϫ ͑␶ ͒ ͔ ␶ ͑ ͒ Q exp Uint k /kBT d k . 16 ϭ ͵ ͑␶ ͒ ͑␶ ͒ ␶ ͑ ͒ Dij PLC k Dij k d k 6 This distribution is for the same temperature, and the same ␶ and solvent as PLC( k), but is for an isotropic phase. Note that ␶ Þ ␶ Piso( k) PLC( k), but that these distributions are related by ␮ ␥ ␥ h ͑␶ ͒ϭϪͩ 0 ͪ i j ͓ ͑␶ ͒͑ 2 ␪k Ϫ ͒ Dij k ␶ Szz k 3 cos 1 P ͑␶ ͒ϭP ͑␶ ͕͒Z/͓W͑␶ ͒Q͔͖. ͑17͒ 4␲ ␲2͑ k͒3 ijz iso k LC k k 8 rij ϩ͑ ͑␶ ͒Ϫ ͑␶ ͒͒͑ 2 ␪k Ϫ 2 ␪k ͔͒ B. The additive potential AP model for U ␤,␥,␶ Sxx k Syy k cos ijx cos ijy . „ … ext „ k… ͑7͒ There have been several suggestions as to how to model ␤ ␥ ␶ 14,15 Uext( , , k). For flexible molecules in particular there It is tempting to assume that the dependence of S␣␣(␶ )on k are advantages in using a general expansion in spherical har- ␶ can also be neglected, but although this is usually done by k monics, and for a uniaxial phase this is those using dipolar couplings to study protein structures it is ϱ likely to lead to major errors in the function P (␶ ) and the LC k ͑␤ ␥ ␶ ͒ϭϪ ␧ ͑␶ ͒ ͑␤ ␥͒ ͑ ͒ structures derived when there is an appreciable degree of Uext , , k ͚ L,m k CL,m , . 18 L,m intramolecular motion present in the molecule. even ␤ ␥ ␶ To proceed further, a probability, PLC( , , k), that the ␤ ␥ The CL,m( , ) are modified spherical harmonics, and the molecule is at an orientation with the director defined by the ␧ (␶ ) are conformationally-dependent coefficients whose ␤ ␥ L,m k polar angles and whilst in a conformation described by magnitudes depend upon the strength of the anisotropic part ␶ the set k is introduced. It is defined in terms of a mean of the solute–solvent interaction. Computer simulations of ␤ ␥ ␶ potential, ULC( , , k), such that solutes in liquid crystalline solvents suggest16 that the expan- ␤ ␥ ␶ ͒ϭ Ϫ1 Ϫ ␤ ␥ ␶ ͒ ͑ ͒ sion in Eq. ͑18͒ can be restricted to Lϭ2, and mϭ0, Ϯ2 PLC͑ , , k Z exp͓ ULC͑ , , k /kBT͔ 8 when the aim is to use it to obtain averages of second-rank with interactions, such as dipolar or quadrupolar couplings. In the ␧ ␶ version of the AP method used here the L,m( k) will be ϭ ͵ ͓Ϫ ͑␤ ␥ ␶ ͒ ͔ ␤ ␤ ␥ ␶ ͑ ͒ ␧ Z exp ULC , , k /kBT sin d d d k . 9 approximated as sums of contributions L,m( j) from rigid molecular subunits, ␤ ␥ ␶ The mean potential can be divided into a part, Uext( , , k), ␶ which vanishes for an isotropic phase, and a part, Uint( k), ␧ ͑␶ ͒ϭ ␧ ͑ ͒ 2 ͑⍀ ͑␶ ͒͒ ͑ ͒ 2,m k ͚ 2,p j D p,m j k , 19 which does not, p, j

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This gives an energy surface of the correct periodicity and symmetry, but it has the disadvantage shared by Fourier ex- ␶ pansions for Uint( k) that it is not possible to vary the position of the energy minima, their relative energies, and the barriers between them in an independent way. This disadvantage can be overcome by adding more terms to the expansion, when ﬁtting a functional form to the results from a calculation of an energy surface. Adding extra terms to Eq. ͑21͒ is usually not feasible when ﬁtting observed data, such as averaged dipolar couplings, to a model for a probability distribution as in Eqs. ͑6͒, ͑8͒, and ͑10͒, because of the limited amount of experimental data available.

D. The direct probability description of conformational distributions ␶ An alternative is to determine the probability PLC( k) directly without going through the intermediate stage of ﬁrst characterizing an energy surface. Such a probabilistic approach was proposed initially by Zannoni17 for obtaining ␤ ␥ ␶ PLC( , , k) from averaged NMR data. The original ␤ ␥ ␶ Zannoni method derived a form for PLC( , , k) by apply- ing the maximum entropy ͑ME͒ principle. This gives FIG. 5. Conformations of diphenylmethane and their symmetry point ME͑␤ ␥ ␶ ͒ϭ Ϫ1 ͓Ϫ ͑␤ ␥ ␶ ͒ ͔ ͑ ͒ groups. PLC , , k ZME exp UME , , k /kBT 22 with 2 ⍀ ␶ where D ( j( k)) is the Wigner function describing the p,m ϭ ͵ ͓Ϫ ͑␤ ␥ ␶ ͒ ͔ ␤ ␤ ␥ ␶ orientation of fragment j in molecular reference axes. ZME exp UME , , k /kBT sin d d d k . ͑23͒ C. Modelling the internal potential U ␶ int„ k… U (␤,␥,␶ ) has the same form as U (␤,␥,␶ ) in Eq. ͑18͒ as a truncated Fourier series ME k ext k except that the coefﬁcients depend directly on the experi- ␶ There are also advantages in expanding Uint( k)ina mental data and hence contain information on both the ori- general way, and the most commonly adopted choice is as a entational order and the conformational distribution. This has truncated Fourier series, such as the advantage of not going beyond the data available, and hence has been described as being the least biased approach. ␶ ͒ϭ ␶ ͒ ͑ ͒ ␤ ␥ ␶ Uint͑ k ͚ Vn cos͑n l , 20 The method has the disadvantage however that PLC( , , k) l,n in the isotropic phase becomes a constant independent of with n taking values to give the correct periodicity of both orientational order, as it should, and conformation, ␶ Uint( k), and the series being kept as short as will represent which it should not. This disadvantage can be removed by ␤ ␥ ␶ the correct overall shape of this bond rotational potential ͑at introducing the constraint that PLC( , , k) become identi- ␶ 18,19 least near to the expected potential minima͒. However, for cal with Piso( k) in the isotropic phase. molecules like diphenylmethane this simple form for The method proposed here offers a different vision of the ␶ ␶ Uint( k) is not able to describe correctly the energy surface. probabilistic approach. The distribution function PLC( k)is Thus, with nϭ2 and 4, which allows for energy minima with modeled directly as a sum ͑for interdependent conforma- Ͻ␶ ␶ Ͻ ͑ ͒ ͒ ͑ ͒ 0° 1 , 2 90°, Eq. 20 predicts equal energies for struc- tions or product for independent states of Gaussian func- ͑ ͒ tures with symmetry C2 and Cs see Fig. 5 . This energy tions. Thus, for rotation about the lth bond between Nl con- degeneracy can be removed by adding additional terms. formations,

Thus, adding a term to account for short-range repulsion, Nl which depends on r , differentiates between the structures ͑␶ ␶0 ͒ϭ ͓Ϫ͕ ͑␶ Ϫ␶0 ͖͒2͔ ij PLC l ,Nl ,kl , l ͚ Al exp kl l l , with C and C symmetries, but by only a small amount that j j jϭ1 j j j 2 s ͑ ͒ could be insufﬁcient for diphenylmethane. A more ﬂexible 24 ͑ ͒ ␶ ␶0 approach is to add terms to Eq. 20 which depend on ϩ where the l are the positions of maximum probability, Al ϭ␶ ϩ␶ ␶ ϭ␶ Ϫ␶ j j 1 2 and Ϫ 1 2 , are their relative amplitudes, and k gives the width of the l j ␶ ␶ ͒ϭ ␶ ϩ ␶ ͒ Uint͑ 1 , 2 V2͑cos 2 1 cos 2 2 distribution. The physical interpretation is that the molecule exists in j conformations, centered on ␶0 , and that there is a ϩV ͑cos 4␶ ϩcos 4␶ ͒ l j 4 1 2 symmetric oscillation at each. The relative amounts of each ϩVϩ cos 2␶ϩϩVϪ cos 2␶Ϫ . ͑21͒ conformer are obtained by integration.

␶ ϭ ␶ ϭ and the HCH plane are 1 63.9° and 2 71.1°. The asym- metry must arise because of the asymmetric environment in the solid state, and is not expected either for an isolated molecule, or for an achiral liquid phase. Quantum chemistry calculations on an isolated diphenylmethane molecule6,7 suggest that only the structures with C2 symmetry are appre- ␶ ␶ ciably populated. The probability function, PLC( 1 , 2) describing these conformations is conveniently expressed ␶ ϭ␶ ϩ␶ by introducing the angular variables ϩ 1 2 and ␶ ϭ␶ Ϫ␶ Ϫ 1 2 , ␶ ␶ ͒ϭ ␶ ͒ ␶ ͒ ͑ ͒ PLC͑ ϩ , Ϫ Pϩ͑ ϩ PϪ͑ Ϫ . 25 ␶ ␶ The correct periodicity of PLC( ϩ , Ϫ) is obtained by introducing indices s, n, and j, 1ϩs͑2Ϫs͒ ␶ ␶ ͒ϭ PLC͑ ϩ , Ϫ ͚ ͚ ͚ sϭ0,1,2 nϭϪ 1/2 , 1/2 jϭϪ 1/2 , 1/2 2 ͓Ϫ 2 2 ͔ ͓Ϫ 2 2 ͔ ͑ ͒ • exp xϩ/2hϩ exp xϪ/2hϪ 26 with 0 xϩϭ␶ϩϩ2 j͑s␲Ϫ␶ϩ͒ ͑27͒ and

xϪϭ␶Ϫϩ2sn␲͑2Ϫs͒. ͑28͒

The maximum probability is constrained to have ␶Ϫϭ0° by the C2 symmetry requirement, and to be at positions along 0 ␶ϩ determined by ␶ϩ. The value of hϩ gives the full width at half maximum ͑FWHM͒ of the Gaussian function ͑related to the amplitude of conformational oscillation in the ␶ϩ direction͒ whilst hϪ is the same along ␶Ϫ . The maps represented in Fig. 7 show ␶ ␶ how the shape of the distribution PLC( ϩ , Ϫ) depends on the relative values of hϩ and hϪ . ␤ ␥ ␶ ␶ The AP method was used to model Uext( , , 1 , 2) ␧R ␧R Ϫ␧R with identical contributions zz and xx yy from each phe- 0 nyl ring. These were varied, together with hϩ , hϪ , ␶ϩ, and any appropriate geometrical parameters, to minimize the tar- get function 1/2 Rϭͭ M Ϫ1͚ ͓D ͑observed͒ϪD ͑calculated͔͒2ͮ ij ij iϽ j ͑29͒ FIG. 6. The three Gaussian functions used to describe the three conforma- with Mϭnumber of independent couplings. tions generated by rotation about the C–C bond in a compound of type

XCH2CH2Y.

A simple example is rotation about the C–C bond in VI. RESULTS AND DISCUSSION compounds of the type XCH2CH2Y, as shown in Fig. 6. The The calculations proceeded ﬁrst with the neglect of the variables used to describe this situation are ␶ , k , k , and g g t effect on the Dij of small amplitude vibrational motion. The At ; these can be varied independently, but with the con- ﬁrst step was to adjust the positions of protons in the aro- straint that P (␶, j,k ,␶ ) is normalized. We will return to a LC j j matic rings to minimize R for the intraring Dij . The spectral more detailed description of this case, with a comparison to ϭ ϭ analysis produces D19,23 D20,22 ( D15,17), and so the dis- the conventional cosine expansion of a potential energy func- tances r19,23 , r20,22 , and r15,17 must be equal. Fixing the ring tion, in the future. geometry to have: ϭ ϭ ϭ ϭ V. APPLICATION OF THE PROBABILISTIC all rCC 1.40 Å, except r12,13 ( r6,7 r9,10 r3,4); APPROACH TO DIPHENYLMETHANE all CCC angles 120° in the rings; ϭ ϭ The diphenylmethane molecules in the crystalline state all rCH 1.085 Å except r11,21 ( r5,16); ␧R ␧R Ϫ␧R ͑ are asymmetric in that the angles between the ring normals the function R was minimized by varying zz , xx yy or

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␶ ␶ ͑ ͒ FIG. 7. The effect of hϩ and hϪ on the distribution PLC( ϩ , Ϫ); a has 0 hϩϭhϪ , and in ͑b͒ hϩϭ1.5 hϪ . The value of ␶ϩ is the same in both cases.

TABLE III. Optimized and assumed geometrical parameters for ͑ ͒ R diphenylmethane-d3 compound 3 and local order parameters, S␣␤ , for the aromatic ring.

Parameters Optimized values FIG. 8. The distribution P (␶ϩ ,␶Ϫ) determined by the probabilistic ap- Ϯ LC r9,10 /Å 1.398 0.002 proach: ͑a1͒ and ͑a2͒ without, and ͑b͒ with vibrational corrections to the Ϯ ͑ ͒ r11,21 /Å 1.080 0.001 dipolar couplings see text for explanation . R Ϯ Scc 0.1140 0.0001 SR ϪSR 0.1336Ϯ0.0001 aa bb R R Ϫ R Assumed valuesa equivalently Szz and Sxx Syy), r9,10 and r11,21 to give the results shown in Table III. The ring geometries were then CH ͑ring͒/Å 1.085 kept fixed in all subsequent calculations. CC ͑ring͒/Å 1.400 The potential energy surface for diphenylmethane has Angles ͑ring͒/Å 120.0 been calculated by a number of computational methods,6,7 r /Å 1.521 1,2 and it was decided to investigate how well two such surfaces aTaken from Ref. 22 and kept fixed. fit the observed dipolar couplings. In both cases the first

TABLE IV. The values of the total rms error, R, and the geometrical parameters obtained by ﬁtting the observed dipolar couplings with the potential energy surface calculated from MNDO and B3LYP/6-31G*.

MNDO B3LYP/6-31G*

␣/° 114.7a 126.3Ϯ0.5 104.4Ϯ0.5 114.7a 118.6Ϯ0.5 119.5Ϯ0.5 ␤/° 105.2a 105.2a 124.7Ϯ0.5 105.2a 105.2a 118.3Ϯ0.5 R/Hz 84.7 29.7 17.3 91.9 5.1 4.7

aTaken from Ref. 22 and kept ﬁxed.

calculations used the same geometry as the quantum chem- A. The effect of allowing for vibrational averaging ␧R ␧R of the dipolar couplings istry calculations, and the only variables were zz and xx Ϫ␧R yy . The values of R obtained, given in Table IV, are The inclusion of vibrational averaging into the deriva- unacceptably large, and so the effect of retaining the calcu- ␶ ␶ tion of PLC( ϩ , Ϫ) is not straightforward. The main prob- lated energy surface but varying the geometry was explored. lem is the inclusion of small-amplitude vibrational motion in The results are also shown in Table IV, and again are unac- addition to the large amplitude bond rotational motion. This ceptable. Note that when the R value is not very high the is a general problem, but it is compounded in the case of value of the ␤ or ␣ angles are unacceptable. averaging of dipolar or quadrupolar couplings by having to Fitting the Dij by the probabilistic approach, varying R R R 0 include the variation of the orientational order as the mol- ␧ , ␧ Ϫ␧ , ␶ , hϩ , hϪ , and the angle ␣ gives a close fit zz xx yy ϩ ecules changes shape during these two kinds of motion. An to the data and the parameters shown in Table V, column A. approximate method is used for diphenylmethane, which is In order to reduce the CPU requirements while keeping an described more fully by Lesot et al.20 This method consists high resolution (1°) in exploring the conformational surface, in calculating a wave function for small amplitude, harmonic the calculations have been carried out by implementing a ‘‘smart,’’ properly normalizing algorithm which catches only vibrational motion for the molecule in the conformation with ␶ ϭ␶ ϭ 1 2 56.5° by the B3LYP/6-31G* method using the the essence of the distribution, without redundancies. It var- 21 ␶ ␶ Ϫ␶ ϩ␶ GAUSSIAN 98 package. This is then used to calculate vibra- ies step by step 1 from 0° to 90° and 2 from 1 to 1 . tion corrections to the Dij , assuming that the order param- The obtained Dij are reported in Table II column A and the ␶ ␶ ͑ ͒ eters are independent of the vibrational motion. The effect of probability PLC( ϩ , Ϫ) obtained is shown in Fig. 8 a1 .It ␶ ϭ␶ ϭ Ϯ the additional averaging over the large amplitude bond rota- has a maximum at 1 2 56.5° 0.5°, compared with 57.0° ͑Ref. 6͒ obtained by the B3LYP/6-31G* calculation, tional motion is then done using the probabilistic method in and a rather flat surface, as seen by the large values of hϩ conjunction with the AP method for calculating how the or- ϭ29.5° and hϪϭ28.7°. The optimized value of ␣ of dering changes with this motion. 121.8°Ϯ0.5° is significantly larger than that of 114.6° deter- The ring geometry was first optimized with the inclusion 6 ϭ mined by the B3LYP/6-31G* method. of vibrational averaging. This produced r11,21 1.074 ␣ ␶ ␶ Ϯ ϭ Ϯ In Ref. 6, a strong correlation between and 1 and 2 0.002 Å, and r9,10 1.403 0.001 Å, which are changed was found, and the calculations with this theoretical relation- by Ͻ0.6%. Then, using all the vibrational modes and vary- ␧R ␧R Ϫ␧R ␶0 ␣ ship were repeated to give the data of column B in Table V. ing zz , xx yy , hϩ , hϪ , ϩ, and produced the results Note that the value of R is only slightly increased and the shown in Table V column C. The error, R, is considerably 0 position of the maximum probability changes by only 2.8° worse than without vibrational averaging. The value of ␶ϩ is ␶ ϭ␶ ϭ Ϯ ␶ ϭ␶ ϭ ( 1 2 59.3° 0.5°). The resulting distribution, shown in not changed significantly, corresponding to 1 2 58.8° Fig. 8͑a2͒, has quite different cross sections, the widths hϩ Ϯ0.5°. and hϪ differing noticeably (hϩϭ20.5°; hϪϭ47.4°) from The larger value of R is probably because the effect of ␶ ␶ the previous ones. an oscillation involving 1 and 2 is counted twice: once as a

␧ TABLE V. The interaction coefﬁcients, 2,m, and the parameters obtained by ﬁtting observed dipolar couplings with the probabilistic approach without vibrational corrections: optimizing the ␣ angle ͑column A͒ and impos- ing to the ␣ angle the behavior found from the B3LYP/6-31G* calculation ͑column B͒; and with vibrational contributions: using all the frequencies ͑column C͒ and without torsional frequencies ͑column D͒.

ABCD ␧ Ϫ1 Ϯ Ϯ Ϯ zz /kJ mol 0.926Ϯ0.001 0.881 0.001 0.882 0.001 0.915 0.001 ␧ ␧ Ϫ1 Ϯ Ϯ Ϯ xx- yy /kJ mol 1.286Ϯ0.001 1.292 0.001 1.277 0.001 1.274 0.001 hϩ /° 29.5Ϯ0.1 20.5Ϯ0.7 28.0Ϯ0.5 34.7Ϯ0.1 hϪ /° 28.7Ϯ1.5 47.4Ϯ0.4 25Ϯ19 21.3Ϯ3.7 0 ␶ϩ/° 113.0Ϯ1.0 118.6Ϯ1.0 120.0Ϯ1.0 110.2Ϯ1.0 ␣/° 121.8Ϯ0.5 120.7Ϯ0.5 121.3Ϯ0.5 R/Hz 0.4 1.0 6.5 1.0

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vibrational mode, and second by allowing the widths of the it suggests that this simple method of representing ␶ ␶ gaussian functions to vary. Note, too, the very large error on PLC( 1 , 2) in terms of Gaussian functions is a good ap- hϪ in column C of Table V, which is another indication that proximation. this model is ﬂawed. Excluding the vibrational modes ␶ ␶ strongly dependent on 1 and 2 from the calculation of the ACKNOWLEDGMENTS vibrational average leads to Rϭ1.0 Hz and the results in col- ␶ ϭ␶ ϭ Ϯ umn D of Table V. A value of 1 2 55.1 0.5° is now This work has been supported by MIUR PRIN ex 40%. obtained. There is a large change in the values of hϩ and The authors gratefully acknowledge Dr. Rosa Saladino for hϪ , which is reﬂected in a large change in the shape of the her help in the spectral analysis and Dr. J. Grunenberg for his ␶ ␶ ͑ ͒ distribution PLC( 1 , 2) as shown in Fig. 8 b . ‘‘remote’’ support and kindness during the progress of the work. J.W.E. thanks the Leverhulme Trust for the award of ␶ ␶ B. Using a jump model for PLC„ 1 , 2… an Emeritus Fellowship. It is often assumed that only conformations corresponding to the positions of energy minima need to be considered 1 J. Kaski, J. Vaara, and J. Jokisaari, J. Am. Chem. Soc. 118,8879͑1996͒. when averaging dipolar or quadrupolar couplings. In the 2 E. de Alba and N. Tjandra, Prog. Nucl. Magn. Reson. Spectrosc. 40,175 ͑2002͒. probabilistic model this corresponds to collapsing the Gaus- 3 0 I. Bertini, C. Luchinat, and G. Parigi, Prog. Nucl. Magn. Reson. Spectrosc. sians into delta functions at the positions ␶ϩ. This approxi- 40,249͑2002͒. mation was explored, including averaging over all the vibra- 4 R. Mahjan, A. Vora, and A. Pathak, in Proceedings of the ILCC 2002, tional modes, the torsional ones too in order to simulate the Edinburgh, 30 June–5 July 2002, p. 187. 5 J. C. Barnes, J. D. Paton, J. R. Damewood, Jr., and K. Mislow, J. Org. torsion as a libration about the minimum. In this way a mini- Chem. 46, 4975 ͑1981͒. ␶ ϭ␶ ϭ 6 mum in R of 19 Hz was found when 1 2 70°. Allowing M. Feigel, J. Mol. Struct. 366,83͑1996͒. for a change of the HCH angle the value of R is reduced 7 T. Strassner, Can. J. Chem. 75,1011͑1997͒. 8 to 1 Hz, but we obtain for that angle an unacceptable value J. W. Emsley, G. R. Luckhurst, and C. P. Stockley, Mol. Phys. 44,565 ͑1981͒. of 93°. 9 J. W. Emsley, in Solid-State NMR Spectroscopy Principles and Applica- tions, edited by M. J. Duer ͑Blackwell Science, New York, 2002͒,p.531. ␶ ␶ 10 C. Calculation of Piso 1 , 2 M. Longeri, G. Chidichimo, and P. Bucci, Org. Magn. Reson. 22,408 „ … ͑1984͒. ␶ ␶ 11 Having obtained PLC( 1 , 2) directly from the experi- G. Celebre, G. De Luca, M. Longeri, and E. Sicilia, J. Chem. Inf. Comput. Sci. 34, 539 ͑1994͒. mental values of the Dij it is straightforward to calculate the ␶ ␶ ͑ ͒ 12 J. W. Emsley and G. R. Luckhurst, Mol. Phys. 41,19͑1980͒. distribution Piso( 1 , 2) from Eq. 17 . These distributions 13 ͑ ␧ ␶ P. Diehl, in NMR of Liquid Crystals, edited by J. W. Emsley Reidel, are expected to differ when the 2,n( k) have a strong depen- Dordrecht, 1985͒, Chap. 7. dence on conformation. This corresponds to the conforma- 14 J. W. Emsley, G. R. Luckhurst, and C. P. Stockley, Proc. R. Soc. London, Ser. A 381,117͑1982͒. tions having very different shapes. Thus, differences of up to 15 14 J. W. Emsley, in Encyclopedia of NMR, edited by D. M. Grant and R. K. 30% were found for the liquid crystal 5CB. For diphenyl- Harris ͑Wiley, Chichester, 1995͒, p. 2781. methane, however, the differences are found to be small 16 G. La Penna, E. K. Foord, J. W. Emsley, and D. J. Tildesley, J. Chem. (Ͻ3%). Phys. 104, 233 ͑1996͒. 17 C. Zannoni, in NMR of Liquid Crystals edited by J. W. Emsley ͑Reidel, Dordrecht, 1985͒, Chap. 2. VII. CONCLUSION 18 D. Catalano, J. W. Emsley, G. La Penna, and C. A. Veracini, J. Chem. ͑ ͒ The application of the probabilistic approach, coupled Phys. 105, 10595 1996 . 19 R. Berardi, F. Spinozzi, and C. Zannoni, Chem. Phys. Lett. 260,633 with the AP model for the conformational dependence of the ͑1996͒. interaction with the liquid crystal molecules, gives a distri- 20 P. Lesot, D. Merlet, J. Courtieu, J. W. Emsley, T. T. Rantala, and J. Jok- bution P (␶ ,␶ ) which is similar to that found by quantum isaari, J. Phys. Chem. 101, 5719 ͑1997͒. LC 1 2 21 chemical calculations. The positions of the maxima in the M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 98, Revision A.9, Gaussian, Inc., Pittsburgh, PA, 1998. distribution are in particularly close agreement. This cannot 22 S. A. Katsyuba, J. Grunenberg, and R. Schmutzler, J. Mol. Struct. 559, be taken to be a proof of the validity of the model used, but 315 ͑2001͒.

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