Direct Measurement of the Angular Pair Correlation Coefficients in

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Direct Measurement of the Angular Pair Correlations Coefficients in Molecular Liquids Using NMR. Benchmarking Force Fields for Atomistic Simulations. Leah Marie Heist, Chi-Duen Poon, Edward Thaddeus Samulski, and Demetri J. Photinos J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b01435 • Publication Date (Web): 27 Mar 2017 Downloaded from http://pubs.acs.org on March 28, 2017

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The Journal of Physical Chemistry

Direct Measurement of the Angular Pair Correlation Coefficients in Molecular Liquids Using NMR. Benchmarking Force Fields for Atomistic Simulations.

Leah M. Heista, Chi-Duen Poona, Edward T. Samulski*a and Demetri J. Photinos*b, a

Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290 b

Department of Materials Science, University of Patras, Patras 26504, Greece

One Sentence Summary: Measuring biased diffusional motion of molecular liquids in a strong magnetic field yields angular pair correlation coefficients and insights into liquid state structure.

Abstract High field deuterium NMR spectroscopy is used to characterize a number of molecular liquids and their mixtures in order to probe the directional part of the intermolecular interactions through the orientational ordering induced in the isotropic liquid phase by the spectrometer magnetic field. The systems studied include benzene, chloroform, hexafluorobenzene and thiophene at various

concentrations

and

in

mixtures.

Dilution

with

the

magnetically

isotropic

tetramethylsilane provides quantification of ordering at “infinite magnetic dilution”, i.e., in the absence of magnetic intermolecular correlations, and thereby allows the identification of the contribution of these correlations to the orientational ordering in neat phases and at various degrees of magnetic dilution. Such contributions are conveyed by angular pair correlation coefficients which, in addition to being accessible to direct NMR measurement, are also possible to evaluate directly from molecular dynamics simulations. By using various force fields,

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simulations provide bench mark quantities for testing and possibly further improving the force field performance, particularly with respect to the directional components of the intermolecular interactions. The latter are critical for the simulation of self-assembly generally and particularly in biological systems.

Introduction Is has been more than half a century since J.D. Bernal reviewed his geometrical model1 of liquid state structure2 and anticipated the potential utility of then nascent computer simulations.3 Over the intervening years his insightful perspectives have continued to be refined4 while the theory of liquids progressed rapidly.5–8 Perhaps the most significant advances in constructing a comprehensive molecular description of liquid state structure and dynamics have come from computer simulations.9,10 The force fields, providing the parameterization of the molecular interactions for these simulations, have evolved considerably: For example, for the prototypical molecular liquid benzene, simulations initially used coarse intermolecular potentials (e.g., MM2) in the early 1980s,11 more refined atom-centered force fields followed in the mid-1990s,12 culminating with explicit inclusion of π electrons in the presently used versions.13 Force fields continue to be optimized and fine-tuned by fitting experimentally accessible properties, often supplemented with the transcription of results from quantum mechanical calculations, within the classical force field format.14 The sets of experimentally accessible properties that are presently used for the optimization of the force fields are limited and typically comprise densities, phase transition enthalpies, diffusion coefficients and radial correlations obtained from X-ray and neutron scattering. Often, those properties are not sufficient to discriminate between force fields and consequently give divergent results when used to simulate other properties of a given liquid.


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It is therefore essential for the improvement of the predictive power and credibility of the molecular simulations, currently tested mainly through the transferability of the force field parameterization to different classes of molecules, to extend the set of properties to be fitted. Such extension would be particularly useful if the added properties are: (i) accessible to direct, precise and unambiguous experimental measurement, (ii) readily accessible to molecular simulation, and (iii) sensitive to the more subtle features of the intermolecular interactions, so as to allow the discrimination between force fields which can be tuned to provide essentially identical results for the gross properties (densities, enthalpies, etc.) of a liquid. In this work we propose a class of such additional quantities and demonstrate that they satisfy these three requirements. Specifically, we show that (i) high field NMR on deuterated liquids and their mixtures with other liquids (not necessarily deuterated) provides a robust methodology for directly extracting the values of integrated, second rank components of the pair correlation function, (ii) these components are directly and accurately obtainable from molecular dynamics simulations. Earlier we communicated that contrasting pair correlations derived from NMR with those from simulations discriminate rather sensitively between force fields.15

Herein we

consider a variety of molecular liquids and mixtures, including the case of benzene, in binary and tertiary mixtures with the highly symmetric diluent molecule, tetramethylsilane (TMS). Results are also presented for thiophene, which provides an example of a less symmetric molecule and, consequently, a richer set of measurable quantities. The important difference reported herein is that the measurable quantities probe the directionality of the molecular interactions, in contrast to the standard properties (densities, evaporation enthalpies, radial distribution functions) which are dominated by the isotropic part of the interactions. This difference is of particular significance


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for

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simulations of self-organization, self-assembly, and delicate quaternary structures in

biological systems, as these are sensitive to the directional part of the force field used.

Very high field nuclear magnetic resonance (NMR) is, by and large, the domain of structural biologists seeking a detailed, atomistic picture of complex biological macromolecules. While the influence of such magnetic fields on liquids is negligibly small, there is a measurable effect: the high field perturbs molecular rotational diffusion revealing incompletely averaged, anisotropic NMR interactions. When deuterium NMR (2H NMR) is employed to study deuterated molecules or natural abundance deuterium in unlabeled molecules, the residual quadrupolar interactions cause a small splitting Δν of the resonance lines in the 2H NMR spectrum.16,17 The observed Δν may be precisely measured and are proportional to the incompletely averaged electric field gradient (EFG) at the deuterium nucleus. That non-zero average stems from the very slight orientational bias experienced by a molecule in the magnetic field resulting in an anisotropic distribution of its C—D bond directions relative to the spectrometer magnetic field. In liquids, intermolecular associations can enhance or detract from the magnetic potential energy of an isolated molecule. The residual quadrupolar interactions reflect such associations and thereby provide an experimental determination of the leading tensor component of the pair correlation function in liquids, unambiguous information about liquid state structure. Hence, the high field NMR spectrometers currently being manufactured (1.2 GHz proton resonance frequency),18 will be able to provide an additional critical assessment tool for optimizing force fields and a facile way to explore subtle aspects of liquid state structure.


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We previously have shown15 that for the prototypical molecular liquids benzene and chloroform, residual quadrupolar interactions provide an experimental determination of the leading tensor component of the pair correlation function. Moreover, these quantities probe the directionality of the molecular interactions unlike the standard physical and thermodynamic properties which are dominated by the isotropic part of the interactions, and therefore can be used to discriminate among and refine simulation force fields. The idea is simple: The standard quantities (densities, transition enthalpies, radial distribution functions…) used for the optimization of force fields in the liquid state are only indirectly influenced by the directional part of the intermolecular interactions even though the molecules are anisotropic in most cases. To probe directly in the isotropic phase a quantity/property that would be sensitive to the anisotropic part of the molecular interactions, one could impose a directional bias on the isotopic liquid and then measure the liquid’s response (induced orientational order) to the bias. Both of these steps can be taken simultaneously in a very high field NMR experiment, wherein the directional bias is provided by the spectrometer magnetic field and the induced orientational order can be obtained quantitatively from the NMR spectra. Naturally, for this type of measurement to be practically feasible, the molecules should posess sufficient magnetic orientability, i.e. have a sufficiently large molecular magnetizability anisotropy in relation to the available magnetic field strength, so that a measurable effect can be observed in the spectrum. Here we extend our deuterium NMR methodology to molecular liquids and mixtures to demonstrate the robustness of this technique.

The paper is organized as follows: In section II we present the experimental method and the results of the measurements. Section III deals with the statistical mechanical derivation of the relation of the measured quantities to the pair correlation functions, specifically the second rank


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correlation factors, in a mixture of molecular liquids. The derived relations are used in section IV to analyze the results of the measurements of section II. In section V, the experimentally determined correlation factors are compared with the calculated values from molecular dynamic simulations. Section VI contains the discussion and conclusions.

II. Deuterium NMR of molecular liquids at high magnetic field. The small splitting Δν (∼1Hz in a 22.3 T magnetic field) of the deuterium resonance in the 2H NMR spectra of simple deuterium labeled liquids is proportional to the incompletely averaged !

electric field gradient (EFG) tensor V at the deuterium nucleus. The incomplete averaging is a result of the orientational bias that the molecules experience from the spectrometer magnetic

! 1 ! field B. The orientation-dependent magnetic energy contribution is − B ⋅ χ ⋅ B , with χ 2 denoting the molecular magnetizability anisotropy tensor. The small, magnetic-field-induced, quadrupole splitting at labeled site s may be readily extracted from 2H NMR spectra and in general form is given by,19

⎡3 3 Δν ( s ) = ν Q( s ) ⎢ 2 ⎣2

(

Bˆ ⋅ zˆs

)

2

1 η (s) − + EFG 2 2

{(

Bˆ ⋅ xˆs

2

)

− Bˆ ⋅ yˆ s

(

2

)

}

⎤ ⎥ ⎦

(1)

where ν Q( s ) is the deuterium quadrupole coupling constant, the unit vector Bˆ denotes the direction of the magnetic field and xˆs , yˆ s , zˆs are the principal axes of the EFG tensor V ( s ) at the deuterated site s. These axes are assigned so that Vz(ssz)s is the largest component, i.e. zˆ is identified with the C-D bond direction, and Vy(ssy)s is the smallest component. Consequently, the

(

)

(s) asymmetry parameter of the EFG tensor, defined as η EFG = Vx(ssx)s − Vy(ssy)s / Vz(ssz)s is positive. The


6

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angular brackets indicate motional averages of the direction of the magnetic field relative to the principal EFG axes at the deuterated site s. For rigid molecules it is often convenient to express these averages relative to a set of molecular axes aˆ , bˆ, cˆ that is common for all the deuterated sites on the molecule. As the molecular systems considered in this study have at least one axis of twofold, or higher, symmetry and at least two mutually orthogonal symmetry planes containing that axis, the molecular frame aˆ , bˆ, cˆ is naturally chosen so that one of its axes is identified with the molecular symmetry axis and the other two are taken perpendicular to the symmetry planes (Figure 1). The expression in Eq (1) can then be put in the form (s) ⎡⎛ 3 ⎤ 2 ⎞ 1⎞ ⎛3 1 η EFG 2 ˆ ⎜ ˆ ˆ ˆ ( ) (cˆ ⋅ xˆ s )2 − (cˆ ⋅ yˆ s )2 ⎟⎟ + ⎥ B ⋅ c − × c ⋅ z − + ⎟ ⎜ ⎢⎜ s 2⎠ ⎝2 2 2 ⎠ ⎢⎝ 2 ⎥ 3 (s) ⎢ 2 ⎛3⎛ ⎞⎥⎥ 2 = νQ ⎢ ( aˆ ⋅ zˆ s ) − bˆ ⋅ zˆ s ⎞⎟ + ⎜ ⎜ ⎟ 2 ⎠ ⎢ 1 Bˆ ⋅ aˆ 2 − Bˆ ⋅ bˆ 2 ⎜ 2 ⎝ ⎟⎥ ( s ) ⎢2 ⎜ η EFG ⎛ ⎟⎥ 2 2 2 2⎞ ˆ ˆ ⎢ ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) a ⋅ x + b ⋅ y − b ⋅ x − a ⋅ y ⎜ ⎟ ⎟⎥ ⎜ s s s s ⎠ ⎠⎦ ⎝ 2 ⎝ ⎣

( )

Δν ( s )

(

(

( ) ( )

)

)

(

) (

(2)

)

where now the motional averages involve the projections of the magnetic field unit vector on the molecular axes aˆ , bˆ, cˆ . Because the orientational order of the molecules in this work originates from their interaction with the magnetic field, the cˆ molecular axis will be chosen to correspond to the principal axis of the magnetizability anisotropy tensor χ of the molecule, which automatically renders the aˆ , bˆ, cˆ directions the principal axes frame of the of χ . With this choice, the two orientational averages in eq(2), hereafter to be referred to as the molecular orientation order parameters S and R, are expressed as

⎛3 ⎜ ⎝2

( Bˆ ⋅ cˆ )

2

1⎞ 3 1 − ⎟ = cos 2 θ − = S , 2⎠ 2 2

2

( Bˆ ⋅ aˆ ) − ( Bˆ ⋅ bˆ )

2

= sin 2 θ cos 2ϕ = R ,


(3)

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where θ is the angle of the magnetic field direction Bˆ relative to the cˆ molecular axis and φ is the angle between the aˆ axis and the projection of Bˆ on the aˆ , bˆ plane.

Figure 1. The molecule-fixed axes systems. A. benzene; B. chloroform; C. thiophene. D shows the angles θ and φ that the magnetic field direction B forms in the molecular a,b,c frame. III. Relation of order parameters to molecular pair correlations in a liquid under high magnetic field. The orienting potential of a molecule having magnetizability anisotropy tensor χ in the magnetic field is

1 1 1 1 ⎛3 ⎞ u (θ , ϕ ) = − B ⋅ χ ⋅ B = − B 2 χ cc ⎜ cos 2 θ − + η ( χ ) sin 2 θ cos 2ϕ ⎟ , 2 2 2 2 ⎝2 ⎠

(4)

where η ( χ ) = ( χ aa − χ bb ) / χ cc denotes the magnetic biaxiality of the molecule and χ aa , χ bb , χ cc denote the principal values of χ , with χ aa + χ bb + χ cc = 0 and the principal axes aˆ , bˆ, cˆ assigned so that χ cc ≥ χ bb ≥ χ aa ). In a liquid mixture consisting of N molecules, of which N A are of species A, N B are of species B, etc, with N = N A + N B + ... , the formal definition of the order parameter S A for a molecule of species A is given by


8

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1⎞ N ⎛3 2 N d ω dr d ω dr .. d ω dr cos θ − ⎟ × ∏ exp ( −ui / kT ) × P0 (ω1 , r1 ; ω2 , r2 ;.....ωN , rN ) 1 ∫ 1 1 2 2 N N ⎜⎝ 2 2 ⎠ i =1 , (5) SA = N N ∫ dω1dr1dω2dr2 ..dωN drN ∏ exp ( −ui / kT ) × P0 (ω1 , r1; ω2 , r2 ;.....ωN , rN ) i =1

! where ω i , ri denote respectively, the orientation (described by the Euler angles θi , ϕi ,ψ i ) and position of molecule i (= 1, 2...N ) ; the molecules are labeled so that the first N A of them (i.e. for

i = 1, 2... N A ) are of species A, the next N B (i.e. for i = ( N A + 1)...( N A + N B ) ) are of species B etc, ! ! ! and P0N (ω1 , r1;ω 2 , r2 ;.....ω N , rN ) is the full (position & orientation) probability distribution of the N

molecules in the liquid, in the absence of the external magnetic field. The term involving explicitly the angle θ1 in the numerator of eq(5) corresponds to the molecule i=1, which is of species A; its replacement by a respective term corresponding to any of the i=2,3,,,NA molecules would obviously lead to an identical outcome. Starting from the above rigorous definition, and noting that for the magnetic field strengths and the molecular magnetiziabilities considered in this study the magnetic energy is of such small magnitude that ui / kT is on the order of 10 −4 , we obtain, on retaining up to linear terms in ui / kT , the first order perturbation result: ⎞ ⎛ B 2 ⎞ ⎛ cc ⎛ 1 χ 02 ⎞ 1 χ 02 ⎞ ⎛ 00 1 χ 02 ⎞ ⎞ cc ⎛ 00 cc ⎛ 00 SA = ⎜ ⎟ ⎜ χ A ⎜1 + xA ⎜ g AA + η A g AA ⎟ ⎟ + xB χ B ⎜ g AB + ηB g AB ⎟ + xC χC ⎜ g AC + ηC g AC ⎟ + ... ⎟ 2 2 2 ⎝ ⎠⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎝ 10kT ⎠⎝ ⎠

(6) Here xJ =NJ/N is the mole fraction of species J(=A,B,C…), χ Jcc and η Jχ are, respectively, the principal value and biaxiality of the molecular magnetizability anisotropy of the same species. 00 02 The correlation factors g AJ and g AJ are integrals of the pair correlations function g AJ (r ; ω1 , ωi ) ,

specifically 00 g AJ =

N 1⎞ ⎛3 1⎞ ⎛3 dω1 ∫ dωi dr ⎜ cos 2 θ1 − ⎟ × ⎜ cos 2 θi − ⎟ × g AJ (r ; ω1 , ωi ) 2 ∫ V 8π 2⎠ ⎝2 2⎠ ⎝2


(7)

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02 and g AJ is obtained from the above expression on replacing the second factor in the integrand by

sin 2 θi cos 2ϕi . Equivalently, these correlation factors can be directly identified with the radial integrals of second rank tensor components of the Wigner matrix expansion of the correlation function20 between a pair of molecules of species A and J, (see p. 180 ref 7) namely: g AJ (r ; ω1 , ωi ) =

∑g

l1l2l m1m2 m n1n2

AJ

l2 * l* ˆ . (l1l2l; n1n2 ; r ) × C (l1l2l; m1m2 m) × Dml11*n1 (ϕ1θψ 1 1 ) × Dm2 n2 (ϕiθψ i i ) × Ym (r )

(8)

In particular,

⎛N⎞ 00 g AJ = ⎜ ⎟ 5π ∫ r 2 g AJ (220;00; r )dr ⎝V ⎠

(9)

and

1 ⎛N⎞ 02 g AJ = ⎜ ⎟ 5π ∫ r 2 {g AJ (220;02; r ) + g *AJ (220;02; r )} dr 2 ⎝V ⎠

(10)

Similarly, starting from the counterpart of eq(5) for the order parameter RA for a molecule of species A we obtain the result ⎞ ⎛ B 2 ⎞⎛ cc ⎛ 2 χ ⎛ 20 1 χ 22 ⎞ ⎞ ⎛ 20 1 χ 22 ⎞ ⎟⎟⎜⎜ x A ⎜⎜ η A + x A ⎜ g AA R A = ⎜⎜ + η A g AA ⎟ ⎟⎟ + x B χ Bcc ⎜ g AB + η B g AB ⎟ + xC χ Ccc ... ⎟⎟ , (11) 2 2 ⎝ ⎠⎠ ⎝ ⎠ ⎝ 10kT ⎠⎝ ⎝ 3 ⎠ 20 22 , g AJ where now the correlation factors g AJ for the A-J pair are obtained from the respective

Wigner expansion second rank components analogously to equations (9, and 10) with the integrands

replaced

by

1 g AJ (220;20; r ) + g *AJ (220;20; r )} { 2

for

20 g AJ

and

by

1 22 g AJ (220;22; r ) + g *AJ (220;22; r ) + g AJ (220;2(−2); r ) + g *AJ (220;2(−2); r )} for g AJ . { 4


10

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It is apparent from equation (6) that the order parameter S A has a concentration independent part, which does not involve molecular pair correlations, [ S A

]0 =

B 2 χ Acc , describing the 10kT

magnetic ordering of the molecule in the absence of magnetic intermolecular correlations (“gas phase”) ordering, here corresponding to magnetic ordering in the liquid phase for infinite dilution by a magnetically neutral diluent). The ratio S A / [ S A ]0 is often16,17,21 conveyed as the ratio of an effective magnetizability anisotropy ⎡⎣ χ Acc ⎤⎦ over the molecular one χ Acc , and this ratio is in turn eff identified with the Kirkwood correlation factor g2, that is S A / [ S A

]0 = ⎡⎣ χ

cc A

⎤⎦ / χ Acc = g 2 . It eff

follows directly from eq(6) that the explicit expression for this ratio in terms of correlation factors among different species and their concentrations is given by:

⎛ χ cc ⎛ 00 1 χ 02 ⎞ ⎞ SA = ⎜1 + ∑ xJ Jcc ⎜ g AJ + η J g AJ ⎟ ⎟ 2 [ S A ]0 ⎝ J = A,B,... χ A ⎝ ⎠⎠

(12)

Similarly, for magnetically biaxial molecules there is a concentration independent part of the

⎛ B 2 χ Acc ⎞ ⎛ 2 χ ⎞ ⎟⎜ η A ⎟ . In analogy with the effective ⎝ 10kT ⎠ ⎝ 3 ⎠

order parameter in eq (11), [ RA ]0 = ⎜

magnetisability, an effective biaxiality, ⎡⎣η Aχ ⎤⎦ , can be defined through its ratio over the eff molecular magnetic biaxiality η Aχ according to

⎛ ⎛ 20 1 χ 22 ⎞ ⎞ . ⎡⎣η Aχ ⎤⎦ / η Aχ = RA / [ RA ]0 = ⎜1 + ∑ xJ (3χ Jcc / 2η Aχ χ Acc ) ⎜ g AJ + η J g AJ ⎟ ⎟ eff 2 ⎝ ⎠⎠ ⎝ J = A, B ,...

(13)

The systems in the present study fall under the following symmetry categories:


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1. Binary mixtures where the molecules of species A are uniaxial, i.e. the magnetic principal axis c is a higher-than-twofold symmetry axis, and there is one additional species which is magnetically isotropic. In this case the expressions for the order parameters in eqs (6) and (11) reduce to ⎛ B 2 χ Acc ⎞ 00 SA = ⎜ ⎟ (1 + xA g AA ) ; RA = 0 ⎝ 10kT ⎠

(14.1)

2. Tertiary mixtures in which the molecules of species A and B are uniaxial and have equal concentrations, xA = xB = x ≤ 1/ 2 , and there is one additional species, C, which is magnetically isotropic. The respective expressions are in this case: ⎛ B 2 χ Acc ⎞ 00 cc cc 00 SA = ⎜ ⎟ 1 + x g AA + ( χ B / χ A ) g AB ⎝ 10kT ⎠

( (

)) ; R

A

=0

(14.2)

3. Binary mixtures with the molecules of species A being biaxial (the magnetic principal axis c is not a symmetry axis) and the other component is magnetically isotropic. The expressions for the two order parameters then reduce to ⎛ B 2 χ Acc ⎞ ⎛ ⎛ B 2 χ Acc ⎞ ⎛ 2 χ ⎛ 00 1 χ 02 ⎞ ⎞ ⎛ 20 1 χ 22 ⎞ ⎞ SA = ⎜ 1 + x g + η g ; R = ⎟⎜ ⎜ ⎟ ⎜ η A + xA ⎜ g AA + η A g AA ⎟ ⎟ A ⎜ AA A AA ⎟ ⎟ A 2 2 ⎝ ⎠⎠ ⎝ ⎠⎠ ⎝ 10kT ⎠ ⎝ ⎝ 10kT ⎠ ⎝ 3

(14.3)

In addition to the explicit dependence of the order parameters S A , RA on temperature and concentration, there is an implicit dependence associated with the temperature and concentration dependence of the correlation factors. This dependence is generally weak for the systems studied, as the measurements are not taken close to a phase transition, and can be adequately n1n2 conveyed by the leading terms in Taylor expansions. Thus, the value of a correlation factor g AJ


12

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at a temperature T and fixed concentration is to a good approximation obtained from its value and first two derivatives at a reference temperature T0 and the same concentration: n1n2 ⎡ ∂g n1n2 ⎤ ⎡ ∂ 2 g AJ ⎤ 1 n1n2 n1n2 g AJ (T ) ≈ g AJ (T0 ) + (T − T0 ) ⎢ AJ ⎥ + (T − T0 ) 2 ⎢ 2 ⎥ ⎣ ∂T ⎦T0 2 ⎣ ∂T ⎦T0

(15)

Similarly the variation with concentration x at fixed temperature is obtained from the pure compound (x=1) values as

g

n1n2 AJ

( x) ≈ g

n1n2 AJ

n1n2 2 n1n2 ⎡ ∂g AJ ⎤ 1 2 ⎡ ∂ g AJ ⎤ ( x = 1) + ( x − 1) ⎢ ⎥ + (x − 1) ⎢ 2 ⎥ ⎣ ∂x ⎦ x =1 2 ⎣ ∂x ⎦ x =1

(16).

As shown in the next section, the quadratic contributions in either of eq(15) or (16) are mostly below the resolution of the experimental measurements.

IV. Evaluation of correlation factors from the measured spectra. In the following we present the results from the study of three mixtures: (i) benzene/hexafluorobenzene 1:1 mixture in TMS, (ii) chloroform/benzene 1:1 mixture in TMS and (iii) thiophene in TMS. Each mixture matches one of the cases laid out in Section III (Equations 14.1, 14.2, 14.3). Table 1 gives the properties used to evaluate the data and calculate the order parameters in Equations 14.1-3. Benzene-d6 has magnetically equivalent deuterons, chloroform-d1 has only one deuteron, and thiophene-d4 has two inequivalent pairs of deuterons; the deuterons 7,8 and 6,9 are labeled in Figure 1.


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Page 14 of 35

Table 1. Constants used in the determination of correlation factors for benzene, chloroform, and thiophene. Thiophene has inequivalent deuterons (see labeling in Figure 1) where deuterons 6 and 9 have their C—D bond adjacent to the sulfur atom.

Molecule

parameter B K T

benzene-d6

χ cc χ aa χ bb ηEFG

benzene-F6 chloroform-d1

reference

a

+ 3.6 × 10 −28 JT −2

a

+ 3.6 × 10 −28 JT −2

a

0.054

22

νQ

187 ± 0.4 kHz

23

χ cc χ cc

- 4.6 x 10-28 JT-2

a

+ 1.5 × 10 −28 JT −2

a

− 0.76 × 10 −28 JT −2 − 0.76 × 10 −28 JT −2

a a

bb

thiophene-d4

Value 22.3 T 1.38 × 10 −23 JT −1 303 K − 7.1 × 10 −28 JT −2

χ ηEFG

0

νQ

167 ± 0.6 kHz

χ cc χ aa χ bb ηχ

− 5.7 × 10 −28 JT −2

0.016

ν Q (7,8)

190.4 kHz

ηEFG (7,8)

0.058

ν Q (6,9)

193.6 kHz

ηEFG (6,9)

0.080

24

a

−2

a

+ 2.9 × 10 −28 JT −2

a

+ 2.8 × 10

−28

JT

(7,8)

(180 ± 34)° u (6,9) u ± 74° a The numerical values for magnetizabilities were calculated at the B3LYP/aug-cc-pVTZ level of theory using the Gaussian program package (Gaussian 09, Revision D.01, 2009)


14

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IV.I Tertiary mixtures The tertiary mixtures studied are 1:1 mixtures diluted with the magnetically isotropic molecule TMS. The expression for the quadrupolar splittings of the deuterated species at temperature T and mole fraction x is given in by

⎛ ⎛ 00 ⎛ χ Bcc ⎞ 00 ⎞ ⎞ Δν A (T , x A ) = ν 0, A (T )⎜1 + x⎜⎜ g AA + ⎜⎜ cc ⎟⎟ g AB ⎟⎟ ⎟ , ⎜ ⎟ ⎝ χA ⎠ ⎝ ⎠⎠ ⎝

(17)

where ν 0, A (T ) is the splitting of the molecule at infinite magnetic dilution and x denotes the equal molecular concentrations of the A and B species; for benzene-d6 (bnz), 2 cc ⎛ − 3 ⎞ ⎛⎜ B χ benz ν 0,benz (T ) = ⎜ ⎟ν Q ⎜ ⎝ 4 ⎠ ⎝ 10kT

⎞ ⎟⎟(1 + η EFG ) and for chloroform-d1 (clfm), ⎠

2 cc ⎛ 3 ⎞ ⎛⎜ B χ clfm ⎞⎟ ν 0,clfm (T ) = ⎜ ⎟ν Q . ⎝ 2 ⎠ ⎜⎝ 10 kT ⎟⎠

Benzene:hexafluorobenzene 1:1 mixture in TMS The molecule of species A in this case is benzene-d6, so that the infinite dilution splitting in the mixture case should represent a benzene molecule in the absence of intermolecular magnetic correlations;ν 0,benz (T = T0 )calc = 1.25 Hz . The concentration dependence of the splittings of benzene in the benzene:hexafluorobenzene mixture normalized to the infinite dilution splitting, Δν/ν0, is shown in Figure 2. In contrast with the binary mixture of benzene in TMS, the Δν/ν0 in the tertiary mixture (Fig. 2 solid line) do not extrapolate to 1. This might mean that even at high dilutions some residual association persists between benzene:hexafluorobenzene pairs. The 2H spectrum of the neat 1:1 mixture of benzene-d6/hexafluorobenzene is shown in Figure 3.


15


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Page 16 of 35

Figure 2. Observed Δν/ν0 versus mol fraction in TMS (x) for benzene and 1:1 benzene/hexafluorobenzene; the respective lines are fits of Equation 16 to the data.

Figure 3. 2H NMR spectrum of neat 1:1 mixture of benzene-d6:hexafluorobenzene (no TMS) recorded at 22.3 T (Bruker Avance III 950 MHz proton NMR spectrometer at temperature, T = 303 K). The outer 13C satellite resonances show the true quadrupolar splittings Δν. 25


16

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Using Equation 17 and the values in Table 1 for neat benzene-d6 in the mixture (where x = 1/2) and the quadrupolar splitting (Δν benz (T = T0 , x = 1 / 2) = 2.01 Hz ) , the value for benzene in the mixture is

g

00 benz −C6 F6

cc ⎡ ⎛ Δν A (T , x A ) ⎞ ⎤⎛ χ benz 00 ⎜ ⎜ ⎟ = ⎢2⎜ − 1⎟ − g benz −benz ⎥ cc ⎜ ⎠ ⎣⎢ ⎝ ν 0, A (T ) ⎦⎥⎝ χ C6 F6

⎞ ⎟ = +2.1 ⎟ ⎠

(18)

00 15 where gbenz ). Interestingly, the concentration dependence of the splitting −benz = −0.16 (from ref.

for this mixture (Figure 2) shows no appreciable deviation from linearity; the data can be fit to a straight line (Figure 2, solid line) with a coefficient of determination (R2) of 0.99; benzene 00 (Figure 2, dashed line) shows some deviation from linearity indicating that g benz −benz depends

slightly on concentration.15 The straight line for the mixture data (Figure 2, solid line) indicates 00 that the (magnetically scaled, see eq (17)) sum of the pair correlation factors g benz −benz and

00 gbenz −C6 F6 shows no concentration dependence. The reason for this observation will be explained

in more detail in Section V when the benzene/hexafluorobenzene complex is discussed.

Chloroform:benzene 1:1 mixture The molecule of species A in this case is chloroform-d1 (the mixture is made with protonated C6H6), so that the infinite dilution splitting in the mixture case should represent a chloroform molecule in the absence of intermolecular magnetic correlations; ν 0,clfm (T = T0 )calc = 0.456 Hz .15 The 2H spectrum of the neat 1:1 mixture of chloforom/benzene is shown in Figure 4.


17


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Page 18 of 35

Figure 4. 2H NMR spectrum of neat 1:1 mixture of chloform-d/benzene (no TMS) recorded at 22.3 T (Bruker Avance III 950 MHz proton NMR spectrometer at temperature, T = 303 K). The quadrupolar splitting of the central resonance is not quite resolved but may be determined from fitting the lineshape to yield Δν = 0.26 Hz.

Using Equation 17 with the appropriate values for the chloroform/benzene mixture, the values in Table 1 for neat chloroform-d1 in the mixture (where x = 1/2), and the quadrupolar splitting

(Δν (T = T , x = 1 / 2) ≈ 0.26 Hz), the pair correlation value for chloroform-d clfm

1

0

g

00 clfm−C6 H 6

cc ⎡ ⎛ Δν A (T , x A ) ⎞ ⎤⎛ χ clfm 00 ⎜ ⎜ ⎟ = ⎢2⎜ − 1⎟ − g clfm−clfm ⎥ cc ⎜ ⎠ ⎣⎢ ⎝ ν 0, A (T ) ⎦⎥⎝ χ C6 H 6

⎞ ⎟ = +0.21 ⎟ ⎠

in the mixture is

(19)

The positive value means that the symmetry axes of the two molecules prefer to be parallel to each other (i.e. the clfm axis aligns normal to the benzene ring plane), but the magnetic field energetically favors the clfm axis to be along the field direction and the benzene axis to be perpendicular to the field. Therefore the chloroform-benzene pair correlations mitigate against the simultaneous magnetic alignment of the two molecules.


18

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From Figure 4 it is clear that, relative to chloroform-d1 in TMS (|Δν| = 0.52 Hz), the magnitude of the unresolved splitting is smaller (|Δν| = 0.26 Hz) and may be expected to be of opposite sign. This qualitative difference confirms that the propensity for magnetic field alignment of the symmetry axis of chloforom-d1 is reduced when mixed with benzene. An idealized configuration for the hydrogen-bonded dimer of chloroform with benzene has the CDCl3 C3-symmetry axis normal to the benzene ring. In that case a factor of two reduction in the chloroform-d1 Δν would be anticipated if the aromatic benzene ring were constrained to its low magnetic potential energy orientation in the field—with its ring plane tangent to B.

IV.II Binary mixtures Thiophene in TMS In the case of thiophene (a biaxial molecule), it is possible to obtain two separate pair correlation 00 20 factors, g thio − thio and g thio − thio . In thiophene-d4 there are two pairs of non-equivalent deuterated

sites, symmetrically located relative to the twofold symmetry axis of the molecule (the molecular axis a; see Figure 1); the c-axis is normal to the ring. From the geometrical data of the molecule (Table 1) we obtain the following expressions for the two classes of splittings (s) where the deuterated sites correspond to s = 6,9 or 7,8,

3 (s) ⎡ 1 1 (s) (s) Δν = ν Q ⎢− S A × 1 + η EFG + RA cos 2u ( s ) 3 − η EFG 2 4 ⎣ 2

(

)

(

)⎤⎥ . ⎦

(20)

The complete expressions for SA and RA , where now the subscript A refers to the thiophene molecule and will be omitted, can be found in Equation 14.3. Due to the smallness of (χ ) ηthio (= 0.016) , the terms

1 χ 02 1 η g and η χ g 22 in these equations can be safely neglected, 2 2


19


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Page 20 of 35

compared to g 00 and g 20 , respectively, as their contribution is below the resolution of the experiment. Accordingly, the simplified equations for the order parameters from Equation 14.3 are

S = [ S ]0 (1 + xg 00 )

;

2 R = [ S ]0 ( η χ + xg 20 ) 3

(21)

with [ S ]0 = B2 χ cc /10kT . Combining Equations 20 and 21, the values of g 00 and g 20 can be determined using the splittings for the two classes of deuterons in thiophene. To determine the concentration dependence, it is convenient to put eq(20) in the following form, using the expressions for the order parameters from eq(21):

Δν ( s )

ν

(s) 0

= 1 + x ( g 00 + r ( s ) g 20 ) / λ ( s ) ,

where

(22)

(s) ν 0( s ) = −3vQ( s ) [S ]0 (1 +ηEFG ) λ (s) / 4 , λ (s) =1+ 2η χ r (s) / 3

and

(s) (s) r ( s ) = cos 2u ( s ) (3 −ηEFG ) / 2 (1 +ηEFG ) . The infinite dilution values are ν 0(6,9) = 1.05 Hz and

ν 0( 7 ,8) = 1.03 Hz . A spectrum of thiophene-d4 in TMS is shown in Figure 5 with the peaks labeled according to the deuteron sites (6,9 or 7,8). The concentration dependence of the frequency ratios are shown in figure 6(a). It is apparent from this plot that values of the two frequency ratios are nearly coincident, their differences falling within the experimental uncertainty, except for low concentrations (x

Direct Measurement of the Angular Pair Correlation Coefficients in

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