F O Goncharov
email: fedor.goncharov.ol@gmail.com

Integrals of spherical harmonics with Fourier exponents in multidimensions

We consider integrals of spherical harmonics with Fourier exponents on the sphere S n , n ≥ 1. Such transforms arise in the framework of the theory of weighted Radon transforms and vector diffraction in electromagnetic fields theory. We give analytic formulas for these integrals, which are exact up to multiplicative constants. These constants depend on choice of basis on the sphere. In addition, we find these constants explicitly for the class of harmonics arising in the framework of the theory of weighted Radon transforms. We also suggest formulas for finding these constants for the general case.

Introduction

We consider the integrals

I m k (p, ρ) def = S n Y m k (θ)e iρ(p,θ) dθ, p ∈ S n , ρ ≥ 0, n ≥ 1, (1)
where

{Y m k | k ∈ N ∪ {0}, m = 1, a k,n+1
} is orthonormal basis of spherical harmonics on S n ⊂ R n+1 (see e.g. [START_REF] Morimoto | Analytic functionals on the sphere[END_REF], [START_REF] Stein | Introduction to Fourier analysis on Euclidean spaces[END_REF]), a k,n is defined as follows:

a k,n+1 = n + k k + n + k -2 k -2 , a0,n = 1, a1,n = n, (2)
where

n k = n! k!(n -k!)
, n, k ∈ N ∪ {0}.

(3)

We recall that spherical harmonics {Y m k } are eigenfunctions of the spherical Laplacian ∆ S n and the following identity holds (see e.g. [START_REF] Morimoto | Analytic functionals on the sphere[END_REF], [START_REF] Stein | Introduction to Fourier analysis on Euclidean spaces[END_REF]):

∆ S n Y m k = -k(n + k -1)Y m k , m = 1, a k,n+1 , (4)
where a k,n is defined in (2). Integrals I m k arise, in particular, in connection with iterative inversions of the weighted Radon transforms in dimension d = n + 1 = 2; see [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF]. In addition, an exact (up to multiplicative coefficient depending on k) analytic formula for (1) was given in [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF]

for d = n + 1 = 2, k = 2j, j ∈ N ∪ {0}.
We recall that weighted Radon transform operator RW is defined (in dimension d = n + 1) as follows (see e.g. [START_REF] Goncharov | An analog of Chang inversion formula for weighted Radon transforms in multidimensions[END_REF], [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF]):

RW f (s, θ) def = xθ=s W (x, θ)f (x) dx, s ∈ R, θ ∈ S n , ( 5
)
where W is the weight function on R n+1 × S n , f is a test-function. The present work is strongly motivated by the fact that integrals I m k also arise in the theory of weighted Radon transforms defined by (5) for higher dimensions d = n + 1 ≥ 3. This issue will be presented in detail in the subsequent work [START_REF] Goncharov | Iterative inversion of weighted Radon transforms in 3D[END_REF].

On the other hand, in [NPF + 06] integrals I m k were considered for the case of n = 2 in connection with vector diffraction in electromagnetic theory and exact analytic formulas were given for this case.

In addition, for the case of dimension n = 2 more general forms of integrals I m k were considered in the recent work [A13]. In particular, the results of [A13] coincide with the results of the present work for the case of dimension n = 2.

In the present work we prove that

I m k (p, ρ) = c(m, k, n)Y m k (p)ρ (1-n)/2 J k+ n-1 2 (ρ), (6)
where Jr(•) is the r-th Bessel function of the first kind, c(m, k, n) is a constant which depends on indexes m, k of spherical harmonic Y m k and on dimension n; see Theorem 1 in Section 2. This result is new for the case of d = n + 1 = 2 for odd k and for d = n + 1 > 3 in general. In the framework of applications to the theory of weighted Radon transforms, integrals I m k arise for the case of even k = 2j, j ∈ N ∪ {0}; see formula (7) of [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF] for n = 1 and subsequent work [START_REF] Goncharov | Iterative inversion of weighted Radon transforms in 3D[END_REF] for n ≥ 2. For k = 2j we find explicitly the constants c(m, 2j, n) arising in (6); see Theorem 1 in Section 2.

It is interesting to note that the constants c(m, 2j, n) are expressed via the eigenvalues of the Minkowski-Funk transform M on S n , where operator M is defined as follows (see e.g. [START_REF] Funk | Über Flächen mit lauter geschlossenen geodätischen Linien[END_REF], [START_REF] Gel'fand | Selected topics in integral geometry[END_REF]):

M[f ](p) = S n , (θp)=0 f (θ) dθ, p ∈ S n , ( 7
)
where f is an even test-function on S n ; see Section 2 for details.

In Section 3 we give proofs of Theorem 1 and Remark 1.

Main results

Theorem 1. Let I i k (p, ρ) be defined by (1). Then: (i) The following formulas hold:

I m k (p, ρ) = c(m, k, n)Y m k (p)ρ (1-n)/2 J k+ n-1 2 (ρ), (8)
I m k (p, -ρ) = (-1) k I m k (p, ρ), (9)
p ∈ S n , ρ ∈ R+ = [0, +∞),
where Jr(ρ) is the standard r-th Bessel function of the first kind, c(m, k, n) depends only on integers m, k, n for fixed orthonormal basis {Y m k }. (ii) In addition, for k = 2j, j ∈ N ∪ {0}, the following formulas hold:

c(m, 2j, n) = 2 (n-1)/2 πΓ j + n 2 λj,n Γ j + 1 2 , ( 10
)
λj,n = 2(-1) j √ π Γ(j + 1 2 ) Γ(j + 1) n-1 , ( 11
)
where c(m, k, n) are the constants arising in (8), Γ(•) is the Gamma function, λj,n are the eigenvalues of the Minkowski-Funk operator M defined in (7).

In the case of n = 1, k = 2j formulas (8)-( 9) arise in formula (7) of [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF]. In the case of n = 2 formulas (8)-( 9) and constants c(m, k, n) for general k were given in formula (1) of [NPF + 06].

We didn't success to find in literature the explicit values for the eigenvalues λj,n of operator M defined in (15) for n > 2.

In particular, formulas (8)-( 11) are very essential for inversion of weighted Radon transforms; see [START_REF] Kunyansky | Generalized and attenuated Radon transforms: restorative approach to the numerical inversion[END_REF] for n = 1 and the subsequent work [START_REF] Goncharov | Iterative inversion of weighted Radon transforms in 3D[END_REF] for n ≥ 2.

Remark 1. Let c(m, k, n) be the constant arising in (8) and function Φ(k, n) be defined as follows:

Φ(k, n) def = a k,n+1 m=1 c(m, k, n), ( 12
)
where a k,n+1 is defined in (2). Then function Φ(k, n) does not depend on the particular basis of spherical harmonics {Y m k } for fixed k, n.

Proofs

Proof of formula (8)

From the Funk-Hecke Theorem (see e.g. [START_REF] Morimoto | Analytic functionals on the sphere[END_REF], Chapter 2, Theorem 2.39) it follows that:

S n Y m k (θ)e iρ(pθ) dθ = Y m k (p)c m k (ρ), m = 1, a k,n+1 . (13)
Formula (8) follows from (13), the following differential equation:

1 ρ n d dρ ρ n dc m k dρ + 1 - µ k,n ρ 2 c m k = 0, c m k (0) = 0 for k ≥ 1, (14)
µ k,n = k(n + k -1), (15)
where function c m k (ρ) arises in the right hand-side of (13), and from the fact that the solution of equation ( 14) with indicated boundary condition is given by the formula:

c k m (ρ) = c(m, k, n)ρ (1-n)/2 J k+ n-1 2 (ρ), (16)
where c(m, k, n) is some constant depending on integers m, k, n; Jr(ρ) is the r-th Bessel function of the first kind (see e.g. [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF]). Formulas (13), (16) imply formula (8).

It remains to prove that formulas ( 14)-( 16) hold. First, we prove that formulas ( 14), (15) hold.

We recall that Laplacian ∆ in R n+1 in spherical coordinates is given by the formula:

∆u = 1 ρ n ρ n d dρ u + 1 ρ 2 ∆ S n u, ρ ∈ (0, +∞), ( 17
)
where u is a test function.

From formulas (1), (4), ( 13), ( 17) it follows that

∆I m k (p, ρ) = Y m k (p) 1 ρ n d dρ ρ n dc m k dρ - µ k,n c m k ρ 2 , ( 18
)
where µ k,n is defined in (15). On the other hand,

∆I m k (p, ρ) = S n Y m k (θ)∆e iρ(pθ) dθ = - S n Y m k (θ) |θ| 2 e iρ(pθ) dθ = -I m k (p, ρ), p ∈ S n , ρ ≥ 0. (19)
Formulas ( 14), (15) follow from (1), ( 18), (19). In particular, the boundary condition in (14) follows from orthogonality of {Y m k } on S n and the following formulas:

I m k (0, p) = S n Y m k (θ) dθ = 0, k ≥ 1, vol(S n )c, k = 0 , (20)
Y 1 0 (p) = c = 0, p ∈ S n , (21)
where I m k is defined in (1), vol(S n ) denotes the standard Euclidean volume of S n . Next, formula (16) is proved as follows. We use the following notation for fixed k, m:

y(t) = c m k (ρ), t = ρ ≥ 0. ( 22
)
Using differential equation ( 14) and the notations from ( 22) we obtain:

ty ′′ (t) + ny ′ (t) + t - µ k,n t y(t) = 0, y(0) = 0, for k ≥ 1. ( 23
)
In order to solve (23) we make the following change of variables:

y(t) = t (1-n)/2 Z(t), t ≥ 0. ( 24
)
Formula (24) implies the following expressions for y ′ (t), y ′′ (t) arising in (23):

y ′ (t) = 1 -n 2 t -(1+n)/2 Z(t) + t (1-n)/2 Z ′ (t), (25)
y ′′ (t) = (n 2 -1) 4 t -(1+n)/2 Z(t) t + (1 -n)t -(1+n)/2 Z ′ (t) + t (1-n)/2 Z ′′ (t), t ≥ 0, (26)
where Z(t) is defined in (24). Using formulas (23), ( 25), (26) we obtain:

tZ ′′ (t) + Z ′ (t) + t - k + n-1 2 2 t Z(t) = 0, t ≥ 0. ( 27
)
Differential equation ( 27) for unknown function Z(t) is known as Bessel differential equation of the first kind with parameter k + (n -1)/2 ∈ R (see e.g. [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF]). The complete solution of ( 27) is given by the following formula:

Z(t) = C1J k+ n-1 2 (t) + C2Y k+ n-1 2 (t), t ≥ 0, (28)
where Jr(t), Yr(t) are r-th Bessel functions of the first and second kind, respectively, C1, C2 are some constants; see e.g. [START_REF] Temme | Special functions: An introduction to the classical functions of mathematical physics[END_REF]. Boundary condition in (23) implies that

Z(t) = C1J k+ n-1 2 (t), t ≥ 0. (29)
Formulas ( 22), ( 24), (29) imply that ( 16) is the complete solution of (23). Formula (8) is proved.

Proof of formula (9)

Formula (9) follows from definition (1) and the following property of the spherical harmonic Y m k :

Y m k (-θ) = (-1) k Y m k (θ), θ ∈ S n . ( 30
)
Property (30) reflects the fact that

Y m k (θ) = Y m k (θ1, . . . , θn+1
) is a homogeneous polynomial of degree k restricted to S n . Formula (9) in Theorem 1 is proved.

Proof of formulas (10), (11)

Formula (10) follows from orthonormality of {Y m k } (in the sense of L 2 (S n )), from formulas (1), ( 8), ( 9), the following formula:

M[Y m 2k ] = λ k,n Y m 2k (31)
and from the following identities:

S n Y m 2k (p) dp +∞ -∞ I m 2k (p, ρ) dρ = 2c(m, 2k, n) S n |Y m 2k (p)| 2 dp +∞ 0 J 2k+ n-1 2 ρ (1-n)/2 dρ (32) = 2c(m, 2k, n) +∞ 0 J 2k+ n-1 2 ρ (1-n)/2 dρ = c(m, 2k, n) 2 (3-n)/2 Γ( 1 2 + k) Γ(k + n 2 ) , S n Y m 2k (p) +∞ -∞ I m 2k (p, ρ) dρ = 2π S n Y m 2k (p) dp S n Y m 2k (θ)δ(pθ) dθ (33) = 2π S n Y m 2k (p)M[Y m 2k ](p) dp = 2πλ k,n S n |Y m 2k | 2 (p) dp = 2πλ k,n , where c(m, 2k, n) arises in (8), δ = δ(s) is 1D Dirac delta function, Y m k is the complex conjugate of Y m k , M[Y m k ] is defined in (7), λ k,n is given in (11), Γ(•)
is the Gamma function. Formula (31) reflects the known property of the Funk-Minkowski transform M that the eigenvalue λ k,n of operator M[•] defined in (7) corresponds to the eigensubspace of harmonic polynomials of degree 2k on R n+1 restricted to S n (see e.g. [START_REF] Gel'fand | Selected topics in integral geometry[END_REF], Chapter 6, p. 24). Note that, in [START_REF] Gel'fand | Selected topics in integral geometry[END_REF] it was proved that formula (31) holds also for all harmonic polynomials in R 3 (i.e. n = 2) restricted to S 2 , however these considerations admit a straightforward generalization to the case of arbitrary dimension n ≥ 1.

Formulas (32), (33) imply formula (10). Now, it remains to find the explicit value for λ k,n in formula (31). We obtain it according to [GGG03] (Chapter 2, page 24), where the case of dimension n = 2 was considered.

In particular, formula (31) holds for any homogeneous harmonic polynomial P 2k of degree 2k in R n+1 , where P 2k is restricted to the sphere S n ⊂ R n+1 ; see [START_REF] Gel'fand | Selected topics in integral geometry[END_REF] (Chapter 2).

We consider the following harmonic polynomial in R n+1 :

P 2k (x) = P 2k (x1, . . . , xn+1) = (xn + ixn+1) 2k , x ∈ R n+1 . ( 34
)
From formula (34), aforementioned results of [START_REF] Gel'fand | Selected topics in integral geometry[END_REF] (Chapter 6) and their generalizations to the case of arbitrary dimension n ≥ 1 it follows that P 2k being restricted to sphere S n is an eigenfunction of operator M which corresponds to the eigenvalue λ k,n .

We consider the spherical coordinates in R n+1 given by the following formulas: Formulas (34), (35) imply that polynomial P 2k being restricted to S n may be rewritten as follows:

P 2k | S n = P 2k (θn, θn-1, . . . , θ1, φ) = e i2kφ n i=2 sin 2k (θi), (36)
where (θn, . . . , θ1, φ) are the coordinates on S n according to (35).

From formulas in (34), ( 35), (36) it follows that:

P 2k | S n = P 2k (π/2, . . . , π/2, 0) = 1. ( 37
)
From formulas (31), ( 37), (36) we obtain:

λ k,n = M[P 2k ](π/2, . . . , π/2, 0) = 2(-1) k n i=2 π 0 sin 2k (θi) dθi = 2(-1) k √ π Γ(k + 1 2 ) Γ(k + 1) n-1 , (38)
which implies (11). Formulas (10), (11) are proved.

Proof of Remark 1

The statement of the remark follows from the following formulas:

a k,n+1 m=1 c(m, k, n) = ρ (1-n)/2 J k+ n-1 2 (ρ) -1 a k,n+1 vol(S n ) S n S n P k,n (pθ)e iρ(pθ) dp dθ, (39)
ρ ≥ 0 such that ρ (1-n)/2 J k+ n-1 2 (ρ) = 0, (40)
P k,n (σθ) = vol(S n ) a k,n+1 a k,n+1 m=1 Y m k (σ)Y m k (θ), σ, θ ∈ S n , (41)
where a k,n+1 is defined in (2), vol(S n ) denotes the standard Euclidean volume of S n , Y m k is the complex conjugate of Y m k , P k,n (t), t ∈ [-1, 1] is a Gegenbauer's polynomial defined by the formula (see [START_REF] Morimoto | Analytic functionals on the sphere[END_REF]):

P k,n (t) def = (-1) k 2 k Γ( n 2 ) Γ(k + n 2 ) 1 (1 -t 2 ) n-2 2 d k dt k (1 -t 2 ) k+ n-2 2 , k ∈ N ∪ {0}, n ≥ 2, (42)
where Γ(•) is the Gamma function.

The integrand in dp dθ in the right hand-side of (39) is independent of basis {Y m k }, which implies the statement of Remark 1.

It remains to prove that formulas (39), (41) hold, where P k,n (t), t ∈ [-1, 1] is defined in (42). Formula (41) is proved in [Mor98] (Chapter 2), which is also the known property of Gegenbauer's polynomials.

From formulas (1), (8), (41) it follows that:

where {Y m k } is the orthonormal basis of spherical harmonics on S n , P k,n (t), t ∈ [-1, 1] is defined in (42). Formulas (41), (43) imply formula (39). Remark 1 is proved.

Aknowledgments

The present work was fulfilled in the framework of research conducted under the direction of Prof. R. G. Novikov.

The author would like also to thank assistant-professor Hendrik De Bie at Ghent University who noticed the first version of this work and provided useful links to his own result on the case odd k in Theorem 1; see [START_REF] De Bie | Spherical harmonics and integration in superspace[END_REF].

  x1 = cos(θn), x2 = sin(θn) cos(θn-1), • • • (35) xn = sin(θn) sin(θn-1) • • • sin(θ2) cos(φ), xn+1 = sin(θn) sin(θn-1) • • • sin(θ2) sin(φ), θn, θn-1, • • • , θ2 ∈ [0, π], φ ∈ [0, 2π).

2

  ρ)Y m k (p) dp = S n S n P k,n (pθ)e iρ(pθ) dθ dp = ρ (1-n)/2 J k+ n-1 )| 2 dp, = ρ (1-n)/2 J k+ n-1 2 (ρ) vol(S n ) a k,n+1 a k,n+1m=1c(m, k, n).


Ewa Banach

Christian Invernizzi

Mathieu Baudin

Reinhard Neier

Diego Carnevale

Columnar self-assembly of N,N',N''-trihexylbenzene-1,3,5-tricarboxamides investigated by means of NMR spectroscopy and computational methods in solution and solid state 1 Columnar self-assembly of N,N',N''-trihexylbenzene-1,3,5-tricarboxamides investigated by means of NMR spectroscopy and computational methods in solution and solid state

Keywords: Columnar Self-Assemblies, NMR Spectroscopy, Magic-Angle Spinning NMR, DFT Calculations, Nuclear Overhauser Effect, Rotating-Frame Overhauser Effect, Chemical Shift Anisotropy, Calculations of Chemical Shifts, CSA Amplification, Dispersion Interactions, B3LYP-D Functional, B97D Functional

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Introduction

Self-assembly processes are very widely spread in nature and play a key role in most chemical systems relevant to life. The striking feature of these processes relies on the fact that a small number of building blocks can undergo self-organization solely under direction of the intermolecular interaction between the building blocks themselves, and result in very complex macromolecular systems. [START_REF] Steed | Supramolecular Chemistry: From Molecules to Nanomaterials[END_REF] From the perspective of supramolecular chemistry, this is an invaluable feature as the whole self-assembly process takes place without external intervention.

A classic example is given by the DNA double helix, where hydrogen bonds and π-π stacking interactions direct the self-assembling of the base pairs. The elucidation of the interplay between the relatively small number of intermolecular interactions that lead to the creation of ordered superstructures starting from much smaller and simpler units is of fundamental importance to understand chemical recognition processes and hence to be able to design selfassembly systems resulting in materials with desired properties. [START_REF] Philp | [END_REF] Columnar self-assemblies based on benzene-1,3,5-tricarboxamide (BTA) motif result from π-π stacking interactions and formation of maximal number of hydrogen bonds between the individual units. [3][4][5][6] These discotic systems [7] are characterized by a helical arrangement of units and have attracted attention as organic electronic elements due to their liquid crystal features. [8] Detailed studies of their aggregation process have highlighted the cooperative nature of these self-assemblies. [9,10] Derivatives of BTA have been investigated as MRI contrast agents, [11] metal complexation agents [12] and utilized in drug-delivery systems. [13] Investigations of BTA-based self-assemblies have focused on the effects of the aliphatic chains, [14,15] introduction of elements of chirality [16] and substitutions on the central aromatic moiety. [17][18][19] The aggregation process of BTA units and their charge transport properties have been investigated by means of quantum chemical calculations. [20][21][22][23] NMR methods based on dipolar interactions and double-quantum approaches have been utilized to characterize structure and dynamics of these systems in the solid state. [24] In spite of all these efforts, and as for most other self-organizing systems, the prediction of the efficiency of the self-assembly processes related to BTA building blocks is still largely based on empirical data and, hence, remains a challenge. [25] NMR spectroscopy is an invaluable tool to probe structure and dynamics on the atomic scale for ordered or disordered systems, both in solution or in solid state. [START_REF] Ernst | Principles of Nuclear Magnetic Resonance in One and Two Dimensions[END_REF] The isotropic tumbling that affects molecules in solution allows for high-resolution spectra with typical linewidths of a few Hz. This is due to the fact that only the isotropic part of the interactions is retained by the averaging due to the fast motion, which is typically much faster than the time scale of the NMR experiment. As a result, isotropic peaks are readily identified, with only fine structures due to scalar J-coupling left to reveal through-bond interconnectivities between the various chemical environments. Nonetheless, the anisotropic parts of the internal interactions play an important role in relaxation phenomena.

The nuclear Overhauser effect (NOE) is widely used to investigate internuclear distances in molecules in solution. [START_REF] Claridge | High-resolution NMR techniques in organic chemistry[END_REF][START_REF] Neuhaus | The nuclear Overhauser effect in structural and conformational analysis[END_REF] This effect relates nuclear spins which are dipolar coupled to one another, i.e., which are in spatial proximity to one another. When a nuclear site within a molecule is perturbed by a radio-frequency (rf) irradiation so as to achieve either inversion or saturation of the population difference across the Zeeman eigenstates, a variation in signal intensity is observed for spins which are dipolar coupled to the irradiated one. When compared to the intensity of a pulse-acquire spectrum, this NOE enhancement may be either positive, null or negative, depending on the correlation time of the molecular species in solution. A widely spread modification of this experiment is the rotating frame Overhauser effect (ROE), [START_REF] Bothner-By | [END_REF]30] where relaxation and polarization transfer take place in the rotating frame of the rf irradiation.

The main advantage of this latter technique is that the enhancements are always positive and may be preferred in cases where the NOEs are close to zero. If either NOE or ROE experiments are conducted in the limit of linear growth, measurements of enhancements can be directly related to internuclear distances by means of the relationship: [START_REF] Claridge | High-resolution NMR techniques in organic chemistry[END_REF]  

16 AB AX X B r r I I  , (1)
where the index A refers to the irradiated spin, X to a spin whose distance from A, rAX, is known and taken as reference, and B is a third spin whose distance from A, rAB, needs to be evaluated.

IX and IB are the NOE/ROE enhancements observed for spins X and B, respectively. This is an invaluable tool for structural studies and may supply experimental evidence that allows discharging a structural hypothesis in favor of another. In cases where also intermolecular internuclear distances need to be taken into account, such as in dimeric structures, or when equivalent environments supply for more than one interatomic distance, such as for CH2 groups, an effective distance that takes into account all possible polarization transfer pathways may be utilized: [31,32]  

16 1 6 6 6 , 1 , 2 , 1 1 1 ... eff i i i i n r r r r              , ( 2
)
where the i-th spin experiences n polarization transfer pathways with n other spins at distances ri,n. These n different pathways are related to direct transfers and indirect effects due to spin diffusion are not considered in this study. [33] An elucidation about the concept of this effective distance r (eff) may be found in Ref. [31].

In contrast to solutions, where the isotropic tumbling motion of molecules averages all interactions to their isotropic components (or to zero if traceless Hamiltonians are involved), in the solid state, the anisotropic part of the internal interactions is retained, and results in broadenings of the NMR lineshapes which are termed inhomogeneous. These result from chemical shift anisotropies (CSA), dipolar couplings, hyperfine couplings with paramagnetic centres and quadrupolar couplings with electric field gradients (EFG). [START_REF] Duer | Solid-State NMR Spectroscopy: Principles and Applications[END_REF] With the exception of 2 nd order quadrupolar broadenings, and provided the spinning rate is large enough, the use of magic-angle spinning (MAS) [START_REF] Andrew | [END_REF]36] allows for an efficient averaging of these inhomogeneities and high-resolution isotropic spectra may be acquired in many cases. Nevertheless, the anisotropic part of the interactions provides very useful information as further means to characterize nuclear environments. Specific techniques have been developed to reintroduce this inhomogeneity in the indirect dimension of 2D experiments. [START_REF] Duer | Solid-State NMR Spectroscopy: Principles and Applications[END_REF] In such a fashion, fast spinning rates ensure direct acquisition of high-resolution spectra where isotropic shifts may be readily available while the anisotropic interaction is recoupled by the action of the rf pulses onto an orthogonal axis. With respect to the anisotropy of the shift interaction, methods have been introduced that are not only capable of recoupling the inhomogeneity averaged out by the fast mechanical rotation, but also amplifying it, as if the experiment was performed at a slower and fictitious spinning rate. [37][38][39][40][41][42] This allows for more precise measurements of the full shielding tensor than those that one can achieve under static conditions or by spinning the sample at lower rates. [38] With respect to the chemical shift interaction, the following convention of NMR parameters is adopted in this study: [START_REF] Haeberlen | Advances in Magnetic Resonance[END_REF] iso

= (xx + yy + zz)/3 = Tr[]/3 (3) CS = zz -iso (4) CS = (yy -xx)/CS (5)
where the isotropic shift iso, the anisotropy CS and asymmetry CS are expressed in terms of the principal components xx, yy and zz of the shielding tensor  in its principal axis frame.

These principal elements are ordered according to zz -iso  xx -iso  yy -iso. The isotropic chemical shift is given by iso = ref -iso, where ref is a reference shielding.

Density functional theory (DFT) [START_REF] Parr | Density-Functional Theory of Atoms and Molecules[END_REF] calculations are widely used nowadays as means to interpret and support experimental evidences. In cases where different structural hypotheses are under consideration, DFT calculations can supply the invaluable means to identify the one that agrees best with the experimental data. With respect to NMR spectroscopy, calculations of parameters such as magnetic shielding tensors, EFG tensors or scalar couplings have proven extremely useful in spectral assignment both in solution and solid state. [START_REF] Bagno | [END_REF][46][47][48][49][50][51][52][53][54] Structural details such as internuclear distances or dihedral angles may directly be extracted from the optimized geometries. Most commonly-used density functionals are characterized by a poor description of van der Waals interactions. [55] These long-range attractive contributions play an important role in supramolecular chemistry and self-assembly processes. Therefore, there has been a considerable effort to develop new functionals capable of describing these dispersion forces. [56] In this framework, we investigate the columnar self-assembly resulting from BTA in both solution and solid state using a combined approach employing NMR spectroscopy and DFT computational methods. One-dimensional (1D) produced by the widely-used B3LYP functional are compared with those produced by the B3LYP-D and B97D functionals to highlight structural differences resulting from the inclusion of dispersion forces. The influence of the orientation of the BTA amide functions and π-π stacking effects on chemical shift parameters of carbon sites is considered.

Results and discussion

Solution study

The structure of N,N',N''-trihexylbenzene-1,3,5-tricarboxamide (BTA) is shown in Figure 1(a). A central C3-symmetrically substituted aromatic ring (discotic unit) is functionalized by three amide groups containing n-hexyl chains. The amide functionalities are linked to the aromatic ring by the carbonyl groups. A simplified scheme depicting a two-unit segment of a columnar self-assembly of BTA is presented in Figure 1(b). The scheme of Figure 1(b) allows the visualization of the main intermolecular interactions that direct the self-assembly process. The parallel orientation of aromatic rings ensures π-π stacking interactions (highlighted in blue), whereas the three amide groups drive the helical arrangement by forming intermolecular Hbonds with neighboring BTA units (dashed red lines). Figure 2(a) shows a 1 H NMR spectrum of a highly-concentrated solution (177 mM) of BTA in CD2Cl2 at 9.4 T (400 MHz). By means of the equation χM = 1/(1+(c/KD) p ) as utilized in concentration-dependent NMR studies of the same BTA assemblies, [19] where c is the total concentration, KD = 39 mM [19] is the dissociation constant and p = 2.2 [19] is a factor that takes into account cooperativity, the molar fraction χM of the free monomeric BTA is evaluated to 0.035, thus ensuring that the self-assembled species are predominant in solution and averaging of NMR observables due to fast chemical exchange may related to the homonuclear scalar interaction that couples this specific environment with the site that has been irradiated.

In order to interpret these ROE enhancements in terms of internuclear distances, a parallel computational study was performed. A monomeric structure of BTA was firstly optimized with the PM7 semiempirical method as implemented in MOPAC2016. [START_REF] Mopac ; James | Stewart Computational Chemistry[END_REF] The aliphatic chains of all BTA units were truncated to propyls so as to reduce computational times.

This optimized geometry was subsequently used to assemble dimeric, trimeric, tetrameric and pentameric columnar systems. The monomeric and pentameric structures supply a computational model to use for calculations of parameters to be compared with experimental data obtained for either a disassembled or assembled system, respectively. In the pentameric case, only the central unit was considered, with the other two units either up or down the column axis having the sole purpose of smoothing the boundary conditions of the column, i.e., of quencing the perturbations due to the different chemical environments experienced by the 1 st and 5 th TBA units which terminate the column. In other words, in our computational study, the central unit of our pentameric assembly is assumed to be representative of a unit of BTA belonging to an infinitely long columnar assembly.

It is important to note at this point that such truncation of a column to a pentameric assembly results in structural parameters such as interplane distances or chemical shifts that are not identical, at equivalent sites, for all units, but rather smear from those of the boundary BTA molecules to those of the central one. This is a consequence of the fact that no periodicity is assumed in our computational study. Again, only the central unit is taken into account in this context as being representative of the columnar assembly as found in either solution or solid state.

All the assembled structures were further optimized at the PM7 level, followed by geometry optimization with DFT methods utilizing the B3LYP functional [START_REF] Becke | [END_REF]59] and 6-31G(d,p) basis set. As the B3LYP method does not take into account dispersion forces and more generally long range non-covalent interactions, the DFT optimizations were also repeated with the B3LYP-D and B97D functionals,[60] ad hoc parametrized to describe these long-range effects. Figure 3(a), (b) and (c) show the pentameric assemblies optimized with the three considered methods.

The columnar arrangement of BTA units with helical pattern of inter-plane hydrogen bonds is particularly evident in the B3LYP case of (a), where an interplane distance of ca. 3.8 Å is found between either the second and third or third and fourth units. In the cases of B3LYP-D and B97D, instead, a more compact pentamer is obtained, with π-π stacking distances between the same units of ca. 3.2 Å in both cases. This is in agreement with previous observations reported in the literature.

[61] It is worth mentioning in this context that X-ray diffraction data related to previous studies on N,N',N''-tris(2-methoxyethyl)benzene-1,3,5-tricarboxamide revealed an interplane distance in the column self-assembly of 3.62 Å. [19] Furthermore, as can be seen in (d), (e) and (f), from views down the column axis of the same structures of (a), (b) and (c), respectively, the alignment of the benzene moieties in the B3LYP case is more regular, with BTA units experiencing a progressive average twist of 58.6 ± 1.0° around the column axis as one moves from plane to plane down the columnar assembly. This arrangement of units is directed by the establishment of the hydrogen-bonding network and results in the helical nature of the column. A measure of this twist may be given by the average H-C-C-H and O=C-C-C-C=O interplane dihedral angles given in Table 1, with the first two sites defining each dihedral angle belonging to the n th unit of the column and the second two to the n th +1 unit. In the cases of B3LYP-D and B97D of Figure 3(e) and (f), in contrast, the progressive average twist of BTA units across the different planes of the columnar assembly, as dictated by the H-bonding network and by π-π stacking interactions, is 62.8 ± 2.0 and 62.7 ± 2.4°, respectively, thus resulting in a less regular alignment of benzene moieties, as viewed down the column axis.

The orientation of the amide functions within the self-assembly may be measured by the dihedral angles H-C-C-C=O as given in Table 2. The structure produced by the B3LYP functional results in amide functions oriented so that this average angle amounts to θ = -34.4 ± 3.6°

whereas the B3LYP-D and B97D functionals result in amide functions closer to coplanarity with the benzene cores, with average dihedral angle amounting to -28.6 ± 6.0 and -25.9 ± 11.4°.

These values reflect the more compact structure obtained when dispersion interactions are taken into account. In this latter case, the larger standard deviation of values confirms the less regular alignment of BTA units obtained with the B3LYP-D and B97D functionals.

Furthermore, one of the amide functions of the boundary BTA unit terminating the column structure (5 th unit in Table 2) produced by the B97D method is oriented discordantly from the other two within the same unit, with H-C-C-C=O dihedral angles amounting to θ = -17.9, -15.5 and +9.6°. This feature can be easily seen in the electron density isosurfaces colorcoded with the electrostatic potential given in Figure ESI1 and represents possible different chemical environments, or structural irregularities, at the boundaries of the assemblies. In contrast, the more regular structure produced by B3LYP results in concordant angles of θ = -29.5, -29.7 and -29.3° for the same unit.

By inspection of the dihedral angles H-C-C-C=O given in Table 2, when moving from either the 1 st or the 5 th terminal BTA units towards the central 3 rd one, one can generally see a progressive adjustment of this structural parameter, with progressively smaller standard deviation of values. This observation indicates that our choice of a pentameric system is effective in 'quenching' the propagation of structural irregularities from the boundaries of the self-assembly up to the considered central unit.

The DFT-optimized monomeric and pentameric structures thus produced were utilized to measure proton-proton interatomic distances. These values were compared with those derived from the experimental ROE measurements of Figures 2. The effective distance of Eq. ( 2)

was utilized so as to take into account for the larger number of polarization transfer pathways only possible in the assembled systems, i.e., for both intra and intermolecular proton-proton interatomic distances. Figure 4 shows the differences between interatomic distances

ROE DFT r r r    as histogram.
The larger these values, the larger the disagreement between the computational model and the experimental measurement. These results indicate that (i) ROE experiments can be used to ascertain whether BTA units in solution have undergone self-assembling or not and (ii) the use of DFT methods that take into account for the effect of non-covalent dispersive interactions is required when interatomic distances in such supramolecular self-assemblies are considered.

In the attempt to obtain ROE measurements related to a non-assembled BTA monomeric state, a more dilute 5 mM solution of BTA in acetone-d6 was considered. At this concentration, the predominant species in CD2Cl2 is the isolated monomeric BTA, for which χM = 0.989. this concentration is inverted. This observation has been confirmed by 1 H- 13 C HSQC spectra (Figure ESI4). The overall inaccuracy of the ROE measurements in the more diluted case may be reasonably ascribed to i) much lower S/N, ii) much longer spin locking pulse (50 ms vs 5 ms used for the 177 mM sample) in the attempt to gain sensitivity. This latter feature results in a less strict fulfillment of the condition of linear growth of the ROE enhancement. Finally, iii) chemical exchange of the NH protons with the residual water of the solvent -more likely to be relevant when BTA is not assembled -introduces leakage of the ROE enhancement towards the water proton bath.

The DFT-optimized BTA structures for monomer, dimer, trimer, tetramer and pentamer have subsequently been used to calculate the chemical shifts for the proton and carbon nuclei at the B3LYP/6-31+G(d,p) level of theory, i.e., with addition of diffuse functions on heavy atoms with respect to the geometry optimizations. In the case of trimers and pentamers, only the central units were considered, for dimeric structures both units were averaged, whereas for the tetramers the two central units were averaged. Within a given unit, all 'equivalent' sites were also averaged. In other words, say, three C=O shifts were averaged to one single shift, three CH2

shifts were averaged to one single CH2 environment, and so on so forth. one single site for the C resonance and three sites for the CH resonance, so as to take into account for the shoulders of this peak. The aliphatic region was excluded from the fitting procedure. Nonetheless, the resulting fit is substantially in good agreement with the experimental spectrum, and produces the anisotropic parameters summarized in Table 3.

Figure 6(d) shows a dipolar-dephasing spectrum which confirms the assignment of the CH resonance at 126.2 ppm as this peak disappears when allowed to diphase under dipolar interactions with its neighboring proton. The disappearance of this peak and of all its spinning sideband family is indicated in figure by a dashed arrow. Importantly, the CP spectra of (a) and (b), along with the simulation of (c), indicate that, as the spinning rate is decreased, spinning sidebands at lower frequency appear in the aliphatic region thus resulting overlapped with the other isotropic aliphatic peaks. This condition prevented the fitting procedure used in Fig. 6(c) to take into account for these lower-frequency spinning sidebands.

An ideal condition would be one where the spinning rate is large enough so as to ensure isotropic peaks to be readily determined and avoid overlaps of resonances or spinning sidebands, but at the same time not large enough so as to result in an adequate number of spinning sidebands for analysis purposes. In order to achieve these advantages and obtain a more reliable characterization of the shielding tensors associated with these three carbon environments, a two-dimensional approach was chosen. The 13 C CP CSA Amplification [39,40] spectrum of the BTA considered in this study is shown in Figure 7(a). The F2 projection results in the conventional 13 C CP spectrum recorded at the true spinning rate of 20 kHz. This projection is analogous to the spectrum of Figure 6(a). The F1 projection instead results in a pure shift CSA amplified dimension. The amplification factor was 8. Therefore, the series of data points that construct the F1 projection of the 2D spectrum represents the intensities of a spinning sideband pattern as if the experiment was performed at a fictitious spinning rate of 20/8 = 2.5 kHz. The CSA patterns related to the individual sites of interest may be readily extracted from the 2D spectrum, as indicated by the dashed lines and arrows, and are shown in (b), (c) and (d). The corresponding fits are overlaid in red. It is worth bearing in mind that the experiment is designed so that the anisotropic part of the shift interaction is amplified whereas the isotropic part is refocused. This feature results in the projections of (b-d) to appear as if they all were associated with sites with iso  = 0 Hz. This has the advantage of reducing the space of variables that needs to be explored by the fitting procedure, as the only parameters that need to be considered are CS  and CS  . These fits are also considerably faster than those of Figure 7(c) as only a number of points that matches the indirect dimension increments need to be simulated, provided in-silico observation is performed stroboscopically with the rotor period. In our case, only 32 points needed to be simulated. Moreover, and in contrast with the fit shown in Fig. 6(c), only one site needs to be included in each fitting procedure as the indirect dimension of the 2D data set allows extraction of single-site CSA patterns. Furthermore, there is no need to simulate the full multi-pulse 2D experiment as a simple pulse-acquire experiment at the fictitious spinning rate suffices. The quality of the fits presented in Figure 7(b-d) is remarkably good. An estimation of the errors associated with these measurements of the CS  and CS  parameters for each analyzed carbon environment is given by the 95% confidence interval as extracted from the rms deviation 2D contour plots shown in (e-g). [41,42,69] The parameters extracted at the minima of these 2D surfaces are given in Table 3 for the three sites of interest.

The CSA Amplification 2D experiment of Figure 7 allowed for a full characterization of the chemical shift interaction, both in its isotropic and anisotropic components. These parameters shed a light on the solid-state BTA system with an atomic-scale resolution and could be used to investigate different structural hypotheses. In our context, we could ascertain whether these MAS NMR measurements are consistent with the existence of columnar self-assemblies in the solid state. As test-bed of our analysis, the hypothesis of a non-assembled system has also been considered. DFT calculations of the NMR parameters performed on the pentameric structures shown in Figure 3 supply a set of parameters that can be related to a columnar self-assembly of BTA units. In order to obtain an analogous set of parameters that instead refers to a non-assembled system, the computational approach shown in Figure ESI6 was adopted. Six BTA units were arbitrarily arranged as faces of a cube with edge length of ca.

14.5 Å. This configuration of molecules is shown in Figure ESI6(a). A schematic representation of the cubic arrangement is also depicted in (b). This initial configuration of BTA units has no physical meaning but rather serves as an arbitrary starting point for a geometry optimization procedure that prevents the units to self-assemble in a columnar superstructure. This system was subsequently allowed to relax by means of a geometry optimization performed at the PM7 semiempirical level. The resulting superstructure was then further refined with DFT methods utilizing the three considered functionals and 6-31G(d,p) basis set. These hexameric arrangements obtained with B3LYP and B97D are shown in Figure ESI6(c) and (d), respectively.

As expected, this procedure produces structures that do not show any columnar arrangement of BTA units and that are meant to mimic a non-assembled system. Both structures were subsequently used for the calculations of the chemical shift parameters. Only the innermost molecule of both arrangements was considered. This molecule is shown in black in Figure ESI6(c) and (d). The corresponding NMR parameters are assumed to be representative of a disordered system in the solid state.

Figures 8(a-c) show the correlations between the CSA parameters measured experimentally with our MAS techniques and those calculated for the structural hypotheses considered in this study, i.e., the columnar pentameric assembly of Fig. 3 and the collapsed cubic arrangement of Fig. ESI3, for B3LYP, B3LYP-D and B97D functionals. As previously discussed, in the case of the columnar pentameric assembly, only the central BTA unit was considered whereas, in the case of non-assembled hexamer of Figure ESI6, only the innermost BTA unit was taken into account. Computational data related to the monomeric BTA is also included for sake of comparison. The error bars on the experimental axes are given in Table 3.

Also the anisotropy of the aliphatic CH2 carbon sites is included in these plots. The ideal 1:1 correlation is indicated by a dashed line. Figure 8(d) shows the correlation coefficients R 2 related to the plots of (a-c), as histogram. Light blue, dark blue and red data refer to the CSA parameter as calculated with B3LYP, B3LYP-D and B97D, respectively. All methods show that the monomeric case results in the lowest qualities of correlation (R 2 = 0.971, 0.984 and 0.975, respectively). This is not surprising as this structural model is meant to represent a BTA unit in solution (or vacuum). More interestingly, our experimentally measured anisotropies correlate slightly better with those calculated on the self-assembled pentameric structure than with those resulting from the non-assembled arrangement of BTA units of Fig. ESI6. Differences in qualities of correlation are found on the third decimal place, with the B3LYP method yielding R 2 = 0.997 and 0.999, B3LYP-D giving R 2 = 0.993 and 0.995, and with B97D method resulting in R 2 = 0.994 and 0.996, for the non-assembled and assembled case, respectively. Data referring to the qualities of correlation for the isotropic shift are shown in grey scale in the histogram of Figure 8(d). In contrast with the shift anisotropy CS  , the isotropic part of the shift interaction iso  seems to be, in this particular case, considerably less informative, with very similar R 2 coefficients for all the structural hypotheses considered.

As previously stated, averaging of both isotropic and anisotropic chemical shifts on 'chemically-equivalent' sites was performed on the selected units of BTA in the calculated structures considered in this study. Within this modus operandi, we have shown that measurements of CSA parameters can be used to probe structural details in columnar selfassemblies of BTA in the solid state. In contrast, isotropic shifts do not supply with the required sensitivity to discriminate between different structural hypotheses in these particular systems.

However, further insights may be discerned by considering the distribution of isotropic shifts within the selected units of BTA in our structural models when those are not subject to averaging over equivalent sites. This is particularly useful for solid samples where crystalpacking effects, or more generally, as in our case, intermolecular interactions can lift the degeneracy of chemical sites in terms of magnetic shielding. Figure 9 shows a series of simulated 13 C spectra which assume all the calculated isotropic shifts (i.e., for the 12 carbon sites without averaging related to three C=O, three C, three CH and three CH2) associated with the selected units of BTA within our structural models. The resulting spectra for the monomeric, pentameric assembly and non-assembled (collapsed cubic arrangement) structures are shown in 9(a), (b) and (c), respectively. Only the computational data produced by the B3LYP method is shown for simplicity. A line broadening of 100 Hz was applied to each Lorentzian function utilized to reconstruct the spectra. This line broadening is similar to that experimentally observed in the MAS 1D spectra of Figure 6. Analogously to the correlations of Figure 5 and8, also the aliphatic CH2 environments have been included in these spectra. In (a), the main four environments can be readily identified, with subtle 'splittings' of ca. 115 Hz observed on the C and CH2 peaks. These 'splittings' represent chemical inequivalences at these sites produced by the geometry optimization procedure. These inequivalences are related to the fact that, in the optimized structure of a monomeric BTA system, two carbonyl groups are pointing up and one is pointing down, with H-C-C-C=O dihedral angles of 14.8, 15.3 and -17.3° for B3LYP and 17.7, 16.1 and -19.4° for B97D. This results in small variations in chemical shifts at the C and CH2 sites. In a real system in solution, free and fast rotation of the substituents about the C-C=O bonds will result in averaged environments (i.e., single peaks) as perceived on the NMR time scale. This feature legitimates the common procedure of averaging the calculated chemicals shifts for chemically-equivalent environments when solution data is analyzed. Figure 9(b) shows the 13 C NMR spectrum resulting from all 12 environments of the central BTA unit of the pentameric self-assembly of Fig. 3. It is interesting to notice that the degree of chemical shift dispersion in the columnar self-assembly is substantially identical to that calculated for the isolated monomeric unit of (a). Tiny splittings of ca. 100 Hz are observed on the C=O and aromatic CH resonances. However, these splittings are comparable with the inhomogeneous linewidth observed experimentally in the spectra of Figure 6(a,b) and therefore justify the averaging of the calculated chemical shifts for the self-assembly as performed in this study. In other words, the chemical shift inequivalences calculated by DFT methods for carbon nuclei in a columnar self-assembly of BTA units are not predicted, as is indeed the case with the only exception of the CH resonance at 126.2 ppm, to be observable in the experimental spectra of

Conclusions

The columnar self-assembly of N,N',N''-trihexylbenzene-1,3,5-tricarboxamide (BTA) units has been investigated in both solution and solid state by means of NMR techniques. A parallel computational study that combined semiempirical and DFT methods has been performed to produce structural parameters to be compared with those measured experimentally, namely, interatomic distances for ROE solution measurements, isotropic chemical shifts to be compared with those obtained from simple 1D solution spectra and anisotropic shifts to be compared with those measured with specifically designed multi-pulse MAS methods in the solid state. We show how a combination of NMR and computational methods allows to shed a light on self-assembly processes in solution by means of simple 1D measurements without the need to follow the assembling process in a multi-sample multi-experiment approach such as that of concentration-dependent NMR studies. More specifically, the analysis of ROE enhancements that take into account both intra and intermolecular interatomic distances allows to ascertain the presence of BTA self-assemblies in solution. DTF methods that include description for non-covalent dispersive interactions are required for such purpose. We also show that both isotropic and anisotropic parts of the shift interaction can be used to characterize BTA-based columnar assemblies in the solid state. With respect to both isotropic and anisotropic chemical shift parameters, the computational analysis of i) the influence of the orientation of the amide functions and ii) effects of dispersive forces, indicates that the CH is the most sensitive carbon environment to monitor columnar self-assembling processes of BTA units.

Experimental and computational details

All solution spectra were acquired in Neuchâtel, at the Neuchâtel Platform of Analytical utilizing the PCM method. [72] The convergence criteria for the DFT optimizations required variations of energy between two successive iterations to be smaller than 3 × 10 -2 kJ/mol.

Frequency calculations were subsequently run at the same level of theory to exclude imaginary vibrational modes. The structures thus obtained were used to calculate the magnetic shielding tensors of both 1 H and 13 C nuclei with the GIAO method [73,74]



, where   D  is the diagonal matrix with the eigenvalues of  as elements and P is the matrix constituted by the eigenvectors of  as columns. A scheme depicting the computational procedure followed in this study is shown in Figure 10. In order to reduce the computational time required for the DFT steps of the scheme, the aliphatic chains of BTA were truncated to propyls. DFT calculations were performed at the EPFL on a Ubuntu Linux 12.04 workstation equipped with 16 2.27 GHz CPUs and 24 GB of RAM.

[60] S. Grimme, J. Comput. Chem., 2006, 27, 1787-1799.

[61] Y. Minenkov, Å. Singstad, G. Occhipinti and V. R. Jensen, Dalton Trans. 2012, 41, 5526.

[62] The mismatch produced by the B3LYP functional in the case of the NH irradiation when the pentameric structure is considered, has to have an affect also the other two measurements for the CH2 and CH irradiations (red data in 4(a)). This is due to the fact that our three ROE experiments have been performed on a system of spins which may be seen as a closed and single one. In fact, when one of these spins is irradiated, a ROE enhancement is measured for the other two. Of these two enhancements, one is given an arbitrary and convenient value and the other is consequently determined. It is completely arbitrary which spin X in Equation ( 1) is chosen as reference, as it is only the ratio between enhancements IX and IB that is needed. By performing irradiation experiments on each of the three spins, we obtain three data sets that probe internuclear distances within this very single and closed three-spin system. In other words, in our case, an overestimation of one internuclear distance must result in a systematic overestimation of the other two. In fact, if an 'offset' distance is systematically subtracted from the red B3LYP data of 4(a), then one would obtain a profile of mismatches very similar to that produced by both B3LYP-D and B97D.

Figures

  be neglected. The NH, CH and NH-CH2 resonances, as indicated in the figure, are well resolved and are therefore ideal sites to irradiate in NOE/ROE experiments. Figures 2(b), (c) and (d) show 1D ROE spectra on the same sample of Figure 2(a), where the CH2 CH and NH resonances are irradiated, respectively. The distortion that affects the CH2 signal at 1.53 ppm in Figure 2(b) is

  Figure 4(a) (b) and (c), show data related to the three DFT methods utilized in this study, i.e., B3LYP, B3LYP-D and B97D, respectively. All results related to the three experiments with irradiations on the NH, CH and CH2 are shown. In these histograms, data shown in blue and red refer to the monomeric and pentameric DFT structures, respectively. With the only exception of the NH experiment as interpreted by the B3LYP method (highlighted by a dashed rectangle), all other cases show that smaller mismatches between the computed model structure and the experimental measurements are obtained when the pentameric structure is considered. In other words, our ROE measurements indicate the BTA as assembled in our sample. The data of Figure4shows that, in the case at hand, and when an appropriate model and functional are assumed, our combined NMR-DFT approach produces errors in the measurement of internuclear distances which are generally smaller than 0.4 Å. Considering the good performance and agreement between B3LYP-D and B97D functionals, the discrepancy obtained for the NH case when the B3LYP method is employed can be rationalized considering an overall overestimation of interplane distances due to lack of description for dispersion forces. With this respect, it is worth bearing in mind that the larger H-C-C-C=O dihedral angles (Table2) produced by B3LYPrelated to a structural 'expansion' of the column -result in the increase in the intramolecular internuclear distance between the NH and the CH proton sites.[62]

  [19] Our choice for acetone, a H-bond acceptor solvent more polar than CD2Cl2, has the purpose of further destabilizing any assembly in favor of the monomer by disrupting the interplane Hbonding network. The corresponding ROE spectra are shown in Figure ESI2. The diluted sample results in much larger error bars in the ROE DFT r r r    profile that prevent the identification of a preferred structure (Figure ESI3). It is worth noting that the order of CH and NH resonances at

Figure 5 (

 5 Figure5(d-f)) correlate very well with those calculated for the monomeric structure, with R 2 = 0.987, 0.987 and 0.994, for B3LYP, B3LYP-D and B97D, respectively. If the same experimental shifts are correlated with those calculated for the pentameric assembly, the correlation factors become R 2 = 0.820, 0.816 and 0.824. In this latter case, the data points related to the CH and NH resonances are significantly brought away from the ideal correlation. These displacements of

Figure 6 (

 6 Figure 6(c) shows a fit of the spectrum in (b) assuming one single site for the C=O resonance,

Figure 6 .

 6 Whenconsidering instead the distribution of chemical shifts observed in (c), resulting from the non-assembled structure of Fig.ESI3, we find a substantially larger differentiation in terms of chemical environments, with spreads of chemical shift of ca. 280 Hz, i.e., about three times larger than the inhomogeneous linewidth experimentally observed in the experimental spectra of Fig.6. Therefore, and in agreement with the analysis performed on the CSA parameter, by considering the spread of the isotropic part of the shift interaction as calculated for the different structural hypotheses considered in this study, a clear evidence of consistency of the experimental solid-state results with the hypothesis of a columnar self-assembled structure can be found.A further series of calculations was performed in order to highlight the effects on the NMR parameters of i) different truncations of the aliphatic chains, ii) the orientation of the amide functions with respect to the aromatic core and iii) non-covalent and dispersive contributions. The corresponding results are shown in FigureESI7 and 8. A brief discussion on this data set is also given in the ESI. The results indicate that the carbon site most affected by variations of the HCCCO dihedral angle and by the 'loosing' of π-π stacking effects is the CH of the aromatic core, with the other carbon sites C and CO experiencing smaller effects. In contrast, very little effect on all considered carbons is produced by variations of the length of the aliphatic chain.

  Chemistry (NPAC), on a Bruker Avance-II spectrometer equipped with a 5 mm narrow-bore magnet operating at 9.4 T (400 MHz for 1 H). All conventional NMR experiments were performed at room temperature, with typical recovery delays of d1 = 2 s and 90° pulse lengths of τp = 9.6 s corresponding to an rf field strength of ν1 = 26 kHz. The 1D ROESY spectra of Figures2 and 3employed spin locking with an rf-filed strength ν1 = 3 kHz and length τSL = 5 and 50 ms, respectively, and sinc-shaped selective π pulses of length τp = 10 ms.All13 C CP MAS spectra of Figure7were recorded in Sion, at the Ecole Polytechnique Fédérale de Lausanne (EPFL), on a wide-bore Bruker 400 spectrometer (9.4 T) with an Avance-III console, using 2.5 mm ZrO2 rotors and a triple-resonance MAS probe. The rf-field amplitude for proton and carbon hard pulses was ν1 = 132 and 109 kHz, respectively, corresponding to 90° flipangle pulse lengths of τp = 1.9 and 2.3 s, respectively. A recycling delay of d1 = 2 s was used in all cases, with a spinning rates of νrot = 20 and 5 kHz. The rf-field strength for the crosspolarization step was ν1 = 70 and 50 kHz for 1 H and13 C, respectively with a contact time τCT = 3.5 ms. The amplitude of the 1 H pulse was ramped from 90 to 100% during CP. The spectra were referenced to TMS using adamantane as external reference. The amplification factor of the CSA Columnar self-assembly of N,N',N''-trihexylbenzene-1,3,5-tricarboxamides investigated by means of NMR spectroscopy and computational methods in solution and solid state 22 Amplification experiment of Figure7was 8, corresponding to a spinning sideband pattern of a fictitious MAS rate of 20/8 = 2.5 kHz mimicked in the indirect dimension, and 576 transients were averaged for each of 32 t1 increments.The structures of BTA in its monomeric and n-meric (n = 2, 3, 4 and 5) forms were preliminarily optimized with the PM7 semiempirical method as implemented in MOPAC2016.[START_REF] Mopac ; James | Stewart Computational Chemistry[END_REF] A further optimization was performed with DFT methods utilizing B3LYP,[START_REF] Becke | [END_REF]59] B3LYP-D and B97D functionals,[60,71] with the 6-31G(d,p) basis set, as implemented in Gaussian09 Revision A.01.[START_REF] Frisch | Gaussian 09, Revision A.01[END_REF] The keyword IOp(3/124=3) was used to include empirical dispersion correction for the B3LYP functional. The influence of solvent effects was investigated

Figure 1 ( 30 Figure 2 (

 1302 Figure 1

Figure 3 (

 3 Figure 3 (a) Pentameric columnar structures of BTA as obtained with DFT methods at the B3LYP/6-31G(d,p) level of theory. (b,c) Structure analogous to that of (a) produced by the functionals B3LYP-D and B97D, respectively. (d-f) View down the helix column axis of the structure in (a-c). Hexyl aliphatic chains have been truncated to propyls to reduce computational times. Hydrogen, carbon, oxygen and nitrogen atoms are shown in white, grey, red and blue, respectively. Cartesian coordinates are given in the ESI.

Figure 4 (

 4 Figure 4 (a) Histogram of differences r = rROE -rDFT between interatomic distances as calculated with B3LYP and those measured from the experiments of Figure 1. (b,c) Histograms analogous to that of (a) for distances taken from structures optimized with the B3LYP-D and B97D functionals. The error bars refer to the experimental data and were calculated with Equation (1) and (2) where the intensity of enhancement of the i-th spin was i

Figure 5 (

 5 Figure 5 (a-c) Calculated variations of proton chemical shifts the CH, NH and CH2 sites as the columnar assembly is progressively increased from one to five units of BTA for B3LYP, B3LYP-D and B97D methods, respectively. (d-f) Correlation between calculated and experimental chemical shifts for B3LYP, B3LYP-D and B97D methods, respectively. Numerical values are given in Tables ESI4-7.

Figure 6 (

 6 Figure 6

Figure 7 (

 7 Figure 7 (a) 13 C CP CSA Amplification MAS NMR spectrum of a BTA powder recorded in a 9.4 T magnet at a spinning rate νrot = 20 kHz. (b-d) Projections of the CO, C and CH sites, respectively, as extracted from the 2D spectrum of (a). Fits are shown in red, overlaid with the experimental data points, corresponding to spinning sideband envelopes as recorded at a fictitious spinning rate of 20/8 = 2.5 kHz. (e-f) 2D plots of the rms deviations of the fits of (b-d), respectively. Resolutions in the anisotropy ΔCS and asymmetry ηCS dimensions were 100 Hz (1 ppm at 9.4 T) and 0.025, respectively, with intensity scale arbitrarily limited to 50.

Figure 8 (

 8 Figure 8 (a) Correlation between chemical shift anisotropies as calculated with the B3LYP functional for the considered structures and those as measured with the experiment of Fig. 7. (b,c) Correlations analogous to those of (a) for the computational data set as produced by the B3LYP-D and B97D functionals. (d) Histogram showing the correlation coefficients R 2 for anisotropy and isotropic shift of (a-c). Values are given in Tables ESI8-11.

Figure 9 (

 9 Figure 9 (a) Simulated 13 C NMR spectrum using the isotropic chemical shifts of all considered carbons in monomeric BTA. (b) Spectrum analogous to that of (a) using carbon chemical shift as calculated for a pentameric columnar assembly. (c) Carbon spectrum resulting from carbon shifts as calculated for the disordered arrangement of BTA unit of Fig. 9(c).The arrows highlight a FWHM of 380 Hz, given by the calculated spread of shifts of 280 Hz and a line broadening of 100 Hz, as used for all Lorentzian lineshapes.

Figure 10

 10 Figure 10Scheme depicting the computational approach adopted in this study. All DFT steps have been also performed with the B3LYP-D and B97D functionals. Optimized structures were used to measure interatomic distances to be compared with those measured by ROE experiments and for the calculations of the full shielding tensors so as to compare isotropic and anisotropic parts of the interactions with those measured experimentally in both solution and solid state.

Acknowledgements

This work was supported by the Swiss National Science Foundation (SNSF), the University of Neuchâtel, the Neuchâtel Platform of Analytical Chemistry (NPAC), the Ecole Polytechnique Fédérale de Lausanne (EPFL), the Swiss Commission for Technology and Innovation (CTI), and

the European Research Council (ERC, contract 'para-water'). DC would like to thank Dr Pascal Miéville for assistance with the solid-state NMR experiments and Dr Daniel Jana for advices in the computational study.

Tables


Magalie Ochs
email: magalie.ochs@lsis.org

Brice Donval
email: brice.donval@telecom-paristech.fr

Philippe Blache
email: philippe.blache@lpl-aix.fr

Virtual Patient for Training Doctors to Break Bad News

Keywords: Embodied conversational agent, virtual reality, virtual patient, training platform

. However, both experienced clinicians and medical trainees consider this task as difficult, daunting, and stressful. Nowadays, training health care professional to break bad news, recommended by the French Haute Autorité de la Santé (HAS), is organized as workshops during which doctors disclose bad news to actors playing the role of patient. In our project, we are developing an embodied conversational agent simulating a patient to train doctors to break bad news. The embodied conversational agent is incorporated in an immersive virtual reality environment (a CAVE) integrating several sensors to detect and recognize in real time the verbal and non-verbal behavior of the doctors interacting with the virtual patient. The virtual patient will adapt its behavior depending on the doctor's verbal and non-verbal behavior. The methodology used to construct the virtual patient behavior model is based on a quantitative and qualitative analysis of corpus of doctors training sessions.

INTRODUCTION

The way doctors deliver bad news has a significant impact on the therapeutic process: disease evolution, adherence with treatment recommendations, litigation possibilities [START_REF] Andrade | Avatarmediated training in the delivery of bad news in a virtual world[END_REF]. However, both experienced clinicians and medical trainees consider this task as difficult, daunting, and stressful. Nowadays, training health care professional to break bad news, recommended by the French Haute Autorité de la Santé (HAS), is organized as workshops during which doctors disclose bad news to actors playing the role of patient. This training solution requires a huge amount of human resources as well as high level of preparation (each 30 mn session requires an hour of preparation), not to speak about funding. It is then difficult to propose the student a large variety of situation or patient profiles. It is then necessary to make use of information technologies applied to learning.

In this project, we aim at developing an embodied conversational agent (ECA) simulating a patient. Such a platform would play a decisive role for institutions involved in training (hospitals, universities): the needs concern potentially thousands of doctors/students. Organizing such training at this scale is not realistic with human actors. A virtual solution would be then an adequate answer.

Our objective is to develop an immersive platform that enables doctors to train to break bad news with a virtual patient. For this purpose, we adopt a multidisciplinary approach in the project gathering computer scientists, linguists, psychologists and doctors. The objective of the paper is to present the global project (motivation, scientific challenges, and methodology). In the following, after a presentation of a state of art in this domain (Section 2), we present the corpus-based approach used to model the virtual patient (Section 3) and we introduce the training platform and its different components (Section 4).

STATE OR ART

For several years, there has been a growing interest in Embodied Conversational Agents (ECAs) to be used as a new type of human-machine interface. ECAs are autonomous entities, able to communicate verbally and nonverbally. They can attract and maintain the attention of users in an interaction, to make the interaction more expressive and more socially adapted. Indeed, several researches have shown that embodied conversational agents are perceived as social entities leading users to show behaviors that would be expected in human-human interactions [START_REF] Krämer | Emotion and motivation in embodied conversational agents[END_REF].

Moreover, recent research showed that virtual agents could help human beings improve their social skills. For instance, in [START_REF] Finkelstein | The effects of culturally-congruent educational technologies on student achievement[END_REF], a virtual agent is used to train kids to adapt their language register to the situation. In the European project TARDIS (Anderson et al., 2013), an ECA endowed the role of a virtual recruiter is used to train young adults to job interview. The E-circus project offers children a role playing environment with expressive virtual agents to learn how to deal with bullying situations at school [START_REF] Weiss | ECIRCUS: Building Voices for Autonomous Speaking Agents[END_REF]. This research shows that embodied conversational agent can be used for social training since users will react to the ECA in a similar way that to another person and the socio-emotional responses of the agents will help them practice and improve their social skills.

Several ECAs embodied the role of virtual patients have already been proposed for use in clinical assessments, interviewing and diagnosis training [START_REF] Cavazza | A virtual patient based on qualitative simulation[END_REF]Andrade et al., 2011;[START_REF] Johnsen | Experiences in Using Immersive Virtual Characters to Educate Medical Communication Skills[END_REF][START_REF] Kenny | Evaluation of Justina: A Virtual Patient with PTSD[END_REF]Lock et al., 2006). Indeed, previous research has shown that doctors demonstrate non-verbal behaviors and respond empathetically to a virtual patient [START_REF] Deladisma | Do medical students respond empathetically to a virtual patient?[END_REF]. In this domain, the research has mainly focused on the anatomical and physiological models of the virtual patient to simulate the effects of medical interventions or on models to simulate particular disorder. For instance, Justina is a virtual patient simulating Post Traumatic Stress Disorder (PTSD) to train medical students' interview skills and diagnostic acumen for patient with such disorder [START_REF] Kenny | Evaluation of Justina: A Virtual Patient with PTSD[END_REF]. DIANA (DIgital ANimated Avatar) is a female virtual character playing the role of a patient with appendicitis (Lock et al., 2006). In the eViP European project (http://www.virtualpatients.eu), the objective is specifically to develop a large amount of virtual patients simulating different pathologies. In our project, we focus on a virtual patient to train doctors to deliver bad news.

A first study [START_REF] Andrade | Avatarmediated training in the delivery of bad news in a virtual world[END_REF] has analyzed the benefits of using a virtual patient to train doctors to deliver bad news. The results show significant improvements of the self-efficacy of the medical trainees. The participants consider the virtual patient as "excellent instructional method for learning how to deliver bad news". The major limit of the proposed system, highlighted by the participants, is the lack of non-verbal behaviors of the patients simulated in the limited environment Second Life (Linden Labs, San Francisco, CA). Our objective in this project is to simulate the non-verbal emotional expression of the virtual patient to improve the believability of the virtual character and the immersive experience of the Most of the embodied conversational agents used for health applications have been integrated in 3D virtual environment on PC. Virtual reality in health domain is particularly used for virtual reality exposure therapy (VRET) for the treatment for anxiety and specific phobias. For instance, people with a fear of public speaking may speak to an audience of virtual characters in virtual reality environment to reduce their anxiety in reality [START_REF] Parsons | Affective outcomes of virtual reality exposure therapy for anxiety and specific phobias: A meta-analysis[END_REF]. In our project, in order to offer an immersive experience to the doctor, we have integrated the virtual patient in a virtual reality environment.

Several researches have been conducted to simulate emotional state through the non-verbal behavior of embodied conversational agent. However, expression of emotions in virtual agents (e.g. Pandzic & Forcheimer 2002) often relies on the six prototypical expressions of emotions (joy, fear, anger, disgust, sadness, and surprise). Moreover, in the context of breaking bad news, the virtual agent has to be able to display specific panel of strong emotions and feelings (such as tears) through particular speech, postures, gestures and facial expressions. To our knowledge, no existing computational model has been developed to simulate the effect of strong negative emotions on the behavior (verbal and non-verbal) of a virtual character.

Moreover, a virtual patient should be able to display verbal and non-verbal reactions appropriately according to the doctor's behavior. During an interaction, the interlocutors in fact coordinate or align their verbal and non-verbal behavior (e.g. feedback, mimicry). According to the Communication Accommodation theory (CAT), the interlocutors adapt the coordination of their behavior to express different social attitudes (Gilis et al., 1991). Recent research in human-machine interaction confirms this hypothesis: the coordination of the virtual agent's behavior on the one of the user or the divergence of its behavior reflects the agent's social attitude (e.g. appreciation, cold, mutual understanding, etc.) (e.g. [START_REF] Bailenson | The independent and interactive effects of embodied agent appearance and behavior on self-report, cognitive, and behavioral markers of copresence in immersive virtual environments[END_REF]). In the project described in this paper, sensors will be used to automatically detect in real-time the verbal and non-verbal behavior of the doctors during their interaction with the virtual patient. These inputs will then be used by the virtual patient to coordinate its behavior on the doctor's one depending on the social attitude to express. The methodology used to define the virtual agent's behavior is based on the analysis of a corpus of doctor-patient interaction to extract rules on its verbal and non-verbal reactions

A CORPUS-BASED APPROACH TO SIMULATE A VIRTUAL PATIENT

For ethical reasons, it is not possible to videotape real breaking bad news situations. Instead, simulations are organized with actors playing the role of the patient. A corpus of such interactions has been collected in different medical institutions (the Institut Paoli Calmette and the hospital of Angers). Simulated patients are actors trained to play the most frequently observed patients reactions (denial, shock…). The actor follows a pre-determined scenario. The doctor (i.e. the trainee) receives details of a medical case before the simulated interaction starts (patient medical history, family background, surgery, diagnosis, etc.). On average, a simulated consultation lasts 30 minutes. The collected corpus is currently composed of 23 videos of patient-doctor interaction with different scenario (e.g. patient aggressive or accommodating). These simulated interactions of the collected corpus are transcribed and annotated (with Elan software -Figure 1) on several levels: at the discourse level (e.g. dialog phases, turn- The coding scheme has been defined by linguists in collaboration with doctors, based on a preliminary analysis of the corpus [START_REF] Saubesty | Multimodal analysis of hand gesture back-channel feedback[END_REF]. The objective of the annotation is to highlight the necessary information to model and implement a virtual patient and reproduce this kind of interaction.

The method used to extract information from the annotated corpus is based on a multidisciplinary approach combining (1) a manually analysis of the data by linguists for a qualitative analysis and (2) automatic requests (using the SPPAS software [START_REF] Bigi | SPPAS -Multi-lingual Approaches to the Automatic Annotation of Speech the Phonetician -International Society of Phonetic Sciences[END_REF]) and datamining algorithms (e.g. [START_REF] Rabatel | Contextual Sequential Pattern Mining[END_REF] on the data for a more quantitative study performed by computer scientists.

VIRTUAL RALITY PLATFORM OF TRAINING

The tool we use to animate the virtual patient is the Greta System [START_REF] Pelachaud | Studies on Gesture Expressivity for a Virtual Agent, Speech Communication, special issue in honor of Björn Granstrom and Rolf Carlson[END_REF]. Greta offers several modules, each dedicated to particular functionality to both design new facial expressions and gestures and to animate in real-time 3D virtual agents in virtual environments. The lexicon of gestures of Greta is enriched by specific gestures and facial expressions of patients identified in the corpus.

The virtual patient has been integrated in the CRVM (Centre de Réalité Virtuelle de Marseille, platform of the ISM partner). The visualization system consists of a high-end platform called "CAVE ™". CAVE ™ is composed of four projection screens: frontal, ground and lateral projections. Each frontal and lateral screen has a projection surface of 3 meters wide by 4 meters high (Figure 2).

The speech of the doctor is recognized by an automatic speech recognition system [START_REF] Nocera | Phoneme lattice based a* search algorithm for speech recognition[END_REF]. We are improving the system by learning the lexicon and language model of the speech recognition system on the transcript corpus of patient-doctor interactions. For this project, we have defined with doctors specialized in this kind of training, a specific use case that reflects a real situation in which doctors may train their social competences in delivering bad news. It is based on real scenario of training (a case of intestinal perforation) including some possible variations (e.g. patient aggressive or accommodating).

The use case has been defined based on a common scenario written by experts at a medical simulation center for the formation of doctors with actors simulating patients. The dialog model of the patient is based on this scenario. We are defining the dialog rules that depend on the profile of the patient (aggressive or accommodating) and the discourse of the trainee (e.g. level of details, use of medical terms, etc.). The dialog rules and more generally the behavior of the virtual patient (e.g. reaction to the doctor's recognized speech, backchannels) is ongoing work that is based on the corpus analysis (as described in the previous section).

To consider the non-verbal behavior of the trainee, on the table in the CAVE, we will install a Kinect to automatically detect some non-verbal signals of the doctors (e.g. the gaze direction, head movements, posture, and facial expressions). The objective is to coordinate or in contrary discoordinate the non-verbal behavior of the virtual patient on those of the trainee depending on the attitude that the virtual patient should convey (e.g. aggressive versus accommodating).

The general architecture of the system is illustrated Figure 3. In this architecture, the speech recognition system [START_REF] Nocera | Phoneme lattice based a* search algorithm for speech recognition[END_REF] receives as input the doctor's speech captured from the microphone and transmits to the scenario controller the ) and non-verbal (socio-emotional reactions) behavior that the virtual patient should adopt to react to the doctor's behavior. This information is defined based on a predefined scenario. The Scenario controller receives as input the verbal response of the virtual patient through the dialog system. This information (virtual patient verbal reaction and socio-emotional state) is transferred to the Non-verbal behavior animation system Greta in the XML standard language FML (Functional Markup Language). The Nonverbal behavior animation system Greta (recently called VIB) computes the animation parameters (Facial Animation Parameters -FAPand Behavioral Animation Parameters -BAP) to animate the face and body of the virtual patient. The scenario controller also informs the Non-verbal behavior animation system Greta when the doctor is speaking in order to generate listener's nonverbal behavior (e.g. head node). The Non-verbal behavior system Greta receives as inputs the detected non-verbal behavior of the doctor (gaze, head movements and facial expressions) to be able to coordinate its non-verbal behavior on the one of the doctor. Moreover, the Non-verbal behavior system Greta contains a textto-speech system (Aylett and Pidcock, 2007) to generate the speech synchronizes with the non-verbal behavior (including the lips animation). The virtual patient is finally animated through different players on different platforms: PC, Virtual reality environment and 3D glasses. Several modules of the architecture are currently under development.

To evaluate the platform, two types of evaluation are considered: the one, intrinsic to the system, aiming at estimating the quality of the conversational virtual patient, in terms of communication skills, the other proposing a measurement of the trainee performance. As for the first evaluation, different measures will be done, according to the component to be evaluated. Speech recognition will be measure with classical techniques (recal/precision), against the general corpus. The virtual patient will be evaluated thanks to user perceptive tests concerning the verbal and non-verbal behavior of the agent: the relevance of its behaviors, the coherence of the attitude, the adaptation to the input information, etc. The trainee evaluation is an entire research subject. An evaluation grid will be defined, starting from the existing one currently used in the hospitals: the « Affective Competency Score » [START_REF] Quest | The validity and reliability of the affective competency score to evaluate death disclosure using standardized patients[END_REF]). ). The ACS will be scored by the trainees to measure their self-efficacy before and after a session with the virtual patient. Professional observers from the CHU will also rate the ACS to evaluate the trainees' performances. Moreover, a presence test questionnaire (such as the Witmer questionnaire [START_REF] Witmer | Measuring presence in virtual environments: A presence questionnaire[END_REF]) will be used to assess the interactive experience of the trainees.

ACKNOWLEDGMENTS

This research is supported by the ANR (Agence Nationale de la Recherche).

Figure 1 :

 1 Figure 1: Corpus annotation with Elan

Figure 2 :

 2 Figure 2: Virtual reality environment for training doctors to break bad news

Figure 3 :

 3 Figure 3: Global architecture of the system