François Fillaux
email: francois.fillaux@upmc.fr

Alain Cousson
email: alain-f.cousson@cea.fr

A neutron diffraction study of the crystal of benzoic acid from 6 to 293 K and a macroscopic-scale quantum theory of the lattice of hydrogen-bonded dimers

Keywords: Benzoic acid, Neutron diffraction, Hydrogen bonding, Tautomerism, tunneling, Quantum mechanics

Measurements via different techniques of the crystal of benzoic acid have led to conflicting conceptions of tautomerism: statistical disorder for diffraction; semiclassical jumps for relaxometry; quantum states for vibrational spectroscopy.

We argue that these conflicts follow from the prejudice that nuclear positions and eigenstates are pre-existing to measurements, what is at variance with the principle of complementarity. We propose a self-contained quantum theory.

First of all, new single-crystal neutron-diffraction data accord with long-range correlation for proton-site occupancies. Then we introduce a macroscopic-scale quantum-state emerging from phonon condensation, for which nuclear positions and eigenstates are indefinite. As to quantum-measurements, an incoming wave (neutron or photon) entangled with the condensate realizes a transitory state, either in the space of static nuclear-coordinates (diffraction), or in that of the symmetry coordinates (spectroscopy and relaxometry). We derive temperaturelaws for proton-site occupancies and for the relaxation rate, which compare favorably with measurements.

Introduction

The crystal of benzoic acid (C 6 H 5 COOH) is a showcase for proton transfer across hydrogen bonds and tautomerism, which are both of importance in many fields across physics, chemistry and biology . Diffraction reveals bistable centrosymmetric cyclic-dimers (C 6 H 5 COOH) 2 linked through hydrogen bonds.

The lattice is comprised of tautomers, say L 1 L 2 (preferred) and R 1 R 2 (less favoured), exclusive of the symmetry-breaking configurations L 1 R 2 and R 1 L 2 (Fig. 1). The ratio R 1 R 2 : L 1 L 2 increases smoothly and reversibly with temperature. Experimental and theoretical investigations enlighten the behavior of hydrogen bonds in a complex environment, with particular emphasis on proton tunneling that is a beacon for the emergence of classical from quantum. At this point in time, there is no widespread agreement on the representation of proton transfer and tautomerism. (i) The coexistence of tautomers evidenced by diffraction is thought of as a random distribution of uncorrelated dimers L 1 L 2 and R 1 R 2 [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF]. (ii) The proton relaxation-rate measured through solid-state-NMR or quasi-elastic neutron scattering (QENS) is represented by semiclassical stochastic jumps of rigid proton-pairs [START_REF] Meier | Structure and dynamics of intramolecular hydrogen bonds in carboxylic acid dimers: A solid state NMR study[END_REF][START_REF] Meyer | Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics[END_REF][START_REF] Skinner | Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF][START_REF] Stöckli | Hydrogen bond dynamics in isotopically substituted benzoic acid dimers[END_REF][START_REF] Meyer | Transitions induced in a double minimum system by interaction with a quantum mechanical heat bath[END_REF][START_REF] Heuer | The dynamics of hydrogens in double well potentials: The transition of the jump rate from the low temperature quantummechanical to the high temperature activated regime[END_REF][START_REF] Sakun | Proton tunneling assisted by the intermolecular vibration excitation. Temperature dependence of the proton spin-lattice relaxation time in benzoic acid powder[END_REF][START_REF] Brougham | Proton transfer dynamics in the hydrogen bond: a direct measurement of the incoherent tunneling rate by NMR and the quantum-to-classical transition[END_REF][START_REF] Neumann | Proton tunneling in benzoic acid crystals at intermediate temperatures: Nuclear magnetic resonance and neutron scattering studies[END_REF][START_REF] Latanowicz | Quantum mechanical and classical dynamics of double proton transfer in the hydrogen bond[END_REF][START_REF] Takeda | An indication of slowing down of hydrogen atom transfer in isotopically mixed hydrogen bonds of benzoic acid crystals[END_REF][START_REF] Latanowicz | Complex dynamics of proton and deuteron transfer in double hydrogen bond of benzoic acid isotopes[END_REF][START_REF] Xue | Isotope effects associated with tunneling and double proton transfezr in the hydrogen bonds of benzoic acid[END_REF][START_REF] Wu | The correspondence between quantum and classical mechanics: an experimental demonstration of the smooth transition between the two regimes[END_REF]. (iii) Vibrational spectra reveal the quantum nature of the crystal, as well as the nonlocal character of proton-states [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF]. (iv) Computational models for the crystal of benzoic acid [START_REF] Plazanet | The vibrational spectrum of crystalline benzoic acid: Inelastic neutron scattering and density functional theory calculations[END_REF][START_REF] Johnson | Dispersion of vibrational modes in benzoic acid crystals[END_REF][START_REF] Takahashi | Interpretation of temperature-dependent low frequency vibrational spectrum of solid-state benzoic acid dimer[END_REF][START_REF] Li | A study into the effect of subtle structural details and disorder on the terahertz spectrum of crystalline benzoic acid[END_REF], or for isolated dimers of formic acid [START_REF] Shida | A reaction surface hamiltonian treatment of the double proton transfer of formic acid dimer[END_REF][START_REF] Kim | Direct dynamics calculation for the double proton transfer in formic acid dimer[END_REF][START_REF] Miura | An ab initio path integral molecular dynamics study of double proton transfer in the formic acid dimer[END_REF][START_REF] Ushiyama | Successive mechanism of double-proton transfer in formic acid dimers: a classical study[END_REF][START_REF] Smedarchina | Calculation of the tunneling splitting in the zero-point level and CO-stretch fundamental of the formic acid dimer[END_REF][START_REF] Smedarchina | Tunneling dynamics of double proton transfer in formic acid and benzoic acid dimers[END_REF][START_REF] Mil´nikov | Ground-state and vibrationally assisted tunneling in the formic acid dimer[END_REF][START_REF] Markwick | Targeted Car-Parinello molecular dynamics: Elucidating double proton transfer in formic acid dimer[END_REF][START_REF] Luckhaus | Concerted hydrogen exchange tunneling in formic acid dimer[END_REF][START_REF] Matanović | Ground and asymmetric CO-stretch excited state tunneling splittings in the formic acid dimer[END_REF][START_REF] Smedarchina | Correlated doubleproton transfer. I. Theory[END_REF][START_REF] Barnes | The effects of asymmetric motions on the tunneling splittings in formic acid dimer[END_REF][START_REF] Barnes | Symmetric double proton tunneling in formic acid dimer: A diabatic basis approach[END_REF][START_REF] Matanović | Generalized approximation to the reaction path: The formic acid dimer case[END_REF][START_REF] Siebrand | Tunneling splitting and level ordering in a CO-stretch fundamental of the formic acid dimer[END_REF][START_REF] Luckhaus | Hydrogen exchange in formic acid dimer: tunneling above the barrier[END_REF][START_REF] Saritha | An electronic structure perspective of the promoter modes in proton transfer reactions[END_REF][START_REF] Jain | Tunneling splittings in formic acid dimer: An adiabatic approximation to the Herring formula[END_REF][START_REF] Pitsevich | Anharmonic analysis of C-H and O-H stretching vibrations of the formic acid dimer[END_REF][START_REF] Ivanov | Quantum free energy landscapes from ab initio path integral metadynamics: Double proton transfer in the formic acid dimer is concerted but not correlated[END_REF], are based on Born-Oppenheimer separation of quantum electrons, on the one hand, and classical nuclei at fixed positions, on the other. Dynamical interconversion of tautomers is represented by two mechanisms as concerted proton transfer in quasi 1-D at low temperature or/and stepwise single-proton transfer in quasi 2-D more likely at elevated temperature.

L 1 L 2 L 1 R 2 R 1 R 2 R 1 L 2 O O C
Here, we seek a self-contained theory bound to unquestionable facts. (i)

The observed symmetry-related selection rules (u-infrared versus g-Raman) are those anticipated for centrosymmetric Bloch-states. (ii) There is no visible ug splitting of the proton stretching-states to suggest any significant intradimer coupling term. (iii) For similar centrosymmetric dimers in the crystal of potassium hydrogen carbonate (KHCO 3 ), the effective oscillator-mass of the proton states measured via inelastic neutron-scattering is practically 1 amu [START_REF] Ikeda | Inelastic neutron scattering study of the localized dynamics of protons in KHCO 3 single crystals[END_REF]. (iv) The observed energy-levels are consistent with double-well operators virtually identical for ν g OH and ν u OH (Table 1). The existing models are in conflict with these observations: (i) excludes interconversion via single-proton transfer; (ii) rules out concerted pairwise interconversion; (iii) excludes coupling terms with heavy atoms. The desired theory should be quantum in nature, what excludes definite trajectories for dimensionless nuclei [START_REF] Leggett | Testing the limits of quantum mechanics: motivation, state of play, prospects[END_REF].

A serious hurdle on the way toward a consistent theory is the diffraction work by Wilson et al. [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF] reporting a significant amount of the less favored tautomer at low temperature (≈ 13% at 20 K). This was attributed to a difference in tautomer entropy, that does not make any contact with solid-state NMR [START_REF] Neumann | Proton tunneling in benzoic acid crystals at intermediate temperatures: Nuclear magnetic resonance and neutron scattering studies[END_REF] and Fig. 1. In order to check this point, we have conducted new diffraction measurements and the data analysis reported in Sec. 2 argues against entropy.

As to the theory, conflicts of representation appear when vibrational states and nuclei are thought of as elements of an objective reality attached to the crystal, irrespective of whether they are observed or not. In this present paper, this standpoint is referred to as "microrealism", by analogy with the notion of macrorealism introduced by Leggett and Garg [START_REF] Leggett | Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?[END_REF]. Within the framework of quantum mechanics, conflicts can be ruled out because outcomes of measurements depend on the measuring apparatus. They are context-dependent and, according to the principle of complementarity [START_REF] Plotnitsky | What is complementarity?: Niels Bohr and the architecture of quantum theory[END_REF], there is no deterministic relationship between positions and energies. Outcomes should be interpreted either in the time-dependent wave-like representation, for spectroscopy and relaxometry, or in the static particle-like representation, for diffraction, knowing that these measurements are mutually exclusive and their representations pertain to orthogonal spaces. We elaborate on a crystal-state pre-existing to measurements, that is consistent with both representations. To this end, we introduce in Sec. 3 the notion of "quantum condensation", in line with our previous report on potassium dihydrogen phosphate (KH 2 PO 4 or KDP) [START_REF] Fillaux | Quantum interferences revealed by neutron diffraction accord with a macroscopic-scale quantum-theory of ferroelectrics KH 2(1-ρ) D 2ρ PO 4[END_REF]. • with error bars: experimental points (this work). ×: experimental points after [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF].

Neutron diffraction

Blue dot dashed with bars: eq. [START_REF] Meier | Structure and dynamics of intramolecular hydrogen bonds in carboxylic acid dimers: A solid state NMR study[END_REF]. Solid plus grey zone: eq. ( 8). Red dashed with bars:

eq. ( 6). The grey zone and the bars represent dispersions of the theoretical curves due to uncertainties for the parameters. hν l = (54 ±6) cm -1 and hν 01 = (172 ±4) cm -1 are deduced from vibrational spectra. and ♦: [START_REF] Wilson | Temperature dependence of proton transfer in 4-chlorobenzoic acid[END_REF] and [START_REF] Meyer | Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics[END_REF], respectively, for p l = 0.

Needle-shaped single-crystals (3 × 3 × 10 mm 3 ) were grown by slowly cooling an ethanol-hexane solution. A specimen wrapped in aluminum was loaded in a

Bond length (Å) are consistent with a two-level system with negligible or even zero entropy. The enthalpy ∆H ρ = (86.8 ± 1.7) K compares with previous estimates based on neutron diffraction, (80 ± 7) K [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF], solid-state NMR, (86.5 ± 1.5) K [START_REF] Neumann | Proton tunneling in benzoic acid crystals at intermediate temperatures: Nuclear magnetic resonance and neutron scattering studies[END_REF], or Raman, (54 ± 6) cm -1 = (81 ± 9) K [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF].

Fig. 3 suggests a linear correlation between CO-bond lengths and ρ L ,ρ R :

R C1O1 (T ) = ρ L (T )R C=O + ρ R (T )R C-O ; R C1O2 (T ) = ρ L (T )R C-O + ρ R (T )R C=O ;
(2)

R C=O = 1.233 Å, R C-O = 1.320 Å, ρ L (T ) = (1 + β) -1 , ρ R (T ) = ρ L (T )β,
β = e -∆Hρ/T . This suggests distinct sublattices with fixed bond-lengths, in line with temperature-independent vibrational frequencies of the CO-modes [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF].

The total probability density at proton-sites is a constant: ρ = ρ L (T ) + ρ R (T ) ≡ 1.00 [START_REF] Hayashi | Disappearances of COOH infrared bands of benzoic acid[END_REF] [55]. This should be confronted with the cross-section anticipated for a static random distribution of uncorrelated occupied and empty sites with coherent scattering lengths b H and 0, respectively. According to Nield and Keen [START_REF] Nield | Oxford series on neutron scattering in condensed matter[END_REF], we can split the scattering length at each site into two parts. The first part is the average scattering length b av H = [b H + 0]/2 and the second is the deviation b d H = ±rb H , where 0 ≤ r ≤ 1/2 is the occupancy of the less favored site. Then, the coherent differential cross-section is comprised of two non vanishing terms. Firstly, Bragg scattering proportional to (1r) 2 b 2 H . Secondly, a constant background of elastic diffuse scattering proportional to (b d H ) 2 = r 2 b 2 H , hidden by the huge incoherent signal. Consequently, the total occupancy, that is a constant equal to 1, should be different from the total probability density ρ deduced from Bragg peak intensities effectively measured, that is expected to decrease from 1 at a very low temperature to 1/4 for r = 1/2 at room temperature. Our data suggest that ρ is independent of r, so ρ L = (1r) and ρ R = r.

It transpires that probability densities at proton sites are strongly correlated and strict periodicity throughout the lattice yields Bragg scattering exclusively.

Protons should be represented by extended density-waves, such that one and the same scatterer with coherent scattering length b H is simultaneously at every site with probability ρ L and ρ R , respectively. Then, Fig. 3 and (1) accord with a system of two tautomer-sublattice states at thermal equilibrium. Similar representations have been reported for various hydrogen bonded crystals [START_REF] Fillaux | Quantum interferences revealed by neutron diffraction accord with a macroscopic-scale quantum-theory of ferroelectrics KH 2(1-ρ) D 2ρ PO 4[END_REF][START_REF] Nelmes | On the structural evidence for a direct proton tunnelling effect in the KH 2 PO 4 -type transition[END_REF][START_REF] Fillaux | Proton transfer across hydrogen bonds: From reaction path to Schrödinger's cat[END_REF][START_REF] Fillaux | Nonlocal protons and deuterons opposed to disorder: a single-crystal neutron diffraction study of KH 0.76 D 0.24 CO 3 and a theoretical framework[END_REF][START_REF] Fillaux | Evidence of macroscopically entangled protons in a mixed crystal of KH p D 1-p CO 3[END_REF][START_REF] Fillaux | Neutron scattering studies of K 3 H(SO 4 ) 2 and K 3 D(SO 4 ) 2 : The particle-in-a box model for the quantum phase transition[END_REF].

On the one hand, our ρ R (T )-values are practically identical to those derived from solid-state NMR [START_REF] Neumann | Proton tunneling in benzoic acid crystals at intermediate temperatures: Nuclear magnetic resonance and neutron scattering studies[END_REF]. On the other hand, they are significantly different from those reported by Wilson et al. below 100 K [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF] (X in Fig. 2), although their R-factors suggest data of reasonably good quality. Since hysteresis was excluded [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF], temperatures claimed to be 50 K and 20 K by these authors were likely not the actual temperatures of the crystal.

Wilson et al. compared two structural models. On the one hand, they refined the mean position of protons (Fig. 2 in [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF]). As the temperature increases, positions shift toward the bond center and thermal functions elongate dramatically along O• • •O. This model is incompatible with the lack of frequency-shift for νOH modes. On the other hand, Wilson et al. considered protons split over two sites, as presented above. They set the total "occupancy", in fact the total probability density, to be a constant equal to 1 and they constrained the isotropic temperature factors to be equal (Fig. 3 in [START_REF] Wilson | Direct determination of the temperature dependence of proton transfer in the benzoic acid dimer by single crystal neutron diffraction[END_REF]). These two constrains amount to suppose strong correlations between proton-site occupancies and to exclude any random distribution. This model was found to be satisfactory, in accordance with our own data analysis. However, this work was not self-consistent, inasmuch as constrained probability densities and temperature factors were in conflict with the misleading entropy value of (0.27 ± 0.08) cal.mol. -1 K -1 implying existence of more than two distinguishable tautomers.

By contrast, our data show that ρ ≡ 1 and ∆S ρ ≈ 0 are model-free.

Quantum theory

A crystal is commonly represented by an ensemble of nuclear oscillators for which quantum correlations are destroyed by mechanisms extrinsic to the quantum theory [START_REF] Leggett | Testing the limits of quantum mechanics: motivation, state of play, prospects[END_REF], such as environment-induced decoherence [START_REF] Joos | The emergence of classical properties through interaction with the environment[END_REF][START_REF] Zurek | Decoherence and the transition from quantum to classical[END_REF][START_REF] Zurek | Decoherence, einselection and the quantum origin of the classical[END_REF][START_REF] Zurek | Quantum darwinism, classical reality, and the randomness of quantum jumps[END_REF]. In the harmonic approximation the molar Helmholtz energy for N degrees of freedom per unit cell is

A i = K + V -T S, = N n=1 N 0 hν n 2 + RT ln 1 -exp - hν n kT ,        (3)
where K, V , S, N 0 are the kinetic energy, the potential energy, the entropy and Avogadro's constant, respectively.

The decoherence scheme does not account for correlated occupancies at proton sites revealed by neutron diffraction (Sec. 2). In fact, for each degree of freedom of an isolated crystal, a superposition of phonons, say ∝ exp i(k.uω(k)t + φ), with continuous distribution of arbitrary phases φ yields a pure state, say |Φ , that complies with the space-group symmetry operators of the crystal. The complexity of the statistical representation is hidden and new properties emerge. The wavefunction Φ ≡ 0 has continuous space-time-translation symmetry [START_REF] Fillaux | Quantum interferences revealed by neutron diffraction accord with a macroscopic-scale quantum-theory of ferroelectrics KH 2(1-ρ) D 2ρ PO 4[END_REF][START_REF] Coleman | Time crystals[END_REF]. Wavevectors, nuclear positions and time are indefinite. The potential energy and the entropy vanish. The eigenenergy for N N 0 free degreesof-freedom at thermal equilibrium is N RT /2. It is single valued. Such a condensate is stable, as compared with the incoherent regime, if N RT /2 < A i . Since this condition is satisfied as T S -→ 0 and not satisfied as T S -→ ∞, there is a dividing temperature for each mode. We suppose equipartition excludes coexistence in the same crystal of a coherent condensate for some degrees of freedom and incoherent phonons for other degrees of freedom, so the transition temperature of the crystal, say T ci , should depend on the mean oscillator-energy hν.

The isolated condensate has no definite property at the microscopic level.

Our knowledge follows from statistics of contextual outcomes of complementary measurements, which are mutually exclusive. An incoming plane-wave (photon or neutron) entangled with the condensate breaks the continuous translationsymmetry and realizes a transitory Bloch-state at On the molar-scale, the condensate as a whole is virtually unaffected by a single realization induced by a measuring instrument, or by the environment, and spontaneous decay of the induced state (Sect. 3.3) destroys quantum correlations with the outgoing wavefunction. The crystal is immune to decoherence.

In addition, the probability for a wave to entangle a state previously realized is insignificant, so the outcome is independent of previous events.

Equation (3) yields T ci > 300 K if hν > 350 cm -1
, what is likely for benzoic acid. We infer that a monodomain crystal at thermal equilibrium can be tentatively represented by a condensate that determines which microstate can be effectively realized and what outcome can be anticipated in the specific context of a measuring apparatus. Diffraction data (Sec. 2) show that Bragg diffraction realizes a superposition of static sublattice-states and the outcome is a linear combination of nuclear-densities for disorder-free sublattices

L 1 L 2 and R 1 R 2 .
There is no evidence of coherent diffuse scattering due to a non-periodic distribution of static positions. Alternatively, vibrational spectroscopy reveals a two-level system, say l = 0 or 1, with energy gap hν l , and eigenstates amenable to double-well operators [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF], which are rationalized in Sec. 3.1. Finally, relaxometry reveals a superposition state realized through off-resonance energy transfer (Sec. 3.3).

Vibrational spectroscopy

It is commonly accepted that vibrational states can be deduced from a scalar potential function of the nuclear coordinates in direct space. However, this deterministic representation is in contradiction with two principles of quantum mechanics. Firstly, Heisenberg's uncertainty principle applied to nuclei at fixed positions means that kinetic momenta and the energy of the system are undetermined. Secondly, Bohr's complementarity principle [START_REF] Plotnitsky | What is complementarity?: Niels Bohr and the architecture of quantum theory[END_REF]: because of the randomness of quantum events, there is no deterministic relationship between positions and energies, which are realized at random via mutually exclusive and unpredictable elastic or inelastic events, respectively. It is, therefore, necessary to consider distinct representations for nuclear positions and energy levels.

An introduction to our line of argument is the textbook case of two harmonic oscillators, say particles 1 and 2, represented by positional coordinates

X 1 , X 2 , coupled via 2K(X 1 -X 2 ). In classical mechanics, the normal coordi- nates X u , X g , diagonalizing the Hamiltonian H(X 1 , X 2 ) = H u (X u ) + H g (X g ),
are linear combinations of X 1 and X 2 . The particles oscillate in the plane {Ξ u ; Ξ g } for eigenfunctions on the one hand, {X 1 ; X 2 } for static positions on the other. There is neither energy-scale nor time-scale for {X 1 ; X 2 }, whereas positions are indefinite for {Ξ u ; Ξ g }. There is no correspondence between the potential operator V u (Ξ u ) + V g (Ξ g ) associated with the vibrational state vectors and the scalar potential V (X 1 , X 2 ) for classical particles. The operator is, therefore, outside the bounds of the Born-Oppenheimer separation. It can be deduced from outcomes of measurements exclusively.

{X 1 ; X 2 } ≡ {X u ; X g }
For the crystal of benzoic acid, we introduce nonlocal coordinates for particles or waves, beyond the bounds of the harmonic approximation. The static positional-parameters along O• • •O bonds of particle-like protons 1 i and 2 i in the periodic lattice of dimers indexed i, 1 ≤ i ≤ N 0 , are represented by orthonormal coordinates x 1 = x 1i , x 2 = x 2i ∀i, in the plane {x 1 ; x 2 } (Fig. 4). The symmetry coordinates for the degrees-of-freedom indexed m, 1 ≤ m ≤ N 0 , are represented by orthogonal coordinates χ u = χ um , χ g = χ gm ∀m, in the plane

{χ u ; χ g } ⊥ {x 1 ; x 2 }. {x 1 ;
x 2 } is specific to static representations of outcomes of elastic events, while {χ u ; χ g } holds for energy levels realized via inelastic events. This plane compares with the coordinates traditionally referred to as "stretching", or νOH, in the classical harmonic approximation. This terminology remains convenient in quantum mechanics, but it does not refer any more to displacements.

The intersection line (dot-dashed in Fig. 4) is the locus for centrosymmetric configurations in x-space. Apart from this locus, configurations breaking C isymmetry, eg R 1 L 2 or L 1 R 2 , are forbidden by space-group symmetry. The C isymmetry is preserved throughout the χ-plane, by construction. The eigenstates of the potential operators in 1-D, say V ξ (χ ξ ), ξ = "u" or "g", are |ψ ξn (χ ξ ) .

The vibrational spectra [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF] are consistent with asymmetric double-wells for bare proton-waves, virtually identical for u and g species (Table 1), which are temperature-independent.

Vibrational spectra and neutron diffraction accord with realization of a combination of state vectors with energy-gap hν l :

|φ = [|L + p l |R ](1 + p l ) -1 ; (4)
where p l = exp(-hν l /kT ) is a deterministic temperature factor. These states are either separable (aka disentangled) for spectroscopy or non-separable (aka entangled) for diffraction or relaxometry.

V L = V 0 (χ u ) + V 0 (χ g ) is the operator -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 2h 01 011 u ( Å ) g ( Å )
-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

h 01 010 u ( Å ) g ( Å )
-0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

h 01 001 u ( Å ) g ( Å ) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 000 u ( Å ) g ( Å ) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6
-0,4 -0,2 0,0 0,2 0,4 0,6

h l +2h 01 111 u ( Å ) g ( Å ) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6
-0,4 -0,2 0,0 0,2 0,4 0,6

h l +h 01 110 u ( Å ) g ( Å ) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6
-0,4 -0,2 0,0 0,2 0,4 0,6

h l +h 01 101 u ( Å ) g ( Å ) -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 h l 100 u ( Å ) g ( Å )
for l = 0, V R = V 1 (χ u ) + V 1 (χ g ) + hν l for l = 1,
|ln u n g = |ln u |ln g are the eigenstates. By convention, the minimum at L ξ is lower than that at R ξ for l = 0 and the energy difference between |01 ξ and |00 ξ is E 01ξ -E 00ξ = hν 01 (Table 1). A l-flip (0 ←→ 1) toggles the sign of the double-well asymmetry, so R ξ is lower than L ξ and E 11ξ -E 10ξ = hν 01 . The eigenstates of interest read: The potential operator in 2 × 1-D and the eigenfunctions Ψ lnung (χ u , χ g ) are sketched in Figs 5 and6, respectively. In previous works [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF][START_REF] Fillaux | Extended tunneling states in the benzoic acid crystal: Infrared and raman spectra of the OH and OD stretching modes[END_REF], the separation of x-space and χ-space was overlooked, so the potential operator was represented as a scalar potential in direct space that is meaningless in quantum mechanics.

|00 ξ = cos θ|ψ Lξ +
Similarly, eigenfunctions represented in x-space were meaningless. In Fig. 5 the

central barrier of V L (≈ 10 4 cm -1
) is surrounded by four minima and we set the origin of the energy-scale to the lowermost state |000 . Superposition of V L and V R yields 8 distinct minima. In Fig. 6, Ψ 000 at zero and Ψ 111 at hν l + 2hν 01 are localized to order ε around L u L g , centred at (-∆χ u /2, -∆χ g /2 This is a graphic illustration of the complementarity principle.

Proton site occupancies

From the energy levels presented in fig. 6, we deduce the different temperature laws shown in fig. 2. Firstly, within the framework of microrealism, the R-site occupancy is

ρ 1 = p l (1 + p l )(1 + p 01 ) 2 + p 01 (1 + p 01 ) 2 + p 2 01 (1 + p l )(1 + p 01 ) 2 , (6)
where p l = p l and p 01 = exp(-hν 01 /kT ) are statistical population factors (see red dash in Fig. 2). Secondly, from condensate viewpoint, measurement-induced lattice configurations depend on the deterministic temperature factors p l and p 01 = p 01 . In Fig. 2 we display two options. (i) The blue dot-dash for the two-level system:

ρ 2 = p l 1 + p l . (7)
(ii) The black solid if the states |l10 and |l01 are not realized via diffraction:

ρ 3 = p l + p 2 01 (1 + p l )(1 + p 2 01 ) . (8)
This is the best option. Although there is no energy scale in x-space, temperature factors transpire from χ-space through probability densities around the contact points

L u L g -L 1 L 2 and R u R g -R 1 R 2 , while probability densi- ties around R u L g or L u R g have no counterpart in x-space. Consequently, it
is impossible to deduce the complete set of eigenstates of the potential operator from nuclear densities and, conversely, the complete energy level scheme is not representative of the probability density effectively measured via diffraction. Eigenstates and probability densities must be determined separately, via complementary measurements.

The site-occupancies are comprised of two terms:

ρ L = a(|Ψ 000 | 2 +p l p 2 01 |Ψ 111 | 2 ); ρ R = a(p l |Ψ 100 | 2 + p 2 01 |Ψ 011 | 2 ); a = [(1 + p l )(1 + p 2 01 )] -1 .
According to the classical words of chemistry, the states |000 and |100 would correspond to COOconfigurations L 1 L 2 and R 1 R 2 (Fig. 1), respectively, while the states |011 and |111 would correspond to transposition of proton pairs from C-O-sites to C=O-sites at a rather modest energy cost. Then, the thermal laws for the mean CO-bond lengths should be different from [START_REF] Hayashi | Infrared spectroscopic evidence for the coexistence of two molecular conformations in crystalline fatty acids[END_REF]. Alternatively, from condensate viewpoint, the COO-groupings are indefinite in the χ-representation. The linear correlation (2) (Fig. 3) suggests that CO-bond lengths realized via diffraction match the proton-site occupancies determined via the contact points, so as to avoid the less favored structures.

From [START_REF] Meyer | Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics[END_REF], we obtain ln ρR ρL = ln(p l + p 2 01 ) that is slightly different from (1). This could explain the residual entropy and the small difference, close to within data statistics, between ∆H ρ and hν l . On the other hand, we anticipate that NMR should realize a linear combination of every eigenstates (see below sec. 3.3) and yield ln ρR ρL = ln( ρ1 1-ρ1 ). In practice, ( 6), [START_REF] Meier | Structure and dynamics of intramolecular hydrogen bonds in carboxylic acid dimers: A solid state NMR study[END_REF] or ( 8) cannot be distinguished conclusively on the ln-scale of van't Hoff plots.

Relaxometry

Off-resonance energy-transfer realizes at the time t = 0 of the event a transitory state at T :

|S = l,nu,ng a l b nung |ln u n g ; (9)
a 2 0 = (1+p l ) -1 ; a 2 1 = p l a 2 0 ; b 2 0,0 = (1+p 01 ) -2 ; b 2 1,0 = b 2 0,1 = p 01 b 2 0,0 ; b 2 1,1 = p 2 01 b 2 0,0 .
Because S ≡ 0, each event realizes the same state and the relaxation statistics for a large number of events is predictable. The critical rate in 2 × 1-D is twice the ground-state splitting in 1-D (ν 0± = 2εν 01 ) scaled by the amplitude of that part of the wavefunction equally delocalized over the four wells. Inspection of Fig. 6 allows us to distinguish three channels:

τ -1 = 4εν 01 2ε 2 + εp 1/2 01 + 2p 3/2 01 . ( 10
)
The first term accounts for decay of nu,ng (a 0 b nung |0n u n g + a 1 b nung |1n u n g ) through the 1-quantum channel 1-hν l , scaled by 4ε 2 . The second term is the 1-

hν 01 channel for a l b nung |ln u n g +a l b n u n g |ln u n g , l = 0 or 1, n u +n g -(n u +n g ) = 1.
) 11
Here the rate follows from spontaneous fluctuations of nuclear positions and the measurement is a very weak process mirroring the unperturbed system. The first term represents "phonon-assisted" interconversion through "incoherenttunneling" of rigid pairs. The other terms model classical jumps above potential barriers for distinct reaction-paths. This scheme accords with the existence of three relaxation channels, but rigid-pairs, low barriers and "phonon-assistedtunneling" are excluded by spectroscopy.

In Fig. 7, [START_REF] Stöckli | Hydrogen bond dynamics in isotopically substituted benzoic acid dimers[END_REF] and [START_REF] Meyer | Transitions induced in a double minimum system by interaction with a quantum mechanical heat bath[END_REF] are similar, but their merits are quite different: two fixed parameters for the former as compared to six tuned parameters, which cannot be measured independently, for the latter. More fundamental, relaxometry is not amenable to microrealism because a snapshot of a stochastic ensemble of pre-existing particles, or wave packets, taken via a non-perturbing wave, is necessarily space-time averaged and should not demonstrate any rate. Alternatively, synchronization of the probe and the instability is intrinsic to quantum measurements of a pre-existing condensate.

In Fig. 7, there is no visible increase of the measured rate [START_REF] Meyer | Transitions induced in a double minimum system by interaction with a quantum mechanical heat bath[END_REF], as compared to [START_REF] Stöckli | Hydrogen bond dynamics in isotopically substituted benzoic acid dimers[END_REF], to suggest incoherent decay and/or emergence of the classical regime at elevated temperatures. In contrast with [START_REF] Meier | Structure and dynamics of intramolecular hydrogen bonds in carboxylic acid dimers: A solid state NMR study[END_REF][START_REF] Meyer | Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics[END_REF][START_REF] Skinner | Theoretical analysis of nuclear magnetic resonance, neutron scattering, and optical experiments[END_REF][START_REF] Stöckli | Hydrogen bond dynamics in isotopically substituted benzoic acid dimers[END_REF][START_REF] Meyer | Transitions induced in a double minimum system by interaction with a quantum mechanical heat bath[END_REF][START_REF] Heuer | The dynamics of hydrogens in double well potentials: The transition of the jump rate from the low temperature quantummechanical to the high temperature activated regime[END_REF][START_REF] Sakun | Proton tunneling assisted by the intermolecular vibration excitation. Temperature dependence of the proton spin-lattice relaxation time in benzoic acid powder[END_REF][START_REF] Brougham | Proton transfer dynamics in the hydrogen bond: a direct measurement of the incoherent tunneling rate by NMR and the quantum-to-classical transition[END_REF][START_REF] Neumann | Proton tunneling in benzoic acid crystals at intermediate temperatures: Nuclear magnetic resonance and neutron scattering studies[END_REF][START_REF] Latanowicz | Quantum mechanical and classical dynamics of double proton transfer in the hydrogen bond[END_REF][START_REF] Takeda | An indication of slowing down of hydrogen atom transfer in isotopically mixed hydrogen bonds of benzoic acid crystals[END_REF][START_REF] Latanowicz | Complex dynamics of proton and deuteron transfer in double hydrogen bond of benzoic acid isotopes[END_REF][START_REF] Xue | Isotope effects associated with tunneling and double proton transfezr in the hydrogen bonds of benzoic acid[END_REF][START_REF] Wu | The correspondence between quantum and classical mechanics: an experimental demonstration of the smooth transition between the two regimes[END_REF], relaxometry confirms that the bulk of the crystal is immune to decoherence.

Discussion

The crystal of benzoic acid is an open system amenable to the framework of quantum mechanics without any bias. The condensate follows from the linear formalism of quantum mechanics. The free energy determines thermal stability.

The measurement-induced vibrational states are fully characterized by hν l , hν 01 and ε, effectively measured. The level-scheme in χ-space is model-free, inasmuch as the potential operator supporting the assignment scheme is of no physical importance. Positional parameters and proton-site occupancies in x-space are also model-free. The theory is self-contained.

The temperature laws [START_REF] Meyer | Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics[END_REF] and [START_REF] Stöckli | Hydrogen bond dynamics in isotopically substituted benzoic acid dimers[END_REF] The main difference is that the eigenstates of KHCO 3 at room temperature and below correspond to the left-hand-side column in Fig. 6, for a single four-well operator (p l = 0). Only |Ψ 000 | 2 and |Ψ 011 | 2 are relevant to diffraction. The difference between ( 6) and ( 8) is enhanced (Fig. 2, and ♦, respectively) and diffraction accords with the latter, in line with contextuality. In addition, the relaxation rate (10) turns into

τ -1 = 4εν 01 εp 1/2 01 + 2p 3/2 01 ( 12
)
that vanishes as T -→ 0, as effectively observed [START_REF] Eckold | Proton disorder and phase transition in KHCO 3[END_REF].

The framework can be also applied to isolated dimers of carboxylic-acids as follows. (i) Superposition of vibrational states yields a molecular condensate.

(ii) Realization of non-separable isoenergetic |L and |R states yields a D 2hstate [START_REF] Fillaux | Quantum entanglement and nonlocal proton transfer dynamics in dimers of formic acid and analogues[END_REF]. (iii) The scheme for bare proton-waves is that depicted in Fig. 6 with hν l = 0. (iv) The ground-state splitting is 2hτ -1 (10). The value 2hτ -1 = (1.6 ± 0.4) × 10 -2 cm -1 for the crystal of benzoic acid at T = 0 compares favourably with those reported for various dimers in supersonic jet-expansions: ≈ 4.6 × 10 -2 cm -1 for benzoic acid [START_REF] Kalkman | Tunneling splittings in the S 0 and S 1 states of the benzoic acid dimer determined by high-resolution UV spectroscopy[END_REF]; ≈ 1.8 × 10 -2 cm -1 for benzoic-formic acid heterodimers [START_REF] Evangelisti | Proton tunneling in heterodimers of carboxylic acids: A rotational study of the benzoic acidformic acid bimolecule[END_REF]; ≈ 1.9 × 10 -2 cm -1 for 3-fluoro-benzoic acid [START_REF] Daly | Gas phase measurements of mono-fluoro-benzoic acids and the dimer of 3-fluoro-benzoic acid[END_REF]; 0.3 × 10 -2 -1.6 × 10 -2 cm -1 for (HCCOH) 2 or (DCOOH) 2 [START_REF] Madeja | High resolution spectroscopy of carboxylic acid in the gas phase: Observation of proton transfer in (DCOOH) 2[END_REF][START_REF] Ortlieb | Proton transfer in (HCOOH) 2 : An IR highresolution spectroscopy studyof the antisymmetric C-O stretch[END_REF][START_REF] Gutberlet | High resolution IR spectroscopy of the carbonyl stretch of (DCOOD) 2[END_REF]; ≈ 0.8 × 10 -2 cm -1 for formic-acetic acid [START_REF] Tayler | Unraveling the spectroscopy of coupled intramolecular tunneling modes: A study of double proton transfer in the formic-acetic acid complex[END_REF]; ≈ 10 -2 cm -1 for formic-propiolic acid [START_REF] Daly | Microwave measurements of proton tunneling and structural parameters for the propiolic acid-formic acid dimer[END_REF] etc.

A superposition of four-well operators with opposite symmetries in 2×1-D could be thus a common feature for dimers of carboxylic acids and the mean observed splitting of ≈ 1.76 × 10 -2 cm -1 suggests that hν 1 and ε are marginally affected by temperature and/or chemical substitution from crystals to isolated dimers.

Within the framework of classical mechanics, the dot-dash (x 1 = x 2 ) in Fig. 4 is the mean reaction-path for C i proton-transfer. The configurations R 1 L 2 and L 1 R 2 at high energy being of no importance in supersonic jets, interconversion

occurs in quasi 1-D. The distance between L 1 L 2 and R 1 R 2 is √ 2d LR .
The mass of the collective variable is 2 amu, or even more if it is correlated with displacements of heavy nuclei. The scalar double-well along the C i locus should compare with cuts of V L or V R along the intersection line (χ u = χ g ), through the central barrier of ≈ 10 4 cm -1 (≈ 30 Kcal.mol -1 ), that is twice that in 1-D (Table 1). This is much greater than 3-12 Kcal.mol -1 computed with quantum chemistry methods for the dimer of formic acid (see Table 2 in [START_REF] Ivanov | Quantum free energy landscapes from ab initio path integral metadynamics: Double proton transfer in the formic acid dimer is concerted but not correlated[END_REF]). This difference is no wonder since the operator holds for bare proton-waves in χ-space while quantum chemistry focuses on adiabatic transfer involving many degrees of freedom in x-space. In fact the condensate framework and quantum chemistry methods account for different systems, although with identical names. They are not comparable. It may happen that the lower adiabatic barrier counterbalances the longer inter-well distance and the greater collective mass, so as to yield, accidentally, a plausible value of the semiclassical tunneling-rate. Nevertheless, the classical view based on the Born-Oppenheimer separation is untenable since the barrier height, the inter-well distance and the collective mass are not in line with spectroscopy. As a matter of fact, the zero-point energy of a νOH transition observed at ≈ 3000 cm -1 yields a lower bound of ≈ 4500 cm -1 (≈ 13.5 Kcal.mol -1 ) for the barrier height in 1-D.

More fundamentally, beyond the great variety of computational models, an energy landscape in x-space is relevant for classical nuclei exclusively [START_REF] Ushiyama | Successive mechanism of double-proton transfer in formic acid dimers: a classical study[END_REF]. For quantum nuclei, x-space is specific to elastic scattering events. There is neither energy-scale, nor potential function, nor spontaneous oscillations, nor locally causal relationship between positions and vibrational states. The dot-dashed locus in Fig. 4 is not a trajectory for the time evolution of a dimer. In contrast with previous works, eg [START_REF] Shida | A reaction surface hamiltonian treatment of the double proton transfer of formic acid dimer[END_REF][START_REF] Mil´nikov | Ground-state and vibrationally assisted tunneling in the formic acid dimer[END_REF][START_REF] Luckhaus | Concerted hydrogen exchange tunneling in formic acid dimer[END_REF][START_REF] Luckhaus | Hydrogen exchange in formic acid dimer: tunneling above the barrier[END_REF], the measurement-induced vibrational states cannot be deduced from a scalar landscape decorated with a quantized

Hamiltonian. Minimum energy-paths [START_REF] Shida | A reaction surface hamiltonian treatment of the double proton transfer of formic acid dimer[END_REF][START_REF] Kim | Direct dynamics calculation for the double proton transfer in formic acid dimer[END_REF][START_REF] Matanović | Ground and asymmetric CO-stretch excited state tunneling splittings in the formic acid dimer[END_REF], averaged trajectories [START_REF] Miura | An ab initio path integral molecular dynamics study of double proton transfer in the formic acid dimer[END_REF][START_REF] Markwick | Targeted Car-Parinello molecular dynamics: Elucidating double proton transfer in formic acid dimer[END_REF][START_REF] Ivanov | Quantum free energy landscapes from ab initio path integral metadynamics: Double proton transfer in the formic acid dimer is concerted but not correlated[END_REF], instanton rates [START_REF] Smedarchina | Tunneling dynamics of double proton transfer in formic acid and benzoic acid dimers[END_REF][START_REF] Mil´nikov | Ground-state and vibrationally assisted tunneling in the formic acid dimer[END_REF][START_REF] Smedarchina | Correlated doubleproton transfer. I. Theory[END_REF] and collective variables do not capture any quantum phenomenon.

In addition to spectroscopic studies of cold molecules in supersonic jets, matter wave interferometry of C 60 [START_REF] Arndt | Waveparticle duality of c 60 molecules[END_REF], C 70 [START_REF] Brezger | Matter-wave interferometer for large molecules[END_REF] and perfluoroalkylated molecules [START_REF] Gerlich | Quantum interference of large organic molecules[END_REF] sublimated at ≈ 10 3 K under high vacuum (∼ 10 By contrast, the reported decoupling of the internal and external degrees of freedom under high vacuum accords with quantum condensation of the internal states, while the suppression of quantum interference as a function of the gas pressure in the vacuum chamber [START_REF] Hornberger | Collisional decoherence observed in matter wave interferometry[END_REF] accords with collision-induced breakdown of the continuous space-time translation symmetry.

Conclusion

The crystal of benzoic acid at thermal equilibrium behaves as a condensate that is energetically favored, as compared to an incoherent mixture of phonons.

Energy levels and nuclear positions (eg chemical bonds or tautomers) are indefinite. This framework can be applied to isolated dimers as well.

Outcomes of incompatible quantum measurements (diffraction versus spectroscopy) pertain to mutually exclusive representations, either in the space of static positional parameters, or in the orthogonal space of the symmetry coordinates for time-dependent eigenfunctions. Experimental studies of these complementary representations are necessary for a comprehensive account of the crystal.

Temperature effects are rationalized within a self-contained framework as follows: (i) measurement-induced realization of a temperature-dependent superposition of the state vectors of a two-level system; (ii) to each level is associated a four-well operator for bare proton-waves in 2 × 1-D that is comprised of identical asymmetric double-wells for each symmetry coordinate; (iii) the doublewells have opposite asymmetries for either state. Then, the temperature laws for proton-site occupancies, revealed by diffraction, or for the relaxation-rate of proton-waves, measured through solid-state NMR or quasi-elastic neutron scattering, and the energy-level scheme, revealed by spectroscopy, witness to the contextuality of outcomes of incompatible measurements. All observations accord with the quantum regime from cryogenic to room temperature.

In classical mechanics, tautomers are thought of as objective entities undergoing dynamical interconversion via nuclear trajectories, reaction paths and transition states. Archetypal mechanisms are concerted proton transfer in quasi 1-D at low temperature or/and stepwise single-proton transfer in quasi 2-D, more likely at elevated temperature. This intuitive scheme does not capture the quantum nature of the dimer of benzoic acid attested by spectroscopy. In the quantum regime, dynamical interconversion is meaningless. On the one hand, nuclear-site occupancies mirror the static tautomerization degree. On the other

V Lξ (χ ξ ) (cm -1 ) ∆χ 0 ( Å) H ξ (cm -1 ) hν 01ξ (cm -1 ) hν 02ξ (cm -1 ) hν 03ξ (cm -1 )
Infrared [START_REF] Fillaux | Quantum proton transfer and interconversion in the benzoic acid crystal: vibrational spectra, mechanism and theory[END_REF] Table 2: Neutron single crystal diffraction data and structure refinement of benzoic acid. λ = 0.8305 Å. Space group P 2 1 /c. σ(I) limit: 3.00.

Refinement on F. The occupation numbers for the proton sites are ρ L and ρ R . The variance for the last digit is given in parentheses.

Crystal -------------------------------------------------

Figure 1 :

 1 Figure 1: Schematic lattice configurations for a crystal of benzoic acid.

Figure 2 :

 2 Figure 2: (Color on line) Temperature laws for the occupancy factor of the less favored proton sites. • with error bars: experimental points (this work). ×: experimental points after [4].

Figure 3 :

 3 Figure 3: CO bond lengths as a function of the occupancy of the less favored proton sites. and with error bars: experimental points (this work). and ♦: after Ref. [[4]]. The straight lines are guides for the eyes.

Figure 4 :

 4 Figure 4: Schematic of the proton static coordinates in the plane (x 1 , x 2 ) perpendicular to that of the symmetry coordinates (χu, χg) for eigenstates. Bullets are distinctive positions discussed in the text. Sticks between bullets are guides for the eye.

Figure 5 :

 5 Figure 5: Perspective view of the measurement-induced potential operator for the hydrogen bonding protons in the crystal of benzoic acid in the state l = 0. χu and χg are the symmetry coordinates of the "stretching" degrees of freedom.

Figure 6 :

 6 Figure 6: Schematic view of the wavefunctions of the measurement-induced "stretching" states. χu and χg are the symmetry coordinates. The weak component of the wave function in one-dimension is enhanced by a factor of 5. The quantum numbers are l, nu, ng, respectively.

## 1 , 3 #

 13 The I/u(I) cutoff below was used for refinement as # well as the _gt R-factors: choose from: rm (reference molecule of known chirality), # ad (anomolous dispersion -Flack), rmad (rm and ad), # syn (from synthesis), unk (unknown) or . (not applicable). Chebychev polynomial,[START_REF] Watkin | [END_REF][START_REF] Prince | Mathematical Techniques in Crystallography and Materials Science[END_REF] [weight] = 1.0/[A~0~*T~0~(x)+A~1~*T~1~(x) ... +A~n-1~]*T~n-1~(x)] where A~i~are the Chebychev coefficients listed below and x= Fcalc/Fmax Method = Robust Weighting (Prince, 1982) W = [weight] * [1-(deltaF/6*sigmaF)^2^]^2Â ~i~are: 0.190 -0.136E-02 -0.409E-02 -0.160E-01 ; _publ_section_references ; Betteridge, P.W., Carruthers, J.R., Cooper, R.I., Prout, K. & Watkin, D.J. (2003). J. Appl. Cryst. 36, 1487. Larson, A.C. (1970) Crystallographic Computing, Ed Ahmed, F.R., Munksgaard, Copenhagen, 291-294. .J., Prout, C.K. & Pearce, L.J. (1996). CAMERON, Chemical Crystallography Laboratory, Oxford, UK. ; # Uequiv = arithmetic mean of Ui i.e. Ueqiv = (U1+U2+U3)/Replace last . with number of unfound hydrogen atomsattached to an atom. # ..._refinement_flags_... # . no refinement constraints S special position constraint on site # G rigid group refinement of site R riding atom # D distance or angle restraint on site T thermal displacement constraints # U Uiso or Uij restraint (rigid bond) P partial occupancy constraint 2189(5) 0.2419(5) 0.01309(10) 0.0281 1.0000 Uani . . . . . . O2 O -0.0946(5) 0.1427(6) 0.06514(10) 0.0279 1.0000 Uani . . . . . . C1 C 0.0960(3) 0.2786(4) 0.05750(7) 0.0193 1.0000 Uani . . . . . . C2 C 0.1772(3) 0.4867(4) 0.10378(7) 0.0165 1.0000 Uani . . . . . . C3 C 0.3853(4) 0.6423(5) 0.09771(9) 0.0249 1.0000 Uani . . . . . . C4 C 0.4591(4) 0.8365(5) 0.14141(9) 0.0264 1.0000 Uani . . . . . .

  H3 0.072(3) 0.097(5) 0.085(3) -0.024(3) 0.041(3) -0.029(3) H4 0.067(3) 0.098(6) 0.112(5) -0.020(4) 0.024(3) -0.037(4) H5 0.093(4) 0.072(4) 0.076(3) -0.036(3) -0.002(3) -0.013(3) H6 0.120(5) 0.093(5) 0.066(3) -0.025(4) 0.045(3) -0.014(

1 ,

 1 Chebychev polynomial,[START_REF] Watkin | [END_REF][START_REF] Prince | Mathematical Techniques in Crystallography and Materials Science[END_REF] [weight] = 1.0/[A~0~*T~0~(x)+A~1~*T~1~(x) ... +A~n-1~]*T~n-1~(x)] where A~i~are the Chebychev coefficients listed below and x= Fcalc/Fmax Method = Robust Weighting (Prince, 1982) W = [weight] * [1-(deltaF/6*sigmaF)2208(10) 0.2394(12) 0.0131(2) 0.0641 1.0000 Uani . . . . . . O2 O -0.0899(9) 0.1408(11) 0.0652(2) 0.0578 1.0000 Uani . . . . . . C1 C 0.0996(7) 0.2702(7) 0.05720(14) 0.0441 1.0000 Uani . . . . . . C2 C 0.1815(6) 0.4756(7) 0.10399(15) 0.0419 1.0000 Uani . . . . . . C3 C 0.3850(8) 0.6260(8) 0.0969(2) 0.0566 1.0000 Uani . . . . . . C4 C 0.4603(8) 0.8164(8) 0.14052(19) 0.0614 1.0000 Uani . . . . . . C5 C 0.3315(8) 0.8541(8) 0.19027(19) 0.0635 1.0000 Uani . . . . . . C6 C 0.1284(8) 0.7025(8) 0.19658(18) 0.0623 1.0000 Uani . . . . . . C7 C 0.0522(7) 0.5158(7) 0.15353(15) 0.0495 1.0000 Uani . . . . . . D3 D 0.4827(10) 0.5965(12) 0.0580(3) 0.0910 1.0000 Uani . . . . . . D4 D 0.6161(11) 0.9351(13) 0.1357(4) 0.1103 1.0000 Uani . . . . . . D5 D 0.3900(12) 1.0007(13) 0.2236(3) 0.0946 1.0000 Uani . . . . . . D6 D 0.0269(13) 0.7328(13) 0.2349(2) 0.1050 1.0000 Uani . . . . . . D7 D -0.1061(10) 0.3937(11) 0.1575(2) 0.0837 1.0000 Uani . . . . . . H10 H 0.137(5) -0.004(4) -0.0368(11) 0.0628 0.5039 Uani . . . . . . H11 H 0.173(4) 0.094(5) -0.0165(10) 0.0650 0.4962 Uani . . . . . .

D3 0 .

 0 083(4) 0.094(4) 0.105(4) -0.032(3) 0.043(3) -0.031(3) D4 0.087(4) 0.099(4) 0.147(6) -0.041(4) 0.027(4) -0.044(4) D5 0.109(4) 0.072(3) 0.099(4) -0.033(3) 0.000(4) -0.007(3) D6 0.145(5) 0.098(5) 0.082(3) -0.035(3) 0.049(3) -0.022(4) D7 0.092(4) 0.086(3) 0.081(3) -0.019(3) 0.042(3) -0.030(3) H10 0.071(13) 0.045(12) 0.076(16) -0.009(9) 0.024(12) -0.010(9)H11 0.071(13) 0.073(17) 0.053(11) -0.015(9) 0.018(9) -0.012[START_REF] Heuer | The dynamics of hydrogens in double well potentials: The transition of the jump rate from the low temperature quantummechanical to the high temperature activated regime[END_REF]

  sin θ|ψ Rξ ; |10 ξ = sin θ|ψ Lξ + cos θ|ψ Rξ ; |01 ξ =sin θ|ψ Lξ + cos θ|ψ Rξ ; |11 ξ = cos θ|ψ Lξsin θ|ψ Rξ . Lξ and ψ Rξ are eigenfunctions for polynomial expansions of the double-wells around L ξ and R ξ , respectively. 2θ = arctan[ν 0± /(ν 01ν 0± )], where hν 0± = (6 ± 1) cm -1 would be the ground-state splitting if hν 01 = 0. Hence cos θ ≈ 1 and sin θ = ε = (1.8 ± 0.3)10 -2 . 00 ξ |χ ξ |00 ξ and 11 ξ |χ ξ |11 ξ are localized to order ε 2 at L ξ , whereas 10 ξ |χ ξ |10 ξ and 01 ξ |χ ξ |01 ξ are localized at R ξ .

	
	
	(5)
	
	ψ

  The scaling amplitude is 2ε. The last term is the 2-hν 01 channel for nu,ng a l b nung |ln u n g , l = 0 or 1, scaled by a factor of 4. Further n-quantum channels are negligible. It turns out that τ -1 is independent of of p l . The rate is distinctive of quantum interferences, via probability amplitudes ε, p

		1/2 01 , p 3/2 01 ,
	while positional coordinates remain indefinite. The notion of particle-like pro-
	tons tunneling back and forth across a barrier is meaningless.
	As opposed to (10), Neumann et al. [15] have fitted NMR and QENS data
	with six phenomenological parameters inspired by the statistical model of Skin-
	ner and Trommsdorff [9]:
	τ -1 exp = (1.72 ± 0.02)10 8 coth[(43 ± 1)/kT ] + 10 10 exp(-120/kT ) + 6.3 × 10 11 exp(-400/kT ).	(

Table 1 :

 1

Double-well operators, distances between minima (∆χ ξ ), barrier heights (H ξ ), and energy levels, for the crystal of Benzoic acid. χ ξ are the symmetry species infrared-(χu) or Raman-active (χg). The oscillator mass is µu = µg = 1 amu. *: observed via inelastic neutron scattering after Ref.

[[23]

].

  265χ u + 0.2860 × 10 6 χ 2

		u u ) +171480 exp(-2.17χ 2	0.70	5005	(172 ± 4)*	2570 ± 20	2840 ± 20
	Raman [22]	270χ g + 0.2829 × 10 6 χ 2 g +171120 exp(-2.15χ 2 g )	0.69	5006	171 ± 4	2602 ± 4	2853 ± 4

175(2) no

 ---------------------------------------------

Method, part 1, Chebychev polynomial, [START_REF] Watkin | [END_REF][START_REF] Prince | Mathematical Techniques in Crystallography and Materials Science[END_REF] [weight] = 1.0/[A~0~*T~0~(x)+A~1~*T~1~(x) ..