Francois Fillaux et al- Experimental Studies and Theory of Nonlinear Rotational Dynamics in the...

8/3/2019 Francois Fillaux et al- Experimental Studies and Theory of Nonlinear Rotational Dynamics in the Quantum Regime:

1/35

PROCEEDINGS OF THE THIRD CONFERENCE

LOCALIZATION & ENERGY TRANSFER IN NONLINEAR

SYSTEMS

EDITORS

LUIS V AZQUEZ

Universidad Complutense and Centro de Astrobiologa CSIC/INTA, Spain

ROBERT S. MACKAY

Warwick University, UK

MARI A PAZ ZORZANO

Centro de Astrobiologa CSIC/INTA, Spain

WORD SCIENTIFIC

New Jersey London Singapore Hong Kong

COPYRIGHT c 2003, IBSN 981-238-296-8

pp 68-101

1


2/35

EXPERIMENTAL STUDIES AND THEORY OFNONLINEAR ROTATIONAL DYNAMICS IN THE

QUANTUM REGIME: THE INTERPLAY OF STRUCTURE,DYNAMICS AND LOCALIZATION IN CRYSTALS

FRANCOIS FILLAUX AND B EATRICE NICOLALADIR-CNRS and Universite Pierre et Marie Curie

2 rue H. Dunant, 94320 Thiais, FranceE-mail: [email protected]

ALAIN COUSSONLaboratoire Leon Brillouin (CEA and CNRS)

CEA Saclay, 91 191, Gif sur Yvette cedex, FranceE-mail: [email protected]

Experimental and theoretical studies of quantum rotation of methyl groups in crys-tals are presented. Tunnelling spectroscopy with the inelastic neutron scatteringtechnique and determination of angular probability densities of methyl groups intheir rotational planes with the single crystal neutron diffraction technique are in-troduced. The importance of temperature and deuteration effects is emphasized.

Three examples are presented. i) In the manganesediacetate tetrahydrate crystal,Mn(CH 3 COO) 2 ;4H 2 O, methyl groups are in three different environments, theirrotational axes have different orientations and they are well separated from eachother. They can be regarded as isolated single rotors and tunnelling excitationsare localized. ii) In the lithiumacetate dihydrate crystal, CH 3 COOLi;2H 2 O, allmethyl groups are equivalent. Close-contact pairs of face-to-face methyl groupsare distributed in a nearly hexagonal planar structure with rather short methyl-methyl distances. The face-to-face methyl groups twisted by 60 perform combinedrotation represented with symmetry adapted coordinates. Ordering of the methylgroups at a low temperature upon deuteration reveals signicant interaction be-tween pairs. However, in the 2D structure arising from methyl-methyl interactionthere is no evidence for collective rotation of the pairs. Tunnelling excitations arelocalized within pairs. iii) In the 4-methylpyridine (C 6 H7 N) crystal the distancesbetween methyl groups are quite similar to those in lithiumacetate. However, inthe tetragonal structure, the face-to-face methyl groups twisted by 90 to eachother cannot perform combined rotation. The corresponding effective potential isvirtually a constant. There is no phase transition upon methyl deuteration. Methylgroups form orthogonal innite chains parallel to a or b and virtually isolated fromeach other. The collective rotation in 1D is represented in the displacive regimewith the quantum sine-Gordon theory. Tunnelling transitions are represented withextended states in an energy-band structure. Additional transitions are attributedto travelling states of a dimensionless pseudo particle: the quantum breather mode.All excitations are totally delocalized.

1


3/35

2

1. Introduction

Light particles moving in a potential with topological degeneracy manifesttheir quantum nature via tunnelling, one of the evidences of the profounddifference between classical and quantal worlds. Owing to the spatial ex-tension of the wave function, the degenerate ground state in the classicalregime splits into sublevels. In the condensed matter, the observation of tunnelling sheds light onto fundamental problems of quantum mechanics ina complex environment.

For light quantum rotors like methyl groups, the topological degeneracyarises from the intrinsic periodicity of the angular coordinate and rotationaltunnelling is observed in many crystals at a low temperature. 1 The upperlimit of the tunnel splitting is the rotational constant B = 2 / 2I r , where I ris the moment of inertia of the rotor. For an isolated and rigid CH 3 grouprotating around its axis of inertia supposed to be xed, B ranges from0.650 to 0.700meV and from 0.325 to 0.350meV for the deuterated analogueCD3 . The magnitude of the tunnel splitting depends on the particle massand potential shape (distances between identical sites, barrier height...).Deuteration is of dramatic consequence to the tunnelling frequency.

In many molecular crystals the frequency range for tunnelling is wellseparate from the density-of-states for optical phonons and dynamical in-teraction with the lattice is very weak. Tunnelling transitions are specicto the rotational coordinates and can be represented with a rather simpleHamiltonian. Therefore, rotational tunnelling is unique to observing non-linear dynamics in the quantum regime. For example, quantum rotationof isolated rotors can be regarded as the solution of the Mathieu equationat the molecular level (see below section 2). 2 Energy localization is charac-teristic of these tunnelling transitions. 3,4 For some systems, methyl-methylinteraction may compete with localization. For example, the rotationaldynamics in the lithiumacetate dihydrate crystal are solutions of the non-linear and nonintegrable Hamiltonian for coupled pairs of rotors (see below3).5,6,8,7,9 Tunnelling excitations remain largely localized in pairs. Finally,collective rotational dynamics in the 4-methylpyridine crystal are solutionsof the sine-Gordon Hamiltonian in the quantum regime (see below sectionsection 4). 10 ,11 ,12 ,13 ,14 ,16 ,15 Tunnelling excitations are totally delocalized

For methyl groups, the great specicity of rotational tunnelling canbe fully exploited with the inelastic neutron scattering (INS) technique,because the cross-section of H atoms for incoherent neutron scattering ismuch greater than for any other atom, by about one order of magnitude. 17

Any intensity arising from the crystal density-of-states (primarily acoustic


4/35

3

phonons in the tunnelling energy range) can be ignored.Nowadays, transitions above 1eV are observed with advanced spec-

trometers available at various neutron sources. However, the observationof tunnelling transitions for deuterated analogues is hampered by the muchweaker cross section of D atoms and by the dramatic decrease of the split-ting upon deuteration. This information is often missing and this is alimitation for modelling rotational dynamics.

Pioneering INS works with rather modest resolution have rst revealedsingle tunnelling transitions and dynamical models for single rotors weresufficient to interpreting the spectra. 3,4 With better spectrometer resolu-tion, multi-component spectra due to dynamical coupling were observed

and different models were proposed to account for local or collective ro-tational dynamics. The diversity of models is partly due to the lack of information and this is a long lasting source of polemics that keep the eldlively. Unfortunately, the hope that advanced methods of quantum chem-istry and molecular dynamics simulation could provide realistic modelling of the effective potentials experienced by methyl rotors is not yet realized. 18 ,19

As long as tunnelling frequencies cannot be calculated with good accuracy(say a few eV), experiments remain the best source of knowledge of thesequantum dynamics. This is a strong incentive to undertaking new exper-iments in order to remove ambiguities, as much as possible. Studies of various isotope derivatives and mixtures, tunnelling spectra measured onsingle crystals, determination of the angular probability density and of thekinetic momentum distribution are among the salient progresses which haveshed a new light on quantum rotation in solids, during the last decade, orso.

This paper is meant to be an introduction to tunnelling spectroscopywith neutrons and to the analysis of model Hamiltonians for quantum ro-tation. We present three prototypical examples of quantum rotors withtotally different dynamics. The impact of advanced techniques and dataanalysis to theoretical developments is emphasized. We hope to convincethe reader that rotational dynamics of methyl groups is a unique bench-mark for quantum mechanics in the solid states. Moreover, this eld of investigation is a lively forum for experimental chemists/physicists on theone hand and theoreticians/mathematicians, on the other. To launch fruit-ful collaborations between such experts from different elds is a major goalof the LOCNET network.


5/35

4

2. The single methyl rotor

The simplest model for methyl rotation is the single rigid rotor in a three-fold potential. The Hamiltonian

H 1 = B 2

2+

V 032

(1 cos3) (1)can be transformed into a Mathieu equation. 2 Eigen states E n and wavefunctions n () depend on two quantum numbers: the principal torsionalquantum number n for the oscillator limit with full degeneracy and a sub-level index , which gives the symmetry of the wave function. As thereis no analytical solution for the eigen values, numerical calculations usingthe basis set of the free rotor are carried out with the variational method(Figure 1). 20 In this simple model, there is only one tunnelling transitionand it is straightforward to estimate threefold potential barriers from mea-surements. As the tunnel splitting varies practically exponentially with thepotential barrier, tunnelling is a very sensitive probe of the local poten-tial experienced by methyl groups in crystals. Experimental values can becomparer with those estimated with various models (for example quantumchemistry methods). 18 ,19

-3 -2 -1 0 1 2 30

5

10 6

5

(3,4)

(1,2)

0

( m e V )

(radian)-3 -2 -1 0 1 2 3

0.00.51.0

0

(radian)

-101

1

-101

2

-101

4

-101

3

Figure 1. Schematic representation of the eigen states and eigen functions for a methylrotor in a threefold potential and molecular models for the acetate entity (upper) and4-methylpyridine (lower).

The effective potential can be decomposed into an internal potentialthat is determined by the molecular frame bearing the methyl group andan external contribution arising from the environment. As internal barriersare negligible for the acetate entity and for the 4-methylpyridine molecule


6/35

5

(see Figure 1), the observed potential barriers are entirely due to the envi-ronment and can be tentatively related to the crystal structure.

A tutorial example is the crystal of manganesediacetate tetrahydrate,Mn(CH 3COO) 2);4H2O, which contains 3 crystallographically inequivalentmethyl groups. 21 The INS spectrum of a powdered sample at 1.5 K reveals 3tunnelling transitions at 1.2, 50 and 137 eV, with equal intensities. 22 Thecorresponding potential barriers (49.0, 17.5 and 11.5 meV, respectively)emphasize the great sensitivity of the tunnelling frequency to the local po-tential. However, effective potentials arising from the crystal environmentat each site are sums of many contributions (atom...atom, electrostatic,multipolar...) that are poorly known and cannot be calculated with good

accuracy. Unavoidable uncertainty precludes a rm assignment scheme forthe tunnelling frequency at each rotor site.

2.1. INS spectra of oriented single crystals

The diffraction techniques are used to determine crystal structures withgreat accuracy. For ordered crystal rather sharp spots of intensity (Bragg-peaks) are observed for well dened orientations of the crystal and detectorwith respect to the incident beam. The diffraction pattern denes the re-ciprocal lattice that is a Fourier transformation of the direct lattice. Theintensity of the peaks is determined by the lattice symmetry and the scat-tering cross-section of the atoms at each site.

X-ray diffraction is the most used technique available in many laborato-ries as a routine facility. The intensity scattered by atoms increases with thenumber of electrons. Consequently, the Bragg-peak intensities are largelydominated by contributions from heavy atoms whilst hydrogen atoms canbe totally hidden. As a general rule, hydrogen atoms of methyl groupscannot be localized accurately with X-ray. However, the crystal structureis a source of information regarding the environment of the methyl groups.

For example, the structure of manganesediacetate tetrahydrate deter-mined with X-ray reveals the orientations of the rotational axes of themethyl groups (see Figure 2). 23 ,24 ,25 Then, with properly oriented singlecrystals, the INS intensity of the tunnelling transitions can be probed as a

function of the orientation of the momentum transfer vectorQ

with respectto the rotational axes. ( Q = k 0 k f with |k 0 | = 2 / 0 and |k f | = 2 / f ,where 0 and f are the incident and scattered wavelengths, respectively.)The intensity is a maximum when Q is perpendicular to the axis of rota-tion and it vanishes if Q is parallel. According to such measurements, thetunnelling frequencies at 1.2, 50 and 137 eV are attributed to sites C, B


7/35

6

and A, respectively.

Figure 2. Schematic representation of the arrangement of the three inequivalent methylgroups in the manganesediacetate tetrahydrate crystal at 14 K, after ref. 21

2.2. The density probability of methyl rotors

The neutron diffraction technique is complementary to X-ray diffraction.The scattering cross section of electrons is negligible and Bragg-peaks arisefrom scattering by nuclei. This technique is unique to determine the posi-tion of H-atoms because the nuclear cross-sections are on the same orderof magnitude for all nuclei. For example, the orientation of the methylgroups and of the water molecules were determined for manganesediacetatetetrahydrate (see Figure 2). 21

With the Fourier difference method, data obtained from single-crystalneutron diffraction provide a full view of the probability density of theH(D)-atoms. For this purpose, once the crystal structure has been deter-mined, Bragg-peak intensities can be calculated for an ideal crystal in whichthe scattering cross section of the H(D)-atoms of the methyl groups is set tozero. The difference from the original pattern contains specic informationon the methyl H(D)-atoms. Further Fourier back-transformation gives theprobability density distribution in direct space (for example, see Figure 3).

Because diffraction arises from coherent scattering by a large number of

atoms in a regular lattice, disorder appears as a perturbation of the latticeperiodicity averaged over space and time. It is impossible to distinguishstatistical and dynamical disorder. However, the dynamical orientationaldisorder of methyl group is so large compared to disorder arising from othervibrations that the angular probability density is quite representative of thewave function and can be somehow related to the potential.


8/35

7

2.2.1. Methyl rotors at low temperature

The Fourier difference maps for the three methyl groups in the manganese-diacetate tetrahydrate crystal at 14 K are presented in Figure 3. The

A B C

-2 -1 0 1 2-2

-1

0

1

2

( )

() -2 -1 0 1 2

-2

-1

0

1

2

( )

() -2 -1 0 1 2

-2

-1

0

1

2

( )

()

-2 -1 0 1 2

-2

-1

0

1

2

( )

()

-2 -1 0 1 2

-2

-1

0

1

2

L

( )

()

-2 -1 0 1 2

-2

-1

0

1

2

( )

()

Figure 3. Top: measured probability density distributions for the three inequivalentmethyl groups in the manganese diacetate tetrahydrate crystal measured at 14 K, afterref. 21. Labels A, B and C refer to those in Figure 2. Bottom: calculated maps (seetext). A: V 3 = 11 .5 meV, u 2r = 0 .05 A2 . B: V 3 = 17 .6 meV, u 2r = 0 .05 A2 , 2L = 0 .06rad 2 . C: V 3 = 49 .0 meV, u 2r = 0 .05 A2 .

different dynamics are immediately recognized by visual examination. Thehighest potential barrier (lowest tunnelling frequency) occurs for the methylgroup C whose protons are quite localized. The more delocalized protonsfor the methyl group A correspond to the lowest potential barrier (highesttunnelling frequency).

The map for the methyl group B is much more intriguing. Whereas thedistributions for A and C have clearly the threefold symmetry anticipated

for rigid methyl groups with xed axes, the probability densities at thethree proton sites are quite different for methyl B. Of course, as each protonis totally delocalized the total probability is the same for each site. Theprobability density is actually a comprise of several contributions arisingfrom both the methyl groups and the crystal lattice. It is necessary toanalyze these maps more thoroughly in order to highlight the rotational


9/35

8

dynamics.One can distinguish three contributions to the probability density: ( i)

the rotational dynamics; ( ii ) the deformation of the methyl groups due tointernal vibrations; ( iii ) the lattice dynamics and/or disorder. Because thetunnelling occurs at a very low frequency, lattice dynamics and internalvibrations can be represented with time/space-averaged distributions. Itis convenient to distinguish isotropic and anisotropic contributions. Theformer, as opposed to the latter, does not disturb the threefold symmetry.

i ) From the tunnelling frequency and Eq. (1) the temperature depen-dent angular probability density is determined by the Boltzmann distribu-tion:

P d (, T ) = n2n ()exp( E n /kT )

nexp(E n /kT )

(2)

ii ) Methyl groups are not perfectly rigid. The 8 degrees of freedom forinternal motions give rise to complex vibrations that can be representedwith a static distribution for the H-atoms. The projection onto the ro-tational plane gives an isotropic distribution with a Gaussian like prole.The variance (mean-square amplitude) of 0.01 A2 is largely temperatureindependent.

iii ) Similarly, lattice vibrations give rise to an isotropic Gaussian distri-bution for the C-atom with a variance of 0.02 A2 at a low temperature.This value may increase signicantly as the temperature is closer to themelting point. The mean square displacements in the rotational plane arefurther multiplied by a factor of 2 due to the distance between the rota-tional plane and the C-atom.

All together, the isotropic mean square displacement due to vibrationsu2iso (T ) is 0.05 A2 at a low temperature and the observed angular distri-bution is a convolution:

P (, T ) = P d (, T ) (, T ) ; (3)with

(, T ) =1

2iso (T )

12

exp ()

2

2iso (T )

(4)

where 2iso (T ) = u2iso (T )/r 2 . The rotational radius for protons r is 1 A.The radial distribution is gaussian in shape and can be represented as:

r (r, T ) =1

u 2iso (T )

12

exp (r r 0)

2

u2iso (T ). (5)


10/35

9

With this rather simple model, it transpires that the probability densi-ties measured for the A and C rotors are directly related to the square of the rotational wave functions, with minor contribution from other internaland lattice dynamics (see Figure 3).

For the B rotor, the dynamical probability density is convoluted withan anisotropic distribution of the angular coordinate L representing theorientational disorder of the methyl axis with respect to one of the potentialminima. The physical picture emerging from this map is quite unforeseen.In the effective potential V (L ), the slow methyl rotational is quenched,one of the minima is much deeper at one proton site than at the others.Indeed, this is not in conict with the threefold symmetry of the effective

rotational potential V 03 () averaged over the fast lattice coordinate L .

2.2.2. Methyl rotors at room temperature

Tunnelling bands are well observed only at very low temperature, usuallybelow 50 K. At higher temperatures, bandwidths increase and transitionsmerge into the huge peak for elastic scattering. The prevailing interpreta-tion is that the quantum dynamics switches to classical diffusion at hightemperature. Nowadays, density distributions obtained at any temperaturecompatible with the stability of the crystal shed light onto the rotationaldynamics at high temperature.

A B C

-2 -1 0 1 2-2

-1

0

1

2

( )

()

-2 -1 0 1 2-2

-1

0

1

2

( )

() -2 -1 0 1 2

-2

-1

0

1

2

( )

()

Figure 4. Measured probability density distributions for the three inequivalent methylgroups in the manganese diacetate tetrahydrate crystal at 300 K after ref. 21.Labels A,B and C refer to those in Figures 2 and 3.

At room temperature, the proton distributions for rotors A and B arealmost isotropic (see Figure 4). Nearly free rotation arises mainly fromthermal population of rotational levels close to, or above, the top of thepotential barrier (see Figure 1). (The rather modest anisotropic features


11/35

10

may arise from convolution with lattice displacements.) For the C rotors,partial localization of the protons due to hindered rotation is still observedand the effective potential barrier still exists at 300 K. In addition, a markedanisotropic contribution from the lattice dynamics is observed. The quan-tum dynamics is likely to survive even at room temperature and mixing of higher rotational states with phonons may take place.

3. Quantum rotational dynamics for pairs of coupled rotors

In the lithiumacetate dihydrate (CH 3COOLi,2H 2O or Liach7) crystal therotational dynamics is better represented with pairs of coupled methyl

groups.

Figure 5. Crystal structure of the methyl deuterated lithium acetate dihydrate afterref. 26. Left: view of the unit cell with thermal ellipsoids. Right: projection onto the(a, b ) plane. The lines along nearest neighbor interactions are guides for eyes. For thesake of clarity, the hydrogen atoms of the water molecules are hidden.

3.1. A honey comb network in 2D of quantum rotors

For the fully hydrogenated derivative the crystal symmetry ( Cmmm ) re-mains unchanged upon cooling from room temperature down to 1 K. 27 ,19

The methyl groups are arranged in innite chains, along the a crystal axis,of face-to-face coaxial pairs parallel to b (see Figure 5). The distances be-


12/35

11

tween consecutive axes of rotation ( a/ 2 3.4A) and between the methyl-carbon atoms within pairs (0 .3b 3.3A) are signicantly shorter than thevan der Waals radii and signicant interactions along the chains and withinpairs are schematically represented as a honey comb like structure in 2D,parallel to the ( a, b) planes. The inter-layer distances of 6.56A between(a, b) planes are much greater and interactions between methyl groups alongthe c axis is negligible.

3.2. A phase transition induced by methyl deuteration

-2 -1 0 1 2-2

-1

0

1

2

a

c

( )

()

-2-1 0

12

100

200

300

400

500

-2-1

01

2

14 K

( )( )

-2 -1 0 1 2-2

-1

0

1

2

a

c

( )

()

-2

-1 01

2

100

200

300

400

500

-2-1

01

2

40 K

( )( )

-2 -1 0 1 2-2

-1

0

1

2

a

c

( )

()

-2-1

01

2

100

200

300

400

500

-2-1 0

12

300 K

( )( )

Figure 6. Probability density distributions of the deuterium atoms in the methyl deuter-ated lithiumacetate dihydrate (CD 3 COOLi,2H 2 O) after ref. 26: at 14 K in the orderedphase Pman , at 40 K in the Cmmm phase, slightly above the transition, and, in thesame phase, at room temperature.

For Liach7 the methyl groups are totally disordered at any tempera-ture and the angular probability density is totally isotropic, even at 1 K.19Quite surprisingly, the methyl deuterated derivative (CD 3COOLi,2H 2O orLiacd3h4) undergoes a phase transition at 17.5 K, from Cmmm to Pmansymmetry. The ordering of the centrosymmetric pairs of methyl groups isclearly observed with the neutron diffraction technique (Figure 5). 28 ,26 Asall other atomic coordinates remain virtually unchanged, there is no signif-icant variation of the effective potential arising from heavy atoms and thephase transition is entirely due to mass effects on the quantum dynamics.

In the ordered phase at 14 K the localization of the probability density


13/35

12

distribution (Figure 6) is consistent with a rather high potential barrier.The two methyl groups in centrosymmetric pairs are indistinguishable andtheir images are superimposed in the Fourier difference. (The three peaksdo not mean that the two methyl groups are in the eclipsed conformation.)At 300 K, the disorder is fully established. At 40 K, there is a mixture of ordered and disordered methyl groups which means that the potential bar-rier does not vanish at the phase transition. The density is a superpositionof angular distributions for delocalized ( 50%) and localized deuteriumatoms ( 50%). The six maxima do not mean that the rotational potentialchanges progressively from threefold to sixfold symmetry and then to freerotation upon increasing the temperature. At 40 K deuterium atoms are

still partially ordered but face-to-face methyl groups are no longer indistin-guishable in the Cmmm phase.

3.3. The tunnelling spectrum

Early measurements of the proton spin-lattice relaxation have revealed al-most free rotation of the methyl groups. 29 However, the tunnelling bandsobserved in the 0.3 meV range (see Figure 7) are quite below the frequencyanticipated for almost free rotation. Moreover, since all methyl groupsin the crystal experience the same effective potential the rather complexspectrum must be interpreted in terms of dynamical correlation betweenindistinguishable quantum rotors.

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30

2

4

6

I n t e n s

i t y

( a . u . )

Energy Transfer (meV)

0.0

0.1

0.2

0.3

0.4

0.5

A

| E E

| AE

| E A

| E E

| AA

E n e r g y

( m e V

)

0.0

0.1

0.2

0.3

B

|1 A a|1 E a

|0 E i

|0 A

i

a

|0 E a

E n e r g y

( m e V

)

Figure 7. Left: the INS tunnelling spectrum of lithium acetate dihydrate at 1.5 K.Right: energy level schemes for a pair of coupled rotors with xed parallel axes (seetext), according to ref. 5 (A) or ref. 26 (B).

In the low-resolution limit, the threefold potential barrier for a singletransition at 0.240meV in Eq. (1) is about 8 meV. Simple examination of Figure 1 reveals that the proton density should be quite localized at a lowtemperature. This is in conict with the full delocalization experimentallyobserved.


14/35

13

For the CD 3 derivative the tunnelling frequency calculated at 33 eVwith the same potential barrier is actually observed at 12 .5eV. 6 Appar-ently, there is a signicant increase of the effective barrier to 11.6meV.This may contribute to the ordering of the CD 3 at low temperature (seebelow section 3.7).

3.4. The coupled pair model

The single rotor is clearly unable to account for the complexity of theobserved spectrum. Coupled rotation of pairs of methyl groups with par-allel axes suggested by the crystal structure was rst represented with the

Hamiltonian depending on the angular coordinates 1 and 2 :7,6

H 2 = B 2

21+

2

22+

V 032

cos31 +V 032

cos32 +W 12

2cos3(1 2)(6)

Each top experiences the same on-site potential V 03 and the couplingdepends only on the phase difference of the two rotors. In contrast to thetextbook case of coupled harmonic oscillators, this Hamiltonian cannot bediagonalized by simple transformation into normal coordinates. 9 On theother hand, numerical calculation of the eigen states and eigen functionsrequires very large basis sets with dimension N 2 , where N is the size of the basis set for each rotor. Early calculations carried out with a verylimited basis sets ( N = 9) 7 were certainly lacking of accuracy and deservesome reservations. Furthermore, as symmetric and antisymmetric states areno longer separate, the interpretation of the energy level scheme is quitecumbersome.

In order to analyze Eq. (6) the potential terms were ignored ( V 03 =W 12 = 0) and energy levels were distinguished according to their symme-try: the ground state ( A1A2 at E = 0), rst excited states correspondingto the transfer of one quantum to one of the two tops ( E 1 A2 and A1E

2

at E = B ) and higher states corresponding to the transfer of one quan-tum to each top ( E 1 E

2 and E

1 E 2 at E = 2 B ).7 Then, the energy level

scheme for non-zero potential terms was obtained by continuation of thezero-potential limit. It was concluded that AE states should occur inall cases (see Figure 7A). However, according to basic quantum mechan-

ics for two coupled and indistinguishable rotors, AE

states are not eigenstates. This speculation is certainly an error and leads to misinterpre-tation of the spectrum. According to the energy level scheme in Figure7A, the INS spectrum was decomposed into four components at 0.283,0.250, 0.214 and 0.030 meV. The estimated values (namely V 03 7.6 meVand W 12 17.0 meV)6 are totally in conict with the crystal structure.


15/35

14

Firstly, the on-site potential barrier quite similar to that for the single rotormust be rejected. Secondly, the negative value for W 12 favors the eclipsedconguration ( 1 = 2), whereas the crystal structure shows the staggeredconguration for face-to-face methyl groups at low temperature (see Figure5). (It is unlikely that the eclipsed conguration would be favored onlyfor the hydrogenated methyl groups.) Thirdly, we are not able to conrmthese calculations. Finally, among the 4 transitions predicted in Figure 7A,three of them are hot transitions and their intensities should decrease ata low temperature. At 1.5 K the relative populations of the |AE levelat 0.250meV and |EE levels at 0.500meV should be 710 2 and510 3 respectively. Therefore, the spectrum should be largely dominatedby the single |AA |AE transition. However, this is not conrmed byexperiments. 8

3.5. Symmetry adapted coordinates

In order to propose a more rigorous analysis of the dynamics of coupledpairs, it is necessary to pay attention to the crystal structure in which theface-to-face methyl groups are indistinguishable and form centrosymmetricpairs. Consequently, the dynamics are better represented with symmetry-adapted coordinates corresponding to in-phase i = ( 1 + 2) / 2 and anti-phase a = ( 1 2) / 2 rotation. In contrast, again, to the harmonic case,the choice of the symmetry adapted coordinates is much more constrained.

It is imposed by the periodicity of the rotors. The coordinates must becompatible with the threefold and sixfold periodicity for in-phase and anti-phase rotation. Any change of the periodicity would give a totally differentenergy level scheme.

In the strong coupling regime {|W 12 | |V 03 | in Eq. (6) }the equilibriumposition of one rotor with respect to the other is well dened and the twodynamics are separable. Then, the Hamiltonian is rewritten as:

H 2ia = H 2i (i ) + H 2a (a ) (7)

with

H 2i (i ) = B2 2

2i+ V 3 i2 cos3i

H 2a (a ) =

B2

2 2a

+ V 3 a2 cos3a +W 12 a

2 cos6a(8)

For in-phase rotation the coupling term vanishes and the dynamics isthat of a single rotor with moment of inertia 2 I r experiencing a threefoldpotential. If the coupling potential is strongly attractive the pair is in theeclipsed conformation and the effective potential is V 3i 2V 03 . Alterna-tively, if the staggered conguration is favored by a repulsive coupling term,


16/35

15

the effective sixfold potential should be very weak and give rise to nearlyfree rotation. In that case, V 03 in Eq. (6) cannot be determined experimen-tally. (This is the price for the separation of the dynamics but this is notreally a penalty since free rotation is always due to a delicate balance be-tween many contributions). For anti-phase rotation, the effective potentialcomprises a contribution from both V 3a and W 12 .

An important consequence of the symmetry adapted coordinate is thatthe rotational constant is divided by a factor 2 compared to the singlerotor. The maximum frequency for free rotation is now in the range 0 .3250.350meV for a pair of CH 3 and 0 .162 0.175meV for a pair of CD 3 .Therefore, the transitions observed around 0.270meV may correspond to

pairs rotating almost freely, in accordance with measurements of the spin-lattice relaxation. 29

The energy level scheme for tunnelling presented in Figure 7B is com-posed of one tunnelling state for in-phase rotation in a three-fold poten-tial ( |0E i ) and three states for anti-phase rotation in a six-fold poten-tial ( |0E a , |1E a and |1A a ). There are four fundamental transitions(|0A |0E i , |0A |0E a , |0A |1E a and |0A |1A a and twohot transitions ( |0E a |1E a and |0E a |1A a ). Even at 1.5 K,the |0E a level at 0.030meV is largely populated (80%) and temperatureeffects on the band intensity is marginal. However, if hot transitions werenot resolved from their parent bands, the attribution of the four transitionspreviously distinguished in the spectrum 7,6 should be straightforward.

3.6. Isotope mixtures

A more focussed view of the rotational dynamics was tentatively soughtfor with mixtures of CH 3 and CD 3 derivatives. 6 The physical and chemicaldifferences for the two analogues are negligible compared to intermolecularinteraction in the crystal, and mixtures are statistical distributions of pairs.If x and (1 x) are the relative concentrations of CD 3 and CH 3 , respec-tively, then, the relative concentrations of pairs CH 3CH3 , CH3CD3 andCD3CD3 are (1 x)2 , 2x(1 x) and x2 , respectively.

According to the energy level scheme in Figure 7A, the totally decoupledmixed pairs have four tunnelling levels: |AH AD , |AH E D , |E H AD and

|E H E D . Since |E H and |E D states are quite separate the splitting arisingfrom the relative rotation of the two tops ( E H E D and E H E D ) can beignored. This decoupling is the normal consequence of the removal of theindistinguishability for mixed pairs. Therefore, the spectrum is composedof degenerate transitions |AH |E H and |AD |E D for decoupledCH3 and CD 3 tops, respectively. If there were no change of the effective


17/35

16

potential upon isotope substitution, the frequency for a decoupled rotorshould be at the barycenter of the spectral prole for the coupled pair.

The tunnelling spectra of mixtures containing small concentrationsof CH3 (from 5 to 20%) in CD 3 environment were decomposed into amain peak at 0.130meV and a weak shoulder at 0.180meV attributed toCH3CD3 and CH 3CH3 pairs, respectively. 6 For an increasing concentra-tion of CH 3 rotors, the intensity at 0.130meV decreases, that at 0.180meVincreases to a maximum value for a concentration of 66% and then de-creases whilst new features centered at 0.250meV appear. Therefore, thereare at least 3 bands instead of the 2 anticipated with the model. The extraband was attributed to CH 3CH3 pairs surrounded by deuterated rotors.The effective potential barrier is increased and the band at 0.250meV isshifted downward to 0.180meV. However, each pair has four nearest neigh-bors methyl groups to which should correspond four different environmentsand effective potentials in isotope mixtures. If the different environmentsare not resolved, a smooth frequency shift should be observed, rather thana change of relative intensities for well separate transitions.

The interpretation of the spectra is hampered by the different crystalstructures for the CH 3 and CD 3 derivatives. The phase transitions maytake place at various temperatures depending on the isotope concentrationand it is difficult to distinguish dynamical and structural effects. In thiscontext, isotope mixtures of CH 3 and CH 2D derivatives are more appro-priate. There is no phase transition (this point deserves further conrma-

tion) and the perturbation of the effective potential, if any, is minimized.Moreover, a great advantage compared to CD 3 is that CH 2D rotors havesignicant scattering cross section. Therefore, tunnelling arising from eachpair CH 3CH3 , CH3CH2D and CH 2DCH2D can be observed. How-ever, owing to the different pair potentials for H. . . H, H. . . D and D. . . Datoms, 13 anti-phase tunnelling is cancelled for CH 2DCH2D pairs.

The spectra presented in Figure 8 provide more detailed informationthan those previously reported. 6 They were decomposed into rather narrowcomponents with relative intensities depending on the isotope concentra-tion. The relative frequency shift for each component is less than 10%,which suggests quite moderate changes of the effective potential. For thesake of continuity, the spectra of the mixtures containing 80% and 100%of CH3 were decomposed into 4 components between 0.150 and 0.300meV,instead of three reported previously. 7,8,6 Therefore, the symmetry adaptedcoordinates in Eqs (7) and (8) and the energy level scheme in Figure 7 arethe most realistic.

At a low concentration in CH 3 (20%) the most intense band at 0.166meV


18/35

17

0

1

2

3

4-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Energy Transfer (meV)Energy Transfer (meV)

20%

CH3

I n

t e n s

i t y

( a

. u

. )

0

2

4

40%

CH3

I n

t e n s

i t y

( a

. u

. )

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.30

2

4

60%

CH3


I n

t e n s

i t y

( a

. u

. )

-0.2 -0.1 0.0 0.1 0.2 0.3

80%

CH3

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3Energy Transfer (meV)

100%

CH3

Figure 8. INS spectra and tentative band decomposition of isotope mixtures of CH 3 COOLi,2H 2 O and CH 2 DCOOLi,2H 2 O at 1.7 K, after ref. 15.

and the weaker band at 0.185meV can be attributed to in-phase rotationof CH2DCH2D (B = 0 .246meV) and CH 3CH2D (B = 0 .281meV), re-spectively. The weak transition at 0.220meV is tentatively assigned toanti-phase tunnelling of CH 3CH2D.

For increasing concentration in CH 3 the band intensity around0.170meV decreases whilst the frequency shifts smoothly upwards to0.191meV for 100% of CH3 rotors. This behavior suggests unresolved tran-sitions due to CH 2DCH2D and CH 3CH3 pairs, respectively. We ex-trapolate the frequency for in-phase tunnelling of CH 2DCH2D pairs to be0.164meV. The band intensity at 0.030meV due to out-of-phase rotationof symmetric CH 3CH3 pairs and the intensities of the three bands at highfrequency increase simultaneously with the concentration in CH 3 .

With the unambiguous assignment of the in-phase tunnelling forCH3CD3 (0.130meV) and CH 2DCH2D (0.164meV), we assign the tran-sition at 0.241meV in Liac h7 to in-phase tunnelling of CH 3CH3 entities.The average of the estimated potential values for pairs containing H atomsis V 3i = (1 .82 0.15)meV. The rather modest dispersion of 10% mayarise from experimental errors, band shape analysis and variations of the


19/35

18

effective potential with the mass.For anti-phase rotation we have virtually no information for the deuter-

ated species. For CH 3CH3 pairs, we assign the doublet at 0.191 and0.216meV to |0E a |1E a and |0A |1E a , respectively. The split-ting of 0.025meV compares to the |0A |0E a transition observed at0.029meV. Similarly, we assign the component at 0.264meV to |0A |1A aand we suppose that the companion transition |0E a |1A a anticipatedat 0.235meV is a part of the main band at 0.241meV. Here we suppose thatall transitions have virtually the same intensity. The best t is obtainedwith the effective potential terms in Eq. (8) V 3a = 0 .19 and W 12 a = 31 .64,in meV units. As anticipated for staggered methyl groups, the on-site po-

tential is almost perfectly cancelled. The condition W 12 a V 3a is for wellseparate in-phase and anti-phase dynamics is largely satised.

3.7. Coupling between pairs and phase transition mechanism

The phase transition is related to long-range ordering of the methyl groupsin the honeycomb-like arrangement of acetate entities in the ( a,b) plane(see Figure 5). Owing to the great stability of the positions of heavy atomsthat remain virtually unchanged through the phase transition, the localpotential arising from the mean crystal eld is not changed. Therefore, thekey parameter for the effective coupling between adjacent pairs of rotors is

the angular mean-amplitude depending on mass and temperature (T ) =2 (T ) 1/ 2 with:

2 (T ) =

/ 3

/ 3

dP (, T ) 2 . (9)

For anti-phase dynamics coupling between adjacent pairs is negligiblecompared to the high sixfold potential barrier. For in-phase rotation, therather weak effective on-site potential can be attributed to interaction be-tween pairs. The Hamiltonian including coupling with the four nearest-neighbors of a pair (see Figure 9) can be written as:

H ci = B2

2

2i+

V 03 i2

(1 cos3i ) +4

n =1

V ci2

[1cos3(in i )] . (10)

In the weak-coupling (between pairs) regime, there is no correlation betweenthe angular coordinate of the pair ( i ) and those of the neighbors ( in ). The


20/35

19

effective potential can be thus expanded as:V 3 i

2 (1 cos3i ) = V 03 i2 (1 cos3i )+

4

n =1

V ci2 (1 cos3i cos3in sin i sin in ) .

(11)

Averaging over the distribution of phase differences gives then the effectivepotential:

V eff 3i V 03i + 4 V ci cos 3 (T ) , (12)With the potential functions determined at low temperature for in-phaserotation, iH 21 for CH3 pairs and iD 11 for CD3 analogues. Then,if the increase of the effective potential barrier from

1.9 to

4.5 meV

upon methyl deuteration is due to the change of (T ) we obtain 4 V ci 9.2meV and V 03 i 3.7meV. The opposite signs mean that the methylorientation favored by the on-site potential is opposite to the orientationfavored by interaction with next nearest neighbors. The effective potentialin Eq. (12) vanishes for 0 22 . This is the key parameter to account forthe phase transition upon deuteration.

for CH3 pairs, V eff 3i 1.8meV. There is no long-range correlation formethyl orientation at any temperature. The density probability derivedfrom the rotational wave functions is markedly localized. Therefore, thedelocalized angular density distribution observed with diffraction meansthat methyl group are statistically distributed among several orientations.

In this case, dynamical and statistical probability densities can be distin-guished. This was largely overlooked in previous calculations with quantumchemistry methods. 19

For CD 3 , the long-range interaction is dominant at 14K and the orien-tation of the CD 3 groups along the a axis is imposed by the interactionbetween methyl groups. In the absence of this methyl-methyl interactionalong a, the orientation of the pairs would be rotated by / 3. As the tem-perature is increased, the rotational levels at 2 meV are populated, themean angular amplitude increases and the long-range ordering decreases.The phase transition takes place when the thermal bath energy is similarto 2 meV or 22 K. This value compares favorably to the temperatureof 17.5 K for the phase transition.

4. Quantum rotational dynamics for innite chains of coupled rotors

In the 4MP crystal the rotation of methyl-groups is nearly free 30 ,31 ,32 butthe dynamics are quite different from those in lithiumacetate dihydrate and


21/35

20

this is a consequence of the different crystal structures for the two systems(compare Figures 9 and 5).

Figure 9. Schematic view of the structure of the 4-methylpyridine crystal ar 10 K, afterref. 33. Left: view of the unit cell. Right: projection onto the ( a, c ) plane showing theinnite chains parallel to a (along the zigzag line) or parallel to b (circles). For the sakeof clarity, all H-atoms are hidden.

In the tetragonal structure ( I 41 /a ) all molecules are equivalent. 34 TheCC rotational axes are parallel to the c axis. The C 2 site symmetry imposedisordering of the methyl groups and even at 10 K the angular probabilitydensity is virtually isotropic. 33 The structure remains unchanged until therst order phase transition at 254 K, close to the melting point at 276.8K.There is no further phase transition for the deuterated crystal. 35

The shortest intermolecular distances of 3.46 A between face-to-facemethyl groups parallel to the c axis and the next shortest methyl-methyldistances of 4.0 A perpendicular to the c axis, are quite similar to dis-tances in lithiumacetate dihydrate ( 3.3 and 3.4 A, respectively, see section3 and Figure 5). However, in contrast to the honey-comb like arrangement,

the equidistant methyl groups in 4-methylpyridine form two equivalent setsof orthogonal innite chains parallel to the a and b crystal axes. The zigzaglines in Figure 9 correspond to chains parallel to a and circles representintersections with the ( a, c) plane of chains parallel to b. There is only oneclose-contact pair in common for nearest orthogonal chains.

Within the tetragonal symmetry, to any particular orientation of a


22/35

21

methyl group corresponds 4 indistinguishable orientations obtained by sym-metry with respect to the molecular plane and by / 2 rotations. Thetwelve equivalent proton sites for a pair give a quasi-isotropic ring for thedensity distribution. Each methyl group experiences a virtually isotropicmean-eld corresponding to the pair potential averaged over all orienta-tions. The effective intra-pair potential is a constant and there is no cou-pling between collective excitations along a or b. Collective rotational dy-namics occur exclusively in 1D.

-2-1

01

2

0

20

40

60

-2-1

01

2

( )( )

0.44 0.46 0.48 0.50 0.52 0.54 0.560

20

40

60

0 . 4

7 2

0 . 5

3 9

0 . 5

1 7

I n

t e n s

i t y

( a

. u

. )


Figure 10. Left: probability density distribution of protons in the rotational plane of 4-methylpyridine at 10 K, after ref. 33. Right: the INS tunnelling spectrum of 4-methylpyridine at 0.5 K after ref. 13

These conclusions emphasize that a supercial examination of the intra-pair distances can be misleading. For example, Ohms and co-workers 34 con-cluded that the crystal structure implies strong correlation between the mu-tual orientations of adjacent methyl-groups which would have to be twistedby 60 with respect to each other and perform combined hindered rotations.Similarly, it is dangerous to utilize calculations of the effective potential,even with advanced quantum chemistry methods, for molecular-dynamicssimulations without caution. Quantum chemistry gives logically a largecoupling potential term for the close-contact pairs if the local symmetryis relaxed. Then, molecular dynamics simulation within the framework of classical mechanics give a false view of the methyl rotation. 36

The tunnelling spectrum of 4-methylpyridine is totally different fromthat of lithium acetate. Early INS measurements with rather modest res-olution have revealed a tunnelling transition at 0.5meV and this wasregarded as a manifestation of nearly free rotation. 30 This was the rstobservation of rotational tunnelling with INS and this remains the highesttunnelling frequency ever reported for a methyl group.


23/35

22

Further INS experiments with better resolution have revealed a complexspectrum composed of an intense band at 0.517(4)meV and weaker bandsat 0.472(4)meV and 0.539(4)meV (see Figure 10). 12 The splitting arisesfrom dynamical correlation and a model based on coupled pairs of rotors,with some similarity with lithiumacetate dihydrate, was rst proposed. 37

However, this is unlikely regarding the crystal structure.

0.30 0.35 0.40 0.45 0.50 0.550

2

4

6

8

10

125%

20%

26%

50%

65%

85%

100%

I n

t e n s

i t y

( a

. u

. )


0.1 0.2 0.3 0.4 0.50.0

0.4

0.8

1.2

1.6

2.0

CH 3CD 3

2.5K

4.5K

6.5K

8.5K

11K

I n

t e n s

i t y

( a

. u

. )


Figure 11. Tunnelling spectra of isotopic mixtures of hydrogenated and deuteratedmolecules, after ref. 12. Left: various concentrations of 4MP h 7 at 2.5 K. Right:temperature effect for the mixture containing 5% of 4MP h 7 and 95% of 4MP d 7 .Dash lines are guides for eyes.

The INS spectra of isotopic mixtures containing hydrogenated anddeuterated molecules (4MP

h7 and 4MP

d7 , respectively) show dramatic

frequency shifts (up to / 30%), depending on concentration and tem-perature (see Figure 11) 12 which cannot be explained with any coupled pairmodel. 38 These experiments establish the collective nature of the methylgroup dynamics and this was represented with the quantum sine-Gordontheory. 12

4.1. The quantum sine-Gordon

This section is a summary of the relevant equations. More details werepresented in refs 12,13,16. The Hamiltonian for an innite chain of coupledrotors can be written as:

H =j

2

2I r 2

2j+

V 0

2(1

cos3i

j) +

V c

2[1

cos3i (

j +1

j)] , (13)

where j is the angular coordinate of the j th rotor in the one-dimensionalchain with parameter L. V 0 is the on-site potential which does not dependon lattice position, and V c is the coupling (strain energy) between neigh-boring rotors. The index i = 1, 2 ... determines the potential periodicity


24/35

23

compatible with the C 3v symmetry of the methyl-groups. In the strongcoupling (or displacive) limit, when j +1 j is sufficiently small, Eq. (13)is equivalent to the sine-Gordon equation:

H j

2

2I r 2

2j+

V 02

(1 cos3ij ) +(3i)2 V c

4(j +1 j )

2 . (14)

If variations of from site to site are small, then the site in-dex can be replaced by a continuous position variable x (see Fig-ure 12). Kinks, rotons (phonons) and breathers are the elementaryexcitations. 39 ,40 ,41 ,42 ,43 ,44 ,45 ,46 ,47 ,48 ,49 ,50 ,51 Since the kink density vanishesat low temperature and phonons are beyond the spectral range under in-vestigation, these excitations can be ignored.

x

4 /32 /3

-4 /3-2 /3

0

kink rotonEnergy surface

breather(doublet) annihilation

(rotons)

kink

antikink

4 /3

2 /3

-4 /3

-2 /3

0

x

Figure 12. Artistic view of the sine-Gordon excitations.

The breather or doublet is a well-dened elementary nonlinear excitationwith long life-time and as such behaves very much like a particle. In theclassical regime, the waveform is:

vB (x, t ) = 4Arct

20 2B

1sin B ( t vx/c 20 )1 v 2 /c 20

cosh ( x

vt )1

2B /

20

d1 v 2 /c 20with

20 =V 0 (3 i ) 2

2I r ;c2

0= V c L

2 (3 i ) 2

2I r= 2

cL2 ;

d = c00 .(15)

The breather envelope travelling at velocity v is oscillating harmonically atfrequency B which may vary continuously from 0 to 0 . Breathers can beviewed as kink-antikink bound states (Figure 12).


25/35

24

In the quantum regime 40 ,41 the classical breather yields a discrete spec-trum of mass at rest qM 0B,l depending on the quantum number l:

qM 0B,l = 2qM 0K sin

l (3i)2

16 1 (3 i ) 2

8

with l = 1 , 2,...

Francois Fillaux et al- Experimental Studies and Theory of Nonlinear Rotational Dynamics in the...

Documents

Transcript of Francois Fillaux et al- Experimental Studies and Theory of Nonlinear Rotational Dynamics in the...