H-Bonds of adipic acid crystals III : measurement of the various anharmonic couplings of the O-H (O-D) stretching bands

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H-Bonds of adipic acid crystals III : measurement of the various anharmonic couplings of the O-H (O-D)

stretching bands

G. Auvert, Y. Maréchal

To cite this version:

G. Auvert, Y. Maréchal. H-Bonds of adipic acid crystals III : measurement of the various anhar- monic couplings of the O-H (O-D) stretching bands. Journal de Physique, 1979, 40 (8), pp.735-747.

�10.1051/jphys:01979004008073500�. �jpa-00209158�

LE JOURNAL DE PHYSIQUE

H-Bonds of adipic ^acid crystals ^{III :}

measurement of the various anharmonic couplings

of the O-H (O-D) stretching ^bands (*)

G. Auvert (**) and Y. Maréchal (***)

Centre d’Etudes Nucléaires de Grenoble, Département de Recherche Fondamentale, ^Section de Résonance Magnétique, ⁸⁵ X, 38041 Grenoble Cedex, ^France

(Reçu ^le 7 février 1979, accepté ^{le 23} avril 1979)

Résumé. 2014 Nous mesurons l’évolution avec la température ^des premiers ^moments de la raie 03BDs(O-H... ^{O ou} O-D ... O) ^{de l’acide} adipique ^{en cristal et nous} calculons les variations théoriques correspondantes ^en supposant que la vibration 03BDs d’une telle liaison isolée est fortement couplée à la vibration 03BD03C3(O-H

O) de la même liaison et aussi à des combinaisons binaires d’autres vibrations. Nous supposons aussi _{que le} moment de transition qui

est à l’origine ^{de cette bande} 03BDs présente ^une anharmonicité électrique couplant aussi 03BDs ^avec 03BD03C3 ^non négligeable,

et nous avançons des arguments pour négliger ^en première approximation ^les interactions harmoniques

de résonance entre deux vibrations _03BDs voisines, interactions qui ^{ne donnent aucune variation} thermique impor-

tante. En comparant ^{théorie et résultats} expérimentaux ^nous pouvons déterminer les valeurs de ^{tous ces} couplages.

Le couplage 03BDs- 03BD03C3 décroît avec la température, ^ce qui correspond ^{à une} élongation ^{de la} distance moyenne O ... O de 0,03 Å ^{entre 10 K et 300 K} que ^nous attribuons à un couplage ^{de 03BD03C3 avec des modes de} vibrations de plus ^basse fréquence de la liaison hydrogène. L’énergie ^{totale des} couplages ^{de résonance} (80 ^cm-1 pour l’acide adipique ^H

et 50 cm-1 pour l’acide adipique D) ^est égale à celle que l’on peut ^{calculer en} supposant que ^ces couplages pro- viennent de la variation du moment d’inertie de l’atome d’hydrogène ^H par rapport ^{à l’atome} d’oxygène ^O quand

la longueur de la liaison O-H change, ^ce qui ^nous suggère que ^ce simple ^mécanisme géométrique ^{est à} l’origine ^des

résonances de Fermi, ^{et nous} permet de calculer simplement ^leur énergie. Nous pouvons aussi ^mesurer la grandeur

de l’anharmonicité électrique, qui ^{semble assez} importante.

Abstract.

We show the evolutions with temperature ^{of the first moments} of the 03BDs bands (O-H... ^{O or O-D... O)}

of adipic ^acid crystals ^{and we} establish theoretical relations giving the values of these first moments with the general assumption ^{that the} 03BDs vibration of a single ^{H-bond is} strongly coupled ^to the 03BD03C3 ^vibration (O-H ... O) ^{of the same}

bond and is also coupled ^to binary combinations or overtones of other vibrations. Both these couplings ^{are anhar-}

monic. We also assume that the transition moment at the origin ^of the 03BDs band shows non negligible ^electrical anharmonicity coupling 03BDs with 03BD03C3 and ^we give arguments ^{for our} neglecting, ^{in a first} approximation, ^harmonic

resonance terms between two neighbouring 03BDs vibrations which do not give important temperature effects. From the comparison ^of experimental and theoretical values of these moments we determine the magnitudes ^{of all}

these couplings. The 03BDs-03BD03C3 coupling ^{is shown to decrease with} temperature, ^which corresponds ^{to an} increase of the average O ... O distance of about 0.03 Å between 10 K and 300 K, which we attribute to a coupling of 03BD03C3

with lower frequency vibrations of the H-bonds. The total energy of anharmonic ^resonance interactions (80 ^cm-1

for H-adipic ^{acid and 50 cm-1 for} D-adipic acid) ^{is shown to} be that which we can calculate if we suppose that these interactions originate from the variation of the moment of inertia of the H atom with respect ^{to the O atom} when the O-H length ^{vibrates. This} strongly suggests ^{that this} simple geometrical ^mechanism might ^{be at the} origin

of Fermi resonances, thus defining ^a simple procedure for their calculation. Finally ^the magnitude of electrical

anharmonicity is also measured and shown to be important.

Classification

Physics ^Abstracts

33.10

33.20E - 35.20G - 78.30

(*) This article is part ^{of a} thesis submitted at the « Université Scientifique ^et Médicale de Grenoble ».

(**) C.E.A.

(***) ^C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004008073500

1. Introduction. ^{- The intense} stretching ^bands

vs(O-H ...0) ^{of H-bonds,} which show characteristic features which are now well-known [1], ^{are a} powerful

source of information on the dynamical properties

of H-bonds, that is, ^on the evolution of these bonds

on a timescale of 10-1°-10-14 s. It seems therefore worthwhile to have clear and precise descriptions ^of the v. ^{modes, in order to be able to} get ^{from the}

corresponding ^IR bands information on these dyna-

mical properties. ^The amplitude of vibration of the H atom in H-bonds is expected ^{to be} relatively important ^{so that} anharmonicity ^{is also} expected

to be important ^{in the} description ^{of these} vs modes.

Indeed it has been shown [2, 3] ^{that two kinds of}

anharmonic couplings should be considered in the

description of these modes. The first one couples vs

with the stretching ^vibration v a( ô-Ii... 0) ^{of the}

H-bond. This kind of coupling ^is fundamental,

as it exists in the ground ^{state of the} vs vibration and can therefore have important consequences in the thermal properties of H-bonds. It has been quanti- tatively ^described [4, 5, 6, 7] ^{and the} energies ^involved

have been estimated in the simple ^{case of} cyclic

H-bonded dimers of carboxylic ^{acids in the} vapour

phase [8, 9], and in the case of simple ^H-bonded

systems ⁱⁿ liquids [10,11,12], gases [13] ^or crystals [14, 15, 16]. The second kind of coupling ^{is due to some}

resonance interactions (often called Fermi resonances) between vg ^and binary combinations of other modes.

As these interactions are only apparent in the first excited state of v,,, ^{which is} thermally inaccessible, this kind of coupling ^{appears more as an inconve-}

nience than as a fundamental property of H-bonds.

Nevertheless it is now widely accepted ^{that these}

two anharmonic couplings ^{have an influence on the} shape of vs [18, 19, 20]. ^Recent measurements of the

integrated transition probabilities ^P of vs ^{bands have}

also revealed unexpected isotope ^effects [21, 22]

which seem to be due to the _presence of a special

kind of electrical anharmonicity [23].

If we wish to have a precise description ^of vs, which could supply information on the dynamical properties of H-bonds it _appears interesting ^{to measure}

with precision ^the magnitudes of these various anhar- monicities. This has never been done, except ^for the measurement of the _Vs-Va mechanical coupling,

and we propose ^to describe in the present ^{work the}

measurement of these anharmonicities from the variations of the first moments of _vs with temperature and after deuteration. We shall therefore first describe the experimental results which ^we obtained on

adipic ^acid crystals ^{and we shall} compare them with those theoretically predicted assuming ^{that the motion}

of the H atom of an H-bond is governed by ^these

various anharmonic couplings. ^{As we discuss} experi-

mental results on H-bonds with a somewhat new

language ^we give ^the theory ^{in more} details than in a preceding ^article concerning ^{formic acid} crystals [24]. The différence between these ^two

articles is that in this previous ^{case we} had discarded anharmonic resonance interactions (Fermi resonances)

which we shall introduce in this paper. We shall however discard harmonic resonance interactions between two neighbouring vs ^modes, justifying ^this

on the basis of previous ^work.

The experimental ^{results which we} shall describe

are those concerning adipic ^acid crystals ^{which have} proved ^{to be} good models of well defined and rather

simple ^{H-bonds. In these} crystals polarized IR spec- troscopy ^has already given interesting information

on vs modes [25]. The H-bonded dimeric cycles . -(COOH)2 ^{of these} crystals ^{are the same as those}

found in the dimers of carboxylic ^acid vapours which we have already ^studied [8, 9]. ^The possibility -

of studying ^these cycles ^at very low temperatures is an illustration of the interesting extension which these crystals allow and which ^we shall relate here.

In the second section we shall describe the experi-

mental variations of the first moments of the v. ^bands

of the crystals ^with temperature ^{and we shall} give ^a qualitative analysis of these variations. In the third section we shall give ^a theoretical description ^{of the}

motion of an H atom in an H-bond which will be

as general ^as possible. ^{In the} fourth section we shall then make hypotheses which will be discussed, in order to compare theory ^and experiments. Finally

the results of this comparison, that is the measure- ment of the magnitudes of the various anharmoni- cities, will be discussed in the fifth section. These last two sections are, in our opinion ^most important

because they give ^{us the} opportunity ^to clearly

define the problems encountered during the elabo- ration of a quantitative theory ^{of the} dynamics ^of

H-bonds and allow us to suggest ^future experiments

which should give ^{us more} precise information on

this dynamics.

2. Expérimental results and qualitative analysis ^of

thèse results.

We shall describe in this section the

experimental variations of the first moments of the Vs bands of H and D-adipic acids with temperature.

In a preceding ^article [25] ^we have shown the shapes

of these bands at 10 K and have described the experi-

mental conditions under which the spectra ^were obtained. As these bands are rather well defined, particularly ^with respect ^{to their} baselines, ^{we can} compare with ^a good precision ^their integrated

transition probabilities ^{P, centres of} gravity 1 ^and

variances at various temperatures. ^These quantities

are defined by ^the equations :

where log 1011 is the absorbance of the v., ^{band at}

wavenumber v, and 1 the thickness of the sample

crossed by ^{the IR beam. In a} preceding ^article [25]

we have shown that the _v. band is mainly polarized

in the _a, c plane and that its component along ^{the b}

axis has an intensity ^{which is} always less than 6 %

of that in the _a, c plane ^{and is too small to be} precisely

studied with ^our thin samples. ^{This is true at all}

temperatures and all the quantities ^{which we shall}

consider are those of the component of vs ^{in the a,}

c plane. ^{In this} plane ^all cycles ^have parallel pro-

jections which allowed us to perform ^rather precise

measurements of the directions of the polarizations

of the different bands. From these measurements we could conclude that the polarization ^{of the} v., band is not constant but varies inside this band

by ^{15o at 10 K} (9- ^{at 300} K) ^{which led us to} suspect strongly that the low frequency modes of the H-bonds

might ^{not be as} simple ^as they ^are usually supposed.

We shall see in this article that we reach the same

conclusion after the analysis of the evolution of the first moments of the _Vs bands with temperature. ^The

experimental values of these quantities ^{P, C-0 and}

are shown at various temperatures ⁱⁿ figures 1, ² and 5. These quantities show characteristic features which we shall briefly describe before analysing

them more precisely.

2. 1 TRANSITION PROBABILITIES.

When looking

at figure ^{1 one can note two} interesting point. ^First

the values of the P’s decrease with temperature by

about 20 % between 10 K and room temperature.

Fig. 1.

Variations of the integrated transition probabilities ^{of the}

v. bands of H and D-adipic acids for crystals having ^{the same}

thickness. Experimental points ^{are shown} by symbols (0). ^The

curves represent ^{the best} interpolations ^of experimental points.

This is true for H-bonds as well as for D-bonds.

As the component of v. along ^{the b axis is} always

less than 6 % ^{that in the} a, ^c plane, this decrease of the P’s ^with temperature ^{cannot be} attributed to

a transfer of intensity between the two components.

Moreover it seems that the integrated ^transition probabilities ^{of the} components ^{of the} vs bands along

b also decrease with temperature. ^{This ensures that} the P’s effectively decrease with temperature. ^The second interesting result is the value found for PH/Pl (an ^{index H or D will} always ^{refer to an H-bond}

or to a D-bond in the ollowing) ^{which is} equal ^{to 2}

at all temperatures. ^{This value is} significantly greater than the value expected ^{for an} harmonic oscillator in vs(J2) ^{which is also} the value for an harmonic oscillator in _Vs having ^a frequency depending ^on

the coordinate of a low-frequency oscillator such as

Va (which ^{leads to a} Vs-Va anharmonic coupling).

This value (2) is however the same as that found in dimeric cycles ^of carboxylic acids in the _gaseous

phase ^{which was} explained ^{with the} supposition ^that

the part ^{of the moment which is at the} origin ^{of the}

0 ---> 1 transition in vs ^{increases when the 0 ... 0}

distance decreases [23]. ^{This is} equivalent ^{to intro-} ducing ^an electrical anharmonicity, because the

development of the transition moment is no longer

linear in the coordinates. It explains why stronger H-bonds have more intense vs transitions (they ^have

shorter 0 ... 0 distances) ^and why ^{the ratio} PH JPD ^is

greater ^than J2 ^{(D-bonds have longer 0}

^{0 dis-}

tances than H-bonds). ^It predicts however that the P’s increase with température ^{if one} supposes that the mean 0

0 distance does ^not _vary with tempe-

rature. An increase of this distance with tempe-

rature seems consequently ^a necessary condition to inverse this tendency. ^This proposition ^{that the}

0...0 mean distance increases with temperature is not so new as it might appear ^at first sight, ^{as it}

has been already considered for various H-bonds [26, 27]. At the end of this section we shall indicate how this supposition ^can be introduced quantitatively. , ^t

2.2 CENTRES oF GRAVITY. - In figure ^{2 the} experi-

mental values found for WH ^and Wo ^{are shown at}

various temperatures. ^{The two} interesting properties concerning these values are that the w’s shift towards

higher values when the temperature increases and that

WD ^{shifts to a} greater ^extent than - ^{This shift,}

which seems to be a general property ^{of H-bonds} [19]

can be easily explained by ^the supposition ^{that the}

mean 0...0 distance increases with temperature, because the mean frequency S) ^{of the} vs vibration increases when the equilibrium 0...0 distance increases [28].

2.3 VARIANCES. - The experimental ^{values of}

the variance a are shown in figure ^{5. As} expected,

these variances which represent ^the half widths of

the best Gaussian functions approximating ^{the true vs}

bands, increase with temperature. ^A surprising ^obser-

Fig. ^2.

Variations of the centres of gravity ^{of the} Vs bands of H and D-adipic ^acids. Experimental points ^{are shown} by symbols (0).

The curves represent ^{the best} interpolation ^of expérimental points.

vation is that they actually weakly increase : for H- bonds as well as for D-bonds the J’s do not increase of more than 13 % between 10 K and 300 K. If we

suppose that vs is anharmonically coupled to Va

through ^a dependence ^{of the} frequency ^{w of the} v.

harmonic vibration on the 0...0 distance [4],

which seems now widely ^{admitted, we can} rapidly

estimate the relative variation of J with T. If Q ^is

the coordinate of the va vibration, the 0 ... 0 distance is equal to Q plus ^a constant term so that QZ which is equal ^to ( (ro - ^co »)2 > ^becomes

if we admit that co varies linearly ^with Q (a ^{non linear}

variation would have the effect of giving ^{an even more} important variation of ^u with T, ^{in a first} approxi- mation). Supposing ^{that the} square of the variance in Q (that ^{is the} quantity S2 ⁼ ( (Q - ( Q »)2 >) ^is

that of an harmonic oscillator having ^a frequency Q,

we see that a should _vary as 2 ^{Z - 1 with tempera-}

ture, ^where Z-1 is 1

exp - hQlkT. Taking

hQ ri 160 cm-l [29] gives ^a relative variation for

a of at least 60-70 % between 10 K and 300 K. Even if this coarse estimation exagerates ^the discrepancy

with experiments because it neglects the contribution due to Fermi resonances which will be shown ^to be

independent ^of temperature, it shows that the experi-

mental variations of the Q’s with temperature ^are

intriguing. ^Our preceding suggestion that Q >

(or ^{the mean 0}

⁰ distance) increases with tempe-

rature cannot explain directly why ^the experimental

J’s increase so weakly ^with temperature, ^{as the J’s} do not depend on Q) but rather depend ^on ( (Q - ( Q »2 > ^{which is} independent of 0.

However the or’s of our coarse estimation also depend

on the extent of the coupling ^{of vs} with v. ^{which is} represented by ^the quantity (dw/dQ ), and this is also true fort Q > (see eq. (21) ^{for a more} precise ^treat- ment). ^The supposition that it is this coupling ^which

decreases with temperature ^{is able to} explain, ^{at least} qualitatively for the moment, why ^{the Q’s so} weakly

increase with temperature. ^{It will also} define the origin

of the dilatation of the 0

0 distance with tempe-

rature. Before introducing ^these suppositions ^{in a}

more quantitative ^{treatment let us} point ^{out this last} supposition is different from that of Romanovski and Sobczyk [26] ^who supposed that the 0...0 distance increases with temperature because the

potential governing the Va vibration has anharmonic terms in _v,,. In their formulation the coupling ^between

vs and Va ^{has no} special ^{reason to} decrease with tem-

perature, leading ^{to a} prediction of the variations of the a’s with temperature ^at variance with experi-

mental results. It is also different from the conclusion of Bournay and Robertson [30] ^on the H-bonds of self associated methanol in solution, ^which attributes the shift of w with T to a non linear variation of co(Q)

with Q. ^This supposition will however also be unable to reproduce the weak variations of the a’s with T in the present ^{case of} adipic ^acid crystals.

3. The 0 ---> 1 transition in v s : general theoretical considérations.

In this section we shall calculate, using general assumptions, ^{the first moments of the}

Vs band of a single H-bond. The moment Mn ^{of order}

n of an optical transition is equal ^to the coefficient of (it)"In ! ^{in the} development of the Fourier trans- form C(t) of the transition probability p(v) ^{of the} sample. ^If log 1011 is the absorbance of the spectrum

at wavenumber v, we have the relations

As a consequence of the fluctuation-dissipation

theorem the quantity C(t) ^{for an} IR transition is :

where Me ^{is the} component along the electric field of the electrical dipole ^moment of the whole set of H-bonds, and X is the Hamiltonian for these H-bonds.

As the H-bonded cycles ^{are well} separated ^{we shall}

consider a single cycle only. ^In this article we shall

even make a more severe restriction and shall consider

a single H-bond whose dipole ^{moment will be} Me

and Hamiltonian Je. In those cycles ^{the resonant}

interaction between two neighbour Vs vibrations can

be appreciable and has been found to be of the order

of 100 cm-1 in the case of carboxylic acids in the

gaseous phase [4, 8, 9], and in the case of oxalic acid

crystals [16]. The manifestation of this interaction

is found in its contribution to the absolute value of (o. However since this interaction is quadratic (or harmonic) in the coordinates of two neighbouring vs vibrations it does not give any special isotope ^effect.

Also the temperature shift of ro with T due to this interaction is expected ^{to be} negligible. ^{We have}

checked that its influence on a is negligible, using preceding theories which took this interaction fully

into account [4, 24]. ^It is clear from _eq. (1) ^{that P}

cannot depend ^{on this} quantity. ^{As we are} considering

the variations of P, ^{5) and 6 with} temperature ^and deuteration, ^we shall therefore omit this interaction,

which will make the following ^treatment simpler.

In a forthcoming ^{article we shall treat the case of}

two interacting ^H-bonds [31] and apply ^{it to a} species (imidazole) ^{where we have measured the} energy of this interaction with precision. We may anticipate

the results of this more elaborate theory by saying

that the introduction of this interaction will not

significantly modify the results described in this article.

Let q be the coordinate defining ^{the vs vibration}

of this single H-bond which is govemed by ^{the Hamil-}

tonian Je. Vs ^{has a} typical frequency ^{of about}

2 500 cm-1. It is coupled ^{to some low} frequency

vibration ( ²⁰⁰ cm-1) ^{of the} H-bond which we

shall define by ^the coordinate Q ^which represents

the Va vibration of this H-bond. It is also coupled

with combinations of modes which are nearly ⁱⁿ

resonance with _vs. These modes belong ^{to the same}

molecule as the H atom of this H-bond and are

defined by ^{the set} of coordinates _qô. The existence of these two kinds of couplings ^{is now} widely accepted [18, 19] ^{so that we have not introduced}

up ^{to now} any special hypothesis. ^The potential corresponding ^{to the last} type of interaction can be written with the general ^{form :}

This part ^{of the} potential ^has the effect of trans-

ferring ^a single excitation in vs ^{to a} binary ^excitation

in _{q /J} and _qô, (or ^{to an overtone 2} qa). ^{If the} energy (J)/J + úJ/J’ of this binary excitation is equal ^{to the fre-}

quency ^co of _vs, which is the condition for resonance to occur, these terms can have appreciable ^effects.

We _may then write the total Hamiltonian of the H-bond considered as :

Mu is the effective mass of the Q ^{mode and} h(q, Q)

is the Hamiltonian describing ^the v, vibration which

parametrically depends ^on Q. ^At present ^{we shall not} precise further the nature of this dependence, ^which

is at the origin ^{of the} Vs-Va coupling, ^{but we shall}

use the property ^{that the} frequency ^of the v, ^{mode is}

much lower than that of the _v. and qô ^{modes to define}

an adiabatic representation ^{for the} ground ^{states of}

these modes which are the only thermally ^accessible

states. It has been indeed shown [4, 7] ^{that non adia-}

batic terms have a negligible ^{effect on these states}

whose wavefunctions tp"(q, qa, Q ) ^{can then} be written :

In eq. (4) ^we implicitely suppose that the low fre- quency vibration Q ^{is a} pure quantum ^{motion. We} also make the reasonable approximation ^{that the} different qô vibrations do not depend ^on Q. ^{In order}

to simplify ^the equations ^{we shall not write the}

wavefunctions Fo of these vibrations and the subscript

ô signifies ^that integration ^{should be} performed ^over

the set of q,, ^{in their} ground ^{states. The uth function} ocu(Q) ^{of the} Va vibration is then an eigenfunction ^of

the Hamiltonian Ho ^{which is} equal ^{to :}

where qo(Q ) ^{is the mean value} of q ^{in its} ground ^state and is defined as go(q, Q ) I q 1 90(q, Q ) )q ^{and ea is}

the total _energy of the qa vibrations in their ground ^{states. The} dipole ^moment of the considered H-bond can

then be written in the form :

In eq. (6) ^{we have} only written the term which is responsible ^{for a 0 --+ 1} transition in v.. Its coefficient

M’(6) ^{can however} depend ^on Q which defines then a special type of electrical anharmonicity ^{which we have} already mentioned. In these conditions the correlation function C(t) ^of eq. (1) ^{is :}

with

lfJ1(q, Q ) is the first excited function of h(q, Q ), ^{It allows us to} define the quantity M(Q ) by the relations :

In order to clarify ^the meaning ^{of the} quantity A(Q) ^{we shall} give its value in the special ^case where v. ^{is an}

harmonic oscillator of mass m and frequency ^{ro which} depends ^on Q. ^{In that} case we see that A(Q) ^is equal ^to

JFi/2 mro(Q) ^{which is a function} practically independent ^of Q. ^{In the} following discussion we shall not consider

specially ^this case, in order to be more general, ^{but we} shall consider it in some digressions ^{which we shall make}

later to precise ^{some ideas.}

When _vs is in its first excited state ({Jl(q, Q ), ^the Hamiltonian goveming the v. vibration is no longer ^the Ho

.

of _eq. (5) ^but Hl, which is defined as :

This equation defines the quantity co’(Q) which will be of fundamental importance ^{in this} theory. ^It obeys

the relation

where

In the special ^{case where} h(q, Q) is the Hamiltonian of an harmonic oscillator in q, ^whose frequency ^go

and equilibrium position qo depend ^on Q, ^{we have} ql(Q) ⁿ qo(Q) (the ^{different states} of an harmonic oscillator have all the same equilibrium positions) ^and (J) + (Q) =- w(Q). These relations which are true in this special

case lead us to neglect ^{in the} general ^{case the last term of} eq. (10), which is indeed not a great approximation

for intermediate strength H-bonds where tunnelling ^{of the H atom can be} neglected. ^{In this} general ^{case we}

obtain from _eq. (7) :

where [A, B] ^{is the} commutator AB-BA. The substitution

has been made, ^so that the brackets ( ) ^without any reference to the coordinate to be integrated ^{denote the}

thermal _average, and we have used the relation :

which can be deduced from eqs. (3) ^and (9). ^{We have then :}

It is of interest to discuss further _eqs. (13) ^and (14)

which are not as complicated ^as they appear ^at first

sight. The first of these equations, ^which gives ^the

value of P, clearly indicates that the integrated ^transi-

tion probability ^of vs ^cannot vary with temperature if no electrical anharmonicity ^is present (that ^{is if} M(Q ) ^{does not} depend ^on Q). ^{We can} altematively

write P as :

The first term in (15) corresponds ^{to the mean value}

of M, whereas the second one describes the fluctua- tions of M around its mean value. This second term is temperature dependent ^{but will} hardly depend ^on

deuteration (except ^{for a} coefficient .J2 ^{coming from}

Ã(Q) ^of eq. (8)) ^because the v. vibration is almost insensitive to deuteration, ^{and the} quantity ^{M’ of}

eqs. (6) ^and (8) ^is independent ^{of the} proton ^mass.

This shows that having ^a ratio for PHI PD equal ^to

2 implies that MH(Q) > ^is greater thane MD(Q) >

which can only ^{occur if the mean value} ( Q > ^{is not}

the same in a H-bond and in a D-bond.

In eqs. (13) ^and (14) ^the expressions for ro and a are somewhat more complicated ^but they ^{can neverthe-}

less be easily analysed. (J) is the sum of two contri- butions which originate ^{from the} Q dependence ^of

both (J) + (which is the difference of energies ^{of the}

two lowest levels of vs) ^{and M} (which is the transition

dipole moment). ^A remarkable property of Zi is that it is independent ^of anharmonic resonance inter- actions even in the _presence of electrical anharmoni-

city. A similar conclusion has been reached _pre-

viously [36] ^{for the case} where electrical anharmoni-

city ^is negligible (that ^{is when} M(Q ) hardly depends

on Q), ^{which leads to the simple result that} W = co’(Q) >. ^{Even when we cannot} neglect ^this

electrical anharmonicity ^{we can} still obtain simple expressions ^{for C-0 if both} (J) + (Q) ^and M(Q ) ^show

linear variations with Q, which is often quite ^a good approximation. ^{In this case we can} then write :

which gives :

where

The quantity s2 which is the _square of the variance

or quadratic fluctuation of Q, ^{will be an} important quantity ^{in the} following discussion. Eq. (17) ^shows

that in the presence of electrical anharmonicity

S not only depends on Q > ^{but also} depends ^on

S2 which is a function of temperature ^{and will conse-}

quently ^{introduce a T} dependence ^{of 0153.}

Eq. (13) shows that U2 is the sum of two indepen-

dent components ut. ^and u;-a. UF is the total contri- bution of anharmonic resonance interactions to the width of _Vs and _Us-a is the contribution of the _Vs-Va

coupling. In eq. (14) ^the quantity

which appears in ^{Op can} be considered as hardly dependent ^on Q (for ^an harmonic oscillator in Vs

it is equal ^{to 3} Fi/2 mro(Q) ^{which we can write as}

The _average value of the quantity w(Q ) - Ct)(0)

will be of the same order of magnitude ^{as the shift}

of ro with T that is, less than 50 cm-1 which gives ^a

small contribution when divided by co(0) which is of the order of 2 500 cm-1). ^{This means that Op will be} hardly temperature dependent. ^{This is not true for}

Us-a which _appears as a quadratic ^{sum of three terms.}

The first _one, which will be shown to be predominant represents the contribution of the Q dependence ^{of w+.}

When electrical anharmonicity ^{can be} neglected (M(Q ) ^almost independent ^of Q ) ^{this term is equal} to (ro+(Q) - w+(Q) »)2) ^{which is equal to} (dw+/dQ)2 S2 if we admit that w+(Q) ^{shows a linear}

variation with Q. This is the expression ^{which we}

have already used in the qualitative discussion of the preceding section. The third term in u;-a ^{is the}

contribution to the width of _Vs of the pure electrical

anharmonicity and will be shown later to vary ^as s2.

The second term in U2 , ^{is a cross term between these}

two anharmonicities and is equal ^{to zero if either co +}

or M do not depend ^on Q. S2 being temperature dependent, Us-a will also be temperature dependent.

From this rapid analysis ^of eqs. (13) ^and (14) ^we

may conclude that it will be relatively easy ^to separated the contributions of the different mechanisms which

are responsible ^{of the} shape ^of vs. Resonance inter-

actions, often called Fermi resonances, have in parti-

cular a contribution which only appears in the width of _v. (P ^{and S are} independent ^{of these} interactions)

and this contribution to the width of v. ^is temperature independent. ^Before discussing ⁱⁿ greater ^details these theoretical considerations, ^{which we} have intro- duced _up to now with the minimum number of hypo- thesis, ^{let us mention} again that in this analysis ^we

have implicitely ^assumed that the low frequency

vibration Q (or ^even the whole set of Q ^{modes in}

the crystal) ^{is a} purely quantum motion, ^described

by ^a Hamiltonian, ^and that it has an influence in the

shape ^of vs through ^its average position ( Q ) ^and quadratic fluctuation _s2. Therefore we have impli- citely ^discarded any modulation of this vibration

by ^a stochastic field (which ^can be the field due to other vibrations) and it is possible ^{that we should}

have to reexamine this implicit hypothesis ^more carefully. Thus, ^{the model cannot} reproduce ^{in detail}

the _vs bands of H-bonds in liquids, where such a

stochastic field is supposed ^{to hâve an} important

contribution [33, 11].

4. Application ^{to H-bonds of} adipic ^acid.

^In

order to compare the experimental results of section 2 with the theoretical considerations of the preceding

section we shall have to formulate further hypotheses

which are specific ^to adipic acid and which we shall

now precise. ^We shall first _suppose that the Va ^vibra-

tion is not very different from that of an harmonic oscillator of frequency ^{Q, so that the term} quadratic

in Q ⁱⁿ eq. (5) ^{will be} equal ^to t Ma Q2 Q 2. ^We

shall also _suppose, in a first approximation, ^{that both} co’(Q) ^and M(Q ) vary linearly ^with Q. ^{We shall}

then use the following dimensionless quantities :

il has been defined in eq. (16). ^{At first} sight ^the quanti-

ties b and b +, ^which correspond respectively to Q >

and dm + /dQ ^{seem to be} independent. ^{This is mot}

true, however, because the quantity dm + /dQ repre- sents the extent of the _Vs-Va coupling ^{and we can} easily

show that ( Q > ^also depends ^{on this} coupling. ^In

order to do this we shall first consider our simplified

case where the _v. vibration is harmonic with a fre- quency ^m depending ^on Q. ^{We have} already ^seen,

from _eq. (10) that, ^{in that} case, ^we had m + (Q) == co(Q)

so that using eq. (5) ^{we can write} Ho ^{as :}

which is a consequence of the particular ^{form of} h(q, Q ) given ^{in that case} by

The origin ^of Q is then defined in such a way that

for Q ^{= 0 the} potential ^surface goveming the v.

and _Va vibrations is minimum. Hence from _eq. (20)

we deduce :

which gives ^{b = b+. In} the discussion below, ^we

shall have to use a somewhat more complicated potential ^for v., (a ^Morse potential) but this will not alter the conclusion that b and b+ are not inde-

pendent (we ^{shall see} that their ratio is constant,

although ^not equal ^to unity ^as it is in that simple ^case

of an harmonic oscillator in vs). ^It might ^also happen that ( Q > depends ^not only ^{on the} Vs-Va coupling

but also on another independent mechanism. This is for instance what happens ^{if we} suppose that thé

potential goveming Va ^has anharmonic terms in

Q [26]. ^{In that} case Q > ^{will be} temperature depen-

dent whereas dco’IdQ ^{will not.} We have however

seen in the preceding section that we shall have difficulties with such an assumption ^to explain ^the

weak variation of the variance u of _vs with tempe-

rature. We shall consequently discard this possi- bility ^and consider that b + is proportional ^{to b.}

Using ^the quantities defined in _eq. (19), ^{we have :}

so that _eqs. (13) ^and (14) ^can be re-written as :

With

These equations ^{are similar to} those established in

previous ^work [24] for formic acid where however anharmonic resonance interactions, ^{such as those}

considered here, ^were neglected. ^{We shall now} try ^to determine the values of b, b+ , J1 ^{from a} comparison

of _eqs. (23) ^with experimental ^curves representing ^the

variations of P, 0153 and a with temperature. ^{We shall} first note that with _eq. (23), ^{P must} increase with tem-

perature ^{if b} and u ^{do not} vary with temperature.

However the expérimental results show that P in fact decreases with temperature (see Fig. 1). ^{Therefore we}

must postulate ^{that b} or Il vary with temperature. ^The variation of b with T seems quite natural for H-bonds where it corresponds ^{to a} variation of the _average 0

0 distance with temperature ^{and we have seen} in the qualitative discussion of the experimental

results that this was quite ^a reasonable assumption.

We have attempted ^to reconstitute the experimental results, using eqs. (23), with this supposition ^that only ^{b and b+} (which, ^{as shown} earlier, ^is propor- tional to b) ^{can be T} dependent ^but that y ^is indepen-

dent of T. We have failed because, if y ^{does not} vary with temperature b should show a far too high ^varia-

tion to explain the decreases of the P’s with tempe-

rature which leads to the impossibility ^of correlating

the variations of the 0153’s with the variations of thé Q’s. We have then been obliged ^to suppose that b, ^{b +} and p ^are temperature dependent. ^{We have seen}

that if we assume that _v. is a pure harmonic vibration whose frequency ^ro depends ^on Q ^{we had b == b+.}

With this assumption ^{alone we} find however some

difficulties in explainin the experimental value of the ratio UiH/côD ( £i ^0.94 2).

We can easily ^overcome these difficulties by sup-

posing that, ^{as in the case of} acetic acid crystals [34],

the potential ^for vs is a Morse potential with anhar-

monicity parameter 5 ^{which we} shall consider ^as

being independent ^of Q ^{in a first} approximation.

We shall then write :

As the _energy levels for such ^a potential ^{are of}

the form

we deduce from eqs. (5) the relation :

which gives

if we neglect ^the Q dependence ^of qo which does not

give any special ^effect apart ^{from a new definition} of the origin ^{of the} Q’s which will be translated by

almost the same amount in H-bonds and D-bonds.

From _eq. (19) ^we then have :

which gives bH N bD J2.

From _eq. (10) and with the neglect ^of the Q depen-

dence of ql (Q ) - qo(Q ) ^{which we have} already

discussed we may write :

Using eq. (19) for the definition of b+ we arrive at the relation :

In order to obtain the correct experimental ^value

for the ratio OJHfWo ^we find that the anharmonicity parameter l5H ^{of eq.} (24) should be of the order of 0.15 so that l50 ^{which is} equal ^to l5H/ J2 ^{should be}

of the order of 0.1. In obtaining this result we have considered that the ratio of the reduced ^masses of the H and D atoms in the H-bonds is of the order of 0.97 J2, ^{and that} 0152JJ(O)/O is of the order of 17

(hQ ^{160 cm-1} [29]). With these values we then find :

In order to compare eqs. (23) ^with experimental

results we shall proceed ^{in the} following way : ^we shall determine at all temperatures the values of b

and y ^which reproduce ^the theoretical curves of

figures ^{1 and 2} around which the experimental points ^{fall. In} figures ^{3 and 4 we} have drawn these values of b(b+) ^{and ,u.} Using eqs. (13) ^and (23) ^we

then calculate the values of the 6’s as a function of temperature ^{and we have drawn the} corresponding

curves in figure ⁵ together ^with experimental points.

It can be seen from this figure ^{that the} expérimental points ^are correctly reproduced, despite ^{their non}

trivial evolution with temperature ^{which we have} discussed in a preceding section. The inflexion point

of the predicted variations of a with T is due to the variation of the quantity

at these temperatures. However the precision ^of

the fit is not sufficient to decide whether this inflexion

Fig. ^3.

Variations of the parameters b and ^{b+ of H and} D-adipic

acids as deduced from curves of figures ^{1 and 2.}

Fig. ^4.

Variations of the electrical anharmonicity parameters ^of H and D-adipic ^{acids as} deduced from figures ^{1 and 2.}

Fig. ^5.

Variations of the variances of the _Vs bands of H and

D-adipic ^acids. Expérimental points ^{are shown} by symbols (0).

The curves are deduced from theory ^{with the} parameters ^of figures ³

and 4.

point really ^{exists or not.} The values of the b, ^{b + and} /1’S ^{allows us to} calculate the variations of the Q’s with T but not their absolute values which also

depend ^on the contribution of resonance interactions in 6. The compatibility ^{of the curves of} figure ^{5 with} experimental points ^{allows us to} determine the contri- butions of these interactions, ^{which are :}

These values are almost the same as those which

we had theoretically predicted [35] by supposing ^that

the origin ^{of these Fermi} resonances is purely geome- trical and originates from the variations of the inertial

moment of the H atom with respect ^{to the 0 atom}

to which it is bounded when the 0-H length ^vibrates (i.e. ^that UF should be 160 cm-1 for H-bonds and 95 cm-1 for D-bonds). ^{We shall return to this} point

in the discussion.

The _accuracy of the determination of the values of b and b+ is high, being better than 10 %. ^{The deter-}

mined values for y ^{are less} precise ^{as a} change ^of

about 25 % would be still acceptable ^{as it would}

still reproduce ^the experimental results within experi-

mental errors. However the relative variations of b, b + andg ^with temperature ^{are rather} precise ^{as with}

all the values which we have tried we could not depart

from the given variations significantly. ^In particular

the choice of the value 160 cm-1 for ha has no real

importance ^as any value between 140 and 200 cm- i would have given almost the same result.

5. Discussion.

The quantitative analysis ^which

we have detailed in the preceding section allowed

us to determine from experimental points ^the magni-

tudes of the anharmonic interaction of _v. with _Va, of the resonance interaction of _Vs with other vibra-

tions, and of the electrical anharmonicity coupling Vs

with _Va. The first question ^{which we} should ask

concerns the validity ^{of such a} determination. The method which we have used consists of calculating

the values of the different parameters ^{which define} these various couplings, ^{from the} experimental ^varia-

tions of P and Ui with temperature ^and deuteration,

and to show that these values explain quite ^{well the}

variations of Q with temperature ^and deuteration.

In other words the theory ^{which we have} proposed

and which required ^the formulation of various successive hypotheses ^{allowed us to establish a}

correlation between the variations of ^P and (J) with temperature and deuteration on one hand and those of a on the other hand. Thus, ^the theory ^{seems to}

show some coherence, and, within the timescale

considered, there should be some validity ^{in the}

dynamical parameters extracted from it, ^{which were}

given in eqs. (30) ^and figures ^{3 and 4.} However, ^it

should be pointed ^{out that our} theory, although

coherent, does not exclude other possibilities.