IR spectral density of H-bonds. Both intrinsic anharmonicity of the fast mode and the H-bond bridge. Part I: Anharmonic coupling parameter and temperature effects

Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21 www.elsevier.com/locate/theochem

IR spectral density of H-bonds. Both intrinsic anharmonicity of the fast mode and the H-bond bridge. Part I: Anharmonic coupling parameter and temperature effects Najeh Rekik b

a,*

, Noureddine Issaoui a, Houcine Ghalla a, Brahim Oujia a, Marek J. Wo´jcik

b

a Laboratoire de Physique Quantique, Faculte´ des Sciences de Monastir, route de Kairouan, 5000 Monastir, Tunisia Laboratory of Molecular Spectroscopy, Faculty of Chemistry, Jagiellonian University, Ingardena 3, 30-060 Krako´w, Poland

Received 8 December 2006; received in revised form 13 June 2007; accepted 19 June 2007 Available online 23 June 2007

Abstract The paper presents extension of a quantum non-adiabatic treatment of H-bonds in which effects of anharmonicities of the high fre! ! quency X AH    Y and the low frequency X AH    Y modes on the tX–H infrared lineshapes of H-bonds systems are considered. The ! ! anharmonic coupling between the high frequency X AH    Y and the low frequency X AH    Y modes is treated within strong anharmonic coupling theory and the relaxation is included following quantum treatment of Ro¨sch and Ratner. The intrinsic anharmonicity of the fast frequency mode is described by a double well potential and of the slow frequency mode by Morse potential. IR spectral density is obtained within the linear response theory by the Fourier transform of the autocorrelation function of the X–H transition dipole moment ! operator. The main feature brought by the anharmonicity of the H-bond bridge X AH    Y is the increase of the average frequency of the tX–H IR band with temperature for asymmetrical H-bonds, and decrease for symmetrical and weakly asymmetrical H-bonds. The numerical results are in fairly good agreement with the experimental behaviour of the first and the second moment of the X–H bands, observed when varying the temperature.  2007 Published by Elsevier B.V. Keywords: Hydrogen bonds; Anharmonicity; Morse potential; Infrared spectral density; Temperature effects; Direct relaxation; Linear response theory; Autocorrelation function

1. Introduction Infrared spectroscopy is a powerful tool in the hydrogen bond research, able to provide complete information about complex dynamics of atoms in hydrogen bonds. Infrared spectroscopy provides a wealthy system of data allowing for deeper understanding of the hydrogen bond nature [1–9]. In IR spectroscopic studies, the most evident effects of hydrogen bonding (H-bond) are red shift of the high fre! quency X AH    Y stretching mode, its intensity increase and band broadening; the latter is often accompanied by presence of peculiar band-shapes. Large increase of the

*

Corresponding author. Fax: +216 73 500 278. E-mail address: [email protected] (N. Rekik).

0166-1280/$ - see front matter  2007 Published by Elsevier B.V. doi:10.1016/j.theochem.2007.06.016

bandwidth, the band asymmetry, the appearance of subsidiary absorption maxima and minima, such as Evans windows, peculiar isotope and temperature effects are challenge for theories of the H-bond. Early physical idea proposed to explain these features has been based on assumption of strong anharmonicity in hydrogen-bonded species. That has been confirmed by ab-initio computations [10–14]. In strong anharmonic coupling theory proposed 30 years ago [15,16] to explain IR spectra, the high frequency mode of the H-bond is assumed to be coupled to the hydrogen bond coordinate X  Y. In addition Fermi resonances [17–20] must play important role in spectra of stronger hydrogen-bonded systems, and in special situations, involving multiple H-bonds, important role is played by Davydov effects [15]. For intermediate and strong H-bonds, the tunnelling effect plays a

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N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

non-negligible role. The theory of instantons was first applied to the tunnelling in H-bonded systems by Kryachko [21,22] and by Smedarchina and Enchev [23]. There are also papers of Benderskii [24,25] which studied tunnelling splittings in vibrational spectra of non-rigid molecules using a perturbative instanton approach. Other papers dealing with the tunnelling in H-bonded systems include 1-D approach of Robertson and Yarwood[26] in which strong anharmonic coupling has been ignored. In an older important paper, Singh and Wood [27] have given a quantum 2-D approach of strong symmetric H-bonds which incorporated a double minimum in the potential energy, an anharmonic coupling between the symmetric and anti-symmetric stretching motions and an electrical anharmonicity for the dipole moment. They ignored the possibility of Fermi resonances and did not take into account the possibility of dissociation in the symmetric coordinate. Using ranges of parameters appropriate to H-bonded systems, they studied combined effects of tunnelling and vibrational interaction on tunnelling, far and normal IR, and combination bands. Unfortunately, their discussion ignored the influence of surrounding. Romanowski and Sobczyk [28] have proposed a model for strong symmetric H-bonds, in which the influence of the surrounding is explicitly taken into account. The stochastic model considered by these authors is very similar to that of Bratos [29]: they considered for the quantum mechanical motion of the proton a symmetric double minimum potential and assumed that this potential is modulated by the Gaussian-like statistical distribution of the symmetric stretching amplitude, i.e., of the H-bond bridge. They obtained an evolution of IR spectra from weak to strong H-bonds. They treated the slow mode like a classical stochastic oscillator, similar as in the Bratos [29] and Robertson and Yarwood models [26]. Abramczyk [30] included tunnelling in symmetric double well potential in the model of Boulil et al. [31]. We must also quote the important work of Ro¨sch [32] who, thirty years ago, proposed a theoretical model in which strong anharmonic coupling, tunnelling in a double well symmetric potential and the interaction with the surrounding were considered on equal footing. Recently, Blaise and Henri-Rousseau [33] reconsidered the model of RR [34] beyond the adiabatic approximation [35], and investigated the influence of quadratic dependence of the angular frequency and the equilibrium position of the fast mode on the H-bond bridge vibration [36]. This model reproduces whole bunch of experimental features of the H-bond spectra, i.e., shift of the frequencies towards lower values, broadening of the band shapes with peculiar sub-bands, isotopic effect and increase of the half-width 1 with temperature according to the coth2 ðcte Þ law [37] but T fails to reproduce increase of the first moment with temperature. This discrepancy is probably due to the fact that potential of the slow mode is assumed to be harmonic though it involves a large amplitude of vibrations leading necessarily to anharmonic curve characterizing the H-bond vibration. Blaise and Henri-Rousseau [33] introduced, for

weak H-bonds, the intrinsic anharmonicity of the slow ! mode X AH    Y by using Morse potential. They have shown that the main feature brought by the anharmonicity of the H-bond bridge is an increase of the average fre! quency of the tS ðX AH    Y Þ IR band upon temperature rising. Their model reproduces successfully experimental behaviour when varying the temperature as well as upon isotopic substitution. In the present paper, we modify our previous model [38] by introducing Morse potential for the slow mode. The main purpose of this treatment is to show that anharmonicity of the slow mode explains experimental changes observed when varying temperature. The model takes into account, within the linear response theory (LRT) [39,40], both anharmonicities, of the fast and slow modes. Anharmonic coupling between these two modes is treated by strong anharmonic coupling theory, and direct damping of the fast mode is introduced in phenomenological way, by assuming that the autocorrelation function decays exponentially in time, as in the fundamental model of RR [34]. The present approach could be applicable to the real medium-strong H-bonds. 2. Model We use standard treatment within the LRT [39,40], which allows to link the infrared spectral density (SD) of ! the X AH    Y mode, I(x) with the autocorrelation function of the transition dipole moment, G(t) through the Fourier transform: Z þ1 Z þ1 ixt IðxÞ / GðtÞe dt ¼ 2Re GðtÞeixt dt ð1Þ 1

0

The autocorrelation function of the dipole moment operator (ACF) is written, in a general way: GðtÞ / trfqð0Þlð0ÞlðtÞg;

ð2Þ

where q(0) is the equilibrium Boltzmann density operator and l(0) the dipole moment operator at initial time, l(t) being the same operator at time (t). Within the Heisenberg representation this last operator is given by:     HTot t HTot t lðtÞ / exp i lð0Þ exp i ð3Þ h h where HTot is the full Hamiltonian of the H-bonded system. 2.1. The full undamped Hamiltonian Let us consider the full Hamiltonian of the H-bond. It is the sum of three Hamiltonians: of the slow mode, of the fast mode and the interaction between the fast and slow modes. We split the Hamiltonian HTot, which describes the fast and slow undamped modes, as follows: HTot ¼ HSlow þ HFast þ Vint

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

The Hamiltonian of fast mode is assumed to be that of a particle moving in an asymmetric double well potential:   h2 o2  ð4Þ HFast ¼   2 þ UðqÞ 2m oq The double well asymmetric potential is in turn considered as an harmonic potential of pulsation xB perturbed by V(q) [38]: 1 UðqÞ ¼ k B q2 þ VðqÞ 2 the perturbation potential is : n o VðqÞ ¼ A exp Bðq  CÞ2

ð5Þ

ð6Þ

and k B ¼ mB x2B , where xB and mB are, respectively, the angular frequency and the reduced mass of the harmonic oscillator. Double well asymmetric potential, used in this work, is treated by the perturbation theory and the repulsive potential generating the barrier of the double well potential is assumed to be Gaussian. Anharmonic oscillators have been extensively studied by this method by many authors [41–48]. Using Morse potential [49] for the slow mode, the Hamiltonian of the slow mode can be written: 2 " rffiffiffiffiffiffiffiffiffiffiffiffi!#2 3 2 P Mx 5 HSlow ¼ 4 þ De 1  exp be Q 2M h  2 " rffiffiffiffiffiffiffiffiffiffiffiffi!#2 3 2 2 h  o Mx 5 ¼ 4 þ De 1  exp be Q 2 2M oQ h  where De is the dissociation energy of the H-bond bridge and be is given by: rffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffi M h   be ¼ x ð7Þ 2De Mx Using Taylor series for the Morse potential, Eq. (4) can be re-written as the sum of the hamiltonian of a harmonic oscillator and of an anharmonic potential V  2  P 1 þ Mx2 Q2 þ V  ðQÞ HSlow ¼ ð8Þ 2M 2 where V is given by: rffiffiffiffiffiffiffiffiffiffiffiffi!n n n 1 X ð1Þ ð2  2Þ Mx be Q V  ðQÞ ¼ De n! h  n¼3

ð9Þ

In the strong anharmonic coupling theory of Mare´chal and Witkowski [15], the effective angular frequency of the fast mode x(Q) is assumed to be strongly dependent on the slow mode stretching coordinate Q, as shown in several experimental correlations [50,51]. Following Mar e´chal and Witkowski we consider linear modulation: xðQÞ ¼ x þ aQ

11

where appears a which is the anharmonic coupling constant between the slow and fast modes. Then, the Hamiltonian coupling the slow and fast mode can be written, according to [15] and taking for mB the value of m to simplify calculations: pffiffiffiffiffiffiffiffiffi 1 Vint ðq; QÞ ¼ a mB k q2 Q þ mB a2 q2 Q2 ð10Þ 2 To simplify calculations we took for mB the value m (the reduced mass of the fast mode). From these equations ! the full Hamiltonian describing the X AH    Y vibration of H-bonded systems becomes:  n o h2 o2 1 HHTot ¼   2 þ k B q2 þ A exp Bðq  CÞ2 2 2m oq   2 2 h o 1  2  þ  þ Mx Q þ V ðQÞ 2M oQ2 2   pffiffiffiffiffiffiffiffi 1 þ a m k q2 Q þ m a2 q2 Q2 ð11Þ 2 2.1.1. Eigenvalues and eigenvectors of the full Hamiltonian For numerical calculations, it is convenient to take for the reduced mass mB the value m. That gives the Hamiltonian (12):  n o h2 o2 1 2 HTot ¼   2 þ m x2B q2 þ A exp Bðq  CÞ 2 2m oq   2 2 h o 1 2 2  Mx   þ Q þ V ðQÞ 2M oQ2 2   pffiffiffiffiffiffiffiffi 1 þ a m k q2 Q þ m a2 q2 Q2 ð12Þ 2 This Hamiltonian may be partitioned into diagonal and non-diagonal parts: HTot ¼ H0 þ Hpert     h2 o2 1  2 2 h2 o2 1 2 2 H0 ¼   2 þ m xB q þ  þ Mx Q 2 2m oq 2M oQ2 2 ð13Þ Let us consider the following eigenvalue equations:     h2 o2 1 1   2 þ m x2B q2 jfkgi ¼ hxB k þ jfkgi 2 2 2m oq     2 2 h o 1 1 2 2   þ Mx q jðnÞi ¼ hx k þ jðnÞi 2 2M oQ2 2

ð14Þ

One may construct the product base according to: fjfkgðnÞig  fjfkgijðnÞig and hðnÞfjgjfkgðrÞi ¼ dj;k dn;r ð15Þ In the partition (13), H0 is diagonal in this product base:      1 1  H0 jfkgðnÞi  hxB K þ þ hx n þ jfkgijðnÞi 2 2 ð16Þ

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From Eqs. (6) and (10), the perturbation of the partition is given by: Hpert ðq:QÞ ¼ VðqÞ ¼ VðqÞ þ V int ðq:QÞ þ V  ðQÞ

ð17Þ

The matrix elements of V(q) in the basis where H0 is diagonal are of the form: hðmÞjhfjgjVðqÞjfKgijðnÞi ¼ hfjgjVðqÞjfkgiom;n

ð18Þ

The right-hand side matrix elements of this last equation have been calculated in a precedent paper [52]. They are: hðmÞjhfjgjVðqÞjfkgijðnÞi pffiffiffiffiffiffiffi   1 X 1 X 1 k!j! ðBÞ½jþk=2 BC 2 X ¼ A ½jþk=2 exp  ½jþkþ1=2 B þ 1 r¼0 v¼0 w¼0 2 ðB þ 1Þ  ð½vþw=2þrÞ  vþw  r B 1 2BC 2   Bþ1 r!v!w! B þ 1 Bþ1 1 1 dm;n  ð½k  r  v=2Þ! ð½j  r  wÞ! In the same basis (15) matrix elements of the anharmonic coupling potential between the slow and fast mode, i.e., Vint(q, Q) and V(Q) are obtained using the propreities of dimensionless lowering and raising operators. Let us look at the corresponding (NN) (NN) truncated matrix representation of the full Hamiltonian (12) in the basis (15), where N and N are, respectively, the dimensions of the fast and the slow basis. Its eigenvalue equation is: hXl jWl i HTot jWl i ¼ 

ð19Þ

Xl is the lth eigenvalue and the corresponding eigenvector Wlæ is given by the expansion : X X

C l;fkgðmÞ jfkgðmÞi ð20Þ jWl i ¼ k

m

Fig. 1. Definition of the fundamental parameters of the double well potential V(q).

Performing tracing on the eigenstates of the total Hamiltonian and using the closeness relation one obtains: XX G ðtÞ / exp fhXm =kT gjhWm jqjWn ij2 m

n

 exp fi½Xm  Xn tg In the presence of direct damping, the quantum ACF may be written [34]: GðtÞ ¼ G ðtÞ expðc tÞ expðidXtÞ

2.2. Theoretical SD

ð23Þ



If one neglects electrical anharmonicity [53–56], the dipole moment operator of the fast mode is represented by expansion:   dl lð0Þ / l þ ð21Þ ðq  q1 Þ dq

where c is the direct damping parameter and dXt is the angular frequency shift (cf. Lamb shift) resulting from the quantum approach to the direct damping. The SD is the Fourier transform of the autocorrelation function: IðxÞ /

XX m

where q1 is the coordinate at the minimum which is not 0 in our present model (see Fig. 1). Neglecting damping, and using Eq. (21) one may write the autocorrelation function of the dipole moment operator as the sum of the time dependent and the time independent terms. The term which plays the physical role within the LRT, is the time dependent term which has the form:      HTot t HTot t  G ðtÞ / tr expfHTot =kTgq exp i q exp i h h ð22Þ



Z

exp fhXm =kT gjhWm jqjWn ij2

n þ1

expðc tÞexpðidXtÞ expfi½Xn  Xm g expðixtÞdt

1

Matrix elements of the dipole moment operator are given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi

h X X C m;fkgðlÞ hWn jqjWm i ¼ 2m xB k l h

pffiffiffiffiffiffiffiffiffiffiffi

pffiffiffii  C ðlÞfkþ1g;n k þ 1 þ C ðlÞfk1g;n k and the SD becomes:

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

13

Fig. 2. Comparison between SDs involving double well potential near symmetry (D = 59 cm1) when the potential for the slow mode is described by: darkened lineshape, harmonic potential; dashed lineshape, Morse curve.

Fig. 3. SDs involving symmetrical double well potential (D = 0 cm1).

14

IðxÞ /

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

 XX h expf hXm =kT gjhWm jqjWn ij2 2m xB m n " # c  c2 þ ½ðXn  Xm Þ  ðx  dXÞ2

Dx ¼ ð24Þ

This SD can be easily computed in a given truncated basis, by using the eigenvalues Eq. (18) and the corresponding expansion coefficients of the eigenvectors Eq. (20). We have calculated a numerical discrete Fast Fourier Transform using a given number of points. The SD must be stable with respect to a change in the dimension N N of the truncated base used for the representation of the full Hamiltonian. In out treatment we have computed the first and second moments of the lineshapes from discretized equations: P xi Iðxi Þ ; hxi ¼ P Iðxi Þ

P 2  2 x Iðxi Þ x ¼ Pi Iðxi Þ

ð25Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hx2 i  hxi2

ð26Þ

3. Numerical results To compute SD Eq. (24) we constructed and diagonalized the total hamiltonian HTot in a truncated basis Eq. (14). The stability of the computed spectra with respect to the size of this basis set was carefully checked. In calculations presented below the stability was usually obtained by taking into account N = 20 levels for the fast and N = 35 levels for the slow mode, so that the full basis included 700 states used for calculations. N is more dependent on the absolute temperature than N since ⁄x00 p p kBT. The stability of the spectra was also checked versus the order of the Taylor expansion of the Morse potential: full numerical stability (self-convergence) was achieved for the order 65 (we used this value in our calculations). Instead of the parameter a, we used the dimen-

Fig. 4. SDs involving double well potential near symmetry (D = 59 cm1).

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

sionless parameter a characterizing the strong anharmonic coupling between the slow and the fast stretching modes, defined by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi a h  a ¼  ð27Þ x 2Mx We calculated representative cases of spectral densities (SDs) at temperature T using Eq. (24). The SD is a function of several parameters. The angular frequencies of the fast and slow modes, used in all the calculations, were 3000 and 150 cm1, the dissociation energy of the H-bond bridge De = 2100 cm1, and be = 0.189. In all computations in the present paper we used the same parameters V, D and D, as in the previous paper [38]. We assumed weak direct damping parameter c = 15 cm1 in order to preserve fine structure of the SD. In the following figures, we plotted SDs in arbitrary intensity units versus wavenumbers given in cm1, to

15

make comparison between harmonic and Morse potential for the slow mode. The influence of the Morse potential for the slow mode on the SDs is shown in Fig. 2. The dashed line shapes correspond to SDs calculated with the Morse potential for the slow mode, whereas the grayed one were calculated with the harmonic potential. All SDs were obtained at room temperature (300 K). Examination of these SDs reveals some discrepancies in the intensities and frequencies of the sub-bands. Introduction of the Morse potential for the slow mode shifts the ! tS ðX AH    Y Þ spectra to higher frequencies. This is a general trend. It results from strong anharmonic coupling mechanism which allows hot bands to appear in the ! tS ðX AH    Y Þ spectrum. A small narrowing of the spectral halfwidth which is brought by the anharmonicity of the H-bond bridge results from a redistribution of subband intensities compared to theoretical SD computed in the harmonic approximation.

Fig. 5. SDs involving weak asymmetric double well potential (D = 455 cm1).

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N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

3.1. Effect of the anharmonic coupling parameter (a) and the damping parameter (c) In Figs. 3–6 we present two sets of line shapes computed by using our model with the 2-D Hamiltonian Eq. (12). In top parts of each figure we present line shapes calculated with the harmonic approximation for the slow mode, and in bottom parts line shapes calculated with the Morse potential. In each figure uppermost SD represents the situation where anharmonic coupling parameter a between the slow and fast modes is zero, and then the parameter is increasing when going down. Fig. 3 illustrates the situation of strictly symmetric double well potential. Fig. 4 corresponds to double well potentials which is near symmetric (gap around 4/1000 of the barrier). Fig. 5 is dealing with a weak asymmetric double well potential (gap less than 10% of the barrier), and Fig. 6 with an asymmetric double well potential (gap around 1/3 of the barrier).

There are interesting forms of simple SDs which occur when anharmonic coupling is zero (a = 0) shown as top lineshapes of the figures. In the symmetrical potential in Fig. 3 there are, as expected, two peaks, one corresponding to the dipolar allowed transitions at room temperature (high frequency peak) and the second one to the allowed dipolar hot band at 300 K (low frequency peak). The frequency gap is related to the splitting resulting from the tunnelling effect involving the first excited states in the right and left wells. When passing to asymmetric potential, the break of symmetry suppresses selection rules even for very weak asymmetry: one observes in Fig. 4, for which the asymmetry parameter is only 4/1000 of the barrier, four transitions occurring in place of two. The lowest frequency transition is very weak because the asymmetry is small. The splitting of components of the allowed doublet is also very small. When asymmetry is increased, as seen by passing from Fig. 4 to Fig. 5, the structure is strongly modified.

Fig. 6. SDs involving asymmetric double well potential (D = 2381.2 cm1).

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

One remains only one transition at 300 K: this is because the asymmetry of the potential introduces localization of the lowest wavefunction either on the right or on the left well. Let us consider what occurs when the anharmonic coupling between the slow and fast modes is increasing.

17

SDs calculated when the potential of the slow mode is given as Morse potential are presented in the bottom parts of the Figs. 3–6. In each figure the anharmonic coupling parameter is increasing from a = 0 to a = 0.2. In Fig. 3, when double well potential is symmetric, the doublet appearing at 300 K when a = 0, is progressively

a γ°= 15 cm-1

1.0

γ°= 24 cm-1 γ°= 30 cm-1

Intensity (a.u)

0.8

Δ =0 cm-1

γ°= 45 cm-1

0.6

0.4

0.2

0.0 1400

1600

1800

2000

2200

2400

2600

2800

3000

Wavenumbers (cm-1)

Common parameters: ω˚=3000 cm -1. ω˚˚=150 cm-1 . V˚=6635 cm-1 . D=0.83 Å. T=300 K. α =0.8

b γ°= 15 cm-1

1.0

γ°= 24 cm-1 γ°= 30 cm-1

Intensity (a.u)

0.8

Δ =59 cm-1

γ°= 45 cm-1

0.6

0.4

0.2

0.0 1400

1600

1800

2000

2200

2400

2600

2800

3000

Wavenumbers (cm-1)

Common parameters: ω˚=3000 cm -1. ω˚˚=150 cm -1 . V˚=6622.4 cm-1 . D=0.828 Å. T=300 K. α =0.8

c γ°= 15 cm-1

1.0

γ°= 24 cm-1 γ°= 30 cm-1

Δ =2381.2 cm-1

Intensity (a.u)

0.8

γ°= 45 cm-1

0.6

0.4

0.2

0.0

1400

1600

1800

2000

2200

2400

2600

2800

3000

Wavenumbers (cm-1)

Common parameters: ω˚=3000 cm -1. ω˚˚=150 cm -1 . V˚=7143.6 cm -1. D=0.803 Å. T=300 K. α =0.8 Fig. 7. Direct damping parameter effect (c).

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broadening with an increase of a, and an asymmetric tail appears involving fine structure. When passing to Fig. 4, presenting line shapes at room temperature with very weak asymmetric double well potential, one observes that increase of the anharmonic coupling causes a broadening of the quadruplet, which leads for a = 0.12 to wide asymmetric SD involving high frequency asymmetric tail. Further increase of the anharmonic coupling leads to narrowing of line shapes and finally, for a = 0.2, to collapse of its structure. In Figs. 5 and 6, one observes evolution of line shapes for increasingly asymmetric double well potentials. In these figures only a single peak appears for the initial uncoupled situation. One observes at room temperature an asymmetric broadening of the line shapes with fine structure when coupling parameter increases to a = 0.16; beyond this value, a subtle narrowing of the line shapes occur. One should note the role played by hot bands, when passing from uncoupled (a = 0) situation to a = 0.2 and when the slow mode is described by the Morse curve, the bands become more complex. Two main conclusions may be inferred from the Figs. 3– 6 when the potential of the slow mode is described by the Morse curve: (i) There is a trend for line shapes to evolve into broadened asymmetric profiles involving fine structures, when the anharmonic coupling parameter increases to about a = 0.12. Beyond this value there is subtle narrowing of line shapes, and finally collapse of structures. (ii) The spectrum becomes more sensitive to the magnitude of the anharmonic coupling, especially concerning intensities, asymmetry and frequencies of subbands. In Fig. 7, we illustrate the effect of the direct damping parameter c on lineshapes calculated with the present

model in three cases: in Fig. 7(a) for symmetric double well potential for the fast mode (D = 0 cm1), in Fig. 7(b) for small asymmetry (D = 59 cm1), and in Fig. 7(c) for high asymmetry (D = 2381.2 cm1). Theoretical lineshapes have been computed at room temperature (300 K), and the direct damping parameter c has been progressively increased (c = 15, 24, 30, 45 cm1). Fig. 7 shows that when the direct damping parameter is increasing there is more broadening of the lineshapes. As a consequence there is a complete lack of fine structure in the lineshapes. This general trend is the same at different temperatures. 3.2. Temperature effects Fig. 8 presents the evolution of the infrared SD with temperature changing from 0 to 300 K. The SDs have been calculated from Eq. (24) for the Morse potential of the slow mode and the symmetric double well potential for the fast mode (D = 0 cm1, V = 6635 cm1 and ˚ ). D = 0.83A Let us consider how temperature effects the first moment Æ xæ (center of the gravity of the spectrum) and the theoretical half-width Dx of the SDs. In Table 1, we compare the temperature effects for the harmonic approximation for the slow mode [38] with those calculated for the Morse potential. The evolution of half-width with temperature is nearly the same for the harmonic and anharmonic case with the same parameters of the double well potential. The position of the band depends however on the potential for the hydrogen bond vibration. In harmonic approximation it is temperature independent, and when the slow mode is described by the Morse potential it changes by 30– 60 cm1 when temperature increases from 0 to 300 K. The change is about 31 cm1 when H-bond becomes increasingly weak. This result is close to experimental data [37], according to which the position of the band, for weak H-bonds, varies by about 40 cm1 in this temperature

Fig. 8. Temperature effect on the SDs when the slow mode is described by a Morse potential.

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

19

Table 1 Temperature effect on the first moment Æxæ and the half-width of the spectra Dx for different values of the barriers V and asymmetric parameters D, when the slow mode is described by a harmonic potential (third and fourth column), a Morse curve (fifth and sixth column) Temperature (K)

Harmonic potential Æxæ

Symmetrical double-well potential (D = 0 cm1)

Double-well potential near asymmetry potential (D = 59 cm1)

Weak asymmetric double-well potential (D = 455 cm1)

Asymmetric double-well potential (D = 2381.2 cm1)

Common parameters: a = 0.16, V 0 2144 10 2125 20 2124 30 2123 40 2123 50 2123 60 2122 70 2122 80 2122 90 2121 100 2121 200 2116 300 2113 400 2111

Dx



Morse potential Æxæ

Dx ˚ = 6635 cm , D = 0 cm , D = 0.83 A, c = 15 cm1 344 2362 338 345 2328 335 345 2319 333 345 2316 333 345 2315 333 345 2314 332 345 2313 332 345 2313 332 345 2312 332 345 2312 332 345 2312 332 347 2307 332 348 2300 333 349 2293 335 1

1

˚, Common parameters: a = 0.16, V = 6622.4 cm1, D = 59 cm1, D = 0.828 A c = 15 cm1 0 2126 343 2328 330 10 2126 343 2328 330 20 2126 343 2328 330 30 2126 343 2327 330 40 2126 343 2326 331 50 2125 343 2325 331 60 2125 343 2324 331 70 2125 343 2322 331 80 2124 344 2321 331 90 2124 344 2320 331 100 2123 344 2319 331 200 2118 346 2308 332 300 2115 347 2299 333 400 2112 348 2290 335 ˚, Common parameters: a = 0.16, V = 6793 cm1, D = 455 cm1, D = 0.824 A c = 15 cm1 0 2149 333 3339 651 10 2149 333 3339 651 20 2149 333 3339 651 30 2149 333 3339 651 40 2149 333 3340 651 50 2148 333 3340 652 60 2148 333 3341 652 70 2148 334 3341 653 80 2148 334 3343 655 90 2147 334 3344 656 100 2147 334 3345 658 200 2143 336 3358 671 300 2139 338 3365 679 400 2135 340 3369 683 ˚, Common parameters: a = 0.16, V = 7143.6 cm1, D = 2381.2 cm1, D = 0.803 A c = 15 cm1 0 2224 295 3334 651 10 2224 295 3334 651 20 2224 295 3334 651 30 2224 295 3334 651 40 2224 295 3334 651 50 2224 295 3334 651 60 2224 295 3335 652 70 2224 295 3336 653 80 2223 295 3337 654 90 2223 295 3338 655 (continued on next page)

20

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21

Table 1 (continued) Temperature (K)

100 200 300 400

! range. The average frequency of the tS ðX AH    Y Þ IR band increases with temperature for asymmetric H-bonds and decreases for symmetric and weakly asymmetric H-bonds. Thus, one may conclude that replacing the harmonic by anharmonic potential for the slow mode, improves considerably reproduction of temperature effect. The Morse potential of the slow mode combined with the mechanism of hot bands, induced by strong anharmonic coupling between the fast and slow modes, affect particu! larly theoretical tS ðX AH    Y Þ average frequency, which closely follows the experimental behaviour. 4. Conclusions In this paper, we proposed, within the linear response theory, a quantum mechanical approach to the ! tS ðX AH    Y Þ IR SD of the medium strong H-bonds. We took into account strong anharmonic coupling between the high and low frequency modes. The anharmonicity has been introduced by using Morse potential for the hydrogen bond vibration and double well potential for the fast mode. We considered direct relaxation. Theoretical SDs were calculated through Fourier transform of the ACF of the dipole moment operator of the high frequency mode. By introducing anharmonicity of the H-bond bridge the present model reproduces well experimental features of line shapes of hydrogen-bonded systems: shift of frequencies towards lower values, broadening of band shapes with peculiar sub-bands and increase of the half-width with temperature. The main features brought by the anharmonicity ! of the H-bond bridge X AH    Y is the increase of the ! average frequency of the tS ðX AH    Y Þ IR band with temperature increase for asymmetric H-bonds and decrease for symmetric and weakly asymmetric H-bonds. The presented approach reproduces qualitatively experimentally observed changes of the first and second moments of the ! tS ðX AH    Y Þ bands with temperature. In the present approach we did not consider indirect relaxation [57]. We also neglected Fermi resonance and Davydov coupling [58–61]. These mechanisms will be included in further work. The approach developed in this paper could be applicable to the real medium-strong H-bonds. It must be stated however that whether this approach is actually valid for this class of hydrogen bonds is still a question because, despite its transparent physics, it relies on a large set of parameters and on a specific description of anharmonicity chosen in a Gaussian form.

Harmonic potential

Morse potential

Æxæ

Dx

Æxæ

Dx

2223 2221 2220 2218

295 297 298 299

3340 3352 3359 3363

657 671 678 683

Acknowledgements We thank Profs. Paul Blaise and Olivier Henri-Rousseau (Universite´ de Perpignan, France) for helpful discussions. References [1] G. Pimentel, A. McCellan, The Hydrogen Bond, Freeman, San Francisco, 1960. [2] O. Henri-Rousseau, P. Blaise, Adv. Chem. Phys. 103 (1998) 1. [3] O. Henri-Rousseau, P. Blaise, in: D. Hadzi (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, New York, 1997, p. 165. [4] D. Hadzˇi, H.W. Thompson (Eds.), Hydrogen Bonding, Pergamon Press, London, 1959. [5] P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond, Recent Developments inTheory and Experiments, vol. I–III, North Holland, 1976.. [6] G.A. Jeffrey, W. Sanger, Hydrogen Bonding in Biological Structures, Springer Verlag, Berlin, 1990. [7] S. Scheiner (Ed.), Hydrogen Bonding. A Theoretical Perspective, Oxford University Press, Oxford, 1997. [8] G.A. Jeffrey, Introduction to Hydrogen Bonding, Oxford University Press, Oxford, 1997. [9] D. Hadzˇi (Ed.), Theoretical Treatments of Hydrogen Bonding, J. Wiley, Chichester, 1997. [10] F. Van Duijneveldt, J. Van Duijneveld-Van de Rijdt, in: D. Hadzi (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, New York, 1997, p. 13. [11] S. Althorpe, D. Clary, J. Chem. Phys. 101 (1994) 3603. [12] D. Sabo, Z. Bacic, T. Burgi, L. Leutwyler, Chem. Phys. Lett. 244 (1995) 283. [13] A. Van Der Avoird, P. Wormer, R. Moszynski, Chem. Rev. 94 (1994) 1931. [14] S. Xantheas, J. Chem. Phys. 104 (1996) 3821. [15] Y. Mare´chal, A. Witkowski, J. Chem. Phys. 48 (1968) 3697. [16] S. Fisher, G. Hofacker, M. Ratner, J. Chem. Phys. 52 (1970) 1934. [17] S. Bratos, D. Hadzi, J. Chem. Phys. 27 (1957) 991. [18] A. Witkowski, M. Wojcik, Chem. Phys. 1 (1973) 9. [19] M.J. Wo´jcik, Mol. Phys. 36 (1978) 1757. [20] K. Belhayara, D. Chamma, O. Henri-Rousseau, J. Mol. Struct. (Theochem.) 648 (2003) 93. [21] E.S. Kryachkom, Chem. Phys. 143 (1990) 359. [22] E. Kryachko, Michael Eckert and Georg Zundel, J. Mol. Struct. 270 (1992) 33. [23] Z. Smedarchina, V. Enchev, J. Photochem. Photobiol. A Chem. 80 (1994) 135. [24] V.A. Benderskii, E.V. Vetoshkin, L. von Laue, H.P. Trommdorff, Chem. Phys. 219 (1997) 143. [25] V.A. Benderskii, E.V. Vetoshkin, I.S. Irgibaeva, H.P. Trommdorff, Chem. Phys. 262 (2000) 393. [26] G. Robertson, J. Yarwood, Chem. Phys. 32 (1978) 267. [27] J. Singh, J. Wood, J. Chem. Phys. 48 (1968) 4567. [28] H. Romanowski, L. Sobczyk, Chem. Phys. 19 (1977) 361.

N. Rekik et al. / Journal of Molecular Structure: THEOCHEM 821 (2007) 9–21 [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

S. Bratos, J. Chem. Phys. 63 (1975) 3499. H. Abramczyk, Chem. Phys. 144 (1990) 305–319. B. Boulil, O. Henri-Rousseau, P. Blaise, Chem. Phys. 126 (1988) 263. N. Ro¨sch, Chem. Phys. 1 (1973) 220. P. Blaise, O. Henri-Rousseau, Chem. Phys. 256 (2000) 85–106. N. Ro¨sch, M. Ratner, J. Chem. Phys. 61 (1974) 3344. P. Blaise, O. Henri-Rousseau, Chem. Phys. 243 (1999) 229. O. Henri-Rousseau, P. Blaise, Chem. Phys. 250 (1999) 249. Y. Mare´chal, in: J. Durig (Ed.), Vibrational Spectra and Structure, vol. 16, Elsevier, Amsterdam, 1987, p. 312. N. Rekik, A. Velescu, P. Blaise, O. Henri-Rousseau, Chem. Phys. 273 (2001) 11. R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. R. Kubo, in: W.E. Brittin, L.G. Dunham (Eds.), Lectures in Theoretical Physics, Interscience, Boulder, 1958. C.M. Bender, T.T. Wu, Phys. Rev. Lett. 21 (1968) 406. C.M. Bender, T.T. Wu, Phys. Rev. Lett. 27 (1971) 461. C.M. Bender, T.T. Wu, Phys. Rev. D 7 (1973) 1620. J. Killingbeck, J. Phys. A 13 (1980) 49. J. Killingbeck, T. Scott, B. Rath, J. Phys. A 33 (2000) 6999. R. Bhattacharya, D. Roy, S. Bhowmick, Phys. Lett. A 244 (1998) 9.

21

[47] A. Dasgupta, D. Roy, R. Bhattacharya, J. Phys. A 40 (2007) 773. [48] L. Powell, B. Crasemann, Quantum Mechanics, Addison-Wesley Pub. (Sd), 1961. [49] P.M. Morse, Phys. Rev. 34 (1929) 57. [50] A. Novak, Struct. Bonding 18 (1974) 177. [51] I. Olovsson, P.-G. Jo¨nsson, in: P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond, Amsterdam, 1976. [52] N. Rekik, A. Velescu, J. Mol. Struct. (Theochem.) 620 (2003) 265. [53] J. Bournay, Y. Mare´chal, Chem. Phys. Lett 27 (1974) 180. [54] L.J. Pejov, V. Stefov, B. Soptrajanov, G. Yukhnevich, E. Tarakanova, J. Mol. Struct. 565–566 (2001) 427. [55] G.V. Yukhnevich, E.G. Tarakanova, J. Mol. Struct. 447 (1998) 257. [56] F. Huisken, E.G. Tarakanova, Andrei A. Vigasin, G.V. Yukhnevich, Chem. Phys. Lett. 245 (1995) 319. [57] N. Rekik, B. Ouari, P. Blaise, O.H. Rousseau, J. Mol. Struct. (Theochem.) 687 (2004) 125–133. [58] O. Henri-Rousseau, D. Chamma, Chem. Phys. 229 (1998) 37. [59] D. Chamma, O. Henri-Rousseau, Chem. Phys. 229 (1998) 51. [60] D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 90. [61] D. Chamma, J. Mol. Struct. (Theochem.) 552 (2000) 187.