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IR spectral density of weak H-bonds involving indirect damping. Part III: application to situations with multiple Fermi resonances
Chemical Physics 293 (2003) 31–40 www.elsevier.com/locate/chemphys
IR spectral density of weak H-bonds involving indirect damping. Part III: application to situations with multiple Fermi resonances Khedidja Belhayara, Didier Chamma *, Olivier Henri-Rousseau Centre dÕEtudes Fondamentales, Universit e de Perpignan, 52 avenue de Villeneuve, F-66860 Perpignan cedex, France Received 10 February 2003; in final form 27 March 2003
Abstract The IR spectral density (SD) of the high frequency stretching mode of weak H-bonded complexes involving Fermi resonances is studied within the linear response theory from a full quantum mechanical point of view. The ! ! anharmonic coupling between the high frequency X–H Y and the low frequency X –H Y modes is treated inside the strong anharmonic coupling theory, whereas Fermi resonances are introduced in the spirit of the work of Witkowski and W ojcik [Chem. Phys. 1 (1973) 9]. The relaxation of the fast and bending modes (direct damping) is treated as usual, whereas the indirect damping is incorporated by aid of our results of part I [Chem. Phys. 293 (2003) 9], dealing with the quantum theory of driven damped quantum harmonic oscillator. The IR SD is obtained by Fourier transform of the autocorrelation function of the dipole moment operator of the fast mode. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Weak hydrogen bonds; Multiple Fermi resonances; Driven damped quantum harmonic oscillator; Effective non-hermitian Hamiltonian; Infrared spectra; Temperature and isotopic effects; Direct and indirect relaxations; Linear response theory; Autocorrelation function
1. Introduction In the area of the infrared lineshape of ! mS ðX–H YÞ stretching mode of H-bonded species, it is generally assumed that Fermi resonances must play an important role [1,2]. Among theories dealing with that [2–7], one may distinguish: (i) the
*
Corresponding author. Tel.: +33-468-662-108; fax: +33-468662-234. E-mail address: [email protected] (D. Chamma).
semi-classical ones [8,9], in which the H-bond bridge is treated classically and the fast mode quantum mechanically, (ii) the peeling off approach of Marechal [10,11], which attempts to remove the Fermi resonances effects in order to get the unperturbed lineshape, (iii) the quantum theories [12], in which both the H-bond bridge and the stretching mode are treated quantum mechanically. The first quantum theory of the ! mS ðX–H YÞ infrared spectral density of weak Hbonds is that of Marechal and Witkowski [13,14], who developed a strong anharmonic coupling
0301-0104/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00272-6
32
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
theory, according to which the high frequency ! mode X–H Y is anharmonically coupled to the ! low frequency mode X –H Y of the H-bond bridge. Using the fact that the frequency of the first mode is very large with respect to that of the second one, Marechal and Witkowski [13,14] applied an adiabatic approximation allowing to separate the slow and fast motions, they also considered the possibility of Davydov coupling, but ignored the possibility of Fermi resonances. In a later paper, Witkowski and W ojcik [12] incorporated in the above model the possibility of Fermi resonances. They considered on an equal footing the strong anharmonic coupling and the Fermi resonance, but ignored the damping. Later, a more general theory has been proposed by Chamma et al. [15,16], who considered the possibility of multiple Fermi resonances of non-resonant situations, took into account the direct dampings, but ignored the indirect relaxation of the H-bond bridge. The authors have shown that Fermi type interactions ought to be very universal since the perturbation of the lineshape remains sensitive even for energy gaps many times larger than the coupling. They have also shown that the relative intensities involved in the SDs are deeply affected by the relative magnitude of the dampings of the fast and the bending modes. There have been also attempts to treat simultaneously Davydov coupling and Fermi resonances, first without damping [17], and later with direct damping only [18,19]. In a precedent paper, we have shown that the quantum harmonic oscillator describing the
damped H-bond bridge may be described by aid of a reduced non-hermitian Hamiltonian. We apply here these results to treat the quantum indirect damping in situation where Fermi resonances are occurring. We shall consider that the damped fast mode is simultaneously coupled to the damped Hbond bridge (through the strong anharmonic coupling) and to one or several damped bending modes (through Fermi resonances). It will then be possible to get, in the Heisenberg picture, the expression of the dipole moment operator of the high frequency mode at time t, knowing its expression at initial time, and then to obtain the autocorrelation function of this operator which, according to the linear response theory, will lead to the infrared spectral density after Fourier transformation. The present model satisfactorily reduces to limit situations appearing in the literature, as shown in Table 1.
2. Theory b of the In order to construct the Hamiltonian H system, we need to define three vibrational modes and their associated parameters, shown in Table 2, ! namely the fast mode X–H Y, the slow mode ! X –H Y and a bending mode. The full Hamilb of a weak H-bond subject to a Fermi tonian H resonance may be written b ¼H bI þ H b II þ H b int : H
ð1Þ
b I describes the slow and fast The Hamiltonian H modes which are coupled together. Within the
Table 1 Some deeply interconnected adiabatic quantum theories dealing with the IR SD of weak H-bonds Authors and references
Strong anharmonic coupling
Marechal and Witkowski [13,14] Witkowski and W ojcik [12] R€ osch and Ratner [20] Boulil et al. [21] Giry et al. [22,23] Belhayara et al. [24] Chamma et al. [15,16] Present work
Fermi coupling
Relaxations
Single
Direct
Multiple
Bending
Indirect
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40 Table 2 Definition of some parameters associated with the three elementary quantum harmonic oscillators ! ! Fast mode X–H Y Slow mode X –H Y Angular frequency Position coordinate Conjugate momentum Reduced mass
x00 b Q Pb m00
x0 b q b p m0
strong anharmonic coupling theory [13,14], its expression is i2 p2 1 h b b bI ¼ 1 b H þ m0 x0 þ b Q q2 2 m0 2 1 Pb 2 1 b 2: þ þ m00 x200 Q ð2Þ 2 m00 2 Here b is the anharmonic coupling constant beb II is the tween the slow and fast modes. Besides, H Hamiltonian describing the bending mode p d2 1 b II ¼ 1 b H þ md x2d b q 2d : 2 md 2
b int ¼ l b H qb q 2d :
ð4Þ y
Let us define by jfkgi, jðmÞi and j½ui the eigenvectors of the zeroth-order quantum harmonic oscillators characterizing, respectively, the fast, the H-bond bridge and the bending modes. Then we may build the following tensorial basis: jW0 ðmÞi ¼ jf0gijðmÞij½0i; jW1 ðmÞi ¼ jf1gijðmÞij½0i; jW2 ðmÞi ¼ jf0gijðmÞij½2i:
Within the adiabatic approximation, which is well verified for weak H-bonds [25], the full Hamiltonian describing an H-bonded system where Fermi coupling, indirect damping of the Hbond bridge, direct relaxation of the fast mode and damping of the bending mode are simultaneously present, is 2 3 b f0g H 0 0 ind 7 bf ¼ 6 b f1g ihc0 H 4 0 5 H h b L Ind ind f0g b b 0 h L H ind þ 2hxd ihcd ð7Þ with b fkg ¼ khx0 þ hx00 fay a þ ka0 ðay þ aÞg H ind
c ðT Þ c ðT Þ 1 i 00 ka20 hx00 1 þ i 00 ; 2x00 2x00 k ¼ 0; 1: ð8Þ
y
Then, using the raising (b ; d ) and lowering (b; d) operators of the fast and bending modes, respectively, and performing the exchange approximation [15,25], the Hamiltonian (4) can be written as: i L hh y 2 2 b int ¼ p ffiffiffi b d þ b d y H with 2 sffiffiffiffiffiffiffiffiffiffiffi l h : ð5Þ L¼ 2 md xd m0 x0
ð6aÞ m ¼ 0; 1; 2; . . . ;
ð6bÞ ð6cÞ
Bending mode xd qd pd md
ð3Þ
b int accounts for the The interaction Hamiltonian H Fermi coupling between the fast mode and the first harmonic of the bending mode, through the anharmonic coupling parameter l:
33
Here c0 and cd are, respectively, the damping parameters of the fast and bending modes, which have been introduced in the spirit of the Green formalism, by considering specific imaginary diagonal energetic terms. ay and a are the raising and lowering operators of the slow mode. a0 is the adimensional anharmonic coupling parameter between the slow and fast modes, given by Eq. (4) of part I [24] and c00 ðT Þ is an effective damping parameter which is given also in part I [24] by Eq. (29). The generalization to multiple Fermi resonances is obvious (see [15]). Within the linear response theory, the infrared spectral density (SD) is given by the Fourier transformation of the autocorrelation function (ACF) GðtÞ of the dipole moment b l of the fast mode: Z þ1 IðxÞ ¼ GðtÞ expðixtÞ dt: ð9Þ 1
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
Fig. 1. SD involving weak H-bonds: connection between the present theory and precedent special ones to which it reduces when Fermi resonances, direct or indirect relaxations are ignored. (a) Situation without damping and Fermi resonance, (b) situation without indirect damping and Fermi resonance, (c) situation without Fermi resonance, (d) situation with Fermi resonance and without damping, (e) situation with Fermi resonance and direct damping, (f) situation with Fermi resonance and both direct and indirect dampings. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , T ¼ 300 K. Fermi resonances parameters: xd ¼ 1350 cm1 , L ¼ 50 cm1 .
At initial time the dipole moment operator is 1 X b l ð0Þ ¼ jf1gij½0ijðmÞihðmÞjh½0jhf0gj: ð10Þ m¼0
At time t, this operator may be obtained within the Heisenberg picture, by aid of an unitary transformation involving the Hamiltonian (7). Since this last Hamiltonian is non-hermitian, its eigenvalues xm are complex and its eigenvalue equation may be written: b f jUm i ¼ h x0 þ x0 a2 x00 icm jUm i; H ð11Þ Ind
m
0
where the imaginary part cm > 0 characterizes the limited life time of the corresponding energy level. After some manipulation, we get for the ACF: XX GðtÞ / expfkmg expfimx00 tg m m n c o exp m 00 t exp ix0m t expfcm tg 2 exp i x0 a20 x00 t jhUm jW1 ðmÞij2 : ð12Þ
Fig. 1 illustrates all possible connections between the different quantum mechanical models published up to now, showing how the SD obtained in the present model transforms when passing from the most general situation to some limit cases.
3. Numerical results and discussion 3.1. Assistance of Fermi resonance by the slow–fast mode anharmonic coupling For the situations where the indirect relaxation may be neglected, Chamma et al. [15,16,18,19] investigated in their previous papers the resonance condition between the first excited state of the fast mode and the excited states of some bending modes. They have shown that this condition does not need to be strictly fulfilled: strong modification of the X–H stretching bandshape can be obtained when 2xd x0 is as large as tenfold the Fermi
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
coupling parameter L. As it appears, the anharmonic coupling between the slow and fast modes assists the coupling between the first excited state of the fast mode and the first harmonic of the bending mode. We show here that this result, which was obtained after neglecting the indirect damping, remains to hold when all the three
35
damping mechanism are taken into account. Inspection of Fig. 2, shows that the SDs are affected by a Fermi coupling of 50 cm1 on a range of angular frequencies xd of 450 cm1 . Moreover, the SDs sensitivity to Fermi resonances does not depend on the relative magnitude of the direct and indirect dampings.
Fig. 2. Influence of Fermi resonances and of the relative magnitude of the direct and indirect damping parameters on the lineshape. The full line SDs are with Fermi resonances and dashed without. Besides, in each frame, the two up SDs are involving situations when the direct damping is greater than the indirect (c0 ¼ 60 cm1 , c00 ¼ 15 cm1 ), whereas it is the inverse for the two down SDs (c0 ¼ 15 cm1 , c00 ¼ 60 cm1 ). The Fermi coupling parameter is the same for all SDs (L ¼ 50 cm1 ) involving Fermi resonances whereas the angular frequency is progressively increased. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , cd ¼ 15 cm1 , T ¼ 300 K.
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
3.2. Influence of the relative magnitude of the different relaxation on the intensities of the subbands Chamma et al. [16] have also shown that in the absence of indirect damping the relative intensities of the sub-bands may be modified by changing the relative magnitude of the relaxation parameter c0 and cd affecting the fast and bending modes. We show here that the introduction of the indirect damping affects also the relative intensities of the sub-bands. That is illustrated in Fig. 3. 3.3. Multiple Fermi coupling and Evans windows The semi-classical theory of the indirect damping by Bratos and Ratajczak [9] including with
Fermi resonances leads to large Evans windows. The quantum theory of the indirect damping leads to more complex situations, where the distinction between windows following from Fermi resonances, and those related to the minima intensity resulting from the Franck–Condon progression is not obvious. Fig. 4 is devoted to the influence of the dampings on the Evans windows. It appears that the dampings become large, there are as many Evans windows as Fermi resonances. The quantum adiabatic description of weak Hbonds, working within the strong anharmonic coupling theory, leads to an abrupt change in the properties of the Hamiltonian of the H-bond bridge just after excitation of high frequency mode. In a first quantum representation, the effective Hamiltonian of the slow mode, which was
Fig. 3. Influence of indirect relaxation on the lineshape. Left column: evolution with c00 of the lineshape involving one Fermi resonance for c0 > cd . Right column: same evolution with c00 but for c0 < cd . The Dirac peaks correspond to the limit situation where c0 ¼ cd ¼ c00 ¼ 0. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , L ¼ 50 cm1 , xd ¼ 1350 cm1 , T ¼ 300 K.
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
37
Fig. 4. Multiple Fermi resonances effects on the lineshape. The number nF of the Fermi resonances which are taken into account in the calculations increases from the top to the bottom of each column. Common parameters: a0 ¼ 1, Lu ¼ 50 cm1 , x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , xd1 ¼ 1350 cm1 , xd2 ¼ 1280 cm1 , xd3 ¼ 1400 cm1 , xd4 ¼ 1450 cm1 , xd5 ¼ 1500 cm1 .
undriven before excitation of the fast mode, becomes driven after excitation. In another equivalent representation, the slow mode jumps from a thermal equilibrium state (before excitation) to a coherent state (after excitation). In both quantum descriptions, which are equivalent, the abrupt change in the behavior of the H-bond bridge leads in the spectral density to a Franck–Condon progression of Dirac delta peaks, when dampings and Fermi resonances are ignored. In the presence of Fermi resonances, the situation becomes more complex and the spectral density involves puzzling Dirac delta peaks. Now, when only the dampings of the fast and bending modes are incorporated in the model, the spectral density is broadened, but keeps a complex structure, in which contrarily to the semi-classical theory of Bratos and Ratajczak [9], it is difficult to distinguish Evans windows. When the indirect damping of the H-bond bridge is also taken into account, it becomes possible to observe in the spectral density as many windows as Fermi resonances. This is a consequence of the fact that the indirect damping enhances the semi-clas-
sical behavior of the driven harmonic oscillator or of the coherent state, the mixing between the two quantum anharmonic couplings being thus partially broken. 3.4. Fermi coupling parameter L As observed by Marechal [10,11] and by some of us [25], for H-bonds involving a single Fermi resonance, the Fermi coupling parameter L may be extracted from the SD in the following way: L2 ¼ ðr2F r20 Þ:
ð13Þ
Here rF and r0 are, respectively, the SD widths with and without Fermi coupling, which are of the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ r ¼ hx2 i hxi2 : The average values involved in this last expression may be computed from the SD by aid of the equations:
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
R xIðxÞ dx hxi ¼ R IðxÞ dx
and
R
x2 IðxÞ dx hx i ¼ R IðxÞ dx 2
ð15Þ after performing a cut-off of the low and high frequencies tails. In Chamma et al. model [25,26], it was shown that Eq. (13) is verified for a single Fermi resonance with direct relaxation of the fast and bending modes, but without indirect damping. Here we show that Eq. (13) may be generalized to multiple Fermi resonances in which the indirect damping is occurring together with the above mentioned direct relaxation. The generalization of Eq. (13) to multiple Fermi couplings is X L2i ¼ ðr2F r20 Þ: ð16Þ i
Table 3 illustrates the validity of Eq. (16) for situations involving 1, 2, 3 or 4 Fermi resonances. Its inspection shows that the expression (16) is suitable in the absence of relaxation (Dirac peaks). On the other hand in presence of dampings, Eq. (16) is only approximately verified, because of the cutoff in the SD. It is important to underline that Eq. (16) may be widely used, its verification being independent from the existence and the nature of the relaxation. 3.5. Temperature and isotopic effects As we have seen above, the features of the ! mS ðX–H YÞ infrared profiles of weak H-bonds
involving Fermi resonances and relaxations of the slow and fast modes and of the bending mode are very complex and very sensitive to weak changes in the basic physical parameters. It is therefore difficult to propose unambiguous assignments. However, it must be remarked that a simultaneous knowledge of the temperature and isotopic effects on the lineshapes partially remove this difficulty. The aim of Fig. 5 is to illustrate this proposal for weak H-bonds, involving simultaneously single or multiple Fermi resonances and the three kinds of relaxations. In this figure are reported 20 lineshapes and their corresponding Dirac delta peaks progression in the absence of the dampings. The two left columns contain SDs for H-species, the far left at 10 K and the middle left at 310 K, whereas the two right columns SDs are the corresponding ones for the D isotope at 10 K (middle right) and 310 K (far right). As references the four SDs without Fermi resonance have been reported at the top of the figure. Three possibilities of Fermi resonances have been considered in the middle of the figure. Following the choice of the angular frequency of the bending mode: (i) the SD of the Hspecies is affected by the Fermi resonances, (ii) the SD of the D-species is affected by the Fermi resonances, where that of the H-species is not, and (iii) the SDs of both H and D-species show the effects of the Fermi resonances. These examples illustrate the possibility to remove some ambiguities in the attempt to explain by Fermi resonances the features of the lineshapes. At last, the bottom
Table 3 Extraction of the Fermi parameterÕs L from the band widths of spectral densities involving Fermi resonances 1 Fermi resonance 1 LP i (cm ) ð i L2i Þ1=2 (cm1 )
1 Fermi resonance
2 Fermi resonances
3 Fermi resonances
4 Fermi resonances
50 50
100 100
50; 50 71
50; 75; 60 108
50; 75; 60; 40 115
Dirac peaks
rF (cm1 ) ðr2F r20 Þ1=2 (cm1 )
260 50
274 100
265 71
277 108
280 115
Lineshapes
rF (cm1 ) ðr2F r20 Þ1=2 (cm1 )
336 49
347 98
340 69
349 106
351 113
The band widths without Fermi resonances are: r0 ¼ 250:81 cm1 for SDs involving Dirac peaks and r0 ¼ 271:23 cm1 for corresponding lineshapes involving damped situations.
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
39
Fig. 5. Temperature and hydrogen–deuterium isotopic effects on the infrared spectral density of H-bonded species involving indirect relaxation and multiple Fermi resonances. X–H Y parameters: a0 ¼ 1, x0 ¼ 3000 cm1 . X–D Y parameters: a0 ¼ 0:75, x0 ¼ 2121 cm1 . Spectral densities with three Fermi resonances (L1 ¼ 50, L2 ¼ 75, L3 ¼ 50 cm1 ) with five L1 ¼ 50, L2 ¼ 75, L3 ¼ 50, L4 ¼ 30, L5 ¼ 60 cm1 . Common parameters: c0 ¼ 60 cm1 , c00 ¼ 15 cm1 , cdi ¼ 15 cm1 .
SDs involving five Fermi couplings exhibit all a strong sensitivity to the Fermi resonances.
4. Conclusion ! In order to get the mS ðX–H YÞ infrared lineshapes, we have given a theory which introduces the indirect damping of the H-bond bridge into the model of weak H-bond involving multiple non-resonant Fermi couplings. The approximations performed in our model are: (i) the adiabatic one, (ii) the harmonic one, for describing the potential of the H-bond bridge, and (iii) the exchange one, for the Fermi coupling. Our numerical investigations lead to the following conclusions: the Fermi resonances are sensitive for a range of angular frequencies which may be greater by one order of magnitude than that of the Fermi cou-
pling; moreover, the relative intensities of the different sub-bands depend on the magnitude of the three damping parameters. Besides the theoretical spectral densities account satisfactorily for the main infrared spectral features observed for Hbond species: (i) Band center is shifted towards low wave numbers. (ii) Important broadening, accompanied by asymmetry and subsidiary structure is equally observed. (iii) Evans windows may occur, the number of which is that of Fermi resonances when damping parameters become large. (iv) Isotopic D substitution results in a narrowing of the lineshapes with respect to that of the Hspecies, as well as in a change in the number and form of the Evans windows. (v) Lineshape width increases with temperature. In a forthcoming paper [27], the rotational structure of the H-bonds will be inserted in this model. Besides, in the future, we will generalize
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
this model by introducing the intrinsic anharmonicities of the slow, fast and bending modes, as well as the possibility of Davydov coupling, beyond the adiabatic and exchange approximations.
[10] [11] [12] [13] [14]
References [1] M. Boczar, M. W ojcik, K. Szczeponek, D. Jamr oz, S. Ikeda, Int. J. Quantum Chem. 90 (2002) 689. [2] O. Henri-Rousseau, P. Blaise, D. Chamma, in: S. Rice, I. Prigogine (Eds.), Adv. Chem. Phys., vol. 121, Wiley, New York, 2001, p. 241. [3] O. Henri-Rousseau, P. Blaise, in: D. Hadzi (Ed.), Theoretical Treatments of Hydrogen Bonding, Wiley, New York, 1997, p. 165. [4] O. Henri-Rousseau, P. Blaise, in: S. Rice, I. Prigogine (Eds.), Adv. Chem. Phys., vol. 103, Wiley, New York, 1998, p. 1. [5] G. Hofacker, Y. Marechal, M. Ratner, in: P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond, North Holland, Amsterdam, 1976, p. 297. [6] Y. Marechal, in: J.R. Durig (Ed.), Vibrational Spectra and Structure, vol. 16, Elsevier, Amsterdam, 1987, p. 311. [7] D. Hadzi, S. Bratos, in: P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond, North Holland, Amsterdam, 1976, p. 567. [8] S. Bratos, J. Chem. Phys. 63 (1975) 3499. [9] S. Bratos, H. Ratajczak, J. Chem. Phys. 76 (1982) 77.
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
Y. Marechal, Chem. Phys. 79 (1983) 85. Y. Marechal, Chem. Phys. 79 (1983) 69. A. Witkowski, M. W ojcik, Chem. Phys. 1 (1973) 9. Y. Marechal, A. Witkowski, J. Chem. Phys. 48 (1968) 3697. Y. Marechal, Ph.D. Thesis, University of Grenoble, France, 1968. O. Henri-Rousseau, D. Chamma, Chem. Phys. 229 (1998) 37. D. Chamma, O. Henri-Rousseau, Chem. Phys. 229 (1998) 51. M. W ojcik, Mol. Phys. 36 (1978) 1757. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 53. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 71. N. R€ osch, M. Ratner, J. Chem. Phys. 61 (1974) 3344. B. Boulil, O. Henri-Rousseau, P. Blaise, Chem. Phys. 126 (1988) 263. M. Giry, B. Boulil, O. Henri-Rousseau, C. R. Acad. Sci. Paris II 316 (1993) 455. B. Boulil, M. Giry, O. Henri-Rousseau, Phys. Status Solidi B 158 (1990) 629. K. Belhayara, P. Blaise, O. Henri-Rousseau, Chem. Phys. 293 (2003) 9. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 91. D. Chamma, J. Mol. Struct. 552 (2000) 187. K. Belhayara, D. Chamma, O. Henri-Rousseau, J. Mol. Struct., submitted.
IR spectral density of weak H-bonds involving indirect damping. Part III: application to situations with multiple Fermi resonances Khedidja Belhayara, Didier Chamma *, Olivier Henri-Rousseau Centre dÕEtudes Fondamentales, Universit e de Perpignan, 52 avenue de Villeneuve, F-66860 Perpignan cedex, France Received 10 February 2003; in final form 27 March 2003
Abstract The IR spectral density (SD) of the high frequency stretching mode of weak H-bonded complexes involving Fermi resonances is studied within the linear response theory from a full quantum mechanical point of view. The ! ! anharmonic coupling between the high frequency X–H Y and the low frequency X –H Y modes is treated inside the strong anharmonic coupling theory, whereas Fermi resonances are introduced in the spirit of the work of Witkowski and W ojcik [Chem. Phys. 1 (1973) 9]. The relaxation of the fast and bending modes (direct damping) is treated as usual, whereas the indirect damping is incorporated by aid of our results of part I [Chem. Phys. 293 (2003) 9], dealing with the quantum theory of driven damped quantum harmonic oscillator. The IR SD is obtained by Fourier transform of the autocorrelation function of the dipole moment operator of the fast mode. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Weak hydrogen bonds; Multiple Fermi resonances; Driven damped quantum harmonic oscillator; Effective non-hermitian Hamiltonian; Infrared spectra; Temperature and isotopic effects; Direct and indirect relaxations; Linear response theory; Autocorrelation function
1. Introduction In the area of the infrared lineshape of ! mS ðX–H YÞ stretching mode of H-bonded species, it is generally assumed that Fermi resonances must play an important role [1,2]. Among theories dealing with that [2–7], one may distinguish: (i) the
*
Corresponding author. Tel.: +33-468-662-108; fax: +33-468662-234. E-mail address: [email protected] (D. Chamma).
semi-classical ones [8,9], in which the H-bond bridge is treated classically and the fast mode quantum mechanically, (ii) the peeling off approach of Marechal [10,11], which attempts to remove the Fermi resonances effects in order to get the unperturbed lineshape, (iii) the quantum theories [12], in which both the H-bond bridge and the stretching mode are treated quantum mechanically. The first quantum theory of the ! mS ðX–H YÞ infrared spectral density of weak Hbonds is that of Marechal and Witkowski [13,14], who developed a strong anharmonic coupling
0301-0104/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00272-6
32
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
theory, according to which the high frequency ! mode X–H Y is anharmonically coupled to the ! low frequency mode X –H Y of the H-bond bridge. Using the fact that the frequency of the first mode is very large with respect to that of the second one, Marechal and Witkowski [13,14] applied an adiabatic approximation allowing to separate the slow and fast motions, they also considered the possibility of Davydov coupling, but ignored the possibility of Fermi resonances. In a later paper, Witkowski and W ojcik [12] incorporated in the above model the possibility of Fermi resonances. They considered on an equal footing the strong anharmonic coupling and the Fermi resonance, but ignored the damping. Later, a more general theory has been proposed by Chamma et al. [15,16], who considered the possibility of multiple Fermi resonances of non-resonant situations, took into account the direct dampings, but ignored the indirect relaxation of the H-bond bridge. The authors have shown that Fermi type interactions ought to be very universal since the perturbation of the lineshape remains sensitive even for energy gaps many times larger than the coupling. They have also shown that the relative intensities involved in the SDs are deeply affected by the relative magnitude of the dampings of the fast and the bending modes. There have been also attempts to treat simultaneously Davydov coupling and Fermi resonances, first without damping [17], and later with direct damping only [18,19]. In a precedent paper, we have shown that the quantum harmonic oscillator describing the
damped H-bond bridge may be described by aid of a reduced non-hermitian Hamiltonian. We apply here these results to treat the quantum indirect damping in situation where Fermi resonances are occurring. We shall consider that the damped fast mode is simultaneously coupled to the damped Hbond bridge (through the strong anharmonic coupling) and to one or several damped bending modes (through Fermi resonances). It will then be possible to get, in the Heisenberg picture, the expression of the dipole moment operator of the high frequency mode at time t, knowing its expression at initial time, and then to obtain the autocorrelation function of this operator which, according to the linear response theory, will lead to the infrared spectral density after Fourier transformation. The present model satisfactorily reduces to limit situations appearing in the literature, as shown in Table 1.
2. Theory b of the In order to construct the Hamiltonian H system, we need to define three vibrational modes and their associated parameters, shown in Table 2, ! namely the fast mode X–H Y, the slow mode ! X –H Y and a bending mode. The full Hamilb of a weak H-bond subject to a Fermi tonian H resonance may be written b ¼H bI þ H b II þ H b int : H
ð1Þ
b I describes the slow and fast The Hamiltonian H modes which are coupled together. Within the
Table 1 Some deeply interconnected adiabatic quantum theories dealing with the IR SD of weak H-bonds Authors and references
Strong anharmonic coupling
Marechal and Witkowski [13,14] Witkowski and W ojcik [12] R€ osch and Ratner [20] Boulil et al. [21] Giry et al. [22,23] Belhayara et al. [24] Chamma et al. [15,16] Present work
Fermi coupling
Relaxations
Single
Direct
Multiple
Bending
Indirect
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40 Table 2 Definition of some parameters associated with the three elementary quantum harmonic oscillators ! ! Fast mode X–H Y Slow mode X –H Y Angular frequency Position coordinate Conjugate momentum Reduced mass
x00 b Q Pb m00
x0 b q b p m0
strong anharmonic coupling theory [13,14], its expression is i2 p2 1 h b b bI ¼ 1 b H þ m0 x0 þ b Q q2 2 m0 2 1 Pb 2 1 b 2: þ þ m00 x200 Q ð2Þ 2 m00 2 Here b is the anharmonic coupling constant beb II is the tween the slow and fast modes. Besides, H Hamiltonian describing the bending mode p d2 1 b II ¼ 1 b H þ md x2d b q 2d : 2 md 2
b int ¼ l b H qb q 2d :
ð4Þ y
Let us define by jfkgi, jðmÞi and j½ui the eigenvectors of the zeroth-order quantum harmonic oscillators characterizing, respectively, the fast, the H-bond bridge and the bending modes. Then we may build the following tensorial basis: jW0 ðmÞi ¼ jf0gijðmÞij½0i; jW1 ðmÞi ¼ jf1gijðmÞij½0i; jW2 ðmÞi ¼ jf0gijðmÞij½2i:
Within the adiabatic approximation, which is well verified for weak H-bonds [25], the full Hamiltonian describing an H-bonded system where Fermi coupling, indirect damping of the Hbond bridge, direct relaxation of the fast mode and damping of the bending mode are simultaneously present, is 2 3 b f0g H 0 0 ind 7 bf ¼ 6 b f1g ihc0 H 4 0 5 H h b L Ind ind f0g b b 0 h L H ind þ 2hxd ihcd ð7Þ with b fkg ¼ khx0 þ hx00 fay a þ ka0 ðay þ aÞg H ind
c ðT Þ c ðT Þ 1 i 00 ka20 hx00 1 þ i 00 ; 2x00 2x00 k ¼ 0; 1: ð8Þ
y
Then, using the raising (b ; d ) and lowering (b; d) operators of the fast and bending modes, respectively, and performing the exchange approximation [15,25], the Hamiltonian (4) can be written as: i L hh y 2 2 b int ¼ p ffiffiffi b d þ b d y H with 2 sffiffiffiffiffiffiffiffiffiffiffi l h : ð5Þ L¼ 2 md xd m0 x0
ð6aÞ m ¼ 0; 1; 2; . . . ;
ð6bÞ ð6cÞ
Bending mode xd qd pd md
ð3Þ
b int accounts for the The interaction Hamiltonian H Fermi coupling between the fast mode and the first harmonic of the bending mode, through the anharmonic coupling parameter l:
33
Here c0 and cd are, respectively, the damping parameters of the fast and bending modes, which have been introduced in the spirit of the Green formalism, by considering specific imaginary diagonal energetic terms. ay and a are the raising and lowering operators of the slow mode. a0 is the adimensional anharmonic coupling parameter between the slow and fast modes, given by Eq. (4) of part I [24] and c00 ðT Þ is an effective damping parameter which is given also in part I [24] by Eq. (29). The generalization to multiple Fermi resonances is obvious (see [15]). Within the linear response theory, the infrared spectral density (SD) is given by the Fourier transformation of the autocorrelation function (ACF) GðtÞ of the dipole moment b l of the fast mode: Z þ1 IðxÞ ¼ GðtÞ expðixtÞ dt: ð9Þ 1
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
Fig. 1. SD involving weak H-bonds: connection between the present theory and precedent special ones to which it reduces when Fermi resonances, direct or indirect relaxations are ignored. (a) Situation without damping and Fermi resonance, (b) situation without indirect damping and Fermi resonance, (c) situation without Fermi resonance, (d) situation with Fermi resonance and without damping, (e) situation with Fermi resonance and direct damping, (f) situation with Fermi resonance and both direct and indirect dampings. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , T ¼ 300 K. Fermi resonances parameters: xd ¼ 1350 cm1 , L ¼ 50 cm1 .
At initial time the dipole moment operator is 1 X b l ð0Þ ¼ jf1gij½0ijðmÞihðmÞjh½0jhf0gj: ð10Þ m¼0
At time t, this operator may be obtained within the Heisenberg picture, by aid of an unitary transformation involving the Hamiltonian (7). Since this last Hamiltonian is non-hermitian, its eigenvalues xm are complex and its eigenvalue equation may be written: b f jUm i ¼ h x0 þ x0 a2 x00 icm jUm i; H ð11Þ Ind
m
0
where the imaginary part cm > 0 characterizes the limited life time of the corresponding energy level. After some manipulation, we get for the ACF: XX GðtÞ / expfkmg expfimx00 tg m m n c o exp m 00 t exp ix0m t expfcm tg 2 exp i x0 a20 x00 t jhUm jW1 ðmÞij2 : ð12Þ
Fig. 1 illustrates all possible connections between the different quantum mechanical models published up to now, showing how the SD obtained in the present model transforms when passing from the most general situation to some limit cases.
3. Numerical results and discussion 3.1. Assistance of Fermi resonance by the slow–fast mode anharmonic coupling For the situations where the indirect relaxation may be neglected, Chamma et al. [15,16,18,19] investigated in their previous papers the resonance condition between the first excited state of the fast mode and the excited states of some bending modes. They have shown that this condition does not need to be strictly fulfilled: strong modification of the X–H stretching bandshape can be obtained when 2xd x0 is as large as tenfold the Fermi
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
coupling parameter L. As it appears, the anharmonic coupling between the slow and fast modes assists the coupling between the first excited state of the fast mode and the first harmonic of the bending mode. We show here that this result, which was obtained after neglecting the indirect damping, remains to hold when all the three
35
damping mechanism are taken into account. Inspection of Fig. 2, shows that the SDs are affected by a Fermi coupling of 50 cm1 on a range of angular frequencies xd of 450 cm1 . Moreover, the SDs sensitivity to Fermi resonances does not depend on the relative magnitude of the direct and indirect dampings.
Fig. 2. Influence of Fermi resonances and of the relative magnitude of the direct and indirect damping parameters on the lineshape. The full line SDs are with Fermi resonances and dashed without. Besides, in each frame, the two up SDs are involving situations when the direct damping is greater than the indirect (c0 ¼ 60 cm1 , c00 ¼ 15 cm1 ), whereas it is the inverse for the two down SDs (c0 ¼ 15 cm1 , c00 ¼ 60 cm1 ). The Fermi coupling parameter is the same for all SDs (L ¼ 50 cm1 ) involving Fermi resonances whereas the angular frequency is progressively increased. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , cd ¼ 15 cm1 , T ¼ 300 K.
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
3.2. Influence of the relative magnitude of the different relaxation on the intensities of the subbands Chamma et al. [16] have also shown that in the absence of indirect damping the relative intensities of the sub-bands may be modified by changing the relative magnitude of the relaxation parameter c0 and cd affecting the fast and bending modes. We show here that the introduction of the indirect damping affects also the relative intensities of the sub-bands. That is illustrated in Fig. 3. 3.3. Multiple Fermi coupling and Evans windows The semi-classical theory of the indirect damping by Bratos and Ratajczak [9] including with
Fermi resonances leads to large Evans windows. The quantum theory of the indirect damping leads to more complex situations, where the distinction between windows following from Fermi resonances, and those related to the minima intensity resulting from the Franck–Condon progression is not obvious. Fig. 4 is devoted to the influence of the dampings on the Evans windows. It appears that the dampings become large, there are as many Evans windows as Fermi resonances. The quantum adiabatic description of weak Hbonds, working within the strong anharmonic coupling theory, leads to an abrupt change in the properties of the Hamiltonian of the H-bond bridge just after excitation of high frequency mode. In a first quantum representation, the effective Hamiltonian of the slow mode, which was
Fig. 3. Influence of indirect relaxation on the lineshape. Left column: evolution with c00 of the lineshape involving one Fermi resonance for c0 > cd . Right column: same evolution with c00 but for c0 < cd . The Dirac peaks correspond to the limit situation where c0 ¼ cd ¼ c00 ¼ 0. Common parameters: a0 ¼ 1, x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , L ¼ 50 cm1 , xd ¼ 1350 cm1 , T ¼ 300 K.
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
37
Fig. 4. Multiple Fermi resonances effects on the lineshape. The number nF of the Fermi resonances which are taken into account in the calculations increases from the top to the bottom of each column. Common parameters: a0 ¼ 1, Lu ¼ 50 cm1 , x0 ¼ 3000 cm1 , x00 ¼ 150 cm1 , xd1 ¼ 1350 cm1 , xd2 ¼ 1280 cm1 , xd3 ¼ 1400 cm1 , xd4 ¼ 1450 cm1 , xd5 ¼ 1500 cm1 .
undriven before excitation of the fast mode, becomes driven after excitation. In another equivalent representation, the slow mode jumps from a thermal equilibrium state (before excitation) to a coherent state (after excitation). In both quantum descriptions, which are equivalent, the abrupt change in the behavior of the H-bond bridge leads in the spectral density to a Franck–Condon progression of Dirac delta peaks, when dampings and Fermi resonances are ignored. In the presence of Fermi resonances, the situation becomes more complex and the spectral density involves puzzling Dirac delta peaks. Now, when only the dampings of the fast and bending modes are incorporated in the model, the spectral density is broadened, but keeps a complex structure, in which contrarily to the semi-classical theory of Bratos and Ratajczak [9], it is difficult to distinguish Evans windows. When the indirect damping of the H-bond bridge is also taken into account, it becomes possible to observe in the spectral density as many windows as Fermi resonances. This is a consequence of the fact that the indirect damping enhances the semi-clas-
sical behavior of the driven harmonic oscillator or of the coherent state, the mixing between the two quantum anharmonic couplings being thus partially broken. 3.4. Fermi coupling parameter L As observed by Marechal [10,11] and by some of us [25], for H-bonds involving a single Fermi resonance, the Fermi coupling parameter L may be extracted from the SD in the following way: L2 ¼ ðr2F r20 Þ:
ð13Þ
Here rF and r0 are, respectively, the SD widths with and without Fermi coupling, which are of the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ r ¼ hx2 i hxi2 : The average values involved in this last expression may be computed from the SD by aid of the equations:
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
R xIðxÞ dx hxi ¼ R IðxÞ dx
and
R
x2 IðxÞ dx hx i ¼ R IðxÞ dx 2
ð15Þ after performing a cut-off of the low and high frequencies tails. In Chamma et al. model [25,26], it was shown that Eq. (13) is verified for a single Fermi resonance with direct relaxation of the fast and bending modes, but without indirect damping. Here we show that Eq. (13) may be generalized to multiple Fermi resonances in which the indirect damping is occurring together with the above mentioned direct relaxation. The generalization of Eq. (13) to multiple Fermi couplings is X L2i ¼ ðr2F r20 Þ: ð16Þ i
Table 3 illustrates the validity of Eq. (16) for situations involving 1, 2, 3 or 4 Fermi resonances. Its inspection shows that the expression (16) is suitable in the absence of relaxation (Dirac peaks). On the other hand in presence of dampings, Eq. (16) is only approximately verified, because of the cutoff in the SD. It is important to underline that Eq. (16) may be widely used, its verification being independent from the existence and the nature of the relaxation. 3.5. Temperature and isotopic effects As we have seen above, the features of the ! mS ðX–H YÞ infrared profiles of weak H-bonds
involving Fermi resonances and relaxations of the slow and fast modes and of the bending mode are very complex and very sensitive to weak changes in the basic physical parameters. It is therefore difficult to propose unambiguous assignments. However, it must be remarked that a simultaneous knowledge of the temperature and isotopic effects on the lineshapes partially remove this difficulty. The aim of Fig. 5 is to illustrate this proposal for weak H-bonds, involving simultaneously single or multiple Fermi resonances and the three kinds of relaxations. In this figure are reported 20 lineshapes and their corresponding Dirac delta peaks progression in the absence of the dampings. The two left columns contain SDs for H-species, the far left at 10 K and the middle left at 310 K, whereas the two right columns SDs are the corresponding ones for the D isotope at 10 K (middle right) and 310 K (far right). As references the four SDs without Fermi resonance have been reported at the top of the figure. Three possibilities of Fermi resonances have been considered in the middle of the figure. Following the choice of the angular frequency of the bending mode: (i) the SD of the Hspecies is affected by the Fermi resonances, (ii) the SD of the D-species is affected by the Fermi resonances, where that of the H-species is not, and (iii) the SDs of both H and D-species show the effects of the Fermi resonances. These examples illustrate the possibility to remove some ambiguities in the attempt to explain by Fermi resonances the features of the lineshapes. At last, the bottom
Table 3 Extraction of the Fermi parameterÕs L from the band widths of spectral densities involving Fermi resonances 1 Fermi resonance 1 LP i (cm ) ð i L2i Þ1=2 (cm1 )
1 Fermi resonance
2 Fermi resonances
3 Fermi resonances
4 Fermi resonances
50 50
100 100
50; 50 71
50; 75; 60 108
50; 75; 60; 40 115
Dirac peaks
rF (cm1 ) ðr2F r20 Þ1=2 (cm1 )
260 50
274 100
265 71
277 108
280 115
Lineshapes
rF (cm1 ) ðr2F r20 Þ1=2 (cm1 )
336 49
347 98
340 69
349 106
351 113
The band widths without Fermi resonances are: r0 ¼ 250:81 cm1 for SDs involving Dirac peaks and r0 ¼ 271:23 cm1 for corresponding lineshapes involving damped situations.
K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
39
Fig. 5. Temperature and hydrogen–deuterium isotopic effects on the infrared spectral density of H-bonded species involving indirect relaxation and multiple Fermi resonances. X–H Y parameters: a0 ¼ 1, x0 ¼ 3000 cm1 . X–D Y parameters: a0 ¼ 0:75, x0 ¼ 2121 cm1 . Spectral densities with three Fermi resonances (L1 ¼ 50, L2 ¼ 75, L3 ¼ 50 cm1 ) with five L1 ¼ 50, L2 ¼ 75, L3 ¼ 50, L4 ¼ 30, L5 ¼ 60 cm1 . Common parameters: c0 ¼ 60 cm1 , c00 ¼ 15 cm1 , cdi ¼ 15 cm1 .
SDs involving five Fermi couplings exhibit all a strong sensitivity to the Fermi resonances.
4. Conclusion ! In order to get the mS ðX–H YÞ infrared lineshapes, we have given a theory which introduces the indirect damping of the H-bond bridge into the model of weak H-bond involving multiple non-resonant Fermi couplings. The approximations performed in our model are: (i) the adiabatic one, (ii) the harmonic one, for describing the potential of the H-bond bridge, and (iii) the exchange one, for the Fermi coupling. Our numerical investigations lead to the following conclusions: the Fermi resonances are sensitive for a range of angular frequencies which may be greater by one order of magnitude than that of the Fermi cou-
pling; moreover, the relative intensities of the different sub-bands depend on the magnitude of the three damping parameters. Besides the theoretical spectral densities account satisfactorily for the main infrared spectral features observed for Hbond species: (i) Band center is shifted towards low wave numbers. (ii) Important broadening, accompanied by asymmetry and subsidiary structure is equally observed. (iii) Evans windows may occur, the number of which is that of Fermi resonances when damping parameters become large. (iv) Isotopic D substitution results in a narrowing of the lineshapes with respect to that of the Hspecies, as well as in a change in the number and form of the Evans windows. (v) Lineshape width increases with temperature. In a forthcoming paper [27], the rotational structure of the H-bonds will be inserted in this model. Besides, in the future, we will generalize
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K. Belhayara et al. / Chemical Physics 293 (2003) 31–40
this model by introducing the intrinsic anharmonicities of the slow, fast and bending modes, as well as the possibility of Davydov coupling, beyond the adiabatic and exchange approximations.
[10] [11] [12] [13] [14]
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