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IR spectral density of weak H-bonds involving indirect damping. Part II: Davydov coupling
Chemical Physics 293 (2003) 23–30 www.elsevier.com/locate/chemphys
IR spectral density of weak H-bonds involving indirect damping. Part II: Davydov coupling Didier Chamma, Adina Velcescu, Paul Blaise *, Olivier Henri-Rousseau Centre dÕEtudes Fondamentales (EA-2986), Universit e de Perpignan, 52 avenue de Villeneuve, 66860 Perpignan cedex, France Received 10 February 2003; in final form 27 May 2003
Abstract A non-perturbative approach of the quantum indirect damping for cyclic dimers of weak H-bonds susceptible to involve Davydov coupling is proposed. The effective non-Hermitian Hamiltonians describing the damped H-bond bridge obtained in Part I [Chem. Phys. (2003) submitted] following the basic quantum model of the indirect damping [Chem. Phys. 126 (1988) 263], is introduced in the model of Marechal and Witkowski [J. Chem. Phys. 48 (1968) 3637]. A result which is equivalent to an infinite order expansion of a precedent perturbative approach [J. Mol. Struct. (Theochem.) 314 (1994) 101] is obtained. It allows us to obtain, within the linear response theory, the infrared lineshape of the mX–H stretching mode. The fine structure of the high frequency tail appears to be more smoothed than that of the low frequency one. 2003 Elsevier Science B.V. All rights reserved. Keywords: Driven damped quantum harmonic oscillator; Effective non-Hermitian Hamiltonians; Spectral density of weak H-bonds; Quantum direct and indirect damping; Davydov coupling
1. Introduction The quantum theories [1–3] of the infrared (IR) lineshapes of the mX–H mode of weak H-bonds are all based on the strong anharmonic coupling theory, according to which the high frequency mode is anharmonically coupled to the H-bond bridge through a dependence of the high frequency on the coordinate of the bridge. For weak H-bonds, it is possible to perform the adiabatic approximation allowing one to separate the fast motion of the high frequency mode from the slow one of the H-bond bridge. Marechal and Witkowski (MW) [4] have firstly considered bare H-bonds. That lead them to describe the harmonic oscillator characterizing the H-bond bridge by effective Hamiltonians, the nature of which depends on the degree of excitation of the fast mode. Later, R€ osch and Ratner [5] have introduced for bare H-bonds, the quantum theory of the ‘‘direct’’ damping, in which the high frequency
*
Corresponding author. Tel.: +33-4-68-662108; fax: +33-4-68-662234. E-mail address: [email protected] (P. Blaise).
0301-0104/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00271-4
24
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
mode relaxes toward the surrounding via dipolar interaction, whereas some of us [6–8], have treated the ‘‘indirect’’ damping of the high frequency mode via that of the H-bond bridge to which it is anharmonically coupled. On the other hand, MW [4] considered also for H-bonded cyclic dimers the possibility of Davydov coupling according to which the excitation of one of the two fast modes belonging to the dimer can be transferred to the other high frequency mode, but they ignored the damping as for bare H-bond. After the MW work, there have been many studies dealing with Davydov coupling, but ignoring relaxation and, among them, one in which Fermi resonances were considered together [9,10]. Flakus [11] was the first to introduce damping, but only in a phenomenological way. Besides, the quantum ‘‘direct’’ relaxation of the high frequency mode, as considered by R€ osch and Ratner [5] for bare H-bonds, has been introduced in the model by some of us [12,13]. Moreover, an attempt [8] has been performed to introduce the quantum indirect damping of bare H-bonds [6] through perturbative expansions using as unperturbed time evolution operator, the reduced time evolution operator of the bare H-bond. Unfortunately, this attempt was unsatisfactory because of the very slow convergence of the expansion. As a matter of fact, the theory of indirect damping for Davydov coupling is yet to be found. This is the purpose of the present paper, where we use the results of Part I [14], in which we have shown that the quantum model [6–8] describing the damped H-bond bridge may be expressed in terms of non-Hermitian Hamiltonians. Unfortunately there are many difficulties to treat simultaneously Davydov coupling, Fermi resonances (beyond the exchange approximation) and relaxation of the H-bond bridge, on which we are working today. However, we consider the present approach as a step toward a comprehensive theory of the IR lineshapes of weak H-bonds which we search to obtain [15–19]. We consider that Part II is in fact the conclusion of a work which started from an helpful discussion with Professor Witkowski, occurring in Paris in 1985, about the necessity to introduce the relaxation (and particularly the indirect one), in the situations of weak H-bonds involving Davydov coupling. Mechanisms
Davydov coupling
[4] [5] [6] [12] [24] Present approach
X
Direct damping
Indirect damping
X X X X
X X X
X X
2. Model and basic equations Now, let us consider the problem of weak H-bonds involving the Davydov effect. For instance, let us look at a cyclic dimer for which, after excitation of one of the mX–H stretching mode, there is the possibility of energy resonant transfer between the two mX–H stretching modes because of the Davydov coupling, as considered by Marechal and Witkowski in their pioneering work [4] on the spectral density of weak Hbonds. We define the following quantities playing a role in the present model: • Coordinate positions of the two degenerate fast modes: q1 and q2 . • Conjugate momentum of the two degenerate fast modes: p1 and p2 . • Reference angular frequency of the two degenerate fast modes: x . • Angular frequency of the two degenerate slow modes: x . • Coordinate positions of the two slow modes: Q1 and Q2 .
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
25
• Conjugate momentum of the two slow modes: P1 and P2 . • Reduced mass of the fast and slow modes: m and M. • Angular frequencies xðQi Þ of the fast modes as a function of Qi . Following MW [4], the effective Hamiltonian describing the two H-bond bridges involved in the cyclic dimer, when the fast mode is in its ground state, is simply the sum of the Hamiltonians of the two corresponding quantum harmonic oscillators. On the other hand, when one of the two fast modes of the dimer is in its first excited state, there is a possibility of resonance with another degenerate situation where the excitation has jumped on the second first excited state. Let jfkr gi, r ¼ 1; 2 be the eigenkets of the two harmonic oscillators characterizing the two identical fast modes of the cyclic dimer, and jðmr Þi the eigenstates of the two degenerate quantum harmonic oscillators describing the two H-bond bridges. Then, one may define the three sets of tensorial products: jW0 ðm; nÞi ¼ jf01 gi jðm1 Þi jf02 gi jðn2 Þi; jWA ðm; nÞi ¼ jf11 gi jðm1 Þi jf02 gi jðn2 Þi;
ð1Þ
jWB ðm; nÞi ¼ jf01 gi jðm1 Þi jf12 gi jðn2 Þi: The first one corresponds to the unexcited situation, whereas the two others correspond to the degenerate situation characterizing the excitation of the first excited state of one of the two degenerate fast modes. Now, follow the MW approach. In the first step, consider separately each part of the cyclic dimer. Then, the adiabatic approximation is performed on each monomer of the cyclic system (see, for instance, paper Part I [14]). Such an approximation leads to describe the H-bond bridge by effective Hamiltonians. For a single one, this effective Hamiltonian is either that of an harmonic oscillator if the fast mode is in its ground state, or that of a driven harmonic oscillator if the fast mode is excited. In the following step, one looks at the cyclic dimer for which one of the two identical fast mode has been excited. Then, because of the symmetry of the cyclic dimer, and of some coupling hV between the two degenerate fast mode excited state a strong non-adiabatic correction may occur through the Davydov resonance exchange between the two moities. Within the basis spanned by the above three tensorial sets (1), the representation of the Hamiltonian governing the weak H-bonds involving the Davydov coupling may be written [12], following Marechal and Witkowski [4], as
Now, let us introduce the direct and indirect damping in the spirit of Part I. Passing to the rising and þ lowering operators aþ 1 , a1 , a2 and a2 of the two harmonic oscillators describing the H-bond bridge, and introducing the dampings of the driven harmonic oscillators according to Part I, we obtain:
ð2Þ
c ceff þ þ þ 2 Hfr;sg ¼ hx 1 i eff ½aþ a þ rða þ a Þa þ sða þ a Þa þ a a
½a h x 1 þ i 1 2 1 1 1 2 2 2 2x 2x
hx þ ihc ðr þ sÞ;
ð3Þ
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D. Chamma et al. / Chemical Physics 293 (2003) 23–30
b a ¼ x
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi h : 2Mx
Here, ceff is a temperature dependent effective damping parameter characterizing the H-bond bridge which is given by Eq. (29) of Part I, whereas c is the direct damping parameter. In the following, we shall have to solve the eigenvalue equation of the full Hamiltonian (2) hcm jUm i: HDav jUm i ¼ ½hxm þ i The ACF of the dipole moment operator l of the fast mode is GDav ðtÞ / trfqlþ ð0ÞlðtÞg; where q is the Boltzmann operator and lðtÞ the dipole moment operator at time t. Of course the dipole moment at time t is related to that at initial time through an Heisenberg transformation involving the Hamiltonian (2). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the same Hamiltonian (2). Again, owing to the block property of the full Hamiltonian and to the fact that, even at room temperature, only the lower part of the energy spectrum plays a physical role so that the Boltzmann operator may be expressed in terms of Hf0;0g . For the cyclic dimer, the dipole moment operator of the fast mode at initial time t ¼ 0 must be written according to [12]: X X 1 lð0Þ ¼ ½jf01 gijf12 gi jf11 gijf02 gi 1m 1n hf01 gjhf02 gj 1m ¼ jðm1 Þihðm1 Þj and 1n ¼ jðn2 Þihðn2 Þj: 2 It may be shown that the ACF of the dipole moment operator of the dimer at temperature T : XXX hx ðm þ nÞhx exp f iðm þ nÞx tg exp f ðm þ nÞceff tg exp
GDav ðtÞ / kT m n v exp ixm expf cm tg exp iðx a2 x Þt expf c tg; ½hUm jW1 ðm; nÞi hUm jW2 ðm; nÞi ½hW1 ðm; nÞjUm i hW2 ðm; nÞjUm i : At last, within linear response theory, the SD is the Fourier transform of this ACF, i.e., Z 1 IDav ðxÞ / 2 Re GDav ðtÞ expf ixtg dt:
ð4Þ
ð5Þ
0
The present quantum model is physically well-behaved with respect to special situations: (i) When the indirect damping is absent (i.e., when ceff ¼ 0), it reduces to that considered by Chamma and HenriRousseau [12] in their approach of weak H-bonds involving Davydov coupling, and direct damping. (ii) When both direct and indirect damping are ignored (i.e., when c ¼ 0 and ceff ¼ 0), it reduces to that of Marechal and Witkowski [4] dealing with Davydov coupling. (iii) When Davydov coupling and the direct damping are vanishing (i.e., when V ¼ 0 and c ¼ 0), it reduces to that of our initial work [6] dealing with the quantum indirect relaxation of single H-bonds. (iv) When Davydov coupling is neglected together with direct and indirect damping (i.e., when V ¼ 0, c ¼ 0 and ceff ¼ 0), it reduces to the crude one of MW [4] involving Dirac delta peaks of Franck–Condon progressions. It may be of interest to observe that the present model may be considered as the result of the infinite order expansion involved in a precedent work dealing with the question [12]. The ACF (4) could be written according to the following equations:
2 GDav ðtÞ ¼ Gg ðtÞ Gþ Gg ðtÞ / tr qU IP ðtÞ ; u ðtÞ þ Gu ðtÞ exp iðx a x þ ic tÞ ; G ð6Þ u ðtÞ / tr qU ðtÞ : In these last equations, reduced time evolution operators are involved. The first one is that of a driven damped quantum harmonic oscillator in an interaction picture [8]
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
U IP ðtÞ ¼ expfia2 e c
t=2
þ UðtÞ a
sinðx tÞg expfia2 x tgeUðtÞa
27
;
where UðtÞ is given by Eq. (A.7) of Part I. On the other hand, the two other time evolution operators are given by a perturbative expansion with respect to the term V C2 , where C2 is the parity operator. This time evolution operators, up to second order, are given by [8]
2 Z t Z t Z t0 i i U ðtÞ ¼ U IP ðtÞ 1 V U IP ðt0 ÞC2 U IP ðt0 Þdt0 U IP ðt0 ÞC2 U IP ðt0 Þ dt0 U IP ðt00 ÞC2 U IP ðt00 Þdt00 : V h h 0 0 0 3. Numerical results The computation of the SDs requires to represent the Hamiltonian (2) in a basis which is the tensorial product of the two subspaces spanning the two harmonic oscillators, labelled 1 and 2 in Eq. (2) and describing the two H-bond bridges of the cyclic dimer. Truncated basis sets were used for each oscillator. Let N be the dimension of one of the two subspaces. The dimension of the tensorial product of the two truncated bases is N 2 . Then, the dimension of the matrix representation of the 2 2 block matrix operator appearing in Eq. (2), will be 2N 2 2N 2 . Practically, for numerical stabilization, it is required to take N ¼ 25, so that the dimension of the matrix representation of the ‘‘2 2 block’’ is 1250 1250. Since the matrix is non-Hermitian, its dimension is in practice 2500 2500, leading to the calculation of 6:5 106 matrix elements. Among cyclic dimers susceptible to involve Davydov coupling there are the dimers of carboxylic acids [20], but also the dimers of compounds involving the N–H bonds [21]. Besides, for the dimers of carboxylic acids, it is generally assumed in the literature that, if Davydov coupling is occurring it is not alone but together with multiple Fermi resonances. However, in the present paper, we shall focus our attention on direct and indirect relaxations appearing in situations where Davydov coupling is occurring without Fermi resonance. The present aim is only methodological. We propose here an approach which is equivalent to the infinite perturbative expansion published 10 years ago and dealing only with Davydov coupling [8]. Moreover, if we ignore here the combined effects of Davydov coupling and Fermi resonances, already studied in precedent works dealing only with direct damping, this is because the indirect damping introduces complexities in the calculations: for cyclic dimers involving indirect relaxation, to which we limit ourselves, as we shall see, we have to diagonalize matrices of dimensions around 2500 2500, whereas for cyclic dimers involving several Fermi resonances (around 10 for instance according to Marechal [22]), the dimension would be around 2:5 104 2:5 104 within the exchange approximation, which is otherwise not well verified. Beyond this last approximation, the dimension would be around 106 106 . As a matter of fact, in order to treat quantum indirect damping in situations where Davydov coupling and multiple Fermi resonances are occurring together, a further progress is necessary; this will require to introduce symmetry, as in the work of MW, in the coupling of the two H-bond bridges of the cyclic dimer with the medium. That will in turn require to introduce symmetry in the Louisell and Walker [23] model of quantum driven damped harmonic oscillator, which is at the basis of the quantum theories of indirect relaxation. Different theoretical lineshapes computed at very low temperature (10 K) and room temperature (300 K), for which the magnitude of the indirect damping parameter is progressively increased, are shown in Figs. 1 and 2. We have chosen for the parameters governing the cyclic dimer in the absence of relaxation values near those used by Marechal and Witkowski [4], i.e., x ¼ 3000 cm 1 , x ¼ 150 cm 1 , V ¼ 105 cm 1 , a ¼ 1 cm 1 . On the other hand, we have used c ¼ 30 cm 1 for all the computations. At last, at 10 K, we have taken c ¼ 0, 7.5, 15, 30, and 60 cm 1 , and at 300 K, we have computed their corresponding values by aid of Eq. (29) of Part I. Figs. 1 and 2 show that when the indirect damping is progressively increased, there is a more sensitive smoothing of the high frequency tail of the lineshapes than of the low frequency one. As a consequence, for
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D. Chamma et al. / Chemical Physics 293 (2003) 23–30
Fig. 1. Davydov coupling with both direct and indirect dampings: spectral densities vs wavenumbers at T ¼ 10 K. The common parameters are: V ¼ 105 cm 1 , a ¼ 1, x ¼ 3000 cm 1 , x ¼ 150 cm 1 .
large indirect damping a fine structure remains on the low frequency tail, whereas there is a complete lack of fine structure on the high frequency tail. This general trend, as verified by other computations that we do not give here, appears to be the same at 10 and 300 K. The above mentioned trend is specific to indirect relaxation, since it is not observed when this relaxation is ignored [12,13], but not specific to Davydov coupling, because also observed for situations involving Fermi resonances. However, this trend completely disappears for bare H-bonds, in which the indirect relaxation induces an asymmetry of the lineshapes together with a tendency to destroy more deeply its high frequency tail. Our results show therefore that the specific role played by indirect damping in the features of the lineshapes of weak H-bonds cannot be ignored in situations where Davydov coupling is occurring. 4. Conclusion Our work is dealing with the IR lineshapes of the mX–H stretching mode of cyclic dimers of weak H-bonds susceptible to involve Davydov coupling. We have given here the first theoretical approach, which is non-perturbative, for the introduction of quantum indirect damping in such situations. The numerical
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
29
Fig. 2. Davydov coupling with both direct and indirect dampings: spectral densities vs wavenumbers at T ¼ 300 K. The common parameters are: V ¼ 105 cm 1 , a ¼ 1, x ¼ 3000 cm 1 , x ¼ 150 cm 1 .
calculation shows that the indirect damping plays a specific role in the features of the lineshapes of weak Hbonds by more favouring the fine structure of the low frequency tail than that of the high frequency one. As a consequence, we think that the indirect damping cannot be ignored in situations where Davydov coupling occurs. We leave for the future the treatment of the general problem of weak H-bonds involving together direct and indirect dampings, or Fermi resonances and Davydov coupling.
References [1] [2] [3] [4] [5] [6]
O. Henri-Rousseau, P. Blaise, in: D. Hadzi (Ed.), Theoretical Treatment of Hydrogen Bonding, Wiley, New York, 1997. O. Henri-Rousseau, P. Blaise, in: I. Prigogine, S.A. Rice (Eds.), Adv. Chem. Phys., Wiley, New York, vol. 103, 1998, p. 1. O. Henri-Rousseau, P. Blaise, D. Chamma, in: I. Prigogine, S.A. Rice (Eds.), Adv. Chem. Phys., Wiley, New York, vol. 121, 2002. Y. Marechal, A. Witkowski, J. Chem. Phys. 48 (1968) 3637. N. R€ osch, M. Ratner, J. Chem. Phys. 61 (1974) 3344. B. Boulil, O. Henri-Rousseau, P. Blaise, Chem. Phys. 126 (1988) 263.
30 [7] [8] [9] [10] [11]
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
D. Chamma et al. / Chemical Physics 293 (2003) 23–30 B. Boulil, J-L. Dejardin, N. El-Ghandour, O. Henri-Rousseau, J. Mol. Struct. (Theochem.) 314 (1994) 83. B. Boulil, P. Blaise, O. Henri-Rousseau, J. Mol. Struct. (Theochem.) 314 (1994) 101. M. Wojcik, Int. J. Quantum Chem. 10 (1976) 747. M. Boczar, M. Wojcik, K. Szczeponek, D. Jamroz, A. Zieba, B. Kawalek, Chem. Phys. 286 (2003) 63. H. Flakus, Acta Phys. Pol. A 53 (1978) 287; Chem. Phys. 50 (1980) 79; Chem. Phys. 62 (1981) 103; J. Mol. Struct. (Theochem.) 187 (1989) 35; J. Mol. Struct. (Theochem.) 285 (1993) 281; J. Mol. Struct. (Theochem.) 318 (1994) 15. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 53. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 71. K. Belharaya, P. Blaise, O. Henri-Rousseau, Chem. Phys. 293 (2003) 9. D. Chamma, O. Henri-Rousseau, Chem. Phys. 229 (1998) 51. D. Chamma, O. Henri-Rousseau, Chem. Phys. 248 (1999) 91. D. Chamma, J. Mol. Struct. 552 (2000) 187. P. Blaise, O. Henri-Rousseau, Chem. Phys. 243 (1999) 229. P. Blaise, O. Henri-Rousseau, Chem. Phys. 256 (2000) 85. Y Marechal, in: J. Durig (Ed.), Vibrational Spectra and Structure, vol. 16, Elsevier, Amsterdam, 1987, p. 312. H. Flakus, A. Machelska, Spectrochim. Acta A: Mol. Biomol. Spectrosc. 58 (2002) 553; H. Flakus, A. Miros, P. Jones, Spectrochim. Acta Part A: Mol. Biomol. Spectrosc. 58 (2002) 225. Y. Marechal, Chem. Phys. 79 (1983) 85. W. Louisell, L. Walker, Phys. Rev. 137 (1965) 204. P. Blaise, O. Henri-Rousseau, A. Grandjean, Chem. Phys. 244 (1999) 405.
IR spectral density of weak H-bonds involving indirect damping. Part II: Davydov coupling Didier Chamma, Adina Velcescu, Paul Blaise *, Olivier Henri-Rousseau Centre dÕEtudes Fondamentales (EA-2986), Universit e de Perpignan, 52 avenue de Villeneuve, 66860 Perpignan cedex, France Received 10 February 2003; in final form 27 May 2003
Abstract A non-perturbative approach of the quantum indirect damping for cyclic dimers of weak H-bonds susceptible to involve Davydov coupling is proposed. The effective non-Hermitian Hamiltonians describing the damped H-bond bridge obtained in Part I [Chem. Phys. (2003) submitted] following the basic quantum model of the indirect damping [Chem. Phys. 126 (1988) 263], is introduced in the model of Marechal and Witkowski [J. Chem. Phys. 48 (1968) 3637]. A result which is equivalent to an infinite order expansion of a precedent perturbative approach [J. Mol. Struct. (Theochem.) 314 (1994) 101] is obtained. It allows us to obtain, within the linear response theory, the infrared lineshape of the mX–H stretching mode. The fine structure of the high frequency tail appears to be more smoothed than that of the low frequency one. 2003 Elsevier Science B.V. All rights reserved. Keywords: Driven damped quantum harmonic oscillator; Effective non-Hermitian Hamiltonians; Spectral density of weak H-bonds; Quantum direct and indirect damping; Davydov coupling
1. Introduction The quantum theories [1–3] of the infrared (IR) lineshapes of the mX–H mode of weak H-bonds are all based on the strong anharmonic coupling theory, according to which the high frequency mode is anharmonically coupled to the H-bond bridge through a dependence of the high frequency on the coordinate of the bridge. For weak H-bonds, it is possible to perform the adiabatic approximation allowing one to separate the fast motion of the high frequency mode from the slow one of the H-bond bridge. Marechal and Witkowski (MW) [4] have firstly considered bare H-bonds. That lead them to describe the harmonic oscillator characterizing the H-bond bridge by effective Hamiltonians, the nature of which depends on the degree of excitation of the fast mode. Later, R€ osch and Ratner [5] have introduced for bare H-bonds, the quantum theory of the ‘‘direct’’ damping, in which the high frequency
*
Corresponding author. Tel.: +33-4-68-662108; fax: +33-4-68-662234. E-mail address: [email protected] (P. Blaise).
0301-0104/$ - see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0301-0104(03)00271-4
24
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
mode relaxes toward the surrounding via dipolar interaction, whereas some of us [6–8], have treated the ‘‘indirect’’ damping of the high frequency mode via that of the H-bond bridge to which it is anharmonically coupled. On the other hand, MW [4] considered also for H-bonded cyclic dimers the possibility of Davydov coupling according to which the excitation of one of the two fast modes belonging to the dimer can be transferred to the other high frequency mode, but they ignored the damping as for bare H-bond. After the MW work, there have been many studies dealing with Davydov coupling, but ignoring relaxation and, among them, one in which Fermi resonances were considered together [9,10]. Flakus [11] was the first to introduce damping, but only in a phenomenological way. Besides, the quantum ‘‘direct’’ relaxation of the high frequency mode, as considered by R€ osch and Ratner [5] for bare H-bonds, has been introduced in the model by some of us [12,13]. Moreover, an attempt [8] has been performed to introduce the quantum indirect damping of bare H-bonds [6] through perturbative expansions using as unperturbed time evolution operator, the reduced time evolution operator of the bare H-bond. Unfortunately, this attempt was unsatisfactory because of the very slow convergence of the expansion. As a matter of fact, the theory of indirect damping for Davydov coupling is yet to be found. This is the purpose of the present paper, where we use the results of Part I [14], in which we have shown that the quantum model [6–8] describing the damped H-bond bridge may be expressed in terms of non-Hermitian Hamiltonians. Unfortunately there are many difficulties to treat simultaneously Davydov coupling, Fermi resonances (beyond the exchange approximation) and relaxation of the H-bond bridge, on which we are working today. However, we consider the present approach as a step toward a comprehensive theory of the IR lineshapes of weak H-bonds which we search to obtain [15–19]. We consider that Part II is in fact the conclusion of a work which started from an helpful discussion with Professor Witkowski, occurring in Paris in 1985, about the necessity to introduce the relaxation (and particularly the indirect one), in the situations of weak H-bonds involving Davydov coupling. Mechanisms
Davydov coupling
[4] [5] [6] [12] [24] Present approach
X
Direct damping
Indirect damping
X X X X
X X X
X X
2. Model and basic equations Now, let us consider the problem of weak H-bonds involving the Davydov effect. For instance, let us look at a cyclic dimer for which, after excitation of one of the mX–H stretching mode, there is the possibility of energy resonant transfer between the two mX–H stretching modes because of the Davydov coupling, as considered by Marechal and Witkowski in their pioneering work [4] on the spectral density of weak Hbonds. We define the following quantities playing a role in the present model: • Coordinate positions of the two degenerate fast modes: q1 and q2 . • Conjugate momentum of the two degenerate fast modes: p1 and p2 . • Reference angular frequency of the two degenerate fast modes: x . • Angular frequency of the two degenerate slow modes: x . • Coordinate positions of the two slow modes: Q1 and Q2 .
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
25
• Conjugate momentum of the two slow modes: P1 and P2 . • Reduced mass of the fast and slow modes: m and M. • Angular frequencies xðQi Þ of the fast modes as a function of Qi . Following MW [4], the effective Hamiltonian describing the two H-bond bridges involved in the cyclic dimer, when the fast mode is in its ground state, is simply the sum of the Hamiltonians of the two corresponding quantum harmonic oscillators. On the other hand, when one of the two fast modes of the dimer is in its first excited state, there is a possibility of resonance with another degenerate situation where the excitation has jumped on the second first excited state. Let jfkr gi, r ¼ 1; 2 be the eigenkets of the two harmonic oscillators characterizing the two identical fast modes of the cyclic dimer, and jðmr Þi the eigenstates of the two degenerate quantum harmonic oscillators describing the two H-bond bridges. Then, one may define the three sets of tensorial products: jW0 ðm; nÞi ¼ jf01 gi jðm1 Þi jf02 gi jðn2 Þi; jWA ðm; nÞi ¼ jf11 gi jðm1 Þi jf02 gi jðn2 Þi;
ð1Þ
jWB ðm; nÞi ¼ jf01 gi jðm1 Þi jf12 gi jðn2 Þi: The first one corresponds to the unexcited situation, whereas the two others correspond to the degenerate situation characterizing the excitation of the first excited state of one of the two degenerate fast modes. Now, follow the MW approach. In the first step, consider separately each part of the cyclic dimer. Then, the adiabatic approximation is performed on each monomer of the cyclic system (see, for instance, paper Part I [14]). Such an approximation leads to describe the H-bond bridge by effective Hamiltonians. For a single one, this effective Hamiltonian is either that of an harmonic oscillator if the fast mode is in its ground state, or that of a driven harmonic oscillator if the fast mode is excited. In the following step, one looks at the cyclic dimer for which one of the two identical fast mode has been excited. Then, because of the symmetry of the cyclic dimer, and of some coupling hV between the two degenerate fast mode excited state a strong non-adiabatic correction may occur through the Davydov resonance exchange between the two moities. Within the basis spanned by the above three tensorial sets (1), the representation of the Hamiltonian governing the weak H-bonds involving the Davydov coupling may be written [12], following Marechal and Witkowski [4], as
Now, let us introduce the direct and indirect damping in the spirit of Part I. Passing to the rising and þ lowering operators aþ 1 , a1 , a2 and a2 of the two harmonic oscillators describing the H-bond bridge, and introducing the dampings of the driven harmonic oscillators according to Part I, we obtain:
ð2Þ
c ceff þ þ þ 2 Hfr;sg ¼ hx 1 i eff ½aþ a þ rða þ a Þa þ sða þ a Þa þ a a
½a h x 1 þ i 1 2 1 1 1 2 2 2 2x 2x
hx þ ihc ðr þ sÞ;
ð3Þ
26
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
b a ¼ x
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi h : 2Mx
Here, ceff is a temperature dependent effective damping parameter characterizing the H-bond bridge which is given by Eq. (29) of Part I, whereas c is the direct damping parameter. In the following, we shall have to solve the eigenvalue equation of the full Hamiltonian (2) hcm jUm i: HDav jUm i ¼ ½hxm þ i The ACF of the dipole moment operator l of the fast mode is GDav ðtÞ / trfqlþ ð0ÞlðtÞg; where q is the Boltzmann operator and lðtÞ the dipole moment operator at time t. Of course the dipole moment at time t is related to that at initial time through an Heisenberg transformation involving the Hamiltonian (2). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the same Hamiltonian (2). Again, owing to the block property of the full Hamiltonian and to the fact that, even at room temperature, only the lower part of the energy spectrum plays a physical role so that the Boltzmann operator may be expressed in terms of Hf0;0g . For the cyclic dimer, the dipole moment operator of the fast mode at initial time t ¼ 0 must be written according to [12]: X X 1 lð0Þ ¼ ½jf01 gijf12 gi jf11 gijf02 gi 1m 1n hf01 gjhf02 gj 1m ¼ jðm1 Þihðm1 Þj and 1n ¼ jðn2 Þihðn2 Þj: 2 It may be shown that the ACF of the dipole moment operator of the dimer at temperature T : XXX hx ðm þ nÞhx exp f iðm þ nÞx tg exp f ðm þ nÞceff tg exp
GDav ðtÞ / kT m n v exp ixm expf cm tg exp iðx a2 x Þt expf c tg; ½hUm jW1 ðm; nÞi hUm jW2 ðm; nÞi ½hW1 ðm; nÞjUm i hW2 ðm; nÞjUm i : At last, within linear response theory, the SD is the Fourier transform of this ACF, i.e., Z 1 IDav ðxÞ / 2 Re GDav ðtÞ expf ixtg dt:
ð4Þ
ð5Þ
0
The present quantum model is physically well-behaved with respect to special situations: (i) When the indirect damping is absent (i.e., when ceff ¼ 0), it reduces to that considered by Chamma and HenriRousseau [12] in their approach of weak H-bonds involving Davydov coupling, and direct damping. (ii) When both direct and indirect damping are ignored (i.e., when c ¼ 0 and ceff ¼ 0), it reduces to that of Marechal and Witkowski [4] dealing with Davydov coupling. (iii) When Davydov coupling and the direct damping are vanishing (i.e., when V ¼ 0 and c ¼ 0), it reduces to that of our initial work [6] dealing with the quantum indirect relaxation of single H-bonds. (iv) When Davydov coupling is neglected together with direct and indirect damping (i.e., when V ¼ 0, c ¼ 0 and ceff ¼ 0), it reduces to the crude one of MW [4] involving Dirac delta peaks of Franck–Condon progressions. It may be of interest to observe that the present model may be considered as the result of the infinite order expansion involved in a precedent work dealing with the question [12]. The ACF (4) could be written according to the following equations:
2 GDav ðtÞ ¼ Gg ðtÞ Gþ Gg ðtÞ / tr qU IP ðtÞ ; u ðtÞ þ Gu ðtÞ exp iðx a x þ ic tÞ ; G ð6Þ u ðtÞ / tr qU ðtÞ : In these last equations, reduced time evolution operators are involved. The first one is that of a driven damped quantum harmonic oscillator in an interaction picture [8]
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
U IP ðtÞ ¼ expfia2 e c
t=2
þ UðtÞ a
sinðx tÞg expfia2 x tgeUðtÞa
27
;
where UðtÞ is given by Eq. (A.7) of Part I. On the other hand, the two other time evolution operators are given by a perturbative expansion with respect to the term V C2 , where C2 is the parity operator. This time evolution operators, up to second order, are given by [8]
2 Z t Z t Z t0 i i U ðtÞ ¼ U IP ðtÞ 1 V U IP ðt0 ÞC2 U IP ðt0 Þdt0 U IP ðt0 ÞC2 U IP ðt0 Þ dt0 U IP ðt00 ÞC2 U IP ðt00 Þdt00 : V h h 0 0 0 3. Numerical results The computation of the SDs requires to represent the Hamiltonian (2) in a basis which is the tensorial product of the two subspaces spanning the two harmonic oscillators, labelled 1 and 2 in Eq. (2) and describing the two H-bond bridges of the cyclic dimer. Truncated basis sets were used for each oscillator. Let N be the dimension of one of the two subspaces. The dimension of the tensorial product of the two truncated bases is N 2 . Then, the dimension of the matrix representation of the 2 2 block matrix operator appearing in Eq. (2), will be 2N 2 2N 2 . Practically, for numerical stabilization, it is required to take N ¼ 25, so that the dimension of the matrix representation of the ‘‘2 2 block’’ is 1250 1250. Since the matrix is non-Hermitian, its dimension is in practice 2500 2500, leading to the calculation of 6:5 106 matrix elements. Among cyclic dimers susceptible to involve Davydov coupling there are the dimers of carboxylic acids [20], but also the dimers of compounds involving the N–H bonds [21]. Besides, for the dimers of carboxylic acids, it is generally assumed in the literature that, if Davydov coupling is occurring it is not alone but together with multiple Fermi resonances. However, in the present paper, we shall focus our attention on direct and indirect relaxations appearing in situations where Davydov coupling is occurring without Fermi resonance. The present aim is only methodological. We propose here an approach which is equivalent to the infinite perturbative expansion published 10 years ago and dealing only with Davydov coupling [8]. Moreover, if we ignore here the combined effects of Davydov coupling and Fermi resonances, already studied in precedent works dealing only with direct damping, this is because the indirect damping introduces complexities in the calculations: for cyclic dimers involving indirect relaxation, to which we limit ourselves, as we shall see, we have to diagonalize matrices of dimensions around 2500 2500, whereas for cyclic dimers involving several Fermi resonances (around 10 for instance according to Marechal [22]), the dimension would be around 2:5 104 2:5 104 within the exchange approximation, which is otherwise not well verified. Beyond this last approximation, the dimension would be around 106 106 . As a matter of fact, in order to treat quantum indirect damping in situations where Davydov coupling and multiple Fermi resonances are occurring together, a further progress is necessary; this will require to introduce symmetry, as in the work of MW, in the coupling of the two H-bond bridges of the cyclic dimer with the medium. That will in turn require to introduce symmetry in the Louisell and Walker [23] model of quantum driven damped harmonic oscillator, which is at the basis of the quantum theories of indirect relaxation. Different theoretical lineshapes computed at very low temperature (10 K) and room temperature (300 K), for which the magnitude of the indirect damping parameter is progressively increased, are shown in Figs. 1 and 2. We have chosen for the parameters governing the cyclic dimer in the absence of relaxation values near those used by Marechal and Witkowski [4], i.e., x ¼ 3000 cm 1 , x ¼ 150 cm 1 , V ¼ 105 cm 1 , a ¼ 1 cm 1 . On the other hand, we have used c ¼ 30 cm 1 for all the computations. At last, at 10 K, we have taken c ¼ 0, 7.5, 15, 30, and 60 cm 1 , and at 300 K, we have computed their corresponding values by aid of Eq. (29) of Part I. Figs. 1 and 2 show that when the indirect damping is progressively increased, there is a more sensitive smoothing of the high frequency tail of the lineshapes than of the low frequency one. As a consequence, for
28
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
Fig. 1. Davydov coupling with both direct and indirect dampings: spectral densities vs wavenumbers at T ¼ 10 K. The common parameters are: V ¼ 105 cm 1 , a ¼ 1, x ¼ 3000 cm 1 , x ¼ 150 cm 1 .
large indirect damping a fine structure remains on the low frequency tail, whereas there is a complete lack of fine structure on the high frequency tail. This general trend, as verified by other computations that we do not give here, appears to be the same at 10 and 300 K. The above mentioned trend is specific to indirect relaxation, since it is not observed when this relaxation is ignored [12,13], but not specific to Davydov coupling, because also observed for situations involving Fermi resonances. However, this trend completely disappears for bare H-bonds, in which the indirect relaxation induces an asymmetry of the lineshapes together with a tendency to destroy more deeply its high frequency tail. Our results show therefore that the specific role played by indirect damping in the features of the lineshapes of weak H-bonds cannot be ignored in situations where Davydov coupling is occurring. 4. Conclusion Our work is dealing with the IR lineshapes of the mX–H stretching mode of cyclic dimers of weak H-bonds susceptible to involve Davydov coupling. We have given here the first theoretical approach, which is non-perturbative, for the introduction of quantum indirect damping in such situations. The numerical
D. Chamma et al. / Chemical Physics 293 (2003) 23–30
29
Fig. 2. Davydov coupling with both direct and indirect dampings: spectral densities vs wavenumbers at T ¼ 300 K. The common parameters are: V ¼ 105 cm 1 , a ¼ 1, x ¼ 3000 cm 1 , x ¼ 150 cm 1 .
calculation shows that the indirect damping plays a specific role in the features of the lineshapes of weak Hbonds by more favouring the fine structure of the low frequency tail than that of the high frequency one. As a consequence, we think that the indirect damping cannot be ignored in situations where Davydov coupling occurs. We leave for the future the treatment of the general problem of weak H-bonds involving together direct and indirect dampings, or Fermi resonances and Davydov coupling.
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