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The interaction of the easily polarizable hydrogen bonds with phonons and polaritons of the thermal bath-far infrared continua
Journal of Molecular Structure (Theochem) 500 (2000) 421–427 www.elsevier.nl/locate/theochem
The interaction of the easily polarizable hydrogen bonds with phonons and polaritons of the thermal bath-far infrared continua A. Hayd, G. Zundel* Institute of Physical Chemistry, University of Munich, Butenandstr., D-81377 Munich, Germany
Abstract B⫹ H…B O B…H⫹ B or AH…B O ⫺ A…H⫹ B hydrogen bonds, respectively, with double minimum proton potential well or broad flat potential well cause continuous absorptions in the infrared spectra. The intensity of these infrared continua is often very large, i.e. their integrated intensity is much larger than that of infrared bands. Short strong hydrogen bonds with broad flat in the average largely symmetrical proton potentials in which the protons fluctuate, show so-called proton polarizabilities. With such systems, the infrared continua extend in the far infrared region, i.e. in the region below 600 cm ⫺1. It is, however known that with such bonds no proton transitions occur in this region. It is demonstrated that the intensity of these infrared continuous absorptions is strongly increased by the coupling of the hydrogen bonds with large proton polarizability with the phonons of the thermal bath around these hydrogen bonds. Further, it is demonstrated that the pronounced absorption in the far infrared region, i.e. the region below 600 cm ⫺1, occurs due to the strong coupling of these hydrogen bonds to the polaritons. The polaritons are the quanta of the thermal bath coupled to the phonons of the thermal bath. 䉷 2000 Elsevier Science B.V. All rights reserved. Keywords: Polaritons; Phonons; Thermal bath-far infrared
1. Introduction Homoconjugated B⫹ H…B O B…H⫹ B hydrogen bonds as well as heteroconjugated AH…B O ⫺ … ⫹ A H B bonds with double minimum proton potential show so-called proton polarizabilities caused by shifts of the proton within these bonds. These proton polarizabilities are about two orders of magnitude larger than the usual polarizabilities due to distortion of electron systems [1–6]. They are observed in the region below 3000 cm ⫺1. These infrared continua arise due to the interaction of these hydrogen bonds with large proton polarizability with their environ* Corresponding author. Present address: Bruno Walter-Straße 2, A-5020 Salzburg, Austria. Tel.: ⫹ 43-662-642311; fax: ⫹ 43-662642311.
ments. One reason for the occurrence of these infrared continua are the interactions of these polarizable hydrogen bonds with the local electrical fields from their environments [2]. Therefore, all polarizable hydrogen bonds having a broad flat proton potential with small barriers or without barriers are more or less strongly polarized by electrical fields. In addition, with short strong hydrogen bonds with almost symmetrical proton potentials continua are found. They extend down to 150 cm ⫺1 to the far infrared region. With such bonds, however no proton transitions occur in the region below 600 cm ⫺1 [3]. It will be demonstrated that these far infrared continua arise due to the interaction of these easily polarizable hydrogen bonds with the polaritons. The polaritons are the quanta of the thermal bath coupled to the phonons of the thermal bath [7].
0166-1280/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(00)00389-4
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A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
Fig. 1. FT-IR spectra region (1400–800 cm ⫺1) of 1:1 MSA ⫹ oxide base mixtures in acetonitrile–chloroform (2:1), concentration 0.5 mol dm ⫺3: at 20⬚C ( - - - -), at 40⬚C (—) and pure base in acetonitrile (– – – ) at room temperature, layer thickness 100 mm: (a) MSA ⫹ DPhSO; (b) MSA ⫹ TPhPO, TPhPO (saturated solution); (c) MSA ⫹ DBzSO, DBzSO (saturated solution); (d) MSA ⫹ DMSO; (e) MSA ⫹ TBPO; and (f) MSA ⫹ TPhAsO, TPhAsO (saturated solution) (from Ref. [12]).
2. Results and discussion Fig. 1 shows the infrared spectra of the methanesulfonic acid (MSA) ⫹ oxide family as an example of a family of systems [8,9]. The systems—their pKa values [10] as far as they are available—and the abbreviations are summarized in Table 1. The pKa values, of these oxides increases within this series. The pKa values of these sulfoxides are used as a
relative measure for their hydrogen bond acceptor strength. With increasing pKa the affinity of the oxides to the proton increases and hence the hydrogen bond acceptor strength of these O atoms increases, too. The family of systems has been already studied in Refs. [11,12]. It has been shown there that the hydrogen bonds between MSA and the oxides are short and strong.
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427 Table 1 Abbreviations and pKa values of the investigated systems (MSA pKa ⫺1:92 System (abbreviation)
Full name of the base
pKa
1. MSA ⫹ DPhSO 2. MSA ⫹ DpToSO 3. MSA ⫹ TPhPO 4. MSA ⫹ MPhSO 5. MSA ⫹ DbzSO 6. MSA ⫹ DMSO 7. MSA ⫹ DBSO 8. MSA ⫹ DtBSO 9. MSA ⫹ TBPO 10. MSA ⫹ TPhAsO
Diphenyl sulfoxide Di-4-tolyl sulfoxide Triphenylphosphine oxide Methylphenyl sulfoxide Dibenzyl sulfoxide Dimethyl sulfoxide Dibutyl sulfoxide Di-tert-butyl sulfoxide Tributylphosphine oxide Triphenylarsine oxide
⫺1.27
0.12 0.45
2.1. The intensity of the infrared continua and the interaction of the polarizable bonds with the phonons In the following paragraphs, the interaction of hydrogen bonds with large proton polarizabilities with the thermal bath is treated. A simplified version of the Schro¨dinger equation of the proton motion in easily polarizable hydrogen bonds in a thermal bath is given by Eq. (1), where d is a damping constant iប
2.91
If the DpKa (pKa of the oxide minus pKa of the acid) is small the proton is present at the acid in a relatively narrow single minimum proton potential. With increasing pKa of the oxide, i.e. increasing DpKa the minimum shifts in the direction of the oxide and the proton potential becomes broad and flat. The proton fluctuates within this minimum with relatively large amplitude and can easily be shifted by external electrical fields. The hydrogen bonds now show large proton polarizability [2,3,6]. This large proton polarizability is indicated by the intense continua [2–6] observed in the infrared spectra in Fig. 1. It is of very large importance that with the almost symmetrical hydrogen bonds the integrated intensity of the continua observed is usually much larger than the intensity of infrared bands. If the DpKa increases still further, the protons become more and more localized at the O atom of the oxides, i.e. the minimum of the proton potential is present there. In Fig. 2 the far infrared spectra of these systems are given. This figure shows that if these systems are more or less symmetrical the continua extend in the far infrared region, down to 150 cm ⫺1. Fig. 3 shows the calculated line spectra of such a short, strong hydrogen bond as a function of the electrical field strength [3]. The lowest proton transition is the 00–10 transition. Fig. 3 shows that this transition cannot contribute to the continuum in the wavenumber region below 600 cm ⫺1. So, there must be another interaction effect of the easily polarizable hydrogen bonds with their environments responsible for these FT-IR continua.
423
2c ប 2 22 c 2c ⫺ ⫹ U xc ⫺ iបdx 2t 2m 2x 2 2x
1
where x is the coordinate of the proton, and U the proton potential. The first term in Eq. (1) takes into account the fluctuation of the local electrical fields. The most important is the third term, which takes into account the interaction of the polarizable hydrogen bonds with the thermal bath. The wave function is given by Eq. (2) [10] X i 2mEn 2 x c x; t C nt exp ⫺ En t ⫹ i ប ប n To solve Eq. (1) we make use of the so-called wavelet ansatz [12–14]. The wavelet transform T0 xn ; b; t is defined by Eq. (3) 1 Z∞ x ⫺ xn c xtf dx 3 T0 xn ; b; t b ⫺∞ b where b is a scaling factor, i.e. a distance characteristic of the interaction of the hydrogen bond with the thermal bath. The wavelet transform is often referred to as a mathematical microscope [14], because it allows the study of properties on any chosen scale b. For the wavelet f x ⫺ xn =b we make the following ansatz: 4 sin zy 2 ⫺A sin zy4 sin zy 2 f y e ⫺A cos zy ⫹e zy zy 4 where y x ⫺ xn =b As the Hamiltonian in Eq. (1) is not Hermitian, the commutation relation for the wavelet transform does not vanish. For Eq. (1) the following commutator is
424
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A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
425
obtained: Tw Tw⫺1
mdb2 ph
5
By an inverse transformation of Eq. (3) using Eq. (4) one obtains: N X x ⫺ xn c x; t T w⫺1 xn ; b; tf 6 b n With this wave function Eq. (1) is solved. Herewith we obtain: • the real and imaginary part of the eigenvalues as a function of the wavenumber; • the eigenfunctions; • the transition and dipole moments etc. as a function of the proton potential, the mass, and d, the parameter describing the coupling to the thermal bath. Infrared continua originated from O⫹ H…O O O…H⫹ O bonds calculated using Eq. (1), are shown in Fig. 4, in the upper part for a mean bond length of ˚ , and in the lower part for a mean bond length of 2.7 A ˚ . The dashed spectra have been obtained without 2.8 A phonon coupling. For the spectra drawn with the long dashed line, the coupling constant is 0.5, and for the spectra drawn with the solid line, the coupling constant is 1.0. These calculated spectra show that the intensity of the infrared continua increases strongly due to the coupling of the hydrogen bonds with large proton polarizability to the phonons of the thermal bath. The stronger the coupling, the more the increase of the intensity. This coupling is taken into account by the third term in Eq. (1). However, also the first term, which takes into account the fluctuation of the local electrical fields, strongly enhances the intensity of the infrared continua, as shown by the spectra drawn with the dashed line. Thus, the coupling of the easily polarizable hydrogen bonds with the phonons of the thermal bath strongly increases the intensity of the infrared continua. This effect is particularly important for longer hydrogen bonds with large proton polariz-
Fig. 3. Relative absorption intensities of the transitions of H5 O⫹ 2 ˚ ) as a function of the electrical field strength F (O–O distance 2.5 A in the hydrogen bond direction. Temperatures: X, 0; B, 100; K, 200; –, 300; × , 400 K. The length of the lines is equal to the relative absorption intensities, for negative electrical fields Iij ⫺F Iij ⫹F (from Ref. [3]).
abilities. In the case of shorter polarizable hydrogen bonds the coupling with the phonons of their local environments is less important, since the proton polarizability of shorter hydrogen bonds is smaller.
Fig. 2. FT-IR spectra (the region 450–150 cm ⫺1) of 1:1 MSA ⫹ oxide base mixtures in acetonitrile–chloroform (2:1), concentration 0.5 mol dm ⫺3: at 20⬚C (- - - - - - -) at ⫺40⬚C (—) and pure base in acetonitrile (– – – ) at room temperature, layer thickness 100 mm: (a) MSA ⫹ DPhSO; (b) MSA ⫹ DpToSO; (c) MSA ⫹ TPhPO; (d) MSA ⫹ MPhSO; (e) MSA ⫹ DBzSO; (f) MSA ⫹ DMSO; (g) MSA ⫹ DBSO; (h) MSA ⫹ DtBSO; (i) MSA ⫹ TBPO; and (j) MSA ⫹ TPhAsO (from Ref. [12]).
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which must be inserted in Eq. (8). Hence, we have two ~ and P: ~ From these equations for the variables F equations one obtains a dispersion relation, i.e. the ~ on the frequency dependence of the wave vector K
v4 ⫺ v2 v2L ⫹ cⴱ2 K 2 ⫹ cⴱ2 K 2 v2T 0
10
where v L is the frequency of the longitudinal optical phonons and c ⴱ the velocity of the light in the medium. The solution of the dispersion relation yields s v v2 ⫺ v2L 11 K ⴱ c v2 ⫺ v2T
Fig. 4. Calculated IR spectra of easily polarizable O⫹ H…O O O…H⫹ O bonds, coupled with the phonons of the thermal bath (- - - -), without coupling; (– – –), coupling constant 0.5; (—), ˚ ; (B) bond length coupling constant 1.0: (A) bond length 2.7 A ˚. 2.8 A
2.2. Calculation of the interaction with the polaritons The polaritons are the quanta of the radiation of the thermal bath coupled to the phonons of the thermal bath [7]. One can describe the dipoles of the thermal bath under the influence of the polarizable hydrogen bonds by the following equation 2 ~ ⫹ v2T P ~ nq F ~ ⫺v2 P mred
7
where v is the frequency of the hydrogen bond vibration, v T the frequency of the transverse optical ~ is the polarization, q is the effective charge phonons, P of the molecules of the thermal bath. ~ of the incoming infrared radiation is The field F described by the wave equation ! 22 22 22 ~ 22 ~ ⫹ 2 ⫹ 2 F 2 D 8 2 2x 2y 2z c e0 2t2 ~ is which follows from the Maxwell equation. D defined by the following equation ~ ⫹P ~ ~ e∞ e0 F D
9
The imaginary part of K is the absorbance. This relation, shows that K becomes imaginary only, if vT ⬍ v ⬍ vL : This is the case when the hydrogen bond vibration v is in between the wavenumber of the longitudinal and the transversal phonons. Hence, the infrared continuum in the region between 500– 100 cm ⫺1 is explained by a coupling of the hydrogen bond vibration with the polaritons. This equation shows that K decreases with decreasing v . This fact explains the decrease of the intensity of the infrared continuum with decreasing wavenumber (Fig. 2). 3. Conclusions The integrated intensity of the infrared continua observed in the spectra of systems with easily polarizable hydrogen bonds is usually much larger than the integrated intensity of the infrared bands. AH…B O⫺A…H⫹ B bonds are very polar structures and therefore their absorbance in the infrared region is very intense. However, there is a second reason for the intense absorbance. It is shown that there is a strong coupling between the easily polarizable hydrogen bonds and the phonons of the thermal bath. Due to this coupling the infrared continua gain additional intensity. In addition, with strong short hydrogen bonds the infrared continua extend to the far infrared region, i.e. down to 150 cm ⫺1. With short strong hydrogen bonds, the absorbance in this region can not be caused by proton transitions. It is shown that this far infrared continua caused by short strong hydrogen bonds arise due to the coupling of the easily polarizable
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
hydrogen bonds with the polaritons of the thermal bath. The polaritons are the quanta of the radiation of the thermal bath coupled to the phonons of the thermal bath. References [1] E.G. Weidemann, G. Zundel, Z. Naturforsch. 25a (1970) 627– 634. [2] R. Janoschek, E.G. Weidemann, H. Pfeiffer, G. Zundel, J. Am. Chem. Soc. 94 (1972) 2378–2396. [3] R. Janoschek, E.G. Weidemann, G. Zundel, J. Chem. Soc., Faraday II 69 (1973) 505–520. [4] G. Zundel, in: P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond-Recent Developments in Theory and Experiments, vol. II, North Holland, Amsterdam, 1976 (chap. 15).
427
[5] G. Zundel, in: I. Prigogine, St. A. Rice (Eds.), Advances in Chemical Physics, vol. CXI (111), Wiley, New York, 1999, pp. 1–217. [6] A. Hayd, E.G. Weidemann, G. Zundel, J. Chem. Phys. 70 (1979) 86–91. [7] Ch. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976, pp. 302–309. [8] P. Huyskens, Th. Zeegers Huyskens, J. Chem. Phys. 61 (1964) 81. [9] P. Huyskens, G. Fernandez, Ind. Chem. Belg. 38 (1973) 1237. [10] S.M. Chackalakal, F.E. Stanford, J. Am. Chem. Soc. 88 (1966) 4815. [11] R. Langner, G. Zundel, J. Chem. Soc., Faraday Trans. 94 (1998) 1805–1811. [12] R. Langner, G. Zundel, J. Chem. Phys. 99 (1995) 12214– 12219.
The interaction of the easily polarizable hydrogen bonds with phonons and polaritons of the thermal bath-far infrared continua A. Hayd, G. Zundel* Institute of Physical Chemistry, University of Munich, Butenandstr., D-81377 Munich, Germany
Abstract B⫹ H…B O B…H⫹ B or AH…B O ⫺ A…H⫹ B hydrogen bonds, respectively, with double minimum proton potential well or broad flat potential well cause continuous absorptions in the infrared spectra. The intensity of these infrared continua is often very large, i.e. their integrated intensity is much larger than that of infrared bands. Short strong hydrogen bonds with broad flat in the average largely symmetrical proton potentials in which the protons fluctuate, show so-called proton polarizabilities. With such systems, the infrared continua extend in the far infrared region, i.e. in the region below 600 cm ⫺1. It is, however known that with such bonds no proton transitions occur in this region. It is demonstrated that the intensity of these infrared continuous absorptions is strongly increased by the coupling of the hydrogen bonds with large proton polarizability with the phonons of the thermal bath around these hydrogen bonds. Further, it is demonstrated that the pronounced absorption in the far infrared region, i.e. the region below 600 cm ⫺1, occurs due to the strong coupling of these hydrogen bonds to the polaritons. The polaritons are the quanta of the thermal bath coupled to the phonons of the thermal bath. 䉷 2000 Elsevier Science B.V. All rights reserved. Keywords: Polaritons; Phonons; Thermal bath-far infrared
1. Introduction Homoconjugated B⫹ H…B O B…H⫹ B hydrogen bonds as well as heteroconjugated AH…B O ⫺ … ⫹ A H B bonds with double minimum proton potential show so-called proton polarizabilities caused by shifts of the proton within these bonds. These proton polarizabilities are about two orders of magnitude larger than the usual polarizabilities due to distortion of electron systems [1–6]. They are observed in the region below 3000 cm ⫺1. These infrared continua arise due to the interaction of these hydrogen bonds with large proton polarizability with their environ* Corresponding author. Present address: Bruno Walter-Straße 2, A-5020 Salzburg, Austria. Tel.: ⫹ 43-662-642311; fax: ⫹ 43-662642311.
ments. One reason for the occurrence of these infrared continua are the interactions of these polarizable hydrogen bonds with the local electrical fields from their environments [2]. Therefore, all polarizable hydrogen bonds having a broad flat proton potential with small barriers or without barriers are more or less strongly polarized by electrical fields. In addition, with short strong hydrogen bonds with almost symmetrical proton potentials continua are found. They extend down to 150 cm ⫺1 to the far infrared region. With such bonds, however no proton transitions occur in the region below 600 cm ⫺1 [3]. It will be demonstrated that these far infrared continua arise due to the interaction of these easily polarizable hydrogen bonds with the polaritons. The polaritons are the quanta of the thermal bath coupled to the phonons of the thermal bath [7].
0166-1280/00/$ - see front matter 䉷 2000 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(00)00389-4
422
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
Fig. 1. FT-IR spectra region (1400–800 cm ⫺1) of 1:1 MSA ⫹ oxide base mixtures in acetonitrile–chloroform (2:1), concentration 0.5 mol dm ⫺3: at 20⬚C ( - - - -), at 40⬚C (—) and pure base in acetonitrile (– – – ) at room temperature, layer thickness 100 mm: (a) MSA ⫹ DPhSO; (b) MSA ⫹ TPhPO, TPhPO (saturated solution); (c) MSA ⫹ DBzSO, DBzSO (saturated solution); (d) MSA ⫹ DMSO; (e) MSA ⫹ TBPO; and (f) MSA ⫹ TPhAsO, TPhAsO (saturated solution) (from Ref. [12]).
2. Results and discussion Fig. 1 shows the infrared spectra of the methanesulfonic acid (MSA) ⫹ oxide family as an example of a family of systems [8,9]. The systems—their pKa values [10] as far as they are available—and the abbreviations are summarized in Table 1. The pKa values, of these oxides increases within this series. The pKa values of these sulfoxides are used as a
relative measure for their hydrogen bond acceptor strength. With increasing pKa the affinity of the oxides to the proton increases and hence the hydrogen bond acceptor strength of these O atoms increases, too. The family of systems has been already studied in Refs. [11,12]. It has been shown there that the hydrogen bonds between MSA and the oxides are short and strong.
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427 Table 1 Abbreviations and pKa values of the investigated systems (MSA pKa ⫺1:92 System (abbreviation)
Full name of the base
pKa
1. MSA ⫹ DPhSO 2. MSA ⫹ DpToSO 3. MSA ⫹ TPhPO 4. MSA ⫹ MPhSO 5. MSA ⫹ DbzSO 6. MSA ⫹ DMSO 7. MSA ⫹ DBSO 8. MSA ⫹ DtBSO 9. MSA ⫹ TBPO 10. MSA ⫹ TPhAsO
Diphenyl sulfoxide Di-4-tolyl sulfoxide Triphenylphosphine oxide Methylphenyl sulfoxide Dibenzyl sulfoxide Dimethyl sulfoxide Dibutyl sulfoxide Di-tert-butyl sulfoxide Tributylphosphine oxide Triphenylarsine oxide
⫺1.27
0.12 0.45
2.1. The intensity of the infrared continua and the interaction of the polarizable bonds with the phonons In the following paragraphs, the interaction of hydrogen bonds with large proton polarizabilities with the thermal bath is treated. A simplified version of the Schro¨dinger equation of the proton motion in easily polarizable hydrogen bonds in a thermal bath is given by Eq. (1), where d is a damping constant iប
2.91
If the DpKa (pKa of the oxide minus pKa of the acid) is small the proton is present at the acid in a relatively narrow single minimum proton potential. With increasing pKa of the oxide, i.e. increasing DpKa the minimum shifts in the direction of the oxide and the proton potential becomes broad and flat. The proton fluctuates within this minimum with relatively large amplitude and can easily be shifted by external electrical fields. The hydrogen bonds now show large proton polarizability [2,3,6]. This large proton polarizability is indicated by the intense continua [2–6] observed in the infrared spectra in Fig. 1. It is of very large importance that with the almost symmetrical hydrogen bonds the integrated intensity of the continua observed is usually much larger than the intensity of infrared bands. If the DpKa increases still further, the protons become more and more localized at the O atom of the oxides, i.e. the minimum of the proton potential is present there. In Fig. 2 the far infrared spectra of these systems are given. This figure shows that if these systems are more or less symmetrical the continua extend in the far infrared region, down to 150 cm ⫺1. Fig. 3 shows the calculated line spectra of such a short, strong hydrogen bond as a function of the electrical field strength [3]. The lowest proton transition is the 00–10 transition. Fig. 3 shows that this transition cannot contribute to the continuum in the wavenumber region below 600 cm ⫺1. So, there must be another interaction effect of the easily polarizable hydrogen bonds with their environments responsible for these FT-IR continua.
423
2c ប 2 22 c 2c ⫺ ⫹ U xc ⫺ iបdx 2t 2m 2x 2 2x
1
where x is the coordinate of the proton, and U the proton potential. The first term in Eq. (1) takes into account the fluctuation of the local electrical fields. The most important is the third term, which takes into account the interaction of the polarizable hydrogen bonds with the thermal bath. The wave function is given by Eq. (2) [10] X i 2mEn 2 x c x; t C nt exp ⫺ En t ⫹ i ប ប n To solve Eq. (1) we make use of the so-called wavelet ansatz [12–14]. The wavelet transform T0 xn ; b; t is defined by Eq. (3) 1 Z∞ x ⫺ xn c xtf dx 3 T0 xn ; b; t b ⫺∞ b where b is a scaling factor, i.e. a distance characteristic of the interaction of the hydrogen bond with the thermal bath. The wavelet transform is often referred to as a mathematical microscope [14], because it allows the study of properties on any chosen scale b. For the wavelet f x ⫺ xn =b we make the following ansatz: 4 sin zy 2 ⫺A sin zy4 sin zy 2 f y e ⫺A cos zy ⫹e zy zy 4 where y x ⫺ xn =b As the Hamiltonian in Eq. (1) is not Hermitian, the commutation relation for the wavelet transform does not vanish. For Eq. (1) the following commutator is
424
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
425
obtained: Tw Tw⫺1
mdb2 ph
5
By an inverse transformation of Eq. (3) using Eq. (4) one obtains: N X x ⫺ xn c x; t T w⫺1 xn ; b; tf 6 b n With this wave function Eq. (1) is solved. Herewith we obtain: • the real and imaginary part of the eigenvalues as a function of the wavenumber; • the eigenfunctions; • the transition and dipole moments etc. as a function of the proton potential, the mass, and d, the parameter describing the coupling to the thermal bath. Infrared continua originated from O⫹ H…O O O…H⫹ O bonds calculated using Eq. (1), are shown in Fig. 4, in the upper part for a mean bond length of ˚ , and in the lower part for a mean bond length of 2.7 A ˚ . The dashed spectra have been obtained without 2.8 A phonon coupling. For the spectra drawn with the long dashed line, the coupling constant is 0.5, and for the spectra drawn with the solid line, the coupling constant is 1.0. These calculated spectra show that the intensity of the infrared continua increases strongly due to the coupling of the hydrogen bonds with large proton polarizability to the phonons of the thermal bath. The stronger the coupling, the more the increase of the intensity. This coupling is taken into account by the third term in Eq. (1). However, also the first term, which takes into account the fluctuation of the local electrical fields, strongly enhances the intensity of the infrared continua, as shown by the spectra drawn with the dashed line. Thus, the coupling of the easily polarizable hydrogen bonds with the phonons of the thermal bath strongly increases the intensity of the infrared continua. This effect is particularly important for longer hydrogen bonds with large proton polariz-
Fig. 3. Relative absorption intensities of the transitions of H5 O⫹ 2 ˚ ) as a function of the electrical field strength F (O–O distance 2.5 A in the hydrogen bond direction. Temperatures: X, 0; B, 100; K, 200; –, 300; × , 400 K. The length of the lines is equal to the relative absorption intensities, for negative electrical fields Iij ⫺F Iij ⫹F (from Ref. [3]).
abilities. In the case of shorter polarizable hydrogen bonds the coupling with the phonons of their local environments is less important, since the proton polarizability of shorter hydrogen bonds is smaller.
Fig. 2. FT-IR spectra (the region 450–150 cm ⫺1) of 1:1 MSA ⫹ oxide base mixtures in acetonitrile–chloroform (2:1), concentration 0.5 mol dm ⫺3: at 20⬚C (- - - - - - -) at ⫺40⬚C (—) and pure base in acetonitrile (– – – ) at room temperature, layer thickness 100 mm: (a) MSA ⫹ DPhSO; (b) MSA ⫹ DpToSO; (c) MSA ⫹ TPhPO; (d) MSA ⫹ MPhSO; (e) MSA ⫹ DBzSO; (f) MSA ⫹ DMSO; (g) MSA ⫹ DBSO; (h) MSA ⫹ DtBSO; (i) MSA ⫹ TBPO; and (j) MSA ⫹ TPhAsO (from Ref. [12]).
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which must be inserted in Eq. (8). Hence, we have two ~ and P: ~ From these equations for the variables F equations one obtains a dispersion relation, i.e. the ~ on the frequency dependence of the wave vector K
v4 ⫺ v2 v2L ⫹ cⴱ2 K 2 ⫹ cⴱ2 K 2 v2T 0
10
where v L is the frequency of the longitudinal optical phonons and c ⴱ the velocity of the light in the medium. The solution of the dispersion relation yields s v v2 ⫺ v2L 11 K ⴱ c v2 ⫺ v2T
Fig. 4. Calculated IR spectra of easily polarizable O⫹ H…O O O…H⫹ O bonds, coupled with the phonons of the thermal bath (- - - -), without coupling; (– – –), coupling constant 0.5; (—), ˚ ; (B) bond length coupling constant 1.0: (A) bond length 2.7 A ˚. 2.8 A
2.2. Calculation of the interaction with the polaritons The polaritons are the quanta of the radiation of the thermal bath coupled to the phonons of the thermal bath [7]. One can describe the dipoles of the thermal bath under the influence of the polarizable hydrogen bonds by the following equation 2 ~ ⫹ v2T P ~ nq F ~ ⫺v2 P mred
7
where v is the frequency of the hydrogen bond vibration, v T the frequency of the transverse optical ~ is the polarization, q is the effective charge phonons, P of the molecules of the thermal bath. ~ of the incoming infrared radiation is The field F described by the wave equation ! 22 22 22 ~ 22 ~ ⫹ 2 ⫹ 2 F 2 D 8 2 2x 2y 2z c e0 2t2 ~ is which follows from the Maxwell equation. D defined by the following equation ~ ⫹P ~ ~ e∞ e0 F D
9
The imaginary part of K is the absorbance. This relation, shows that K becomes imaginary only, if vT ⬍ v ⬍ vL : This is the case when the hydrogen bond vibration v is in between the wavenumber of the longitudinal and the transversal phonons. Hence, the infrared continuum in the region between 500– 100 cm ⫺1 is explained by a coupling of the hydrogen bond vibration with the polaritons. This equation shows that K decreases with decreasing v . This fact explains the decrease of the intensity of the infrared continuum with decreasing wavenumber (Fig. 2). 3. Conclusions The integrated intensity of the infrared continua observed in the spectra of systems with easily polarizable hydrogen bonds is usually much larger than the integrated intensity of the infrared bands. AH…B O⫺A…H⫹ B bonds are very polar structures and therefore their absorbance in the infrared region is very intense. However, there is a second reason for the intense absorbance. It is shown that there is a strong coupling between the easily polarizable hydrogen bonds and the phonons of the thermal bath. Due to this coupling the infrared continua gain additional intensity. In addition, with strong short hydrogen bonds the infrared continua extend to the far infrared region, i.e. down to 150 cm ⫺1. With short strong hydrogen bonds, the absorbance in this region can not be caused by proton transitions. It is shown that this far infrared continua caused by short strong hydrogen bonds arise due to the coupling of the easily polarizable
A. Hayd, G. Zundel / Journal of Molecular Structure (Theochem) 500 (2000) 421–427
hydrogen bonds with the polaritons of the thermal bath. The polaritons are the quanta of the radiation of the thermal bath coupled to the phonons of the thermal bath. References [1] E.G. Weidemann, G. Zundel, Z. Naturforsch. 25a (1970) 627– 634. [2] R. Janoschek, E.G. Weidemann, H. Pfeiffer, G. Zundel, J. Am. Chem. Soc. 94 (1972) 2378–2396. [3] R. Janoschek, E.G. Weidemann, G. Zundel, J. Chem. Soc., Faraday II 69 (1973) 505–520. [4] G. Zundel, in: P. Schuster, G. Zundel, C. Sandorfy (Eds.), The Hydrogen Bond-Recent Developments in Theory and Experiments, vol. II, North Holland, Amsterdam, 1976 (chap. 15).
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[5] G. Zundel, in: I. Prigogine, St. A. Rice (Eds.), Advances in Chemical Physics, vol. CXI (111), Wiley, New York, 1999, pp. 1–217. [6] A. Hayd, E.G. Weidemann, G. Zundel, J. Chem. Phys. 70 (1979) 86–91. [7] Ch. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976, pp. 302–309. [8] P. Huyskens, Th. Zeegers Huyskens, J. Chem. Phys. 61 (1964) 81. [9] P. Huyskens, G. Fernandez, Ind. Chem. Belg. 38 (1973) 1237. [10] S.M. Chackalakal, F.E. Stanford, J. Am. Chem. Soc. 88 (1966) 4815. [11] R. Langner, G. Zundel, J. Chem. Soc., Faraday Trans. 94 (1998) 1805–1811. [12] R. Langner, G. Zundel, J. Chem. Phys. 99 (1995) 12214– 12219.