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Linear Response Theory Applications to IR Spectra of H-Bonded Cyclic Dimers Taking into Account the Surrounding. Updating Contributions Involving Davydov Coupling, Fermi Resonances and Electrical Anharmonicity

Paul Blaise and Olivier Henri-Rousseau

Laboratory of Mathematics and Physics, 52 Av. Paul Alduy, 66100 Perpignan, France

1.1 Introduction

This chapter is devoted to the application of the Henri-Rousseau and Blaise model [1] which has incorporated quantum mechanically the damping of the H-bond bridge into the Maréchal and Witkowski model [2] to the experimental infrared (IR) lineshapes of cyclic centrosymmetric dimers. In Figure 1.1, are depicted for example linear and cyclic H-bonded carboxylic acids.

One may distinguish the length of O-H bond and one of the H-bond. In Figure 1.2 are recapitulated the connections between the present applied theory and diverse older ones.

1.2 Dimer Strong Anharmonic Coupling Theory

1.2.1 Different Theoretical Situations

1.2.1.1 Strong Anharmonic Coupling Within Adiabatic Approximation For Monomer

Let us consider a single H-bonded system where and are nucleophilic substituents such as oxygen or nitrogen (See Figure 1.3). Define and as the operators corresponding to the lengths of X-H and X-Y bonds. Besides, both these lengths are oscillating, the first one at high frequency and the last one H-bond bridge at low frequency.

Now suppose that a strong anharmonic coupling may occur between the X-H high-frequency mode and the XY low-frequency mode .

Within the strong anharmonic coupling theory, it is assumed a linear dependence of the high-frequency mode on the H-bond bridge coordinate , according to:

where is the angular frequency of a isolated X-H bond and some parameter.

Figure 1.1 (a) H-bond monomer and the coordinates. (b) H-bond dimer and the coordinates.

Figure 1.2 Connections between the present theory and different older models.

The full Hamiltonian may be partitioned as follows:

The Hamiltonian of the slow mode may be viewed as either harmonic or anharmonic (Morse-like)

Here, is the momentum coordinate of the slow mode of reduced mass and angular frequency , whereas is the dissociation energy of the Morse curve.

The Hamiltonian is corresponding to the (X-H) high-frequency mode. Within the harmonic approximation and strong anharmonic coupling theory, it is:

whereas is the momentum coordinates for the fast mode.

The eigenvalue equations of the fast and slow harmonic modes are given respectively, neglecting the zero-point energy of the fast mode by:

Within the adiabatic approximation the full Hamiltonian becomes simply:

where

Figure 1.4 represents the absorption mechanism generating a coherent state.

It is possible to generalize the above approach by introducing together with the coupling of the fast mode to the H-bond bridge, another coupling of the fast mode with some bending mode according to:

with, by taking the H-bond bridge potential as Morse-like (See Table 1.1).

Figure 1.3 Coordinates of single H-bonded system.

Figure 1.4 Physics of the absorption mechanism. The ground state of the slow mode H-bond bridge (corresponding to the ground state situation of the fast mode) becomes a coherent state after excitation towards the first excited state of the fast mode.

Table 1.1 Different sorts of Hamiltonians.

where and are respectively the position and momentum coordinates of the bending mode having as angular frequency and as reduced mass.

1.2.1.2 Introduction of Fermi Resonances

Now, there is the possibility to introduce Fermi resonance [11] in this physical model as it is illustrated in Figure 1.5.

There is a coupling characterized by the parameter between the two situations evoked in Figure 1.5.

In the absence of damping, the full Hamiltonian involving Fermi resonances is:

Here, the three first right-hand side Hamiltonians are the components of the bare H-bond Hamiltonians without Fermi resonance given respectively by equations given in Table 1.1. Besides, the Hamiltonian corresponding to the bending mode and the interaction between the fast and bending modes are respectively:

Figure 1.5 Fermi resonances interaction coupling parameters between two situations of the fast, slow, and bending modes.

Source: Henri-Rousseau and Blaise 2008 [18]/John Wiley & Sons.

where and are respectively the position and momentum coordinates of the bending mode of reduced mass and its angular frequency, whereas is the coupling parameter between the fast and bending modes. The eigenvalue equations of the harmonic Hamiltonians corresponding respectively to the fast and slow modes are respectively given by equations given by Eqs. (1.4) whereas that dealing with the bending modes is, ignoring the zero-point energy:

Now, within the adiabatic approximation. The Hamiltonian (1.6) becomes:

The different Hamiltonians are given as follows:

Here, is the anharmonic coupling parameter involved in the Fermi resonance which is a function of .

As a consequence of the above equations, the full Hamiltonian describing the fast mode coupled to the H-bond bridge (via the strong anharmonic coupling theory) and the bending mode (via the Fermi resonance process) may be written within the tensorial basis (1.10) according to [12]:

1.2.1.3 H-Bonded Centrosymmetric Dimer

Now, look at an H-bonded dimer. It will take place in a Davydov coupling [13]. Within the anharmonic coupling, the physics of the system may be viewed in Figure 1.6.

It may be observed that because of the symmetry of the dimer, there is a operator (with ), which exchanges the coordinates of the two slow modes H-bond bridges of the cyclic dimer according to:

Ignoring for the present time the interaction between the two moieties and assuming that, within each moiety, the adiabatic approximation may be performed as for a single H-bond, the Hamiltonian of the symmetric dimer embedded in the thermal bath, is:

Figure 1.6 Davydov coupling interactions.

Source: Henri-Rousseau and Blaise 2008 [18]/John Wiley & Sons.

In Eq. 1.13, the two first right-hand side terms are the adiabatic Hamiltonians of each moiety. They are given by an expression of the same form which is:

are the eigenkets of the Hamiltonians of the fast modes harmonic oscillators, whereas the Hamiltonians of each moiety are respectively:

Here, the last term is the interacting coupling with the thermal bath that we shall ignore in the present simplified exposition. The Hamiltonian of the cyclic dimer involving Davydov coupling between the first excited state of the high-frequency oscillator of one moiety and the excited state of the oscillator of the other moiety and vice versa is,

The Davydov coupling Hamiltonian appearing in this equation may be written either simply or as a function of the two slow modes coordinates [14]:

where is a dimensionless parameter governing the linear dependence of the Davydov coupling operator on the H-bond bridge coordinates.

When ignoring the coupling, then, within the following basis:

the Davydov Hamiltonian (1.14) takes the matrix form:

with respectively:

Then, owing to the symmetry properties given by Eqs. (1.12), it appears that the parity operator exchanges the two last Hamiltonians:

To diagonalize the Davydov Hamiltonian, one may perform the following basis change.

Then the Davydov Hamiltonian becomes:

Moreover, to make tractable the action of the operator, it is suitable to pass to the symmetrized coordinates and their conjugate momenta according to Figure 1.7.

In Table 1.2, are given the symmetrized coordinates in the Davydov coupling model.

Recall here, the improvement brought by Rekik et al. [15-17], by introducing...