Vibrational excitons

Interactions between molecules, in particular dipole-dipole coupling, lead to collective, vibrational motions of the molecules, which are called vibrational excitons.⁹ Similar to phonons, which are coupled motions of the atoms in solids, vibrational excitons𝑙can also be characterized by a frequency, ˜𝜈

𝑙, and a wavevector,K𝑙.

The wavelength,𝜆, of infrared light is much larger than the molecular dimensions.

Therefore, only vibrational excitons with𝐾

𝑙 =2𝜋/𝜆≈ 0 can be excited with infrared absorption spectroscopy. For other modes, molecules in different unit cells would not vibrate in phase and the sum of their transition dipole moments would cancel. Therefore, only𝑛infrared-active exciton modes exist for an adsorption structure with𝑛molecules per unit cell. [106]

In the following, two different models are presented, which describe the infrared spectroscopy of adsorbate layers while considering vibrational excitons. The first model treats the interactions between the molecules with first-order perturbation theory and is especially useful to understand the frequency shifts of vibrational excitons compared to the uncoupled vibrations. The second model takes the effective change of the infrared

9The definition of vibrational excitons as coupled vibrations should not to be confused with the definition of excitons in solid matter physics.

30

2.4 Infrared spectroscopy of adsorbates

cross section due to dipole-dipole interactions between the molecules into account.

Static and dynamic frequency shifts

The exciton model presented in this section [107,108] is extremely useful to understand the nature of the vibrational frequency shift compared to the vibration of uncoupled molecules.

Since the coupling between the molecules is typically weak, the model is based on first-order perturbation theory. The unperturbed, vibrational wavefunction of an individual, uncoupled molecule𝑖is given by𝜙

𝑖in the vibrational ground state and by𝜙⁰

𝑖

in a vibrationally excited state (e.g.,𝑣 =1). Therefore, the zeroth-order wavefunctions of the total system in the ground state and the vibrationally excited state are given as:

𝜓0=𝜙

The ground state wavefunction,𝜓

0, is a product of the vibrational ground state wavefunc-tions of𝑁 uncoupled molecules, whereas ˜𝜈

0characterizes their unperturbed frequencies.

The excited state wavefunction,𝜓

𝑙, is a linear combination of𝑁 products, in which a single molecule is vibrationally excited. Note that 𝑁 excitons and thus 𝑁 excited state wavefunctions exist, which are described by the expansion coefficients𝐶

𝑙𝑖. The perturbation operator describes the sum over all dipole-dipole interactions between the molecules:

𝑗 are the EDMFs of molecules𝑖and 𝑗, respectively. 𝑓

𝑖 𝑗is the orientation-and distance-dependent interaction term for the dipole-dipole interaction between molecules𝑖and 𝑗(see also Section 2.5.2):

𝑓𝑖 𝑗 = 1

e𝑖 and e𝑗 are the unit vectors that describe the orientations of the two molecules, whereasr𝑖 𝑗is the unit vector of the intermolecular separation vector,R𝑖 𝑗, that connects molecules𝑖and𝑗. 𝑅

𝑖 𝑗is the corresponding intermolecular distance. The implementation of multipole interactions beyond the dipole-dipole interaction in Eq. 2.61 would be straightforward (see also Section 2.5.2). [108] However, only the leading dipole-dipole

Chapter 2 Scientific background

interaction term will be discussed here for simplicity.

With Eqs. 2.59 and 2.61, the first-order energy correction to the vibrational ground state is given by:¹⁰

𝐸⁽¹⁾

𝑖iis the permanent dipole moment in the vibrational ground state.

The first-order correction of the excited state must be calculated with perturbation theory for𝑁-fold degenerate zeroth-order states. With Eqs. 2.60 and 2.61, the diagonal elements of the corresponding Hamiltonian matrix are given by:

𝐻⁰

𝑖iis the permanent dipole moment in𝑣 =1. Equation 2.64 can be greatly simplified by subtracting the energy correction of the vibrational ground state (Eq. 2.63) from all diagonal elements:

Δ𝐻⁰ The off-diagonal elements take a much simpler form:¹¹

𝐻⁰

𝑖iis the transition dipole moment.

To find the excitonic frequency shifts and transition dipole moments, the Hamiltonian matrix,𝐻⁰, defined by the matrix elements in Eqs. 2.65 and 2.66, must be diagonalized.

The eigenvalues of𝐻⁰,Δ𝐸⁽¹⁾

𝑙 , can be related to the frequency shifts of the vibrational excitons relative to the frequency of the unperturbed molecules:

Δ𝜈˜

The eigenvectors of 𝐻⁰, C𝑙, define the expansion coefficients of the excitonic wave-functions (Eq. 2.60). The corresponding integrated absorption cross sections along the

10Note thath𝜙

2.4 Infrared spectroscopy of adsorbates direction of the transition dipole moments are given by: [108]

¯

Within the oriented gas model (Eq. 2.53), the integrated absorbance for an exciton absorption line at frequency ˜𝜈

𝑙=𝜈˜

0+Δ˜𝜈

𝑙is therefore given as:

˜

There are two different contributions to the frequency shift in Eq. 2.67, arising from the diagonal and off-diagonal matrix elements. The diagonal elements in Eq. 2.65 describe the difference in the interaction energies of a single molecule 𝑘 excited from 𝑣 = 0 to𝑣 =1 with all other molecules in𝑣 = 0 via their permanent dipole moments. This frequency shift is called the static frequency shift because it only involves interactions between permanent multipole moments. The off-diagonal elements in Eq. 2.66 can be interpreted as the coupling between the transition dipole moments of the two molecules.

This coupling explains the formation of the vibrational excitons and leads to different vibrational frequency shifts for different collective exciton modes. Only these dynamic contributions can cause a frequency splitting, whereas the static shift is often similar for different excitons. The frequency splitting caused by the dynamic frequency shift is also often called Davydov splitting or correlation-field splitting. [41,42]

The dynamic frequency splitting, as observed in the absorption spectra of the CO monolayer in the(2×1) structure (see Section 2.1), depends strongly on the magnitude of the transition dipole moment. Dai and Ewing showed experimentally that the dynamic frequency shift vanishes for small transition dipole moments, as for the 𝑣 = 0 → 2 absorption line, and for molecules that are separated by large distances, as for¹³C¹⁶O diluted in¹²C¹⁶O (and vice versa). [107] These observations are of great importance to the VEP experiments in the present work, where molecules in highly vibrationally excited states are surrounded by molecules in much lower vibrational states, leading to inefficient dynamic coupling. Therefore, only static frequency shifts are expected for LIF experiments,¹²whereas both static and dynamic frequency shifts can be observed in infrared absorption experiments.

12Except for the𝑣 =1→0 transition, which cannot be probed with the present experimental setup because the𝑣 =0→1 transition is excited.

Chapter 2 Scientific background

Effective absorption strength for collective vibrations

Although the exciton model presented in the previous section gives insight into the frequency shifts, it cannot describe the effective change of the infrared absorption that arises due to the collective motion of the molecules. To envision this, consider an array of perpendicularly oriented molecules. For the single infrared-active exciton mode (K=0), all molecules vibrate in phase with each other when they are excited with infrared light.

Therefore, each molecule will experience an electric field due to the in-phase motion of the surrounding molecules, leading to an induced dipole moment that opposes the original transition dipole moment.

Due to the strong dependence of the effective reduction of the infrared absorption on the intermolecular distances, this effect is strongly coverage-depent. The coverage dependence of the absorption strength of CO on various metal surfaces is well stud-ied, [109–113] and theoretical models exist, which consider the effect of the molecular polarizability during the collective motion. [114–116] However, all these models assume that the dipole moments of the CO molecules are perpendicular to the surface and form (1×1)structures. While this is well-justified for CO on metal surfaces, CO on NaCl(100) is not always oriented perpendicular to the surface and shows a(2×1)structure at low temperatures (see also Section 2.1).

Snigur and Rozenbaum developed a model that can describe the effective change of the infrared absorption due to dipole-dipole interactions in an adsorbate layer with𝑛different molecules per unit cell that can have arbitrary orientations. [117] In addition, analytical solutions to the equations can be derived for simple adsorption structures, such as a slightly simplified version of the(2×1)structure in the CO/NaCl(100) monolayer. [118, 119] The model by Snigur and Rozenbaum [117] will be presented in the following and is applied in Chapter 7.

Similar to the quantum mechanical model in the previous section, also in the model by Snigur and Rozenbaum, the collective motion is characterized by an interaction matrix, 𝑊𝑗 𝑗⁰(see Eqs. 2.70 to 2.72). Here 𝑗and 𝑗⁰denote molecules within the same unit cell, whereas the summation runs over the position vectors,R, of all unit cells. 𝑒

𝛼 𝑗 and𝑒

𝛽 𝑗⁰

are the𝑥, 𝑦 and𝑧 components of the orientation unit vectors, e𝑗 ande𝑗⁰, of the two

34

2.4 Infrared spectroscopy of adsorbates

𝑗 , 𝑗⁰in terms of the orientation vectors gives:

𝑊𝑗 𝑗⁰ =∑︁

Comparison of Eq. 2.73 and Eq. 2.62 shows that Eq. 2.73 essentially describes the summation over the dipole-dipole interaction terms 4𝜋 𝜀

0𝑓

𝑖 𝑗between all molecules in real space. Therefore, the matrix elements𝑊

𝑗 𝑗⁰in Eq. 2.73 are essentially identical to the off-diagonal matrix elements𝐻⁰

𝑖 𝑗in Eq. 2.66, except for a constant factor. A minute difference between the models is, however, that the model by Snigur and Rozenbaum is based on the molecules in a single unit cell assuming a periodic structure, while individual molecules are considered in the other model. Diagonalization of𝑊gives𝑛 eigenvalues,𝑊

𝑙, and eigenvectors,C𝑙, which characterize the exciton modes, labelled by the index𝑙. The integrated absorbance of a given exciton absorption line (in units of angular frequency) is given by:

˜

𝑒 are the vibrational and electronic polarizabilities of the molecules parallel to the bond axis. P𝑙 =Í

𝑗𝐶

𝑙 𝑗e𝑗is a vector that is proportional to the transition dipole moment vector. 𝜔

0is the unperturbed frequency of the molecule. Other quantities are defined as for Eq. 2.53. Conversion from angular frequencies to wavenumbers

Chapter 2 Scientific background

The vibrational polarizability describes the dynamic polarizability associated with the 𝑣 =0→1 transition and can be related to the integrated cross section and the transition along the bond axis via: [25,91]

¯

With the relations in Eq. 2.76 in mind, note the strong similarity between Eqs. 2.69 and 2.75, differing only by 1/(1+ 𝜒

𝑒𝑊

𝑙)². This term essentially describes the effective change in the infrared cross section due to collective vibrational motion, quantified by 𝑊𝑙, leading to the induction of additional transition dipole moments via the electronic polarizability,𝜒

𝑒. Also note that this term cannot only lead to a reduction of the effective cross section, but also to an effective increase when𝑊

𝑙 <0. [117]

Within this model, also the vibrational frequencies of the exciton absorption lines can be calculated according to: [117]

˜

For CO on NaCl(100), electrostatic interactions and dispersion interactions are the most important long-range interactions whereas repulsive interactions dominate at short distances. [120,121] Although all these interactions are relevant to describe the absolute binding strength of CO to NaCl(100), electrostatic interactions depend strongly on molecular orientation and the CO bond distance and therefore strongly influence the vibrational dynamics. Therefore, only electrostatic interactions will be discussed in this section. For an extensive introduction to other interactions, the reader is referred to Ref. [122].

2.5.1 Molecules in an external electric field

This section describes the electrostatic interaction between a molecule placed in an external, non-homogeneous electric field. It is mainly based on Ref. [122,123] and

36

Im Dokument Vibrational Energy Transfer Between CO Molecules on a NaCl(100) Surface Studied by Infrared Fluorescence Spectroscopy (Seite 42-48)