Effect of vibrational excitation on IR spectra

For the experiments presented in the second part of this work, the “signal”, as de-scribed in section 1.1, is the absorption of an IR laser beam and the predominant cause for spectral change is vibrational excitation. The spectral change observed is then owed to mechanical as well as electrical anharmonicity of the oscillators observed. Mechani-cal anharmonicity, i.e. the deviation of the restoring forceFfrom strict proportionality to the displacement∆xof the oscillator from its equilibrium positionx0, influences the spectral position of the absorption of an oscillator. It also alters the intensity of the transition as it changes the wave functions of the oscillator involved in the transition.

Electrical anharmonicity, i.e. the deviation of the dipole momentµ from proportion-ality to the displacement of the oscillator, on the other hand, is the cause of possibly changing intensity in absorption of an oscillator, as the intensity depends on the tran-sition dipole moment.^[107–110]

The usual theoretical treatment^{[111, 112]} starts by diagonalizing the potential en-ergy V of second order in the displacements ∆r^α_k of the αth Cartesian coordinate of

Chapter 3. Intramolecular vibrational energy transport

Figure 3.9:Visualization of the modes observed using the geometry used for calculating an-harmonicity constants. Atoms are colored using the common convention (carbon – black, hydrogen – white, nitrogen – blue, oxygen – red), while displacement ar-rows are colored as follows: cyan – azulene ring distortion mode, yellow – amide II mode, purple – amide I mode, green – azide asymmetric stretching mode.

thekth atom from the equilibrium geometry of the molecule V ≈ ¹

2

∑

k,l,α,β

f_kl^αβ∆r^α_k∆r_l^β with α, β= _{x, y, z}

to obtain dimensionless normal coordinatesq_i leading to V ≈ ¹

2

∑

i

ωiqi

whereωiis the normal frequency of theith normal mode. Adding higher terms to the

3.3. Investigation via UV pump IR probe spectroscopy

then introduces the above-mentioned mechanical anharmonicity. Similarly, electrical anharmonicity^{[108, 112]} originates from the higher order terms of the expansion of the dipole momentµ

in normal coordinatesqi.

From the experimental perspective, transition energies have usually received greater attention than intensities. The empirical formula[107, 109–111, 113, 114] of the en-ergy E of a vibrational state (v₁, ...), reminiscent of the vibrational Dunham expan-sion^[115],

where v_i is the vibrational quantum number of the ith oscillator relative to the mini-mum energy of the electronic state, leads to transition energies^[2]

˜

that are shifted depending on the quantum numbers of the observed mode (xii, diag-onal anharmonicity) as well as the other modes of the molecule (xik with i 6= _k, off-diagonalanharmonicity). ˜ν_i⁰ = _ωi+2x_ii+ ¹_2∑k6=ix_ik is the fundamental frequency, i.e.

the frequency of the transition of the ith mode from its ground – to its first excited state with all other modes being in their ground states. It should be noted that addi-tional coupling constants may be necessary for the satisfactory treatment of degenerate modes.^[113] Using a perturbative treatment, it is then possible to derive the constants of anharmonicity x_ij from the third and fourth order potential constants φ_ijk and φ_ijkl

Chapter 3. Intramolecular vibrational energy transport

from above.[109–111, 114] To second order, the corresponding expressions are[111, 116, 117]: x_ii = ¹

whereB^eq_α are the rotational constants andI_α^eqthe moments of inertia taken at the equi-librium geometry of the molecule. c0 is the speed of light in vacuum, andζ^α_ik are the Coriolis constants to be obtained from L^α_km, the elements of the matrixL. The latter is the matrix transforming Cartesian displacement coordinates∆rm^α of theαcoordinate of themth atom to normal coordinatesQ_kas follows:

∆r^α_m =

∑

In the case of a Fermi resonance, divergent terms need to be dropped from Ωim and Ωikm and energies of the resonant states may be obtained by explicitly diagonalizing the appropriate Hamiltonian.^{[116, 117]} Note that the weighting factors Ωim and Ωikm

defined here do not follow the notation used in Ref. [116].

If changes in intensity upon vibrational excitation as well as higher-order shifts are negligible, the mere knowledge of the constants of anharmonicityxijis sufficient to

3.3. Investigation via UV pump IR probe spectroscopy simulate a difference spectrum and thus help explain the underlying energy distribu-tion within an ensemble of molecules (see subsecdistribu-tion 3.5.1).[2, 30, 118, 119] In the general case of a medium-sized organic molecule, individual vibrational states will mostly not be discernible in the IR difference spectrum and hence it is not directly evident which modes are excited and contribute to anharmonic shifting. For the case of statistical intramolecular equilibrium, a nearly thermal distribution can be assumed (see subsec-tion 3.5.2), and it has been shown that fitting a thermal spectrum to the experimental data can be successful^{[2, 118]}.

For the experiments considered later in this work, however, the starting condition is obviously a molecule in which the vibrational energy is fairly well localized in an azulene moiety and IVR will only gradually lead to a situation of randomized energy distribution among all modes. Since a population of excited states of the “sensor”

modes themselves seems unlikely due to their relatively high frequencies (carbonyl – and azide modes), it has been asserted that the spectral shift of their absorptions is caused mainly by those modes which “spatially overlap” with the sensor mode.^{[8, 98]}

Spatial overlap is to be understood as modes having their greatest amplitudes of dis-placement located at the same atoms, increasing the coupling constants of those modes with the sensor mode.^[98] In a sense, the observed modes can thus be thought of as lo-cal heat sensors.^[72] It must be emphasized that albeit in most cases researchers have in the past resolved to interpret reduced optical density at the band origin itself, there is no per se or a priori direct proportionality of this quantity to energy or temperature (if that concept is to be considered adequate at all) in the vicinity of a functional group.