2. Molecular spectroscopy: theoretical background and datasets
2.1. Levels and transitions
Electromagnetic radiation can be characterised either in terms of the wavelength or frequency. Conventionally the wavelength λ [nm/µm] is used in the UV-‐VIS-‐
NIR, the wavenumber 𝜈𝜈 [cm-‐1] in the NIR-‐MIR-‐FIR, and the frequency ν [MHz/GHz] in the microwave regions, Figure 2.1. These units are related as:
𝜈𝜈 = 𝜈𝜈 𝑐𝑐 = 1
𝜆𝜆
Figure 2.1. Electromagnetic spectrum
According to the quantum mechanics, atoms and molecules can exist in certain discrete states that are different with respect to the electron configuration, angular momentum, parity and energy. Transitions between different energy states are associated with the absorption and emission of the electromagnetic radiation by the molecule. The actually observed resulting spectrum may consist of both individual lines corresponding to specific transitions and continuous bands, depending on the density of transitions on wavelength scale and typical Doppler widths.
The use of the word “line” to describe an experimentally observed transition goes back to the early days of observations of visible spectra with spectroscopes.
In those instruments, the lines observed in, for example, the spectrum of a sodium flame are images, formed at various wavelengths, of the entrance slit.
Although modern observations are usually in the form of a plot of some measure of the intensity of the transition vs. wavelength, frequency or wavenumber, peaks in such a spectrum are referred to as “lines”.
Using somewhat simplified approach, the full energy state a molecule can be divided into electronic, rotational, and vibrational constituents:
E = Evib + Erot + Eelec (2.1) The simplest way describe the vibrational energy state is to consider an example of a diatomic molecule. The bond between the two atoms in a diatomic molecule can be viewed as a spring, which, as the internuclear distance r changes from the equilibrium value re, exerts a force that tends to restore atoms to the equilibrium position. Such a system, assuming the parabolic shape of the potential energy curve, is approximated by a harmonic oscillator. For the vibrational motion of a harmonic oscillator, the vibrational energy term is given by:
Evib = ħω (v +!!) (2.2) where ħ = !!!, h is the Plank constant h=6.626×10-‐34 J s, ω is the angular frequency ( ω = 2𝜋𝜋∙ ν ), v is a vibrational quantum number (v = 0,1,2,…), so that energy levels of the ideal oscillator are equally spaced (Figure 2.2 a-‐b).
Figure 2.2. (a) Vibration of heteronuclear diatomic molecule (HCl) (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function (d)
This idealized approximation is valid for low vibrational energy levels only. For real molecules, the potential energy rises sharply at small values of r, whereas for large r the bond stretches until it ultimately breaks and the dissociation occurs. The real potential is not symmetric (Figure 2.2c,d). The energy of vibrational transition is on the order of 0.1 eV, with the corresponding wavelengths in the IR spectral range. Because the vibrational energy level spacing is relatively large (typically of the order of 103 cm-‐1) compared to the thermal energy, most molecules at room temperature are in their lowest vibrational energy level and light absorption normally occurs from v = 0.
A selection rule (transition rule) formally constrains the possible transitions of a system from one state to another. Selection rules have been derived for electronic, vibrational, and rotational transitions. The strength or energy of the interaction between a charge distribution and an electric field depends on the dipole moment of the charge distribution in a molecule.
To obtain the strength of the interaction that causes transitions between states (each characterized by its wave function), the transition dipole moment integral is used rather than the dipole moment. If transition dipole moment integral is zero, then the interaction energy is zero and no transition occurs or is possible between the two states. Such a transition is said to be forbidden (electric-‐dipole
forbidden). If transition dipole moment integral is large, then the probability of a transition is large /Herzberg, 1988/.
For a purely vibrational transition, the selection rule requires the change of the dipole moment during the vibration. This oscillating dipole moment produces an electric field that can interact with the oscillating electric and magnetic fields of the electromagnetic radiation. Heteronuclear diatomic molecules such as NO, HCl, and CO absorb infrared radiation and undergo vibrational transitions, contrary to homonuclear diatomic molecules such as N2 and 02, whose dipole moments remain constant during vibration.
Thus, the vibrational transitions occur for ∆v = ±1, ±2, ±3, … . For the harmonic oscillator, only the transitions with ∆v = ±1 are allowed (fundamental modes).
The anharmonicity of the real molecules leads to weaker overtone transitions with ∆v = ±2, ±3, ±4, … . For linear diatomic molecules only one mode of vibration is possible (stretching of the bond length). For polyatomic molecules, several modes of vibration are possible, resulting in changes of the bond lengths and angles between the bonds (Figure 2.3a)
Figure 2.3. (a) Internal vibrations of the bonds in the H2O molecule, (b) rotational motion of H2O, and (c) translation of the H2O molecule.
For a given electronic configuration and a given vibrational level (v value), it is also possible to discriminate the rotation of the molecule with respect to particular axis (Figure 2.3b). The rotational energy is given as:
Erot = B∙J∙ (J + 1) (2.3) B= ħ2/2Θ
where B is the rotational constant of a molecule (assumed to be a rigid rotator with constant internuclear distances) for a particular rotation mode, connected
with the moment of inertia Θ for the given rotational axis, and J is the rotational quantum number (J = 0, 1, 2,…). Again, this approximation is limited to small values of J, since bond lengths of real polyatomic molecules are subject to stretching due to the centrifugal force, resulting in increase of Θ for higher J values.
The difference in the angular momentum quantum number of initial and final states is given as ∆J =±1, since the photon exchanged with the atom or molecule has a spin of unity. The rotational transition can be observed when ∆J = 0, ±1.
The transitions are denoted as P-‐branch (∆J = -‐1), and R-‐branch (∆J = +1). For Q-‐
branch transitions, observed simultaneously with an electronic transition, ∆J = 0.
The energy difference between two consecutive rotational states (∆J =+1) is:
∆Erot = EJ+1 -‐ EJ = B[ (J+1)(J+2) -‐ J(J+1) ] = 2B(J+1)
The spacing between rotational energy levels increases with J. Energy level spacings are small compared to those of vibrational transitions, typically of the order of 10 cm-‐1 (10-‐3 eV) in the lower levels; this corresponds to absorption in the microwave region. The resulting observed sequence of rotational transitions (called “band”) consists of a series of equally spaced lines.
At room temperature conditions, the thermal energy available is sufficient to populate the energy levels above J = 0.
The Eelec component characterizes the energy of excited electronic configurations of the molecule. Energy difference ∆E associated with electronic transitions is of the order of 1 eV, corresponding to wavelengths in the visible or UV parts of the spectrum. The change in the electronic energy may occur simultaneously with the changes in vibrational and rotational energy (molecular bond length and interaction potential changes with the electronic configuration). As a result, observed electronic transitions may have a vibrational structure with a rotational “fine structure”.
Electronic states notation arises from the symmetry group theory, and involves consideration of the angular momentum of the molecules (electronic and rotational) and their electronic spin. The detailed explanation of concept is available in classical monograph by /Herzberg, 1988/.
The energy levels are thus described by a sum of the corresponding terms:
E = Eelec + ħω ( v + !! ) + B⋅J⋅( J+1) (2.4) Real molecular spectra have a complex structure and involve electronic, vibrational-‐rotational (also called ro-‐vibrational) and pure rotational transitions at typical ambient temperatures. The simple example for transitions between two vibrational levels v´´=0 and v´=1 belonging to the same electronic energy level and separated in the rotational levels J´´ and J´ is shown in Figure 2.4.
The photon energy of allowed transitions is given by the difference of energy of two consecutive states:
∆𝐸𝐸= 𝐸𝐸(v′,𝐽𝐽′) −𝐸𝐸(v′′,𝐽𝐽′′)=
=ℏ𝜔𝜔 v!+!! +𝐵𝐵!∙𝐽𝐽!∙ 𝐽𝐽! +1 −ℏ𝜔𝜔 v!!+!! +𝐵𝐵!!∙𝐽𝐽!!∙ 𝐽𝐽′!+1 (2.6) Assuming B’ ≈ B’’ = B, Δ v = +1
∆𝐸𝐸= ℏ!! +𝐵𝐵∙ 𝐽𝐽!∙ 𝐽𝐽!+1 −𝐽𝐽!!∙ 𝐽𝐽′!+1 (2.6a) and for P, R and Q branches
∆J = +1, J’=J’’+1, ∆𝐸𝐸≈ ℎ𝜈𝜈!+2𝐵𝐵∙ 𝐽𝐽′!+1
∆J = -‐1, J’=J’’-‐1, ∆𝐸𝐸≈ ℎ𝜈𝜈!−2𝐵𝐵∙𝐽𝐽!! (2.6b)
∆J = 0, J’=J’, ∆𝐸𝐸= ℎ𝜈𝜈!+(𝐵𝐵! −𝐵𝐵′′)𝐽𝐽!∙ 𝐽𝐽! +1
Schematically, P, Q and R branches are shown in Figure 2.4. Since difference between rotational constants is not very big, components of the Q-‐branch spectral lines are closely spaced. The Frank-‐Condon principle defines intensity distribution in the branches /Herzberg, 1988/.
Figure 2.4. An energy level diagram showing some of the transitions involved in the IR ro-‐vibrational spectrum of a linear molecule
Resolved ro-‐vibrational molecular spectra are of particular interest for remote sensing applications because they allow to distinguish constituents due to their characteristic spectral features.
Homonuclear molecules (like N2 and O2) do not have strong absorption features in the thermal infrared, making the atmosphere virtually transparent for sunlight
in this region. In contrast, heteronuclear diatomic molecules and most polyatomic molecules have strong resolved rotational spectral features in the IR, which makes this range useful for spectroscopic observations.
The principles considered above for diatomic molecules are generally valid for polyatomic molecules, but their spectra are far more complicated.
Nonlinear polyatomic molecules consisting of n atoms have three rotational constants with respect to the principal axes and 3n-‐6 vibrational degrees of freedom (3n-‐5 for linear polyatomic molecules). The resulting spectrum may contain multiple absorption bands, as well as overtone bands (∆v > 1) and combination bands (absorptions corresponding to the sum of two or more of the fundamental vibrations).