Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering Energetics: The Bohr frequency condition: E( ) E(M) E(M * ) E( ' ) E( ) E(M * ) E(M) h 3 basic modes of motion (translation, vibration, rotation) and 1 internal mode (electronic) play an important role in chemistry because they are ways in which molecules store energy. Spectra of molecules are more complex than for atoms because molecules can have rotational and vibrational states. Absorption, emission, scattering allow to probe for vibrational, rotational and electronic transitions. Emission and absorption spectroscopy give same information about energy level separation.
Microwave absorption/emission rotational spectroscopy Infrared absorption/emission vibrational spectroscopy UV, VIS, IR absorption/emission electronic spectroscopy Raman spectroscopy = analysis of frequencies present in radiation scattered (about 1 on 10 7 of incident photons) by molecules. Applied to rotational and vibration transitions: = 0 < 0 > 0 Rayleigh radiation Stokes radiation anti-stokes radiation Selection rules: restrictions on * M, e.g. H(1s) H(p) H(s) l 1 Kinetics, lifetimes, transition rates (A,B), population inversion.
Intensities of spectral lines The transmittance of a sample for a given frequency is: T I I0 where I = transmitted intensity and I 0 = incident intensity. Empirically, the transmittance varies with the length d of the sample and the molar concentration c (mol/l) of the absorbing species in accord with the Beer-Lambert law: I I 0 10 cd = molar absorption coefficient (or extinction coefficient). It depends on the frequency of the incident radiation. Its units: l m -1 mol -1 = cm mol -1. Hence, it can be viewed as a molar cross section for absorption. The absorbance (or optical density) of a sample for a given frequency is: A logi 0 I or A logt. The Beer-Lambert law becomes: A = cd. The law suggests that to achieve sufficient absorption, path lengths through gaseous samples must be very long, of the order of meters, because concentrations are very low. Long path lengths are achieved by multiple passage of the beam between parallel mirrors at each end of the sample cavity. Conversely, path lengths through liquid samples can be significantly shorter, of the order of mm or cm.
Rate of absorption and transition dipole moment Classically, for a molecule to interact with the EM field and absorb or emit a photon of frequency, it must posses, at least transiently, a dipole oscillating at that frequency. Quantum mechanics: Hamiltonian is: (1) Ĥ Ĥ 0 Ĥ (t) with Ĥ ( 1) (t) = time dependent perturbation = EM field (oscillating electric field) interacting with the electric dipole moment, : (1) Ĥ (t) E0 cos t with = angular frequency E 0 = amplitude The rate of transition wfi of population of the state f due to a transition from state i (painful demo.) is proportional to the square modulus of the matrix element of the perturbation between the two states: w fi H (1) fi E 0 Therefore, the rate of transition, and hence the intensity of absorption of the incident radiation is proportional to the square of the transition dipole moment: fi * f i d f i The size of the transition dipole can be regarded as a measure of the charge redistribution that accompanies a transition. The rate of transition (= rate of change of probability of finding the molecule in the upper state) is also proportional to E, and therefore the intensity of the incident radiation.
Absorption intensities: Einstein s coefficients Einstein identifies three contributions to the transitions between states. Stimulated absorption = transition from a low energy state to one of higher energy induced by a photon. Einstein wrote the transition rate as: w B B = Einstein coefficient of stimulated absorption characterizing the transition, d = energy density of EMR in the frequency range to +d, where = frequency of the transition. When the molecule is exposed to black-body radiation, is given by the Planck distribution (at thermal equilibrium). B is large strong transitions strong absorption. Stimulated emission = radiation induces an upper state to undergo a transition to a lower state, and hence to generate a photon of frequency : w' B' B = Einstein coefficient of stimulated emission. Only radiation of the same frequency as the transition can stimulate this. Spontaneous emission = an excited state undergoes a transition to a lower state by generating a photon of frequency. The rate is independent on frequency and intensity of EMR present, hence: w' A B' A = Einstein coefficient of spontaneous emission.
Total rate of absorption: W N w NB Total rate of emission: W ' N' w' N' A B' where N = population of lower state, and N = population of the upper state. At thermal equilibrium: NB N' A B' Solving for the energy density: N' A A/B NB N'B' N/N' B' /B Using the Boltzmann expression for the ratio of populations of states of energies E and E in the last step: N' N h /k T e B with h E' E hence: e A/B h /k B T B' /B 8h Compare with Planck s distribution: = e B' B We conclude: 3 8h therefore: A B 3 c 3 h/k B T 3 /c 1 1) The knowledge of one coefficient leads to the two others. ) The radiation field stimulates both absorption and emission; the two processes are coherent. 3) Spontaneous emission (A) increases with radiation frequency (laser). Hence, it can be ignored (A=0) at the relatively low frequencies of rotational and vibrational transitions. In this case, the net absorption rate is proportional to the population difference of the two states involved in the transition: Wnet W W ' NB N'B' N N' B
Away from equilibrium, a system will return to equilibrium by spontaneous emission leading to a reduction in population N of the excited state by a first order process: dn(t) dt A N(t) Integration leads to: At t / N (t) N(0) e N(0) e. = 1/A = radiative life-time of the excited state (~10-9 to 10-3 s). From this, the units of A are s -1, hence the units of B and B are m 3 J -1 s - = m kg -1.
Selection rules and transitions moments Not all possible transitions are permissible; hence a spectrum does not arise from the transition of any initial energy level to any other (atoms and molecules). Selection rule = a statement about which transitions are allowed / forbidden. Derived by identifying the transitions that conserve angular momentum when a photon (a spin-1 particle) is emitted or absorbed. The form they take depends on the type of transition. We saw previously that the necessary (transient) electric dipole moment of a molecule is expressed in terms of the transition dipole moment, fi, between states i and f : fi * f ˆ d i f ˆ i The size of the transition dipole can be regarded as a measure of the charge redistribution that accompanies a transition; hence a transition will be active only if the accompanying charge redistribution is dipolar: 1s s: spherical migration of charge; no dipole moment associated: the transition is forbidden. 1s p: dipole moment associated: transition is allowed.
We saw previously that the transition rate is proportional to follows that the coefficient of stimulated absorption (and emission), and therefore the intensity of the transition, is also proportional to fi. A detailed analysis gives: B 6 fi 0 Hence, to identify the selection rules, we must establish the conditions for which fi is nonzero. Physically, the transition dipole moment is a measure of the kick that a transition gives or receives from the electromagnetic field. A gross selection rule specifies the general features a molecule must have if it is to have a spectrum of a given kind. For instance, we shall see that a molecule gives a rotational spectrum only if it has a permanent electric dipole moment. A detailed study of the transition moment leads to the specific selection rules that express the allowed transitions in terms of the changes in quantum numbers. fi. It
Linewidths A number of effects (inherent or not) contribute to the widths of spectroscopic lines and limit the resolution. Doppler broadening Doppler effect = the radiation is shifted in frequency when the source is moving towards or away from the observer. Important for gas-phase rotational spectroscopy, since molecules can rotate freely only in gases. If a source of frequency moves with speed v relative to an observer, the observer detects radiation of frequency: receding 1 v / c 1 v / c 1/ approaching 1 v / c 1 v / c 1/ For nonrelativistic speeds (v << c): receding approaching 1 v / c 1 v / c Detected spectral line = absorption/emission profile from all Doppler shifts. Reflects the distribution of molecular velocities parallel to the line of sight. Shown to be a Gaussian curve.
One can show that for molecules of mass m and temperature T, the observed width of the line at half-height (FWHM in terms of ) is: obs k c B Tln m 1/ For a molecule like N at room temperature, /.3 10. For a typical rotational transition frequency of 30 GHz, the linewidth is about 70 khz. 6 Lifetime broadening Even if Doppler is minimized, there exists a residual broadening due to quantum mechanical effects. True for samples in gas-, liquid- and condensed-phases When solving Schrödinger s equation for a time-dependent system, we find that it is impossible to specify the energy levels exactly. This is reminiscent of the Heisenberg uncertainty principle that can be written: E t /. Specifically, if = lifetime of an excited state, the transition energy is blurred by E: E
In terms of wavenumbers, the spread can be written as (with E h c): 5.3 cm / ps No excited state has an infinite lifetime; therefore, all states are subject to some lifetime broadening. The shorter the lifetime of the states involved in a transition, the broader the corresponding spectral lines. 1 Two processes are responsible for the finite lifetime of excited states. 1) Collisional linewidth: due to collisional deactivation, dominant for low frequency transitions. col = collisional lifetime, mean time between collisions col = 1/z, where z = collision frequency kinetic model of gases: z P (pressure) hence the collisional linewidth E is P. col / col ) Natural linewidth of a transition: the rate of spontaneous emission cannot be changed; hence it is a natural limit to the lifetime of an excited state. In that case: rad = radiative lifetime, 10-9 to 10-3 s (defined earlier as) rad = 1/A and A 3 hence the natural linewidth E is 3. rad / rad
Low frequency transitions (e.g. microwave transitions of rotational spectroscopy) have very small natural linewidths, and collisional and Doppler line-broadening processes are dominant. The natural lifetimes of electronic transitions are very much shorter than for vibrational and rotational transitions, so the natural linewidths of electronic transitions are much greater than those of vibrational and rotational transitions. For example, a typical electronic excited state natural lifetime is about 10-8 s (10 ns), corresponding to a natural width of about 5 10-4 cm -1 (15 MHz). A typical rotational state natural lifetime is about 10 3 s, corresponding to a natural linewidth of only 5 10-15 cm -1 (of the order of 10-4 Hz). Strategy for discussing molecular spectra For extracting the information molecular spectra contain, we will adopt the following procedure: 1) Find the expression for the energy levels of molecules. ) Calculate the transition frequencies. 3) Apply the selection rules. 4) Predict the appearance of the spectrum by taking into account the transition moments and the populations of the states.