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1.2.3 Vibronic States: Homogeneous and Inhomogeneous Lineshape
ОглавлениеThe absorption and emission processes have usually a dependence on the frequency ν of the exciting electromagnetic field. In particular, the amplitude of the absorption coefficient or the intensity of the emitted radiation changes by changing ν. This effect is related to the specific features of the centers interacting with the electromagnetic field through the distribution of the density of their electronic states as a function of ν or of the electron’s energy [8, 9]. In a given experiment, where the intensity of transmitted or emitted light is measured as a function of the frequency, a profile of I(ν) is recorded. This profile is usually called lineshape of the spectrum. In this context, it is possible to distinguish two contributions to the lineshape: homogeneous and inhomogeneous. This classification is related to the origin of the physical process contributing to the spectral width of the lineshape [2, 6, 11, 13]. It is now useful to consider a wider class of systems interacting with the radiation: the molecules. In this case, in addition to the degrees of freedom of the electrons, the degrees of freedom of the nuclei are present. In particular, not only the electrons can move and change their energy, but also the nuclei can move relative to each other, adding a contribution to the total energy of the system. The features affecting the lineshape can now be listed, collecting them in the above‐cited two classes [8, 13].
Homogeneous contributions
1 Once in the excited state, the electron should return to the starting state in a time that corresponds to the emission lifetime. This time, based on the Heisenberg uncertainty principle, is linked to an indetermination in the energy value of the excited state. Such effect gives rise to a spread in the energy of the transition and, as a consequence, a width of the lineshape. In addition, in the case of a gas of atoms or molecules, the collisions between the particles give rise to changes in the permanence time in the excited state and, therefore, impose changes to the lifetime and width of the lineshape.
2 When the system is a molecule, the vibrational degrees of freedom affect the distribution of energy levels and give rise to specific lineshapes (see Eq. 1.92).
Inhomogeneous contributions
1 When the atoms or molecules of a given species are embedded in an environment, the different spatial distributions of the other atoms and molecules originate local electric fields that cause a different energy levels’ separation and distribution among the centers under investigation.
2 When atoms or molecules move in space, their thermal velocity with respect to an observer fixed in space is randomly spread according to the Maxwell–Boltzmann distribution, giving rise to frequency variation of the absorbed or emitted light due to the Doppler effect.
Typically, the homogeneous effect is “intrinsic” to a given lineshape and the inhomogeneous effect induces a replica of the intrinsic lineshape, overall giving a broader lineshape that is the envelope of the many replicas. Neglecting the vibrations, the more common homogeneous lineshape is the Lorentzian, whereas the inhomogeneous lineshape is the Gaussian. The intermediate lineshape that is a Gaussian convolution of Lorentzian lineshapes is known as Voigt lineshape. The analytic forms of these functions are:
(1.77)
this lineshape is centered at ν 0 and is characterized by a full width at half maximum (FWHM) of the amplitude equal to 2A;
this is centered at ν 0 and has FWHM ; finally, the Voigt function can be defined as
(1.79)
where the Gaussian and Lorentzian contributions have been introduced. The final shape of the Voigt function depends on the balance between the FWHM of the two composing contributions.
To go deeper into the homogeneous lineshape features, the case of a molecular system has to be considered. It is then necessary to determine its electronic states in detail. First of all, the molecule is a many‐body system constituted by the electrons, the nuclei (it is useful to consider this unit for spectroscopic aims, associating the constituent parts: protons and neutrons, motion) and their motion and interaction. The more general Hamiltonian for this system is
(1.80)
where the subscripts label electrons’ (e) and nuclei’s (n) kinetic and potential energies, in order: the electrons kinetic energy; the nuclei kinetic energy; the electron–electron interaction potential energy; the electron–nuclei interaction potential energy; the nuclei–nuclei interaction potential energy. This Hamiltonian is really complex and it can be simplified considering that the proton mass is >103 times larger than the electron mass. As a consequence, it can be assumed that the electrons’ motion is much faster than the nuclei’s motion and that each electron explores a slowly varying configuration of the nuclei. Such “adiabatic” approximation is usually known as Born–Oppenheimer approximation [5, 8, 15, 17]. It assumes that the nuclei are practically fixed in space during the electrons’ motion, influencing the latter's energy statically. On the other side, the nuclei experience the electrons’ motion as an average value since the latter instantaneously adapt themselves to the given nuclear configuration. Such rapid rearrangement imposes that the nuclear energy is essentially related to the instantaneous nuclei positions. This enables to define a nuclear configuration and associate to it the overall electrons–nuclei system energy. Under this approximation, a configurational energy can be introduced
where the first three terms, depending on electrons’ motion, are an averaged (over the electrons’ coordinates and motion) contribution for a given spatial configuration of nuclei. The modified Hamiltonian of the overall system is then
(1.82)
where the configurational potential energy and the kinetic energy of the nuclei are explicitly reported. In the harmonic approximation, U conf is simplified as a quadratic function of the generalized coordinate Q [15, 18]. This is the internuclear distance in the most simple diatomic molecule or a normal coordinate in the case of a polyatomic molecule [5, 7]. It can be written
where the equilibrium position (minimum of the potential energy) Q = 0 has been chosen for simplicity and m and ω are the mass and frequency of the oscillation mode whereas U g is a constant representing the minimum energy of the given electronic configuration (related to spin and orbital motion of electrons). The complete Hamiltonian then becomes
(1.84)
that has the harmonic oscillator form [9]. The solution of the Schrodinger equation using this Hamiltonian gives eigenvalues and eigenfunctions of the nuclear coordinate [9]. The total energy of the system based on (1.81) will include the electrons’ energy. A representation of this energy is shown in Figure 1.5.
Figure 1.5 On the left: Schematic representation of the configuration total energy for the ground (lower) and the excited (upper) electronic states of a molecule. The continuous lines refer to the total energy, the horizontal dashed lines refer to the nuclei vibrational energy levels, marked by their quantum numbers. The equilibrium configuration coordinates are Q = 0 and Q = Q e for the ground and excited states, respectively. The zero‐phonon line is highlighted by the double arrow. On the right: The excitation (E abs) and emission (E em) pathways are reported by continuous arrows; ΔQ and ΔU highlight the change in configuration coordinate and potential energy, respectively; the dash‐dotted arrows mark the nuclear relaxation processes.
Each parabola schematizes the energy of the electronic configuration. In the lower parabola, representing the ground state of the system, the electrons have given spin and orbital angular momenta. In the upper parabola, schematizing the first excited state, the spin and orbital angular momenta have in general changed. The parabolic approximation gives the shape of the total energy, and the solutions of the harmonic oscillator define the possible quantized energy for the nuclear motion (represented by dashed lines in the figure). This means that not all the energy values are possible but just those marked by the dashed lines. Anyway, the energy of vibration is much lower than the energy of electrons’ interaction, and almost a continuous change can be considered within the given parabola, schematizing the electron configuration.
Considering the excited energy level, it is characterized, in general, by a configuration coordinate of equilibrium Q = Qe different with respect to the electronic ground state. Assuming again a harmonic approximation, the total energy of the excited state can be written as
where E is the absorption energy and the linear electron–phonon coupling approximation has been applied, represented by the term with F [15, 18]. The parameter F shows that the configuration coordinate Q of the minimum potential energy in the ground state and that in the excited state are different due to the connection between this energy and the electronic configuration (analytically, (1.85) is the formula of a parabola whose vertex position in the energy‐Q space is different from that of the parabola in (1.83) but they have the same concavity). By changing the electronic configuration, the potential energy changes and also the nuclear configuration adapts itself to a novel equilibrium position. By finding the minima of (1.83) and (1.85), it is demonstrated that the difference in the abscissa of minima is
(1.86)
In the linear electron–phonon approximation, the oscillation frequency ω of nuclei is the same in the ground and in the excited state. This is described by parabolic potentials with the same concavity in both states. As reported in Figure 1.5, this also implies that moving by ΔQ far from the minima, the same energy change ΔU is found in both states. By the scheme reported in the above figure, it is found that
where E abs and E em are the absorption and emission energies and the energy difference (E abs − E em) between the maximum of the absorption profile and that of the emission is called Stokes shift. Usually, it is considered that the electronic transition occurs in a much shorter time than the nuclear motion, so the nuclear configuration coordinate is unchanged during both the absorption and emission processes [5]. This statement is known as Franck–Condon principle [5]. From (1.83), it is found that for Q = Q e the configuration energy change of the ground state is
(1.88)
The same absolute value is found for the excited state. This value corresponds to the relaxation energy after the absorption or the emission processes. In particular, based on the Franck–Condon principle, in any transition, the nuclei’s coordinate and momentum are unchanged and a sudden electronic configuration change occurs [5]. Once the electronic state has changed, the nuclei relax toward the minimum energy of vibration compatible with the system’s temperature. At 0 K, this transition is toward the minimum vibrational energy, that is the minimum of the parabolas describing the energy of nuclei (dash‐dotted arrows in Figure 1.5). The Stokes shift marks the presence of a non‐null electron–phonon coupling. If the associated energy difference is zero, the two electronic states have the minima of the potential curves for the same value of Q, the two parabolas are vertically aligned, and transition between the same vibrational levels occurs without energy difference between absorption and emission. To go deeper into this aspect, the Born–Oppenheimer approximation is considered again. The wavefunction ψ that solves the Schrodinger equation can be factorized, separating the nuclei’s and the electrons’ contributions
(1.89)
where φ, ϑ refer to the electronic and nuclear wavefunctions, respectively, the first being parametrically dependent on Q, the nuclear coordinate, and the latter being independent on the electronic coordinate r. Furthermore, the electrons’ wavefunction, on the basis of the Condon approximation, depends on the average value of the nuclear coordinate [5, 15]
To evaluate the probability of transition between the two states reported in Figure 1.5, the dipole matrix element introduced in (1.41) should be considered. In particular, the value
(1.91)
has to be determined between the ground state designated by the energy E 1 and the excited state E 2. By considering (1.90) and the separation of nuclear and electronic coordinates, it is found that [5, 15]:
where the indices 1 and 2 refer to the lower and higher electronic energy levels and the indices n and m refer to the vibrational levels of the nuclei. It is worth underlining that the overall energy of the considered molecular system is the combination of the electrons’ and nuclei’s interactions. The latter is determined by the vibrational state marked by the quantum numbers reported in Figure 1.5. Overall, the transition involves electronic states and nuclear vibrational quantum states; so, the transition is called vibronic transition [5]. Equation (1.92) shows that the first factor gives the amplitude of the probability, being linked to the oscillator strength, and the second factor, |M nm |2, is responsible for the shape of the band for the given transition of absorption or emission. This is known as the Franck–Condon factor [5, 15]. In particular, since it is related to the harmonic oscillator solutions of the Schrodinger equation, this factor is null if n ≠ m for a given oscillator [9]. But because the solutions considered pertain to different equilibrium configuration of oscillators, with the same frequency, the orthonormal wavefunctions of the harmonic oscillators are eigenstates of the “same” oscillator but are referred to the different equilibrium (central) positions Q = 0, Q = Q e, respectively; so their integrals M nm could in general differ from zero. Furthermore, once a wavefunction is chosen, with fixed n, it can be decomposed by the full set of considering all possible values of m, because the latter set is a basis for the space of wavefunctions [5]. It is then found that the overall transition probability from a starting state to any of the excited vibronic states is given by
(1.93)
the summation being equal to 1 due to the orthonormality condition of used wavefunctions [5, 9, 15]. This finding explains that the transition probability from a given vibrational state is dependent on the electronic part of (1.92) whereas the nuclear part is responsible for the shape. This result gives origin to the lineshape of the absorption or emission band, since the same considerations can be done inverting the initial state and because |M nm |2 = |M mn |2. In particular, the homogeneous lineshape for a molecular species is dependent on the electron–phonon coupling and on its strength through |M nm |2. It is also found that absorption and emission lineshapes are symmetric with respect to the transition energy individuated by the M 00 element, known as the zero‐phonon line (ZPL, reported symbolically in Figure 1.5) [5, 15]. This transition is from the electronic ground state without vibration excitation to the excited electronic state without vibration excitation and vice versa, and it coincides for absorption and emission. In general, the vibration quantum ℏω is much larger than the thermal energy kT at room temperature; so, only the ground vibrational level, n = 0, is populated in the ground electronic state. In this case, the most relevant terms for the evaluation of the transition are |M 0m |2 and for the absorption process at energy E = E 00 + m ℏω, one can write
(1.94)
giving the amplitude of absorption for the transition from the ground vibrational level of the electronic ground state to the mth vibrational level of the excited electronic state, where E 00 is the zero‐phonon line energy. Analogously, for the emission process, one can write
(1.95)
considering that at room temperature, after the absorption process, the system relaxes to the lowest vibrational level in the excited state, as shown by the dash‐dotted arrow in Figure 1.5, and then it goes back to the electronic ground state, occupying one of the many vibrational levels depending on the |M n0|2 factor. On the basis of the wavefunctions of the harmonic oscillator, it can be demonstrated that [5, 15, 18, 19]
where the Huang–Rhys factor S has been inserted [19]
connecting the Stokes shift of the absorption and emission transitions to the vibrational energy of the oscillator considered. Equation (1.96) is a Poisson distribution with variance S, standard deviation , and average value S [20]. The Huang–Rhys factor measures the nuclear relaxation energy ΔU in units of the vibrational quantum of energy ℏω. In particular, the larger relaxation occurs in the presence of a strong electron–phonon coupling, implying that when the electron is excited, a large modification of the equilibrium position of nuclei occurs, with the ensuing relative shifts of the parabola describing the energy of the ground and of the excited state, as schematized in Figure 1.5 [15, 18]. It is interesting to observe that on increasing the Huang–Rhys factor, the lineshape of the considered transition, given by the distribution of |M 0m |2, changes from strongly asymmetric (0 < S < 1) with predominance of the zero‐phonon line, to gently asymmetric (1 < S < 6) with residual of the ZPL, to symmetric and almost Gaussian (S > 10) with small contribution of the ZPL [15]. The Huang–Rhys factor can be determined from the Stokes shift once the vibration frequency is known using (1.87) and (1.97). Furthermore, S can be determined from the area of the ZPL and the total area of the absorption (or emission) bands. In fact, starting from (1.96), it can be shown that
(1.98)
Finally, for large values of S, the variance of the Poisson distribution is equal to S and the spectral variance in terms of ℏω will be S(ℏω)2. From the Gaussian profile reported by (1.78), it is then determined that
(1.99)
and the Huang–Rhys factor could be experimentally estimated. To conclude these considerations on the homogeneous lineshape, a more detailed treatment should include the many possible vibration degrees of freedom of a polyatomic molecule and replicas of the considered features have to be inserted with different Huang–Rhys factors for each mode [8, 18, 21].
