Vibrational and rotational sublevels

Figure 4.18. The principle of the Stokes shift experiment. The Stokes shift is the energy difference between the absorbed photon (solid arrow) and emitted photons (dashed arrows). The ground state and excited state jwtential energy surfaces are represented as shallow parabolas for clarity, vibrational and rotational sublevels are not shown. The electronic states are displaced on the nuclear coordinate to indicate that the configurations of the ground and excited state molecule and their respective solvation environments are different. A fluorophore is excited from its ground state to the first excited state (solid arrow). If in the excited state it relaxes to its equilibrium configuration, with associated solvation, it may emit light (dashed arrow 3) that corresponds to the steady-state fluorescence, with full Stokes shift. If the emission is monitored under conditions in which the excited state has not fully relaxed (dashed arrows 1 and 2), the emission maximum will occur at a higher energy and the Stokes shift will be smaller.

Figure 5.9 An electronic transition occurs over a band of energy due to the multiple vibrational and rotational sublevels associated with each electronic state. This schematic depicts four of the many possible transitions that occur. The length of the arrow is proportional to the energy required for the transition, so a molecular electronic transition consists of many closely spaced transitions, resulting in a band of energy absorbed rather than a discrete line absorption.

Fig.1. Schematic representation of the vibrational and rotational sublevels of two electronic states of a molecule...

The rate constant k in Eq. (5.21) is a molecular analog of the Einstein A coefficient for spontaneous emission. As we reasoned in Sect. 5.1, this rate constant should not depend on how the excited electronic state is prepared, as long as the vibrational and rotational sublevels within the state reach thermal equilibrium among themselves. 

Since electronic transitions may occur between states with many vibrational and rotational sublevels, and since these may also be affected by sample-solvent interactions, UV and visible spectra of solutions do not generally give sharp lines, but broad bands, as shown in Figure 3. Generally, the peak wavelength, is specified for analytical purposes. The absorbances obey the Beer-Lambert law, which is described in Topic E2. 

Figure 2.4 Schematic energy level diagram for molecules. Each electronic energy level, Eo.ig,. has associated vibrational sublevels Vi 2 3 and rotational sublevels ii 2 3. .. ...

Since most chemiluminescent reactions produce excited diatomic products rather than excited atoms, molecular species are prime visible chemical laser candidates. However, each molecular electronic state contains a complete set of vibrational and rotational levels. Consequently, the population density of an excited electronic level is diluted over many sublevels. This can be illustrated by the expression for... 

That is, transitions are normally between states of the same spin. Other selection rules may relate to the geometrical symmetry of the molecule. The molecular transitions seen in the visible and ultraviolet regions of the spectrum must be transitions from one rotational-vibrational-electronic state to another. We shall consider this in detail for a hypothetical diatomic molecule for which the ground state and excited state potential curves are those shown in Figure 10.8. For both electronic state potential energy curves there are sets of vibrational states and rotational sublevels. Notice that the equilibrium distance is not the same for both curves and that the curvature (i.e., the force constant) is not the same either. Thus, there are a different vibrational frequency and a different rotational constant for each electronic state. This has to be taken into account in working out the transition frequencies. 

Figure 1.5. Schematic representation of potential energy curves and vibrational levels of a molecule. (For reasons of clarity rotational sublevels are not shown.)...

Molecular spectra are not solely derived from single electronic transitions between the ground and excited states. Quantised transitions do occur between vibrational states within each electronic state and between rotational sublevels. As we have seen, the wavelength of each absorption is dependent on the difference between the energy levels. Some transitions require less energy and consequently appear at longer wavelengths. 

In some cases, the UV /VIS spectra will show the different energies associated with the vibrational sublevels. For example, simple molecules in the gas phase often show the vibrational levels superimposed on the electronic transitions, as seen in Fig. 5.11, the gas phase spectrum for benzene. The sharp peaks on top of the broad bands are called vibrational fine structure . This fine structure is usually lost at high temperatures in the gas phase due to population of higher vibrational energy levels in the ground state electronic level, with the result that many more lines are seen. Molecules in solution (such as the spectrum shown in Fig. 5.1) usually do not exhibit vibrational structure due to interactions between the solvent and the solute molecules. Compare the gas phase spectrum of benzene (Fig. 5.11) to the solution spectrum for benzene (Fig. 5.12) and note the loss of much of the fine structure in solution. Fine structure due to rotational sublevels is never observed in routine UV/VIS spectra the resolution of commercial instrumentation is not high enough to separate these lines. 

The intensity of the transition is also weighted by the sublevel population density n E), the number of molecules per unit volume in the initial vibrational sublevel at energy E. The collisional and rotational broadening at room temperature spans the vibrational level spacing, so this sub-level population density has the form appropriate for Boltzmann (thermal) equilibrium (Fig. 4b) or n E) decreases sharply (falling exponentially) with energy above the bottom of the potential well. The product... 

These collisions are instantaneous, (iii) The collision process is described by an S matrix. Its diagonal elements depend on rotational, but not vibrational, quantum numbers and its non-diagonal matrix elements vanish, (iv) Only a limited number of rotational sublevels are explicitely considered. In turn, the theory by Burshtein et al. conserves the assumptions (i,ii) but employs a different description of the collision process. The S matrix method is replaced by kinetic theoretical methods. The theory contains a collision strength parameter caracterizing the nature of the collision process. Starting from these premises these theories explain successfully the collapse of the Q-branch of the isotropic Raman spectrum through the motional narrowing (Fig.1). 

Let an atom (or molecule) have several sublevels in the ground state. For an atom, there may be, for example, fine and hyperfine ground-state sublevels (and for a molecule, vibration-rotational sublevels of the ground state). Atoms and molecules in a sublevel for which the transition frequency to an excited state is lower than the laser frequency are reflected from an atomic mirror. Thus, if a beam of atoms or molecules distributed among several ground-state sublevels is incident upon the mirror, the reflected beam will contain only atoms in a single chosen quantum state. 

Apatin, V. M., Krivtzun, V. M., Kuritzyn, Yu. A., Makarov, G. N., and Pak, I. (1983). Diode laser study of IR multiphoton-induced depletion of rotational sublevels of the ground vibrational state of SFe molecules cooled in a pulsed free jet. Optics Communications, 47, 251-256. 

Table III. Average deviations of calculated from experimental frequencies in the vibrationally excited Rano n bands of the N2 molecule, as obtained with the CCSD, 4R- and 8R-RMR CCSD methods and cc-pVTZ basis set The last column gives the range of experimentally available rotational / values, /min /inax9 d the difference between the maximum and minimum deviations for this range of rotational sublevels is enclosed in parentheses.

Dimers. It is well known that H2 pairs form bound states which are called van der Waals molecules. The discussions above based on the isotropic interaction approximation have shown that for the (H2)2 dimer a single vibrational state, the ground state (n = 0), exists which has two rotational levels f = 0 and 1). If the van der Waals molecule rotates faster ( > 1), centrifugal forces tear the molecule apart so that bound states no longer exist. However, two prominent predissociating states exist which may be considered rotational dimer states in the continuum (/ = 2 and 3). The effect of the anisotropy of the interaction is to split these levels into a number of sublevels. 

Show that if the overlap between torsional-vibration wave functions corresponding to oscillation about different equilibrium configurations is neglected, the perturbation-theory secular equation (1.207) for internal rotation in ethane has the same form as the secular equation for the Hiickel MOs of the cyclopropenyl system, thereby justifying (5.96)-(5.98). Write down an expression (in terms of the Hamiltonian and the wave functions) for the energy splitting between sublevels of each torsional level. 

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