Populations of Ground and Excited States

The Boltzmann equation (Equation 18.2) shows that, under equilibrium conditions, the ratio of the number (n) of ground-state molecules (A ) to those in an excited state (A ) depends on the energy gap E between the states, the Boltzmann constant k (1.38 x 10 J-K ), and the absolute temperature T(K). 

For the energy of excitation discussed above (3.98 x 10 J) and a temperature of 20°C (293 K), the ratio of excited states to ground states is n(A )/n(A ) = In other words, there is only 

For the energy of excitation discussed above (3.98 x 10 J) and a temperature of 20°C (293 K), the ratio of excited states to ground states is n(A )/n(A ) = 10 . In other words, there is only one excited-state atom or molecule for every 5 x 10 molecules in the ground state Thus, the chances of observing a natural or stimulated emission are vanishingly small because there are almost no atoms or molecules in the excited state. For every photon entering such an assembly of molecules, there are billions of chances that it will be absorbed and only one that it will induce emission. Any photons so emitted would meet only ground-state molecules and not another excited-state atom or molecule required to start a cascade. Thus, an incipient cascade would stop before it could develop. 

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After excitation to the S4 state, pyrrole depopulates via a series of conical intersections between valence and Rydberg states in a very short time. The average adiabatic population of ground and excited states as a function of time is presented in O Fig. 33-7. 

Fig. 4.3. Left column relative vibrational population in ground and excited state of H2 with Tvib (X) as parameter. Right column relative vibrational population in the excited state for H2 and D2 as a function of Tvib (A )...

Stokes (o)q+(0 ) scattering processes. The partial derivative factor, (3aij/9Qk)e evaluated at the equilibrium position of the normal coordinate comprises a necessary condition for Raman activity of the normal mode Q. Raman effects occur only for those normal modes that cause the molecule to undergo a net change in polarizability during the course of the vibration. While equation (7) implies that both Stokes and anti-Stokes components should appear with equal intensity, a quantum mechanical derivation shows that the Stokes/ anti-Stokes intensity ratio is proporti onal to the Boltzmann factor (7), and can be used to determine the molecular temperature of a collection of molecules. The statistical derivation is based upon the thermal population of ground and excited molecular vibrational states according to a Boltzmann distribution. 

A ligand-held interpretation of the quantum yields of photosubstitution reactions of metal complexes has appeared. Similarly, a molecular orbital approach, based on the relative changes in metal-ligand overlap populations between ground and excited states, has been described to determine the relative ligand photolabilizations in the excited states of co-ordination compounds. The results complement the predictions of Adamson s qualitative rules. 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

Pre-saturation In this technique prior to data acquisition, a highly selective low-power rf pulse irradiates the solvent signals for 0.5 to 2 s to saturate them. No irradiation should occur during the data acquisition. This method relies on the phenomenon that nuclei which have equal populations in the ground and excited states are unable to relax and do not contribute to the FID after pulse irradiation. This is an effective pulse sequence of NOESY-type pre-saturation that consists of three 900 pulses RD - 900 - tx - 900 - tm - 90° - FID, where RD is the relaxation delay and t and tm are the presaturation times. 

From a microscopic point of view of the absorption process, we can assume a simple two energy level quantum system for which N and N are the ground and excited state population densities (the atoms per unit volume in each state). The... 

Oxidized Fe2S2 ferredoxins, containing two equivalent iron atoms, with J = 400 cm , show sharper NMR lines with respect to the monomeric iron model provided by oxidized rubredoxin (107-109), due to the decreased Boltzmann population of the paramagnetic excited states. For reduced ferredoxins (Si = 5/2, S2 = 2), with J = 200 cm , the ground state is paramagnetic (S = 1/2) (110). A smaller decrease in linewidth is expected. However, the fast electron relaxation rates of the iron(II) ion cause both ions to relax faster, and the linewidths in the dimer are sharp. 

The stability of related isomers towards ring opening in both ground and excited state can be compared using electronic overlap populations of the C4a-C4b bonds. Thus in the ground state for 44 n = 0.7274 while for n = 0.7201 reflecting the greater stability of 44 relative to 45. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

Reversal-temperature measurements of the Na and Cr lines in simple molecular gases, shock-heated to 2000-3000°K and to 0,2-2 atmospheres agree excellently with temperatures calculated from the measured shock velocity. Thus in these cases, collision processes are rapid enough to maintain effective equilibrium between ground and excited state populations despite radiatio n losses. In some shock tube work, however, the reversal temperature is initially above the equilibrium value, probably owing to delay in dissociation of the molecules, so that the temperature in translation and in internal degrees of freedom of the molecules is initially too high... 

The photocyclodehydrogenation of thienyl ethylenes is well-defined when both thiophene rings are bound via a C(2) atom to the ethylenic bond as in (70). In other cases, however, more cyclization products are possible. To predict the photocyclization mode for heterohelicenes the F s rule fails in many cases, because correction factors for the hetero atoms in the Huckel MO calculation have to be introduced and the systems are not well comparable with carbocyclic diaryl ethylenes. A better reaction parameter in these cases is the Mulliken overlap population (nrs)51), introduced by Muszkat52) for these cases. The overlap populations of the atoms r and s in ground and excited state (nIiS and n s), are calculated using the extended Huckel method. Cyclizations should not occur when nr>s and An s (= nr>s — n s) have negative values. (This method can also be used for diaryl olefins, but in these cases calculation of F s is more simple.). 

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