Vibrational Ground Level

The o-vibrational state of a molecular ID-system is described by die Hamiltonian  

Starting with fulfilling the normalization condition for the trial wave-function, we have  

Going now to compute the trial vibrational energy, we calculate successively  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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Figure 9.1 Ground-state (g) and excited-state (1,2) diabatic potential energy surfaces of model I. Indicated are the vibrational ground level of g) and the vibrational levels of the uncoupled (l/i2 = 0) excited states 1) and 2) represents the characteristic electronic excitation energy.

Nascent NH(X) from 5vi and 6vi forms in about equal amounts in the symmetric F and F3 spin-rotation states [4]. The population of the F2 state is about 3 to 4% [17]. More than 96% of the energy liberated is released by translation. The rotational distributions correspond to NH temperatures of 280 50 K for dissociation from 5vi and 570 60 K for dissociation from 6v-, and the population of the vibrational ground level is >95% yielding an average increase of the internal energy by only 200 cm"V The decomposition from 5vi and 6vi is similar to that of DN3 in the IRMPD experiments [4] see above. 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

We can use the energy level diagram in Figure 10.14 to explain an absorbance spectrum. The thick lines labeled Eq and Ei represent the analyte s ground (lowest) electronic state and its first electronic excited state. Superimposed on each electronic energy level is a series of lines representing vibrational energy levels. 

Another form of radiationless relaxation is internal conversion, in which a molecule in the ground vibrational level of an excited electronic state passes directly into a high vibrational energy level of a lower energy electronic state of the same spin state. By a combination of internal conversions and vibrational relaxations, a molecule in an excited electronic state may return to the ground electronic state without emitting a photon. A related form of radiationless relaxation is external conversion in which excess energy is transferred to the solvent or another component in the sample matrix. 

Figure 7.18 shows sets of vibrational energy levels associated with two electronic states between which we shall assume an electronic transition is allowed. The vibrational levels of the upper and lower states are labelled by the quantum numbers v and u", respectively. We shall be discussing absorption as well as emission processes and it will be assumed, unless otherwise stated, that the lower state is the ground state. 

We have seen in Section 6.1.3.2 that, for diatomic molecules, vibrational energy levels, other than those with v = 1, in the ground electronic state are very often obtained not from... 

Figure 10.6 Graph of the Boltzmann distribution function for the CO molecule in the ground electronic state for (a), the vibrational energy levels and (b), the rotational energy levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations.

Possible explanations for this difference in behavior may involve complex formation between the nitrobenzenes and ammonia, reaction in a vibrationally excited level of the ground state, or, preferably, a recognition that in these cases the reaction partner is an electrically neutral species present in high... 

S0, S], S2,. .. represent so-called singlet states in which all the electrons have paired spins, and T1( T2,. .. represent triplet states in which two electrons have unpaired spins. The energy levels of both ground (S0) and activated states (Sj, S2,. ..) are subdivided into vibrational and rotational energy levels. The vibrational energy levels are shown in Figure 11.2. Differences in rotational levels are very small and can be ignored for the present discussion. 

Fig. 2 Jablonski energy level diagram illustrating possible transitions, where solid lines represent absorption processes and dotted lines represent scattering processes. Key A, IR absorption B, near-IR absorption of an overtone C, Rayleigh scattering D, Stokes Raman transition and E, anti-Stokes Raman transition. S0 is the singlet ground state, S, the lowest singlet excited state, and v represents vibrational energy levels within each electronic state.

The multiplicity of excitations possible are shown more clearly in Figure 9.16, in which the Morse curves have been omitted for clarity. Initially, the electron resides in a (quantized) vibrational energy level on the ground-state Morse curve. This is the case for electrons on the far left of Figure 9.16, where the initial vibrational level is v" = 0. When the electron is photo-excited, it is excited vertically (because of the Franck-Condon principle) and enters one of the vibrational levels in the first excited state. The only vibrational level it cannot enter is the one with the same vibrational quantum number, so the electron cannot photo-excite from v" = 0 to v = 0, but must go to v = 1 or, if the energy of the photon is sufficient, to v = 1, v = 2, or an even higher vibrational state. 

An electron excites from a vibrational energy level in the lower, ground-state Morse curve, to a vibrational level in the excited-state Morse curve. 

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