Energy space

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

Eor transition metals the splitting of the d orbitals in a ligand field is most readily done using EHT. In all other semi-empirical methods, the orbital energies depend on the electron occupation. HyperChem s molecular orbital calculations give orbital energy spacings that differ from simple crystal field theory predictions. The total molecular wavefunction is an antisymmetrized product of the occupied molecular orbitals. The virtual set of orbitals are the residue of SCE calculations, in that they are deemed least suitable to describe the molecular wavefunction. 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Dimethylenefuran-, 3,4-dimethylenethiophene- and 3,4-dimethylenepyr-rolediyl radicals as non-Kekule molecules with tunable singlet-triplet energy spacings 97ACR238. 

As shown in Figure 10.1, the energy spacings between the vibrational levels in the two electronic levels are not the same. This is to be expected... 

Here Zj = gq/gj and ZE = [Pg.110]

Figure 8. Photoelectron spectrum (PES) and Penning ionization electron spectrum (PIES) of nitric oxide radical. Average vibrational energy spacing of the first band amounts to 285 and 284 cm", respectively (104).

The nonadiabatic transition state theory given in the Section II.C, namely, Eq. (17), can be applied to the electron-transfer problem [28]. Since the electron transfer theory should be formulated in the free energy space, we introduce the... 

The energy of the scattered radiation is less than that of the incident radiation for the Stokes line of the Raman spectrum and the energy of the scattered radiation is more than that of the incident radiation for the anti-Stokes line. The energy increase or decrease from the excitation is related to the vibrational energy spacing... 

This corresponds to an EPR-silent sample that gives no detectable ESR spectrum at X-band frequencies because it possesses a zero-field splitting larger than the Zeeman interaction (see Chapter 6), and the energy spacing between the two lowest levels is too large to be spanned by a microwave quantum at X-band. Nevertheless, higher frequencies are able to induce transitions. Since... 

Mankind began exploring use of all possible sources of energy, space and many elements (some 25,000 years ago). 

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