Vibrational function

The time evolved wavefunction, according to this first rule, can be expressed in terms of the vibrational functions fPfi 4ind energies Efi of the N2 ion as... 

From a practical standpoint, simple harmonic vibration functions are related to the circular frequencies of the rotating or moving components. Therefore, these frequencies are some multiple of the basic running speed of the machine-train, which is expressed in revolutions per minute (rpm) or cycles per minute (cpm). Determining these frequencies is the first basic step in analyzing the operating condition of the machine-train. 

Translational functions in eq. (100) cancel out because of the approximation adopted, the vibrational functions also cancel. The logarithm of the ratio of rotation functions possesses a constant value of —.246. The contribution of the exponential term is considerably higher and therefore determines the value of the disproportionation constant. Theoretical values of log K listed in Table X... 

The third explanation is based on vertical excitations from the vibration functions outside the potential energy curves of the os-stilbene (Figure 9.5, mode d). 

The vibrational functions, corresponding to the energy eigenvalues of equation 3.11 or vibrational eigenfunctions, are given by... 

The corresponding vibrational function in the optical continuum frequency field has the form... 

The intensity of the observed line is determined by the dipole transition moment. Thus, in order to evaluate the intensities, knowledge of the electron and vibrational functions is required. Furthermore, it is important to take into account that the nuclear part of the dipole operator has the form. 

If the vibrational functions are described within the harmonic oscillator approximation, it can be shown that the <%vf I (Ra - Ra e) I %vj> integrals vanish unless vf = vi +1, vi -1 (and that these integrals are proportional to (vi +1)1/2 and (vi)1/2 in the respective cases). Even when %vf and %vi are rather non-harmonic, it turns out that such Av = 1 transitions have the largest <%vf I (Ra - Ra e) I %vj> integrals and therefore the highest infrared intensities. For these reasons, transitions that correspond to Av = 1 are called "fundamental" those with Av = 2 are called "first overtone" transitions. 

If in the total wave function ( ), one electron orbital ( ) is factored out from the vibrational function (X) and the spin function (S), we get... 

In that case one vibrational function will be orthogonal to all the others, and transitions can only take place between two electronic states which have the same vibrational state, as shown in Figure 5-3. However, if these parameters are not completely identical, the... 

It is assumed that target states p are indexed for each value of q such that a smooth diabatic energy function Ep(q) is defined. This requires careful analysis of avoided crossings. The functions should be a complete set of vibrational functions for the target potential Vp = Ep, including functions that represent the vibrational continuum. All vibrational basis functions are truncated at q = qd, without restricting their boundary values. The radial functions fra should be complete for r < a. 

In certain cases, the formulae for p and q may be simplified further. If the spectroscopic constants for the interacting states are similar, many of the vibrational matrix elements will reduce to zero. Consider the y = 0 level of a given 2II state. The principal matrix element will be with the v = 0 level of 2 2 and the value <2II[B 12S> between vibrational functions will be approximately the B value of 2n. By the orthogonality rules, the remaining matrix elements should be zero. This has led Van Vleck to suggest... 

The solution of the Schiodinger equation for the nuclear motion furnishes the vibrational levels and wave-unctions corresponding to a given electronic state, usually the ground state. If we further consider electronic excited states, band structures of electronic spectra may be determined. As well known, the square of the overlap between the vibrational functions of both stat will give us the so-called iVanck-Condon factors which are a measure of the intensity of the transition between the vibrational levels of both states... 

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