Vibrational matrix elements

At that time the permanent electric dipolar moment po of HCl had already been estimated to be 3.59 X10 C m [128], but Dunham made no use of this value hence we leave po in symbolic form. One or other value of coefficient p2 depends on a ratio (l/p(x) 0)/(2 p(x) 0) of pure vibrational matrix elements of electric dipolar moment between the vibrational ground state and vibrationaUy excited state V = 1 or 2. We compare these data with an extended radial function derived from 33 expectation values and matrix elements in a comprehensive statistical treatment [129],... 

Table 4. Band origin, pure vibrational matrix element and strength of vibration-rotational...

To convert these data into radial functions, one might apply algebraic expressions for vibrational matrix elements of x to various powers, of form such as... 

Adiabatic and Non-Condon. The non-Condon approaches, as mentioned in Section 10c, retain some interaction between the electronic and vibrational matrix elements. As a general conclusion, two primary results emerge from all these treatments. First, the temperature dependence of the transition probability is still mainly determined by the vibrational levels. This follows since the... 

Further symmetries arise when identical molecules, or molecules of high symmetry are involved several examples of practical interest will be considered below. The vibrational matrix elements for transitions between molecular states (si, s2) —> (s), s 2)... 

The dependence of these vibrational matrix elements B on the rotational states, 71, j[, etc., is sometimes so weak that it can be suppressed. 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

For the rotovibrational bands, the induced dipole components Bc of Eqs. 6.13 through 6.18 become the vibrational matrix element [281], Eq. 4.20. Furthermore, each XifoAL component now occurs twice, once with molecule 1 vibrating and once with molecule 2 vibrating in the final state. For like molecular pairs, this may be taken into account by removing the factors of 1/2 in Eqs. 6.13, 6.14, 6.16, and 6.17. For dissimilar molecular pairs, the factors 1/2 are absent from the equations quoted. 

For the case of induced absorption of H2-He pairs, the vibrational matrix elements of potential and dipole function are well known [151, 294]. The spectral moments Mq- M2 have been computed for the main induction components with the corrections for the vibrational dependences... 

The orbital and vibrational matrix elements may now be examined for symmetry requirements other than g and u. The requirement may best be expressed as the group-theoretical condition... 

The evaluation of the vibrational matrix elements is quite straightforward however, the problem has been formulated in terms of the amplitude of vibration with respect to the centre of gravity of the oscillator, so that a transformation of the usual oscillator wave functions is required. For a diatomic harmonic-oscillator... 

For polyatomic molecules equation (18) is employed with the vibrational matrix elements modified as described above. For vibrational exchange, in equation (18) the single vibrational matrix element is replaced by the product of the squares of the matrix elements for each molecule. In general, the theory leads to collision probabilities which are in good agreement with experiment. 

Connection with vibrational lifetime on surfaces. The decay of molecular vibrations in the excitation of the electron-hole pairs of metallic surfaces have been identified with the mechanisms of vibration excitation by tunneling electrons [42]. Intuitively this may seem so. Indeed, an excited vibration may couple to the surface electronic excitations through the same electron-vibration matrix elements of Eqs. (2) and (4). The surface... 

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 matrix elements connecting 2 = 1/2 and 3/2 states are more complicated, because these states have different sets of vibrational wave functions, and there is no simple expression for the vibrational matrix elements for these highly anharmonic potential functions. These matrix elements are therefore treated as phenomenological spectroscopic parameters, QVtV>, where v and v refer to the 2 = 1/2 and 3/2 states respectively. The addition of centrifugal distortion constants further complicates the analysis [211]. 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Substituting wave functions of Eq. (18) into the matrix element Eq. (16) and isolating the electron-vibrational matrix element we get... 

Let us single out the electron part of the electron-vibrational matrix element. To this end let us consider its integrand versus the distance between the nuclei R ... 

Sileo, R.N. and Cool, T.A. (1976) Overtone emission spectroscopy of HF and DF vibrational matrix elements and dipole moment function. J. Chem. Phys., 65, 117-133. 

Effect of Diatom Stretching Dependence. The features of the poten-tial energy surface most central to a discussion of its effect on the predissociation process are not the individual radial strength functions V j((R), but rather the vibrational matrix elements (integrated over the diatom bond length) of the full potential... 

Then the vibrational matrix elements of He(R) are expressed in terms of Rn centroids,... 

Figure 3.10 Perturbations between the 6pw and llj>7r 1n Rydberg states of H2. Two Rydberg series converging to different vibrational levels of the X2E state interact via nonzero Av — 1 vibrational matrix elements of the d/dR operator.

Values for many of the parameters in Heff cannot be determined from a spectrum, regardless of the quality or quantity of the spectroscopic data, because of correlation effects. When two parameters enter into the effective Hamiltonian with identical functional forms, only their sum can be determined empirically. Sometimes it is possible to calculate, either ab initio or semiempirically, the value of one second-order parameter, thereby permitting the other correlated parameter to be evaluated from the spectrum. Often, although the parameter definition specifies a summation over an infinite number of states, the largest part or the explicitly vibration-dependent part of the parameter may be evaluated from an empirically determined electronic matrix element times a sum over calculable vibrational matrix elements and energy denominators (Wicke, et al, 1972). 

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