Wave functions Vibrational

Moiseyev N, Friesner R A and Wyatt R E 1986 Natural expansion of vibrational wave functions RRGM with residue algebra J. Chem. Phys. 85 331... 

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

As discussed above, the permutational symmetry of the total wave function requires the proper combination of its various contributions. These are summarized in Tables V-Xn for all isotopomers of Lis. Note that the conclusions hold provided that the various wave functions have the appropriate symmetries. If, for some reason, one of the components fails to meet such a requirement, then the symmetry of the total wave function will fail too. For example, even if the vibrational wave functions are properly assigned, the total wave... 

Drukker, K., Hammes-Schiffer, S. An analytical derivation of MC-SCF vibrational wave functions for the quantum dynamical simulation of multiple proton transfer reactions Initial application to protonated water chains. J. Chem. Phys. 107 (1997) 363-374. 

A property such as a vibrational wave function of, say, H2O may or may not preserve an element of symmetry. If it preserves the element, carrying out the corresponding symmetry operation, for example (t , has no effect on the wave function, which we write as... 

The symmetry properties of a fundamental vibrational wave function are the same as those of the corresponding normal coordinate Q. For example, when the C3 operation is carried out on Qi, the normal coordinate for Vj, it is transformed into Q[, where... 

Figure 1.13 shows the potential function, vibrational wave functions and energy levels for a harmonic oscillator. Just as for rotation it is convenient to use term values instead of energy levels. Vibrational term values G(v) invariably have dimensions of wavenumber, so we have, from Equation (1.69),... 

The transition moment (Equation 2.13) for a transition between lower and upper states with vibrational wave functions and j/[ respectively is given by... 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

If hd+ (i ) and pxi+ (k) denote the Fourier transforms of the indicated rotational and vibrational wave functions, the expression for the differential cross-section is... 

Figure 3 UMP2/6-311G potential energy profile (a) and hyperfine coupling constants (b) of CH3 versus s. Vibrational wave functions are normalized to 5.

The intensity of a vibronic transition depends upon the square of the overlap integral of the vibrational wave functions,... 

The modulation of the charge of the adsorbed atom by the vibrations of heavy particles leads to a number of additional effects. In particular, it changes the electron and vibrational wave functions and the electrostatic energy of the adatom. These effects may also influence the transition probability and its dependence on the electrode potential. 

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