Vibrational eigenfunctions

Vibrational Eigenfunctions Using a Modified Single Lanczos Method Application to Acetylene (HCCH). 

The probability of a transition v" v is determined by the Franck-Condon factor, which is proportional to the squared overlap integral of both vibrational eigenfunctions in the upper and lower state. 

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

Mathematically, this corresponds to the time for which the relative phase of two neighboring vibrational eigenfunctions accumulates to be 2jt. For a harmonic potential, is independent of the quantum number n. This means that the WP moving in a harmonic potential recovers its original shape every T vib-... 

The amplitudes and phases of the vibrational eigenfunctions within the WP can be retrieved by the irradiation of another optical pulse referred to as a probe pulse. 

Fig. 14.4. Vibrational eigenfunctions, multiplied by the X —> A transition dipole function, for the gerade states of H20(X). All wavefunctions are symmetric with...

The knowledge of these quantities, coupled with the equation of motion, allows to find the vibrational eigenfunctions and eigenvalues, in the harmonic approximation. In order to compute the IFCs, a variation-perturbation approach to the DFT has been used. 

Fig. 5.2. Electronic excitation of LiH wavepacket from the outer classical turning point ( 6 a0) of the ground X1L7+ state. The B 1J7 X 1S+ transition is considered and the initial wavepacket is a 3ao shifted ground vibrational eigenfunction of LiH. The potential energy curves and the transition dipole moment of LiH are taken from the ab initio work of Partridge and Langhoff [55]...

Fig. 5.7. Pump-dump control of NaK molecule using two quadratically chirped pulses. The initial state is taken as the ground vibrational eigenfunction of the ground state X1S+ and this is excited by a quadratically chirped pulse to the excited state A1E+. The excited wavepacket is dumped at the outer turning point t cs 230 fs by the second quadratically chirped pulse. The laser parameters used are...

Solution of this equation provides a set of vibrational eigenfunctions i and their associated energies E , where the subscripts n are all integers— the vibrational quantum numbers. The vibrational energy levels are found to be ... 

These form a sequence of equidistant energy levels separated by A = /ivg and bounded by the potential energy curve V = Vik rf, as shown in Fig. 2.26. The lowest (ground-state) level with n = 0 has an energy 0 = Vi/ivg, known as the zero-point energy. The vibrational eigenfunctions are usually interpreted in terms of probabihty density functions I i j I of the type shown in Fig. 2.26. 

The observed levels in Table 3.6 may be obtained from the diabatic potentials represented by the Te,ue,ujexe, and Re constants, which generate the deperturbed levels via Tv = Te + we (v + h) — wexe ( electronic matrix element, He = 365 cm, and vibrational overlap factors calculated using the vibrational eigenfunctions of the deperturbed diabatic potentials. Similarly, the observed levels may be computed from the adiabatic potentials, a Lorentzian interaction term [Eq. (3.3.14)] We(R) with b = 0.1014... 

Although centrifugal distortion is not a perturbation effect, a derivation of the form of the centrifugal distortion terms in Heff provides an excellent illustration of the Van Vleck transformation. If the vibrational eigenfunctions of the nonrotating molecular potential, V(R) rather than [V(R) + J(J + 1)H2/2/j,R2, are chosen as the vibrational basis set, then the rotational constant becomes an operator,... 

The unavailability of an RKR-like inversion (hence the impossiblity of obtaining the potential energy surface, V(Q), and exact vibrational eigenfunctions directly from experimental data) makes it convenient to use products of simple harmonic or Morse-oscillator basis functions as vibrational basis states... 

Here, i is the dipole moment in the electronic ground state, / is the vibrational eigenfunction given by Eq. 1.21, and v and v are the vibrational quantum numbers before and after the transition, respectively. The activity of the normal vibration whose normal coordinate is Qa is being determined. By resolving the dipole moment into the three components in the x, y, and z directions, we obtain the result... 

Let us consider the fundamentals in which transitions occur from u = 0 to u" = 1. It is evident from the form of the vibrational eigenfunction (Eq. 1.22) that /o(Ga) is invariant under any symmetry operation, whereas the symmetry of Qa) is the same as that of Qa. Thus, the integral does not vanish when the symmetry of for example,... 

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