Adiabatic vibrational basis

For 7 1.0, which corresponds to the case for the SiO G and I 1II states, the mixing of the vibrational basis functions is large in both diabatic and adiabatic descriptions. 

The coupled equations have also been solved using the adiabatic electronic basis set for the H2 E+ states (Glass-Maujean, et al, 1983). In this basis set, the d/dR derivative, acting on the unknown md vibrational functions, appears in the second term on the right-hand side of the coupled equations, namely,... 

The coupled-equation numerical vibrational wavefunctions are difficult to visualize. A major difficulty is that their number of nodes is not simply related to a vibrational quantum number hence prior expectations about the node count cannot be used to establish correspondences between observed and calculated levels. Johnson (1978) has presented a method for counting the number of nodes of the calculated wave functions to avoid inadvertently skipping over an eigenstate. The xnad(7i) basis functions for the H2 XD+ states have been expressed in the form of Eq. (4.4.44) as a linear combination of the known xtt adiabatic vibrational functions. These x n R) functions were used to compute... 

For a given (i,Q) adiabatic potential, a coupled basis set of vibrational basis functions is used for each value and the S matrix elements... 

Fig. 4 Snapshots of the probability distribution evolution for two CO molecules on a copper(lOO) surface subject to non-adiabatic coupling. The dynamics is initiated with two quanta of vibrational excitation in one of the CO molecules, here the qi mode. The labels full and fact, refer to the choice of vibrational basis to represent the reduced density matrix in eqn (19). The left panels show the system in the absence of intermode coupling and the strong intermode coupling regime is depicted in the right panels. Reproduced with permission from ref. 97.

The obtained PES forms the basis for the subsequent dynamical calculation, which starts with determining the MEP. The next step is to use the vibrationally adiabatic approximation for those PES degrees of freedom whose typical frequencies a>j are greater than a>o and a>. Namely, for the high-frequency modes the vibrationally adiabatic potential [Miller 1983] is introduced,... 

The term H e is the electron correlation operator, the term H p corresponds to phonon-phonon interaction and H l corresponds to electron-phonon interaction. If we analyze the last term H l we see that when using crude approximation this corresponds to such phonons that force constant in eq. (17) is given as a second derivative of electron-nuclei interaction with respect to normal coordinates. Because we used crude adiabatic approximation in which minimum of the energy is at the point Rg, this is also reflected by basis set used. Therefore this approximation does not properly describes the physical vibrations i.e. if we move the nuclei, electrons are distributed according to the minimum of energy at point Rg and they do not feel correspondingly the R dependence. The perturbation term H) which corresponds to electron-phonon interaction is too large... 

As an example we can take the excited states of NO. It has been shown that there are two excited states of the same symmetry ( 11) whose vibrational levels are best interpreted on the basis of diabatic curves which cross as in Fig. 1 (75-7 7). One of these states (B) arises from the electron excitation to an antibonding valence molecular orbital and the other (C) from excitation to a Rydberg orbital. The Born-Oppenheimer adiabatic curves cannot cross (by virtue of the non-crossing rule which is to be discussed in a later section) and must fullow the dashed curves shown in the figure. 

Analysis of the vibrational normal modes obtained at the HF/6-31G(d,p) level of theory in terms of adiabatic modes provides the basis for a quantitative dissection of the... 

The motion of a molecule is characterized by several steps of adiabaticity. The fast electronic motion is accompanied by slow vibrational motion, and the latter motion is faster than the rotational motion of the molecule. On this basis the total wavefunction can be written as a product... 

The above analysis is based on the Born-Openheimer approximation in which the adiabatic electron wave functions are frozen at the bottom of the corresponding minimum. An important advantage of this approach is that we work with the limited size for the electron basis functions and not with the infinite basis of the vibrational states. This makes the problem solvable in simple terms. 

---