Born-Oppenheimer electronic states adiabatic

Since HF has a closed-shell electronic structure and no low-lying excited electronic states. HF-HF collisions may be treated quite adequately within the framework of the Born-Oppenheimer electronic adiabatic approximation. In this treatment (4) the electronic and coulombic energies for fixed nuclei provide a potential energy V for internuclear motion, and the collision dynamics is equivalent to a four-body problem. After removal of the center-of-mass coordinates, the Schroedinger equation becomes nine-dimensional. This nine-dimensional partial differential... 

The Born-Oppenheimer approximation states that the electrons are able to adjust themselves instantaneously to tlie motions of the nuclei. The motions of the nuclei are in this approximation therefore not able to induce electronic transitions, an assumption that is also known as tire adiabatic approximation. The electrons thus create an effective electronic potential in which the nuclei move, and for a given electronic state tire valuation in the electronic energy with respect to the nuclear configuration defines a potential energy surface for the electronic state. The electronic Schrodinger equation can be written as... 

H. We understood H to be complete and including electronic as well as nuclear degrees of freedom, and in which case the states are the true nonadiabatlc vlbronic eigenstates of the system and hence the properties are the exact ones. Nothing prevents us, however, to introduce the adiabatic approximation and to assume the wave functions to be products of electronic and nuclear (vibrational) parts. In this case, the Born-Oppenheimer electronic plus vibrational properties will appear. We can even reduce the accuracy to the extent that we adopt the electronic Hamiltonian, work with the spectrum of electronic states, and thus extract the electronic part of the properties. In all these cases, the SOS property expressions remain unchanged. 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

The electronic contributions to the g factors arise in second-order perturbation theory from the perturbation of the electronic motion by the vibrational or rotational motion of the nuclei [19,26]. This non-adiabatic coupling of nuclear and electronic motion, which exemplifies a breakdown of the Born-Oppenheimer approximation, leads to a mixing of the electronic ground state with excited electronic states of appropriate symmetry. The electronic contribution to the vibrational g factor of a diatomic molecule is then given as a sum-over-excited-states expression... 

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. 

The most extensive potential obtained so far with experimental confirmation is that of Le Roy and Van Kranendonk for the Hj — rare gas complexes 134). These systems have been found to be very amenable to an adiabatic model in which there is an effective X—Hj potential for each vibrational-rotational state of (c.f. the Born Oppenheimer approximation of a vibrational potential for each electronic state). The situation for Ar—Hj is shown in Fig. 14, and it appears that although the levels with = 1) are in the dissociation continuum they nevertheless are quasi bound and give spectroscopically sharp lines. 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

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