Electronically adiabatic

Ben]amin I, Barbara P F, Gertner B J and Hynes J T 1995 Nonequilibrium free energy functions, recombination dynamics, and vibrational relaxation of tjin acetonitrile molecular dynamics of charge flow in the electronically adiabatic limit J. Phys. Chem. 99 7557-67... 

The total orbital wave function for this system is given by an electronically adiabatic n-state Bom-Huang expansion [2,3] in terms of this electronic basis set as... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Then, let the two real electronically adiabatic wave functions be written as [4]... 

At this stage, we would like to emphasize that the same transformation has to be applied to the electronic adiabatic basis set in order not to affect the total wave function of both the elecbons and the nuclei. Thus if is the electronic basis set that is attached to 4> then and are related to each other as... 

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... 

In the meantime other experiments have also improved our range of observational results. For example, Watts et al. carried out experiments very similar to the NO/Ag(lll) experiments described above.32 A critical difference in this work was the substitution of Cu(110) in place of the Ag(lll). Despite the chemically distinct metal surface, nearly identical results were obtained as those in Refs. 24 and 25, including surface temperature and incidence energy dependence. While it is not unlikely that the bond softening of NO is similar on Ag(lll) and Cu(110), there is no a priori reason to believe that these two metals would exhibit the same incidence energy and surface temperature dependence in vibrational excitation experiments. More importantly, there has not been a theoretical attempt to explain why these two chemically distinct systems behave so similarly within the context of electronically adiabatic models. 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

Note that since the profile of the lower adiabatic potential energy surface for the proton depends on the coordinates of the medium molecules, the zeroth-order states and the diabatic potential energy surfaces depend also on the coordinates of the medium molecules. The double adiabatic approximation is essentially used here the electrons adiabatically follow the motion of all nuclei, while the proton zeroth-order states adiabatically follow the change of the positions of the medium molecules. 

For P(r), one usually has two choices (1) the electronic adiabatic approximation, or (2) the SCF method. In the adiabatic approximation, the velocity of the excess electron is assumed to be small compared with that of molecular electrons. Then, electronic polarization of the medium does not contribute to binding. Jortner (1962, 1964) questioned the validity of this approximation for ehor eam, since the binding energy of the excess electron (-1-2 eV) is not insignificant compared with that of the medium electrons. He used the SCF method, in which all electrons are treated on equal footing. The resultant potential V(r) is now given by (see Eq. 6.10)... 

This is the electronic adiabatic Schrodinger equation and in the case of a single coordinate x Eq. (B.14) takes the following form ... 

As shown in Eq. (92), the gauge field Abc is simply related to the non-adiabatic coupling elements ikhm. For an infinite set of electronic adiabatic states [N = oo in Eq. (90)], Ft,c = 0. This important results seems to have been first established... 

The electronically adiabatic wave functions v(/f ad(r q ) are defined as eigenfunctions of the electronic Hamiltonian Hel with electronically adiabatic potential energies ad(qjJ as their eigenvalues ... 

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