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The atom-atom model for ion-pair formation in molecular collisions = MESO-m3c-mod

In this module we give an account of an atom-atom model for ion-pair formation in molecular collisions. In short, this simple classical model describes the electronic states of the system as coupled covalent or ionic states. Because the states are coupled, the potential energy curves cross and a non-adiabatic transition between the states is possible [The transition is explained in a macroscopic <i>Theoretical methods</i> module (link type: `elaborated in/explained in/project/wider range'; target: MACRO-m3c-diab#diabatic] . Thus the ion pair formation is induced by the crossing of the potential curves or (hyper)surfaces, i.e. ionisation takes place via surface hopping. In collisions between alkali atoms and halogen molecules, this electron jump is the first step of the harpoon reaction [The harpoon reaction is described in Herschbach (link type: `detailed in/context/external'; target: Herschbach)], [The context is sketched in the mesoscopic <i>Situation</i> module (link type: `context/project'; target: MESO-m2a)] .

Crossing potentials

Let us consider the ion-pair formation process
\begin{displaymath}\mathrm{A + B \to A^{+} + B^{-}},
\end{displaymath}

where A is an electropositive particle (such as an alkali atom) and B an electronegative one (such as a halogen atom or molecule). In the diabatic representation [The diabtactic representation is elaborated in a macroscopic module (link type: `elaborated in/project/wider range'; target: MACRO-m3c-diab)] , the states of the system are described as either purely ionic ( A+ + B-) or purely covalent ( A + B). According to the model, the electron transfer is due to the configuration interaction between the purely ionic and the purely covalent state at the point where these states cross, e.g. at the point where the energies of these states are equal. The crossing of the two potentials is caused by the fact that the energy of the ionic state at infinite separation is of the order of one eV higher then the energy of the covalent state, whereas the ionic potential at smaller distances will be lower than the covalent one because of the attractive Coulomb potential.

The typical shape of the A-B intermolecular potential is shown in Fig. MESO-m3c-mod-F1. The chemi-ionization process is seen to be endothermic with an energy

\begin{displaymath}{\Delta E} = I(\mathrm{A})-EA(\mathrm{B}),
\end{displaymath}

where I(A) represents the ionization potential of the electropositive particle
and EA (B) the electron affinity of the electronegative particle. The ionic and covalent ground-state potential curves cross at the intermolecular distance Rc.

The crossing distance Rc is approximately given by

\begin{displaymath}R_c \approx \frac{e^2}{I(\rm A) - EA(\rm B)}
\end{displaymath}

This approximation is quite good for large crossing distances (Rc of about 5 Å) as in that case the induction and dispersion forces can be neglected: At large distances the two potentials can be represented by V=0 for the covalent state and $V(R) = \Delta E - \frac{e^2}{R}$ for the ionic state.
  
Figure MESO-m3c-F1: The typical shape of the A-B intermolecular potential

The groundstates of the system are described by two curves. There exist many analytical expressions to describe the ionic alkali-halide potential curve. [Recently, a review has been given by Patel and Gohel (link type: `elaborated in/external; target: Patel-m1]. For the ionic-potential curve we have chosen a Rittner potential [The formula for the potential is input from Rittner's publication (link type: `input/external'; target: Rittner)] of the form:

 
Uion(R) = $\displaystyle - \frac{e^2}{R} - \frac{e^2 ( \alpha_{\zs\M^+} + \alpha
_{\zs\X_2^-} )}{2(R^4+a^4)}-
\frac{2e^2 \alpha_{\zs\M}\alpha
_{\zs\X_2^-} }{R^7}$
    $\displaystyle -\frac{C_\mathrm{ion}}{R^6}+A_\mathrm{ion}\,\mathrm{e}^{-R/\rho_\mathrm{ion}}+{\Delta }E.$
The covalent potential is given by:
 
$\displaystyle U_\mathrm{cov}(R)=-(C_\mathrm{cov}/R^6)+A_\mathrm{cov}\,\mathrm{e}^{-R/\rho_\mathrm{cov}}.$    
The covalent potential consists of a Van der Waals term and the repulsive term, for the ionic potential we add a coulombic term, a screened polarization term, dipole-dipole interaction and at last the endothermicity ${\Delta }E$.

species and the multiplicity of the ionic and covalent ground states are the same. At the crossing point the transitions between the covalent and the ionic configurations are induced by a strong coupling between the state [The coupling is explained in a macroscopic module (link type: `elaborated in/explained in/project/wider range' target: MACRO-m3c-diab#diabatic)] . The Landau-Zener theory provides a simple expression for this transition as a function of the radial velocity at the crossing point. At a collision, ionization of the neutral particles takes place via such a non-adiabatic transition by electron transfer.

Trajectories

The potential curves of Fig. MESO-m3c-mod-F1 clearly show the two collision trajectories leading to chemi-ionization. Incoming on the covalent curve and outgoing on the ionic one, the particles follow either the ionic potential curve inside the crossing distance with a covalent-ionic> transition at the first passing of Rc followed by a covalent-covalent crossing, or the covalent potential curve inside the crossing distance Rc (with a covalent-covalent transition followed by a covalent-ionic one). The former process will be indicated by ``ionic'' scattering, the latter one by ``covalent'' scattering. In terms of an electron jump between the two interactants, ionic scattering means that the electron has jumped from A to B when these first approached to the crossing distance Rc of each other and that it remains there during the transition that takes place the second time the particles are at Rc. In covalent scattering there is no charge transfer in a diabatic transition at the first passing of the crossing distance, but the electron jumps when the separating interactants are again at Rc.

Landau-Zener theory

In he two-dimensional case the transition probability between different configurations has been given firstly by Landau, Zener and Stueckelberg (commonly called the Landau-Zener theory and abbreviated here as L-Z theory). This theory has been criticized and extended by different authors [Nikitin's survey of the LZ theory (link type: (detailed in/external; target: E.E. Nikitin, Chemische Elementar Prozesse (Springer, 

Berlin, 1968) p. 43.)] . Transitions between multi-dimensional surfaces have been treated by Teller [To the details on Teller's work (link type: `detailed in/external' target: Teller)]. Further work has been done during the last decade by Nikitin [To the details on Nikitin's work (link type: `detailed in/external' target: Nikitin)].

The probability Pb for a covalent-ionic transition at a single passage of the crossing at Rc is given by the Landau-Zener formula:

 \begin{displaymath}P_\br=\exp\left( \frac{-2\pi H_{12}^2}{\hbar v_\rr\Big\vert\frac{\dr}{\dr
R}(H_{11}-H_{22})\Big\vert _{R_\crm} }\right),
\end{displaymath}
where $\dr H_{11}/\dr R$ and $\dr H_{22}/\dr R$ are the slopes of the diabatic potentials at Rc and H12 is half the energy difference of the adiabatic potential curves at Rc. For a given collision energy, the radial velocity at the crossing point vr depends on the collision parameter b. The probability of a covalent-covalent or ionic-ionic transition is (1-Pb). Thus both the ionic and the covalent scattering trajectories leading to ionization have the equal probability Pb(1-Pb).

At thermal energies the probability of the electron jump will be close to unity, except for the largest Rc values. At higher energies, however, the probability for a transition from covalent to ionic at the crossing point decreases, while the covalent-covalent transition probability increases: at centre of mass energies larger than ${\Delta }E$ the probability that only one electron jump takes place is non-negligible, such that ion pair formation can occur due to an electron jump at the first crossing while passing the second time no change occurs or vice versa. [The context is sketched in the mesoscopic <i>Situation</i> module (link: `context/project'; target: MESO-m2a)].

From the potential-energy curves and the transition probability, the total and differential [The details on the calculation method are given in a mesoscopic <i>Theoretical methods</i> module (link type: `detailed in/focused on in/project'; target: MESO-m3c-defl] cross sections for ion-pair formation via the different trajectories can be calculated via the deflection function.

Restrictions of the model

In the simplified atom-atom model, it is assumed that the system has an isotropic, static two-body potential; vibrational and rotational degrees of freedom are neglected. The potential associated to harpoon reaction

\begin{displaymath}\mathrm{M + XY \to M^+ + (X...Y)^- \to MX + Y}
\end{displaymath}
for example should be described using hypersurfaces.

Calculations on three-atomic systems are very difficult because they require a knowledge of the complete energy surfaces involved. Moreover, the situation is complicated very much because multiple passing of the crossing region is possible in contrast with the two-atom case, where the crossing region is passed only twice during the collision. An attempt has been made by Bjerre and Nikitin [Details on this attempt in their article (link type: `detailed in/external; target: Bjerre-m*] to calculate cross sections for quenching of Na* by N2. This quenching is also supposed to take place via crossings of potential surfaces. They made trajectory calculations on three-dimensional crossing surfaces, determining the transition probabilities at the crossings by an extended L-Z theory. Such calculations are very computer-time consuming and, because the knowledge of the potential energy surfaces is very poor, one may not expect results with more than only qualitative meaning.

It is known now that the quantitative validity of the L-Z theory is very limited. Moreover, this theory in its simple form is applicable neither to three-particle collisions nor to endothermic processes. Indeed it is found to be impossible to fit the energy dependence of the measured cross section with that predicted by the L-Z theory. However, one can try to use the results of the theory as a kind of ``adiabatic criterion'' to determine the position and also the order of magnitude of the maximum cross section of this type of inelastic processes.