Diabatic functions

This is a post Born-Oppenheimer scheme. It retains the essential idea of separability but the eleetronie base functions are diabatic functions. These functions are obtained from one Hamiltonian operator, namely the electronic operator defined in eq. (5). 

Now let us inquire into the physical meaning of the two conditions Hu -H22 and H12 (= H21) = 0. If we consider the basis (< )1 and 2) of the secular equation (Eq. [2]) as the diabatic components of the adiabatic electronic eigenfunction (a diabatic function describes the energy of a particular spin-coupling or atomic orbital occupancies,14 while the adiabatic function represents the surface of the real state), the crossing condition (real or avoided) is fulfilled when the two diabatic components and ( >2 cross each other, and this happens when H11 = H22, that is, when the energy of the two diabatic potentials (H11 is... 

For the two-state Hamiltonian the adiabatic functions can be easily transformed into diabatic functions. In the latter representation the Hamiltonian is not diagonal, and the model is completely specified by the matrix Hik as a function of R. If, besides, the semiclassical approximation is adopted there will be only two essential functions AH(t) = Hn(t) — and... 

A number of chemical processes can be represented by stepwise solutions of equation (5), by choosing particular pathways (or reaction coordinates ) in the laboratory space of the classical PCB. In actual calculations, a finitedimensional manifold defined by selected electronic diabatic functions may suffice to obtain qualitative and semi-quantitative information. (The reader is referred to Refs. [10-13] for specific examples.) We refer to this as the quantum-classical (QC) model. The classical PCB can be used to localize possible reaction events by manipulating charges in laboratory space. 

The most general representation for the quantum states of the complete molecular system of electrons and nuclei in terms of diabatic functions is given as... 

Our goal now is to show, in an example, what happens to adiabatic states (eigenstates of (/ )), if two diabatic energy eurves (mean values of the Hamiltonian with the diabatic functions) do cross. Although we are not aiming at an accurate deseription of the NaCl molecule (we prefer simplicity and generality), we will try to construct a toy (a model) that mimies this particular system. 

In polyatomic systems there is a serious problem with the adiabatic basis (this is why the diabatic functions are preferred). As we will see later (p. 264). the adiabatic electronic wave function is multivalued, and the corresponding rovibrational wave function, having to compensate for this (because the total wave function must be single-valued), also has to be multi-valued. 

Note that <[)/ are electronic wavefunctions that are not eigenfunctions of H. In every realistic case (except in diatomic molecules) where the sum over states J is truncated, the derivative coupling cannot vanish completely for every but it can become negligibly small. Making f/y very small corresponds to choosing the electronic wavefunctions so that they are always smooth functions of the nuclear coordinates. Physically diabatic functions maintain the character of the states. For example, assume that a diabatic state <[)] corresponds to a covalent configuration and another diabatic state 2 corresponds to an ionic configuration. Before a nonadiabatic transition occurs, the adiabatic states will be the same, v /i = tj)/ and v(72 = 2. After a nonadiabatic... 

As proved in Ref [63], this follo vs trivially for a polyatomic molecule if one adopts as diabatic functions a set of (j-independent functions for vhich Vc]) = 0. This can be seen if we use Equation 1.53 to prove another very useful result [66, 67] ... 

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