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Thus, the Hessian &$,. is not diagonalized by the transformation matrix of the ground state, A;i. In the following we will assume for simplicity that ground and excited states can be described by the same normal mode coordinates. We will allow, however, for state dependent normal mode frequencies, wa,<. 74) Here and in the following, we will drop the electronic state index at the normal coordinates, qg,t - qe,< = qt and introduce the abbreviation q:“’ = Aqe,c. Typical PES along some normal coordinate valid for the ground and the excited state are plotted in Fig.
1 10)). (Figure courtesy of H. Naundorf) First, let us consider the standard approach of quantum chemistry. Suppose we have performed a search for stationary points and transitions states on the multi-dimensional PES (geometry optimization). For simplicity we assume that there are two minima separated by a single transition state as shown in Fig. 12. This situation may correspond to an isomerization reaction occurring, for example, in the course of intramolecular proton transfer (cf. Chapter7).
Forthe second termonehas 6 , (nz)Ca, ( g z ) x Ca, (0)) = 2 , if both electrons have the same spin; otherwise the summation gives zero. 30) obtained by summing over all indices i and j and multiplying the result by 1/2. Instead of summing over ( i , j ) we can equivalently take the summation with respect to all Ne1/2 occupied spatial orbitals. This finally gives for the ground state energy to be minimized Xu, Xu, X,, N,I/~ C (HeI) = 2 a h a a -t Em, N4/2 C [2Jab - K a b ] . 137) ab From the first term in Eq.