The normalisation integral

In this equation and k denote the standard Coulomb and exchange operators involving core orbital (pi, D and D(ji v) stand for the normalisation integral for the SC wavefunction and the elements of the first-order density matrix in the space of the SC orbitals, and and k, are generalised Coulomb and exchange operators with matrix elements = XikMq )AXp K u Xq) = At least Voutofthe Af... 

As a last step we wish to evaluate the normalisation integral in the sphere... 

The normalisation integral which forms the denominator of eqn ( 2.A.12) is the simplest case of a whole class of many-electron overlap integrals which occur in the reduction of the energy integral which fornis the numerator of eqn ( 2. A. 12)... 

Of the terms in the expansion of the normalisation integral associated with an... 

It now only remains to evaluate each of the six different terms to generate the value of the normalisation integral. 

If is a single determinant of orthonormal orbitals, we can use the earlier result for the normalisation integral of a single determinant. The expression above is the same as the normalisation integral except that integration over the coordinates of the first (now unlabelled) electron is missing, the result is... 

In this case, the straightforward approach which we have used for the normalisation integral and the one-electron terms would work, but it turns out to be very unwieldy to extract the general result from a simple case. If we try to evaluate the result for the simplest manageable case explicitly, n = 3 generating six terms, the outcome is not typical because only individual orbital overlaps remain after integration over the coordinates of the two electrons involved in the repulsion integral. The next case n = 4 with twenty-four terms is too unwieldy to handle explicitly. 

Androulakis et al. (2004) introduced the time-integrated flux indicator, which is based on the normalised integral of the quantity defined in Eq. (4.2). The purpose of the time-integrated element flux analysis is to establish a global insight into the reaction pathways. It is clear, however, that in many cases, inspection of the... 

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